FOUNDATIONS OF p-ADIC HODGE THEORYbhattb/almost_purity_2011/gabber...FOUNDATIONS OF p-ADIC HODGE THEORY OFER GABBER AND LORENZO RAMERO January 21, 2011 Fifth Release Ofer Gabber I.H.E.S

FOUNDATIONS OF p-ADIC HODGE THEORY

OFER GABBER AND LORENZO RAMERO

January 21, 2011

Fifth Release

Ofer GabberI.H.E.S.Le Bois-Marie35, route de ChartresF-91440 Bures-sur-Yvettee-mail address: [email protected]

Lorenzo RameroUniversite de Lille ILaboratoire de MathematiquesF-59655 Villeneuve d’Ascq Cedexe-mail address: [email protected] page: http://math.univ-lille1.fr/∼ramero

Acknowledgements We thank Niels Borne, Lutz Geissler, and W.-P. Heidorn for pointing out some mistakes inearlier drafts.

1

2 OFER GABBER AND LORENZO RAMERO

CONTENTS

0. Introduction 31. Categories 41.1. Basic category theory 41.2. Tensor categories and abelian categories 151.3. 2-categories 321.4. Fibrations 391.5. Sieves and descent theory 451.6. Profinite groups and Galois categories 592. Sites and topoi 662.1. Topologies and sites 672.2. Topoi 882.3. Algebra on a topos 972.4. Cohomology on a topos 1133. Monoids and polyhedra 1203.1. Monoids 1203.2. Integral monoids 1373.3. Polyhedral cones 1473.4. Fine and saturated monoids 1613.5. Fans 1743.6. Special subdivisions 1904. Complements of commutative and homological algebra 2064.1. Complexes in an abelian category 2074.2. Simplicial objects 2204.3. Injective modules, flat modules and indecomposable modules 2454.4. Graded rings, filtered rings and differential graded algebras 2584.5. Some homotopical algebra 2744.6. Witt vectors, Fontaine rings and divided power algebras 2874.7. Regular rings 3064.8. Excellent rings 3195. Local cohomology 3345.1. Cohomology in a ringed space 3345.2. Quasi-coherent modules 3445.3. Duality for quasi-coherent modules 3515.4. Depth and cohomology with supports 3575.5. Depth and associated primes 3695.6. Duality over coherent schemes 3815.7. Schemes over a valuation ring 3945.8. Local duality 4055.9. Hochster’s theorem and Stanley’s theorem 4156. Logarithmic geometry 4276.1. Log topoi 4276.2. Log schemes 4386.3. Logarithmic differentials and smooth morphisms 4536.4. Logarithmic blow up of a coherent ideal 4676.5. Regular log schemes 4886.6. Resolution of singularities of regular log schemes 5056.7. Local properties of the fibres of a smooth morphism 5237. Etale coverings of schemes and log schemes 530

FOUNDATIONS OF p-ADIC HODGE THEORY 3

7.1. Acyclic morphisms of schemes 5307.2. Local asphericity of smooth morphisms of schemes 5447.3. Etale coverings of log schemes 5577.4. Local acyclicity of smooth morphisms of log schemes 5768. The almost purity toolbox 5898.1. Non-flat almost structures 5898.2. Almost pure pairs 6128.3. Normalized lengths 6298.4. Formal schemes 6518.5. Algebras with an action of Frobenius 6678.6. Finite group actions on almost algebras 6788.7. Complements : locally measurable algebras 6869. The almost purity theorem 7039.1. Semistable relative curves 7069.2. Almost purity : the smooth case 7159.3. Model algebras 7229.4. Almost purity : the case of model algebras 7389.5. Purity of the special fibre 7549.6. Almost purity : the log regular case 770References 789

0. INTRODUCTION

The aim of this project is to provide a complete – and as self-contained as possible – proofof the so-called ”almost purity” theorem, originally proved by Faltings in [34]. In a secondstage, we plan to apply almost purity to establish general comparison theorems between theetale and the de Rham cohomologies of a scheme defined over the field of fractions of a rankone valuation ring K+ of mixed characteristic.

In our formulation, the theorem states that a certain pair (X,Z), consisting of an affine K+-scheme X and a closed subset Z ⊂ X is almost pure (see definition 8.2.25), in analogy withthe notion that is found in [44].

The proof follows the general strategy pioneered by Faltings, but our theorem is stronger thanhis, since we allow non-discrete valuation rings K+. The price to pay for this extra generalityis that one has to extend to this non-noetherian context a certain number of standard toolsand results from commutative algebra and algebraic geometry. Especially, chapter 5 gives arather thorough treatment of local cohomology, for rings and schemes that are not necessarilynoetherian.

The complete proof is found in chapter 9. The case where R := OX(X) has Krull dimensionone had already been dealt with in our previous work [36]. Almost purity in dimension twois theorem 9.1.31. Theorem 9.2.16 takes care of the case of a smooth K+-scheme of Krulldimension strictly greater than three; the main features of this case are the use of the Frobeniusendomorphism of the ring R/pR, and a notion of normalized length for arbitrary R-modules.

In truth, one does not really need to consider separately the case of dimension > 3, since theargument given in section 9.4 works uniformly for every dimension ≥ 3. However, the case ofdimension > 3 is considerably easier, and at the same time illuminates the more difficult caseof dimension 3. For the latter, one constructs a ring A(R)+ with a surjection onto the p-adiccompletionR∧ ofR, and an endomorphism σR that lifts the Frobenius endomorphism ofR/pR.This construction was originally found by Fontaine, who exploited it in case R is the ring ofintegers of a local field of characteristic zero.


1. CATEGORIES

1.1. Basic category theory. The purpose of this section is to fix some notation that shall beused throughout this work, and to collect, for ease of reference, a few well known generalitieson categories and functors, which are frequently used. Our main reference on general nonsenseis the treatise [10], and another good reference is the more recent [51]. For homological algebra,we mostly refer to [75].

Sooner or later, any honest discussion of categories and topoi gets tangled up with somefoundational issues revolving around the manipulation of large sets. For this reason, to beable to move on solid ground, it is essential to select from the outset a definite set-theoreticalframework (among the several currently available), and stick to it unwaveringly.

Thus, throughout this work we will accept the so-called Zermelo-Fraenkel system of axiomsfor set theory. (In this version of set theory, everything is a set, and there is no primitive notionof class, in contrast to other axiomatisations.)

Additionally, following [3, Exp.I, §0], we shall assume that, for every set S, there exists auniverse V such that S ∈ V. (For the notion of universe, the reader may also see [10, §1.1].)

Throughout this section, we fix some universe U. A set S is U-small (resp. essentially U-small), if S ∈ U (resp. if S has the cardinality of a U-small set). If the context is not ambiguous,we shall just write small, instead of U-small.

1.1.1. Recall that a category C is the datum of a set Ob(C ) of objects and, for every A,B ∈Ob(C ), a set of morphisms from A to B, denoted :

HomC (A,B).

This datum must fulfill a list of standard axioms, which we omit. For any A ∈ Ob(C ), we write1A for the identity morphism of A. We also often use the notation :

EndC (A) := HomC (A,A).

Likewise, AutC (A) ⊂ EndC (A) is the group of automorphisms of the object A. Furthermore,we denote by Morph(C ) the set of all morphisms in C .

The category of all small sets shall be denoted U-Set or just Set, if there is no need toemphasize the chosen universe.

We say that the category C is small, if both Ob(C ) and Morph(C ) are small sets. Somewhatweaker is the condition that C has small Hom-sets, i.e. HomC (A,B) ∈ U for every A,B ∈Ob(C ). The collection of all small categories, together with the functors between them, formsa category U-Cat. Unless we have to deal with more than one universe, we shall usually omitthe prefix U, and write just Cat. If A and B are any two categories, we denote by

Fun(A ,B)

the set of all functors A → B. If A is small and B has small Hom-sets, then Fun(A ,B) ∈ U;especially, Cat is a category with small Hom-sets. Moreover, there is a natural fully faithfulimbedding :

Set→ Cat.

Indeed, to any set S one may assign its discrete category also denoted S, i.e. the unique categorysuch that Ob(S) = S and Morph(S) = 1s | s ∈ S. If S and S ′ are two discrete categories,the datum of a functor S → S ′ is clearly the same as a map of sets Ob(S)→ Ob(S ′).

1.1.2. The opposite category C o is the category with Ob(C o) = Ob(C ), and such that :

HomC o(A,B) := HomC (B,A) for every A,B ∈ Ob(C ).

Given an object A of C , sometimes we denote, for emphasis, by Ao the same object, viewedas an element of Ob(C o); likewise, given a morphism f : A → B in C , we write f o for the


corresponding morphism Bo → Ao in C o. Furthermore, any functor F : A → B induces afunctor F o : A o → Bo, in the obvious way.

1.1.3. A morphism f : A→ B in C is said to be a monomorphism if the induced map :

HomC (X, f) : HomC (X,A)→ HomC (X,B) : g 7→ f g

is injective, for everyX ∈ Ob(C ). Dually, we say that f is an epimorphism if f o is a monomor-phism in C o. Obviously, an isomorphism is both a monomorphism and an epimorphism. Theconverse does not necessarily hold, in an arbitrary category.

Two monomorphisms f : A → B and f ′ : A′ → B are equivalent, if there exists anisomorphism h : A → A′ such that f = f ′ h. A subobject of B is defined as an equivalenceclass of monomorphisms A→ B. Dually, a quotient of B is a subobject of Bo in C o.

One says that C is well-powered if, for every A ∈ Ob(C ), the set :

Sub(A)

of all subobjects of A is essentially small. Dually, C is co-well-powered, if C o is well-powered.

Definition 1.1.4. Let F : A → B be a functor.

(i) We say that F is faithful (resp. full, resp. fully faithful), if it induces injective (resp.surjective, resp. bijective) maps :

HomA (A,A′)→ HomB(FA, FA′) : f 7→ Ff

for every A,A′ ∈ Ob(A ).(ii) We say that F reflects monomorphisms (resp. reflects epimorphisms, resp. is conser-

vative) if the following holds. For every morphism f : A→ A′ in A , if the morphismFf of B is a monomorphism (resp. epimorphism, resp. isomorphism), then the sameholds for f .

(iii) We say that F is essentially surjective if every object of B is isomorphic to an objectof the form FA, for some A ∈ Ob(A ).

(iv) We say that F is an equivalence, if it is fully faithful and essentially surjective.

1.1.5. Let A , B be two categories, F,G : A → B two functors. A natural transformation

(1.1.6) α : F ⇒ G

from F to G is a family of morphisms (αA : FA → GA | A ∈ Ob(A )) such that, for everymorphism f : A→ B in A , the diagram :

FAαA //

Ff

GA

Gf

FBαB // GB

commutes. If αA is an isomorphism for every A ∈ A , we say that α is a natural isomorphismof functors. For instance, the rule that assigns to any object A the identity morphism 1FA :FA → FA, defines a natural isomorphism 1F : F ⇒ F . A natural transformation (1.1.6) isalso indicated by a diagram of the type :

AF ))

G

55 α B.


1.1.7. The natural transformations between functors A → B can be composed; namely, if α :F ⇒ G and β : G⇒ H are two such transformations, we obtain a new natural transformation

β α : F ⇒ H by the rule : A 7→ βA αA for every A ∈ Ob(A ).

With this composition, Fun(A ,B) is the set of objects of a category which we shall denote

Fun(A ,B).

There is also a second composition law for natural transformations : if C is another category,H,K : B → C two functors, and β : H ⇒ K a natural transformation, we get a naturaltransformation

β ∗ α : H F ⇒ K G : A 7→ βGA H(αA) = K(αA) βFA for every A ∈ Ob(A )

called the Godement product of α and β ([10, Prop.1.3.4]). Especially, if H : B → C (resp.H : C → A ) is any functor, we write H ∗ α (resp. α ∗H) instead of 1H ∗ α (resp. α ∗ 1H).

These two composition laws are related as follows. Suppose that A , B and C are threecategories, F1, G1, H1 : A → B and F2, G2, H2 : B → C are six functors, and we have fournatural transformations

αi : Fi ⇒ Gi βi : Gi ⇒ Hi (i = 1, 2).

Then we have the identity :

(β2 ∗ β1) (α2 ∗ α1) = (β2 α2) ∗ (β1 α1).

The proof is left as an exercise for the reader (see [10, Prop.1.3.5]).

1.1.8. Let A and B be two categories, F : A → B a functor. Recall that a functor G : B →A is said to be left adjoint to F if there exist bijections

ϑA,B : HomA (GB,A)∼→ HomB(B,FA) for every A ∈ Ob(A ) and B ∈ Ob(B)

and these bijections are natural in both A and B. Then one says that F is right adjoint to G,and that (G,F ) is an adjoint pair of functors.

Especially, to any object B of B (resp. A of A ), the adjoint pair (G,F ) assigns a morphismϑGB,B(1GB) : B → FGB (resp. ϑ−1

A,FA(1A) : GFA→ A), whence a natural transformation

(1.1.9) η : 1B ⇒ F G (resp. ε : G F ⇒ 1A )

called the unit (resp. counit) of the adjunction. These transformations are related by the trian-gular identities expressed by the commutative diagrams :

Fη∗F +3

1F FFFFFFFFF

FFFFFFFFF FGF

F∗ε

GG∗η +3

1G FFFFFFFFF

FFFFFFFFF GFG

ε∗G

F G.

Conversely, the existence of natural transformations ε and η as in (1.1.9), which satisfy theabove triangular identities, implies thatG is left adjoint to F ([10, Th.3.1.5] or [51, Prop.1.5.4]).

Remark 1.1.10. Let (G,F ) be an adjoint pair as in (1.1.8), with unit η and counit ε.(i) Suppose that (H2, H1) is another adjoint pair, with H1 : B → C , H2 : C → B (for some

category C ), and let ηH (resp. εH) be a unit (resp. a counit) for this second adjoint pair. Thenclearly (G H2, H1 F ) is an adjoint pair, and the transformation

ε := ε (G ∗ εH ∗ F ) (resp. η := (H1 ∗ η ∗H2) ηH)

is a counit (resp. a unit) for this adjunction. We say that η and ε are the unit and counit inducedby (η, ε) and (ηH , εH) .


(ii) Suppose that (G′, F ′) is another adjoint pair, with F ′ : A → B, G′ : B → A , andlet η′ (resp. ε′) be a unit (resp. a counit) for this second adjoint pair. Suppose moreover thatτ : F ⇒ F ′ is a natural transformation. Then we obtain an adjoint transformation τ † : G′ ⇒ G,by the composition :

G′BG′(ηB)−−−−→ G′FGB

G′(τGB)−−−−−→ G′F ′GBε′GB−−−→ GB.

Conversely, from such τ † we can recover τ , hence the rule τ 7→ τ † establishes a natural bijectionfrom the set of natural transformations F ⇒ F ′, to the set of natural transformations G′ ⇒ G.Notice that this correspondence depends not only on (G,F ) and (G′, F ′), but also on (η, ε) and(η′, ε′).

(iii) Moreover, using the triangular identities of (1.1.8), it is easily seen that the diagram :

G′ F G′∗τ +3

τ†∗F

G′ F ′

ε′

G F ε +3 1A .

commutes.(iv) Furthermore, suppose (G′′, F ′′) is another adjoint pair with F ′′ : A → B, G′′ : B →

A and η′′, ε′′ are given units and counit for this pair. Let also ω : F ′ ⇒ F ′′ be a naturaltransformation; then we may use (η′, ε′) and (η′′, ε′′) to define ω†, and we have :

(ω τ)† = τ † ω†

provided (η, ε) and (η′′, ε′′) are used to define the left-hand side.(v) Lastly, let (H1, H2) and the unit and counit ηH , εH be as in (i), and suppose moreover that

we have another adjoint pair (H ′1, H′2) where H ′1 : B → C and H ′2 : C → B, with respective

unit η′H and counit ε′H , and furthermore we are given a natural transformation ν : H1 ⇒ H ′1.Then we get as in (ii) the natural transformation ν† : H ′2 ⇒ H2, and we have the identity

(ν ∗ τ)† = τ † ∗ ν†

provided the left hand-side is defined via (η, ε) and via the unit and counit for the adjoint pair(H ′2 G′, H ′1 F ′) induced by (η′, ε′) and (η′H , ε

′H).

(vi) Especially, if we use (η, ε) and (η, ε) to define (H2 ∗ τ)†, we have :

(H2 ∗ τ)† = τ † ∗H1

(all these assertions are exercises for the reader : see also [41, §I.6]).

Proposition 1.1.11. Let F : A → B be a functor.

(i) The following conditions are equivalent :(a) F is fully faithful and has a fully faithful left adjoint.(b) F is an equivalence.

(ii) Suppose that F admits a left adjoint G : B → A , and let η : 1B ⇒ F G andε : G F ⇒ 1A be a unit and respectively counit for the adjoint pair (G,F ). Then :(a) F (resp. G) is faithful if and only if εX (resp. ηY ) is an epimorphism for every

X ∈ Ob(A ) (resp. a monomorphism for every Y ∈ Ob(B)).(b) F (resp. G) is fully faithful if and only if ε (resp. η) is an isomorphism of functors.

(iii) Suppose that F admits both a left adjoint G : B → A and a right adjoint H : B →A . Then G is fully faithful if and only if H is fully faithful.

Proof. These assertions are [10, Prop.3.4.1, 3.4.2, 3.4.3]; see also [51, Prop.1.5.6].


1.1.12. A standard construction associates to any object X ∈ Ob(C ) a category :

C/X

as follows. The objects of C/X are all the pairs (A,ϕ) whereA ∈ Ob(C ) and ϕ : A→ X is anymorphism of C . For any ϕ : A→ X , and ψ : B → X , the morphisms HomC/X((A,ϕ), (B,ψ))are all the commutative diagrams :

A //

ϕ @@@@@@@ B

ψ~~~~~~~~~~

X

of morphisms of C , with composition of morphisms induced by the composition law of C . Anobject (resp. a morphism) of C/X is also called an X-object (resp. an X-morphism) of C .Dually, one defines

X/C := (C o/Xo)o

i.e. the objects of X/C are the pairs (A,ϕ) with A ∈ Ob(C ) and ϕ : X → A any morphism ofC . We have an obvious faithful source functor :

(1.1.13) C/X → C (A,ϕ) 7→ A

and likewise one obtains a target functor X/C → C . Moreover, any morphism f : X → Y inC induces functors :

(1.1.14)f∗ : C/X → C/Y : (A, g : A→ X) 7→ (A, f∗g := f g : A→ Y )

f ∗ : Y/C → X/C : (B, h : Y → B) 7→ (B, f ∗h := h f : X → B).

Furthermore, given a functor F : C → B, any X ∈ Ob(C ) induces functors :

(1.1.15)F|X : C/X → B/FX : (A, g) 7→ (FA, Fg)

X|F : X/C → FX/B : (B, h) 7→ (FB,Fh).

1.1.16. The categories C /X and X/C are special cases of the following more general con-struction. Let F : A → B be any functor. For any B ∈ Ob(B), we define FA /B as thecategory whose objects are all the pairs (A, f), where A ∈ Ob(A ) and f : FA → B is amorphism in B. The morphisms g : (A, f) → (A′, f ′) are the morphisms g : A → A′ in Asuch that f ′ Fg = f . There are well-defined functors :

F/B : FA /B → B/B : (A, f) 7→ (FA, f) and ιB : FA /B → A : (A, f) 7→ A.

Dually, we define :

B/FA := (F oA o/Bo)o

and likewise one has natural functors :

B/F : B/FA → B/B and ιB : B/FA → A .

Obviously, the category C/X (resp. X/C ) is the same as 1C C /X (resp. X/1C C ).Any morphism g : B′ → B induces functors :

g/FA : B/FA → B′/FA : (A, f) 7→ (A, f g)

FA /g : FA /B′ → FA /B : (A, f) 7→ (A, g f).


1.1.17. Let C be a category; the categories of the form C/X and X/C can be faithfully embed-ded in a single category Morph(C ). The objects of Morph(C ) are all the data (A,B, ϕ), whereA,B ∈ Ob(C ) and ϕ : A→ B is any morphism of C . If f : A→ B and f ′ : A′ → B′ are twosuch morphisms, the set HomMorph(C )((A,B, f), (A′, B′, f ′)) consists of all the commutativediagrams of morphisms of C :

(1.1.18)A

f //

g

B

g′

A′

f ′ // B′

with composition of morphisms induced by the composition law of C , in the obvious way.There are two natural source and target functors :

Cs←− Morph(C )

t−→ C

such that s(A → B) := A, t(A → B) := B for any object A → B of Morph(C ), ands(1.1.18) = g, t(1.1.18) = g′. Especially, the functor (1.1.13) is the restriction of s to thesubcategory C/X .

1.1.19. Let C be a category; a very important construction associated to C is the category

C ∧U := Fun(C o,U-Set)

whose objects are called the U-presheaves on C . The morphisms in C ∧U are the natural trans-formations of functors. We usually drop the subscript U, unless we have to deal with more thanone universe. Let us just remark that, if U′ is another universe, and U ⊂ U′, the natural inclusionof categories :

C ∧U → C ∧U′

is fully faithful. If C has small Hom-sets (see (1.1.1)), there is a natural functor

h : C → C ∧

(the Yoneda embedding) which assigns to any X ∈ Ob(C ) the functor

hX : C o → Set Y 7→ HomC (Y,X) for every Y ∈ Ob(C )

and to any morphism f : X → X ′ in C , the natural transformation hf : hX ⇒ hX′ such that

hf,Y (g) := f g for every Y ∈ Ob(C ) and every g ∈ HomC (Y,X).

Proposition 1.1.20 (Yoneda’s lemma). With the notation of (1.1.19), we have :(i) The functor h is fully faithful.

(ii) Moreover, for every F ∈ Ob(C ∧), and every X ∈ Ob(C ), there is a natural bijection

F (X)∼→ HomC∧(hX , F )

functorial in both X and F .

Proof. Clearly it suffices to check (ii). However, the sought bijection is obtained explicitly asfollows. To a given a ∈ F (X), we assign the natural transformation τa : hX ⇒ F such that

τa,Y (f) := Ff(a) for every Y ∈ Ob(C ) and every f ∈ hX(Y ).

Conversely, to a given natural transformation τ : hX ⇒ F we assign τX(1X) ∈ F (X). As anexercise, the reader can check that these rules establish mutually inverse bijections. The functo-riality in F is immediate, and the functoriality in the argument X amounts to the commutativity


of the diagram

HomC∧(hX′ , F )∼ //

HomC∧ (hϕ,F )

F (X ′)

F (ϕ)

HomC∧(hX , F )

∼ // F (X)

for every morphism ϕ : X → X ′ in C : the verification shall also be left to the reader.

An object F of C ∧ is a representable presheaf , if it is isomorphic to hX , for some X ∈Ob(C ). Then, we also say that F is representable in C .

1.1.21. We wish to explain some standard constructions of presheaves that are in constant usethroughout this work. Namely, let I be a small category, C a category with small Hom-sets,and X any object of C . We denote by cX : I → C the constant functor associated to X :

cX(i) := X for every i ∈ Ob(I) cX(ϕ) := 1X for every ϕ ∈ Morph(I).

Any morphism f : X ′ → X induces a natural transformation

cf : cX′ ⇒ cX by the rule : (cf )i := f for every i ∈ I.

Definition 1.1.22. With the notation of (1.1.21), let F : I → C be any functor.(i) The limit of F is the presheaf on C denoted

limIF : C o → Set

and defined as follows. For any X ∈ Ob(C ), the set limI F (X) consists of all thenatural transformations cX ⇒ F ; and any morphism f : X ′ → X induces the map

limIF (f) : lim

IF (X)→ lim

IF (X ′) τ 7→ τ cf for every τ : cX ⇒ F .

(ii) Dually, the colimit of F is the presheaf on C o

colimI

F := limIoF o.

(iii) We say that C is complete (resp. cocomplete) if, for every small category I and everyfunctor F : I → C , the limit (resp. the colimit) of F is representable in C (resp. inC o).

Remark 1.1.23. (i) In the situation of (1.1.21), let F, F ′ : I → C be two functors, andg : F ⇒ F ′ a natural transformation; we deduce a morphism of presheaves on C

limIg : lim

IF → lim

IF ′ τ 7→ g τ for every X ∈ Ob(C ) and every τ : cX ⇒ F .

(ii) Likewise, g induces a morphism of presheaves on C o

colimI

g : colimI

F ′ → colimI

F τ o 7→ τ o go for every X ∈ Ob(C ) and τ : cX ⇒ F .

(iii) With the notation of (ii), suppose that the colimits of F and F ′ are representable byobjectsCo

F , respectivelyCoF ′ of C o. Then the colimit of g corresponds to a morphismCo

F ′ → CoF

in C o, i.e. a morphism CF → CF ′ in C .(iv) If ϕ : I ′ → I and H : C → C ′ are any two functors, we obtain a natural transformation

limϕH : lim

IF ⇒ (lim

I′H F ϕ) Ho

by ruling that limϕH(X)(τ) := H ∗ τ ∗ ϕ for every X ∈ Ob(C ) and every τ : cX ⇒ F . Thereader may spell out the corresponding assertion for colimits.


(v) Let I a small category, F : C → C ′ and H : I → C two functors, and suppose the limitof H is representable by an object L of C , so there exists an isomorphism of presheaves

hL∼→ lim

IH

under which, the image of 1L ∈ hL(L) corresponds to a natural transformation τ : cL ⇒ Hfrom the constant functor cL : I → C , to H . There follows a natural transformation

F ∗ τ : cFL ⇒ F Hof functors I → C ′. In this situation, we say that F commutes with the limit of H , if F ∗ τinduces an isomorphism of functors (defined as in (i))

limIF ∗ τ : hFL

∼→ limIcL

∼→ limIF H

in which case FL represents the limit of F H . Likewise, we say that F commutes with thecolimit of H , if the dual condition holds, i.e. if F o commutes with the limit of Ho.

Example 1.1.24. (i) For i = 1, 2, let fi : A → Bi be two morphisms in a category C withsmall Hom-sets; the push-out or coproduct of f1 and f2 is the colimit of the functor F : I → C ,defined as follows. The set Ob(I) consists of three objects s, t1, t2 and Morph(I) consists oftwo morphisms ϕi : s→ ti for i = 1, 2 (in addition to the identity morphisms of the objects ofI); the functor is given by the rule : Fs := A, Fti := Bi and Fϕi := fi (for i = 1, 2).

If C ∈ Ob(C ) represents the coproduct of f1 and f2, we have two morphisms f ′i : Bi → C(i = 1, 2) such that f ′1 f1 = f ′2 f2; in this case, we say that the commutative diagram :

Af1 //

f2

B1

f ′1

B2

f ′2 // C

is cocartesian. Dually one defines the fibre product or pull-back of two morphisms gi : Ai → B.If D ∈ Ob(C ) represents this fibre product, we have morphisms g′i : D → Ai such thatg1 g′1 = g2 g′2, and we say that the diagram :

Dg′1 //

g′2

A1

g1

A2

g2 // B

is cartesian. The coproduct of f1 and f2 is usually called the coproduct of B1 and B2 overA, denoted B1 q(f1,f2) B2, or simply B1 qA B2, unless the notation gives rise to ambiguities.Likewise one writes A1 ×(g1,g2) A2, or just A1 ×B A2 for the fibre product of g1 and g2.

(ii) As a special case, if B := B1 = B2, the coproduct BqAB is also called the coequalizerof f1 and f2, and is sometimes denoted Coequal(f1, f2). Likewise, if A := A1 = A2, the fibreproduct A×B A is also called the equalizer of g1 and g2, sometimes denoted Equal(g1, g2).

(iii) Furthermore, let f : A→ B be any morphism in C ; notice that the identity morphismsof A and B induce natural morphisms of presheaves

πf : Coequal(f, f)→ B ιf : A→ Equal(f, f).

and it is easily seen that f is a monomorphism (resp. an epimorphism) if and only if ιf (resp.πf ) is an isomorphism in C ∧ (here we abuse notation, by writing A and B instead of the corre-sponding presheaves hA and hB).

(iv) Let I ∈ U be any small set, and B := (Bi | i ∈ I) any family of objects of C . Wemay regard I as a discrete category (see (1.1.1)), and then the rule i 7→ Bi yields a functor


I → C , whose limit (resp. colimit) is called the product (resp. coproduct) of the family B, andis denoted

∏i∈I Bi (resp.

∐i∈I Bi).

(v) In the situation of (1.1.21), let f : F ⇒ cX be a natural transformation, and g : Y → Xany morphism in C . Suppose that all fibre products are representable in C , and consider thefunctor

F ×X Y : I → C i 7→ F (i)×(fi,g) Y

(which means that for every i ∈ Ob(I) we pick an object of C representing the above fibreproduct, and to a morphism ϕ : i→ j in I , we attach the morphism F (ϕ)×X 1Y , with obviousnotation.) The projections induce a commutative diagram

F ×X Y +3

F

f

cY

cg +3 cX .

Suppose moreover that the colimit of F is representable by an object Co of C o, and for everyX ∈ Ob(C ), and every f , g as above, the colimit of F ×X Y is representable by an object Co

Y

of C o. By remark 1.1.23(iii) we deduce a natural morphism in C :

(1.1.25) CY → C ×X Y.

We say that the colimit of F is universal if the resulting morphism (1.1.25) is an isomorphismfor everyX ∈ C and every f and g as above. In this case, clearly C×X Y represents the colimitof F ×X Y .

(vi) Let f : X → Y be a morphism in C , and suppose that, for every other morphismY ′ → Y , the fibre product X ×Y Y ′ is representable in C . In this case, we say that f is auniversal monomorphism (resp. a universal epimorphism) if f ×X Y : X ×Y Y ′ → Y ′ is amonomorphism (resp. an epimorphism) for every morphism Y ′ → Y in C .

(vii) Suppose that all fibre products are representable in C ; in this case, notice that, in viewof (iii), the morphism f is a universal epimorphism if and only if f is an epimorphism and thecoequalizer of f and f is a universal colimit.

(viii) Suppose that C is complete and well-powered (see (1.1.3)). Then, for every morphismf : X → Y we may define the image of f , which is a monomorphism :

Im(f)→ Y

defined as the smallest of the family F of subobjects Y ′ → Y such that f factors through a(necessarily unique) morphism X → Y ′. Indeed choose, for every equivalence class c ∈ F ,a representing monomorphism Yc → Y , and let I be the full subcategory of C/Y such thatOb(I) = Yc → Y | c ∈ F; then I is a small category, and the image of f is more preciselythe limit of the inclusion functor ι : I → C/Y (which is representable, since C is complete).

As an exercise, the reader can show that the resulting morphism X → Im(f) is an epimor-phism.

Example 1.1.26. Let C be a category, and X ∈ Ob(C ).(i) We say that X is an initial (resp. final) object of C if HomC (X, Y ) (resp. HomC (Y,X))

consists of exactly one element, for every Y ∈ Ob(C ).(ii) It is easily seen that an initial object X of C represents the empty coproduct in C , i.e. the

coproduct of an empty family of objects of C , as in example 1.1.24(iv). Dually, a final objectrepresents the empty product in C .

(iii) We say thatX is disconnected, if there existA,B ∈ Ob(C ), neither of which is an initialobject of C , and such that X represents the coproduct A q B. We say that X is connected, ifX is not disconnected.


Example 1.1.27. (i) The category Set is complete and cocomplete, and all colimits in Set areuniversal. Hence, all epimorphisms of Set are universal, in view of example 1.1.24(vii).

(ii) Also the category Cat is complete and cocomplete. For instance, for any pair of functors

AF−−→ C

G←−− B

the fibre product (in the category Cat) of F and G is the category :

A ×(F,G) B

whose set of objects is Ob(A )×Ob(C ) Ob(B); the morphisms (A,B)→ (A′, B′) are the pairs(f, g) where f : A → A′ (resp. g : B → B′) is a morphism in A (resp. in B), such thatFf = Gg. If the notation is not ambiguous, we may also denote this category by A ×C B.In case B is a subcategory of C and G is the natural inclusion functor, we also write F−1Binstead of A ×C B. See [10, Prop.5.1.7] for a proof of the cocompleteness of Cat.

1.1.28. Let I , C be two small categories, F : I → C a functor, and G any presheaf on C . Wededuce a functor

HomC∧(G, h F ) : I → Set i 7→ HomC∧(G, hF (i)) for every i ∈ I

and by inspecting the definitions, we find a natural isomorphism :

(1.1.29) limI

HomC∧(G, h F )∼→ HomC∧(G, lim

IF )

(more precisely, the limit on the left is represented by the set on the right). Likewise, we have anatural isomorphism :

(1.1.30) limIo

HomC∧(h F,G)∼→ HomC∧(colim

IF,G).

1.1.31. Suppose that F : A → B is right adjoint to a functor G : B → A . Then it is easilyseen that F commutes with all representable limits of A , in the sense of remark 1.1.23(v). Du-ally,G commutes with all representable colimits of B. It is also easy to check that F transformsmonomorphisms into monomorphisms, and G transforms epimorphisms into epimorphisms.

Conversely, we have the following :

Theorem 1.1.32. Let A be a complete category, F : A → B a functor, and suppose that Aand B have small Hom-sets (see (1.1.1)). The following conditions are equivalent :

(a) F admits a left adjoint.(b) F commutes with all the limits of A , and every object B of B admits a solution set,

i.e. an essentially small subset SB ⊂ Ob(A ) such that, for every A ∈ Ob(A ), everymorphism f : B → FA admits a factorization of the form f = Fhg, where h : A′ →A is a morphism in A with A′ ∈ SB, and g : B → FA′ is a morphism in B.

Proof. This is [10, Th.3.3.3]. Basically, one would like to construct a left adjoint G explicitlyas follows. For any B ∈ Ob(B) and any morphism f : B′ → B, set :

GB := limB/FA

ιB Gf := limf/FA

1A .

(notation of (1.1.16)). The problem with this is that the categories B/FA and f/FA are notnecessarily small, so the above limits do not always exist. The idea is to replace the categoryB/FA by its full subcategory EB, whose objects are all the morphismsB → FAwithA ∈ SB;then it is easily seen that this is isomorphic to a small category, and the limit over EB of therestriction of ιB yields the sought adjoint : see loc.cit. for the details.

Of course, the “dual” of theorem 1.1.32 yields a criterion for the existence of right adjoints.


1.1.33. The notion of Kan extension of a given functor yields another frequently used methodto produce (left or right) adjoints, which applies to the following situation. Let f : A → B bea functor, and C a third category. Then we deduce a natural functor

f ∗ : Fun(B,C )→ Fun(A ,C ) : G 7→ G f (α : G⇒ G′) 7→ α ∗ f.

Proposition 1.1.34. In the situation of (1.1.33), suppose that A is small, B and C have smallHom-sets, and C is cocomplete (resp. complete). Then f ∗ admits a left (resp. right) adjoint.

Proof. Indeed, if C is cocomplete, a left adjoint f! : Fun(A ,C )→ Fun(B,C ) to f ∗ is givenexplicitly as follows. For a given functor F : A → C , define f!F by the rule :

B 7→ colimfA /B

F ιB ϕ 7→ (colimfA /ϕ

1C : colimfA /B

F ιB → colimfA /B′

F ιB′)

for everyB ∈ Ob(B) and every morphism ϕ : B → B′ in B (notation of (1.1.16)). This makessense, since – under the current assumptions – the categories fA /B and fA /ϕ are small.

The construction of a right adjoint, in case C is complete, is dual to the foregoing, by virtueof the isomorphism of categories :

Fun(D ,C )∼→ Fun(Do,C o)o for every category D .

See [10, Th.3.7.2] or [51, Th.2.3.3] for the detailed verifications.

1.1.35. As an application, suppose that B is a small category, and C a category with smallHom-sets; any functor f : B → C induces a functor

f ∗U : C ∧U → B∧U F 7→ F f o

and it is easily seen that f ∗U commutes with all limits and all colimits. From proposition 1.1.34we obtain both a left and a right adjoint for f ∗U, denoted respectively :

fU! : B∧U → C ∧U and fU∗ : B∧U → C ∧U .

As usual, we drop the subscript U, unless the omission may cause ambiguities.Notice that the diagram of functors :

(1.1.36)B

hB //

f

B∧

f!

C

hC // C ∧

(whose horizontal arrows are the Yoneda imbeddings) is essentially commutative, i.e. the twocompositions f! hB and hC f are isomorphic functors. Indeed – by of proposition 1.1.20(ii)– for every B ∈ Ob(B), the objects f!hB and hfB both represent the functor

C ∧ → Set : F 7→ F (fB).

Definition 1.1.37. Let C be a category.(i) We say that C is finite if both Ob(C ) and Morph(C ) are finite sets.

(ii) We say that C is path-connected, if Ob(C ) 6= ∅ and every two objects X , Y can beconnected by a finite sequence of morphisms in C , of arbitrary length :

X → Z1 ← Z2 → · · · ← Zn → Y

(iii) We say that C is directed, if for every X, Y ∈ Ob(C ) there exist Z ∈ Ob(C ) andmorphisms X → Z, Y → Z in C . We say that C is codirected, if C o is directed.

(iv) We say that C is locally directed (resp. locally codirected) if, for every X ∈ Ob(C ),the category X/C is directed (resp. codirected).


(v) We say that C is pseudo-filtered if it is locally directed, and the following holds. Forany X, Y ∈ Ob(C ), and any two morphisms f, g : X → Y , there exists Z ∈ Ob(C )and a morphism h : Y → Z such that h f = h g.

(vi) We say that C is filtered, if it is pseudo-filtered and path-connected (in which case, Cis also directed). We say that C is cofiltered if C o is filtered (in which case, C is alsocodirected).

(vii) A limit limC F is path-connected (resp. codirected, resp. locally codirected, resp.cofiltered, resp. finite), if C is path-connected (resp. codirected, resp. locally codi-rected, resp. cofiltered, resp. finite). Dually, one defines path-connected, directed,locally directed, filtered and finite colimits.

(viii) We say that a functor F : A → B is left exact, if F commutes with all finite limits,in the sense of remark 1.1.23(v). Dually, F is right exact if it commutes with all finitecolimits. Finally, F is exact if it is both left and right exact.

Remark 1.1.38. (i) Notice that, if I is a filtered category, and i is any object of I , the categoryi/I is again filtered; dually, if I is cofiltered, the category I/i is again cofiltered. Furthermore,let F : I → A be any functor; there follow functors F t : i/I → A and F s : I/i → A(notation of (1.1.17)), and we have natural identifications :

colimI

F∼→ colim

i/IF t lim

IF∼→ lim

I/iF s.

(ii) Let C be a small category. There is a natural decomposition in Cat :

C∼→∐i∈I

Ci

where each Ci is a path-connected category, and I is a small set. This decomposition inducesnatural isomorphisms in A ∧ :

colimC

F∼→∐i∈I

colimCi

F ei limCF∼→∏i∈I

limCiF ei

for any functor F : C → A , where ei : Ci → C is the natural inclusion functor, for every i ∈ I .The details shall be left to the reader. These simple observations often simplify the calculationof limits and colimits.

(iii) Let C be a complete category (more generally, a category in which all fibre productsare representable), and F : C → B a left exact functor. If follows formally from example1.1.24(iii) that F transforms monomorphisms into monomorphisms. If F is also conservative,then a morphism ϕ : X → Y in C is a monomorphism if and only if the same holds for Fϕ.

(iv) Dually, if F is right exact, and all coproducts are representable in C , example 1.1.24(iii)implies that F transforms epimorphism into epimorphisms, and if F is also conservative, thena morphism ϕ : X → Y in T is an epimorphism if and only if the same holds for Fϕ.

(v) In the situation of (1.1.33), suppose that the category fA /B is finite for every B ∈Ob(B); then, by inspecting the proof of proposition 1.1.34, we see that f ∗ admits a left adjoint,provided all finite colimits of C are representable and C has small Hom-sets. Likewise, we canweaken the condition for the existence of the right adjoint : for the latter, it suffices that all finitelimits of C are representable and C has small Hom-sets.

1.2. Tensor categories and abelian categories. In this section we assemble some basic defi-nitions and results that pertain to abelian categories and other related classes of categories withextra structure.

Definition 1.2.1. A tensor category is a datum C := (C ,⊗,Φ,Ψ) consisting of a category C ,a functor

⊗ : C × C → C : (X, Y ) 7→ X ⊗ Y


and natural isomorphisms :

ΦX,Y,Z : X ⊗ (Y ⊗ Z)∼→ (X ⊗ Y )⊗ Z ΨX,Y : X ⊗ Y ∼→ Y ⊗X

for every X, Y, Z ∈ Ob(C ), called respectively the associativity and commutativity constraintsof C , that satisfy the following axioms.

(a) Coherence axiom : the diagram

X ⊗ (Y ⊗ (Z ⊗ T ))ΦX,Y,Z⊗T //

X⊗ΦY,Z,T

(X ⊗ Y )⊗ (Z ⊗ T )ΦX⊗Y,Z,T // ((X ⊗ Y )⊗ Z)⊗ T

X ⊗ ((Y ⊗ Z)⊗ T )ΦX,Y⊗Z,T // (X ⊗ (Y ⊗ Z))⊗ T

ΦX,Y,Z⊗TOO

commutes, for every X, Y, Z, T ∈ Ob(C ).(b) Compatibility axiom : the diagram

X ⊗ (Y ⊗ Z)ΦX,Y,Z //

X⊗ΨY,Z

(X ⊗ Y )⊗ ZΨX⊗Y,Z // Z ⊗ (X ⊗ Y )

ΦZ,X,Y

X ⊗ (Z ⊗ Y )ΦX,Z,Y // (X ⊗ Z)⊗ Y

ΨX,Z⊗Y // (Z ⊗X)⊗ Y

commutes, for every X, Y, Z ∈ Ob(C ).(c) Commutation axiom : we have ΨY,X ΨX,Y = 1X⊗Y for every X, Y ∈ Ob(C )

(d) Unit axiom : there exists an object U ∈ Ob(C ) and an isomorphism u : U∼→ U ⊗ U

such that the functor

(1.2.2) C → C : X 7→ U ⊗Xis an equivalence. One says that (U, u) is a unit object of C .

Lemma 1.2.3. Let C := (C ,⊗,Φ,Ψ) be any tensor category. The diagram

X ⊗ (Y ⊗ Z)ΦX,Y,z //

X⊗ΨY,Z

(X ⊗ Y )⊗ ZΨX,Y ⊗Z // (Y ⊗X)⊗ Z

X ⊗ (Z ⊗ Y )ΦX,Z,Y // (X ⊗ Z)⊗ Y

ΨX⊗Z,Y // Y ⊗ (X ⊗ Z)

ΦY,X,Z

OO

commutes, for every X, Y, Z ∈ Ob(C ).

Proof. To ease notation, we shall omit the tensor symbol ⊗ between objects, and we shall dropthe subscript from Φ and Ψ when we display a diagram. It suffices to consider the diagram

Y (XZ)Φ //

Y⊗Ψ

(Y X)Z

Ψ

Y (ZX)Φ // (Y Z)X

Ψ⊗X // (ZY )X Z(Y X)Φoo

(XZ)YΨ⊗Y //

Ψ

BB(ZX)Y

Ψ

OO

Z(XY )

Z⊗Ψ

OO

Φoo (XY )ZΨoo

Ψ⊗Z

\\999999999999999999

X(ZY )

Φ

OO

X(Y Z)X⊗Ψoo

Φ

OO

whose two triangular subdiagrams commute by naturality of Ψ, and whose three rectangularsubdiagrams commute by compatibility.


Definition 1.2.4. Let C := (C ,⊗,Φ,Ψ) and C ′ := (C ′,⊗′,Φ′,Ψ′) be two tensor categories.A tensor functor C → C ′ is a pair (F, c) consisting of a functor F : C → C ′ and a naturalisomorphism

cX,Y : FX ⊗ FY ∼→ F (X ⊗ Y ) for all X, Y ∈ Ob(C )

such that the following holds.(a) For every objects X, Y, Z of C , the diagram

FX ⊗ (FY ⊗ FZ)FX⊗cY Z //

Φ′FX,FY,FZ

FX ⊗ F (Y ⊗ Z)cX,Y⊗Z // F (X ⊗ (Y ⊗ Z))

F (ΦX,Y,Z)

(FX ⊗ FY )⊗ FZ

cX,Y ⊗FZ // F (X ⊗ Y )⊗ FZcX⊗Y,Z // F ((X ⊗ Y )⊗ Z)

commutes.(b) For all objects X , Y of C , the diagram

FX ⊗ FYcX,Y //

ΨFX,FY )

F (X ⊗ Y )

F (ΨX,Y )

FY ⊗ FX

cY,X // F (Y ⊗X)

commutes.(c) If (U, u) is a unit object of C , then (FU, c−1

U,U Fu) is a unit object of C ′.

Remark 1.2.5. (i) Lemma 1.2.3 illustrates a general principle valid in every tensor categor C :namely, say that X1, . . . , Xn is a sequence of distinct objects of C , and X ′ and X ′′ are obtainedfrom these two sequences by taking tensor products several times, and in any order, in whichcase we say that X ′ and X ′′ have no repetitions. Now, there will be usually various ways tocombine the associativity and commutativity constraints, in order to exhibit some isomorphismX ′

∼→ X ′′. However, the resulting isomorphism shall be independent of the way in which it isexpressed as such a combination. This follows from a theorem of Mac Lane. To formalize thisresult, one could observe that, for any set Σ, there exists a universal tensor category TΣ “gener-ated by Σ”, i.e. such that – for any other tensor category C – any mapping Σ→ Ob(C ) extendsuniquely, up to isomorphism, to a tensor functor TΣ → C . Then Mac Lane’s theorem says that,for every set Σ, and every object X ∈ Ob(TΣ) that has no repetitions, the group AutTΣ

(X)is trivial. Instead of relying on such a general result, we shall make ad hoc verifications, as inthe proof of lemma 1.2.3 and of the forthcoming proposition 1.2.6. However, in view of thisprinciple, in the following we shall often omit a detailed description of the isomorphism thatwe choose to connect two given objects that are thus related : the reader will be able in anycase to produce one isomorphism, and the principle says that these choices cannot be source ofambiguities.

(ii) Let D be any category, and C a tensor category. Then notice that Fun(D ,C ) inheritsfrom C a natural tensor category structure : we leave to the reader the task of spelling out thedetails. Moreover notice that, if (F, c) : C 1 → C 2 and (F ′, c′) : C 2 → C 3 are any two tensorfunctors between tensor categories, then the composition

(F ′ F, (F ′ ∗ c) (c′ ∗ (F × F ))) : C 1 → C 3

is again a tensor functor.

Proposition 1.2.6. Let (U, u) be a unit object of a tensor category C . Then there exists a uniquenatural isomorphism

uX : X∼→ U ⊗X for every X ∈ Ob(C )


such that uU = u, and such that the diagrams

X ⊗ YuX⊗Y //

uX⊗Y ))SSSSSSSSSSSSSSS U ⊗ (X ⊗ Y )

ΦU,X,Y

X ⊗ YuX⊗Y //

X⊗uY

(U ⊗X)⊗ YΨU,X⊗Y

(U ⊗X)⊗ Y X ⊗ (U ⊗ Y )ΦX,U,Y // (X ⊗ U)⊗ Y

commute for every X, Y ∈ Ob(C ).

Proof. Since (1.2.2) is an equivalence, there exists a unique isomorphism uX fitting into thecommutative diagram

UXu⊗X //

U⊗uX ((PPPPPPPPPPPPP (UU)X

U(UX).

Φ

OO

With this definition, the naturality of the rule : X 7→ uX is clear. In order to check the commu-tativity of the first diagram, it suffices to show that

(1.2.7) (U ⊗ ΦU,X,Y ) (U ⊗ uXY ) = U ⊗ (uX ⊗ Y ).

However, set Θ := (ΦU,U,X ⊗ Y ) ΦU,UX,Y ; we have :

Θ (U ⊗ (uX ⊗ Y )) = (ΦU,U,X ⊗ Y ) ((U ⊗ uX)⊗ Y ) ΦU,X,Y

= ((u⊗X)⊗ Y ) ΦU,X,Y

= ΦUU,X,Y (u⊗ (XY ))

= ΦUU,X,Y ΦU,U,XY (U ⊗ uXY )

= Θ (U ⊗ ΦU,X,Y ) (U ⊗ uXY )

where the first and third identities hold by naturality of Φ, the second and fourth by the definitionof uX and respectively uX⊗Y , and the fifth by coherence. Since Θ is an isomorphism, we get(1.2.7). Next, in light of the foregoing, the second diagram commutes if and only if the diagram

(1.2.8)

XYuXY //

X⊗uY ''PPPPPPPPPPPPP U(XY )Φ // (UX)Y

Ψ⊗Y

X(UY )Φ // (XU)Y

commutes, and it suffices to check that U ⊗ (1.2.8) commutes. To this aim, we consider thediagram

U(XY )U⊗(X⊗uY )

//

u⊗(XY )

U⊗uXY

''OOOOOOOOOOOU(X(UY ))

U⊗Φ // U((XU)Y )

U⊗(Ψ⊗Y )

U(U(XY ))U⊗Φ //

Φ

wwoooooooooooU((UX)Y )

Φ

(UU)(XY )Φ // ((UU)X)Y (U(UX))Y.

Φ⊗Yoo

whose lower subdiagram commutes by the coherence axiom for (U,U,X, Y ), whose uppersubdiagram is equivalent to (1.2.8), since Ψ−1

U,X = ΨX,U , and whose triangular subdiagram


commutes by definition of uXY . Hence, we are reduced to showing that the outer rectangularsubdiagram of the above diagram commutes. However, we have a commutative diagram :

U(X(UY ))

Φ

U(XY )U⊗(X⊗uY )oo

u⊗(XY )//

Φ

(UU)(XY )

Φ

(UX)(UY )

Ψ⊗(UY )

(UX)Y(UX)⊗uYoo

(u⊗X)⊗Y//

Ψ⊗Y

((UU)X)Y

Ψ⊗Y

(XU)(UY ) (XU)Y(X⊗u)⊗Y

//(XU)⊗uYoo (X(UU))Y

X(UY )X⊗(U⊗uY )

uullllllllllllllX⊗(u⊗Y )

))RRRRRRRRRRRRRR

Φ

OO

X(U(UY ))X⊗Φ //

Φ

OO

X((UU)Y )

Φ

OO

so we are further reduced to checking the commutativity of the diagram :

U(X(UY ))U⊗Φ //

Φ

wwoooooooooooU((XU)Y )

U⊗(Ψ⊗Y )// U((UX)Y )

Φ

''OOOOOOOOOOO

(UX)(UY )

Ψ⊗(UY )

U((UY )X)

U⊗Ψ

OO

Φ

U(U(Y X))U⊗Φoo

U⊗(U⊗Ψ)//

Φ

U(U(XY ))

U⊗Φ

OO

Φ

(U(UX))Y

Φ⊗Y

(XU)(UY ) (U(UY ))X

Ψ Φ⊗X ''OOOOOOOOOOOO

(UU)(Y X)(UU)⊗Ψ

//

Φ

(UU)(XY )Φ // ((UU)X)Y

Ψ⊗Ywwooooooooooo

X(U(UY ))

Φ

ggOOOOOOOOOOO

X⊗Φ ''OOOOOOOOOOOO((UU)Y )X

Ψ

(X(UU))Y

X((UU)Y ).

Φ

55kkkkkkkkkkkkkkk

However, the leftmost and the lower triangular subdiagrams commute by compatibility, and theupper rightmost subdiagram commutes by coherence. The lower leftmost subdiagram com-mutes by naturality of Ψ, and the central square sudiagram commutes by naturality of Φ. Theremaining central subdiagram commutes by coherence, and the top rectangular subdiagram isof the form U ⊗D, where D is a diagram that commutes by virtue of lemma 1.2.3.

The uniqueness of uX is clear by inspecting the second diagram, with Y = U .

Remark 1.2.9. (i) Keep the notation of proposition 1.2.6. As a consequence of the naturalityof uX , we get a commutative diagram

XuX //

uX

U ⊗Xu⊗uX

U ⊗XuU⊗X // U ⊗ (U ⊗X)

for every object X of C . In other words :

uU⊗X = U ⊗ uX for every X ∈ Ob(C ).


(ii) Let X = Y = U in the second diagram of proposition 1.2.6; we obtain a commutativediagram

U ⊗ U u⊗U //

U⊗u

(U ⊗ U)⊗ UΨU,U⊗U

U ⊗ (U ⊗ U)ΦU,U,U // (U ⊗ U)⊗ U

Since u = uU , we may combine with (i), to deduce that ΨU,U ⊗ U = 1(U⊗U)⊗U = 1U⊗U ⊗ U .By naturality of Ψ, it follows that U ⊗ ΨU,U = U ⊗ 1U⊗U , and since (1.2.2) is an equivalence,we conclude that

ΨU,U = 1U⊗U .

Example 1.2.10. Let C be any category with small Hom-sets, in which finite products arerepresentable. For every pair (X, Y ) of objects of C , pick an object X ⊗ Y representing theirproduct, and fix also two projections pX,Y : X ⊗ Y → X , qX,Y : X ⊗ Y → Y inducing anisomorphism of functors

hX⊗Y∼→ hX×hY : ϕ 7→ (pX,Y ϕ, qX,Y ϕ) for every Z ∈ Ob(C ) and ϕ ∈ hX×Y (Z)

(notation of (1.1.19)). If (X ′, Y ′) is another such pair, and (g, h) : (X, Y ) → (X ′, Y ′) anymorphism in C × C , then there exists a unique morphism f : X ⊗ Y → X ′ ⊗ Y ′ such that

(pX′,Y ′ f, qX′,Y ′ f) = (g pX,Y , h pX,Y )

and we set g ⊗ h := f . These rules define a functor ⊗ : C × C → C . For every three objectsX, Y, Z there is a natural isomorphismX⊗(Y ⊗Z)

∼→ (X⊗Y )⊗Z that yields an associativityconstraint for ⊗; namely, we let

ΦX,Y,Z := (pX,Y⊗Z ⊗ (pY,Z ⊗ qX,Y⊗Z))⊗ (qY,Z qX,Y⊗Z).

Likewise, we get a commutativity constraint by setting

ΨX,Y := qX,Y ⊗ pX,Y for every X, Y ∈ Ob(C ).

The verifications of the axioms of definition 1.2.1 are lengthy but straightforward, and shallbe left to the reader. If U is any final object of C (example 1.1.26(i)), then there exists aunique morphism u : U → U ⊗ U which is easily seen to be an isomorphism, and the pair(U, u) yields a unit for ⊗. In this way, any category with finite products and small Hom-sets isnaturally endowed with a structure of tensor category. Notice that, for this tensor structure onC (and indeed, for most of the tensor categories that are found in applications), the existence offunctorial isomorphisms uX fulfilling the conditions of proposition 1.2.6, is self-evident.

Definition 1.2.11. Let (C ,⊗,Φ,Ψ) be a tensor category, X ∈ Ob(C ) any object, and supposethat the functor

−⊗X : C → C Y 7→ Y ⊗Xadmits a right adjoint :

Hom(X,−) : C → C Y 7→Hom(X, Y ).

Then, we call Hom(X,−) the internal Hom functor for the object X .

Remark 1.2.12. (i) As usual, the internal Hom functor is determined up to unique isomor-phism, if it exists. The counit of adjunction is a morphism of C

evX,Y : Hom(X, Y )⊗X → Y

called the evaluation morphism.


(ii) Suppose that every object of C admits an internal Hom functor; then we say briefly thatC admits an internal Hom functor, and clearly we get a functor

C o × C → C : (X, Y ) 7→Hom(X, Y ) for every X, Y ∈ Ob(C ).

Moreover, for every X, Y, Z ∈ Ob(C ) the composition

(Hom(X, Y )⊗Hom(Y, Z))⊗X ∼ // (Hom(X, Y )⊗X)⊗Hom(Y, Z)

evX,Y ⊗Hom(Y,Z)

ZevY,Z←−−−−Hom(Y, Z)⊗ Y Y ⊗Hom(Y, Z)

∼oo

corresponds, by adjunction, to a unique composition morphism

Hom(X, Y )⊗Hom(Y, Z)→Hom(X,Z).

(iii) In the situation of (ii), notice that the functor C → C given by the rule : Z 7→Hom(X,Hom(Y, Z)), for every Z ∈ Ob(C ), is right adjoint to the functor given by therule : Z 7→ (Z ⊗X)⊗ Y ∼→ T ⊗ (X ⊗ Y ). There follows a natural isomorphism

Hom(X,Hom(Y, Z))∼→Hom(X ⊗ Y, Z) for every X, Y, Z ∈ Ob(C ).

(iv) Moreover, for any unit (U, u) of C , we get natural bijections :

HomC (U,Hom(X, Y ))∼→ HomC (U ⊗X, Y )

∼→ HomC (X, Y ) for every X, Y ∈ Ob(C ).

Also, for every object Y of C , denote by uY : Y∼→ U⊗Y the isomorphism given by proposition

1.2.6; for every X ∈ Ob(X), it induces natural bijections

HomC (Y,Hom(U,X))∼→ HomC (Y ⊗ U,X)

HomC (uY ΨU,Y ,X)−−−−−−−−−−−−→ HomC (Y,X)

which correspond, via the Yoneda embedding, to a natural isomorphism

Hom(U,X)∼→ X for every X ∈ Ob(C ).

(v) Let X, Y, Z be any three objects of C ; the natural transformation

HomC (W ⊗X, Y )→ HomC (W ⊗ (X ⊗ Z), Y ⊗ Z) ϕ 7→ (ϕ⊗ Z) ΦW,X,Z

corresponds, via the Yoneda imbedding, to a unique morphism

tX,Y,Z : Hom(X, Y )→Hom(X ⊗ Z, Y ⊗ Z).

The reader can check that tX,Y,Z also corresponds, by adjunction, to the morphism

(evX,Y ⊗ Z) ΦHom(X,Y ),X,Z : Hom(X, Y )⊗ (X ⊗ Z)→ Y ⊗ Z.

(vi) In the situation of remark 1.2.5(ii), suppose that C admits an internal Hom functor; thenit is easily seen that the resulting tensor category Fun(D ,C ) inherits as well an internal Homfunctor, in the obvious way.

The formalism of tensor categories provides the language to deal uniformly with the notionsof algebras and their modules that occur in various concrete settings.

Definition 1.2.13. Let (C ,⊗,Φ,Ψ) be a tensor category, A and B any two objects of C .(i) A left A-module (resp. a right B-module) is a datum (X,µX), consisting of an object

X of C , and a morphism in C :

µX : A⊗X → X (resp. µX : X ⊗B → X)

called the scalar multiplication of X .


(ii) A morphism of left A-modules (X,µX)→ (X ′, µX′) is a morphism f : X → X ′ in Cwhich makes commute the diagram :

A⊗XµX //

1A⊗f

X

f

A⊗X ′

µX′ // X ′.

One defines likewise morphisms of right B-modules.(iii) An (A,B)-bimodule is a datum (X,µlX , µ

rX) such that (X,µlX) is a left A-module,

(X,µrX) is a right B-module, and the scalar multiplications commute, i.e. the diagram

A⊗ (X ⊗B)1A⊗µrX //

ΦA,X,B

A⊗XµlX

(A⊗X)⊗BµlX⊗1B // X ⊗B

µrX // X

commutes. Of course, a morphism of (A,B)-bimodules must be compatible with bothleft and right multiplication.

We denote by A-Modl (resp. B-Modr, resp. (A,B)-Mod) the category of left A-modules(resp. right B-modules, resp. (A,B)-bimodules). For any two left A-modules (resp. rightB-modules, resp. (A,B)-bimodules) X and X ′, we shall write

HomAl(X,X′) ( resp. HomBr(X,X

′) ) ( resp. Hom(A,B)(X,X′) )

for the set of morphisms of left A-modules (resp. of right B-modules, resp. of (A,B)-bimodules) X → X ′.

1.2.14. In the situation of definition 1.2.13, notice that

HomBr(X,X′) = Equal( HomC (X,X ′)

α //

β// HomC (X ⊗B,X ′) )

where α (resp. β) is given by the rule :

f 7→ f µX ( resp. f 7→ µX′ (f ⊗B) ) for every f ∈ HomC (X,X ′)

and similarly for left A-modules. Now, suppose that all equalizers in C are representable, andthat C admits an internal Hom functor; then we may define

HomBr(X,X′) := Equal( Hom(X,X ′)

α //

β// Hom(X ⊗B,X ′) )

where α := Hom(µX , X′) and β := Hom(X ⊗ B, µX′) tX,X′,B (notation of remark

1.2.12(v)). Then, it is easily seen that the bijections of remark 1.2.12(iv) induce natural identi-fications

HomC (U,HomBr(X,X′))

∼→ HomBr(X,X′) for every X,X ′ ∈ Ob(Br-Mod).

Likewise we may represent in C the set of morphisms between two left A-modules, and two(A,B)-modules (details left to the reader).


1.2.15. Let C be a tensor category as in (1.2.14) and A,B,C any three objects of C ; supposethat (X,µlX , µ

rX) is an (A,B)-bimodule, and (X ′, µlX′ , µ

rX′) a (C,B)-bimodule. Then we claim

that H := HomBr((X,µrX), (X ′, µrX′)) is naturally a (C,A)-bimodule. For this, we have to

exhibit natural morphismsC ⊗H

µl−−→Hµr←−−H ⊗ A

fulfilling the condition of definition 1.2.13(iii). However, by adjunction, the datum of µl is thesame as that of a morphism C → Hom(H ,H ), and since the functor Hom(X,−) is leftexact, the latter is the same as a morphism

C → Equal( Hom(H ,Hom(X,X ′))Hom(H ,α)

//

Hom(H ,β)// Hom(H ,Hom(X ⊗B,X ′))

which in turn – by remark 1.2.12(iii) – corresponds to a morphism

C → Equal( Hom(H ⊗X,X ′)Hom(H ⊗α,X′)

//

Hom(H ⊗β,X′)// Hom(H ⊗ (X ⊗B), X ′)

and again, the latter is the same as an element of

Equal( HomC (C ⊗ (H ⊗X), X ′)HomC (C⊗(H ⊗α),X′)

//

HomC (C⊗(H ⊗β),X′)// HomC (C ⊗ (H ⊗ (X ⊗B)), X ′).

By unwinding the definition, it is easily seen that the composition

µl : C ⊗ (H ⊗X)evX,X′−−−−−→ C ⊗X ′

µlX′−−−→ X ′

lies in the above equalizer, and it provides a left C-module structure for H . Likewise, µr shallbe the morphism corresponding to the composition

µr : (H ⊗X)⊗ A ∼→H ⊗ (A⊗X)H ⊗µlX−−−−−→H ⊗X

evX,X′−−−−−→ X ′.

Then, the condition that (H , µl, µr) is a bimodule, comes down to the commutativity of thediagram

C ⊗ ((H ⊗X)⊗ A)1C⊗µr //

C⊗ΦH ,X,A

C ⊗X ′

µlX′

C ⊗ (H ⊗ (X ⊗ A))C⊗(H ⊗(µlXΨX,A))

// C ⊗ (H ⊗X)µl // X ′

which is immediate (details left to the reader). We have thus obtained a bifunctor :

(1.2.16) HomBr(−,−) : (A,B)-Modo × (C,B)-Mod→ (C,A)-Mod.

Likewise, we may define a bifunctor :

HomAl(−,−) : (A,B)-Modo × (A,C)-Mod→ (B,C)-Mod.

1.2.17. Keep the situation of (1.2.15), and suppose moreover that all coequalizers in C arerepresentable. Fix an (A,B)-bimodule (X,µlX , µ

rX); the functor

(C,B)-Mod→ (C,A)-Mod : X ′ 7→HomBr(X,X′)

admits a left adjoint, the tensor product

(C,A)-Mod→ (C,B)-Mod : (X ′, µlX′ , µlX′) 7→ (X ′, µlX′ , µ

rX′)⊗A (X,µlX , µ

rX)


given by the coequalizer (in C ) of the morphisms :

X ′ ⊗ (A⊗X)1X′⊗µlX //

(µrX′⊗1X)ΦX′,A,X

// X ′ ⊗X

with scalar multiplications induced by µrX and µlX′ . Likewise, we have a functor :

(A,C)-Mod→ (B,C)-Mod : (X ′, µlX′ , µlX′) 7→ (X,µlX , µ

rX)⊗A (X ′, µlX′ , µ

lX′)

which admits a similar description, and is left adjoint to the functor X ′ 7→ HomAl(X,X′)

(verifications left to the reader).

1.2.18. Let (U, u) be unit object for C . Notice that, for any object A of C , the rule (Y, µY ) 7→(Y, u−1

Y , µY ) (where uY : Y → U ⊗ Y is the natural isomorphism supplied by proposition1.2.6), induces a faithful functor A-Modr → (U,A)-Mod. Letting C := U in (1.2.17), we seethat any (A,B)-bimodule X also determines a functor :

A-Modr → B-Modr : Y 7→ Y ⊗A Xand likewise for left A-modules.

Example 1.2.19. If C = Set is the category of sets (regarded as a tensor category as in example1.2.21), then a left A-module is just a set X ′ with a left action of A, i.e. a map of sets

A×X ′ → X ′ : (a, x) 7→ a · x.An (A,A)-bimodule X is a set with both left and right actions of A, such that (a · x) · a′ =a · (x · a′) for every a, a′ ∈ A and every x ∈ X . With this notation, the tensor productX ′ ⊗A X is the quotient (X ′ × X)/∼, where ∼ is the smallest equivalence relation such that(x′a, x) ∼ (x′, ax) for every x ∈ X , x′ ∈ X ′ and a ∈ A.

Definition 1.2.20. Let C := (C ,⊗,Φ,Ψ) be a tensor category, and (U, u) a unit for C .(i) A C -semigroup is a datum (M,µM) consisting of an object M of C and a morphism

µM : M⊗M →M , the multiplication law of M , such that (M,µM , µM) is a (M,M)-bimodule. A morphism of C -semigroups is a morphism ϕ : M → M ′ in C , suchthat

µM ′ (ϕ⊗ ϕ) = ϕ µM .(ii) A C -monoid is a datum M := (M,µM , 1M), where (M,µM) is a semigroup, and

1M : U →M is a morphism in C , called the unit of M , such that

µM (1M ⊗ 1M) uM = 1M = µM (1M ⊗ 1M) uMwhere uM : M

∼→ U ⊗M is the natural isomorphism provided by proposition 1.2.6.We say that M is commutative, if

µM = µM ΨM,M .

A morphism of monoids M → M ′ is a morphism of semigroups ϕ : M → M ′ suchthat ϕ 1M = 1M ′ .

Example 1.2.21. (i) Let C be any category with small Hom-sets, in which finite productsare representable, endow C with the tensor category structure described in example 1.2.10,and pick a final object 1C of C . Then, a C -monoid is a datum M := (M,µM , 1M), whereµM : M ×M → M and 1M : 1C → M are morphisms of C , and the axioms for µM and 1Mcan be rephrased as requiring that, for every object X of C , the set M(X) := HomC (X,M),endowed with the composition law :

M(X)×M(X)∼→ HomC (X,M ×M)

HomC (X,µM )−−−−−−−−−→M(X) (m,m′) 7→ m ·m′


is a (usual) monoid, with unit Im 1M(X) ∈ M(X). Of course, M is commutative, if and onlyif m ·m′ = m′ ·m for all objects X of C and every m,m′ ∈M(X).

(ii) In the situation of (i), a C -monoid shall also be called simply a C -monoid. The categoryof C -monoids admits an initial object which is also a final object, namely 1C := (1C , µ1,11),where µ1 is the (unique) morphism 1C ×1C → 1C . (Most of the above can be repeated with thetheory of semigroups replaced by any ”algebraic theory” in the sense of [11, Def.3.3.1] : e.g. inthis way one can define C -groups, C -rings, and so on.)

1.2.22. Let C and U be as in definition 1.2.20, and M := (M,µM , 1M) a C -monoid; of course,we are especially interested in the M -modules which are compatible with the unit and multipli-cation law of M . Hence we define a left M -module as a left M -module (S, µS) such that thefollowing diagrams commute :

U ⊗ S1M⊗1S

U ⊗ S M ⊗ (M ⊗ S)1M⊗µS //

ΦM,M,S

M ⊗ SµS

M ⊗ S

µS // S

uS

OO

(M ⊗M)⊗ S µM⊗1S // M ⊗ SµS // S

where uS is the isomorphism given by proposition 1.2.6. Likewise we define right M -modules,and (M,N)-bimodules, if N is a second C -monoid; especially, (M,M)-bimodules shall alsobe called simply M -bimodules.

A morphism of left M -modules (S, µS) → (S ′, µS′) is just a morphism of left M -modules,and likewise for right modules and bimodules. For instance, M is a M -bimodule in a naturalway, and an ideal ofM is a sub-M -bimodule I ofM . We denote byM -Modl (resp. M -Modr,resp. M -Mod) the category of left (resp. right, resp. bi-) M -modules; more generally, if N isa second C -monoid, we have the category (M,N)-Mod of the corresponding bimodules.

Example 1.2.23. Take C := Set, regarded as a tensor category, as in example 1.2.10. Then aC -monoid is just a usual monoid M , and a left M -module is a datum (S, µS) consisting of a setS and a scalar multiplication M × S → S : (m, s) 7→ m · s such that

1 · s = s and x · (y · s) = (x · y) · s for every x, y ∈M and every s ∈ S.A morphism ϕ : (S, µS)→ (T, µT ) of M -modules is then a map of sets S → T such that

x · ϕ(s) = ϕ(x · s) for every x ∈M and every s ∈ SLikewise, an ideal of M is a subset I ⊂M such that a ·x, x ·a ∈ I whenever a ∈ I and x ∈M .

Remark 1.2.24. Let C be a complete and cocomplete category, whose colimits are universal(see example 1.1.24(v)), and M a C -monoid (see example 1.2.21(ii)).

(i) The categories M -Modl, M -Modr and (M,N)-Mod are complete and cocomplete,and the forgetful functor M -Modl → C (resp. the same for right modules and bimodules)commutes with all limits and colimits.

(ii) Notice also that the forgetful functor M -Modl → C is conservative. Together with (i)and remark 1.1.38(iii,iv), this implies that a morphism of left M -modules is a monomorphism(resp. an epimorphism) if and only if the same holds for the underlying morphism in C (andlikewise for right modules and bimodules).

(iii) For each of these categories, the initial object is just the initial object ∅C of C , endowedwith the trivial scalar multiplication. Likewise, the final object is the final object 1C of C , withscalar multiplication given by the unique morphism M × 1C → 1C . Moreover, the forgetfulfunctorM -Modl → C admits a left adjoint, that assigns to any Σ ∈ Ob(C ) the freeM -moduleM (Σ) generated by Σ; as an object of C , the latter is just M × Σ, and the scalar multiplicationis derived from the composition law of M , in the obvious way.


For instance, for any n ∈ N, and any left (or right or bi-) M -module S, we denote as usualby S⊕n the coproduct of n copies of S.

Remark 1.2.25. Let C be a tensor category, M , N , P , Q four C -monoids.(i) Let S be a (M,N)-bimodule, S ′ a (P ,N)-bimodule and S ′′ a (P ,M)-bimodule. Then

it is easily seen that the (P,M)-bimodule (resp. the (P,N)-bimodule) HomNr(S, S′) (resp.

S ′′ ⊗M S) is actually a (P ,M)-bimodule (resp. a (P ,N)-bimodule) and the adjunction of(1.2.17) restricts to an adjunction between the corresponding categories of bimodules : thedetails shall be left to the reader.

(ii) We have as well the analogue of the usual associativity constraints. Namely, for ev-ery (M,N)-bimodule S, every (N,P )-bimodule S ′ and every (P ,Q)-bimodule S ′′, there is anatural isomorphism

(S ⊗N S ′)⊗P S ′′∼→ S ⊗N (S ′ ⊗P S ′′) in (M,Q)-Mod

and natural isomorphisms M ⊗M S∼→ S

∼→ S ⊗N N in (M,N)-Mod.(iii) Also, if M is commutative, every left (or right) M -module is naturally a (M,M)-

bimodule, and we have a commutative constraint

S ⊗M S ′∼→ S ′ ⊗M S for all left (or right) M -modules.

And taking into account (ii), it is easily seen that (M -Modl,⊗M) is a tensor category.

1.2.26. Let ϕ : M1 →M2 be a morphism of C -monoids; we have the (M1,M2)-bimodule :

M1,2 := (M2, µM2 (ϕ⊗ 1M2), µM2).

Letting X := M1,2 in (1.2.18), we obtain a base change functor for right modules :

M1-Modr →M2-Modr : X 7→ X ⊗M1 M2 := X ⊗M1 M1,2.

The base change is left adjoint to the restrictions of scalars associated to ϕ, i.e. the functor :

M2-Modr →M1-Modl : (X,µX) 7→ (X,µX)(ϕ) := (X,µX (1X ⊗ ϕ))

(verifications left to the reader). The same can be repeated, as usual, for left modules; forbimodules, one must take the tensor product on both sides : X 7→M2 ⊗M1 X ⊗M1 M2.

Example 1.2.27. Take C = Set, and let M be any monoid, Σ any set, and M (Σ) the freeM -module generated by Σ. From the isomorphism

M (Σ) ⊗M 1∼→ 1(Σ) = Σ

we see that the cardinality of Σ is an invariant, called the rank of the free M -module M (Σ), andwhich we denote rkMM

(Σ).

Definition 1.2.28. (i) A non-empty category A is additive, if the following holds :(a) For every A,B ∈ Ob(A ), the set HomA (A,B) is small and carries an abelian group

structure (especially, it is not empty).(b) For every A,B,C ∈ Ob(A ), the composition law

HomA (A,B)× HomA (B,C)→ HomA (A,C)

is a bilinear pairing.(ii) A functor F : A → B between additive categories is additive if it induces group

homomorphisms

HomA (X, Y )→ HomB(FX,FY ) ϕ 7→ Fϕ

for every X, Y ∈ Ob(A ).


Remark 1.2.29. Let A be any additive category.(i) If A ∈ Ob(A ) is any object, denote by 0A the neutral element of the abelian group

EndA (A). Suppose that the equalizer of the pair of morphisms 1A,0A : A→ A is representableby an object 0 of A (see example 1.1.24(ii)). Then, the datum of a morphism B → 0 is thesame as that of a morphism ϕ : B → A that factors through 0A. By the bilinearity of theHom-pairing, the latter condition holds if and only if ϕ is the neutral element of HomA (B,A).We conclude that 0 is a final object in A . Dually, if the coequalizer of the pair (1A,0A) isrepresentable by some object 0′ of A , then 0′ is initial in A . Moreover, if A admits a finalobject 0, then it is easily seen that the unique morphismA→ 0 is also the coequalizer of the pair(1A,0A), so 0 is also an initial object. Conversely, if A admits an initial object, then this objectis also final in A , and for any two objects A,B of A , the neutral element 0A,B of HomA (A,B)is the unique morphism that factors through 0. We say that 0 is a zero object for A .

(ii) Suppose that A admits a zero object 0, and moreover that the product A1 × A2 isrepresentable in A for given A1, A2 ∈ Ob(A ). Denote by pi : A1 × A2 → Ai (i = 1, 2) theprojections; then, there are unique morphisms ei : Ai → A1 × A2 (i = 1, 2) such that

(1.2.30) pi ei = 1Ai for i = 1, 2 and pi ej = 0Aj ,Ai for i 6= j.

Notice that

(1.2.31) e1 p1 + e2 p2 = 1A1×A1 .

Indeed, we have

pi (e1 p1 + e2 p2) = (pi e1 p1) + (pi e2 p2) = pi i = 1, 2

by bilinearity of the Hom pairing, and 1A1×A2 is the unique endomorphism ϕ of A1 × A2 suchthat pi ϕ = pi for i = 1, 2. It follows that A1 × A2 also represents the coproduct A1 q A2.Indeed, say that fi : Ai → B, for i = 1, 2, are two morphisms to another object B of A , andset f := f1 p1 + f2 p2 : A1 × A2 → B; it is easily seen that f ei = fi for i = 1, 2, and byvirtue of (1.2.31), the morphism f is the unique one that satisfies these identities. Conversely,if the coproduct of A1 and A2 is representable, a similar argument shows that also A1 × A2 isrepresentable. We say that A1 × A2 is a biproduct of A1 and A2, and denote it by A1 ⊕ A2.

(iii) Notice that the morphisms (pi, ei | i = 1, 2) with the identities (1.2.30) and (1.2.31)characterize A1 ⊕ A2 up to unique isomorphism. Namely, say that B is another object of A ,for which exist morphisms p′i : B → Ai and e′i : Ai → B (i = 1, 2) such that p′i e′i = 1Ai fori = 1, 2, and p′i e′j = 0AjAi for i 6= j, and moreover e′1 p′1 + e′2 p′2 = 1B. Then the pair(e′1, e

′2) (resp. (p′1, p

′2)) induces a morphism e′ : A1 ⊕ A2 → B (resp. p′ : B → A1 ⊕ A2) with

pj p′ e′ ei = p′j e′i = pj ei for i, j = 1, 2

which – by virtue of the universal properties of the biproduct – implies that p′ e′ = 1A1⊕A2 .Likewise, we may compute

e′ p′ = (e′ e1 p1 + e′ e2 p2) (e1 p1 p′ + e2 p2 p′)= e′ e1 p1 p′ + e′ e2 p2 p′

= e′1 p′1 + e′2 p′2 = 1B

whence the contention.(iv) Suppose that B1 and B2 are any other two objects of A such that B1 ⊕ B2 is also

representable; given two morphisms f1 : A1 → B1 and f2 : A2 → B2, we denote by f1 ⊕ f2 :A1 ⊕ A2 → B1 ⊕B2 the unique morphism such that

pB,i (f1 ⊕ f2) eA,i = fi for i = 1, 2, and pB,i (f1 ⊕ f2) eA,j = 0Aj ,Bi for i 6= j.


Notice that

(1.2.32) f1 ⊕ f2 = (f1 ⊕ 0A2,B2) + (0A1,B1 ⊕ f2)

(where the sum is taken in the abelian group HomA (A1⊕A2, B1⊕B2)); indeed, by bilinearityof the Hom pairing, it is easily seen that the right-hand side of (1.2.32) also satisfies the sameidentities above that define f1 ⊕ f2.

(v) If f : A → B is any morphism of A , then we define the kernel (resp. cokernel) of f asthe equalizer (resp. coequalizer)

Ker f := Equal(f,0A,B) Coker f := Coequal(f,0A,B).

Suppose that Ker f and Coker f are representable in A for every such f , and denote by

ιf : Ker f → A πf : B → Coker f

the natural morphisms. Notice that ιf is a monomorphism, and πf an epimorphism. Notice alsothat f factors uniquely as a composition

(1.2.33) Aπιf−−−→ Coker ιf

βf−−→ Ker πfιπf−−→ B.

(vi) Suppose that A admits a zero object 0, and let f : A → B be any morphism; bydefinition Ker f is the presheaf such that

Ker f(C) = g : B → C | g f = g 0A,B = 0A,C.If f is a monomorphism, the identity g f = 0A,C = 0B,C f implies that g = 0B,C , so Ker fis represented by 0. Dually, if f is an epimorphism, then Coker f is represented by 0.

Remark 1.2.34. Let A , B be any two additive categories that admit a zero object, and F :A → B a functor.

(i) If F is additive, remark 1.2.29(iii) immediately implies that F transforms representablebiproducts into representable biproducts. The latter assertion still holds in case F is not nec-essarily additive, but is either left or right exact. Indeed, suppose that F is left exact, letA1 ⊕ A2 be any biproduct, and let pi, ei be the morphisms described in remark 1.2.29(ii); byleft exactness, F (A1 ⊕ FA2) represents the product of FA1 and FA2, and any isomorphismFA1 ⊕ FA2

∼→ F (A1 ⊕ FA2) identifies Fp1 and Fp2 with the natural projections. Moreover,F transforms the final object of A into the final object of B (see example 1.1.26(ii)); then, byinspecting the argument in remark 1.2.29(ii), it is easily seen that F identifies as well Fei withthe natural injections FAi → FA1 ⊕ FA2, for i = 1, 2, so the assertion follows from remark1.2.29(iii). A similar argument works in case F is right exact.

(ii) Suppose moreover, that all biproducts of A are representable. Then we claim that theabelian group structure on HomA (A,B) is determined by the category A , i.e. if B is any otheradditive category, and F : A → B is any equivalence of categories, then F induces groupisomorphisms (and not just bijections) on Hom sets. Indeed, let A and B be any two objects ofA , and denote by ∆A : A → A⊕ A (resp. µB : B ⊕ B → B) the unique morphism such thatpi ∆A = 1A (resp. µB ei = 1B) for i = 1, 2. Then we have

f1 + f2 = µB (f1 ⊕ f2) ∆A for every f1, f2 : A→ B

where f1+f2 denotes the sum in the abelian group HomA (A,B). Indeed, since clearly 0A1,B1⊕0A2,B2 = 0A1⊕A2,B1⊕B2 , identity (1.2.32) reduces to checking that f1 = µB (f1⊕0A2,B2)∆A

(and likewise for f2), which follows easily from (1.2.31) : details left to the reader.(iii) Combining (i) and (ii) we see that, if all biproducts of A are representable, and F is

either left or right exact, then F is additive. More generally, we see that for F to be additive, itsuffices that F sends the zero object of A to the zero object of B, and F commutes with thebiproducts of the form A⊕ A, for every A ∈ Ob(A ).


Definition 1.2.35. An abelian category is an additive category A such that the following holds:(a) All the kernels and cokernels of A are representable (especially, A admits an initial

and final object 0).(b) For every morphism f of A , the morphism βf of (1.2.33) is an isomorphism.(c) The product of any finite family of objects of A is representable in A (see example

1.1.24(iv)).

Remark 1.2.36. Let A be any additive category.(i) For any other additive category B, let us denote by Add(B,A ) the full subcategory of

Fun(B,A ) whose objects are the additive functors. Notice that, if C is a small category, thenFun(C ,A ) is an additive category; indeed, if τ, σ : F ⇒ G are two natural transformationsbetween functors F,G : C → A , then we obtain a natural transformation τ + σ from F toG, by the rule : (τ + σ)X := τX + σX for every X ∈ Ob(C ) (where the sum denotes theaddition law of HomA (FX,GX)). Clearly, this rule yields an abelian group structure, and thecomposition of natural transformation defines a bilinear pairing (τ, τ ′) 7→ τ τ ′ on the resultinggroups of natural transformations (verification left to the reader).

(ii) Especially, if B is a small additive category, then also Add(B,A ) is an additive cate-gory. Moreover, if A is an abelian category, then Fun(C ,A ) is an abelian category, for everysmall category C : details left to the reader.

(iii) By definition, for everyA,B ∈ Ob(A ), the set hA(B) carries an abelian group structure,such that the presheaf hA factors through an additive functor h†A : A o → Z-Mod from A o tothe category of abelian groups, and the forgetful functor Z-Mod → Set. Hence, the Yonedaimbedding factors through a fully faithful group-valued Yoneda imbedding

h† : A → Add(A o,Z-Mod).

In view of (ii), we conclude that every small additive category is a full subcategory of an abeliancategory. Moreover, Yoneda’s lemma extends verbatim to the group-valued case : namely, byinspecting the proof of proposition 1.1.20, we see that, for everyA ∈ Ob(A ) and every additivefunctor F : A o → Z-Mod there are natural isomorphisms of abelian groups

(1.2.37) F (A)∼→ HomAdd(A o,Z-Mod)(hA, F ).

(iv) Suppose that f : A → B is any functor between small additive categories; then thearguments of (1.1.35) extend verbatim to the present situation : namely, the induced functor

f ∗ : Fun(Bo,Z-Mod)→ Fun(A o,Z-Mod)

admits both left and right adjoints, denoted respectively f! and f∗, and we have :

Proposition 1.2.38. In the situation of remark 1.2.36(iv), suppose that f is additive. Then :(i) Both f ∗, f! and f∗ are additive functors, and restrict to functors

Add(Bo,Z-Mod)f∗ // Add(A o,Z-Mod).

f! f∗oo

(ii) The resulting diagram of functors

Ah† //

f

Add(A o,Z-Mod)

f!

Bh† // Add(Bo,Z-Mod)

is essentially commutative.


Proof. (i): Since every left (resp. right) adjoint functor is right (resp. left) exact, remark1.2.34(iii) says that f ∗, f∗ and f! are additive. Next, a simple inspection shows that f ∗ trans-forms additive functors into additive functors. Let now F : A o → Z-Mod be an additivefunctor, B ∈ Ob(B) any object, and set G := f!F ; from the proof of proposition 1.1.34, wesee that

GB = colimψ:B→fA

FA

where the colimit ranges over the small category fA o/B of all pairs (A,ψ) consisting of anobject A of A , and a morphism ψ : B → fA in B. Denote by 0A and 0B the zero objects ofA and B; we wish to show that G is additive, and according to remark 1.2.34(iii), it suffices tocheck that G(0B) = 0, and that the natural morphism G(B ⊕B)→ GB ⊕GB (deduced fromthe projections pi : B ⊕B → B) is an isomorphism, for every B ∈ Ob(B).

However, notice that the functor ι0B: fA o/0B → A o is an isomorphism of categories

(notation of (1.1.16)); whence a natural isomorphism

G(0B)∼→ colim

A oF∼→ F (0A ) = 0

where the last identity holds, since F is additive. Next, for any B1, B2 ∈ Ob(B) consider thefunctor

Φ : (fA o/B1)×(fA o/B2)→ fA o/B1⊕B2 ((A1, ψ1), (A2, ψ2)) 7→ (A1⊕A2, ψ1⊕ψ2).

Claim 1.2.39. For any category C , and any functor H : fA o/B1 ⊕ B2 → C , the functor Φinduces an isomorphism

colim(fA o/B1)×(fA o/B2)

H Φ∼→ colim

fA o/(B1⊕B2)H.

Proof of the claim. Let X be any object of C , and τ : H Φ⇒ H cX a natural transformation,where cX : fA o/B1 ⊕ B2 → C is the constant functor with value X; since Φ is faithful, itsuffices to check that τ extends uniquely to a natural transformation τ ′ : H ⇒ cX .

However, let (A,ψ : B1 ⊕ B2 → fA) be any object of fA o/B1 ⊕ B2, and denote byψi : Bi → fA (for i = 1, 2) the composition of ψ with the natural monomorphism ei : Bi →B1 ⊕B2. Since f is additive, the morphism µA : A⊕ A→ A defines a morphism

µA : (A,ψ)→ (A⊕ A,ψ1 ⊕ ψ2) in fA o/B1 ⊕B2

and we may setτ ′(A,ψ) := τ(A⊕A,ψ1⊕ψ2) H(µoA).

Suppose that βo : (A,ψ) → (A′, ψ′) is any morphism in fA o/B1 ⊕ B2; there follows acommutative diagram in B

Bψ′1⊕ψ′2 //

ψ1⊕ψ2 ((QQQQQQQQQQQQQQQ fA′ ⊕ fA′µfA′ //

fβ⊕fβ

fA′

fβ

fA⊕ fA

µfA // fA

which allows to compute

τ ′(A′,ψ′) H(βo) = τ(A′⊕A′,ψ′1⊕ψ′2) H(µoA′) H(βo)

= τ(A′⊕A′,ψ′1⊕ψ′2) H(βo ⊕ βo) H(µoA)

= τ(A⊕A,ψ1⊕ψ2) H(µoA) = τ ′(A,ψ)

so τ ′ is indeed a natural transformation, and clearly it is the unique one extending τ . ♦


In light of claim 1.2.39, we are reduced to checking that the natural morphism

colim(fA o/B1)×(fA o/B2)

F ιB1⊕B2 Φ→ GB1 ⊕GB2

is an isomorphism, for any B1, B2 ∈ Ob(B). The latter assertion follows easily by inspectingthe definitions, since F is additive. Lastly, a similar argument shows that f∗F is additive,whenever the same holds for F : the reader can spell out the proof as an exercise.

(ii): One may argue as in (1.1.35) : in view of (1.2.37), we see that, for every object A of A ,both h†fA and f!h

†A represent the same functor : details left to the reader.

Definition 1.2.40. An abelian tensor category is a tensor category (C ,⊗,Φ,Ψ) such that C isan abelian category, and the functor ⊗ induces bilinear pairings

HomC (A,B)× HomC (A′, B′)→ HomC (A⊗ A′, B ⊗B′) : (f, g) 7→ f ⊗ gfor every A,A′, B,B′ ∈ Ob(C ).

Remark 1.2.41. Let (A ,⊗,Φ,Ψ) be a tensor category, such that A is abelian. If A admits aninternal Hom functor, then the functor −⊗A is right exact, and the functor Hom(A,−) is leftexact for every A ∈ Ob(A ), so both are additive, by virtue of remark 1.2.34(iii). Especially,A is an abelian tensor category, in this case.

Lemma 1.2.42. Let A be any abelian category, an Σ ⊂ Ob(A ) a small subset. We have :(i) There exists a small full abelian subcategory B of A such that Σ ⊂ Ob(B).

(ii) If A is small, there exists a complete and cocomplete abelian tensor category (C ,⊗)with internal Hom functor, and a fully faithful additive functor A → C .

Proof. (i): Let B0 be the full subcategory of A such that Ob(B0) = Σ; clearly B0 is small.Next, for any subcategory D of A , denote by D ′ a subcategory of A obtained as follows. Forevery morphism ϕ of D , we pick objects in A representing the kernel and cokernel of ϕ, andfor any two objects of D , we pick an object in A representing their product; let Σ′ ⊂ Ob(A )be the resulting subset. Then D ′ is the full subcategory of A such that Ob(D ′) = Ob(D)∪Σ′.It is easily seen that D ′ is small, whenever the same holds for D . Then we set inductivelyBi+1 := B′i for every i ∈ N. The full subcategory B of A with Ob(B) =

⋃i∈N Ob(Bi) is

still small, and it is abelian, by construction.(ii): We let C := Fun(A ,Z-Mod). Then C is an abelian category, by virtue of remark

1.2.36(ii), and since Z-Mod is complete and cocomplete, the same holds for C ; moreover, thestandard tensor product of abelian groups defines a tensor category structure with internal Homfunctor on Z-Mod, and the latter is inherited by C (remarks 1.2.5(ii) and 1.2.12(vi)). It is clearthat these two structures amount to an abelian tensor category, and the group-valued Yonedaimbedding is the sought fully faithful functor.

1.2.43. Let A be a small abelian category, and h† : A → A † := Fun(A o,Z-Mod) the fullyfaithful group-valued Yoneda imbedding. For every abelian group G, denote by GA : A o →Z-Mod the constant functor with value G : so, GA (A) := A for every A ∈ Ob(A ), andGA (ϕ) := 1A for every morphism ϕ in A . Since A † is an abelian tensor category (see theproof of lemma 1.2.42(ii)), we may form the tensor product

G⊗Z A := GA ⊗ h†A for every A ∈ Ob(A ).

We claim that, if G is finitely generated, G ⊗Z A lies in the essential image of h†. Indeed, thisis clear if G is free of finite rank, since in that case G ⊗Z A is a finite direct sum of copies ofA; in the general case, we may write G as a cokernel of a map L1 → L2 of free abelian groupsof finite rank, and since the functor − ⊗ h†A is right exact, we see that G ⊗Z A is the cokernelof a morphism of A , so it is represented by an object of A . Moreover, if ϕ : G → H is any


morphism of abelian groups, we have an obvious induced morphism ϕA : GA → HA , whencea morphism ϕ⊗Z A := ϕA ⊗ h†A.

Thus, after replacing G⊗Z A by an isomorphic object, we obtain a well defined functor

(1.2.44) Z-Modfg ×A → A (G,A) 7→ G⊗Z A

where Z-Modfg is the full subcategory of Z-Mod whose objects are the finitely generatedabelian groups. This functor is not unique, but any two such functors are naturally isomorphic.

Remark 1.2.45. Keep the notation of (1.2.43); we have :(i) From the construction, it is clear that (1.2.44) is a biadditive functor, i.e., for every abelian

group G, and every A ∈ Ob(A ), the restrictions G⊗− and −⊗ A of (1.2.44) are additive.(ii) Suppose that A is cocomplete; since the tensor product is right exact, it follows easily

that (1.2.44) extends to the whole of Z-Mod : details left to the reader.(iii) On the other hand, using Zorn’s lemma, (1.2.44) can be defined even in case A is not

small : again, we leave the details to the reader.

1.3. 2-categories. In dealing with categories, the notion of equivalence is much more centralthan the notion of isomorphism. On the other hand, equivalence of categories is usually notpreserved by the standard categorical operations discussed thus far. For instance, consider thefollowing :

Example 1.3.1. Let C be the category with Ob(C ) = a, b, and whose only morphisms are1a, 1b and u : a → b, v : b → a. Then necessarily u v = 1b and v u = 1a. Let Ca (resp.Cb) be the unique subcategory of C with Ob(Ca) = a (resp. Ob(Cb) = b). Clearly bothinclusion functors Ca → C ← Cb are equivalences. However, Ca ×C Cb is the empty category;especially, this fibre product is not equivalent to C = C ×C C .

It is therefore natural to seek a new framework for the manipulation of categories and functors“up to equivalences”, and thus more consonant with the very spirit of category theory. Preciselysuch a framework is provided by the theory of 2-categories, which we proceed to present.

1.3.2. The category Cat, together with the category structure on the sets Fun(−,−) (as in(1.1.7)), provides the first example of a 2-category. The latter is the datum of :

• A set Ob(A ), whose elements are called the objects of A .• For every A,B ∈ Ob(A ), a category A (A,B). The objects of A (A,B) are called

1-cells or arrows, and are designated by the usual arrow notation f : A → B. Givenf, g ∈ Ob(A (A,B)), we shall write f ⇒ g to denote a morphism from f to g inA (A,B). Such morphisms are called 2-cells. The composition of 2-cells α : f ⇒ gand β : g ⇒ h shall be denoted by β α : f ⇒ h.• For every A,B,C ∈ Ob(A ), a composition bifunctor :

cABC : A (A,B)×A (B,C)→ A (A,C).

Given 1-cells Af−→ B

g−→ C, we write g f := cABC(f, g).Given two 2-cells α : f ⇒ g and β : h ⇒ k, respectively in A (A,B) and A (B,C),we use the notation

β ∗ α := cABC(α, β) : h f ⇒ k g.

Also, if h is any 1-cell of A (B,C), we usually write h ∗α instead of 1h ∗α. Likewise,we set β ∗ f := β ∗ 1f , for every 1-cell f in A (A,B).• For every element A ∈ Ob(A ), a unit functor :

uA : 1→ A (A,A)


where 1 := (∗,1∗) is the terminal element of Cat. Hence uA is the datum of an object:

1A ∈ Ob(A (A,A))

and its identity endomorphism, which we shall denote by iA : 1A → 1A.The bifunctors cABC are required to satisfy an associativity axiom, which says that the diagram:

A (A,B)×A (B,C)×A (C,D)1×cBCD //

cABC×1

A (A,B)×A (B,D)

cABD

A (A,C)×A (C,D)cACD // A (A,D)

commutes for every A,B,C,D ∈ Ob(A ). Likewise, the functor uA is required to satisfy a unitaxiom; namely, the diagram :

1×A (A,B)

uA×1A (A,B)

A (A,B)∼oo ∼ // A (A,B)× 1

1A (A,B)×uB

A (A,A)×A (A,B)cAAB // A (A,B) A (A,B)×A (B,B)

cABBoo

commutes for every A,B ∈ Ob(A ).

1.3.3. In a 2-category A , it makes sense to speak of adjoint pair of 1-cells, or of the Kanextension of a 1-cell : see [10, Def.7.1.2, 7.1.3] for the definitions. In the same vein, we say thata 1-cell f : A → B of A is an equivalence if there exists a 1-cell g : B → A and invertible2-cells g f ⇒ 1A and f g ⇒ 1B.

Definition 1.3.4. ([10, Def.7.5.1]) Let A and B be two 2-categories. A pseudo-functor F :A → B is the datum of :

• For every A ∈ Ob(A ), an object FA ∈ Ob(B).• For every A,B ∈ Ob(A ), a functor :

FAB : A (A,B)→ B(FA, FB).

We shall often omit the subscript, and write only Ff instead of FABf : FA → FB,for a 1-cell f : A→ B.• For every A,B,C ∈ Ob(A ), a natural isomorphism γABC between two functors

A (A,B) × A (B,C) → B(FA, FC) as indicated by the (not necessarily commu-tative) diagram :

A (A,B)×A (B,C)cABC //

FAB×FBC

vvvv 7?γABC

A (A,C)

FAC

B(FA, FB)×B(FB,FC) cFA,FB,FC// B(FA, FC).

To ease notation, for every (f, g) ∈ A (A,B) × A (B,C), we shall write γf,g insteadof (γABC)(f,g) : cFA,FB,FC(FABf, FBCg)→ FAC(cABC(f, g)).• For every A ∈ Ob(A ), a natural isomorphism δA between functors 1→ B(FA, FA),

as indicated by the (not necessarily commutative) diagram :

1uA //

AIδA

A (A,A)

FAA

1 uFA// B(FA, FA).


The system (δ•, γ•••) is called the coherence constraint for F . This datum is required to satisfy:• A composition axiom, which says that the diagram :

Fh Fg Ff1Fh∗γf,g +3

γg,h∗1Ff

Fh (F (g f))

γgf,h

F (h g) Ffγf,hg +3 F (h g f)

commutes for every sequence of arrows : Af−→ B

g−→ Ch−→ D in A .

• A unit axiom, which says that the diagrams :

Ff 1FA

1Ff

1Ff∗δA +3 Ff F1A

γ1A,f

1FB Ff1Ff

δB1Ff +3 F (1B) Ffγf,1B

Ff1Ff +3 F (f 1A) Ff

1Ff +3 F (1B f)

commute for every arrow f : A → B (where, to ease notation, we have written δAinstead of (δA)∗ : 1FA → FAA1A, and likewise for δB).

Definition 1.3.5. Consider two pseudo-functors F,G : A → B between 2-categories A , B.A pseudo-natural transformation α : F ⇒ G is the datum of :

• For every object A of A , a 1-cell αA : FA→ GA.• For every pair of objects A,B of A , a natural isomorphism τAB between two functors

A (A,B)→ B(FA,GB), as indicated by the (not necessarily commutative) diagram:

A (A,B)

GAB

FAB //

vvvv 7?τAB

B(FA, FB)

B(1B,αB)

B(GA,GB)

B(αA,1B)// B(FA,GB)

where B(αA,1B) is the functor obtained by fixing αA in the first argument of thecomposition bifunctor cFA,GA,GB : B(FA,GA)×B(GA,GB)→ B(FA,GB), andlikewise for B(1B, αB). The datum τ•• is called the coherence constraint for α.

This datum is required to satisfy the following coherence axioms (in which we denote by(δF , γF ) and (δG, γG) the coherence constraints for F and respectively G) :

• For every A ∈ Ob(A ), the diagram :

αA1αA +3

1αA

1GA αAδGA ∗1αA +3 G(1A) αA

τ1A

αA 1FA1αA∗δ

FA +3 αA F (1A)

commutes.• For each pair of arrows A

f−→ Bg−→ C in A , the diagram :

Gg Gf αA1Gg∗τf +3

γGf,g∗1αA

Gg αB Ffτg∗1f +3 αC Fg Ff

1αC ∗γFf,g

G(g f) αA

τgf +3 αC F (g f)

commutes.


If α : F ⇒ G and β : G ⇒ H are two pseudo-natural transformations, we may define thecomposition β α : F ⇒ H which is the pseudo-natural transformation given by the ruleA 7→ βA αA for every A ∈ Ob(A ). The coherence constraint of β α is given by the rule :

(A,B) 7→ (βB ∗ ταAB) (τβAB ∗ αA) for every A,B ∈ Ob(A )

where τα•• (resp. τβ••) denotes the coherence constraint of α (resp. of β).

Example 1.3.6. (i) Any category A can be regarded as a 2-category in a natural way : namely,for any two objects A and B of A one lets A (A,B) be the discrete category HomA (A,B);hence the only 2-cells of A are the identities 1f : f ⇒ f , for every morphism f : A → B.The composition bifunctor cABC is of course given (on 1-cells) by the composition law formorphisms of A . Likewise, the functor uA assigns to every objectA its identity endomorphism.

(ii) In the same vein, every functor between usual categories, is a pseudo-functor between thecorresponding 2-categories as in (i); of course, the coherence constraint consists of identities.Finally, every natural transformation of usual functors can be regarded naturally as a pseudo-natural transformation between the corresponding pseudo-functors.

(iii) As it has already been mentioned, the category Cat is naturally a 2-category. Namely,for any three small categories A , B and C , the 1-cells in Cat(A ,B) are the functors from Ato B, and the 2-cells are the natural transformations between these functors. The compositionlaw cA BC is defined on 1-cells by the usual composition of functors, and on 2-cells as in (1.1.7).

(iv) The standard constructions on categories admit analogues for 2-categories. However, ifA is a 2-category, there are several inequivalent candidates for the opposite 2-category A o :one can reverse the 1-cells, one can reverse the 2-cells, i.e. replace the categories A (A,B) bytheir opposites, or do both. We leave to the reader the task of spelling out the definition(s).

(v) Likewise, if X is any object of A , one may define a 2-category A/X . Its objects are thearrows in A of the form f : A → X . If g : B → X is another such arrow, A/X(f, g) is thesubcategory of A (f, g) whose objects are all the arrows h : A → B such that g h = f , andwhose morphisms are all the 2-cells α of A (f, g) such that 1g ∗ α = 1f . The composition andunit functors for A/X are the restrictions of the corresponding ones for A .

(vi) Consider 2-categories A , B. For every object B of B, one may define the constantpseudo-functor with value B : this is the pseudo-functor

FB : A → B such that FB(A) := B FB(f) := 1B FB(α) := iB

for every A ∈ Ob(A ), every 1-cell f , and every 2-cell α of A . The coherence constraint forFB consists of identities. Given a pseudo-functor F : A → B, a pseudo-cone on F with vertexB is a pseudo-natural transformation FB ⇒ F . Especially, every 1-cell f : B → B′ induces apseudo-cone :

Ff : FB ⇒ FB′ : (Ff )A := f for every A ∈ Ob(A )

whose coherence constraint consists of identities.Dually, a pseudo-cocone on F with vertex B is a pseudo-natural transformation F ⇒ FB,

and Ff can thus be viewed as a pseudo-cocone on F with vertex B′.

Definition 1.3.7. Consider two pseudo-functors F,G : A → B between 2-categories A , B,and two pseudo-natural transformations α, β : F ⇒ G. A modification Ξ : α β is a family :

ΞA : αA ⇒ βA

of 2-cells of B, for every object A of A . Such a family is required to satisfy the followingcondition. For every pair of 1-cells f, g : A → A′ of A , and every 2-cell γ : f ⇒ g, theequality

(ΞA′ ∗ Fγ) ταAA′,f = τβAA′,g (Gγ ∗ ΞA)

holds in B, where τα•• (resp. τβ••) denotes the coherence constraint for α (resp. for β).


1.3.8. If Ξ,Θ : α β are two modifications between pseudo-natural transformations α, β :F ⇒ G of pseudo-functors F,G : A → B, we may define the composition

Ξ Θ : α β : A 7→ ΞA ΘA for every A ∈ Ob(A).

We may then consider the category :

PsNat(F,G)

whose objects are the pseudo-natural transformations F ⇒ G, and whose morphisms are themodifications α β between them. For instance, PsNat(FB, F ) (resp. PsNat(F, FB)) is thecategory of pseudo-cones (resp. pseudo-cocones) on F with vertex B (example 1.3.6(vi)).

1.3.9. Let A ,B be two 2-categories; using the modification, we can endow the categoryPsFun(A ,B) of pseudo-functors A → B, with a natural structure of 2-category. Namely,for any two pseudo-functors F,G : A → B, the 1-cells F → G of PsFun(A ,B) are thepseudo-natural transformations F ⇒ G of two such functors; of course, for fixed F and G, thecategory structure on the set of 1-cells F → G is precisely the one of PsNat(F,G), i.e. the2-cells of PsFun(A ,B) are the modifications. The composition functor

PsNat(F,G)× PsNat(G,H)→ PsNat(F,H)

assigns, to any two modifications Ξ : α β and Ξ′ : α′ β′, the modification

Ξ′ ∗ Ξ : α′ α β′ β A 7→ Ξ′A ∗ ΞA for every A ∈ Ob(A ).

With this notation, we say that a pseudo-natural transformation of pseudo-functors A → Bis a pseudo-natural equivalence if it is an equivalence in the 2-category PsFun(A ,B) (in thesense of (1.3.3)).

Definition 1.3.10. Let A and B be two 2-categories, and F : A → B a pseudo-functor.(i) We say that F is a 2-equivalence from A to B if the following holds :

• For every A,B ∈ Ob(A ), the functor FAB is an equivalence (notation of defini-tion 1.3.4).• For every A′ ∈ Ob(B) there exists A ∈ Ob(A ) and an equivalence FA→ A′.

(ii) Let G : B → A be a pseudo-functor. We say that G is right 2-adjoint to F if thefollowing holds.• For every A ∈ Ob(A ) and B ∈ Ob(B) there exists an equivalence of categories

ϑAB : HomB(FA,B)→ HomA (A,GB).

• For every pair of 1-cells f : A′ → A in A , g : B → B′ in B, there exists a naturalisomorphism of functors

HomB(FA,B)HomB(Ff,g)

//

ϑAB

vvvv 7?τf,g

HomB(FA′, B′)

ϑA′B′

HomA (A,GB)HomA (f,Gg)

// HomA (A′, GB′).

• For every pair of compositions of 1-cells A′′f ′−→ A′

f−→ A in A , Bg−→ B′

g′−→ B′′,we have the identity

(τf ′,g′ ∗ HomB(Ff, g)) (HomA (f ′, Gg′) ∗ τf,g) = τff ′,g′g.

In this case, we also say that (F,G) is a 2-adjoint pair of pseudo-functors.


Remark 1.3.11. Let F : A → B be a pseudo-functor.(i) The reader may show that F is a 2-equivalence if and only if there exists a pseudo-functor

G : B → A and two pseudo-natural equivalences 1A ⇒ G F and F G⇒ 1B.(ii) Suppose that F admits a right 2-adjoint G : A → B, and let ϑ•• and τ•• be the

corresponding data as in definition 1.3.10(ii). Using these data, one can define a unit η : 1A ⇒F G and counit ε : G F ⇒ 1B, that are pseudo-natural transformations fulfilling triangularidentities as in (1.1.8).

(iii) Conversely, the existence of pseudo-natural transformation ε, η as in (ii), fulfilling theabove mentioned triangular identities, implies that G is right 2-adjoint to F (details left to thereader).

Definition 1.3.12. Let A be a 2-category, F : A → B a pseudo-functor.(i) A 2-limit of F is a pair :

2-limA

F := (L, π)

consisting of an object L of B and a pseudo-cone π : FL ⇒ F , such that the functor :

B(B,L)→ PsNat(FB, F ) : f 7→ π Ff

is an equivalence of categories, for every B ∈ Ob(B).(ii) Dually, a 2-colimit of F is a pair :

2-colimA

F := (L, π)

consisting of an object L of B and a pseudo-cocone π : F ⇒ FL, such that the functor:

B(L,B)→ PsNat(F, FB) : f 7→ Ff πis an equivalence of categories, for every B ∈ Ob(B).

As usual, if the 2-limit exists, it is unique up to (non-unique) equivalence : if (L′, π′) is another2-limit, there exists an equivalence h : L→ L′ and an isomorphism β : π′Fh

∼→ π; moreover,the pair (h, β) is unique up to unique isomorphism, in a suitable sense, that the reader may spellout, as an exercise. A similar remark holds for 2-colimits.

(iii) We say that B is 2-complete (resp. 2-cocomplete) if, for every small 2-category A ,every pseudo-functor A → B admits a 2-limit (resp. a 2-colimit).

Remark 1.3.13. To be in keeping with the terminology of [10, Ch.VII], we should write pseudo-bilimit instead of 2-limit (and likewise for 2-colimit). The term “2-limit” denotes in loc.cit. arelated notion, which makes it unique up to isomorphism, not just up to equivalence. However,the notion introduced in definition 1.3.12 is the one that occurs most frequently in applications.

The following lemma 1.3.14 indicates that the framework of 2-categories does indeed providean adequate answer to the issues raised in (1.3).

Lemma 1.3.14. Let A be a 2-category, F,G : A → B two pseudo-functors, ω : F ⇒ G apseudo-natural equivalence, and suppose that the 2-limit of F exists. Then the same holds forthe 2-limit of G, and there is a natural equivalence in B :

2-limA

F∼→ 2-lim

AG.

More precisely, if (L, π) is a pair with L ∈ Ob(B) and a pseudo-cone π : FL ⇒ F representingthe 2-limit of F , then the pair (L, ω π) represents the 2-limit of G.

A dual assertion holds for 2-colimits.

Proof. It is easily seen that the rule α 7→ ω α induces an equivalence of categories :

PsNat(FB, F )→ PsNat(FB, G).

The claim is an immediate consequence.


Proposition 1.3.15. For every small category B, the 2-category Cat/B is 2-complete and2-cocomplete.

Proof. We only show 2-completeness, and we leave the proof of 2-cocompleteness as an exer-cise for the reader. Notice that, in the special case where B is the terminal object of Cat, theassertion means that Cat is 2-complete and 2-cocomplete. Let :

F : A → Cat : a 7→ Fa for every a ∈ Ob(A )

be a pseudo-functor from a small 2-category A ; we define a category LF as follows :• The objects of LF are all the systems (X•, ξ

X• ), where Xa ∈ Ob(Fa) for every a ∈

Ob(A ) and ξXf : Fab(f)(Xa)∼→ Xb is an isomorphism in Fb, for every morphism

f : a→ b in A . The data (X•, ξX• ) are required to fulfill the following conditions.

(a) If f : a→ b and g : b→ c are any two morphisms in A , the diagram :

Fbc(g) Fab(f)(Xa)Fbc(g)(ξ

Xf )//

γ(f,g)(Xa)

Fbc(g)(Xb)

ξXg

Fac(g f)(Xa)ξXgf (Xa)

// Xc

commutes, where γ denotes the coherence constraint of F .(b) Moreover, ξX1a = 1Xa for every a ∈ Ob(A ), where 1a is the image of the unit

functor ua : 1→ A .• The morphisms (X•, ξ

X• ) → (Y•, ξ

Y• ) in LF are the systems of morphisms t• := (ta :

Xa → Ya | a ∈ Ob(A )) such that the diagram :

Fab(f)(Xa)Fab(f)(ta)

//

ξXf

Fab(f)(Ya)

ξYf

Xbtb // Yb

commutes for every morphism f : a→ b in A .Next, we define a pseudo-cone π : FLF

⇒ F as follows.• For every a ∈ Ob(A ), we let πa : LF → Fa be the functor given by the rule :

(X•, ξX• ) 7→ Xa and t• 7→ ta.

• For every morphism f : a → b in A , we let τf : Fab(f) πa ⇒ πb be the naturaltransformation given by the rule :

(τf )(X•,ξX• ) := ξ•f .

The verification that (π•, τ•) satisfies the coherence axioms for a pseudo-natural transformation,is straightforward, using the foregoing conditions (a) and (b). We claim that (LF , π) is a 2-limitof F . Indeed, suppose that C is another small category, and α : FC ⇒ F is a pseudo-cone withvertex C . By definition, α is the datum of functors αa : C → Fa for every a ∈ Ob(A ), andnatural isomorphisms σf : Fab(f) αa ⇒ αb for every morphism f : a→ b in A fulfilling theusual coherence axioms. To such datum we attach a functor Lα : C → LF by the followingrule. For every X ∈ Ob(C ), we set :

Lα(X) := (α•X, (σ•)X)

and to every morphism g : X → Y we assign the compatible system (αag | a ∈ Ob(A )). It isthen immediate to see that π FLα = α, and LπFG = G for every functor G : C → LF .


1.3.16. Let A be any 2-category. We shall say that a diagram of objects and arrows in A :

(1.3.17)A

f //

g

B

h

Ck // D

is essentially commutative, if there exists an invertible 2-cell α : h f ⇒ k g. Let L bethe small category with Ob(L) := 0, 1, 2, and whose set of arrows consists of the identitymorphisms, and two more arrows 1 → 0 and 2 → 0. An essentially commutative diagram(1.3.17) can be regarded as a pseudo-cone π with vertex A, on the functor F : L → A suchthat F (0) := D, F (1) := B, F (2) := C, F (1 → 0) := h and F (2 → 0) := k. We say that(1.3.17) is 2-cartesian if (A, π) is a 2-limit of the functor F .

For instance, let A := Cat; by inspecting the proof of proposition 1.3.15, we see that(1.3.17) is a 2-cartesian diagram, if and only if the functors f and g and the 2-cell α induce anequivalence from the small category A, to the category whose objects are all data of the formX := (b, c, ξ), where b ∈ Ob(B), c ∈ Ob(C), and ξ : h(b)

∼→ k(c) is an isomorphism. IfX ′ := (b′, c′, ξ′) is another such datum, the morphisms X → X ′ are the pairs (ϕ, ψ), whereϕ : b→ b′ (resp. ψ : c→ c′) is a morphism in B (resp. in C), and ξ′ h(ϕ) = k(ϕ′) ξ.

This category shall be called the 2-fibre product of the functors k and h, and shall be denoted

B2×

(h,k)C

or sometimes, just B2×D C, if there is no danger of ambiguity.

1.4. Fibrations. We keep the assumptions and notation of section 1.1; especially, Cat is syn-onimous with U-Cat. Let ϕ : A → B be a functor, f : A′ → A a morphism in A , andset

g := ϕf : ϕA′ → ϕA.

We say that f is ϕ-cartesian, or – slightly abusively – that f is B-cartesian, if the inducedcommutative diagram of sets (notation of (1.1.14)) :

HomA (X,A′)f∗ //

ϕ

HomA (X,A)

ϕ

HomB(ϕX,ϕA′)

g∗ // HomB(ϕX,ϕA)

is cartesian for every X ∈ Ob(A ). In this case, one also says that f is an inverse image of Aover g, or – slightly abusively – that A′ is an inverse image of A over g. One verifies easily thatthe composition of two B-cartesian morphisms is again B-cartesian.

Definition 1.4.1. Let ϕ : A → B be a functor, and B any object of B.(i) The fibre of ϕ over B is the category ϕ−1B whose objects are all the A ∈ Ob(A ) such

that ϕA = B, and whose morphisms f : A′ → A are the elements of HomA (A′, A)such that ϕf = 1B. We denote by :

ιB : ϕ−1B → A

the natural faithful imbedding of ϕ−1B into A .(ii) We say that ϕ is a fibration if, for every morphism g : B′ → B in B, and every

A ∈ Ob(ϕ−1B), there exists an inverse image f : A′ → A of A over g. In this case,we also say that A is a fibred B-category.

Example 1.4.2. Let C be any category, X any object of C .


(i) The functor (1.1.13) is a fibration, and all the morphisms in C/X are C -cartesian. Theeasy verification shall be left to the reader.

(ii) The source functor s : Morph(C ) → C is a fibration (notation of (1.1.17)); moreprecisely, the s-cartesian morphisms are the square diagrams (1.1.18) in which g′ is anisomorphism. For any B ∈ Ob(B), the fibre s−1B is the category B/C .

(iii) Suppose that all fibre products are representable in C . Then also the target functort : Morph(C ) → C is a fibration; more precisely, the t-cartesian morphisms are thesquare diagrams (1.1.18) which are cartesian (i.e. fibred). For any B ∈ Ob(B), thefibre t−1B is the category C/B.

Definition 1.4.3. Let A , A ′ and B be three small categories, ϕ : A → B and ϕ′ : A ′ → Btwo functors, which we regard as objects of Cat/B; let also F : A → A ′ be a B-functor,i.e. a morphism ϕ → ϕ′ in Cat/B. We say that F is cartesian (resp. strongly cartesian), if itsends B-cartesian morphisms of A (resp. all morphisms of A ), to B-cartesian morphisms inA ′. We denote by :

CartB(A ,A ′)

the category whose objects are the strongly cartesian B-functors F : A → A ′, and whosemorphisms are the natural transformations :

(1.4.4) α : F ⇒ G such that ϕ′ ∗ α = 1ϕ.

Remark 1.4.5. The reader might prefer to reserve the notation CartB(A ,A ′) for the largercategory of all cartesian functors. However, in our applications only the categories of stronglycartesian functors arise, hence we may economize a few adverbs.

Besides, in many cases we shall consider B-categories A whose every morphism is B-cartesian (this happens, e.g. when A is a subcategory of B); in such situations, obviouslyevery cartesian functor A → A ′ is strongly cartesian.

1.4.6. Let ϕ : A → B be a fibration, and g : B′ → B any morphism in B. Suppose we havechosen, for every object A ∈ Ob(ϕ−1B), an inverse image :

gA : g∗A→ A

of A over g (so g∗A ∈ Ob(ϕ−1B′) and ϕ(gA) = g). Then the rule A 7→ g∗A extends naturallyto a functor :

g∗ : ϕ−1B → ϕ−1B′.

Namely, by definition, for every morphism h : A′ → A in ϕ−1B there exists a unique morphismg∗h : g∗A′ → g∗A, such that :

(1.4.7) h gA′ = gA g∗h

and if k : A′′ → A′ is another morphism in ϕ−1B, the uniqueness of g∗k implies that :

g∗k g∗h = g∗(k h).

Moreover, (1.4.7) also means that the rule A 7→ gA defines a natural transformation :

g• : ιB′ g∗ ⇒ ιB

(notation of definition 1.4.1(i)).


1.4.8. Let ϕ : A → B be a fibration between small categories, and B′′ h−→ B′g−→ B two

morphisms in B. Proceeding as in (1.4.6), we may attach to g and h three functors :

g∗ : ϕ−1B → ϕ−1B′ h∗ : ϕ−1B′ → ϕ−1B′′ (g h)∗ : ϕ−1B → ϕ−1B′′

as well as natural transformations :

g• : ιB′ g∗ ⇒ ιB h• : ιB′′ h∗ ⇒ ιB′ (g h)• : ιB′′ (g h)∗ ⇒ ιB

and by inspecting the constructions, one easily finds a unique natural isomorphism :

γh,g : h∗ g∗ ⇒ (g h)∗

which fits into a commutative diagram :

ιB′′ h∗ g∗h•∗g∗ +3

ιB′′∗γh,g

ιB′ g∗

g•

ιB′′ (g h)∗

(gh)• +3 ιB.

It follows that the rule which assigns to each B ∈ Ob(B) the small category ϕ−1B, to eachmorphism g in B a chosen functor g∗, and to each pair of morphisms (g, h) as above, the naturalisomorphism γh,g, defines a pseudo-functor

(1.4.9) c : Bo → Cat.

Notice as well that, for every B ∈ Ob(B), we may choose (1B)∗ to be the identity functor ofϕ−1B; then the isomorphism

(δB)∗ : ϕ−1B → ϕ−1B

required by definition 1.3.4, shall also be the identity. (Here we view Bo as a 2-category,as explained in example 1.3.6(i); also, the natural 2-category structure on Cat is the one ofexample 1.3.6(iii)). The datum of a pseudo-functor c as in (1.4.9) is called a cleavage (infrench: “clivage”) for the fibration ϕ.

Example 1.4.10. Let B be any category.(i) Let ϕ : A → B be a fibration, and ψ : C → B any functor. Then the induced

functor A ×(ϕ,ψ) C → C is also a fibration. The cartesian morphisms of A ×(ϕ,ψ) Care the pairs (f, g) where f is a cartesian morphism of A , g is a morphism of C , andϕ(f) = ψ(g).

(ii) A composition of fibrations is again a fibration. Also, for i = 1, 2, let Ai → B be twofibrations; combining with (i) we see that A1 ×B A2 → B is also a fibration.

(iii) Let F be a presheaf on B. To F we may attach a fibration ϕF : AF → B as follows.The objects of AF are all the pairs (X, s), where X ∈ Ob(B) and s ∈ F (X). A mor-phism (X, s)→ (X ′, s′) is a morphism f : X → X ′ in B, such that F (f)(s′) = s. Thefunctor ϕF is defined by the rule : ϕF (X, s) = X for every (X, s) ∈ Ob(AF ). For anyX ∈ Ob(B), the fibre ϕ−1

F (X) is (naturally isomorphic to) the discrete category F (X)(see (1.1.1)). Notice also that every morphism in AF is cartesian. This construction isa special case of (1.4.15).

(iv) Let F and G be presheaves on B. With the notation of (iii), there is a natural bijection:

HomC∧(F,G)∼→ CartB(AF ,AG).

Namely, to a morphism ψ : F → G one assigns the functor Aψ : AF → AG such thatAψ(X, s) = (X,ψ(X)(s)) for every (X, s) ∈ Ob(AF ).

(v) For instance, if B has small Hom-sets, and F is representable by an object X of B,then one checks easily that AF is naturally isomorphic to C/X and the fibration AF isequivalent to that of example 1.4.2(i).


(vi) Furthermore, for i = 1, 2, let Gi → F be two morphisms of presheaves on B; then wehave a natural isomorphism of fibrations over B :

AG1 ×AF AG2

∼→ AG1×FG2

(details left to the reader).

1.4.11. Let A ′′ → B be another small B-category, and F : A → A ′ any B-functor; weobtain a natural functor :

CartB(F,A ′′) : CartB(A ′,A ′′)→ CartB(A ,A ′′) G 7→ G F.To any morphism α : G⇒ G′ in CartB(A ′,A ′′), the functor CartB(F,A ′′) assigns the naturaltransformation α ∗ F : G F ⇒ G′ F .

Likewise, any cartesian B-functor G : A ′ → A ′′ induces a functor :

CartB(A , G) : CartB(A ,A ′)→ CartB(A ,A ′′) F 7→ G Fwhich assigns to any morphism β : F ⇒ F ′ in CartB(A ,A ′), the natural transformationG ∗ β : G F ⇒ G F ′.

Furthermore, in case ϕ′ : A ′ → B is a fibration, we have a natural equivalence of categories

(1.4.12) CartA (A , p′) : CartA (A ,A ×(ϕ,ϕ′) A ′)∼→ CartB(A ,A ′)

(where p′ : A ×(ϕ,ϕ′) A ′ → A ′ is the natural projection functor).

Proposition 1.4.13. Let ϕ : A → B and ϕ′ : A ′ → B be two fibrations, and F : A → A ′ acartesian B-functor. We have :

(i) The following conditions are equivalent :(a) F is a fibrewise equivalence, i.e. for any B ∈ Ob(B) the restriction ϕ−1B →

ϕ′−1B of the functor F , is an equivalence.(b) F is a B-equivalence, i.e. an equivalence in the 2-category U′-Cat/B (in the

sense of (1.3.3)) where U′ is any universe relative to which the categories A , A ′

and B are small.(ii) If the equivalent conditions of (i) hold, the functor CartB(C , F ) is an equivalence, for

every B-category C → B.

Proof. Condition (b) means that F admits a quasi-inverse G which is a B-functor. The proof issimilar to that of [11, Prop.8.4.2], and shall be left as an exercise for the reader.

Definition 1.4.14. Let B be any small category.(i) A split fibration over B is a pair (ϕ, c) consisting of a fibration ϕ : A → B and a

cleavage c for ϕ, such that c is a functor.(ii) For i = 1, 2, let (ϕi : Ai → B, ci) be two split fibrations, and F : A1 → A2 a cartesian

functor. For every B ∈ Ob(B), let FB : ϕ−11 B → ϕ−1

2 B denote the restriction of F .We say that F is a split cartesian functor (ϕ1, c1) → (ϕ2, c2) if the rule : B 7→ FBdefines a natural transformation c1 ⇒ c2.

1.4.15. Let B be a small category. Clearly, the collection of all fibrations A → B (resp. ofsplit fibrations (ϕ : A → B, c)) with A small, forms a category Fib(B) (resp. sFib(B))whose morphisms are the cartesian functors (resp. the split cartesian functors). Furthermore,we have an obvious forgetful functor :

(1.4.16) sFib(B)→ Fib(B) (ϕ, c) 7→ ϕ.

On the other hand, we have as well a natural functor

C : Fib(B)→ sFib(B)


defined as follows. For a fibration ϕ : A → B, we let C(A /B) be the category whose objectsare all the pairs (B,G), where B ∈ Ob(B) and G : B/B → A is a cartesian functor (recallthat B/B is fibred over B, by example 1.4.2(i)). The morphisms (B,G) → (B′, G′) are thepairs (f, α), where f : B → B′ is a morphism of B, and α : G ⇒ G′ f∗ is a morphismof CartB(B/B,A ). The rule (B,G) 7→ B defines a functor C(ϕ) : C(A /B) → B, and itis easily seen that C(ϕ) is a fibration; indeed, for any morphism f : B′ → B and any object(B,G) of C(A /B), the pair (B′, G f∗) is an inverse image of (B,G) over f (details left tothe reader). Notice that, with this choice of inverse images, the resulting cleavage for C(ϕ) isactually a functor. Notice as well that the rule : (A → B) 7→ C(A /B) is functorial in thefibration A . Namely, every cartesian functor F : A1 → A2 of fibred B-categories induces acartesian functor C(F/B) : C(A1/B) → C(A2/B) via the rule : (B,G) 7→ (B,F G) forevery object (B,G) of C(A1/B), and if F ′ : A2 → A3 is another cartesian functor of fiberedB-categories, we have the identity

C(F ′ F/B) = C(F ′/B) C(F/B).

Let us spell out the cleavage of C(ϕ) : this is the functor

CartB(B/−,A ) : Bo → U′-Cat

(for a suitable universe U′) that assigns to any B ∈ Ob(Bo) the category CartB(B/B,A ), andto any morphism B′

g−→ B in B the functor :

CartB(g∗,A ) : CartB(B/B,A )→ CartB(B/B′,A ).

Moreover, let F : A1 → A2 be a cartesian functor of fibred B-categories. Then we obtain anatural transformation of cleavages :

CartB(B/−,A1)⇒ CartB(B/−,A2) B 7→ CartB(B/B, F ).

Summing up, this shows that the rule (ϕ : A → B) 7→ (C(ϕ),CartB(B/−,A )) defines thesought functor C.

Theorem 1.4.17. With the notation of (1.4.15), we have :(i) The functor C is right 2-adjoint to (1.4.16) (see definition 1.3.10(ii)).

(ii) The counit of this pair of 2-adjoint functors is a pseudo-natural equivalence.

Proof. According to remark 1.3.11(iii), it suffices to exhibit a unit and a counit fulfilling trian-gular identities as in (1.1.8). To this aim, let ϕ : A → B be a fibration, and consider the naturalcartesian functor of B-categories

ev• : C(A /B)→ A

that assigns to every object (B,G) of C(A /B) its evaluation G1B ∈ Ob(ϕ−1B), and to anymorphism (f, α) : (B,G)→ (B′, G′), the morphism αB : G1B → G′f .

Claim 1.4.18. The functor ev• is a B-equivalence.

Proof of the claim. It suffices to show that ev• is a fibrewise equivalence (proposition 1.4.13(i)).However, for every B ∈ Ob(B), an inverse equivalence βB : ϕ−1B → CartB(B/B,A ) can beconstructed explicitly by choosing, for a fixed A ∈ Ob(ϕ−1B), and every object g : B′ → B ofB/B, an inverse image gA : g∗A→ A of A over g. These choices determine βB on objects, viathe rule :

βB(A) : Ob(B/B)→ Ob(A ) : g 7→ g∗A.

Then the image of βB(A) is uniquely determined on every morphism of B/B, by the universalproperty of gA (cp. (1.4.6)). By the same token, one sees that βB(A)(f) is B-cartesian, forevery morphism f of B/B, i.e. βB(A) is a cartesian functor. Finally, the rule A 7→ βB(A) isfunctorial on ϕ−1B : for the details, see [11, Prop.8.2.7]. ♦


Clearly every cartesian functor F : A1 → A2 of fibred B-categories yields a commutativediagram of B-categories :

C(A1/B)C(F/B)

//

ev•

C(A2/B)

ev•

A1F // A2

hence ev• is our candidate pseudo-natural counit, and claim 1.4.18 implies that (ii) holds forthis choice of counit. Next, let (ϕ : A → B, c) be any split fibration. To define a unit, we needto exhibit a natural split cartesian functor :

(1.4.19) (ϕ, c)→ (C(A /B),CartB(B/−,A )).

This is obtained as follows. Let A be any object of A , and set B := ϕ(A); to A we assignthe unique cartesian functor FA determined on objects by the rule : FA(f) := f ∗A, for everyf ∈ Ob(B/B); of course, f ∗ denotes the pull-back functor provided by the split cleavagec. It is easily seen that the rule A 7→ FA extends to a unique split cartesian functor (1.4.19).Lastly, we entrust to the reader the verification that the unit and counit thus defined do fulfill thetriangular identities.

Remark 1.4.20. Let ϕ : A → B be a fibration. Since the equivalence ev• of claim 1.4.18 isindependent of choices, one might hope that the rule : B 7→ (ev• : C(ϕ)−1B → ϕ−1B) extendsto a natural isomorphism between functors Bo → U′-Cat. However, since (1.4.9) is only apseudo-functor, the best one can achieve is a pseudo-natural equivalence :

ev : CartB(B/−,A )⇒ c

which will be uniquely determined, for every given choice of a cleavage c for the fibration ϕ.

Proposition 1.4.21. Let F : A → B be a functor, C → B a fibration, and suppose that,for every B ∈ Ob(B), the induced functor CartB(F/B,C ) is an equivalence (notation of(1.1.16)). Then the functor CartB(F,C ) is an equivalence.

Proof. Let D denote the category whose objects are all the pairs (B,G), where B ∈ Ob(B)and G : FA /B → C is a strictly cartesian functor. The morphisms in D are defined as forthe category C(C ) introduced in the proof of theorem 1.4.17. It is easily seen that the rule :(B,G) 7→ G determines a fibration D → B. Moreover, the natural transformation

CartB(F/−,C ) : CartB(B/−,C )⇒ CartB(FA /−,C )

induces a cartesian functor of B-categories G : C(C )→ D , whence a commutative diagram :

(1.4.22)

CartB(B,C )CartB(F,C )

//

α

CartB(A ,C )

β

CartB(B,C(C ))

CartB(B,G)// CartB(B,D).

Namely, α assigns to any strongly cartesian B-functor F : B → C the strongly cartesianfunctor α(F ) : B → C(C ), given by the rule :

B 7→ (F sB : B/B → C ) for every B ∈ Ob(B)

where sB : B/B → B is the functor (1.1.13). Likewise, β sends a strongly cartesian functorG : A → C to the strongly cartesian functor β(G) : B → D given by the rule :

B 7→ (F ιB : FA /B → C ) for every B ∈ Ob(B)

where ιB : FA /B → A is defined as in (1.1.16).


Our assumption then means that G is a fibrewise equivalence, in which case, proposition1.4.13(ii) ensures that the bottom arrow of (1.4.22) is an equivalence. We leave to the readerthe verification that both vertical arrows are equivalences as well, after which the assertionfollows.

1.5. Sieves and descent theory. This section develops the basics of descent theory, in thegeneral framework of fibred categories. For later use, it is convenient to introduce the notionof n-faithful functor, for all integers n ≤ 2. Namely : if n < 0, every functor is n-faithful;a functor F : A → B (between any two categories A and B) is 0-faithful, if it is faithful(definition 1.1.4); F is 1-faithful, if it is fully faithful; finally, we say that F is 2-faithful, if it isan equivalence.

Definition 1.5.1. Let B and C be two categories, F : B → C a functor.(i) A sieve of C is a full subcategory S of C such that the following holds. If A ∈

Ob(S ), and B → A is any morphism in C , then B ∈ Ob(S ).(ii) If S ⊂ Ob(C ) is any subset, there is a smallest sieve SS of C such that S ⊂ Ob(SS);

we call SS the sieve generated by S.(iii) If S is a sieve of C , the inverse image of S under F is the full subcategory F−1S of

B with Ob(F−1S ) = B ∈ Ob(B) | FB ∈ Ob(S ) (notice that F−1S is a sieve).(iv) If f : X → Y is any morphism in C , and S is any sieve of C/Y , we shall write

S ×Y f for the inverse image of S under the functor f∗ (notation of (1.1.14)).

Example 1.5.2. For instance, suppose that S is the sieve of C/Y generated by a family :

Yi → Y | i ∈ I ⊂ Ob(C/Y ).

If the fibre product Xi := X ×Y Yi is representable in C for every i ∈ I , then S ×Y f is thesieve generated by the family of induced projections Xi → X | i ∈ I ⊂ Ob(C/X).

1.5.3. Let C be a category with small Hom-sets, X an object of C , and S a sieve of thecategory C/X; we define the presheaf hS on C by ruling that

hS (Y ) := f ∈ HomC (Y,X) | (Y, f) ∈ Ob(S ) for every Y ∈ Ob(C ).

For a given morphism f : Y ′ → Y in C , the map hS (f) is just the restriction of HomC (f,X).Hence hS is a subobject of hX (notation of (1.1.19)), and indeed the rule S 7→ hS sets upa natural bijection between the subobjects of hX in C ∧ and the sieves of C/X . The inversemapping sends a subobject F of hX to the full subcategory SF of C/X such that :

Ob(SF ) =⋃

Y ∈Ob(C )

(Y, f) | f ∈ F (Y ).

In the notation of (1.1.16), this is naturally isomorphic to the category hC /F and it is easy tocheck that it is indeed a sieve of C/X .

Example 1.5.4. In the situation of (1.5.3), let S := Xi → X | i ∈ I be any family ofmorphisms in C . Then S generates the sieve S if and only if :

hS =⋃i∈I

Im(hXi → hX).

(Notice that the above union is well defined even in case I is not small.)

In the same way, one sees that the sieves of C are in natural bijection with the subobjects ofthe final object 1C of C ∧. Moreover, it is easy to check that, for every sieve S of C/X , andevery morphism f : Y → X in C , the above correspondence induces a natural identification ofsubobjects of hY :

hS×Xf = hS ×hX hY .


Since the Yoneda imbedding is fully faithful, we shall often abuse notation, to identify an objectX of C with the corresponding representable presheaf hX ; with this notation, we may writehS×Xf = hS ×X Y .

1.5.5. Let C be a small category, and S the sieve of C generated by a subset S ⊂ Ob(C ). Saythat S = Si | i ∈ I for a set I of the universe U; for every i ∈ I , there is a faithful imbeddingεi : C/Si → S , and for every pair (i, j) ∈ I × I , we define

C/Sij := C/Si ×(εi,εj) C/Sj

(notation of (1.1.27)). Hence, the objects of C/Sij are all the triples (X, gi, gj), where X ∈Ob(C ) and gl ∈ HomC (X,Sl) for l = i, j. The natural projections :

π1ij∗ : C/Sij → C/Si π0

ij∗ : C/Sij → C/Sj

are faithful imbeddings. We deduce a natural diagram of categories :∐(i,j)∈I×I

C/Sij∂0 //

∂1

//∐i∈I

C/Siε−→ S

where :∂0 :=

∐(i,j)∈I×I

π0ij∗ ∂1 :=

∐(i,j)∈I×I

π1ij∗ ε :=

∐i∈I

εi.

Remark 1.5.6. Notice that, under the current assumptions, the product Sij := Si × Sj is notnecessarily representable in C . In case it is, we may consider another category, also denotedC/Sij , namely the category of Sij-objects of C (as in (1.1.12)). The latter is naturally isomor-phic to the category with the same name introduced in (1.5.5). Moreover, under this naturalisomorphism, the projections π0

ij∗ and π1ij∗ are identified with the functors induced by the natu-

ral morphisms π0ij : Sij → Sj and respectively π1

ij : Sij → Si. Hence, in this case, the notationof (1.5.5) is compatible with (1.1.14).

Lemma 1.5.7. With the notation of (1.5.5), the functor ε induces an isomorphism between Sand the coequalizer (in the category Cat) of the pair of functors (∂0, ∂1).

Proof. Let A be any other object of Cat, and F :∐

i∈I C/Si → A a functor such that F ∂0 =F ∂1. We have to show that F factors uniquely through ε. To this aim, we construct explicitlya functor G : S → A such that G ε = F . First of all, by the universal property of thecoproduct, F is the same as a family of functors (Fi : C/Si → A | i ∈ I), and the assumptionon F amounts to the system of identities :

(1.5.8) Fi π1ij∗ = Fj π0

ij∗ for every i, j ∈ I.Hence, let X ∈ Ob(S ); by assumption there exists i ∈ I and a morphism f : X → Si in C , sowe may set GX := Fif . In case g : X → Sj is another morphism in C , we deduce an objecth := (X, f, g) ∈ Ob(C/Sij), so f = π1

ij∗h and g = π0ij∗h; then (1.5.8) shows that Fif = Fjg,

i.e. GX is well-defined.Next, let ϕ : X → Y be any morphism in S ; choose i ∈ I and a morphism fY : Y → Si,

and set fX := fY ϕ. We let Gϕ := Fi(ϕ : fX → fY ). Arguing as in the foregoing,one verifies easily that Gϕ is independent of all the choices, and then it follows easily thatG(ψ ϕ) = Gψ Gϕ for every other morphism ψ : Y → Z in S . It is also clear thatG1X = 1GX , whence the contention.

Example 1.5.9. Let B be any category and X any object of B.(i) Let ϕ : A → B be a fibration, and S any sieve of A . By restriction, ϕ induces a

functor ϕ|S : S → B, and it is straightforward to see that ϕ|S is again a fibration.


(ii) In the situation of example 1.4.10(iii), suppose that F = hS for some sieve S ofB/X (notation of (1.5.3)). Then AF is naturally isomorphic to S , and ϕF is naturallyidentified to the restriction S → B of the functor (1.1.13).

1.5.10. In the situation of (1.5.5), suppose that the set of generators S is the whole of Ob(S );in this case, the augmentation ε can also be used to produce the following 2-categorical presen-tation of S . Consider the functor

GS : S → Cat/C : Y 7→ C/Y (Zf−→ Y ) 7→ (C /Z

f∗−−→ C /Y ).

We have a natural coconeε : GS ⇒ FS

where FS is the constant functor S → Cat/C with value S , and εX : C /X → S is thefaithful imbedding as in (1.5.5), for every X ∈ Ob(S ). We may then state :

Lemma 1.5.11. The functor ε induces an equivalence of categories :

2-colimS

GS∼→ S .

Proof. For a given small C -category A , a pseudo-cocone ϕ• : GS ⇒ FA is the datum of asystem of functors

ϕX : C /X → A for every X ∈ Ob(S )

and natural isomorphisms

τf : ϕZ ⇒ ϕY f∗ for every (f : Z → Y ) ∈ Morph(S )

such that

(1.5.9) τgf = (τg ∗ f∗) τf for every composition Zf−→ Y

g−→ X in S .

To such ϕ• we associate a functor ϕ† : S → A , by the rule :

ϕ†(X) := ϕX(1X) for every X ∈ Ob(S ).

Let g : Y → X be any morphism in S , and denote by (g/X : g → 1X) ∈ HomC /X(g,1X)the element corresponding to g; we have a morphism τg(1Y ) : ϕY (1Y )→ ϕX(g∗1Y ) = ϕX(g),and we set

ϕ†(g) := ϕX(g/X) τg(1Y ) : ϕ†(Y )→ ϕ†(X).

Let us check that ϕ† is indeed a functor on S : for any two morphisms f : Z → Y andg : Y → X we may compute

ϕ†(g) ϕ†(f) =ϕX(g/X) τg(1Y ) ϕY (f/Y ) τf (1Z)

=ϕX(g/X) ϕX(f/X) τg(f) τf (1Z)

=ϕX(g f/X) τgf (1Z)

=ϕ†(g f)

where the second identity follows from the naturality of τg, and the third follows from (1.5.9).It is easily seen that the rule : ϕ• 7→ ϕ† defines a functor

PsNat(GS ,FA )→ Fun(S ,A )

such that (Fψ ε)† = ψ for every functor ψ : S → A . Lastly, for every ϕ• as above, thepseudo-cocone Fϕ† ε is naturally isomorphic to ϕ•, so the claim follows (details left to thereader).

Definition 1.5.12. Let ϕ : A → B be a fibration between small categories, B an object of B,S a sieve of B/B, and denote by ιS : S → B/B the fully faithful imbedding.


(i) For i ∈ 0, 1, 2, we say that S is a sieve of ϕ-i-descent, if the restriction functor :

CartB(ιS ,A ) : CartB(B/B,A )→ CartB(S ,A )

is i-faithful. (Here B/B is fibred over B as in example 1.4.2(i).)(ii) For i ∈ 0, 1, 2, we say that S is a sieve of universal ϕ-i-descent if, for every mor-

phism f : B′ → B of B, the sieve S ×B f is of ϕ-i-descent (notation of definition1.5.1(iv)).

(iii) Let f : B′ → B be a morphism in B. We say that f is a morphism of ϕ-i-descent (resp.a morphism of universal ϕ-i-descent), if the sieve generated by f is of ϕ-i-descent(resp. of universal ϕ-i-descent).

Example 1.5.13. (i) Consider the fibration s : Morph(C )→ C of example 1.4.2(ii). A sieve ofs-1-descent (resp. of universal s-1-descent) is also called an epimorphic sieve (resp. a universalepimorphic sieve), and a sieve of s-2-descent (resp. of universal s-2-descent) is also called astrict epimorphic sieve (resp. a universal strict epimorphic sieve).

(ii) Notice that, for every X ∈ Ob(C ) there is a natural equivalence of categories :

X/C∼→ CartC (C/X,Morph(C )).

Namely, to a morphism f : X → Y in C one assigns the cartesian functor f∗ : C/X → C/Y ⊂Morph(C ) (notation of (1.1.14)). An essential inverse for this equivalence is given by the rule :F 7→ F (1X) for every cartesian functor F : C/X → Morph(C ).

(iii) More generally, for every sieve S ⊂ C/X there is a natural faithful functor :

(1.5.14) hS /hC → CartC (S ,Morph(C )).

Indeed, denote by h|S : S∼→ hC /hS the isomorphism of categories provided by (1.5.3).

Then (1.5.14) assigns to any object ϕ : hS → hY of hS /hC the functor

(hC /ϕ) h|S : S → hC/hY (g : Z → X) 7→ ϕ h|S (g)

(notation of (1.1.16), and the Yoneda imbedding identifies the category hC/hY with C/Y ⊂Morph(C )). Under the faithful imbedding (1.5.14) and the equivalence of (ii), the restrictionfunctor CartC (ιS ,Morph(C )) corresponds to the composition :

X/Ch−→ hX/hC

i∗−−→ hS /hC .

where i : hS → hX is the natural inclusion of presheaves.

1.5.15. In the situation of definition 1.5.12, suppose that S is the sieve generated by a set ofobjects Si → B | i ∈ I ⊂ Ob(B/B). There follows a natural diagram of categories (notationof (1.5.5)) :

CartB(S ,A )ε∗−→∏i∈I

CartB(B/Si,A )∂∗0 //

∂∗1

//∏

(i,j)∈I×I

CartB(B/Sij,A )

where ε∗ := CartB(ε,A ) and ∂∗i := CartB(∂i,A ), for i = 0, 1. With this notation, lemma1.5.7 easily implies that ε∗ induces an isomorphism between CartB(S ,A ) and the equalizer(in the category Cat) of the pair of functors (∂∗0 , ∂

∗1).

Example 1.5.16. A family of morphisms (fi : Xi → X | i ∈ I) in a category C is called anepimorphic (resp. strict epimorphic, resp. universal strict epimorphic) family if it generates asieve of C/X with the same property. In view of (1.5.15) and example 1.5.13(ii), we see thatsuch a family is epimorphic (resp. strict epimorphic) if and only if the following holds. Forevery Y ∈ Ob(C ), the natural map∏

i∈I

fi∗ : HomC (X, Y )→∏i∈I

HomC (Xi, Y )


is injective (resp. and its image consists of all the systems of morphisms (gi : Xi → Y | i ∈ I)such that, for every Z ∈ Ob(C ), every i, j ∈ I , and every pair of morphisms hi : Z → Xi,hj : Z → Xj with fi hi = fj hj , we have gi hi = gj hj).

We say that the family (fi | i ∈ I) is effective epimorphic, if it is strict epimorphic, andmoreover the fibre products Xi ×X Xj are representable in C for every i, j ∈ I . This is thesame as saying that the map

∏i∈I fi∗ identifies HomC (X, Y ) with the equalizer of the two

natural maps ∏i∈I

HomC (Xi, Y ) ////∏

(i,j)∈I×I

HomC (Xi ×X Xj, Y ).

A family (fi | i ∈ I) as above is called universal epimorphic (resp. universal effective epimor-phic) if (a) the fibre products Yi := Xi ×X Y are representable in C , for every i ∈ I and everymorphism Y → X in C , and (b) all the resulting families (Yi → Y | i ∈ I) are still epimorphic(resp. effective epimorphic).

1.5.17. We would like to exploit the presentation (1.5.15) of CartB(S ,A ), in order to translatedefinition 1.5.12 in terms of the fibre categories ϕ−1Si and ϕ−1Sij . The problem is that such atranslation must be carried out via a pseudo-natural equivalence (namely ev), and such equiva-lences do not respect a presentation as above in terms of equalizers in the category Cat. Whatwe need is to upgrade our presentation of S to a new one, which is preserved by pseudo-naturaltransformations. This is achieved as follows. Resume the general situation of (1.5.5). For everyi, j, k ∈ I , set C/Sijk := C/Sij ×C C/Sk. We have a natural diagram of categories :

(1.5.18)∐

(i,j,k)∈I3

C/Sijk∂0 //

∂2

//∂1//∐

(i,j)∈I2

C/Sij∂0 //

∂1

//∐i∈I

C/Siε−→ S

where ∂0 is the coproduct of the natural projections π0ijk∗ : C/Sijk → C/Sjk for every i, j, k ∈ I ,

and likewise ∂1 (resp. ∂2) is the coproduct of the projections π1ijk∗ : C/Sijk → C/Sik (resp.

π2ijk∗ : C/Sijk → C/Sij). We can view (1.5.18) as an augmented 2-truncated semi-simplicial

object in Cat/C , i.e. a functor :

FS : (Σ∧2)o → Cat/C .

from the opposite of the category Σ∧2 whose objects are the ordered sets ∅, 0, 0, 1 and0, 1, 2, and whose morphisms are the non-decreasing injective maps (this is a subcategory ofthe category ∆∧2 of (4.2)).

Remark 1.5.19. Suppose that finite products are representable in B, and for every i, j, k ∈ I ,set Sij := Si × Sj , and Sijk := Sij × Sk. Just as in remark 1.5.6, the category C/Sijk of Sijk-objects of C is naturally isomorphic to the category with the same name introduced in (1.5.17),and under this isomorphism, the functors π0

ijk∗ are identified with the functors arising from thenatural projections π0

ijk : Sijk → Sjk (and likewise for π1ijk∗ and π2

ijk∗). For this reason, evenin case these products are not representable in C , we shall abuse notation, and write X → Sij(resp. X → Sijk) to signify an object of C/Sij (resp. of C/Sijk).

With this notation, denote by Σ2 the full subcategory of Σ∧2 whose objects are the non-emptysets; we have the following 2-category analogue of lemma 1.5.7 :

Proposition 1.5.20. The augmentation ε induces an equivalence of categories :

2-colimΣo2

FS∼→ S .


Proof. Here we regard Σo2 (resp. Cat/C ) as a 2-category, as in example 1.3.6(i) (resp. example

1.3.6(iii),(v)). Now, for a given small C -category A , a pseudo-cocone ϕ• : FS ⇒ FA is adatum (ϕ•, α•, β•) consisting of a system of C -functors :

ϕ0 :∐i∈I

C/Si → A ϕ1 :∐

(i,j)∈I2

C/Sij → A ϕ2 :∐

(i,j,k)∈I3

C/Sijk → A

together with natural C -isomorphisms (i.e. invertible 2-cells in Cat/C ) :

αi : ϕ0 ∂i ⇒ ϕ1 βj : ϕ1 ∂j ⇒ ϕ2 (i = 0, 1 j = 0, 1, 2)

related by the simplicial identities :

β1 (α0 ∗ ∂1) = β0 (α0 ∗ ∂0)

β2 (α0 ∗ ∂2) = β0 (α1 ∗ ∂0)

β2 (α1 ∗ ∂2) = β1 (α1 ∗ ∂1).

We consider the natural C -isomorphism :

ωϕ := α−11 α0 : ϕ0 ∂0 ⇒ ϕ0 ∂1.

The above simplicial identities yield :

ωϕ ∗ ∂0 = (α0 ∗ ∂2)−1 β−12 β1 (α0 ∗ ∂1)

ωϕ ∗ ∂1 = (α1 ∗ ∂2)−1 β−12 β1 (α0 ∗ ∂1)

which in turn imply the (cocycle) identity :

(ωϕ ∗ ∂2) (ωϕ ∗ ∂0) = ωϕ ∗ ∂1.

Hence, let us denote by d(A ) the category whose objects are all pairs (ϕ, ω) consisting of aC -functor ϕ :

∐i∈I C/Si → A and a natural C -isomorphism ω : ϕ ∂0 ⇒ ϕ ∂1 fulfilling the

cocycle identity : (ω ∗ ∂2) (ω ∗ ∂0) = ω ∗ ∂1; the morphisms α : (ϕ, ω) → (ϕ′, ω′) in d(A )are the natural C -transformations (i.e. 2-cells in Cat/C ) α : ϕ⇒ ϕ′, such that the diagram :

ϕ ∂0

α∗∂0

ω +3 ϕ ∂1

α∗∂1

ϕ′ ∂0

ω′ +3 ϕ′ ∂1

commutes. It is easily seen that the rule (ϕ•, α•, β•) 7→ (ϕ := α−11 α0, ωϕ) extends to a functor:

dA : PsNat(FS ,FA )→ d(A ).

The functor dA is an equivalence of C -categories; indeed, a quasi-inverse C -functor :

eA : d(A )→ PsNat(FS ,FA )

can be constructed as follows. Given any object (ϕ, ω) of d(A ), set :

ϕ0 := ϕ ϕ1 := ϕ ∂0 ϕ2 := ϕ1 ∂0

α0 := 1ϕ1 α1 := ω−1 β0 := 1ϕ2 =: β1 β2 := α1 ∗ ∂0.

Using the cocycle identity for ω, one verifies easily that the datum eA (ϕ, ω) := (ϕ•, α•, β•)is a pseudo-cocone; moreover, the construction is obviously functorial in (ϕ, ω), and there arenatural isomorphisms dA eA ⇒ 1 and eA dA ⇒ 1.

Next, consider the C -category R whose objects are the same as the objects of∐

i∈I C/Si,and with morphisms given by the rule :

HomR(X → Si, Y → Sj) := HomC (X, Y )


for any pair of objects (X → Si, Y → Sj). We have obvious C -functors

ε′ :∐i∈I

C/Si → R and δ : R → S : (X → Si) 7→ X

where ε′ is the identity on objects, and it is clear that δ is an equivalence of C -categories (i.e.it is an equivalence, when regarded as a 1-cell in Cat/C ). Moreover, there is a natural C -isomorphism :

i : ε′ ∂0 ⇒ ε′ ∂1

which assigns to every object X → Sij of C/Sij the identity map i(X→Sij) := 1X . It is easilyseen that the pair (ε′, i) is an object of d(R), whence a pseudo-cocone

ε′• := eR(ε′, i) : FS ⇒ FR .

Clearly ε = δ ε′, hence it suffices to show that the composed functor :

(1.5.21) Cat/C (R,A )→ PsNat(FS ,FA )→ d(A ) : G 7→ dA (FG ε′•)is an isomorphism for every small C -category A . By inspecting the construction, we see thatthe latter assigns to every C -functor G : R → A the pair (G ε′, G ∗ i). To conclude, we shallexhibit an inverse for (1.5.21).

Namely, let (ϕ, ω) be any object of d(A ); we set :

Gf := ϕf for every f ∈ Ob(R).

Now, suppose that fX : X → Si and fY : Y → Sj are two objects of R, and h : X → Y isan element of HomR(fX , fY ); the pair (fX , f

′X := fY h) determines a morphism f ′X × fX :

X → Sji with ∂0(f ′X × fX) = fX and ∂1(f ′X × fX) = f ′X (notation of remark 1.5.19); hencewe may define Gh as the composition :

Gh : GfXωf ′X×fX−−−−−→ ϕf ′X

ϕ(h/Sj)−−−−→ ϕfY = GfY

where h/Sj denotes h, regarded as an element of HomC /Sj(f′X , fY ).

Next, let fZ : Z → Sk be another object of R, and g : Y → Z any element of HomR(fY , fZ);we have to verify thatG(gh) = GgGh. As in the foregoing, we deduce morphismsX → Sj ,X → Sk and Y → Sk, as well as their products X → Sji, X → Ski, and Y → Skj , whence adiagram :

(1.5.22)

GfX := ϕ(X → Si)ω(X→Sji) //

ω(X→Ski) **UUUUUUUUUUUUUUUUUϕ(X → Sj)

ϕ(h/Sj) //

ω(X→Skj)

GfY := ϕ(Y → Sj)

ω(Y→Skj)

ϕ(X → Sk)

ϕ(h/Sk)//

ϕ(gh/Sk) **UUUUUUUUUUUUUUUUUϕ(Y → Sk)

ϕ(g/Sk)

GfZ := ϕ(Z → Sk).

The sought identity amounts to asserting that (1.5.22) commutes, which can be easily verified,using the cocycle condition for ω, and the obvious identities :

∂0(h/Skj) = h/Sj and ∂1(h/Skj) = h/Sk.

Finally, notice that G1f = G(1f 1f ) = G1f G1f , for every f ∈ Ob(R); since G1f is anisomorphism, it follows that G1f = 1Gf .

Hence G is a functor R → A ; furthermore, the rule (ϕ, ω) 7→ G is clearly functorial, and asimple inspection shows that G ε′ = ϕ and G ∗ i = ω. Conversely, if we apply the foregoingprocedure to a pair of the form (G ε′, G ∗ i), we obtain back the functor G.


1.5.23. Resume the situation of (1.5.15), and notice that all the categories appearing in (1.5.18)are fibred over B : indeed, every morphism in each of these categories is cartesian, hence allthe functors appearing in (1.5.18) are cartesian. Let us consider now the functor :

CartB(−,A ) : (Cat/B)o → Cat

that assigns to every B-category C the category CartB(C ,A ). With the notation of (1.5.15),we deduce a functor :

CartB(FS ,A ) : Σ∧2 → Cat

and in light of the foregoing observations, proposition 1.5.20 easily implies that CartB(ε,A )induces an equivalence of categories :

(1.5.24) 2-limΣ2

CartB(FS ,A )∼→ CartB(S ,A ).

Next, suppose that the fibre products Sij := Si × Sj and Sijk := Sij × Sk are representable inB (see remark 1.5.19); in this case, we may compose with the pseudo-equivalence ev of remark1.4.20 : combining with lemma 1.3.14 we finally obtain an equivalence between the categoryCartB(S ,A ), and the 2-limit of the pseudo-functor d := ev CartB(FS ,A ) : Σ2 → Cat :∏

i∈Iϕ−1Si

∂0//

∂1//∏

(i,j)∈I2

ϕ−1Sij∂0//

∂2//∂1 //∏

(i,j,k)∈I3

ϕ−1Sijk.

Of course, the coface operators ∂s on∏

i∈I ϕ−1Si decompose as products of pull-back functors:

π0∗ij : ϕ−1Sj → ϕ−1Sij π1∗

ij : ϕ−1Si → ϕ−1Sij

attached – via the chosen cleavage c of ϕ – to the projections π0ij : Sij → Sj and π1

ij : Sij → Si(and likewise for the components πt∗ijk of the other coface operators).

1.5.25. By inspecting the proof of proposition 1.3.15, we may give the following explicit de-scription of this 2-limit. Namely, it is the category whose objects are the data

X := (Xi, Xij, Xijk, ξui , ξ

sij, ξ

tijk | i, j, k ∈ I; s ∈ 0, 1;u, t ∈ 0, 1, 2)

where :

Xi ∈ Ob(ϕ−1Si) Xij ∈ Ob(ϕ−1Sij) Xijk ∈ Ob(ϕ−1Sijk) for every i, j, k ∈ Iand for every i, j, k ∈ I :

ξ0i : (π1

ikπ1ijk)∗Xi

∼→ Xijk ξ0ij : π0∗

ij Xj∼→ Xij ξ0

ijk : π0∗ijkXjk

∼→ Xijk

ξ1j : (π1

jkπ0ijk)∗Xj

∼→ Xijk ξ1ij : π1∗

ij Xi∼→ Xij ξ1

ijk : π1∗ijkXik

∼→ Xijk

ξ2k : (π0

jkπ0ijk)∗Xk

∼→ Xijk ξ2ijk : π2∗

ijkXij∼→ Xijk

are isomorphisms related by the cosimplicial identities :

ξ0ijk π0∗

ijkξ0jk = ξ2

k γ00X ξ1

ijk π1∗ijkξ

0ik = ξ2

k γ01X

ξ2ijk π2∗

ijkξ0ij = ξ1

j γ02X ξ0

ijk π0∗ijkξ

1jk = ξ1

j γ10X

ξ2ijk π2∗

ijkξ1ij = ξ0

i γ12X ξ1

ijk π1∗ijkξ

1ik = ξ0

i γ11X

where γst := γd(∂s),d(∂t) : d(∂s) d(∂t) ⇒ d(∂s ∂t) denotes the coherence constraint of thecleavage c, for any pair of arrows (∂s, ∂t) in the category Σ2. The morphisms X → Y in thiscategory are the systems of morphisms :

(Xi → Yi, Xij → Yij, Xijk → Yijk | i, j, k ∈ I)

that are compatible in the obvious way with the various isomorphisms. However, one mayargue as in the proof of proposition 1.5.20, to replace this category by an equivalent one which


admits a handier description : given a datum X , one can make up an isomorphic datum X∗ :=(Xi, X

∗ij, X

∗ijk, ηi, ηij, ηijk), by the rule :

X∗ij := π1∗ij Xi X∗ijk := (π1

ik π1ijk)∗Xi

η0i := 1 η1

ij := 1 η0ijk := η1

j γ10X

η1j := (ξ0

i )−1 ξ1

j η0ij := (ξ1

ij)−1 ξ0

ij η1ijk := γ11

X

η2k := (ξ0

i )−1 ξ2

k η2ijk := γ12

X .

The cosimplicial identities for this new object are subsumed into a single cocycle identity forωij := η0

ij . Summing up, we arrive at the following description of our 2-limit :• The objects are all the systems X := (Xi, ω

Xij | i, j ∈ I) where Xi ∈ Ob(ϕ−1Si) for

every i ∈ I , andωXij : π0∗

ij Xj∼→ π1∗

ij Xi

is an isomorphism in ϕ−1Sij , for every i, j ∈ I , fulfilling the cocycle identity :

p2∗ijkω

Xij p0∗

ijkωXjk = p1∗

ijkωXik for every i, j, k ∈ I

where, for every i, j, k ∈ I , and t = 0, 1, 2 we have set :

pt∗ijkωX•• := γ1t

X πt∗ijkωX•• (γ0tX )−1.

• The morphisms X → Y are the systems of morphisms (fi : Xi → Yi | i ∈ I) with :

(1.5.26) ωYij π0∗ij fj = π1∗

ij fi ωXij for every i, j ∈ I.

1.5.27. We shall call any pair (X•, ω•) of the above form, a descent datum for the fibration ϕ,relative to the family S := (πi : Si → B | i ∈ I) and the cleavage c. The category of suchdescent data shall be denoted:

Desc(ϕ, S, c).

Sometimes we may also denote it by Desc(A , S, c), if the notation is not ambiguous. Of course,two different choices of cleavage lead to equivalent categories of descent data, so usually weomit mentioning explicitly c, and write simply Desc(ϕ, S) or Desc(A , S). The foregoing dis-cussion can be summarized, by saying that there is a commutative diagram of categories :

(1.5.28)

CartB(B/B,A )CartB(ιS ,A )

//

evB

CartB(S ,A )

δS

ϕ−1BρS // Desc(ϕ, S, c)

whose vertical arrows are equivalences, and where ρS is determined by c. Explicitly, ρS assignsto every object C of ϕ−1B the pair (C•, ω

C• ) where Ci := π∗iC, and ωCij is defined as the

composition :

π0∗ij π∗jC

γ(π0ij,πj)

−−−−−→∼

(πj π0ij)∗C = (πi π1

ij)∗C

γ−1

(π1ij,πi)−−−−→∼

π1∗ij π∗iC

where γ(π1ij ,πi)

and γ(π0ij ,πj)

are the coherence constraints for the cleavage c (see (1.4.8)). Thedescent datum (X•, ω•) is said to be effective, if it lies in the essential image of ρS .

We also have an obvious functor :

pS : Desc(ϕ, S)→∏i∈I

ϕ−1Si (Xi, ωij | i, j ∈ I) 7→ (Xi | i ∈ I)

such that :pS ρS =

∏i∈I

π∗i : ϕ−1B →∏i∈I

ϕ−1Si.


1.5.29. Furthermore, for every morphism f : B′ → B in B, set

S ×B f := (πi ×B B′ : Si ×B B′ → B′ | i ∈ I)

which is a generating family for the sieve S ×B f (notation of (1.1.12)); then we deduce apseudo-natural transformation of pseudo-functors (B/B)o → Cat :

ρ : c ioB ⇒ Desc(ϕ, S ×B −, c) (f : B′ → B) 7→ ρS×Bf

(where iB : B/B → B is the functor (1.1.13)) fitting into a commutative diagram :

CartB(B/−,A ) ioBCartB(ιS×B−,A )

+3

ev∗ioB

CartB(S ×B −,A )

δS×B−

c ioBρ +3 Desc(ϕ, S ×B −, c)

using which, one can figure out the pseudo-functoriality of the rule : f 7→ Desc(ϕ, S ×B f, c).Namely, every pair of objects f : C → B and f ′ : C ′ → B, and any morphism h : C ′ → C inB/B, yield a commutative diagram :

Sj ×B C ′

hj :=Sj×Bh

Sij ×B C ′π0ijoo

π1ij //

hij :=Sij×Bh

Si ×B C ′πi //

hi:=Si×Bh

C ′

h

Sj ×B C Sij ×B C

π0ij

ooπ1ij

// Si ×B C πi// C.

Hence one obtains a functor :

Desc(ϕ, h, c) : Desc(ϕ, S ×B f, c)→ Desc(ϕ, S ×B f ′, c)

by the rule :(Xi, ω

Xij | i, j ∈ I) 7→ (h∗iXi, ω

Xij | i, j ∈ I)

where ωXij is the isomorphism that makes commute the diagram :

h∗ijπ0∗ij Xj

γ(hij ,π

0ij

)

//

h∗ijωij

(π0ij hij)∗Xj π0∗

ij h∗jXj

γ(π0ij,hj)

oo

ωXij

h∗ijπ1∗ij Xi

γ(hij ,π

1ij

)

// (π1ij hij)∗Xi π1∗

ij h∗iXi

γ(π1ij,hi)

oo

and if f ′′ : C ′′ → B is a third object, with a morphism g : C ′′ → C ′, we have a naturalisomorphism of functors :

Desc(ϕ, g, c) Desc(ϕ, h, c)⇒ Desc(ϕ, h g, c)

which is induced by the cleavage c, in the obvious fashion.

Theorem 1.5.30. For i = 1, 2, let ϕi : Ai → B be two fibrations, F : A1 → A2 a cartesianfunctor of B-categories, B an object of B and S a sieve of B/B generated by the family(Si → B | i ∈ I). We assume that Sij and Sijk are representable in B, for every i, j, k ∈ I (seeremark 1.5.19); then we have :

(i) For n ∈ 0, 1, 2 and every i, j, k ∈ I , suppose that(a) S is a sieve both of ϕ1-n-descent and of ϕ2-(n− 1)-descent.(b) The restriction Fi : ϕ−1

1 Si → ϕ−12 Si of F is n-faithful.

(c) The restriction Fij : ϕ−11 Sij → ϕ−1

2 Sij of F is (n− 1)-faithful.(d) The restriction Fijk : ϕ−1

1 Sijk → ϕ−12 Sijk of F is (n− 2)-faithful.

Then the restriction FB : ϕ−11 B → ϕ−1

2 B of F is n-faithful.


(ii) Suppose that the functors Fij are fully faithful, and the functors Fijk are faithful, forevery i, j, k ∈ I . Then the natural commutative diagram

CartB(S ,A1)CartB(S ,F )

//

CartB(S ,A2)

∏i∈I ϕ

−11 Si

∏i∈I Fi //

∏i∈I ϕ

−12 Si

is 2-cartesian.

Proof. (i): In view of theorem 1.4.17, we may assume that both A1 and A2 are split fibrations(with a suitable choice of cleavages), and F is a split cartesian functor. Recall that the lattercondition means the following. For every morphism g : X → Y in B, the induced diagram

ϕ−11 Y

g∗ //

F

ϕ−11 X

F

ϕ−12 Y

g∗ // ϕ−12 X

commutes (where the horizontal arrows are the pull-back functor given by the chosen cleav-ages). In this situation, we have a commutative diagram

(1.5.31)

ϕ−11 B

ρS //

FB

Desc(ϕ1, S)

F

ϕ−1

2 BρS // Desc(ϕ2, S)

whose right vertical arrow is the functor given by the rule :

X := (Xi, ωij | i, j ∈ I) 7→ F (X) := (FiXi, Fijωij | i, j ∈ I).

for every object X of Desc(ϕ1, S). By assumption, the top horizontal arrow of (1.5.31) is n-faithful, and the bottom horizontal arrow is (n − 1)-faithful. We need to prove that the leftvertical arrow is n-faithful, and it is easily seen that this will follow, once we have shown thatthe same holds for the right vertical arrow.

Suppose first that n = 0; we have to show that F is faithful. However, let X and Y be twoobjects of Desc(ϕ1, S), and h1, h2 : X → Y two morphisms. By definition, ht (for t = 1, 2) isa compatible system (ht,i : Xi → Yi | i ∈ I), where each ht,i is a morphism in ϕ−1

1 Si. Then,F (ht) is the compatible system (Fiht,i | i ∈ I). Thus, the condition F (h1) = F (h2) translatesthe system of identities Fih1,i = Fih2,i for every i ∈ I . By assumption, each Fi is faithful,therefore h1 = h2, as stated.

For n = 1, assumption (d) is empty, (b) means that Fi is fully faithful, and (c) means thatFij is faithful for every i, j ∈ I . In light of the previous case, we have only to show that Fis full. Hence, let X, Y be as in the foregoing, and (hi : FiXi → FiYi | i ∈ I) a morphismF (X)→ F (Y ) in Desc(ϕ2, S). By assumption, for every i ∈ I we may find a unique morphismfi : Xi → Yi such that Fifi = hi. It remains only to check that the system (fi | i ∈ I) fulfillscondition (1.5.26), and since the functors Fij are faithful, it suffices to verify that Fij(1.5.26)holds. However, since F is split cartesian, we have :

Fij π0∗ij fj = π0∗

ij Fjfj Fij π1∗ij fi = π1∗

ij Fifi

hence we reduce to showing that (FijωYij )π0∗

ij hj = π1∗ij hi(FijωXij ), which holds by assumption.


Next, we consider assertion (ii) : the contention is that the functors F and :

p1,S : Desc(ϕ1, S)→∏i∈I

ϕ−11 Si

as in (1.5.27), induce an equivalence (p1,S, F ) between Desc(ϕ1, S) and the category C con-sisting of all data of the form G := (Gi, Hi, αi, ω

Hij | i, j ∈ I), where Gi ∈ Ob(ϕ−1

1 Si),Hi ∈ Ob(ϕ−1

2 Si), αi : FiGi∼→ Hi are isomorphisms in ϕ−1

2 Si, and H := (Hi, ωHij | i, j ∈ I) is

an object of Desc(ϕ2, S). However, given an object as above, set :

ωH′

ij := (π1∗ij α−1i ) ωHij (π0∗

ij αj).

Since ϕ2 is a split fibration, one verifies easily that the datumH ′ := (H ′i := FiGi, ωH′ij | i, j ∈ I)

is an object of Desc(ϕ2, S) isomorphic to H , and the new datum (Fi, H′i,1H′i , ω

H′ij | i, j ∈ I)

is isomorphic to G; hence C is equivalent to the category C ′ whose objects are all data of theform (Gi, ωij | i, j ∈ I) where Gi ∈ Ob(ϕ−1

1 Si), and (FiGi, ωij | i, j ∈ I) is an object ofDesc(ϕ2, S). By assumption Fij is fully faithful, and F is a split cartesian functor; hence wemay find unique isomorphisms ωGij : π0∗

ij Gj∼→ π1∗

ij Gi such that ωij = FijωGij . We claim that

the datum (Gi, ωGij | i, j ∈ I) is an object of Desc(ϕ1, S), i.e. the isomorphisms ωGij satisfy the

cocycle condition

(1.5.32) π2∗ijkω

Gij π0∗

ijkωGjk = π1∗

ijkωGik for every i, j, k ∈ I.

To check this identity, since by assumption the functors Fijk are faithful, it suffices to see thatFijk(1.5.32) holds, which is clear, since the cocycle condition holds for the isomorphisms ωij(and since F is split cartesian). This shows that (p1,S, F ) is essentially surjective. Next, sincethe functor p1,S is faithful, the same holds for (p1,S, F ). Finally, let

G := (Gi, ωij | i, j ∈ I) G′ := (G′i, ω′ij | i, j ∈ I)

be two objects of C ′. A morphism G → G′ consists of a system (αi : Gi → G′i | i ∈ I) ofmorphisms such that (Fiαi | i ∈ I) is a morphism

(FiGi, ωij | i, j ∈ I)→ (FiG′i, ω′ij | i, j ∈ I)

in Desc(ϕ2, S). To show that (p1S, F ) is full, and since we know already that this functor isessentially surjective, we may assume that there exist (Gi, ω

Gij | i, j ∈ I), (G′i, ω

G′ij | i, j ∈ I) in

Desc(ϕ1, S) such that ωij = FijωGij and ω′ij = Fijω

G′ij for every i, j ∈ I; in this case, it suffices

to verify the identity

(1.5.33) ωG′

ij π0∗ij αj = π1∗

ij αi ωGij for every i, j ∈ I.Again, the faithfulness of Fij reduces to checking that Fij(1.5.33) holds, which is clear, sinceF is split cartesian.

Lastly, notice that the case n = 2 of assertion (i) is a formal consequence of (ii).

1.5.34. In the situation of (1.5.27), let S be the sieve generated by the family S, and g : B′ →B any morphism in B. We let :

B′i := B′ ×B Si B′ij := B′ ×B Sij B′ijk := B′ ×B Sijk for every i, j, k ∈ Iand denote gi : B′i → Si, gij : B′ij → Sij and gijk : B′ijk → Sijk the induced projections.

Corollary 1.5.35. With the notation of (1.5.34), let n ∈ 0, 1, 2. The following holds :(i) S is a sieve of ϕ-n-descent, if and only if ρS is n-faithful (see (1.5.28)).

(ii) Suppose that :(a) S is a sieve of universal ϕ-n-descent.(b) The pull-back functors g∗i : ϕ−1Si → ϕ−1B′i are n-faithful.


(c) The pull-back functors g∗ij : ϕ−1Sij → ϕ−1B′ij are (n− 1)-faithful.(d) The pull-back functors g∗ijk : ϕ−1Sijk → ϕ−1B′ijk are (n− 2)-faithful.

Then the pull-back functor g∗ is n-faithful.(iii) Suppose that the functors g∗ij are fully faithful, and the functors g∗ijk are faithful, for

every i, j, k ∈ I . Then the natural essentially commutative diagram :

Desc(ϕ, S)Desc(ϕ,g)

//

pS

Desc(ϕ, S ×B B′)pS×BB′

∏i∈I ϕ

−1Si

∏i∈I g

∗i //

∏i∈I ϕ

−1B′i

is 2-cartesian (see (1.3.16)).

Proof. (i) follows by inspecting (1.5.28).(ii): Thanks to theorem 1.4.17, we may assume that ϕ is a split fibration. Now, set

C := Morph(B) A1 := C ×(t,ϕ) A A2 := C ×(s,ϕ) A

where s, t : C → B are the source and target functors (see (1.1.17)). The natural projectionsϕi : Ai → C (for i = 1, 2) are two fibrations (see example 1.4.10(i)). Moreover, t induces afunctor t|g : C/g → B/B (notation of (1.1.15)) and we let S/g := t−1

|g S , which is the sieve ofC/g generated by the cartesian diagrams

Di :

B′igi //

Si

B′

g // B

for every i ∈ I.

Notice that the products Dij := Di ×Dj are represented in C/g by the diagrams

B′ijgij //

Sij

B′

g // B

for every i, j ∈ I

and likewise one may represent the triple products Dijk := Dij ×Dk.By definition, the objects of A1 (resp. A2) are the pairs (h : X → Y, a), where h is a

morphism in B and a ∈ Ob(ϕ−1Y ) (resp. a ∈ Ob(ϕ−1X)). A morphism of A1 (resp. of A2)

(h : X → Y, a)→ (h′ : X ′ → Y ′, a′)

is a datum (f1, f2, t), where f1 : X → X ′ and f2 : Y → Y ′ are morphisms in B withf2 h = h′ f1, and t : a→ f ∗2a

′ (resp. t : a→ f ∗1a′) is a morphism in ϕ−1Y (resp. in ϕ−1X).

Now, we define a functor F : A1 → A2 of C -categories, by the rule :• (h, a) 7→ (h, h∗a) for every (h, a) ∈ Ob(A1).• (f1, f2, t) 7→ (f1, f2, h

∗t) for every morphism (f1, f2, t) of A1 as above. Notice that,if t : a → f ∗2a

′ is a morphism in ϕ−1Y , then h∗t : h∗a → h∗f ∗2a′ = f ∗1h

′∗a′ is amorphism of ϕ−1Y ′, since ϕ is a split fibration.

Notice that a morphism (f1, f2, t) of either A1 or A2 is cartesian if and only if t is an iso-morphism; especially, it is clear that F is a cartesian functor. Moreover, for every objecth : X → Y of C , the restriction ϕ−1

1 h → ϕ−12 h of F is isomorphic to the pull-back functor

h∗ : ϕ−1Y → ϕ−1h. Especially, conditions (b)–(d) say that the restriction Fi : ϕ−11 gi → ϕ−1

2 gi(resp. Fij : ϕ−1

1 gij → ϕ−12 gij , resp. Fijk : ϕ−1

1 gijk → ϕ−12 gijk) are n-faithful (resp. (n − 1)-

faithful, resp. (n− 2)-faithful). In light of theorem 1.5.30(i), we are then reduced to showing


Claim 1.5.36. S /g is a sieve both of ϕ1-n-descent and of ϕ2-n-descent.

Proof of the claim. Let D be any (small) category; we remark first that a functor D → A1

is the same as a pair of functors (H : D → A , K : D → C ) such that ϕ H = t K, andlikewise one can describe the functors D → A2. Then, it is easily seen that the functors

CartB(B/B,A )→ CartC (C /g,A1) G 7→ (G t, t)

CartB(B/B′,A )→ CartC (C /g,A2) G 7→ (G s, s)

are equivalences, and induce equivalencesCartB(S ,A )→ CartC (S /g,A1)

CartB(S ×B g,A )→ CartC (S /g,A2)

(details left to the reader). The claim follows immediately.

1.5.37. Quite generally, if ϕ : A → B is a fibration over a category B that admits fibreproducts, the descent data for ϕ (relative to a fixed cleavage c) also form a fibration :

Dϕ : ϕ-Desc→ Morph(B).

Namely, for every morphism f : T ′ → T of B, the fibre over f is the category Desc(ϕ, f)of all descent data (f, A, ξ) relative to the family f, and the cleavage c, so A is an object ofϕ−1T ′ and ξ : p∗1A

∼→ p∗2A is an isomorphism in the category ϕ−1(T ′ ×T T ′) satisfying theusual cocycle condition (here p1, p2 : T ′×T T ′ → T ′ denote the two natural morphisms). Giventwo objects A := (f : T ′ → T,A, ξ), A′ := (g : W ′ → W,A′, ζ) of ϕ-Desc, the morphismsA→ A′ are the data (h, α) consisting of a commutative diagram :

W ′ h //

g

T ′

f

W // T

and a morphism α : A→ A′ such that ϕ(α) = h and p∗1(α) ξ = ζ p∗2(α).We have a natural cartesian functor of fibrations :

A ×(ϕ,t) Morph(B)d //

p ((RRRRRRRRRRRRRϕ-Desc

Dϕxxrrrrrrrrrrr

Morph(B)

where t : Morph(B) → B is the target functor (see (1.1.17) and example 1.1.27(ii)). Namely,to any pair (T, f : S → ϕT ) with T ∈ Ob(A ) and f ∈ Ob(Morph(B)), one assigns thecanonical descent datum d(T, f) := ρf(T ) in Desc(ϕ, f) associated to the pair as in (1.5.28).

Corollary 1.5.38. In the situation of (1.5.37), let f : B′ → B be a morphism of B, and S asieve of B/B, generated by a family (Si → B | i ∈ I). Let n ∈ 0, 1, 2, and suppose that :

(a) S is a sieve of universal ϕ-n-descent.(b) For every i ∈ I , the morphism Si ×B f is of ϕ-n-descent.(c) For every i, j ∈ I , the morphism Sij ×B f is of ϕ-(n− 1)-descent.(d) For every i, j, k ∈ I , the morphism Sijk ×B f is of ϕ-(n− 2)-descent.

Then f is a morphism of ϕ-n-descent.

Proof. In view of corollary 1.5.35(i), it is easily seen that a morphism g : T ′ → T in B is of ϕ-n-descent if and only if the restriction ϕ−1T → Dϕ−1g of d is n-faithful. Set C := Morph(B);as in the proof of corollary 1.5.35(ii), the functor t induces a functor t|f : C /f → B/B, andwe let S/f := t−1

|f S . With this notation, theorem 1.5.30(i) reduces to showing :


Claim 1.5.39. The sieve S/f is both of p-n-descent and of Dϕ-n-descent.

Proof of the claim. By claim 1.5.36, it is already known that S/f is of p-n-descent. To showthat S/f is of Dϕ-n-descent, we consider the commutative diagram

CartB(B/B′,A ) //

CartC (C/f, ϕ-Desc)

CartB(S ×B f,A ) // CartC (S/f, ϕ-Desc)

whose left (resp. right) vertical arrow is induced by the inclusion S ×B f → B/B′ (resp.S/f → S×Bf ) and whose top horizontal arrow is defined as follows. Given a cartesian functorG : B/B′ → A , we let DG : C/f → ϕ-Desc be the unique cartesian functor determined onthe objects of C/f by the rule : T ′

g //

h

T

B′

f // B

7→ d(G(h), g).

We leave to the reader the verification the rule G 7→ DG extends to a well defined functor,and then there exists a unique (similarly defined) bottom horizontal arrow that makes commutethe foregoing diagram. Moreover, both horizontal arrow thus obtained are equivalences ofcategories. The claim follows.

1.6. Profinite groups and Galois categories. Quite generally, for any profinite group P , letP -Set denote the category of discrete finite sets, endowed with a continuous left action of P(the morphisms in P -Set are the P -equivariant maps). Any continuous group homomorphismω : P → Q of profinite groups induces a restriction functor

Res(ω) : Q-Set→ P -Set

in the obvious way. In case the notation is not ambiguous, one writes also ResPQ for this functor.For any two profinite groups P and Q, we denote by

Homcont(P,Q)

the set of all continuous group homomorphisms P → Q. If ϕ1, ϕ2 are two such group homo-morphisms, we say that ϕ1 is conjugate to ϕ2, and we write ϕ1 ∼ ϕ2, if there exists an innerautomorphism ω of G, such that ϕ2 = ω ϕ1. Clearly the trivial map π → G (whose image isthe neutral element of G), is the unique element of a distinguished conjugacy class.

1.6.1. Let P be any profinite group; for any (discrete) finite group G, consider the pointed setHomcont(P,G)/∼ of conjugacy classes of continuous group homomorphisms P → G. This isalso denoted

H1cont(P,G)

and called the first non-abelian continuous cohomology group of P with coefficients in G (so Gis regarded as a P -module with trivial P -action). Clearly the formation ofH1(P,G) is covarianton the argument G, and controvariant for continuous homomorphisms of profinite groups.

Lemma 1.6.2. Letϕ : P → P ′ be a continuous homomorphism of profinite groups, and supposethat the induced map of pointed sets :

H1cont(P

′, G)→ H1cont(P,G) : f 7→ f ϕ

is bijective, for every finite group G. Then ϕ is an isomorphism of topological groups.


Proof. First we show that ϕ is injective. Indeed, let x ∈ P be any element; we may find an opennormal subgroup H ⊂ P such that x /∈ H; taking G := P/H , we deduce that the projectionP → P/H factors through ϕ and a group homomorphism f : P ′ → P/H , hence x /∈ Kerϕ, asclaimed. Moreover, let H ′ := Ker f ; clearly H ′∩P = H , so the topology of P is induced fromthat of P ′. It remains only to show that ϕ is surjective; to this aim, we consider any continuoussurjection f ′ : P ′ → G′ onto a finite group, and it suffices to show that the restriction of f ′

to ϕP is still surjective. Indeed, let G be the image of ϕP in G′, denote by i : G → G′ theinclusion map, and let f : P → G be the unique continuous map such that i f = f ′ ϕ; byassumption, there exists a continuous group homomorphism g : P ′ → G such that f = g ϕ.On the other hand, (i g) ϕ = f ′ ϕ, hence the conjugacy class of i g equals the conjugacyclass of f ′, especially i g is surjective, hence the same holds for i, as required.

1.6.3. Let G be a profinite group, and H ⊂ G an open subgroup. It is easily seen that H is alsoa profinite group, with the topology induced from G. Morever, the restriction functor

ResHG : G-Set→ H-Set

admits a left adjointIndGH : H-Set→ G-Set.

Namely, to any finite set Σ with a continuous left action of H , one assigns the set IndGHΣ :=G× Σ/∼, where ∼ is the equivalence relation such that

(gh, σ) ∼ (g, hσ) for every g ∈ G, h ∈ H and σ ∈ Σ.

The left G-action on IndGHΣ is given by the rule : (g′, (g, σ)) 7→ (g′g, σ) for every g, g′ ∈ G andσ ∈ Σ. It is easily seen that this action is continuous, and the reader may check that the functorIndGH is indeed left adjoint to ResHG .

1.6.4. Moreover, let 1 denote the final object of H-Set; notice that IndGH1 = G/H , the set oforbits of G under its right translation action by H . Hence, for any finite set Σ with continuousH-action, the unique map tΣ : Σ→ 1 yields a G-equivariant map

IndGHtΣ : IndGH → G/H

and therefore IndGH factors through a functor

(1.6.5) H-Set→ G-Set/(G/H).

It is easily seen that (1.6.5) is an equivalence. Indeed, one obtains a natural quasi-inverse, bythe rule : (f : Σ→ G/H) 7→ f−1(H). The detailed verification shall be left to the reader.

Definition 1.6.6. ([42, Exp.V, Def.5.1]) Let C be a category, and F : C → Set a functor.(i) We say that C is a Galois category, if C is equivalent to P -Set, for some profinite

group P . We denote Galois the category whose objects are all the Galois categories,and whose morphisms are the exact functors between Galois categories.

(ii) We say that F is a fibre functor, if F is exact and conservative, and F (X) is a finite setfor every X ∈ Ob(C ).

(iii) We denote fibre.Fun the 2-category of fibre functors, defined as follows :(a) The objects are all the pairs (C , F ) consisting of a Galois category C and a fibre

functor F for C .(b) The 1-cells (C1, F1) → (C2, F2) are all the pairs (G, β) consisting of an exact

functor G : C1 → C2 and an isomorphism of functors β : F1∼→ F2 G.

(c) And for every pair of 1-cells (G′, β′), (G, β) : (C1, F1) → (C2, F2), the 2-cells(G′, β′)→ (G, β) are the isomorphisms γ : G′

∼→ G such that (F2 ∗ γ) β′ = β.


Composition of 1-cells and 2-cells is defined in the obvious way. We shall also denotesimply by G a 1-cell (G, β) as in (b), such that F1 = F2 G and β is the identityautomorphism of F1.

1.6.7. Notice that any Galois category C admits a fibre functor : indeed, if P is any profinitegroup, the forgetful functor

fP : P -Set→ Set

fulfills the conditions of definition 1.6.6(ii), therefore the same holds for the functor fP β,if β : C → P -Set is any equivalence. For any Galois category C and any fibre functorF : C → Set, we denote

π1(C , F )

the group of automorphisms of F , and we call it the fundamental group of C pointed at F . Bydefinition, for every X ∈ Ob(C ), the finite set F (X) is endowed with a natural left action ofπ1(C , F ). For every X ∈ Ob(C ) and every ξ ∈ F (X), the stabilizer HX,ξ ⊂ π1(C , F ) is asubgroup of finite index, and we endow π1(C , F ) with the coarsest group topology for whichall such HX,ξ are open subgroups. The resulting topological group π1(C , F ) is profinite, and itsnatural left action on every F (X) is continuous. Thus, F upgrades to a functor denoted

F † : C∼→ π1(C , F )-Set.

A basic result states that F † is an equivalence ([42, Exp.V, Th.4.1]).

Example 1.6.8. Let P be any profinite group. Then there is an obvious injective map

P → π1(P -Set, fP )

and [42, Exp.V, Th.4.1] implies that this map is an isomorphism of profinite groups. In otherwords, the group P can be recovered, up to unique isomorphism, from the category P -Settogether with its forgetful functor fP .

Remark 1.6.9. Let C , C ′ be two Galois categories, and F : C → Set a fibre functor.(i) Any exact functor G : C ′ → C induces a continuous group homomorphism :

π1(G) : π1(C , F )→ π1(C ′, F G) ω 7→ ω ∗G.

(ii) Furthermore, any isomorphism β : F ′∼→ F of fibre functors of C induces an isomor-

phism of profinite groups :

π1(β) : π1(C ′, F )∼→ π1(C ′, F ) ω 7→ β−1 ω β

(see [42, Exp.V, §4] for all these generalities).(iii) Let now (G, β) : (C1, F1)→ (C2, F2) be a 1-cell of fibre.Fun. Combining (i) and (ii),

we deduce a natural continuous group homomorphism

π1(G, β) : π1(C2, F2)π1(G)−−−−→ π1(C1, F2 G)

π1(β)−−−−→ π1(C1, F1).

Proposition 1.6.10. With the notation of remark 1.6.9, the rule that assigns :• To any object (C , F ) of fibre.Fun, the profinite group π1(C , F )• To any 1-cell (G, β) of fibre.Fun, the continuous map π1(G, β)

defines a pseudo-functorπ1 : fibre.Fun→ pf .Grpo

from the 2-category of fibre functors, to the opposite of the category of profinite groups (andcontinuous group homomorphisms).


Proof. (Here we regard pf .Grp as a 2-category with trivial 2-cells : see example 1.3.6(i)). Let

(C1, F1)(G,βG)−−−−−→ (C2, F2)

(H,βH)−−−−−→ (C3, F3)

be any pair of (composable) 1-cells; functoriality on 1-cells amounts to the identity :

π1(G, βG) π1(H, βH) = π1(H G, (βH ∗G) βG)

whose detailed verification we leave to the reader. Next, let γ : (G′, β′) → (G, β) be a 2-cell between 1-cells (G′, β′), (G, β) : (C1, F1) → (C2, F2); we have to check that π1(G, β) =π1(G′, β′). This identity boils down to the commutativity of the diagram :

π1(C2, F2)π1(G′)

//

π1(G)

π1(C1, F2 G′)

π1(β′)

π1(C2, F2 G)π1(β)

//

π1(F2∗γ)44iiiiiiiiiiiiiiiiπ1(C1, F1).

However, the commutativity of the lower triangular subdiagram is clear, hence we are reducedto checking the commutativity of the upper triangular subdiagram; the latter is a special case ofthe following more general :

Claim 1.6.11. Let C and C ′ be two Galois categories, G,G′ : C ′ → C two exact functors,β : G′

∼→ G an isomorphism, and F : C → Set a fibre functor. Then the induced diagram ofprofinite groups

π1(C , F )π1(G)

wwoooooooooooπ1(G′)

''PPPPPPPPPPP

π1(C ′, F G)π1(F∗β)

// π1(C ′, F G′)commutes.

Proof of the claim. Left to the reader.

Example 1.6.12. Let ω : P → Q be a continuous group homomorphism between profinitegroups. Clearly fP Res(ω) = fQ, and it is easily seen that the resulting diagram

Pω //

Q

π1(P -Set, fP )

π1(Res(ω))// π1(Q-Set, fQ)

commutes, where the vertical arrows are the natural identifications given by example 1.6.8 : theverification is left as an exercise to the reader.

1.6.13. Let P := (Pi | i ∈ I) be a cofiltered system of profinite groups, with continuoustransition maps, and denote by P the limit of this system, in the category of groups. Then P isnaturally a closed subgroup ofQ :=

∏i∈I Pi, and the topology T induced by the inclusion map

P → Q makes it into a compact and complete topological group. Moreover, since the topologyof Q is profinite, the same holds for the topology T (details left to the reader). It is then easilyseen that the resulting topological group (P,T ) is the limit of the system P in the category ofprofinite groups.

Proposition 1.6.14. In the situation of (1.6.13), the natural functor

2-colimi∈I

Pi-Set→ P -Set

is an equivalence.


Proof. The functor is obviously faithful; let us show that it is also full. Indeed, let j ∈ I be anyindex, Σ,Σ′ two objects of Pj-Set, and ϕ : Σ→ Σ′ a P -equivariant map; we need to show thatϕ is already Pi-equivariant, for some index i ∈ I . To this aim, we may as usual replace I byI/j, and assume that j is the final element of I . We may also find an open normal subgroupHj ⊂ Pj that acts trivially on both Σ and Σ′. Then, for every i ∈ I , we let Hi ⊂ Pi be thepreimage of Pj , and we set P i := Pi/Hi. Let also P := P/H , where H ⊂ P is the preimageof Hj; by construction, ϕ is P -equivariant. Clearly, we may find i ∈ I such that the image of Pin the finite group P j equals the image of P i, and for such index i, the induced map P → P i isan isomorphism. Especially, ϕ is Pi-equivariant, as sought.

Lastly, we show essential surjectivity. Indeed, let Σ be any object of P -Set; we have to showthat the P -action on Σ is the restriction of a continuous Pi-action, for a suitable i ∈ I . However,we may find a normal open subgroupH ⊂ P that acts trivially on Σ. Then there exists a normalopen subgroup L ⊂ Q (notation of (1.6.13)) such that P ∩ L ⊂ H . We may also assume thatthere exists a finite subset J ⊂ I and for every i ∈ J an open normal subgroup Li ⊂ Pi suchthat L =

∏i∈J Li ×

∏i∈I\J Pi. Pick an index j ∈ I that admits morphisms fi : j → i in I , for

every i ∈ J , and let L′i ⊂ Pj denote the preimage of Li under the corresponding map Pj → Pi.Finally, set Hj :=

⋂i∈J L

′i. By construction, H contains the preimage of Hj in P , and we may

therefore assume that H is this preimage. We may replace as usual I by I/j, and assume that jis the final element of I . Then, for every i ∈ I , we let Hi denote the preimage of Hj in Pi, andwe set P i := Pi/Hi. Set as well P := P/H . Clearly, there exists i ∈ I such that the image ofthe induced map P → P j equals the image of P i; for such index i, the induced map P → P i isan isomorphism. Thus, Σ the restriction of an object of Pi-Set, as wished.

1.6.15. We consider now a situation that generalizes slightly that of (1.6.13). Namely, let I bea small filtered category, and

(C•, F•) : I → fibre.Fun i 7→ (Ci, Fi)

a pseudo-functor. By proposition 1.6.10, the composition of π1 and (C•, F•) is a functor

π1(C•, F•) : Io → pf .Grp i 7→ Pi := π1(Ci, Fi).

Let P denote the limit (in pf .Grp) of the cofiltered system P•, and set

C := 2-colimI

C•

where the 2-colimit is formed in the 2-category of small categories. We may then state :

Corollary 1.6.16. In the situation of (1.6.15), there exists a natural equivalence :

C∼→ P -Set.

Proof. Recall that (C•, F•) is the datum of isomorphisms

βϕ : Fj∼→ Fi Cϕ for every morphism ϕ : j → i in I

and 2-cells :

τψ,ϕ : (Cψϕ, βψϕ)∼→ (Cψ, βψ) (Cϕ, βϕ) for every composition j

ϕ−→ iψ−→ k

that – by definition – satisfy the identities :

(1.6.17) (βψ ∗ Cϕ) βϕ = (Fk ∗ τψ,ϕ) βψϕ for every composition jϕ−→ i

ψ−→ k

as well as the composition identities :

((Cµ, βµ) ∗ τψ,ϕ) τµ,ψϕ = (τµ,ψ ∗ (Cϕ, βϕ)) τµψ,ϕ for compositions jϕ−→ i

ψ−→ kµ−→ l.

Let P•-Set : I → Cat denote the functor given by the rule : i 7→ Pi-Set for every i ∈ Ob(I),and ϕ 7→ Res(Pϕ), where Pϕ := π1(Cϕ, βϕ) for every morphism ϕ of I . In view of proposition


1.6.14 and lemma 1.3.14, it suffices to show that the rule : i 7→ F †i for every i ∈ Ob(I) (notationof (1.6.7)), extends to a pseudo-natural isomorphism

F †• : C•∼→ P•-Set.

Indeed, let ϕ : j → i be any morphism of I , and X any object of Cj; we remark that thebijection

βϕ(X) : Res(Pϕ)(F †jX)∼→ F †i Cϕ(X)

is Pi-equivariant; the proof amounts to unwinding the definitions, and shall be left to the reader.Hence we get an isomorphism of functors β†ϕ : Res(Pϕ) F †j

∼→ F †i Cϕ, and from (1.6.17) wededuce the identities :

(β†ψ ∗ Cϕ) (Res(Pϕ) ∗ β†ϕ) = (F †k ∗ τψ,ϕ) β†ψϕ for every composition jϕ−→ i

ψ−→ k.

The latter show that the system β†• fulfills the coherence axiom for a pseudo-natural transforma-tion, as required.

Remark 1.6.18. (i) Keep the situation of (1.6.15), and let a• : C• ⇒ C be the universalpseudo-cocone induced by the pseudo-functor C•. We may regard the pseudo-functor (C•, F•)as a pseudo-cocone F• : C• ⇒ Set whose vertex is the category Set. Then, by the universalproperty of colimits, we get a functor F : C → Set and an isomorphism

σ• : F ∗ a•∼→ F•.

On the other hand, let r• : P•-Set ⇒ P -Set be the natural cocone (so ri is the restrictionfunctor corresponding to the natural map P → Pi, for every i ∈ Ob(I)); the equivalence G ofcorollary 1.6.16 is deduced from the pseudo-cocone r• F †• : C• → P -Set (where F †• is as inthe proof of corollary 1.6.16), so we have an isomorphism of pseudo-functors

t• : G ∗ a•∼→ r• F †•

whence an isomorphism

fP ∗ t• : (fP G) ∗ a•∼→ fP ∗ (r• F †• ) = F•

(notation of (1.6.7)). There follows an isomorphism F ∗ a•∼→ (fP G) ∗ a•; by the universal

property of the 2-colimit, the latter must come from a unique isomorphism ϑ : F∼→ fP G.

Especially, we see that F is also a fibre functor, and we get a pseudo-cocone

(a•, σ•) : (C•, F•)⇒ (C , F ).

It is now immediate that (C , F ) is the 2-colimit (in the 2-category fibre.Fun) of the pseudo-functor (C•, F•), and (a•, σ•) is the corresponding universal pseudo-cocone.

(ii) Likewise, r• may be regarded as a universal pseudo-cocone

r• : (P•-Set, fP•)⇒ (P -Set, fP )

(with trivial coherence constraint), and the coherence constraint β†• as in the proof of corollary1.6.16 yields a pseudo-natural equivalence

F †• : (C•, F•)∼→ (P•-Set, fP•)

as well as an isomorphism(G, ϑ) ∗ (a•, σ•)

∼→ r• F †• .Thus, for every i ∈ Ob(I) we get a 2-cell of fibre.Fun :

(G, ϑ) (ai, σi)→ ri F †i


whence – by proposition 1.6.10 – a commutative diagram of profinite groups :

P //

π1(G,ϑ)

Pi

π1(F †i )

π1(C , F )π1(ai,σi) // π1(Ci, Fi).

1.6.19. Let (C , F ) be a fibre functor, and X a connected object of C (example 1.1.26(iii));pick any ξ ∈ F (X), and let Hξ ⊂ π1(C , F ) be the stabilizer of ξ for the natural left action ofπ1(C , F ) on F (X). For every object f : Y → X of C/X , we set

Fξ(f) := F (f)−1(ξ) ⊂ F (Y ).

It is clear that the rule f 7→ Fξ(f) yields a functor F †ξ : C/X → Hξ-Set, which we call thesubfunctor of F|X selected by ξ.

Proposition 1.6.20. In the situation of (1.6.19), we have :(i) C/X is also a Galois category, and Fξ := fHξ F

†ξ is a fibre functor for C/X .

(ii) The functor F †ξ induces a natural isomorphism of profinite groups :

π1(F †ξ ) : Hξ∼→ π1(C/X, Fξ).

Proof. The fibre functor F induces an equivalence of categories

C/X∼→ π1(C , F )-Set/F (X).

On the other hand, sinceX is connected, there exists a unique isomorphism ω : F (X)∼→ G/Hξ

of G-sets such that ω(ξ) = Hξ, and then the discussion of (1.6.4) yields an equivalence

π1(C , F )-Set/F (X)∼→ Hξ-Set.

A simple inspection shows that the resulting equivalence C/X∼→ Hξ-Set is none else than the

functor F †ξ , so the assertion follows from remark 1.6.9(i) and example 1.6.8.

1.6.21. Let (C , F ) be a fibre functor, and let us now fix a cleavage c : C o → Cat for the fibredcategory t : Morph(C )→ C (see example 1.4.2(iii)). Also, let I be a small cofiltered category,and X• : I → C a functor such that Xi is a connected object of C , for every i ∈ Ob(I); wepick an element

ξ• ∈ limIF X•.

In other words, ξ• := (ξi ∈ F (Xi) | i ∈ I) is a compatible system of elements such that

F (ϕ)(ξj) = ξi for every morphism ϕ : j → i in I .

For every i ∈ I , we denote by Hi ⊂ π1(C , F ) the stabilizer of ξi for the left action of π1(C , F )on F (Xi). Clearly, any morphism j → i induces an inclusion Hj ⊂ Hi. Furthermore, let

F †i : C/Xi → Hi-Set for every i ∈ Ibe the subfunctor selected by ξi, and set Fi := fHi F

†i . Let ϕ : j → i be a morphism of I; to

the corresponding morphism Xϕ : Xj → Xi, the cleavage c associates a pull-back functor

X∗ϕ : C/Xi → C/Xj.

Especially, for any object Y ∈ Ob(C /Xi) we have the cartesian diagram in C :

X∗ϕ(Y ) //

Y

Xj

Xϕ // Xi


whence, since F is exact, a natural bijection :

F (X∗ϕ(Y ))∼→ F (Y )×F (Xi) F (Xj)

which in turns yields a bijection :

F †j (X∗ϕ(Y ))∼→ F (Y )×F (Xi) ξj = F †i (Y )× ξj.

That is, we have a natural isomorphism of functors :

(1.6.22) α†ϕ : F †j X∗ϕ∼→ Res

HjHi F †i

and since the X∗ϕ are exact functors, it is easily seen that the isomorphisms αϕ := fj ∗ α†ϕ yielda pseudo-functor

(1.6.23) I → fibre.Fun i 7→ (C/Xi, Fi) ϕ 7→ (X∗ϕ, αϕ).

Moreover, α†ϕ can be seen as a 2-cell of fibre.Fun : F †j (X∗ϕ, αϕ)∼→ ResHiHj F

†i , whence a

commutative diagram of profinite groups :

Hj//

π1(F †j )

Hi

π1(F †i )

π1(C/Xj, Fj)π1(X∗ϕ,αϕ)

// π1(C/Xi, Fi)

whose top horizontal arrow is the inclusion map. Especially, notice that the map π1(X∗ϕ, αϕ)does not depend on the chosen cleavage; this can also be seen by remarking that any two cleav-ages c, c′ are related by a pseudo-natural isomorphism c

∼→ c′ (details left to the reader).

1.6.24. Let (C/X, Fξ) be the 2-colimit of the pseudo-functor (1.6.23), as in remark 1.6.18(i),and fix a corresponding universal pseudo-cocone (a•, σ•) : (C/X•, F•) ⇒ (C/X, Fξ). We maythen state :

Corollary 1.6.25. In the situation of (1.6.24), there exists a natural isomorphism of profinitegroups :

H :=⋂

i∈Ob(I)

Hi∼→ π1(C/X, Fξ)

which fits into a commutative diagram :

(1.6.26)

H //

Hi

π1(F †i )

π1(C/X, Fξ)π1(ai,σi) // π1(C/Xi, Fi)

for every i ∈ Ob(I)

whose top horizontal arrow is the inclusion map.

Proof. By corollary 1.6.16, we have an isomorphism of π1(C/X, Fξ) with the limit of the cofil-tered system (π1(C/X,Fi) | i ∈ Ob(I)); on the other hand, the discussion of (1.6.21) shows thatthe latter system is naturally isomorphic to the system (Hi | i ∈ Ob(I)). Lastly, the commuta-tivity of (1.6.26) follows from remark 1.6.18(ii).

2. SITES AND TOPOI

In this chapter, we assemble some generalities concerning sites and topoi. The main referencefor this material is [3]. As in the previous sections, we fix a universe U, and small means U-small throughout. Especially, a presheaf on any category takes its values in U, unless explicitlystated otherwise.


2.1. Topologies and sites. Let C be a small category; we wish to begin with a closer investi-gation of the category C ∧ of presheaves on C , introduced in (1.1.19). First, we remark that C ∧

is complete and cocomplete, and for every X ∈ Ob(C ), the functor

C ∧ → Set : F 7→ F (X)

commutes with all limits and all colimits (in other words, the limits and colimits in C ∧ are com-puted argumentwise) : see [10, Th.2.5.14 and Cor.2.15.4]. As a corollary, a morphism in C ∧ isan isomorphism if and only if it is both a monomorphism and an epimorphism, since the sameholds in the category Set. Likewise, a morphism of presheaves F → G is a monomorphism(resp. an epimorphism) if and only if the induced map of sets F (X)→ G(X) is injective (resp.surjective) for everyX ∈ Ob(C ) : see [10, Cor.2.15.3]. Furthermore, the filtered colimits in C ∧

commute with all finite limits, again because the same holds in Set ([10, Th.2.13.4]). For thesame reason, all colimits and all epimorphisms are universal in C ∧ (see example 1.1.24(v,vii)).

It is also easily seen that C ∧ is well-powered : indeed, for every presheaf F on C , and everyX ∈ Ob(C ), the set of subsets of F (X) is small, and a subobject of F is just a compatiblesystem of subsets F ′(X) ⊂ F (X), for X ranging over the small set of objects of C . Likewiseone sees that C ∧ is co-well-powered.

Hence, for every morphism f : F → G in C ∧, the image of f is well defined (see example1.1.24(viii)); more concretely, Im(f) ⊂ G is the subobject defined by the rule :

X 7→ Im(F (X)→ G(X)) for every X ∈ Ob(C ).

Denote by ∗ a final object of Set (i.e. any choice of a set with a single element); the initial(resp. final) object of C ∧ is the presheaf ∅C (resp. 1C ) such that ∅C (X) = ∅ (resp. 1C (X) =∗) for every X ∈ Ob(C ).

Lemma 2.1.1. Let C be a small category, F a presheaf on C . We have a natural isomorphism:

colimhC /F

h/F∼→ 1F in the category C ∧/F

where h : C → C ∧U is the Yoneda embedding (notation of (1.1.16) and (1.1.19)).

Proof. To begin with, notice that, under the current assumptions, hC /F is a small category. Lets : C ∧/F → C ∧ be the source functor as in (1.1.13); let G be any presheaf on C ; according to(1.1.30) we have a natural bijection :

HomC∧(colimhC /F

s h/F,G)∼→ S := lim

(hC /F )oHomC∧(s h/F,G).

By inspecting the definitions, we see that the elements of S are in natural bijection with thecompatible systems of the form (fσ ∈ G(X) | X ∈ Ob(C ), σ ∈ F (X)), such that (Gψ)(fσ) =f(Fψ)(σ) for every morphism ψ : X ′ → X in C . Such a system defines a unique naturaltransformation F ⇒ G, hence F and the above colimit represent the same presheaf on thecategory (C ∧U′)

o, and the assertion follows.

As a special case of lemma 2.1.1, consider any object X of a small category C , any sieveS of C/X , and take F := hS (notation of (1.5.3)); recalling the isomorphism of categorieshC /F

∼→ S , we deduce a natural isomorphism in C ∧ :

(2.1.2) colimS

h s∼→ hS

where s : S → C is the restriction of the functor (1.1.13).

Proposition 2.1.3. In the situation of (1.1.35), let F be any presheaf on B. We have :


(i) f!F is naturally isomorphic to the presheaf on C given by the rule :

(2.1.4) Y 7→ colim(Y/fB)o

F ιoY ϕ 7→ ( colim(ϕ/fB)o

1B : colim(Y/fB)o

F ιoY → colim(Y ′/fB)o

F ιoY ′)

for every Y ∈ Ob(C ) and every morphism ϕ : Y ′ → Y in C (notation of (1.1.16)).(ii) f∗F is naturally isomorphic to the presheaf on C given by the rule :

Y 7→ lim(fB/Y )o

F ιoY ϕ 7→ ( lim(fB/ϕ)o

1B : lim(fB/Y )o

F ιoY → lim(fB/Y ′)o

F ιoY ′)

for every Y ∈ Ob(C ) and every morphism ϕ : Y ′ → Y in C .(iii) If all the finite limits of B are representable, and f is left exact, then the functor f! is

left exact.

Proof. (i), (ii): The above expressions are derived directly from the proof of proposition 1.1.34.(iii): Under these assumptions, the category Y/fB is cofiltered for every Y ∈ Ob(C ), hence

(Y/fB)o is filtered. However, the filtered colimits in the category Set commute with all finitelimits, so the assertion follows from (i).

2.1.5. In the situation of (1.1.35), let V be a universe such that U ⊂ V. As a first corollary ofproposition 2.1.3(i,ii) we get an essentially commutative diagram of categories

C ∧U

B∧UfU! //fU∗oo

C ∧U

C ∧U B∧V

fV! //fV∗oo C ∧V

whose vertical arrows are the inclusion functors. Let us remark :

Lemma 2.1.6. Let f : B → C be a functor between small categories. We have :

(i) f is fully faithful if and only if the same holds for f!, if and only if the same holds forf∗.

(ii) Suppose that f admits a right adjoint g : C → B. Then there are natural isomor-phisms of functors :

f ∗∼→ g! f∗

∼→ g∗.

Proof. (i): By proposition 1.1.11(iii), f! is fully faithful if and only if the same holds for f∗.Now, suppose that f is fully faithful. We have to show that the unit of adjunction F → f ∗f!F

is an isomorphism, for every F ∈ Ob(B∧) (proposition 1.1.11(ii)). Since both f ∗ and f!

commute with arbitrary colimits, lemma 2.1.1 reduces to the case where F = hY for someY ∈ Ob(B) In this case, taking into account (1.1.36), we see that the unit of adjunction is themap given by the composition :

hY (Z) = HomB(Z, Y )ω−→ HomC (fZ, fY ) = (f ∗hfY )(Z) for every Z ∈ Ob(B)

where ω is the map given by f , which is a bijective by assumption, whence the claim.Conversely, if f! is fully faithful, then (1.1.36) and the full faithfulness of the Yoneda imbed-

dings, imply that f is fully faithful.(ii): It suffices to show the first stated isomorphism, since the second one will follow by

adjunction. However, let X ∈ Ob(B), F ∈ Ob(C ∧), and let s : hC /F → C be the functorgiven by the rule : (hY → F ) 7→ Y . Taking into account lemma 2.1.1 and (1.1.36), we may


compute :f ∗F (X) = F (fX)

∼→ colimhC /F

HomC (f(X), s)

∼→ colimhC /F

HomB(X, g s)∼→ colim

hC /F(g! h s)(X)

∼→ g!F (X)

whence the contention.

Example 2.1.7. (i) Let C be a category, X any object of C , denote by ιX : C/X → C thefunctor of (1.1.13), and suppose that C is V-small, for some universe V containing U. Byinspecting (2.1.4) we obtain a natural isomorphism :

(2.1.8) (ιX)V!F (Y )∼→ (a, ϕ) | ϕ ∈ HomC (Y,X), a ∈ F (ϕ)

for every V-presheaf F on C/X and every Y ∈ Ob(C ). If ψ : Z → Y is any morphism of C ,the corresponding map (ιX)V!F (Y )→ (ιX)V!F (Z) is given by the rule :

(2.1.9) (a, ϕ) 7→ (F (ψ)(a), ϕ ψ) for every (a, ϕ) ∈ (ιX)V!F (Y ).

(ii) Especially, if C has small Hom-sets and F is a U-presheaf on C/X , then we see that(ιX)V!F is a U-presheaf. Since the inclusion (C/X)∧U → (C/X)∧V is fully faithful, we deducethat the target of (2.1.8) can be used to define a left adjoint (ιX)U! : (C/X)∧U → C ∧U to (ιX)∗U. (Asusual the latter shall often be denoted just ιX!). We also deduce from (2.1.8) that ιX! transformsmonomorphisms to monomorphisms, and more generally, that it commutes with fibre products.On the other hand, it does not generally preserve final objects, hence it is not generally exact.

(iii) More precisely, (2.1.8) yields a natural isomorphism :

ιX!(1C/X)∼→ hX .

It follows that ιX! is the composition of a functor

eX : (C/X)∧ → C ∧/hX

and the functor ιhX : C ∧/hX → C ∧. Let F be any presheaf on C/X; in view of (2.1.9), we seethat the corresponding morphism eX(F ) : ιX!F → hX is given by the rule : (a, ϕ) 7→ ϕ forevery Y ∈ Ob(C ) and every (a, ϕ) ∈ ιX!F (Y ). We claim that eX is an equivalence. Indeed, letG be another presheaf on C/X , and f : eX(F )→ eX(G) a morphism in C ∧/hX ; the foregoingdescription of eX(F ) shows that f is the datum of a system of maps fϕ : F (ϕ)→ G(ϕ), for ϕ :Y → X ranging over the objects of C/X , subject to the condition that fϕψ F (ψ) = G(ψ)fϕfor every morphism ψ : Z → Y of X-objects. Such a datum is obviously nothing else than amorphism F → G in (C/X)∧, so this shows already that eX is fully faithful.

Next, the datum of a morphism g : F → hX in C ∧ amounts to a compatible system ofpartitions F (Y ) =

⋃ϕ F (Y )ϕ, for every Y ∈ Ob(C ), with ϕ ranging over HomC (Y,X);

namely, F (Y )ϕ := g(Y )−1(ϕ) for every such ϕ; the rule ϕ 7→ F (ιX(Y ))ϕ then defines apresheaf G on C/X with a natural isomorphism ιX(G)

∼→ F , all of which shows that eX isessentially surjective, as claimed. The quasi-inverse just constructed, can be described morecompactly as the functor that assigns to g : F → hX the presheaf given by the rule :

(Yϕ−→ X) 7→ HomC∧/hX ((hY

hϕ−−→ hX), g).

Definition 2.1.10. Let C be a category.(i) A topology on C is the datum, for every X ∈ Ob(C ), of a set J(X) of sieves of C/X ,

fulfilling the following conditions :


(a) (Stability under base change) For every morphism f : Y → X of C , and everyS ∈ J(X), the sieve S ×X f lies in J(Y ).

(b) (Local character) Let X be any object of C , and S , S ′ two sieves of C/X , withS ∈ J(X). Suppose that, for every object f : Y → X of S , the sieve S ′ ×X flies in J(Y ). Then S ′ ∈ J(X).

(c) For every X ∈ Ob(C ), we have C/X ∈ J(X).(ii) In the situation of (i), the elements of J(X) shall be called the sieves covering X .

Moreover, say that S is the sieve of C/X generated by a family (fi : Xi → X | i ∈ I)of morphisms. If S is a sieve covering X , we say that the family (fi | i ∈ I) coversX , or that it is a covering family of X .

(iii) The datum (C , J) of a category C and a topology J := (J(X) | X ∈ Ob(C )) on C iscalled a site, and then C is also called the category underlying the site (C , J). We saythat (C , J) is a small site, if C is a small category.

(iv) The set of all topologies on C is partially ordered by inclusion: given two topologies J1

and J2 on C , we say that J1 is finer than J2, if J2(X) ⊂ J1(X) for every X ∈ Ob(C ).

Remark 2.1.11. Let (C , J) be any site.(i) It is easily seen that any finite intersection of sieves covering an objectX , again coversX .

Indeed, say that S1 and S2 are two sieves covering X; set S := S1 ∩S2 and let f : Y → Xbe any object of S1. Then S ×X f = S2 ×X f .

(ii) Also, any sieve of C/X containing a covering sieve is again a covering sieve. Indeed, ifS ⊂ S ′, then S ′ ×X f = C /Y for every object f : Y → X of S .

(iii) Let (fi : Yi → X | i ∈ I) be a family of objects of C/X that generates a sieve Scovering X , and for every i ∈ I , let (gij : Zij → Yi | i ∈ Ji) a family of objects of C /Yi thatgenerates a sieve Si covering Yi. Then the family (fi gij : Zij → X | i ∈ I, j ∈ Ji) generatesa sieve S ′ covering X . Indeed, say that f : Y → X lies in S , and pick i ∈ I such that ffactors through fi and a morphism g : Y → Yi; then it is easily seen that Si ×Yi g ⊂ S ′ ×X f .

Example 2.1.12. Let F : A → B be a fibration between two categories. For every X ∈Ob(B), let JF (X) denote the set of all sieves S ⊂ B/X of universal F -2-descent. (To makethis definition, we have to choose a universe V such that A and B are V-small, but clearlythe resulting JF does not depend on V.) Then we claim that JF is a topology on B. Indeed,it is clear that JF fulfills conditions (a) and (c) of definition 2.1.10(i). In order to conclude, itsuffices therefore to show the following :

Lemma 2.1.13. In the situation of example 2.1.12, let S ′ ⊂ S be two sieves of B/X , andsuppose that, for every object f : Y → X of S , the sieve S ′ ×X f lies in JF (Y ). ThenS ′ ∈ JF (X) if and only if S ∈ JF (X).

Proof. Up to replacing U by a larger universe, we may assume that B and C are small. Noticethen that our assumptions are preserved under any base change X ′ → X in B, hence it sufficesto show that S ′ is a sieve of F -2-descent if and only if the same holds for S . To this aim, forevery small category A we construct a functor

ϑ : CartB(S ′,A )→ 2-limS

CartB(GS ,A )

where GS : S → Cat/B is the functor introduced in 1.5.10. Indeed, let ϕ : S ′ → A be acartesian functor; for every object f : Y → X of S , denote

B/Yjf←−− S ′ ×X f

if−−→ S ′

the natural functors. By assumption, there exist cartesian functors ϕf : B/Y → A , and naturalisomorphisms

αf : ϕf jf ⇒ ϕ if


for every such f . Moreover, for every morphism g : Z → Y in S , there exists a uniqueisomorphism of functors

ωf,g : ϕf g∗ ⇒ ϕfg

fitting into the commutative diagram :

ϕf g∗ jfgωf,g∗jfg

ϕf jf igαf∗ig

ϕfg jfgαfg +3 ϕ ifg ϕ if ig.

The uniqueness of ωf,g implies that the datum (ϕf , ωf,g | f ∈ Ob(S ), g ∈ Morph(S )) definesa pseudo-natural transformation ϕ• : GS ⇒ FA . Moreover, if τ : ϕ ⇒ ϕ′ is any naturaltransformation of cartesian functors S ′ → A , there exists, for every f ∈ Ob(S ), a uniquenatural transformation τf : ϕf ⇒ ϕ′f fitting into the commutative diagram :

ϕf jfτf∗jf +3

αf

ϕ′f jfα′f

ϕ ifτ∗jf +3 ϕ′ if .

In other words, the rule ϕ 7→ ϕ• defines a functor ϑ as sought.Notice next that, if f : Y → X lies in Ob(S ′), then S ′ ×X f = B/Y ; it follows that the

restriction to S ′ of the functor GS agrees with the functor GS ′ , and this restriction operationinduces a functor

ρ : 2-limS

CartB(GS ,A )→ 2-limS ′

CartB(GS ′ ,A ).

Summing up, we arrive at the essentially commutative diagram :

CartB(S ,A )CartB(ι,A )

//

ω

CartB(S ′,A )

ϑ

sshhhhhhhhhhhhhhhhhhh

ω′

2-lim

SCartB(GS ,A )

ρ // 2-limS ′

CartB(GS ′ ,A )

in which ι : S ′ → S is the inclusion functor, and where ω and ω′ are equivalences deducedfrom lemma 1.5.11. A little diagram chase shows that CartB(ι,A ) must then also be an equiv-alence, whence the contention.

2.1.14. Suppose now that C has small Hom-sets. Then, in view of the discussion in (1.5.3), atopology can also be defined by assigning, to any object X of C , a family J ′(X) of subobjectsof hX , such that :

(a) For every X ∈ Ob(C ), every R ∈ J ′(X), and every morphism Y → X in C , the fibreproduct R×X Y lies in J ′(Y ).

(b) Say that X ∈ Ob(C ), and let R, R′ be two subobjects of hX , such that R ∈ J ′(X).Suppose that, for every Y ∈ Ob(C ), and every morphism f : hY → R, we haveR′ ×X Y ∈ J ′(Y ). Then R′ ∈ J ′(X).

(c) hX ∈ J ′(X) for every X ∈ Ob(C ).In this case, the elements of J ′(X) are naturally called the subobjects covering X . This view-point is adopted in the following :

Definition 2.1.15. Let V be a universe, C := (C , J) be a site, and suppose that the category Chas V-small Hom-sets. Let also F ∈ Ob(C ∧V ).


(i) We say that F is a separated V-presheaf (resp. a V-sheaf) on C, if for every X ∈Ob(C ) and every subobject R covering X , the induced morphism :

F (X) = HomC∧V(hX , F )→ HomC∧V

(R,F )

is injective (resp. is bijective). For V = U, we shall just say separated presheaf insteadof separated U-presheaf, and likewise for sheaves.

(ii) The full subcategory of C ∧V consisting of all V-sheaves (resp. all separated presheaves)on C is denoted C∼V (resp. Csep

V ). For V = U, this category will be usually denotedsimply C∼ (resp. Csep).

2.1.16. In the situation of definition 2.1.15(i), say that R = hS for some sieve S covering X ,and let (Si → X | i ∈ I) be a generating family for S . Combining lemma 1.5.7 and examples1.4.10(iv), 1.5.9(ii), we get a natural isomorphism :

HomC∧(R,F )∼→ Equal

∏i∈I

CartC (C/Si,AF ) // //∏

(i,j)∈I×I

CartC (C/Sij,AF )

which – again by example 1.4.10(iv,v,vi) – we may rewrite more simply as :


∏i∈I

F (Si) ////∏

(i,j)∈I×I

HomC∧(hSi ×hX hSj , F )

.

The above equalizer can be described explicitly as follows. It consists of all the systems

(ai | i ∈ I) with ai ∈ F (Si) for every i ∈ I

such that, for every i, j ∈ I , every object Y → X of C/X , and every pair (gi : Y → Si, gj :Y → Sj) of morphisms in C/X , we have :

(2.1.17) F (gi)(ai) = F (gj)(aj).

If the fibred product Sij := Si ×X Sj is representable in C , the latter expression takes the morefamiliar form :


∏i∈I

F (Si) ////∏

(i,j)∈I×I

F (Sij)

.

Remark 2.1.18. Let C be a category with small Hom-sets.(i) The arguments in (2.1.16) yield also the following. A presheaf F on C is separated

(resp. is a sheaf) on C, if and only if every covering sieve of C is a sieve of 1-descent (resp. of2-descent) for the fibration ϕF : AF → C of example 1.4.10(iii).

(ii) Let F be a presheaf on C . We deduce from (i) and example 2.1.12 that the topologyJF := JϕF is the finest on C for which F is a sheaf. A subobjectR ⊂ hX (for anyX ∈ Ob(C ))lies in JF (X) if and only if the natural map F (X ′) → HomC∧(R ×X X ′, F ) is bijective forevery morphism X ′ → X in C .

(iii) More generally, if (Fi | i ∈ I) is any family of presheaves on C , we see that there existsa finest topology for which each Fi is a sheaf : namely, the intersection of the topologies JFi asin (ii).

(iv) As an important special case, we deduce the existence of a finest topology J on C suchthat all representable presheaves are sheaves on (C , J). This topology is called the canonicaltopology on C . The foregoing shows that a sieve of C/X is universal strict epimorphic (seeexample 1.5.13) if and only if it covers X in the canonical topology of C .


2.1.19. Suppose furthermore, that C is small; then, directly from definition 2.1.15 (and from(1.1.29)), we see that the category C∼ is complete, and the fully faithful inclusion C∼ → C ∧

commutes with all limits; moreover, given a presheaf F on C , it is possible to construct asolution set for F relative to this functor, and therefore one may apply theorem 1.1.32 to producea left adjoint. However, a more direct and explicit construction of the left adjoint can be given;the latter also provides some additional information which is hard to extract from the formermethod. Namely, we have :

Theorem 2.1.20. In the situation of (2.1.19), the following holds :(i) The inclusion functor i : C∼ → C ∧ admits a left adjoint

(2.1.21) C ∧ → C∼ F 7→ F a.

For every presheaf F , we call F a the sheaf associated to F .(ii) Morever, (2.1.21) is an exact functor.

Proof. We begin with the following :

Claim 2.1.22. Let F be a separated presheaf on C , and S1 ⊂ S2 two sieves covering someX ∈ Ob(C ). Then the natural map HomC∧(hS2 , F )→ HomC∧(hS1 , F ) is injective.

Proof of the claim. We may find a generating family (fi : Si → X | i ∈ I2) for S2, and asubset I1 ⊂ I2, such that (fi | i ∈ I1) generates S1. Let s, s′ : hS2 → F , whose images agreein HomC∧(hS1 , F ). By (2.1.16), s and s′ correspond to families (si | i ∈ I2), (s′i | i ∈ I2) withsi, s

′i ∈ F (Si) for every i ∈ I2, fulfilling the system of identities (2.1.17), and the foregoing

condition means that si = s′i for every i ∈ I1. We need to show that si = s′i for every i ∈ I2.Hence, let i ∈ I2 be any element; by assumption, the natural map

F (Si)→ HomC∧(hS1 ×X Si, F )

is injective. However, the objects of S1×X fi are all the morphisms gi : Y → Si in C such thatfi gi = fj gj for some j ∈ I1 and some gj : Y → Sj in C . If we apply the identities (2.1.17)to these maps gi, gj , we deduce that :

F (gi)(si) = F (gj)(sj) = F (gj)(s′j) = F (gi)(s

′i).

In other words, si and s′i have the same image in HomC∧(hS1 ×X Si, F ), hence they agree, asclaimed. ♦

Next, for every X ∈ Ob(C ), denote by J(X) the full subcategory of Cat/(C/X) such thatOb(J(X)) = J(X). Notice that J(X) is small and cofiltered, for every such X . Define afunctor h : J(X)→ C ∧ by the rule : S 7→ hS for every S ∈ J(X); to an inclusion of sievesS ′ ⊂ S there corresponds the natural monomorphism hS ′ → hS of subobjects of hX .

For a given presheaf F on C , set

F+(X) := colimJ(X)o

HomC∧(ho, F ).

For a morphism f : Y → X in C and a sieve S ∈ J(X), the natural projection hS×XY → hS

induces a map :HomC∧(hS , F )→ HomC∧(hS ×X Y, F )

whence a map F+(X)→ F+(Y ), after taking colimits. In other words, we have a functor :

(2.1.23) C ∧ → C ∧ F 7→ F+.

with a natural transformation F (X)→ F+(X), since C/X ∈ J(X) for every X ∈ Ob(C ).

Claim 2.1.24. (i) The functor (2.1.23) is left exact.(ii) For every F ∈ Ob(C ∧), the presheaf F+ is separated.

(iii) If F is a separated presheaf on C , then F+ is a sheaf on C∼.


Proof of the claim. (i) is clear, since J(X)o is filtered for every X ∈ Ob(C ).(ii): Let s, s′ ∈ F+(X), and suppose that the images of s and s′ agree in HomC∧(hS , F

+),for some S ∈ J(X). We may find a sieve T ∈ J(X) such that s and s′ come from elementss, s′ ∈ HomC∧(hT , F ). Let (gi : Si → X | i ∈ I) be a family of generators for S ; in viewof (2.1.16), the images of s and s′ agree in F+(Si) for every i ∈ I . The latter means that, forevery i ∈ I , there exists Si ∈ J(Si), refining T ×X gi, such that the images of s and s′ agree inHomC∧(hSi

, F ). For every i ∈ I , let (giλ : Tiλ → Si | λ ∈ Λi) be a family of generators for Si,and consider the sieve T ′ of C/X generated by (gi giλ : Tiλ → X | i ∈ I, λ ∈ Λi). Then T ′

coversX (remark 2.1.11(iii)) and refines T , and the images of s and s′ agree in HomC∧(hT ′ , F )(as one sees easily, again by virtue of (2.1.16)). This shows that s = s′, whence the contention.

(iii): In view of (ii), it suffices to show that the natural map F+(X) → HomC∧(hS , F+) is

surjective for every S ∈ J(X). Hence, say that s ∈ HomC∧(hS , F+), and let (Si | i ∈ I) be

a generating family for S . By (2.1.16), s corresponds to a system (si ∈ F+(Si) | i ∈ I) suchthat the following holds. For every i, j ∈ I , and every pair of morphisms ui : Y → Si anduj : Y → Sj in C/X , we have

(2.1.25) F+(ui)(si) = F+(uj)(sj).

For every i ∈ I , let Si ∈ J(Si) such that si is the image of some si ∈ HomC∧(hSi, F ). For

every ui, uj as above, set Sij := (Si ×Si ui) ∩ (Sj ×Sj uj); since F is separated, (2.1.25) andclaim 2.1.22 imply that the images of si and sj agree in HomC∧(hSij

, F ), for every i, j ∈ I .However, for every i ∈ I , let (giλ : Tiλ → Si | λ ∈ Λi) be a generating family for Si;

then si corresponds to a compatible system of sections siλ ∈ F (Tiλ), and Sij is the sieve of allmorphisms f : Z → Y such that

ui f = giλ f ′i and uj f = gjµ f ′jfor some λ ∈ Λi, µ ∈ Λj and some f ′i : Z → Tiλ, f ′j : Z → Tjµ, so by construction we have

(2.1.26) F (f ′i)(siλ) = F (f ′j)(sjµ) for every i, j ∈ I and λ ∈ Λi, µ ∈ Λµ.

Finally, let T be the sieve of C/X generated by (gi giλ : Tiλ → X | i ∈ I, λ ∈ Λi); then Tcovers X (remark 2.1.11(iii)), and (2.1.26) shows that the system (F (giλ)(siλ) | i ∈ I, λ ∈ Λi)defines an element of HomC∧(hT , F ), whose image in F+(X) agrees with s. ♦

From claim 2.1.24 we see that the rule : F 7→ F a := (F+)+ defines a left exact functorC ∧ → C∼, with natural transformations ηF : F ⇒ i(F a) for every F ∈ Ob(C ∧) and εG :(iG)a ⇒ G for every G ∈ Ob(C∼) fulfilling the triangular identities of (1.1.8). The theoremfollows.

Remark 2.1.27. Let C := (C , J) be a small site.(i) It has already been remarked that C∼ is complete, and from theorem 2.1.20 we also

deduce that C∼ is cocomplete, and the functor (2.1.21) (resp. the inclusion functor i : C∼ →C ∧) commutes with all colimits (resp. with all limits); more precisely, if F : I → C∼ is anyfunctor from a small category I , we have a natural isomorphism in C∼ (resp. in C ∧) :

colimI

F∼→ (colim

Ii F )a (resp. i(lim

IF )

∼→ limIi F ).

Especially, limits in C∼ are computed argumentwise (see (1.1.31)). Moreover, it follows thatall colimits and all epimorphisms are universal in C∼ (see example 1.1.24(v,vii)), and filteredcolimits in C∼ commute with finite limits, since the same holds in C ∧.

(ii) Furthermore, C∼ is well-powered and co-well-powered, since the same holds for C ∧. Es-pecially, every morphism f : F → G in C∼ admits a well defined image (example 1.1.24(viii)).Such an image can be constructed explicitly as the subobject (Im i(f))a (details left to thereader).


(iii) By composing with the Yoneda embedding, we obtain a functor

ha : C → C∼ : X 7→ (hX)a for every X ∈ Ob(C )

and lemma 2.1.1 yields a natural isomorphism :

colimhC /F

haX∼→ F for every sheaf F on C.

(iv) From the proof of claim 2.1.24 it is clear that the functor

C ∧ → Csep F 7→ F sep := Im(F → F+)

is left adjoint to the inclusion Csep → C ∧. Moreover, we have a natural identification :

F a ∼→ (F sep)+ for every F ∈ Ob(C ∧).

(v) Let V be a universe such that U ⊂ V; from the definitions, it is clear that the fully faithfulinclusion C ∧U ⊂ C ∧V restricts to a fully faithful inclusion

C∼U ⊂ C∼V .

Moreover, by inspecting the proof of theorem 2.1.20, we deduce an essentially commutativediagram of categories :

C ∧U //

C∼U

C ∧V // C∼V

whose vertical arows are the inclusion functors, and whose horizontal arrows are the functorsF 7→ F a.

In practice, one often encounters sites that are not small, but which share many of the prop-erties of small sites. These more general situations are encompassed by the following :

Definition 2.1.28. Let C := (C , J) be a site.(i) A topologically generating family for C is a subset G ⊂ Ob(C ), such that, for every

X ∈ Ob(C ), the family

G/X :=⋃Y ∈G

HomC (Y,X) ⊂ Ob(C/X)

generates a sieve covering X .(ii) We say that C is a U-site, if C has small Hom-sets, and C admits a small topologically

generating family. In this case, we also say that J is a U-topology on C .

2.1.29. Let C =: (C , J) be a U-site, and G a small topologically generating family for C.For every X ∈ Ob(C ), denote by JG(X) ⊂ J(X) the set of all sieves covering X which aregenerated by a subset of G/X (notation of definition 2.1.28(i)).

Lemma 2.1.30. With the notation of (2.1.29), for every X ∈ Ob(C ) the following holds :(i) JG(X) is a small set.

(ii) JG(X) is a cofinal subset of the set J(X) (where the latter is partially ordered byinclusion of sieves).

Proof. (i) is left to the reader.(ii): Let S be any sieve covering X , and say that S is generated by a family (fi : Si →

X | i ∈ I) of objects of C/X (indexed by some not necessarily small set I). Let S ′ be the sievegenerated by ⋃

i∈I

fi g | g ∈ G/Si.


It is easily seen that S ′ ⊂ S and S ′ ∈ JG(X).

Remark 2.1.31. (i) In the situation of (2.1.29), let V be a universe with U ⊂ V, and such that Cis a V-small site. For every X ∈ Ob(C ), denote by JG(X) the full subcategory of J(X) suchthat Ob(JG(X)) = JG(X), and let hG : JG(X) → C ∧ be the restriction of h (notation of theproof of theorem 2.1.20). Lemma 2.1.30 implies that the natural map :

colimJG(X)o

HomC∧(hoG, F )→ F+(X)

is bijective. Therefore, if F is a U-presheaf, F+(X) is essentially U-small, and then the sameholds for F a(X). In other words, the restriction to C ∧U of the functor C ∧V → C∼V : F 7→ F a,factors through C∼.

(ii) We deduce that theorem 2.1.20 holds, more generally, when C is an arbitrary U-site.Likewise, a simple inspection shows that remark 2.1.27(i,iv,v) holds when C is only assumedto be a U-site.

Proposition 2.1.32. Let C := (C , J) be a U-site. The following holds :

(i) A morphism in C∼ is an isomorphism if and only if it is both a monomorphism and anepimorphism.

(ii) All epimorphisms in C∼ are universal effective.

Proof. (i): Let ϕ : F → G be a monomorphism in C∼; then the morphism of presheavesi(ϕ) : iF → iG is also a monomorphism, and it is easily seen that the cocartesian diagram

D :

iFi(ϕ)

//

i(ϕ)

iG

α

iG // iGqiF iG

is also cartesian, hence the same holds for the induced diagram of sheaves Da. If moreover, ϕis an epimorphism, then αa is an isomorphism, hence the same holds for ϕ = (i(ϕ))a.

(ii): Let f : F → G be an epimorphism in C∼; in view of remarks 2.1.27(i) and 2.1.31(ii),it suffices to show that f is effective. However, set G′ := Im(i(f)), and let pi : F ×G F → F(for i = 1, 2) be the two projections. Suppose ϕ : F → X is a morphism in C∼ such thatϕp1 = ϕp2; since i(F ×GF ) = iF ×iG iF = iF ×G′ iF , the morphism i(ϕ) factors througha (unique) morphism ψ : G′ → X . On the other hand, it is easily seen that (G′)a = G, hence ϕfactors through the morphism ψa : G→ X .

Remark 2.1.33. Proposition 2.1.32(i) implies that every morphism ϕ : X → Y in C∼ factorsuniquely (up to unique isomorphism) as the composition of an epimorphism followed by amonomorphism. Indeed; such a factorization is provided by the natural morphismsX → Im(ϕ)

and Im(ϕ) → Y (see example 1.1.24(viii)). If Xϕ′−−→ Z → Y is another such factorization,

then by definition ϕ′ factors through a unique monomorphism ψ : Im(ϕ) → Z. However, ψ isan epimorphism, since the same holds for ϕ′. Hence ψ is an isomorphism.

Proposition 2.1.34. Let (C , J) be a U-site, X ∈ Ob(C ), and R any subobject of hX . Thefollowing conditions are equivalent :

(a) The inclusion map i : R→ hX induces an isomorphism on associated sheaves

ia : Ra ∼→ haX .

(b) R covers X .


Proof. (b)⇒(a) : By definition, the natural map Ra(X) → HomC∧(R,Ra) is bijective, hencethere exists a morphism f : hX → Ra in C ∧ whose composition with i is the unit of adjunctionR → Ra. Therefore, fa : haX → Ra is a left inverse for ia. On the other hand, we have acommutative diagram :

HomC∧(haX , haX) //

HomC∧(Ra, haX)

haX(X) // HomC∧(R, haX)

whose bottom and vertical arrows are bijective, so that the same holds also for the top arrow.Set g := ia fa, and notice that g ia = ia, therefore g must be the identity of haX , whence thecontention.

(a)⇒(b): Let ηX : hX → haX be the unit of adjunction, and set j := (ia)−1 ηX : hX → Ra.By remarks 2.1.27(iv) and 2.1.31(ii), we may find a covering subobject i1 : R1 → hX , and amorphism j1 : R1 → Rsep whose image in HomC∧(R1, R

a) equals j i1. Denote by η′X : hX →hsepX the unit of adjunction; by construction, the two morphisms

isep j1, η′X i1 : R1 → hsep

X

have the same image in HomC∧(R1, haX). This means that there exists a subobject i2 : R2 → R1

covering X , such that isep j1 i2 = η′X i1 i2.Next, let Y• := (Yλ → X | λ ∈ Λ) be a generating family for the sieve of C /X corresponding

to R2. There follows, for every λ ∈ Λ, a commutative diagram :

(2.1.35)

hYλjλ //

iλ

Rsep

isep

hX

η′X // hsepX .

Then, for every λ ∈ Λ there exists a covering subobject sλ : Rλ → hYλ such that jλ lifts to sometλ : Rλ → R, and we pick a generating family (Zλµ → Yλ | µ ∈ Λλ) for the sieve of C /Yλcorresponding to Rλ; after replacing Y• by the resulting family (Zλµ → X | λ ∈ Λ, µ ∈ Λλ)(which still covers X , by virtue of remark 2.1.11(iii)), we may assume that (2.1.35) lifts to acommutative diagram

hYλtλ //

iλ

R

hX

η′X // hsepX .

for every λ ∈ Λ. Then there exists a covering subobject s′λ : R′λ → hYλ such that i tλ s′λ =iλ s′λ in HomC∧(R′λ, hX). Finally, set

R′ :=⋃λ∈Λ

Im(iλ s′λ : R′λ → hX)

(notice thatR′ ∈ Ob(C ∧U ) even in case Λ is not a small set). It is easily seen thatR′ is a coveringsubobject of X , and the inclusion map R′ → hX factors through R, so R covers X as well.

Definition 2.1.36. Let (C , J) be a site, such that C has small Hom-sets, and let ϕ : F → G bea morphism in C ∧.

(i) We say that ϕ is a covering morphism if, for every X ∈ Ob(C ) and every morphismhX → G in C ∧, the image of the induced morphism F ×G hX → hX is a coveringsubobject of X .


(ii) We say that ϕ a bicovering morphism if both ϕ and the morphismG→ G×FG inducedby ϕ, are covering morphisms.

Example 2.1.37. In the situation of definition 2.1.36, let S := Xi | i ∈ I be a family ofmorphisms in C , and pick a universe V containing U, such that I is V-small. Using example1.5.4, it is easily seen that S covers X if and only if the induced morphism in C ∧V∐

i∈I

hXi → hX

is a covering morphism.

Corollary 2.1.38. Let C := (C , J) be a U-site, and ϕ : F → G a morphism in C ∧. Then ϕ isa covering (resp. bicovering) morphism if and only if ϕa : F a → Ga is an epimorphism (resp.is an isomorphism) in C∼.

Proof. Let V be a universe with U ⊂ V, and such that C is V-small. Clearly ϕ is a covering(resp. bicovering) morphism in C ∧U if and only if the same holds for the image of ϕ under thefully faithful inclusion C ∧U ⊂ C ∧V . Hence, we may replace U by V, and assume that C is a smallsite.

Now, suppose that ϕa is an epimorphism; letX be any object of C , and hX → G a morphism.Since the epimorphisms of C∼ are universal (remark 2.1.27(i)), the induced morphism

(ϕ×G X)a : (F ×G hX)a → haX

is an epimorphism. LetR ⊂ hX be the image of ϕ×GhX ; then the induced morphismRa → haXis both a monomorphism and an epimorphism, so it is an isomorphism, by proposition 2.1.32(i).Hence R is a covering subobject, according to proposition 2.1.34.

Conversely, suppose that ϕ is a covering morphism. By remark 2.1.27(iii), G is the colimitof a family (haXi | i ∈ I) for certain Xi ∈ Ob(C ). By definition, the image Ri of the inducedmorphism ϕ ×G Xi : F ×G hXi → hXi covers Xi, for every i ∈ I . Now, the induced mor-phism F ×G hXi → Ri is an epimorphism, and the morphism Ra

i → haXi is an isomorphism(proposition 2.1.34), hence (ϕ×G Xi)

a is an epimorphism, and then the same holds for

colimi∈I

(ϕ×G Xi)a : colim

i∈IF a ×Ga haXi → colim

i∈IhaXi = Ga

which is isomorphic to ϕa, since the colimits of C∼ are universal (remark 2.1.27(i)); so ϕa is anepimorphism.

Next, if ϕ is a bicovering morphism, the foregoing shows that ϕa is an epimorphism, and theinduced morphism Ga → Ga×Fa Ga is both an epimorphism and a monomorphism, hence it isan isomorphism (proposition 2.1.32(i)), therefore ϕa is monomorphism (remark 1.1.38(iii)).So finally, ϕa is an isomorphism, again by proposition 2.1.32(i). Conversely, if ϕa is anisomorphism, the reader may show in the same way, that both ϕ and the induced morphismG→ G×F G are covering morphisms.

Definition 2.1.39. Let C = (C , J) and C ′ = (C ′, J ′) be two sites, and g : C → C ′ a functoron the underlying categories.

(i) We say that g is continuous for the topologies J and J ′, if the following holds. Forevery universe V such that C and C ′ have V-small Hom-sets, and every V-sheaf F onC ′, the V-presheaf g∗VF is a V-sheaf on C (notation of (1.1.35)). In this case, g clearlyinduces a functor

gV∗ : C ′∼V → C∼V


such that the diagram of functors :

C ′∼VgV∗ //

iC′

C∼V

iC

C ′∧Vg∗V // C ∧V

commutes (where the vertical arrows are the natural fully faithful embeddings).(ii) We say that g is cocontinuous for the topologies J and J ′ if the following holds. For

every X ∈ Ob(C ), and every covering sieve S ′ ∈ J ′(gX), the sieve g−1|X S ′ covers X

(notation of definition 1.5.1(iii) and (1.1.15)).

Lemma 2.1.40. In the situation of definition 2.1.39, the following conditions are equivalent :(a) g is a continuous functor.(b) There exists a universe V, such that C is V-small, C ′ is a V-site, and for every V-sheaf

F on C ′, the V-presheaf g∗VF is a V-sheaf.

Proof. Obviously, (a)⇒(b). For the converse, we notice :

Claim 2.1.41. Let V be a universe such that C and C ′ are V-small. The following conditionsare equivalent :

(c) For every V-sheaf F on C ′, the V-presheaf g∗VF is a V-sheaf.(d) For every X ∈ Ob(C ), and every covering subobject R ⊂ hX , the induced morphism

gV!R→ hg(X) is a bicovering morphism in C ′∧V (notation of (1.1.35)).

Proof of the claim. (c)⇒(d): By assumption, for every sheaf F on C ′, the natural map

g∗VF (X)→ HomC∧(R, g∗VF )

is bijective. By adjunction, it follows that the natural map F (g(X))→ HomC∧(gV!R,F ) is alsobijective, therefore the morphism (gV!R)a → hag(X) is an isomorphism, whence (d), in light ofcorollary 2.1.38. The proof that (d)⇒(c) is analogous, and shall be left as an exercise for thereader. ♦

Hence, suppose that (b) holds, and let V′ be another universe such that V ⊂ V′; it followseasily from claim 2.1.41 and remark 2.1.27(v) that condition (b) holds also for V′ instead of V.

Let V′ be any universe such that C and C ′ have V′-small Hom-sets, and pick any universeV′′ such that V ∪ V′ ⊂ V′′. By the foregoing, for every V′′-sheaf F , the V′′-presheaf g∗V′′F is aV′′-sheaf; if F is a V-sheaf, then obviously we conclude that g∗VF is a V-sheaf.

Lemma 2.1.42. In the situation of definition 2.1.39, consider the following conditions :(a) g is continuous.(b) For every covering family (Xi → X | i ∈ I) in C, the family (gXi → gX | i ∈ I)

covers gX in C ′.(c) For every small covering family (Xi → X | i ∈ I) in C, the family (gXi → gX | i ∈ I)

covers gX in C ′.(d) For every universe V such that C and C ′ have V-small sets, g induces a functor

gsepV∗ : C ′sep

V → CsepV


C ′sepV

gV∗ //

iC′

CsepV

iC

C ′∧Vg∗V // C ∧V


commutes (where the vertical arrows are the natural fully faithful embeddings).Then (a)⇒(b)⇔(d)⇒(c). Moreover, if all fibre products are representable in C , and g commuteswith fibre products, then (b)⇒(a). Furthermore, if C is a U-site, then (c)⇒(b).

Proof. Obviously (b)⇒(c).(a)⇒(b): After replacing U by a larger universe, we may assume that I is a small set and both

C and C ′ are small. Let F be any sheaf on C ′; by assumption, g∗F is a sheaf on C, hence thenatural map :

F (gX) = g∗F (X)→∏i∈I

g∗F (Xi) =∏i∈I

F (gXi)

is injective (by (2.1.16)). This means that the induced morphism

(2.1.43)∐i∈I

hagXi → hagX

is an epimorphism in C∼. Then the assertion follows from corollary 2.1.38 and example 2.1.37.(d)⇒(b) is similar : we may assume that I , C and C ′ are small. If F is any separated presheaf

on C ′, then by assumption g∗F is separated on C, and arguing as in the foregoing, we deducethat the induced morphism qi∈Ihsep

gXi→ hsep

gX is an epimorphism in Csep, and then (2.1.43) is anepimorphism in C∼, so we conclude, again by corollary 2.1.38 and example 2.1.37.

(b)⇒(d): let F be a separated V-presheaf on C ′, and (Xi → X | i ∈ I) any covering familyin C; in view of (2.1.16), it suffices to show that the induced map

F (gX) = g∗VF (X)→∏i∈I

g∗VF (Xi) =∏i∈I

F (gXi)

is injective. But this is clear, since (gXi → gX | i ∈ I) is a covering family in C ′.Next, suppose that (b) holds, the fibre products in C are representable, and g commutes with

all fibre products. For every i, j ∈ I , set Xij := Xi ×X Xj . To show that (a) holds, it suffices –in view of (2.1.16) – to prove :

Claim 2.1.44. The natural map

g∗F (X)→ HomC∧

(Coequal

(∐i,j∈I

hXij // //∐i∈I

hXi

), g∗F

)is bijective.

Proof of the claim. Since g! is right exact, and due to (1.1.36), this is the same as the naturalmap

F (gX)→ HomC∧

(Coequal

(∐i,j∈I

hgXij // //∐i∈I

hgXi

), F

).

However, by assumption gXij = gXi×gX gXj , and then the claim follows by applying (2.1.16)to the covering family (gXi → gX | i ∈ I). ♦

Lastly, suppose that C is a U-site; in order to show that (c)⇒(b), we remark more precisely :

Claim 2.1.45. Let C be a U-site, F := (ϕi : Xi → X | i ∈ I) any covering family. Then thereexists a small set J ⊂ I such that the subfamily (ϕi | i ∈ J) covers X .

Proof of the claim. Let S ⊂ C/X be the sieve generated by S . By lemma 2.1.30, we may finda small covering family F ′ := (ψi : X ′i → X | i ∈ I ′) (i.e. such that I ′ is small), that generatesa sieve S ′ ⊂ S . Then, for every i ∈ I ′ we may find γ(i) ∈ I such that ψi factors throughϕγ(i). The subset J := γI ′ will do. ♦


Let F and J be as in claim 2.1.45; then the sieve gS generated by (g(ϕi) | i ∈ I) containsthe sieve gS ′ generated by (g(ϕi) | i ∈ J). Especially, if gS ′ is a covering sieve, the sameholds for gS , whence the contention.

2.1.46. In the situation of definition 2.1.39, let V be a universe with U ⊂ V, such that C is aV-site, and C ′ has V-small Hom-sets. Then we may define a functor

g∗V : C ′∼V → C∼V F 7→ (g∗ iC′F )a

(where iC′ : C ′∼V → C ∧V is the forgetful functor). As usual, when V = U we often omit thesubscript U. With this notation, we have :

Lemma 2.1.47. In the situation of definition 2.1.39, let V be a universe with U ⊂ V, such thatC is V-small, and C ′ has V-small Hom-sets. Then the following conditions are equivalent :

(a) g is a cocontinuous functor.(b) For every V-sheaf F on C, the V-presheaf g∗F is a V-sheaf on C ′.

(Notation of (1.1.35).) When these conditions hold, the restriction of gV∗ is a functor

gV∗ : C∼V → C ′∼V

which is right adjoint to g∗V.

Proof. After replacing V by U, we may assume that C is small, and C ′ has small Hom-sets. Tobegin with, we remark :

Claim 2.1.48. Let X be any object of C , and T ′ ⊂ C ′/gX a covering sieve; to ease notation,set T := g−1

|X T ′. Then :hT = g∗hT ′ ×g∗hgX hX .

Proof of the claim. Left to the reader. ♦

(b)⇒(a): The assumption implies that, for every sheaf F on C, every X ∈ Ob(C ), and everycovering sieve S ′ ⊂ C ′/gX , the natural map

g∗F (gX)→ HomC ′∧(hS ′ , g∗F )

is bijective. By adjunction, the same then holds for the natural map

HomC∧(g∗hgX , F )→ HomC∧(g∗hS ′ , F )

so the induced morphism g∗hS ′ → g∗hgX is bicovering (proposition 2.1.34). Also, this mor-phism is a monomorphism (since g∗ commutes with all limits); therefore, after base changealong the unit of adjunction hX → g∗hgX , we deduce a covering monomorphism

g∗hS ′ ×g∗hgX hX → hX .

Then the contention follows from claim 2.1.48.(a)⇒(b): Let F be any sheaf on C; we have to show that the natural map

g∗F (Y )→ HomC ′∧(hS , g∗F )

is bijective, for every Y ∈ Ob(C ′), and every sieve S ′ covering Y . By adjunction (and bycorollary 2.1.38), this is the same as saying that the monomorphism g∗hS ′ → g∗hY is a coveringmorphism. Hence, it suffices to show that, for every morphism ϕ : hX → g∗hY in C ∧, theinduced subobject g∗hS ′ ×g∗hY hX covers X . However, by adjunction ϕ corresponds to amorphism hgX → hY , from which we obtain the subobject hS ′ ×hY hgX covering gX . Thenthe assertion follows from claim 2.1.48, applied to T ′ := hS ′ ×hY hgX .

Finally, the assertion concerning the left adjoint g∗U is immediate from the definitions.

Lemma 2.1.49. Let C ′ := (C ′, J ′) be a U-site, C := (C , J) a small site, and g : C → C ′ acontinuous functor. Then the following holds :


(i) The composition

g∗U : C∼U → C ′∼U : F 7→ (g! iCF )a

provides a left adjoint to gU∗. (Here iC : C∼U → C ∧U is the forgetful functor.)(ii) The natural diagram of functors :

Cg //

ha

C ′

ha

C∼U

g∗U // C ′∼U

is essentially commutative. (Notation of (2.1.19).)(iii) Suppose moreover that all the finite limits in C are representable, and that g is left

exact. Then g∗U is exact.

Proof. The first assertion is straightforward, and (ii) is reduced to the corresponding assertionfor (1.1.36), which has already been remarked. Next, since the functor F 7→ F a is left exact onC ′∧, assertion (iii) follows from proposition 2.1.3(iii).

2.1.50. Let (C , J) be a small site, (C ′, J ′) a U-site, u, v : C → C ′ two functors, such that u iscontinuous and v is cocontinuous. Let also V be a universe such that U ⊂ V. Then it followseasily from (2.1.5), lemmata 2.1.47 and 2.1.49(i), and remark 2.1.27(v) that we have essentiallycommutative diagrams of categories :

C∼Uu∗U //

C ′∼U

C∼Uv∗U //

C ′∼U

C∼V

u∗V // C ′∼V C∼Vv∗V // C ′∼V

whose vertical arrows are the inclusion functors. More generally, the diagram for v∗U is welldefined and essentially commuative, whenever C is a U-site, and C ′ has small Hom-sets.

Lemma 2.1.51. Let C := (C , J) and C ′ := (C ′, J ′) be two sites, and v : C → C ′, u : C ′ → Ctwo functors, such that v is left adjoint to u. The following conditions are equivalent :

(a) u is continuous.(b) v is cocontinuous.

Moreover, when these conditions hold, then for every universe V such that C and C ′ are V-small, we have natural isomorphisms of functors :

uV∗∼→ vV∗ u∗V

∼→ v∗V.

Proof. In view of lemma 2.1.40, we may replace U by a larger universe, after which we mayassume that C and C ′ are small sites. In this case, the lemma follows from lemma 2.1.6.

Lemma 2.1.52. Let (C , J) be a small site, (C ′, J ′) a U-site, u : C → C ′ a continuous andcocontinuous functor. Then we have :

(i) u∗ = u∗ and this functor admits the left adjoint u∗ and the right adjoint u∗.(ii) u∗ is fully faithful if and only if the same holds for u∗.

(iii) If u is fully faithful, then the same holds for u∗. The converse holds, provided thetopologies J and J ′ are coarser than the canonical topologies.

Proof. (i) is clear by inspecting the definitions. Assertion (ii) follows from (i) and proposition1.1.11(iii). Next, suppose that u is fully faithful; then the same holds for u∗ (lemma 2.1.6(i)),so the claim follows from (ii). Finally, suppose that u∗ is fully faithful, and both J and J ′ arecoarser than the canonical topologies on C and C ′. In such case, the Yoneda imbedding for


C (resp. C ′) realizes C (resp. C ′) as a full subcategory of C∼ (resp. of C ′∼). Then, from(1.1.36) and the explicit expression of u∗ provided by lemma 2.1.49(i), we deduce that u isfully faithful.

Definition 2.1.53. Let C := (C , J) be a site, B any category, and g : B → C a functor. Picka universe V such that B is V-small and C is a V-site. According to remark 2.1.18(iii), there isa finest topology Jg on B such that, for every V-sheaf F on C, the V-presheaf g∗F is a V-sheafon (B, Jg). By lemma 2.1.40, the topology Jg is independent of the chosen universe V. We callJg the topology induced by g on B. Clearly g is continuous for the sites (B, Jg) and C.

We have the following characterization of the induced topology.

Lemma 2.1.54. In the situation of definition 2.1.53, let X be any object of B, and R ⊂ hX anysubobject in B∧. The following conditions are equivalent :

(i) R ∈ Jg(X).(ii) For every morphism Y → X in B, the induced morphism gV!(R ×X Y ) → hg(Y ) is a

bicovering morphism in C ∧V (for the topology J on C ).

Proof. (i)⇒(ii) by virtue of claim 2.1.24. The converse follows easily from remark 2.1.18(ii)and corollary 2.1.38. (Details left to the reader.)

Example 2.1.55. (i) Resume the situation of example 2.1.7, and let f : Y → X be any objectof C/X; it is easily seen that the rule S 7→ (ιX)−1

|f S establishes a bijection between the sievesof (C/X)/f and the sieves of C/Y (notation of definition 1.5.1(iii) and (1.1.15)). Also, forevery subobject R ⊂ hf of the presheaf hf on C/X , the presheaf ιX!R is a subobject of hY .More precisely, for a sieve S of (C/X)/f and a sieve T of C/Y , we have the equivalence :

(2.1.56) hT = ιX!hS ⇔ S = (ιX)−1|f T .

(ii) Suppose now that J is a topology on C , and set C := (C , J). Let us endow C/X withthe topology JX induced by ιX , and let us pick a universe V such that C is V-small; since (ιX)V!

commutes with fibre products, the criterion of lemma 2.1.54 says that a subobjectR of an objectf : Y → X of C/X is a covering subobject, if and only if the induced morphism (ιX)V!R→ hYcovers Y . In view of (2.1.56), it follows easily that ιX is both continuous and cocontinuous.

(iii) Moreover, if J is a U-topology, JX is a U-topology as well : indeed, if G ⊂ Ob(C ) isa small topologically generating family for C, then G/X ⊂ Ob(C/X) is a small topologicallygenerating family for (C/X, JX).

(iv) Next, suppose that C is a U-site; then by example 2.1.7(ii) and remark 2.1.31(ii), itfollows that we may define a functor

ι∗X : (C/X, JX)∼U → C∼U

by the same rule as in lemma 2.1.49(i). Moreover, say that V is a universe with U ⊂ V, andsuch that C is V-small; then, remark 2.1.27(v) (and again remark 2.1.31(ii)) implies that thisfunctor is (isomorphic to) the restriction of (ιX)∗V, and since the inclusion functor C∼U → C∼V isfully faithful, we deduce that it is also a left adjoint to ιX∗.

(v) Furthermore, recalling example 2.1.7(iii), we see that ι∗X factors through a functor

eX : (C/X, JX)∼ → C∼/haX

and the functor ιhaX : C∼/haX → C∼ from (1.1.13). Indeed, a direct inspection shows that wehave a natural isomorphism :

eX(F )∼→ (eX iF )a

with eX as in example 2.1.7(iii), and where i : (C/X, JX)∼ → (C/X)∧ is the forgetful functor.(vi) Let g : Y → Z be any morphism in C ; then we have the sites (C/Y, JY ) and (C/Z, JZ)

as in (ii), as well as the functor g∗ : C/Y → C/Z of (1.1.14). Since ιZ g∗ = ιY , we


deduce easily from (ii) that g∗ is continuous for the topologies JY and JZ , and more precisely,JY is the topology induced by g∗ on C/Y . Notice also the natural isomorphism of categoriesC/Y

∼→ (C/Z)/g, in terms of which, the functor g∗ can be viewed as a functor ιX of the typeoccuring in example 2.1.7(i) (where we takeX := g, regarded now as an object of C/Z). Hence,all of the foregoing applies to g∗ as well; especially, g∗ is cocontinuous, and if C is a U-site,then (g∗)∗ : (C/Z, JZ)∼ → (C/Y, JY )∼ admits a left adjoint (g∗)

∗.

Proposition 2.1.57. With the notation of example 2.1.55(v), the functor eX is an equivalence.

Proof. To begin with, let us remark the following :

Claim 2.1.58. Let F be a presheaf on C/X , and set G := ιX!F . We have :(i) If F is a sheaf for the topology JX , then the natural map

(2.1.59) hX ×hsepXGsep → hX ×h+

XG+

is an isomorphism.(ii) F is a sheaf for the topology JX , if and only if the diagram

D :

GεG //

eX(F )

Ga

eX(F )a

hX

εX // haX

is cartesian in C ∧. (Here εG and εX are the units of adjunction.)

Proof of the claim. (i): The map is clearly a monomorphism, hence it suffices to show that itis an epimorphism. Thus, let Y be any object of C , and consider any pair (ϕ, σ) consisting ofelements ϕ ∈ hX(Y ) and σ ∈ G+(Y ) whose images ϕ and σ in hsep

X (Y ) coincide. Thereforeϕ : Y → X is a morphism in C , and there exists a covering subobject R ⊂ hY for the topologyJ , such that σ is the image of an element σR ∈ HomC∧(R,G). Let (ψi : Yi → Y | i ∈ I)be a family of morphisms in C that generates R; then σR is given by a compatible system(σi | i ∈ I), where σi ∈ G(Yi) for every i ∈ I . According to example 2.1.7(i), σi is the same asa pair (ai, ϕi), where ϕi : Yi → X is an object of C/X , and ai ∈ F (ϕi), for every i ∈ I . Theidentity σ = ϕ means that, after replacing R by a smaller covering subobject, we have

ϕ ψi = ϕi for every i ∈ I .

It follows that R = ιX!R′ for a covering subobject R′ ⊂ hϕ in the topology JX (example

2.1.55(ii)), and the compatible system (ai | i ∈ I) defines an element of Hom(C/X)∧(R′, F ).Since F is a sheaf, the latter is the image of a (unique) a ∈ F (ϕ). The pair (a, ϕ) yieldsa section σ ∈ G(Y ), and (ϕ, σ) gives an element of hX ×hsep

XGsep(Y ) whose image under

(2.1.59) is the original (ϕ, σ).(ii): In view of lemma 2.1.1, the diagram D is cartesian if and only if, for every object

ϕ : Y → X of C/X , the morphism εG induces a bijection :

(2.1.60) HomC∧/hX (hϕ, eX(F ))∼→ HomC∧/haX

(εX hϕ, eX(F )a)

and notice that, by example 2.1.7(iii), the source of (2.1.60) is just F (ϕ). Now, suppose firstthat D is cartesian, and let R ⊂ hϕ be a covering subobject for the topology JX . We havenatural isomorphisms :

Hom(C/X)∧(R,F )∼→ HomC∧/hX (eX(R), eX(F ))

∼→ HomC∧/haX(εX eX(R), eX(F )a).

However, a morphism εX eX(R) → eX(F )a is just a haX-morphism ιX!R → Ga; since ιX!Rcovers Y for the topology J (example 2.1.55(ii)), any such morphism extends uniquely to ahaX-morphism hϕ → Ga, and in view of the bijection (2.1.60), the latter comes from a unique


element of F (ϕ). This shows that the natural map F (ϕ) → Hom(C/X)∧(R,F ) is bijective, i.e.F is a sheaf, as claimed.

Conversely, suppose that F is a sheaf; we decompose D into two subdiagrams :

G //

eX(F )

G+ //

eX(F )+

Ga

eX(F )a

hX // h+

X// haX

and it suffices to show that both of these subdiagrams are cartesian. However, denote by E theleft subdiagram of D , and notice that the right subdiagram is none else that E +; by claim 2.1.24,we know that, if E is cartesian, the same holds for E +. Thus, we are reduced to showing that Eis cartesian, and in view of (i), this is the same as checking that the same holds for the similardiagram

Gε′G //

eX(F )

Gsep

eX(F )sep

hX

ε′X // hsepX .

Now, let ϕ : Y → X be any object of C/X , and β : hY → Gsep a given hsepX -morphism.

Since ε′G is an epimorphism, β lifts to an hsepX -morphism β′ : hY → G; then we may find a

covering subobject j : R′ → hY , such that the restriction β′ j : R′ → G is an hX-morphism.By example 2.1.55(i,ii), R′ = ιX!R for some subobject R ⊂ hϕ covering ϕ. We have naturalisomorphisms :

(2.1.61) HomC∧/hX (R′, G)∼→ Hom(C/X)∧(R,F )

∼→ F (ϕ)

so that β′ j extends to a hX-morphism γ : hY → G. By construction, ε′G γ j = ε′G β′ j,whence ε′G γ = ε′G β′ = β, since Gsep is separated, and j is a covering morphism. Thisshows that the natural map

F (ϕ)→ HomC∧/h+X

(ε′X hϕ, eX(F )+)

is surjective. Lastly, suppose that ε′G β1 = ε′G β2 for two given hX-morphisms hY → G;then we may find a covering subobject ιX!R ⊂ hY such that β1 and β2 restrict to the samehX-morphism ιX!R→ G. Then again, (2.1.61) says that β1 = β2, as required. ♦

Now, let H → haX be any object of C∼/haX , and set H ′ := H ×haX hX ; by example 2.1.7(iii),we may find an object F of (C/X)∧ such that eX(F ) = (H ′ → hX). Since clearly H ′a = H ,claim 2.1.58 shows that F is a sheaf for the topology JX , and clearly eX(F ) ' H . Lastly, saythat H1 and H2 are two haX-object of C∼, define H ′1, H ′2 as in the foregoing, and pick sheavesF1, F2 in (C/X, JX)∼ with eX(Fi) ' H ′i (for i = 1, 2); then we have natural bijections :

HomC∧/haX(H1, H2)

∼→ HomC∧/hX (H ′1, H′2)∼→ Hom(C/X)∧(F1, F2)

so eX is also fully faithful.

The functor ι∗X given explicitly in example 2.1.55(iv) is a special case of a construction whichgeneralizes lemma 2.1.49 to any continuous functor between two U-sites. To explain this, wenotice the following :

Proposition 2.1.62. Let C := (C , J) be a U-site, G a small topologically generating family forC. Denote by G the full subcategory of C with Ob(G ) = G, and endow G with the topology J ′

induced by the inclusion functor u : G → C . Then :(i) u is cocontinuous.

(ii) The induced functor u∗ : C∼ → (G , J ′)∼ is an equivalence.


Proof. For every presheaf F on C , denote by εF : u!u∗F → F the counit of adjunction.

Claim 2.1.63. (i) εF is a bicovering morphism, for every F ∈ Ob(C ∧).(ii) The functor u∗ is fully faithful.

Proof of the claim. (i): First we show that εF is a covering morphism. To this aim, recall thatboth u∗ and u! commute with all colimits, hence the same holds for the counit of adjunction.In view of lemma 2.1.1 and corollary 2.1.38, we may then assume that F = hX for someX ∈ Ob(C ). In this case, by assumption there exists a family (Yi | i ∈ I) of objects of G , anda covering morphism f :

∐i∈I huYi → hX in C ∧. By adjunction (and by (1.1.36)), f factors

(uniquely) through εF , so the latter must be a covering morphism as well, again by corollary2.1.38.

The claim amounts now to the following. Let F ∈ Ob(C ∧), X ∈ Ob(C ), and p, q : hX →u!u∗F two morphisms such that εF p = εF q; then Equal(p, q) is a covering subobject of hX .

However, pick again a covering morphism f as in the foregoing; for every i ∈ I , the restrictionfi : Yi → X of f satisfies the identity εF pfi = εF q fi. Again by adjunction and (1.1.36),it follows that p fi = q fi, i.e. f factors through Equal(p, q), whence the contention, in viewof corollary 2.1.38.

(ii): In view of lemma 2.1.49(i), it is easily seen that, for every sheaf F on C, the morphism(εiCF )a is the counit of adjunction u∗ u∗F → F . By (i) and corollary 2.1.38, the latter is anisomorphism, so u∗ is fully faithful (proposition 1.1.11(ii)). ♦

(i): Let Y be an object of G and S ′ a sieve covering uY ; we need to show that u−1|Y S ′ covers

Y in the induced topology of G . Since u is fully faithful, the counit of adjunction u∗huY → hYis an isomorphism, hence the latter amounts to showing that u∗hS ′ is a covering subobject of hYfor the induced topology (claim 2.1.48). This in turns means that, for every morphism X → Yin G , the induced morphism u!(u

∗hS ′×hY hX)→ huX is a bicovering morphism in C ∧ (lemma2.1.54). However :

u!(u∗hS ′ ×hY hX) = u!u

∗(hS ′ ×huY huX)

hence claim 2.1.63(i) reduces to showing that the natural morphism hS ′ ×huY huX → huX isbicovering, which is clear.

From (i) and lemma 2.1.52(ii) we deduce that u∗ is fully faithful. Combining with claim2.1.63(ii) and proposition 1.1.11(i), we deduce that u∗ is an equivalence, as stated.

Corollary 2.1.64. Let C := (C , J) and C ′ := (C ′, J ′) be two U-sites, g : C → C ′ a functor.We have :

(i) If g is continuous, the following holds :(a) For every universe V such that U ⊂ V, the functor gV∗ : C ′∼V → C∼V admits a left

adjoint g∗V : C∼V → C ′∼V .(b) For every pair of universes V ⊂ V′ containing U, we have an essentially commu-

tative diagram of categories :

C∼Vg∗V //

C ′∼V

C∼V′

g∗V′ // C ′∼V′

whose vertical arrows are the inclusion functors.(c) Suppose moreover that all the finite limits in C are representable, and that g is left

exact. Then g∗U is exact.(ii) If g is cocontinuous, the following holds :


(a) For every universe V such that U ⊂ V, the functor g∗V : C ′∼V → C∼V admits a rightadjoint gV∗ : C∼V → C ′∼V .

(b) For every pair of universes V ⊂ V′ containing U, we have an essentially commu-tative diagram of categories :

C∼VgV∗ //

C ′∼V

C∼V′

gV′∗ // C ′∼V′

whose vertical arrows are the inclusion functors.(iii) If g is both continuous and cocontinuous, we have a natural isomorphism :

gV∗∼→ g∗V

for every universe V with U ⊂ V.

Proof. (i.a): We choose a small topologically generating family G for C, and define the site(G , J ′) and the continuous functor u : (G , J ′) → C as in proposition 2.1.62. By applyinglemma 2.1.49(i) to the continuous functor h := g u, we deduce that hV∗ = uV∗ gV∗ admits aleft adjoint. Then the assertion follows from proposition 2.1.62(ii).

(i.b): More precisely, g∗V = h∗V uV∗. Thus, the assertion follows from (2.1.50).(i.c): In view of (i.b), the assertion can be checked after enlarging the universe U, so that we

may assume that C is small, in which case we conclude by lemma 2.1.49(iii).(ii.a): Let u and h be as in the foregoing; from proposition 2.1.62(i) we deduce that h is

cocontinuous, hence h∗V = u∗V g∗V admits a right adjoint; however u∗V = uV∗ (lemma 2.1.52(i)),and the latter is an equivalence (proposition 2.1.62(ii)), whence the contention.

(ii.b): More precisely, gV∗ = hV∗ u∗V, hence the assertion follows from (2.1.50).(iii): In view of (i.b) and (ii.b), in order to prove the assertion, we may assume that C and C ′

are V-small, and this case is already covered by lemma 2.1.52(i).

Example 2.1.65. (i) In the situation of example 2.1.55(iv), by corollary 2.1.64(ii,iii), the func-tor ιX∗ = ι∗X : C∼ → (C/X, JX)∼ admits also a right adjoint ιX∗. We introduce a specialnotation and terminology for these functors :

• The functor ιX∗ shall be also denoted j∗X , and called the functor of restriction to X .• The functor ιX∗ shall be denoted jX∗, and called the direct image functor.• The functor ι∗X shall be denoted jX!, and called the functor of extension by empty.

Thus, j∗X is right adjoint to jX!, and left adjoint to jX∗.(ii) Likewise, let g : Y → Z be any morphism in C ; by example 2.1.55(vi) and corollary

2.1.64(ii,iii), the functor (g∗)∗ admits also a right adjoint (g∗)∗. In agreement with (i), we shalllet :

j∗g := (g∗)∗ jg∗ := (g∗)∗ jg! := (g∗)∗.

Clearly j∗g j∗Z = j∗Y , and we have isomorphisms of functors : jZ∗jg∗∼→ jY ∗ and jZ!jg!

∼→ jY !.(iii) Moreover, under the equivalence eX of proposition 2.1.57, the functor j∗X is identified

to the functor :C∼ → C∼/haX U 7→ X × U

and jX! is identified to the functor C∼/haX → C∼ of (1.1.13). Likewise, g∗ is identified to thefunctor C∼/haZ → C∼/haZ given by the rule : (U → haZ) 7→ (U ×haZ h

aY ), and g! is identified to

the functor (hag)∗ : C∼/haY → C∼/haZ .


2.2. Topoi. A U-topos T is a category with small Hom-sets which is equivalent to the categoryof sheaves on a small site. We shall usually write “topos” instead of “U-topos”, unless this maygive rise to ambiguities. Denote by CT the canonical topology on T , and say that T = C∼, fora small site C := (C , J); then the set haX | X ∈ Ob(C ) is a small topological generatingfamily for the site (T,CT ), especially, the latter is a U-site. Moreover, the Yoneda embeddinginduces an equivalence

(2.2.1) T → (T,CT )∼

between T and the category of sheaves on the site (T,CT ). The Hom-sets in the category(T,CT )∼ are not small in general; however, they are essentially small, therefore (T,CT )∼ isisomorphic to a U-topos. In view of this, the objects of T may be thought of as sheaves on(T,CT ), and one uses often the suggestive notation :

X(S) := HomT (S,X) for any two objects X , S of T .

The elements ofX(S) are also called the S-sections ofX . The final object of T shall be denoted1T , and X(1T ) is also called the set of global sections of X .

Remark 2.2.2. (i) By proposition 2.1.62(ii), if C is a U-site, then C∼ is a U-topos.(ii) Remark 2.1.27(i,ii) can be summarized by saying that every U-topos T is a complete and

cocomplete, well-powered and co-well-powered category. Morever, every epimorphism in T isuniversal effective, every colimit is universal, and all filtered colimits in T commute with finitelimits.

Definition 2.2.3. (i) Let C = (C , J) and C ′ = (C ′, J ′) be two sites. A morphism of sitesC ′ → C is the datum of a continuous functor g : C → C ′, such that the left adjoint g∗U of theinduced functor gU∗ is exact (notation of definition 2.1.39(i)).

(ii) A morphism of topoi f : T → S is a datum (f ∗, f∗, η), where

f∗ : T → S f ∗ : S → T

are two functors such that f ∗ is left exact and left adjoint to f∗, and η : 1S ⇒ f∗f∗ is a unit of

the adjunction (these are sometimes called geometric morphisms : see [12, Def.2.12.1]).(iii) Let f := (f ∗, f∗, ηf ) : T ′′ → T ′ and g := (g∗, g∗, ηg) : T ′ → T be two morphisms of

topoi. The composition g f is the morphism

(f ∗ g∗, g∗ f∗, (g∗ ∗ ηf ∗ g∗) ηg) : T ′′ → T

(see remark 1.1.10(i)). The reader may verify that this composition law is associative.(iv) Let f, g : T → S be two morphisms of topoi. A natural transformation τ : f ⇒ g is

just a natural transformation of functors τ∗ : f∗ ⇒ g∗. Notice that, in view of remark 1.1.10(ii),the datum of τ∗ is the same as the datum of a natural transformation τ ∗ : g∗ ⇒ f ∗.

Remark 2.2.4. (i) By corollary 2.1.64(i.c), if C and C ′ are U-sites, and all finite limits of C arerepresentable, every left exact continuous functor C → C ′ defines a morphism of sitesC ′ → C.

(ii) More generally, let L be the category whose objects are all morphisms X → gY inC ′, where X (resp. Y ) ranges over the objects of C ′ (resp. of C ). The morphisms (β : X →gY ) → (β′ : X ′ → gY ′) in L are the pairs (ϕ, ψ) where ϕ : X → X ′ (resp. ψ : Y → Y ′) isa morphism in C ′ (resp. in C ) and β′ ϕ = ϕ′ β. There is an obvious fibration p : L → C ′,such that p(X → gY ) := X for every object (X → gY ) in L , and one can prove that acontinuous functor g : C → C ′ defines a morphism of sites, if and only if the fibration p islocally cofiltered (see [4, Exp.V, Def.8.1.1]; the necessity is a special case of [4, Exp.V, lemme8.1.11]). This shows that the definition of morphism of sites depends only on g, and not on theuniverse U.


(iii) Every morphism of U-sites g : C ′ → C induces a morphism of topoi g∼ : C ′∼ → C∼,and conversely, a morphism f : T → S of topoi determines a morphism of sitesCf : (T,CT )→(S,CS) for the corresponding canonical topologies.

(iv) The topoi form a 2-category, denoted

Topos

whose objects are all the topoi, whose 1-cells are the morphisms of topoi, and whose 2-cells arethe natural transformations between such morphisms, as in definition 2.2.3(ii,iv).

Proposition 2.2.5. Let T , T ′ be two topoi, f : T → T ′ be a functor.

(i) If f commutes with all colimits, and all fibre products in T , the following holds :(a) f is continuous for the canonical topologies on T and T ′.(b) There exists a morphism of topoi ϕ : T ′ → T , unique up to unique isomorphism,

such that ϕ∗ = f .(ii) If f is exact, the following holds :

(a) f is conservative if and only if it reflects epimorphisms.(b) For every morphism ϕ in T , the natural morphism f(Imϕ) → Im(fϕ) is an

isomorphism.

Proof. (i.a): Let S := gi : Xi → X | i ∈ I be a small covering family of morphisms in(T,CT ); by lemma 2.1.42, it suffices to show that fS := f(gi) | i ∈ I is a covering familyin (T ′, CT ′). However, our assumption implies that the induced morphism qi∈IXi → X isan epimorphism, hence the same holds for the induced morphism qi∈IfXi → fX in T ′ (byexample 1.1.24(iii)). But all epimorphisms are universal effective in T ′ (remark 2.2.2(ii)), hence(f(gi) | i ∈ I) is a universal effective family, as required.

(i.b): By corollary 2.1.64(i.a,i.c), for every universe V such that U ⊂ V, the functor f givesrise to a morphism of topoi

(f ∗V, fV∗, ηV) : (T ′, CT ′)∼V → (T,CT )∼V

(where η is any choice of unit of adjunction), and it remains only to check that f ∗U is isomorphicto f , under the natural identifications hU : T

∼→ (T,CT )∼U , h′U : T ′∼→ (T ′, CT ′)

∼U . In view of

corollary 2.1.64(i.b), it suffices to show that there exists a universe V such that the diagram

Tf //

hV

T ′

h′V

(T,CT )∼Vf∗V // (T ′, CT ′)

∼V

is essentially commutative (where hV and h′V are the Yoneda embeddings). By lemma 2.1.49(i)and (1.1.36), the latter holds whenever T is V-small.

(ii.b) follows easily from remark 2.1.33.(ii.a): The condition is necessary, due to example 1.1.24(iii). Conversely, assume that f

reflects epimorphisms, and suppose that ϕ is a morphism in T such that f(ϕ) is an isomorphism.It follows already that ϕ is an epimorphism, hence it suffices to show that ϕ is a monomorphism(proposition 2.1.32(i)). This is the same as showing that the natural map ιϕ : Y → Equal(ϕ, ϕ)is an isomorphism (again by example 1.1.24(iii)); however, ιϕ is always a monomorphism, andsince f is left exact, f(ιϕ) is the natural morphism fY → Equal(fϕ, fϕ), which we know tobe an isomorphism, so ιϕ is an epimorphism, under our assumption.

Example 2.2.6. Let T be a topos.


(i) Say that T = (C , J)∼ for some small site (C , J). Then the category C ∧ is also a topos([3, Exp.IV, §2.6]), and since the functor F 7→ F a of (2.1.19) is left exact, it determines amorphism of topoi T → C ∧. (On the other hand, the category T∧ is too large to be a U-topos.)

(ii) If f := (f ∗, f∗, η) : T → S is any morphism of topoi, we have an essentially commuta-tive diagram of categories :

T //

f∗

T∧

f∧∗

S // S∧

whose horizontal arrows are the Yoneda embeddings, and where f∧∗ = Fun(f ∗o,Set).(iii) Let U be any object of T . By applying the discussion of example 2.1.65 to the U-site

(T,CT ), we deduce that the category T/U (notation of (1.1.2)) is a topos, and the natural functor

j∗U : T → T/U X 7→ X|U := (X × U → U)

induces a morphisms of U-sites (T,CT ) → (T/U,CT/U), whence a morphism of topoi jU :T/U → T , unique up to unique isomorphism. Moreover, we also obtain a left adjoint jU ! :T/U → T for j∗U , given explicitly by the rule : (X → U) 7→ X for any U -object X of T .In case U is a subobject of the final object, the morphism jU is called an open subtopos of T .Likewise, any morphism g : Y → Z in T determines, up to unique isomorphism, a morphismof topoi jg : T/Y → T/Z, with an isomorphism jZ jg

∼→ jY of morphisms of topoi.(iv) Let U be a subobject of 1T . We denote by CU the full subcategory of T such that

Ob(CU) = X ∈ Ob(T ) | j∗UX = 1T/U.

Then CU is a topos, called the complement of U in T , and the inclusion functor i∗ : CU → Tadmits a left adjoint i∗ : T → CU , namely, the functor which assigns to every X ∈ Ob(T ) thepush-out X|CU in the cocartesian diagram :

X × UpX //

pU

X

U // X|CU

where pX and pU are the natural projections. Moreover, i∗ is an exact functor, hence the adjointpair (i∗, i∗) defines a morphism of topoi CU → T , unique up to unique isomorphism. (See [3,Exp.IV, Prop.9.3.4].)

(v) Another basic example is the global sections functor Γ : T → Set, defined as :

U 7→ Γ(T, U) := U(1T ).

Γ admits a left adjoint :Set→ T : S 7→ ST := S × 1T

(the coproduct of S copies of 1T ) and for every set S, one calls ST the constant sheaf with valueS. This pair of adjoint functors defines a morphism of topoi Γ : T → Set ([3, Exp.IV, §4.3]).One can chek that there exists a unique such morphism of topoi, up to unique isomorphism.

Example 2.2.7. Let T be a topos.(i) We say that T is connected (resp. disconnected) if the same holds for the final object 1T

of T (see example 1.1.26(iii)). Notice that, if U is any object of T , then U is connected if andonly if the same holds for the topos T/U .

(ii) T is connected if and only if the natural map

(2.2.8) Z/2Z→ Γ(T,Z/2ZT )


is surjective (notation of example 2.2.6(v)). Indeed, Z/2ZT is the coproduct 1T q 1T , and anydecomposition α : 1T

∼→ X0 q X1 determines a section of 1T q 1T , namely the compositionα′ of α with the coproduct of the unique morphisms X0 → 1T and X1 → 1T . Conversely,let β′ : 1T → Z/2ZT be a global section, and denote by j0, j1 : 1T → 1T q 1T the naturalmorphisms (so j0, j1 is the image of the map (2.2.8)). We define Xi for i = 0, 1, as the fibreproducts in the cartesian diagrams :

Xi//

1T

ji

1Tβ′ // Z/2ZT .

Since all colimits in T are universal (remark 2.2.2(ii)), the induced morphism β : X0qX1 → 1Tis an isomorphism, and it is easily seen that the rules α 7→ α′ and β′ 7→ β establish mutuallyinverse bijections (details left to the reader).

Remark 2.2.9. (i) Let f : T ′ → T be a morphism of topoi, U an object of T , and supposethat s : T ′ → T/U is a morphism of topoi with an isomorphism jU s

∼→ f . Hence, we have afunctorial isomorphism s∗(j∗UX)

∼→ f ∗X for everyX ∈ Ob(T ), and especially, s∗1U = 1T ′; let∆U : 1U → j∗UjU ! = j∗U(U) be the unit of adjunction (i.e. the diagonal morphism in T/U ); thenσ := s∗(∆U) : 1T ′ → f ∗U is an element of Γ(T ′, f ∗U). Moreover, for every object ϕ : X → Uin T/U , we have a cartesian diagram in T/U :

D :

XΓϕ //

ϕ

ϕ

@@@@@@@@ X × U

j∗Uϕ

j∗UX

wwwwwwwww

U

U

1U

??~~~~~~~~

∆U

// U × Uj∗UU

ccGGGGGGGGG

where Γϕ is the graph of ϕ, and s∗D is isomorphic to the cartesian diagram (in T ) :

E :

s∗ϕ //

f ∗X

f∗ϕ

1T ′σ // f ∗U.

This shows that s∗ – and therefore also s – is determined, up to unique isomorphism, by σ.(ii) Conversely, if σ ∈ Γ(T ′, f ∗U) is any global section, then we may define a functor

s∗ : T/U → T ′ by means of the cartesian diagram E , and clearly s∗ j∗U = f ∗. Let us alsodefine a functor t! : T ′ → T ′/f ∗U , by the rule :

t!(Y ) = σ uY for every Y ∈ Ob(T ′)

where uY : Y → 1T ′ is the unique morphism in T ′. By inspecting the diagram E , we deducenatural bijections :

HomT ′(Y, s∗ϕ)

∼→ HomT ′/f∗U(t!Y, f∗ϕ) for every Y ∈ Ob(T ′) and ϕ ∈ Ob(T/U).

Since f ∗ is exact, it follows easily that s∗ is left exact (indeed, s∗ commutes with all the limitswith which f ∗ commutes). Moreover, since all colimits are universal in T (see (2.1.27)(i)), itis easily seen that s∗ commutes with all colimits. Then, by proposition 2.2.5(i.b), the functors∗ determines a morphism of topoi s : T ′ → T/U , unique up to unique isomorphism, and anisomorphism jU s

∼→ f .


(iii) Summing up, the constructions of (i) and (ii) establish a natural bijection betweenΓ(T ′, f ∗U) and the set of isomorphism classes of morphisms of topoi s : T ′ → T/U suchthat jU s is isomorphic to f .

Definition 2.2.10. Let T be a topos.(i) A point of T (or a T -point) is a morphism of topoi Set→ T . If ξ = (ξ∗, ξ∗) is a point,

and F is any object of T , the set ξ∗F is usually denoted by Fξ.(ii) If ξ is a point of T , and f : T → S is any morphism of topoi, we denote by f(ξ) the

S-point f ξ.(iii) A neighborhood of ξ is a pair (U, a), where U ∈ Ob(T ), and a ∈ Uξ. A morphism of

neighborhoods (U, a)→ (U ′, a′) is a morphism f : U → U ′ in T such that fξ(a) = a′.The category of all neighborhoods of ξ shall be denoted Nbd(ξ).

(iv) We say that a set Ω of T -points is conservative, if the functor

T → U′-Set F 7→∏ξ∈Ω

Fξ

is conservative (definition 1.1.4(ii)). (Here U′ is a universe such that U ⊂ U′ andΩ ∈ U′.) We say that T has enough points, if T admits a conservative set of points.

2.2.11. It follows from remark 2.2.9(iii), that a neighborhood (U, a) of ξ is the same as thedatum of an isomorphism class of a point ξU of the topos T/U which lifts ξ, i.e. such thatξ ' jU ξU . Moreover, say that ξU ′ is another lifting of ξ, corresponding to a neighborhood(U ′, a′) of ξ; then, by inspecting the constructions of remark 2.2.9(i,ii) we see that, under thisidentification, a morphism (U, a)→ (U ′, a′) of neighborhoods of ξ corresponds to the datum ofa morphism ϕ : U → U ′ in T and an isomorphism of T -points :

jϕ ξU∼→ ξU ′

where jϕ is the morphism of topoi T/U → T/U ′ induced by ϕ. Thus, let Lift(ξ) denote thecategory whose objects are the triples (U, ξU , ωU), where U ∈ Ob(T ), ξU is a T/U -point liftingξ, and ωU : jU ξU

∼→ ξ is an isomorphism of T -points (here we fix, for every object U and everymorphism ϕ of T , a choice of the morphisms jU and jϕ : recall that the latter are determined byU and respectively ϕ, up to unique isomorphism). The morphisms (U, ξU , ωU) → (V, ξV , ωV )are the pairs (ϕ, ωϕ), where ϕ : U → V is a morphism in T , and

ωϕ : jϕ ξU∼→ ξV

is an isomorphism of T/V -points, such that :

ωV ∗ (jV ∗ ∗ ωϕ∗) = ωU∗.

The composition of such a morphism with another one (ψ, ωψ) : (V, ξV , ωV ) → (W, ξW , ωW )is the pair (ψ ϕ, ωψϕ) determined by the commutative diagram of T/W -points :

(2.2.12)

jψ jϕ ξUjψ∗ωϕ //

cψ,ϕ∗ξU

jψ ξVωψ

jψϕ ξU

ωψϕ // ξW .

where cψ,ϕ : jψ jϕ∼→ jψϕ is the unique isomorphism. As an exercise, the reader may verify

the associativity of this composition law. The foregoing shows that we have an equivalence ofcategories :

(2.2.13) Nξ : Lift(ξ)∼→ Nbd(ξ) (U, ξU , ωU) 7→ (U, aU)


where aU ∈ Uξ is the unique morphism which fits in the commutative diagram :

ξ∗1TaU //

ω∗U (1T )

ξ∗(U)

ω∗U (U)

ξ∗U j∗U(1T )

ξ∗U (∆U )// ξ∗U j∗U(U)

(notation of remark 2.2.9(i)). To check the functoriality of (2.2.13), let (ϕ, ωϕ) be as in (2.2.11),and set ϑV := jϕ ξU ; we can write :

ϑ∗V (∆V ) = ξ∗U j∗ϕ(∆V ) = ξ∗U(Γϕ) = ξ∗U(j∗Uϕ) ξ∗U(∆U) = ϑ∗V (ϕ) ξ∗U(∆U)

from which it follows easily that aV = ϕξ(aU), as required.

Remark 2.2.14. (i) Let T be a topos with enough points, and U any object of T ; as a con-sequence of the discussion in (2.2.11), we deduce that also T/U has enough points. Moreprecisely, say that Ω is a conservative set of T -points, and denote by j−1

U Ω the set of all T/U -points ξ such that jU ξ ∈ Ω; we claim that j−1

U Ω is a conservative set of points. Indeed, letϕ : X → Y be a morphism in T/U such that ϕξ is an epimorphism for every ξ ∈ j−1

U Ω; byproposition 2.2.5(ii.a), it suffices to show that ϕ is an epimorphism, and the latter will follow, ifwe show that the same holds for jU !ϕ.

Thus, suppose by way of contradiction, that jU !ϕ is not an epimorphism; then there existsa point ξ ∈ Ω, and y ∈ (jU !Y )ξ, such that y does not lie in the image of (jU !ϕ)ξ. Let a :=(jU !πX)ξ(y) ∈ Uξ, where πX : X → 1T/U is the unique morphism in T/U . By (2.2.13), we mayfind a lifting (U, ξU , ωU) of ξ such that

(2.2.15) Nξ(U, ξU , ωU) = (U, a).

After replacing ξ by jU ξU , we may assume that ωU is the identity of ξ, in which case (2.2.15)means that a = ξ∗U∆U , where ∆U : 1T/U → j∗UjU !1T/U is the unit of adjunction (i.e the diagonalU → U × U ). We have a commutative diagram of sets :

ξ∗UXξ∗Uϕ //

ξ∗U∆X

ξ∗UY

ξ∗U∆Y

(jU !X)ξ

(jU !ϕ)ξ // (jU !Y )ξ

where ∆X : X → j∗UjU !X is the unit of adjunction, and likewise for ∆Y . More plainly,∆Y : Y × 1T/U → Y × (j∗jU !1T/U) is the product 1Y × ∆U , and under this identification,ξ∗U∆Y is the mapping ξ∗UY → ξ∗UY × (j∗jU !1T/U) given by the rule : z 7→ (z, a) for everyz ∈ ξ∗UY . Especially, we see that y lies in the image of ξ∗U∆Y . But by assumption, the map ξ∗Uϕis surjective, hence y lies in the image of (jU !ξ)ξ, a contradiction.

(ii) The localization morphisms jU of example 2.2.6(iii), and the notion of point of a topos,form the basis for a technique to study the local properties of objects in a topos. Indeed, supposethat P(T, F, ϕ) is a property of sequences F := (F1, . . . , Fn) of objects in a topos T , and ofmorphisms ϕ := (ϕ1, . . . , ϕm) in T ; we say that P can be checked on stalks, if the followingtwo conditions hold for every topos T , every F ∈ Ob(T )n, and every ϕ ∈ Morph(T )m.

(a) P(T, F, ϕ) implies P(Set, Fξ, ϕξ) for every T -point ξ. (Here Fξ := (F1ξ, . . . , Fnξ),and likewise for ϕξ.)

(b) If Ω is a conservative set of T -points such that P(Set, Fξ, ϕξ) holds for every ξ ∈ Ω,then P(T, F, ϕ) holds.

Then we have the following :


Lemma 2.2.16. Let T be a topos with enough points, (Uλ → 1T | λ ∈ Λ) a family of morphismsin T covering the final object (in the canonical topology CT ), and P a property that can bechecked on stalks. Then, for any F ∈ Ob(T )n and ϕ ∈ Morph(T )m, we have P(T, F, ϕ) if andonly if P(T/Uλ, F|Uλ , ϕ|Uλ) holds for every λ ∈ Λ.

Proof. Suppose first that P(T, F, ϕ) holds. If η is any T/Uλ-point, then ξ := jUλ η is a T -point,and (j∗UλF )η = Fξ, (and likewise for ϕ, with obvious notation) hence P(Set, (j∗UλF )η, (j

∗Uλϕ)η)

holds. Since P can be checked on stalks, remark 2.2.14(i) then implies that P(T/Uλ, F|Uλ , ϕ|Uλ)holds.

Conversely, suppose that P(T/Uλ, F|Uλ , ϕ|Uλ) holds for every λ ∈ Λ, and let ξ be any T -point. By claim 2.1.45 we may assume that Λ is a small set; then for every T -point ξ, the set(qλ∈ΛUλ)ξ = qλ∈Λξ

∗Uλ covers the final object of Set, i.e. is not empty. Pick λ ∈ Λ such thatξ∗Uλ 6= ∅; by the discussion of (2.2.11) we see that ξ lifts to a T/Uλ-point η. Since P can bechecked on stalks, P(Set, (F|Uλ)η, (ϕ|Uλ)η) holds, and this means that P(Set, Fξ, ϕξ) holds, soP(T, F, ϕ) holds.

2.2.17. Let X be any (small) set; we consider the functor

FX : Lift(ξ)o → Set (U, ξU , ωU) 7→ ξ∗UξU∗X.

For a given morphism (ϕ, ωϕ) : (U, ξU , ωU) → (V, ξV , ωV ) in Lift(ξ), set ϑV := jϕ ξU ; bydefinition 2.2.3(iv), ωϕ is a natural isomorphism of functors ωϕ∗ : ϑV ∗

∼→ ξV ∗, which determines(and is determined by) a natural isomorphism of functors ω∗ϕ : ξ∗V

∼→ ϑ∗V . The morphismFX(ϕ, ωϕ) is then defined by the commutative diagram :

ξ∗V ϑV ∗(X)ω∗ϕ∗ϑV ∗(X)

//

ξ∗V ∗ωϕ∗(X)

ϑ∗V ϑV ∗(X)

ξ∗U∗εϕ∗ξU∗(X)

ξ∗V ξV ∗(X)

FX(ϕ,ωϕ)// ξ∗UξU∗(X)

where εϕ : j∗ϕjϕ∗ ⇒ 1Set is the counit of adjunction. With the notation of (2.2.11), to verifythat FX is a well defined functor, amounts to checking that FX(ψ ϕ, ωψϕ) = FX(ϕ, ωϕ) FX(ψ, ωψ). To this aim, we may assume that either ωϕ is the identity of jϕ ξU (which thencoincides with ξV ), or else that ϕ is the identity (in which case U coincides with V , and jϕ is theidentity of T/U ), and likewise for ωψ and ψ. We have then four cases to consider separately; wewill proceed somewhat briskly, since these verifications are unenlightening and rather tedious.

First, suppose that both ωϕ and ωψ are identities. In this case, we have :

FX(ϕ, ωϕ) = ξ∗U ∗ εϕ ∗ ξU∗X FX(ψ, ωψ) = ξ∗V ∗ εψ ∗ ξV ∗X = ξ∗U ∗ j∗ϕ ∗ εψ ∗ jϕ∗ ∗ ξU∗X

and we remark as well that ωψϕ = (cψ,ϕ ∗ ξU)−1. Then the sought identity translates thecommutativity of the diagram ξ∗U ∗D ∗ ξU∗, where D is the diagram :

j∗ϕ j∗ψ jψϕ∗c∗−1ψ,ϕ ∗jψϕ∗ //

j∗ϕ∗j∗ψ∗c−1ψ,ϕ∗

j∗ψϕ jψϕ∗εψϕ

j∗ϕ j∗ψ jψ∗ jϕ∗

j∗ϕ∗εψ∗j∗ϕ // j∗ϕ jϕ∗εϕ // 1.

Now, notice that the bottom row is the counit of adjunction defined by the morphism jψ jϕ;then the commutativity of D is just a special case of remark 1.1.10(iii).

Next, suppose that ϕ is the identity of U = V , and ωψ is the identity of jψ ξV = ξW . In thiscase, ωψϕ = jψ ∗ ωϕ, and FX(ψ, ωψ) is the same as in the previous case, whereas :

FX(ϕ, ωϕ) = (ω∗ϕ ∗ ξU∗) (ξ∗V ∗ ωϕ∗)−1.


Then the sought identity translates the commutativity of the diagram :

ξ∗W ξW∗ξ∗V ∗εψ∗ξV ∗

ξ∗W jψ∗ ξU∗ξ∗W ∗jψ∗∗ωψ∗oo

ω∗ϕ∗j∗ψ∗jψ∗∗ξU∗ //

ξ∗V ∗εψ∗ξU∗

ξ∗U j∗ψ jψ∗ ξU∗ξ∗U∗εψ∗ξU∗

ξ∗V ξV ∗ ξ∗V ξU∗ξ∗V ∗ωϕ∗oo

ω∗ϕ∗ξU∗ // ξ∗U ξU∗.

which is an immediate consequence of the naturality of ωϕ∗ and ω∗ϕ.The case where ωψ is the identity of ξW and ψ is the identity of V = W , is similar to the

previous one. The remaining case where both ϕ and ψ are identities is easy, and shall be left tothe reader.

Remark 2.2.18. (i) Let ξ be a point of the topos T ; we remark that the category Nbd(ξ)is cofiltered; indeed, if (U1, a1) and (U2, a2) are two neighborhoods, the natural projectionsU12 := U1 × U2 → Ui (for i = 1, 2) give morphisms of neighborhoods (U12, (a1, a2)) →(Ui, ai). (Notice that (U1 × U2)ξ = U1,ξ × U2,ξ.) Likewise, since ξ∗ commutes with equalizers,for any two morphisms ϕ, ϕ′ : (U1, a1) → (U2, a2), we have a morphism of neighborhoodsψ : (Equal(ϕ, ϕ′), a1)→ (U1, a1), such that ϕ ψ = ϕ′ ψ.

(ii) Denote by ι : Nbd(ξ) → T the functor given by the rule : (U, a) 7→ U . Then there is anatural transformation of functors :

τξ : HomT (ιo, F )⇒ cFξ : Nbd(ξ)o → Set

where cFξ denotes the constant functor with value Fξ. Indeed, if (U, a) is any neighborhood ofξ, we define τξ(U, a) : HomT (U, F ) → Fξ by the rule : s 7→ sξ(a) for every s : U → F . Thenwe claim that τξ induces a natural bijection :

(2.2.19) colimNbd(ξ)o

HomT (ιo, F )∼→ Fξ.

Indeed, every element of the above colimit is represented by a triple (U, a, β), where (U, a)is a neighborhood of ξ, and β ∈ HomT (U, F ); two such triples (U1, a1, β1) and (U2, a2, β2) areidentified if and only if there exist morphisms ϕi : (V, b) → (Ui, ai) in Nbd(ξ) (for i = 1, 2),such that ϕ1 β1 = ϕ2 β2. It is then easily seen that every such triple is equivalent to a uniquetriple of the form (F, a,1F ), which gets mapped to a under τξ(F, a), whence the assertion.

(iii) For every object (U, ξU , ωU) of Lift(ξ), let εU : ξ∗UξU∗X → X be the counit of adjunc-tion coming from the morphism of topoi ξU ; we claim that the rule (U, ξU , ωU) 7→ εU defines anatural transformation

(2.2.20) FX ⇒ cX

where cX denotes the constant functor with value X . Indeed, let (ϕ, ωϕ) : (U, ξU , ωU) →(V, ξV , ωV ) be a morphism in Lift(ξ); we need to show that εU FX(ϕ, ωϕ) = εV . Thisamounts to checking the identity :

εU (ξ∗U ∗ εϕ ∗ ξU∗) (ω∗ϕ ∗ ϑV ∗) = εV (ξ∗V ∗ ωϕ∗)where ϑV := jϕ ξU . Now, notice that ε′V := εU (ξ∗U ∗ εϕ ∗ ξU∗) : ϑ∗V ϑV ∗ → 1 is the counit ofadjunction given by the morphism of topoi ϑV . On the other hand, remark 1.1.10(ii) says that :

ω∗ϕ = (εV ∗ ϑ∗V ) (ξ∗V ∗ ωϕ∗ ∗ ϑ∗V ) (ξ∗V ∗ η′V )

where η′V : 1 → ϑV ∗ϑ∗V is the unit of adjunction given by ϑV . The naturality of εV implies the

identity :εV (ξ∗V ∗ ξV ∗ ∗ ε′V ) = ε′V (εV ∗ ϑ∗V ∗ ϑV ∗).

Hence we come down to showing that

(ξ∗V ∗ ξV ∗ ∗ ε′V ) (ξ∗V ∗ ωϕ∗ ∗ ϑ∗V ∗ ϑV ∗) (ξ∗V ∗ η′V ∗ ϑV ∗) = ξ∗V ∗ ωϕ∗.


The latter in turn will follow, once we know that

(2.2.21) (ξV ∗ ∗ ε′V ) (ωϕ∗ ∗ ϑ∗V ∗ ϑV ∗) (η′V ∗ ϑV ∗) = ωϕ∗.

However, the naturality of ωϕ∗ says that

(2.2.22) (ωϕ∗ ∗ ϑ∗V ∗ ϑV ∗) (η′V ∗ ϑV ∗) = (ξV ∗ ∗ η′V ) ωϕ∗Then (2.2.21) follows easily from (2.2.22) and the triangular identities of (1.1.8).

Lemma 2.2.23. The natural transformation (2.2.20) induces a natural isomorphism :

colimLift(ξ)o

FX∼→ X for every (small) set X .

Proof. Set C := Morph(Lift(ξ)o). We define yet another functor

G : C → Set

by the rule :

((U ′, ξU ′ , ωU ′)(ψ,ωψ)−−−−→ (U, ξU , ωU)) 7→ G(ψ, ωψ) := Γ(ψ, ξU∗X)

(here we regard ψ : U ′ → U as an object of T/U ). Recall that a morphism (β, ωβ) → (ψ, ωψ)in C is a commutative diagram of morphisms in Lift(ξ)

D :

(U ′, ξU ′ , ωU ′)(ϕ′,ωϕ′ ) //

(ψ,ωψ)

(V ′, ξV ′ , ωV ′)

(β,ωβ)

(U, ξU , ωU)

(ϕ,ωϕ)// (V, ξV , ωV ).

To such a morphism, we assign the map G(D) : G(β, ωβ)→ G(ψ, ωψ) defined as follows. First,we may regard ϕ′ as a morphism jϕ!(ψ)→ β in T/V . Denote by

∆ψ : ψ → j∗ϕjϕ!(ψ)

the unit of adjunction. Then, for every s : β → ξV ∗X , the section G(D)(s) : ψ → ξU∗X is thecomposition :

ψj∗ϕ(sϕ′ )∆ψ−−−−−−−−→ j∗ϕξV ∗X

j∗ϕ∗ω−1ϕ∗X−−−−−−→ j∗ϕjϕ∗ξU∗X

εϕ∗ξU∗X−−−−−−→ ξU∗X.

With some patience, the reader may check that G is really a well defined functor. Next, wedefine a natural transformation :

(2.2.24) G⇒ FX s

where s : C → Lift(ξ)o is the source functor of (1.1.17). Indeed, notice that, for every lifting(U, ξU , ωU) of ξ, we have an isomorphism of categories :

Lift(ξ)/(U, ξU , ωU)∼→ Lift(ξU) (ψ, ωψ) 7→ (ψ, ξU ′ , ωψ)

(where (ψ, ωψ) is as in the foregoing). On the other hand, by composing the equivalence NξU

of (2.2.13), and the natural transformation τξU of remark 2.2.18(ii) (for F := ξU∗X), we obtaina natural transformation :

τξU ∗NξU (ψ, ξU ′ , ωψ) : Γ(ψ, ξU∗X)→ ξ∗UξU∗X

which yields (2.2.24). By inspecting the construction, and in view of (2.2.19), we get a naturalisomorphism :

colimC

G∼→ colim

Lift(ξ)oF.

Now, notice that the functor

(2.2.25) Lift(ξ)o → C (U, ξU , ωU) 7→ 1(U,ξU ,ωU )


is cofinal. A simple inspection reveals that the composition of G with (2.2.25) is the constantfunctor with value X , whence the contention.

2.2.26. Suppose now that T = C∼ for some small site C := (C , J). For a point ξ of T , wemay define another category Nbd(ξ, C), whose objects are the pairs (U, a) where U ∈ Ob(C )and a ∈ (haU)ξ; the morphisms (U, a)→ (U ′, a′) are the morphisms f : U → U ′ in C such that(haf )ξ(a) = a′. (Notation of remark 2.1.27(iii).) The rule (U, a) 7→ (haU , a) defines a functor

(2.2.27) Nbd(ξ, C)→ Nbd(ξ).

Proposition 2.2.28. In the situation of (2.2.26), we have :(i) The category Nbd(ξ, C) is cofiltered.

(ii) The functor (2.2.27) is cofinal.

Proof. To begin with, since ξ∗ commutes with all colimits, remark 2.1.27(iii) implies easily that,for every object (U, a) of Nbd(ξ) there exists an object (V, b) of Nbd(ξ, C) and a morphism(haV , b)→ (U, a) in Nbd(ξ). Hence, it suffices to show (i).

Thus, let (U1, a1) and (U2, a2) be two objects of Nbd(ξ, C); since Nbd(ξ) is cofiltered, wemay find an object (F, b) of Nbd(ξ) and morphisms ϕi : (F, b) → (haUi , ai) (for i = 1, 2). Bythe foregoing, we may also assume that (F, b) = (haV , b) for some object (V, b) of Nbd(ξ, C),in which case ϕi ∈ haUi(V ) for i = 1, 2. By remark 2.1.27(iv), we may find a sieve S coveringV such that ϕi ∈ HomC∧(hS , h

sepUi

) for both i = 1, 2.On the other hand, (2.1.2) and proposition 2.1.34 yield a natural isomorphism :

colimS

ha s∼→ haV .

Therefore, since ξ∗ commutes with all colimits, we may find (f : S → V ) ∈ Ob(S ) andc ∈ (haS)ξ such that haf : (haS, c)→ (haV , b) is a morphism of neighborhoods of ξ.

For i = 1, 2, denote by ϕS,i ∈ hsepU (S) the image of ϕi (under the map induced by the natural

morphism hS → hS coming from (2.1.2)). Pick any ϕS,i ∈ HomC (S, V ) in the preimage ofϕS,i; then ϕS,i defines a morphism (S, c)→ (Ui, ai) in Nbd(ξ, C).

Next, suppose that ϕ1, ϕ2 : (U, a) → (U ′, a′) are two morphisms in Nbd(ξ, C); argu-ing as in the foregoing, we may find an object (V, b) of Nbd(ξ, C), and ψ ∈ hsep

U (V ) =HomC∧(hsep

V , hsepU ) whose image in haU(V ) yields a morphism ψa : (haV , b) → (haU , a) in

Nbd(ξ), and such that ϕsep1 ψ = ϕsep

2 ψ in hsepU ′ (V ). We may then find a covering sub-

object i : R→ hV , such that ϕ1 (ψ i) = ϕ2 (ψ i) in HomC∧(R, h′U). Again, by combining(2.1.2) and proposition 2.1.34, we deduce that there exists a morphism β : (V ′, b′) → (V, b) inNbd(ξ, C), such that ϕ1 (ψ β) = ϕ2 (ψ β). This completes the proof of (i).

As a corollary of proposition 2.2.28 and of remark (2.2.18)(ii), we deduce, for every sheaf Fon C, a natural isomorphism :

colimNbd(ξ,C)

F ιoC∼→ Fξ

where ιC : Nbd(ξ, C)→ C is the functor given by the rule (U, a) 7→ U on every object (U, a).

2.3. Algebra on a topos. Let T be any topos, and endow T with the structure of tensor categoryas explained in example 1.2.10 (so, the tensor functor is given by fixed choices of products forevery pairs of objects of T , and any final object 1T can be taken for unit object of (T,⊗)). Wenotice that (T,⊗) admits an internal Hom functor (see remark 1.2.12(ii)). Indeed, let X and X ′

be any two objects of T . It is easily seen that the presheaf on T :

U 7→ HomT/U(X ′|U , X|U) = HomT (X ′×U,X)

is actually a sheaf on (T,CT ) (notation of example 2.2.6(iii)), so it is an object of T , denoted :

HomT (X ′, X).


The functor :T → T : X 7→HomT (X ′, X)

is right adjoint to the functor T → T : Y 7→ Y ×X ′, so it is an internal Hom functor for X ′.If f : T → S is a morphism of topoi, and Y ∈ Ob(S), we have a natural isomorphism in S :

(2.3.1) HomS(Y, f∗X)∼→ f∗HomT (f ∗Y,X)

which, on every U ∈ Ob(S), induces the natural bijection :

HomS(Y ×U, f∗X)∼→ HomT (f ∗Y ×f ∗U,X)

given by the adjunction (f ∗, f∗). By general nonsense, from (2.3.1) we derive a natural mor-phism in S :

f∗HomT (X ′, X)→HomS(f∗X′, f∗X)

and in T :

f ∗HomS(Y ′, Y )ϑf−→HomT (f ∗Y ′, f ∗Y ) for any Y, Y ′ ∈ Ob(S).

Moreover, if g : U → T is another morphism of topoi, the diagram :

(2.3.2)

g∗f ∗HomS(Y ′, Y )g∗ϑf //

ϑfg **UUUUUUUUUUUUUUUUg∗HomT (f ∗Y ′, f ∗Y )

ϑgttiiiiiiiiiiiiiiiii

HomU(g∗f ∗Y ′, g∗f ∗Y )

commutes, up to a natural isomorphism.

2.3.3. Let A be any object of T , and (X,µX) a left A-module for the tensor category structureon T as in (2.3); for every object U of T we obtain a left A|U -module (on the topos T/U : seeexample 2.2.6(iii)), by the rule :

(X,µX)|U := (X|U , µX × 1U).

If (X ′, µX′) is another left A-module, it is easily seen that the presheaf on T :

U 7→ HomA|U -Modl((X,µX)|U , (X′, µX′)|U)

is actually a sheaf for the canonical topology, so it is an object of T , denoted :

HomAl((X,µX), (X ′, µX′))

(or just HomAl(X,X′), if the notation is not ambiguous). The same considerations can be

repeated for the sets of morphisms of right B-modules, and of (A,B)-bimodules, so one getsobjects HomBr(X,X

′) and Hom(A,B)(X,X′). By a simple inspection, we see that these

objects are naturally isomorphic to the objects denoted in the same way in (1.2.14), so thenotation is not in conflict with loc.cit.; it also follows that HomAl(X,X

′) is the equalizer oftwo morphisms in T :

HomT (X,X ′) //// HomT (A×X,X ′) .

In the same vein, let A,B,C ∈ Ob(T ) be any three objects, S an (A,B)-bimodule, S ′ a(C,B)-bimodule, and S ′′ a (C,A)-bimodule. Then the (C,B)-bimodule HomBr(S, S

′) and the(C,B)-bimodule S ′′ ⊗A S (see (1.2.17)) are the sheaves on (T,CT ) associated to the presheafgiven by the rules : U 7→ HomB|U,r(S|U , S

′|U), and respectively : U 7→ S ′(U) ⊗M(U) S(U) for

every object U of T . Furthermore, the general theory of monoids, their modules and their tensorproducts, developed in section 1.2 is available in the present situation, so we have a well definednotion of T -monoid (see example 1.2.21(ii) and remark 1.2.24). Via the equivalence (2.2.1), aT -monoid M is also the same as a sheaf of monoids M on the site (T,CT ), and a left (resp.


right, resp. bi-) M -module is the same as the datum of a sheaf S in (T,CT ), such that S(U) isa left (resp. right, resp. bi-) M(U)-module, for every object U of T .

2.3.4. Let f : T1 → T2 be a morphism of topoi, A1 an object of T1, and (X,µX) a left A1-module. Since f∗ is left exact, we have a natural isomorphism : f∗(A1 × X)

∼→ f∗A1 × f∗X ,so we obtain a left f∗A1-module :

f∗(X,µX) := (f∗X, f∗µX)

which we denote just f∗X , unless the notation is ambiguous. Likewise, since f ∗ is left exact,from any object A2 of T2, and any left A2-module (Y, µY ), we obtain a left f ∗A2-module :

f ∗(Y, µY ) := (f ∗Y, f ∗µY ).

The same considerations apply of course, also to right modules and to bimodules. Furthermore,let Ai, Bi, Ci be three objects of Ti (for i = 1, 2); since f∗ is left exact, for any (A1, B1)-bimodule X and any (C1, A1)-bimodule X ′ we have a natural morphism of (f∗C1, f∗B1)-bimodules :

f∗X′ ⊗f∗A1 f∗X → f∗(X

′ ⊗A1 X)

and since f ∗ is exact, for any (A2, B2)-bimodule Y and any (C2, A2)-bimodule Y ′ we have anatural isomorphism :

f ∗Y ′ ⊗f∗A2 f∗Y

∼→ f ∗(Y ′ ⊗A2 Y )

of (f ∗C2, f∗B2)-bimodules.

2.3.5. The constructions of the previous paragraphs also apply to presheaves on T : this can beseen, e.g. as follows. Pick a universe V such that T is V-small; then T∧V is a V-topos. Hence,if A is any V-presheaf on T , and X,X ′ ∈ Ob(T∧V ) two left A-modules, we may constructHomAl(X,X

′) as an object in T∧V . Now, if A,X,X ′ lie in the full subcategory T∧U of T∧V , it iseasily seen that also HomAl(X,X

′) lies in T∧U . Likewise, if A,B,C are two U-presheaves onT , we may define X ′ ⊗A X in T∧U , for any (A,B)-bimodule X and (C,A)-bimodule X ′, andthis tensor product will still be left adjoint to the Hom-functor for presheaves on T .

Moreover, we have a natural morphism of topoi iV : (T,CT )∼V → T∧V , given by the forgetfulfunctor (and its left adjoint F 7→ F a) (see example 2.2.6(i)). The restriction of iV∗ to the fullsubcategory T∧U factors through the inclusion T → (T,CT )∼V , therefore, the discussion of (2.3.4)specializes to show that, for every U-presheaf A on T , and every A-module X ∈ Ob(T∧U ),the object Xa ∈ Ob(T ) is naturally an Aa-module. Also, for any (A,B)-bimodule X and(C,A)-bimodule X ′, such that A,B,C,X,X ′ are U-small, we have a natural isomorphism of(Ca, Ba)-bimodules :

X ′a ⊗Aa Xa ∼→ (X ′ ⊗A X)a.

The following definition gathers some further notions – specific to monoids over a topos – whichshall be used in this work.

Definition 2.3.6. Let T be a topos, M a T -monoid, S a left (resp. right, resp. bi-) M -module.(i) S is said to be of finite type, if there exists a covering family (Uλ → 1T | λ ∈ Λ) of the

final object of T , and for every λ ∈ Λ an integer nλ ∈ N and an epimorphism of left(resp. right, resp. bi-) M |Uλ-modules : M⊕nλ

|Uλ → S|Uλ .(ii) S is finitely presented, if there exists a covering family (Uλ → 1T | λ ∈ Λ) of the

final object of T , and for every λ ∈ Λ integers mλ, nλ ∈ N and morphisms fλ, gλ :M⊕mλ|Uλ → M⊕nλ

|Uλ whose coequalizer – in the category of left (resp. right, resp. bi-)M |Uλ-modules – is isomorphic to S|Uλ .

(iii) S is said to be coherent, if it is of finite type, and for every object U in T , everysubmodule of finite type of S|U is finitely presented.


(iv) S is said to be invertible, if there exists a covering family (Uλ → 1T | λ ∈ Λ), andfor every λ ∈ Λ, an isomorphism M |Uλ

∼→ S|Uλ of left (resp. right, resp. bi-) M |U -modules. (Thus, every invertible module is finitely presented.)

(v) An ideal I ⊂ M is said to be invertible, (resp. of finite type, resp. finitely presented,resp. coherent) if it is such, when regarded as an M -bimodule.

Example 2.3.7. (i) Take again T = Set. Then an M -module S is of finite type if and only ifthere exists a finite subset Σ ⊂ S, such that S = M · Σ, with obvious notation. In this case,we say that Σ is a finite system of generators of S (and we also say that S is finitely generated;likewise, an ideal of finite type is also called finitely generated). We say that S is cyclic, ifS = M · s for some s ∈ S.

(ii) If S and S ′ are two M -modules, the coproduct S ⊕ S ′ is the disjoint union of S andS ′, with scalar multiplication given by the disjoint union of the laws µS and µS′ . The productS × S ′ is the cartesian product of the underlying sets, with scalar multiplication given by therule : x · (s, s′) := (x · s, x · s′) for every x ∈M , s,∈ S, s′ ∈ S ′.

For future use, let us also make the :

Definition 2.3.8. Let T be any topos, P a T -monoid, and (N,+, 0) any monoid.(i) We say that P is N -graded, if it admits a morphism of monoids π : P → NT , where

NT is the constant sheaf of monoids arising from N (the coproduct of copies of thefinal object 1T indexed by N ). For every n ∈ N we let P n := π−1(nT ), the preimageof the global section corresponding to n. Then

P =∐n∈N

P n

the coproduct of the objects P n, and the multiplication law of P restricts to a mapP n × Pm → P n+m, for every n,m ∈ N . Especially, each P n is a P 0-module, and Pis also the direct sum of the P n, in the category of P 0-modules. The morphism π iscalled the grading of P .

(ii) In the situation of (i), let S be a left (resp. right, resp. bi-) P -module. We say that S isN -graded, if it admits a morphism of P -modules πS : S → NT , where NT is regardedas a P -bimodule via the grading π of P . Then S is the coproduct S =

∐n∈N Sn, where

Sn := π−1S (nT ), and the scalar multiplication of S restricts to morphisms P n × Sm →

Sn+m, for every n,m ∈ N . The morphism πS is called the grading of S.(iii) A morphism P → Q of N -graded T -monoids is a morphism of monoids that respects

the gradings, with obvious meaning. Likewise one defines morphisms of N -gradedP -modules.

Example 2.3.9. Take T = Set, and let M be any commutative monoid. Then we claim thatthe only invertible object in the tensor category M -Modl is M ; i.e. if S and S ′ are any two(M,M)-bimodules, then S ⊗M S ′ 'M if and only if S and S ′ are both isomorphic to M .

Indeed, let ϕ : S ⊗M S ′∼→ M be an isomorphism, and choose s0 ∈ S, s′0 ∈ S ′ such that

ϕ(s0 ⊗ s′0) = 1. Consider the morphisms of left M -modules :

Mα−→ S

β−→M Mα′−−→ S ′

β′−−→M

such that :

α(m) = m · s0 β(s) = ϕ(s⊗ s′0) α′(m) = m · s′0 β′(s′) = ϕ(s0 ⊗ s′)for every m ∈ M , s ∈ S, s′ ∈ S ′; we notice that β α = 1M = β′ α′. There follows naturalmorphisms :

S ′γ−→M ⊗M S ′

α⊗MS′−−−−→ S ⊗M S ′β⊗MS′−−−−→M ⊗M S ′

γ−1

−−→ S ′


whose composition is the identity 1S′ . However, it is easily seen that ϕ (α ⊗M S ′) γ = β′

and γ−1 (β ⊗M S ′) ϕ−1 = α′, thus α′ β′ = 1S′ , hence both α′ and β′ are isomorphisms,and the same holds for α and β.

Example 2.3.10. Let M be a T -monoid, and L a M -bimodule. For every n ∈ N, let L ⊗n :=L ⊗M · · · ⊗M L , the n-fold tensor power of L . The N-graded M -bimodule

Tens•ML :=∐n∈N

L ⊗n

is naturally a N-graded T -monoid, with composition law induced by the natural morphismsL ⊗n ⊗M L ⊗m ∼→ L ⊗n+m, for every n,m ∈ N. (Here we set L ⊗0 := M .) If L is invertible,Tens•ML is a commutative N-graded T -monoid, which we also denote Sym•ML .

Remark 2.3.11. (i) Let f : T → S be a morphism of topoi, M := (M,µM) a T -semigroup,and N := (N,µN) a S-semigroup. Then clearly f∗M := (f∗M, f∗µM) is a S-semigroup, andf ∗N := (f ∗N, f ∗µN) is a T -semigroup.

(ii) Furthermore, if 1M : 1T → M (resp. 1N : 1S → N ) is a unit for M (resp. for N ), thennotice that f∗1T = 1S (resp. f ∗1S = 1T ), since the final object is the empty product; it followsthat f∗1M (resp. f ∗1S) is a unit for f∗M (resp. for f ∗N ).

(iii) Obviously, if M (resp. N ) is commutative, the same holds for f∗M (resp. f ∗N ).(iv) If X is a left M -module, then f∗X is a left f∗M -module, and if Y is a left N -module,

then f ∗Y is a left f ∗N -module. The same holds for right modules and bimodules.(v) Moreover, let εM : f ∗f∗M → M (resp. ηN : N → f∗f

∗N ) be the counit (resp. unit) ofadjunction. Then the counit (resp. unit) :

εX : f ∗f∗X → X(εM ) (resp. ηY : Y → f∗f∗Y(ηN ))

is a morphism of f ∗f∗M -modules (resp. of N -modules) (notation of (1.2.26)). (Details left tothe reader.)

(vi) Let ϕ : f ∗N →M be a morphism of T -monoids. Then the functor

N -Modl →M -Modl : Y 7→M ⊗f∗N f ∗Yis left adjoint to the functor :

M -Modl → N -Modl : X 7→ f∗X(ηN ).

(And likewise for right modules and bimodules : details left to the reader.)(vii) The considerations of (2.3.5) also apply to monoids : we get that, for any presheaf of

monoids M := (M,µM , 1M) on T , the datum Ma := (Ma, µaM , 1aM) is a T -monoid, and we

have a well defined functor :

M -Modl →Ma-Modl X 7→ Xa.

(And as usual, the same applies to right modules and bimodules.)

2.3.12. Let T be a topos, U any object of T , and M a T -monoid. As a special case of re-mark 2.3.11(i), we have the T/U -monoid j∗UM = M |U , and if we take ϕ := 1j∗UM in remark2.3.11(vi), we deduce that the functor

j∗U : M -Modl →M |U -Modl Y 7→ Y|U

admits the right adjoint jU∗. Now, suppose that X → U is any left M |U -module. The scalarmultiplication of X is a U -morphism µX : M × X → X and jU !µX is the same morphism,seen as a morphism in T (notation of example 2.2.6(iii)). In other words, jU ! induces a faithfulfunctor on left modules, also denoted :

jU ! : M |U -Modl →M -Modl.


It is easily seen that this functor is left adjoint to the foregoing functor j∗U . Especially, thisfunctor is right exact; it is not generally left exact, since it does not preserve the final object(unless U = 1T ). However, it does commute with fibre products, and therefore transformsmonomorphisms into monomorphisms. All this holds also for right modules and bimodules.

2.3.13. Let T be any category as in example 1.2.21(i), denote by 1T a final object of T , and byM any T -monoid. A pointed left M -module is a datum

(S, 0S)

consisting of a left M -module S and a morphism of M -modules 0S : 1T → S, where 0 isthe final object of M -Modl. Often we shall write S instead of (S, 0S), unless this may giverise to ambiguities. As usual, a morphism ϕ : S → T of pointed modules is a morphism ofM -modules, such that 0T = ϕ 0S . In other words, the resulting category is just 0/M -Modl,and shall be denoted M -Modl.

Likewise one may define the category M -Modr of right M -modules, and (M,N)-Modof pointed bimodules, for given T -monoids M and N .

Remark 2.3.14. Let T be a category as in remark 1.2.24.(i) For reasons that will become readily apparent, for many purposes the categories of pointed

modules are more useful than the non-pointed variant of (1.2.22). In any case, we have a faithfulfunctor :

(2.3.15) M -Modl →M -Modl S 7→ S := (S ⊕ 0, 0S)

where 0S : 0 → S ⊕ 0 is the obvious inclusion map. Thus, we may – and often will, withoutfurther comment – regard any M -module as a pointed module, in a natural way. (The same canof course be repeated for right modules and bimodules.)

(ii) In turn, when dealing with pointed M -modules, things often work out nicer if M itselfis a pointed T -monoid. The latter is the datum (M, 0M) of a T -monoid M and a morphism ofM -modules 0M : 0 → M . A morphism of pointed T -monoids is of course just a morphismf : M → M ′ of T -monoids, such that f 0M = 0M ′ . As customary, we shall often just writeM instead of (M, 0M), unless we wish to stress that M is pointed.

(iii) Let (M, 0M) be a pointed T -monoid; a pointed left (M, 0M)-module is a pointed leftM -module S, such that 0 · s = 0 for every s ∈ S. A morphism of pointed left (M, 0M)-modules is just a morphism of pointed leftM -modules. As usual, these gadgets form a category(M, 0M)-Modl. Similarly we have the right and bi-module variant of this definition.

(iv) The forgetful functor from the category of pointed T -monoids to the category of T -monoids, admits a left adjoint :

M 7→ (M, 0M).

Namely, M is the M -module M ⊕ 0, the zero map 0M : 0→M ⊕ 0 is the obvious inclusion,and the scalar multiplicationM×M →M is extended to a multiplication law µ : M×M →M in the unique way for which (M, µ, 0M0) is a pointed monoid. The unit of adjunctionM →M is the obvious inclusion map.

(v) If M is a (non-pointed) monoid, the restriction of scalars

(M, 0M)-Modl →M -Modl

is an isomorphism of categories. Namely, any pointed left M -module S is naturally a pointedleft M-module : the given scalar multiplication M ×S → S extends to a scalar multiplicationM × S → S whose restriction 0 × S → S factors through the zero section 0S (and likewisefor right modules and bimodules).

(vi) Let T be a topos. The notions introduced thus far for non-pointed T -monoids, also admitpointed variants. Thus, a pointed module (S, 0S) is said to be of finite type if the same holds for


S, and S is finitely presented if, locally on T , it is the coequalizer of two morphisms betweenfree M -modules of finite type.

Example 2.3.16. Take T := Set, and let M be any monoid; then a pointed left M -module isjust a left M -module S endowed with a distinguished zero element 0 ∈ S, such that m · 0 = 0for every m ∈ M . A morphism ϕ : S → S ′ of pointed left M -modules is just a morphism ofleft M -modules such that ϕ(0) = 0 (and similarly for right modules and bimodules.)

Likewise, a pointed monoid is endowed with a distinguished zero element, denoted 0 as usual,such that 0 · x = 0 for every x ∈M .

Remark 2.3.17. Let T be a category as in remark 1.2.24, and M a T -monoid.(i) Regardless of whether M is pointed or not, the category M -Modl is also complete

and cocomplete; for instance, if (S, 0S) and (S ′, 0S′) are two pointed modules, the coproduct(S ′′, 0S′′) := (S, 0S) ⊕ (S ′, 0S′) is defined by the push-out (in the category M -Modl) of thecocartesian diagram :

0⊕ 00S⊕0S′ //

S ⊕ S ′

0

0S′′ // S ′′.

Likewise, if ϕ′ : S ′ → S and ϕ′′ : S ′′ → S are two morphisms in M -Modl, the fibre productS ′ ×S S ′′ in the category M -Modl is naturally pointed, and represents the fibre product in thecategory of pointed modules. All this holds also for right modules and bimodules.

(ii) The forgetful functor M -Modl → T := 1T/T to the category of pointed objects ofT , commutes with all limits, since it is a right adjoint; it also commutes with all colimits. Thisforgetful functor admits a left adjoint, that assigns to any Σ ∈ Ob(T ) the free pointedM -moduleM (Σ) . If M is pointed, the latter is defined as the push-out in the cocartesian diagram

1T × Σ //

M × Σ

1T // M (Σ)

and if M is not pointed, one defines it via the equivalence of remark 2.3.14(v) : by a simpleinspection we find that in this case M (Σ) = (M (Σ)), where M (Σ) is the free (unpointed)M -module, as in remark 1.2.24(iii).

Notice as well that the forgetful functors T → T andM -Modl →M -Modl both commutewith all connected colimits, hence the same also holds for the forgetful functorM -Modl → T .(See definition 1.1.37(vii).) The same can be repeated for right modules and bimodules.

(iii) Moreover, if ϕ : S → S ′ is any morphism in M -Modl, we may define Kerϕ andCokerϕ (in the categoryM -Modl); namely, the kernel is the limit of the diagram S

ϕ−→ S ′ ← 0

and the cokernel is the colimit of the diagram 0 ← Sϕ−→ S ′. Especially, if S is a submodule of

S ′, we have a well defined quotient S ′/S of pointed left M -modules. Furthermore, we say thata sequence of morphisms of pointed left M -modules :

0→ S ′ϕ−→ S

ψ−→ S ′′ → 0

is right exact, if ψ induces an isomorphism Cokerϕ∼→ S ′′; we say that it is left exact, if ϕ

induces an isomorphism S ′∼→ Kerψ, and it is short exact if it is both left and right exact.

(Again, all this can be repeated also for right modules and bimodules.)

Example 2.3.18. Take T = Set, and let M be a pointed or not-pointed monoid. Then theargument from example 1.2.27 can be repeated for the free pointed M -modules : if Σ is any set,


we haveM (Σ) ⊗M 1

∼→ 1(Σ) = Σ

where Σ is the disjoint union of Σ and the final object of Set (a set with one element). Hence,the cardinality of Σ is an invariant, called the rank of the free pointed M -module M (Σ) , anddenoted rkMM

(Σ) .

2.3.19. Let T be a topos, (M, 0M), (N, 0N) and (P , 0P ) three pointed T -monoids, S, (resp. S ′)a pointed (M,N)-bimodule (resp. (P ,N)-bimodule); we denote

Hom(N,0N )r(S, S′)

the set of all morphisms of pointed right N -modules S → S ′. As usual, the presheaf

Hom(N,0N )r(S, S′) : U 7→ Hom(N,0N )r|U (S|U , S

′|U)

(with obvious notation) is a sheaf on (T,CT ), hence it is represented by an object of T . Indeed,this object is also the fibre product in the cartesian diagram :

Hom(N,0N )r(S, S′) //

HomNr(S, S′)

0∗S

HomNr(0, 0)0S′∗ // HomNr(0, S

′).

Especially, Hom(N,0N )r(S, S′) is naturally a (P ,M)-bimodule, and moreover, it is pointed : its

zero section represents the unique morphism S → S ′ which factors through 0.Notice also that, for every pointed (P ,M)-bimodule S ′′, the tensor product S ′′ ⊗M S is

naturally pointed, and as in the non-pointed case, the functor

(2.3.20) (P ,M)-Mod → (P ,N)-Mod : S ′′ 7→ S ′′ ⊗M S

is left adjoint to the functor

(P ,N)-Mod → (P ,M)-Mod : S ′ 7→Hom(N,0N )r(S, S′).

By general nonsense, the functor (2.3.20) is right exact; especially, for any right exact sequenceT ′ → T → T ′′ → 0 of pointed (P ,M)-bimodules, the induced sequence

T ′ ⊗M S → T ⊗M S → T ′′ ⊗M S → 0

is again right exact.

Remark 2.3.21. Suppose M , N and P are non-pointed T -monoids, S is a (M,N)-bimoduleand S ′′ a (P ,M)-bimodule.

(i) If S and S ′′ are pointed, one may define a tensor product S ′′ ⊗M S in the category(P ,N)-Mod, if one regards S as a pointed (M, N)-bimodule, and S ′′ as a (P ,M)-bimodule as in remark 2.3.14(v); then one sets simply S ′′ ⊗M S := S ′′ ⊗M S, which is thenviewed as a pointed (P ,N)-bimodule. In this way one obtains a left adjoint to the correspondinginternal Hom-functor HomN from pointed (P ,N)-bimodules to pointed (P ,M)-bimodules(details left to the reader).

(ii) Finally, if neither S nor S ′′ is pointed, notice the natural isomorphism :

(S ′′ ⊗M S)∼→ S ′′ ⊗M S in the category (P ,N)-Mod.

Definition 2.3.22. In the situation of (2.3.19), let P = N := (1T ), and notice that – with thesechoices of P and N – a pointed (P ,M)-bimodule (resp. a pointed (M,N)-bimodule) is just aright pointed M -module (resp. a left pointed M -module), and a pointed (P ,N)-module is justa pointed object of T .


(i) We say that S is a flat pointed leftM -module (or briefly, that S isM -flat), if the functor(2.3.20) transforms short exact sequences of right pointed M -modules, into short exactsequences of pointed T -objects. Likewise, we define flat pointed right M -modules.

(ii) Let ϕ : M → M ′ be a morphism of pointed T -monoids. We say that ϕ is flat, if M ′ isa flat left M -module, for the module structure induced by ϕ.

Remark 2.3.23. (i) In the situation of remark 2.3.11(i), suppose that M := (M, 0M) is apointed T -monoid and N := (N, 0N) a pointed S-monoid. By arguing as in remark 2.3.11(ii),we see that f ∗N := (f ∗N, f ∗0N) is a pointed T -monoid, and f∗M := (f∗M, f∗0M) is a pointedS-monoid.

(ii) Likewise, if (X, 0X) is a pointed left M -module, and (Y, 0Y ) a pointed left N -module,the f∗(X, 0X) := (f∗X, f∗0X) is a pointed f∗M -module, and f ∗(Y, 0Y ) := (f ∗Y, f ∗0Y ) is apointed f ∗N -module (and likewise for right modules and bimodules).

(iii) Also, just as in remark 2.3.11(vii), the associated sheaf functor F 7→ F a transformsa presheaf M of pointed monoids on T , into a pointed T -monoid Ma, and sends pointed left(resp. right, resp. bi-) M -modules to pointed left (resp. right, resp. bi-) Ma-modules.

(iv) Moreover, if ϕ : f ∗N → M is a morphism of pointed T -monoids, then – in view of thediscussion of (2.3.19) – the adjunction of remark 2.3.11(vi) extends to pointed modules : weleave the details to the reader.

(v) Furthermore, in the situation of (2.3.12), we may also define a functor

jU ! : M |U -Modl →M -Modl

which will be a left adjoint to j∗U . Indeed, let (X, 0X) be a left pointed M |U -module; the functorfrom (2.3.12) yields a morphism jU !0X of (non-pointed) M -modules, and we define jU !(X, 0X)

to be the push-out (in the category M -Modl) of the diagram 0 ← jU !0XjU !0X−−−−→ jU !X . The

latter is endowed with a natural morphism 0 → jU !(X, 0X), so we have a well defined pointedleftM -module. We leave to the reader the verification that the resulting functor, called extensionby zero, is indeed left adjoint to the restriction functor.

(vi) It is convenient to extend definition 2.3.22 to non-pointed modules and monoids :namely, if S is a non-pointed left M -module, we shall say that S is flat, if the same holds forthe pointed left M-module S. Likewise, we say that a morphism ϕ : M → N of non-pointedT -monoids is flat, if the same holds for ϕ.

Lemma 2.3.24. Let T be a topos, U any object of T , and denote by i∗ : CU → T the inclusionfunctor of the complement of U in T (see example 2.2.6(iv)). Let also M , N , P be three pointedT -monoids. Then the following holds :

(i) The functor jU ! of extension by zero is faithful, and transforms exact sequences ofpointed left M |U -modules, into exact sequences of pointed left M -modules (and like-wise for right modules and bimodules).

(ii) For every pointed (M,N)-bimodule S and every pointed (P |U ,M |U)-bimodule S ′, thenatural morphism of pointed (P ,N)-modules

jU !(S′ ⊗M|U S|U)→ jU !S

′ ⊗M S

is an isomorphism.(iii) If S is flat pointed left M |U -module, then jU !S is a flat pointed left M -module (and

likewise for right modules).(iv) For every pointed (M,N)-bimodule S, and every pointed (i∗P , i∗M)-bimodule S ′, the

natural morphism of pointed (P ,N)-bimodules

i∗S′ ⊗M S → i∗(S

′ ⊗i∗M i∗S)

is an isomorphism.


(v) If S is a flat pointed left i∗M -module, then i∗S is a flat left pointed M -module (andlikewise for right modules).

(vi) If S is a flat pointed left M -module, then S|U is a flat left pointed M |U -module.

Proof. (i): Let us show first that jU ! is faithful. Indeed, suppose that ϕ, ψ : S → S ′ are twomorphisms of left pointed M |U -modules, such that jU !ϕ = jU !ψ. We need to show that ϕ = ψ.Let p : S ′ → S ′′ be the coequalizer of ϕ and ψ; then jU !p is the coequalizer of jU !ϕ and jU !ψ(since jU ! is right exact); hence we are reduced to showing that a morphism p : S ′ → S ′′ is anisomorphism if and only if the same holds for jU !p. This follows from remark 1.2.24(ii) and thefollowing more general :

Claim 2.3.25. Let ϕ : X → X ′, A → X , A → B be three morphisms in T . Then ϕ is amonomorphism (resp. an epimorphism) if and only if the same holds for the induced morphismϕqA B : X qA B → X ′ qA B.

Proof of the claim. We may assume that T = C∼ for some small site C := (C , J). Thenϕ qA B = (iϕ qiA iB)a, where i : C∼ → C ∧ is the forgetful functor. Since the functorF 7→ F a is exact, we are reduced to the case where T = C ∧, and in this case the assertion canbe checked argumentwise, i.e. we may assume that T = Set, where the claim is obvious. ♦

Next, we already know that jU ! transforms right exact sequences into right exact sequences.To conclude, it suffices then to check that jU ! transforms monomorphisms into monomor-phisms.To this aim, we apply again remark 1.2.24(ii) and claim 2.3.25.

(ii) is proved by general nonsense, and (iii) is an immediate consequence of (i) and (ii) : weleave the details to the reader.

(iv): By (2.3.4), we have j∗U(i∗S′ ⊗M S) ' 0 ⊗j∗UM j∗US ' 1T/U , hence i∗S ′ ⊗M S ∈

Ob(CU). Notice now that, for every object X of CU , the counit of adjunction i∗i∗X → X is anisomorphism (proposition 1.1.11(ii)); by the triangular identities of (1.1.8), it follows that thesame holds for the unit of adjunction i∗X → i∗i

∗i∗X . Especially, the natural morphism :

i∗S′ ⊗M S → i∗i

∗(i∗S′ ⊗M S)

∼→ i∗(i∗i∗S

′ ⊗i∗M i∗S)∼→ i∗(S

′ ⊗i∗M i∗S).

is an isomorphism. The latter is the morphism of assertion (iv).(v) follows easily from (iv) and its proof.(vi): In view of (i), it suffices to show that the functor S ′ 7→ jU !(S

′ ⊗M|U S|U) transformsexact sequences into exact sequences. The latter follows easily from (ii).

Proposition 2.3.26. Let P(T,M, S) be the property : “S is a flat pointed left M -module” (fora monoid M on a topos T ). Then P can be checked on stalks. (See remark 2.2.14(ii).)

Proof. Suppose first that Sξ is a flat leftM ξ-module for every ξ in a conservative set of T -points;let ϕ : X → X ′ be a monomorphism of pointed right M -modules; by (2.3.4) we have a naturalisomorphism

(ϕ⊗M S)ξ∼→ ϕξ ⊗Mξ

Sξ

in the category of pointed sets, and our assumption implies that these morphisms are monomor-phisms. Since an arbitrary product of monomorphisms is a monomorphism, remark 1.1.38(iii)shows that ϕ⊗M S is also a monomorphism, whence the contention.

Next, suppose that S is a flat pointed left M -module. We have to show that the functor

(2.3.27) S ′ 7→ S ′ ⊗MξSξ

from pointed right M ξ-modules to pointed sets, preserves monomorphisms.However, let (U, ξU , ωU) be any lifting of ξ (see (2.2.11)); in view of (2.3.4), we have

PU(S ′) := (ξ∗UξU∗S′)⊗Aξ Sξ ' (ξ∗UξU∗S

′)⊗ξ∗UA|U ξ∗US|U ' ξ∗U(ξU∗S

′ ⊗A|U S|U)


and then lemma 2.3.24(vi) implies that the functor S ′ → PU(S ′) preserves monomorphisms.By lemma 2.2.23, remark 2.2.18(i) and (2.2.13), the functor (2.3.27) is a filtered colimit of suchfunctors PU , hence it preserves monomorphisms as well.

2.3.28. We wish now to introduce a few notions that pertain to the special class of commutativeT -monoids. When T = Set, these notions are well known, and we wish to explain quickly thatthey generalize without problems, to arbitrary topoi.

To begin with, for every category T as in example 1.2.21(i), we denote by MndT (resp.MndT) the category of commutative unitary non-pointed (resp. pointed) T -monoids; in caseT = Set, we shall usually drop the subcript, and write just Mnd (resp. Mnd). Notice that, ifM is any (pointed or not pointed) commutative T -monoid, every left or rightM -module is aM -bimodule in a natural way, hence we shall denote indifferently by M -Mod (resp. M -Mod)the category of non-pointed (resp. pointed) left or right M -modules.

The following lemma is a special case of a result that holds more generally, for every ”alge-braic theory” in the sense of [11, Def.3.3.1] (see [11, Prop.3.4.1, Prop.3.4.2]).

Lemma 2.3.29. Let T be a topos. We have :(i) The category MndT admits arbitrary limits and colimits.

(ii) In the category MndT , filtered colimits commute with all finite limits.(iii) The forgetful functor ι : MndT → T that assigns to a monoid its underlying object of

T , commutes with all limits, and with all filtered colimits.

Proof. (iii): Commutation with limits holds because ι admits a left adjoint : namely, to an objectΣ of T one assigns the free monoid N(Σ)

T generated by Σ, defined as the sheaf associated to thepresheaf of monoids

U 7→ N(Σ(U)) for every U ∈ Ob(T )

where N is the additive monoid of natural numbers (see remark 2.3.11(vii)). One verifies easilythat this T -monoid represents the functor

M 7→ HomT (Σ,M) MndT → Set.

Moreover, if I is any small category, and F : I → MndT any functor, one checks easilythat the limit of ι F can be endowed with a unique composition law (indeed, the limit of thecomposition laws of the monoids Fi), such that the resulting monoid represents the limit of F .

A similar argument also shows that MndT admits arbitrary filtered colimits, and that ι com-mutes with filtered colimits. It is likewise easy to show that the product of two T -monoids Mand N is also the coproduct of M and N . To complete the proof of (i), it suffices thereforeto show that any two maps f, g : M → N admit a coequalizer; the latter is obtained as thecoequalizer N ′ (in the category T ) of the two morphisms :

M ×NµN(g×1N )

//µN(f×1N )

// N.

We leave to the reader the verification that the composition law of N descends to a (necessarilyunique) composition law on N ′.

(ii) follows from (iii) and the fact that the same assertion holds in T (remark 2.2.2(iii)).

Example 2.3.30. (i) For instance, if T = Set, the product M1 ×M2 of any two commutativemonoids is representable in Mnd; its underlying set is the cartesian product of M1 and M2, andthe composition law is the obvious one.

(ii) As usual, the kernel Kerϕ (resp. cokernel Cokerϕ) of a map of T -monoids ϕ : M → Nis defined as the fibre product (resp. push-out) of the diagram of T -monoids

Mϕ−→ N ← 1T (resp. 1T ←M

ϕ−→ N ).


Especially, if M ⊂ N , one defines in this way the quotient N/M .(iii) Also, if T = Set, and ϕ1 : M → M1, ϕ2 : M → M2 are two maps in Mnd, the

push-out M1 qM M2 can be described as follows. As a set, it is the quotient (M1 ×M2)/∼,where ∼ denotes the minimal equivalence relation such that

(m1,m2 · ϕ2(m)) ∼ (m1 · ϕ1(m),m2) for every m ∈M , m1 ∈M1, m2 ∈M2

and the composition law is the unique one such that the projection M1 ×M2 → M1 qM M2 isa map of monoids. We deduce the following :

Lemma 2.3.31. Let G be an abelian group. The following holds :(i) If ϕ : M → N and ψ : M → G are two morphisms of monoids (in the topos T = Set),

GqM N is the quotient (G×N)/≈, where ≈ is the equivalence relation such that :

(g, n) ≈ (g′, n′) ⇔ (ψ(a) · g, ϕ(b) · n) = (ψ(b) · g′, ϕ(a) · n′) for some a, b ∈M.

(ii) If ϕ : G → M and ψ : G → N are two morphisms of monoids, the set underlyingMqGN is the set-theoretic quotient (M×N)/G for theG-action defined via (ϕ, ψ−1).

(iii) Especially, if M is a monoid and G is a submonoid of M , then the set underlying M/Gis the set-theoretic quotient of M by the translation action of G.

Proof. (i): One checks easily that the relation ≈ thus defined is transitive. Let ∼ be the equiv-lence relation defined as in example 2.3.30(iii). Clearly :

(g, n · ψ(m)) ≈ (g · ϕ(m), n) for every g ∈ G, n ∈ N and m ∈Mhence (g, n) ∼ (g′, n′) implies (g, n) ≈ (g′, n′). Conversely, suppose that (ψ(a) · g, ϕ(b) · n) =(ψ(b) · g′, ϕ(a) · n′) for some g ∈ G, n ∈ N and a, b ∈M . Then :

(g, n) = (g, ϕ(a) · ϕ(a)−1 · n) ∼ (ψ(a) · g, ϕ(a)−1 · n) = (ψ(b) · g′, ϕ(a)−1 · n)

as well as : (g′, n′) = (g′, ϕ(b) · ϕ(a)−1 · n) ∼ (ψ(b) · g′, ϕ(a)−1 · n). Hence (g, n) ∼ (g′, n′)and the claim follows.

(ii) follows directly from example 2.3.30(iii), and (iii) is a special case of (ii).

2.3.32. Let T be a topos. For any T -ring R, we let R-Mod be the category of R-modules(defined in the usual way); especially, we may consider the T -ring ZT (the constant sheaf withvalue Z : see example 2.2.6(v)). Then ZT -Mod is the category of abelian T -groups. Theforgetful functor ZT -Mod→MndT admits a right adjoint :

MndT → ZT -Mod : M 7→M×.

The latter can be defined as the fibre product in the cartesian diagram :

M× //

M ×MµM

1T

1M // M.

For i = 1, 2, let pi : M ×M → M be the projections, and p′i : M× → M the restriction of pi;for every U ∈ Ob(T ), the image of p′i(U) : M×(U) → M(U) consists of all sections x whichare invertible, i.e. for which there exists y ∈ M(U) such that µM(x, y) = 1M . It is easily seenthat such inverse is unique, hence p′i is a monomorphism, p′1 and p′2 define the same subobjectof M , and this subobject M× is the largest abelian T -group contained in M . We say that M issharp, if M× = 1T . The inclusion functor, from the full subcategory of sharp T -monoids, toMndT , admits a left adjoint

M 7→M ] := M/M×.

We call M ] the sharpening of M .


2.3.33. Let S be a submonoid of a commutative T -monoid M , and FS : MndT → Set thefunctor that assigns to any commutative T -monoid N the set of all morphisms f : M → Nsuch that f(S) ⊂ N×. We claim that FS is representable by a T -monoid S−1M .

In case T = Set, one may realize S−1M as the quotient (S ×M)/∼ for the equivalencerelation such that (s1, x1) ∼ (s2, x2) if and only if there exists t ∈ S such that ts1x2 = ts2x1.The composition law of S−1M is the obvious one; then the class of a pair (s, x) is denotednaturally by s−1x. This construction can be repeated on a general topos : letting X := S ×M ,the foregoing equivalence relation can be encoded as the equalizerR of two mapsX×X×S →M , and the quotient under this equivalence relation shall be represented by the coequalizer oftwo other maps R→ X; the reader may spell out the details, if he wishes. Equivalently, S−1Mcan be realized as the sheaf on (T,CT ) associated to the presheaf :

T →Mnd : U 7→ S(U)−1M(U)

(see remark 2.3.11(vii)). The natural morphism M → S−1M is called the localization map.For T = Set, and f ∈M any element, we shall also use the standard notation :

Mf := S−1f M where Sf := fn | n ∈ N.

Lemma 2.3.34. Let f1 : M → N1 and f2 : M → N2 be morphisms of T -monoids, S ⊂ M ,Si ⊂ N i (i = 1, 2) three submonoids, such that fi(S) ⊂ Si for i = 1, 2. Then the naturalmorphism :

(S1 · S2)−1(N1 qM N2)→ S−11 N1 qS−1M S−1

2 N2

is an isomorphism.

Proof. One checks easily that both these T -monoids represent the functor MndT → Set thatassigns to any T -monoid P the pairs of morphisms (g1, g2) where gi : N i → P satisfiesgi(Si) ⊂ P×, for i = 1, 2, and g1 f1 = g2 f2. The details are left to the reader.

2.3.35. The forgetful functor Z-ModT → MndT from abelian T -groups to commutative T -monoids, admits a left adjoint

M 7→Mgp := M−1M.

A commutative T -monoid M is said to be integral if the unit of adjunction M → Mgp is amonomorphism. The functor M 7→ Mgp commutes with all colimits, since all left adjoints do;it does not commute with arbitrary limits (see example 2.3.36(v)).

We denote by Int.MndT the full subcategory of MndT consisting of all integral monoids;when T = Set, we omit the subscript, and write just Int.Mnd. The natural inclusion ι :Int.MndT →MndT admits a left adjoint :

MndT → Int.MndT : M 7→M int.

Namely, M int is the image (in the category T ) of the unit of adjunction M → Mgp. It followseasily that the category Int.MndT is cocomplete, since the colimit of a family (Mλ | λ ∈ Λ)of integral monoids is represented by

(colimλ∈Λ

ιMλ)int.

Likewise, Int.MndT is complete, and limits commute with the forgetful functor to T ; to checkthis, it suffices to show that

L := limλ∈Λ

ι(Mλ)

is integral. However, by lemma 2.3.29(iii) we have L ⊂∏

λ∈Λ Mλ ⊂∏

λ∈ΛMgpλ , whence the

claim.


Example 2.3.36. (i) Take T = Set; if M is any monoid, and a ∈ M is any element, we saythat a is regular, if the map M → M given by the rule : x 7→ a · x is injective. It is easily seenthat M is integral if and only if every element of M is regular.

(ii) For an arbitrary topos T , notice that the T -monoid Ga associated to a presheaf of groupsG on T , is a T -group : indeed, the condition G× = G implies (Ga)× = Ga, since the functorF 7→ F a is exact. More precisely, for every presheaf M of monoids on T , we have a naturalisomorphism :

(Mgp)a∼→ (Ma)gp for every T -monoid M

since both functors are left adjoint to the forgetful functor from T -groups to presheaves ofmonoids on T .

(iii) It follows from (ii) that a T -monoid M is integral if and only if M(U) is an integralmonoid, for every U ∈ Ob(T ). Indeed, if M is integral, then M(U) ⊂ Mgp(U) for everysuch U , so M(U) is integral. Conversely, by definition Mgp is the sheaf associated to thepresheaf U 7→M(U)gp; now, ifM(U) is integral, we haveM(U) ⊂M(U)gp, and consequentlyM ⊂Mgp, since the functor F 7→ F a is exact.

(iv) We also deduce from (ii) that the functor M 7→ Ma sends presheaves of integralmonoids, to integral T -monoids. Therefore we have a natural isomorphism :

(2.3.37) (M int)a∼→ (Ma)int

as both functors are left adjoint to the forgetful functor from integral T -monoids, to presheavesof monoids on T . In the same vein, it is easily seen that the forgetful functor Int.MndT → Tcommutes with filtered colimits : indeed, (2.3.37) and lemma 2.3.29(iii) reduce the assertionto showing that the colimit of a filtered system of presheaves of integral monoids is integral,which can be verified directly.

(v) Take T = Set, and let ϕ : M → N be an injective map of monoids; if N (hence M ) isintegral, one sees easily that the induced map ϕgp : Mgp → Ngp is also injective. This may fail,when N is not integral : for instance, if M is any integral monoid, and N := M is the pointedmonoid associated to M as in remark 2.3.14(iv), then for the natural inclusion i : M →M wehave igp = 0, since (M)

gp = 1.

Lemma 2.3.38. Let T be a topos, M be an integral T -monoid, and N ⊂ M a T -submonoid.Then M/N is an integral T -monoid.

Proof. In light of example 2.3.36(iv), we are reduced to the case where T = Set. Moreover,since the natural morphismM/N → N−1M/Ngp is an isomorphism, we may assume thatN isan abelian group. Now, notice that (M/N)gp = Mgp/N since the functor P 7→ P gp commuteswith colimits. On the other hand, M/N is the set-theoretic quotient of M by the translationaction of N (lemma 2.3.31(iii)). This shows that the unit of adjunction M/N → (M/N)gp isinjective, as required.

2.3.39. Let M be an integral monoid. Classically, one says that M is saturated, if we have :

M = a ∈Mgp | an ∈M for some integer n > 0.In order to globalize the class of saturated monoid to arbitrary topoi, we make the following :

Definition 2.3.40. Let T be a topos, ϕ : M → N a morphism of integral T -monoids.(i) We say that ϕ is exact if the diagram of commutative T -monoids

Dϕ :

Mϕ //

N

Mgp ϕgp

// Ngp


is cartesian (where the vertical arrows are the natural morphisms).(ii) For any integer k > 0, the k-Frobenius map ofM is the endomorphism kM ofM given

by the rule : x 7→ xk for every U ∈ Ob(T ) and every x ∈ M(U). We say that M isk-saturated, if kM is an exact morphism.

(iii) We say that M is saturated, if M is integral and k-saturated for every integer k > 0.

We denote by Sat.MndT the full subcategory of Int.MndT whose objects are the saturatedT -monoids. As usual, when T = Set, we shall drop the subscript, and just write Sat.Mnd forthis category. The above definition (and several of the related results in section 3.2) is borrowedfrom [74].

Remark 2.3.41. (i) Clearly, when T = Set, definition 2.3.40(iii) recovers the classical no-tion of saturated monoid. Again, for usual monoids, it is easily seen that the forgetful functorSat.Mnd → Int.Mnd admits a left adjoint, that assigns to any integral monoid M its satu-ration M sat. The latter is the monoid consisting of all elements x ∈Mgp such that xk ∈M forsome integer k > 0; especially, the torsion subgroup of Mgp is always contained in M sat. Theeasy verification is left to the reader. Clearly, M is saturated if and only if M = M sat. Moregenerally, the unit of adjunction M →M sat is just the inclusion map.

(ii) For a general topos T , and a morphism ϕ as in definition 2.3.40(i), notice that ϕ is exact ifand only if the induced map of monoids ϕ(U) : M(U)→ N(U) is exact for every U ∈ Ob(T ).Indeed, if Dϕ is cartesian, then the same holds for the induced diagram Dϕ(U) of monoids;since the natural map M(U)gp → Mgp(U) is injective (and likewise for N ), it follows easilythat the diagram of monoids Dϕ(U) is cartesian, i.e. ϕ(U) is exact. For the converse, notice thatDϕ is of the form (hDϕ)a, where h : T → T∧ is the Yoneda embedding, and F 7→ F a denotesthe associated sheaf functor T∧ → (T,CT )∼ = T ; the assumption means that hD is a cartesiandiagram in T∧, hence D is exact in T , since the associated sheaf functor is exact.

(iii) Example 2.3.36(iii) and (ii) imply that a T -monoid M is saturated, if and only if M(U)is a saturated monoid, for every U ∈ Ob(T ). We also remark that, in view of example 2.3.36(ii),the functor F 7→ F a takes presheaves of k-saturated (resp. saturated) monoids, to k-saturated(resp. saturated) T -monoids : indeed, if η : M → Mgp is the unit of adjunction for a presheafof monoids M , then ηa : Ma → (Mgp)a = (Ma)gp is the unit of adjunction for the associatedT -monoid, hence it is clear the functor F 7→ F a preserves exact morphisms.

(iv) It follows easily that the inclusion functor Sat.MndT → Int.MndT admits a leftadjoint, namely the functor

Int.MndT → Sat.MndT : M 7→M sat

that assigns to M the sheaf associated to the presheaf U → M ′(U) := M(U)sat on T (noticethat the functor M 7→ M ′ from presheaves of integral monoids, to presheaves of saturatedmonoids, is left adjoint to the inclusion functor). Just as in example 2.3.36(iv), we deduce anatural isomorphism

(2.3.42) (M sat)a∼→ (Ma)sat for every T -monoid M

since both functors are left adjoint to the forgetful functor from Sat.MndT , to presheaves ofintegral monoids on T .

(v) By the usual general nonsense, the saturation functor commutes with all colimits. More-over, the considerations of (2.3.35) can be repeated for saturated monoids : first, the categorySat.MndT is cocomplete, and arguing as in example 2.3.36(iv), one checks that filtered col-imits commute with the forgetful functor Sat.MndT → T ; next, if F : Λ → Sat.MndT is afunctor from a small category Λ, then for each integer k > 0, the induced diagram of integral


monoids

limΛ

DkF :

limΛF limΛ kF //

limΛF

lim

ΛF gp limΛ k

gpF // lim

ΛF gp

is cartesian; since the natural morphism

(limΛF )gp → lim

ΛF gp

is a monomorphism, it follows easily that the limit of F is saturated, hence Sat.MndT iscomplete, and furthermore all limits commute with the forgetful functor to T .

2.3.43. In view of remark 2.3.11(i,ii,iii), a morphism of topoi f : T → S induces functors :

(2.3.44) f∗ : MndT →MndS f ∗ : MndS →MndT

and one verifies easily that (2.3.44) is an adjoint pair of functors.

Lemma 2.3.45. Let f : T → S be a morphism of topoi, M an S-monoid. We have :

(i) If M is integral (resp. saturated), f ∗M is an integral (resp. saturated) T -monoid.(ii) More precisely, there is a natural isomorphism :

f ∗(M int)∼→ (f ∗M)int (resp. f ∗(M sat)

∼→ (f ∗M)sat, if M is integral).

(iii) If ϕ is an exact morphism of integral S-monoids, then f ∗ϕ is an exact morphism ofintegral T -monoids.

Proof. To begin with, notice that the adjoint pair (f ∗, f∗) of (2.3.44) restricts to a correspondingadjoint pair of functors between the categories of abelian T -groups and abelian S-groups (sincethe condition G = G× for monoids, is preserved by any left exact functor).

There follows a natural isomorphism :

(f ∗M)gp ∼→ f ∗(Mgp) for every S-monoid M

since both functors are left adjoint to the functor f∗ from abelian T -groups to S-monoids. Now,if M is an integral S-module, and η : M → Mgp is the unit of adjunction, it is easily seen thatf ∗η : f ∗M → (f ∗M)gp is also the unit of adjunction. From this and proposition 2.2.5(ii.b), wededuce the assertion concerning f ∗(M int).

By the same token, we get assertion (iii) of the lemma. Especially, if M is saturated, then thesame holds for f ∗M . The assertion concerning f ∗(M sat) follows by the usual argument.

Lemma 2.3.46. (i) The functor f ∗ of (2.3.44) commutes with all finite limits and all colimits.

(ii) Let P(T,M) be the property “M is an integral (resp. saturated) T -monoid” (for atopos T ). Then P can be checked on stalks. (See remark 2.2.14(ii).)

Proof. (i): Concerning finite limits, in light of lemma 2.3.29(iii) we are reduced to the assertionthat f ∗ : S → T is left exact, which holds by definition. Next f ∗ commutes with colimits,because it is a left adjoint.

(ii): A T -monoid M is integral if and only if the unit of adjunction η : M → M int is an iso-morphism. However, (M int)ξ

∼→ (M ξ)int, in view of lemma 2.3.45(ii), and ηξ : M ξ → (M ξ)

int

is the unit of adjunction. The assertion is an immediate consequence. The same argumentapplies as well to saturated T -monoids.


Example 2.3.47. (i) For instance, the unique morphism of topoi Γ : T → Set (see example2.2.6(iii)) induces a pair of adjoint functors :

(2.3.48) MndT →Mnd : M 7→ Γ(T,M) and Mnd→MndT : P 7→ PT

where PT is the constant sheaf of monoids on (T,CT ) with value P .(ii) Specializing lemma 2.3.45(ii) to this adjoint pair, we obtain natural isomorphisms :

(2.3.49) (MT )int ∼→ (M int)T (MT )sat ∼→ (M sat)T

of functors Mnd→ Int.MndT and Int.Mnd→ Sat.MndT . Especially, if M is an integral(resp. saturated) monoid, then the constant T -monoid MT is integral (resp. saturated).

(iii) If ξ is any T -point, notice also that the stalkMT,ξ is isomorphic toM , since ξ is a sectionof Γ : T → Set.

2.3.50. Let T be a topos, R a T -ring. We have a forgetful functor R-Alg → MndT thatassigns to a (unitary, commutative) R-algebra (A,+, ·, 1A) its multiplicative T -monoid (A, ·).If T = Set, this functor admits a left adjoint Mnd → R-Alg : M 7→ R[M ]. Explicitly,R[M ] =

⊕x∈M xR, and the multiplication law is uniquely determined by the rule :

xa · yb := (x · y)ab for every x, y ∈M and a, b ∈ R.

For a general topos T , the above construction globalizes to give a left adjoint

(2.3.51) MndT → R-Alg : M 7→ R[M ].

The latter is the sheaf on (T,CT ) associated to the presheaf U 7→ R(U)[M(U)], for everyU ∈ Ob(T ). The functor (2.3.51) commutes with arbitrary colimits (since it is a left adjoint);especially, if M → M1 and M → M2 are two morphisms of monoids, we have a naturalidentification :

(2.3.52) R[M1 qM M2]∼→ R[M1]⊗R[M ] R[M2].

By inspecting the universal properties, we also get a natural isomorphism :

(2.3.53) S−1R[M ]∼→ R[S−1M ]

for every monoid M and every submonoid S ⊂M .

2.3.54. Likewise, if M is any T -monoid, let R[M ]-Mod denote as usual the category of mod-ules over the T -ring R[M ]; we have a forgetful functor R[M ]-Mod → M -Mod. WhenT = Set, this functor admits a left adjoint M -Mod → R[M ]-Mod : S 7→ R[S]. Explic-itly, R[S] is the free R-module with basis given by S, and the R[M ]-module structure on R[S]is determined by the rule:

xa · sb := µS(x, s)ab for every x ∈M , s ∈ S and a, b ∈ R.

For a general topos T , this construction globalizes to give a left adjoint

M -Mod→ R[M ]-Mod : (S, µS) 7→ R[S]

which is defined as the sheaf associated to the presheaf U 7→ R(U)[S(U)] in T∧.

2.4. Cohomology on a topos. In this section we introduce the cohomology with values in asheaf of (not necessarily abelian) groups over a topos. Then we explain some basic notionsconcerning the points of the etale and Zariski topoi of a scheme, and we conclude with theproof of Hilbert’s theorem 90 (lemma 2.4.26(iv)).

Definition 2.4.1. Let T be a topos, and G a T -group.


(i) A left G-torsor is a left G-module (X,µX), inducing an isomorphism

(µX , pX) : G×X → X ×X(where pX : G×X → X is the natural projection) and such that there exists a coveringmorphism U → 1T in T for which X(U) 6= ∅. This is the same as saying that theunique morphism X → 1T is an epimorphism.

(ii) A morphism of left G-torsors is just a morphism of the underlying G-modules. Like-wise, we define right G-torsors, G-bitorsors, and morphisms between them. We let:

H1(T,G)

be the set of isomorphism classes of right G-torsors.(iii) A (left or right or bi-) G-torsor (X,µX) is said to be trivial, if Γ(T,X) 6= ∅.

Remark 2.4.2. (i) In the situation of definition 2.4.1, notice that H1(T,G) always contains adistinguished element, namely the class of the trivial G-torsor (G, µG).

(ii) Conversely, suppose that (X,µX) is a trivial left G-torsor, and say that σ ∈ Γ(T,X);then we have a cartesian diagram :

Gµσ //

X

1X×σ

G×X(µX ,pX)

// X ×Xwhich shows that (X,µX) is isomorphic to (G, µG) (and likewise for right G-torsors).

(iii) Notice that every morphism f : (X,µX) → (X ′, µX′) of G-torsors is an isomorphism.Indeed, the assertion can be checked locally on T (i.e., after pull-back by a covering morphismU → 1T ). Then we may assume that X admits a global section σ ∈ Γ(T,X), in which caseσ′ := σ f ∈ Γ(T,X ′). Then, arguing as in (ii), we get a commutative diagram :

Gµσ

~~~~~~~~~~ µσ′

AAAAAAAA

Xf // X ′

where both µσ and µσ′ are isomorphisms, and then the same holds for f .(iv) The tensor product of a G-bitorsor and a left G-bitorsor is a left G-torsor. Indeed the

assertion can be checked locally on T , so we are reduced to checking that the tensor product ofa trivial G-bitorsors and a trivial left G-torsor is the trivial left G-torsor, which is obvious.

(v) Likewise, ifG1 → G2 is any morphism of T -groups, and X is a left G1-torsor, it is easilyseen that the base change G2⊗G1 X yields a left G2-torsor (and the same holds for right torsorsand bitorsors). Hence the rule G 7→ H1(T,G) is a functor from the category of T -groups, tothe category of pointed sets. One can check that H1(T,G) is an essentially small set (see [38,Chap.III, §3.6.6.1]).

(vi) Let f : T → S be a morphism of topoi; if X is a left G-torsor, the f∗G-module f∗X isnot necessarily a f∗G-torsor, since we may not be able to find a covering morphism U → 1Ssuch that f∗X(U) 6= ∅. On the other hand, if H a S-group and Y a left H-torsor, then it iseasily seen that f ∗Y is a left f ∗H-torsor.

2.4.3. Let f : T ′ → T be a morphism of topoi, and G a T ′-monoid; we define a U-presheafR1f∧∗ G on T , by the rule :

U 7→ H1(T ′/f ∗U,G|f∗U).

(More precisely, since this set is only essentially small, we replace it by an isomorphic smallset). If ϕ : U → V is any morphism in T , and X is any G|f∗V -torsor, then X ×f∗V f ∗U isa G|f∗U -torsor, whose isomorphism class depends only on the isomorphism class of X; this


defines the map R1f∧∗ G(ϕ), and it is clear that R1f∧∗ G(ϕ ψ) = R1f∧∗ G(ψ) R1f∧∗ G(ϕ), forany other morphism ψ : W → U in T . Finally, we denote by :

R1f∗G

the sheaf on (T,CT ) associated to the presheafR1f∧∗ G. Notice that the objectR1f∗G is pointed,i.e. it is endowed with a natural global section :

τf,G : 1T → R1f∗G

namely, the morphism associated to the morphism of presheaves 1T → R1f∧∗ Gwhich, for everyU ∈ Ob(T ), singles out the isomorphism class τf,G(U) ∈ R1f∧∗ G(U) of the trivialG|f∗U -torsor.

2.4.4. Let g : T ′′ → T ′ be another morphism of topoi, and G a T ′′-group. Notice that :

f∧∗ R1g∧∗G = R1(f g)∧∗G

hence the natural morphism (in T ′∧) R1f∧∗ G → R1f∗G induces a morphism R1(f g)∧∗G →f∧∗ R

1g∗G in T∧, which yields, after taking associated sheaves, a morphism in T :

(2.4.5) R1(f g)∗G→ f∗R1g∗G.

One sees easily that this is a morphism of pointed objects of T , i.e. the image of the globalsection τfg,G under this map, is the global section f∗τg,G.

Next, suppose that U ∈ Ob(T ) and X is any g∗G|f∗U -torsor (on T ′/f ∗U ); we may formthe g∗g∗G|g∗f∗U -torsor g∗X , and then base change along the natural morphism g∗g∗G → G, toobtain the G|g∗f∗U -torsor G ⊗g∗g∗G g∗X . This rule yields a map R1f∧∗ (g∗G) → R1(f g)∧∗G,and after taking associated sheaves, a natural morphism of pointed objects :

(2.4.6) R1f∗(g∗G)→ R1(f g)∗G.

Remark 2.4.7. As a special case, let h : S ′ → S be a morphism of topoi, H a S ′-group. If wetake T ′′ := S ′, T ′ := S, g := h and f : S → Set the (essentially) unique morphism of topoi,(2.4.6) and (2.4.5) boil down to maps of pointed sets :

(2.4.8) H1(S, h∗H)→ H1(S ′, H)→ Γ(S,R1h∗H).

These considerations are summarized in the following :

Theorem 2.4.9. In the situation of (2.4.4), there exists a natural exact sequence of pointedobjects of T :

1T → R1f∗(g∗G)→ R1(f g)∗G→ f∗R1g∗G.

Proof. The assertion means that (2.4.6) identifies R1f∗(g∗G) with the subobject :

R1(f g)∗G×f∗R1g∗G f∗τg,G

(briefly : the preimage of the trivial global section). We begin with the following :

Claim 2.4.10. In the situation of remark 2.4.7, the sequence of maps (2.4.8) identifies thepointed set H1(S, h∗H) with the preimage of the trivial global section τh,H of R1h∗H .

Proof of the claim. Notice first that a global section Y of R1h∧∗H maps to the trivial sectionτh,H of R1h∗H if and only if there exists a covering morphism U → 1S in (S,CS), such thatY (h∗U) 6= ∅. Thus, let X be a right h∗H-torsor; the image in H1(S ′, H) of its isomorphismclass is the class of the H-torsor Y := h∗X ⊗h∗h∗H H . The latter defines a global section ofR1h∗H . However, by definition there exists a covering morphism U → 1S such that X(U) 6=∅, hence also h∗X(h∗U) 6= ∅, and therefore Y (h∗U) 6= ∅. This shows that the image ofH1(S, h∗H) lies in the preimage of τh,H .


Moreover, notice that h∗Y (U) 6= ∅, hence h∗Y is a h∗H-torsor. Now, let εH : h∗h∗H → H(resp. ηh∗H : h∗H → h∗h

∗h∗H) be the counit (resp. unit of adjunction); we have a naturalmorphism α : h∗X → Y(εH) of h∗h∗H-modules, whence a morphism :

h∗α : h∗h∗X → h∗Y(h∗εH)

of h∗h∗h∗H-modules. On the other hand, the unit of adjunction ηX : X → h∗h∗X(ηh∗H) is a

morphism of h∗H-modules (remark 2.3.11(v)). Since h∗εH ηh∗H = 1h∗H (see (1.1.8)), thecomposition h∗α ηX is a morphism of h∗H-modules, hence it is an isomorphism, by remark2.4.2(iii). This implies that the first map of (2.4.8) is injective.

Conversely, suppose that the class of a H-torsor X ′ gets mapped to τh,H ; we need to showthat the class of X ′ lies in the image of H1(S, h∗H). However, the assumption means that thereexists a covering morphism U → 1S such that X ′(h∗U) 6= ∅; by adjunction we deduce thath∗X

′(U) 6= ∅, hence h∗X ′ is a h∗H-torsor. In order to conclude, it suffices to show that theimage in H1(S ′, H) of the class of h∗X ′ is the class of X .

Now, the counit of adjunction h∗h∗X′ → X ′ is a morphism of h∗h∗H-modules (remark

2.3.11(v)); by adjunction it induces a map h∗h∗X ′ ⊗h∗h∗H H → H of H-torsors, which mustbe an isomorphism, according to remark 2.4.2(iii). ♦

If we apply claim 2.4.10 with S := T ′/f ∗U , S ′ := T ′′/(g∗f ∗U) and h := g/(g∗f ∗U), for Uranging over the objects of T , we deduce an exact sequence of presheaves of pointed sets :

1T → R1f∧∗ (g∗H)→ R1(f g)∧∗G→ f∧∗ R1g∗G

from which the theorem follows, after taking associated sheaves.

2.4.11. Let f : T ′ → T be a morphism of topoi, U an object of T , G a T ′-group, p : X → U aright G|U -torsor. Then, for every object V of T we have an induced sequence of maps of sets

X(f ∗V )p∗−→ U(f ∗V )

∂−→ R1f∧∗ G(V )

where p∗ is deduced from p, and for every σ ∈ U(f ∗V ), we let ∂(σ) be the isomorphism class ofthe rightG|f∗V -torsor (X×U f ∗V → f ∗V ). Clearly the image of p∗ is precisely the preimage of(the isomorphism class of) the trivial G|f∗V -torsor. After taking associated sheaves, we deducea natural sequence of morphisms in T :

(2.4.12) f∗Xp−→ f∗U

∂−→ R1f∗G

such that the preimage of the global section τf,G is precisely the image in Γ(T ′, U) of the set ofglobal sections of X .

2.4.13. A ringed topos is a pair (T,OT ) consisting of a topos T and a (unitary, associative)T -ring OT , called the structure ring of T . A morphism f : (T,OT ) → (S,OS) of ringed topoiis the datum of a morphism of topoi f : T → S and a morphism of T -rings :

f \ : f ∗OS → OT .

We denote, as usual, by O×T ⊂ OT the subobject representing the invertible sections of OT . Forevery object U of T , and every s ∈ OT (U), let D(s) ⊂ U be the subobject such that :

HomT (V,D(s)) := ϕ ∈ U(V ) | ϕ∗s ∈ O×T (V ).We say that (T,OT ) is locally ringed, if D(0) = ∅T (the initial object of T ), and moreover

D(s) ∪D(1− s) = U for every U ∈ Ob(T ), and every s ∈ OT (f).

A morphism of locally ringed topoi f : (T,OT )→ (S,OS) is a morphism of ringed topoi suchthat

f ∗D(s) = D(f \(U)(f ∗s)) for every U ∈ Ob(S) and every s ∈ OS(U).


If T has enough points, then (T,OT ) is locally ringed if and only if the stalks OT,ξ of thestructure ring at all the points ξ of T are local rings. Likewise, a morphism f : (T,OT ) →(S,OS) of ringed topoi is locally ringed if and only if, for every T -point ξ, the induced mapOS,f(ξ) → OT,ξ is a local ring homomorphism.

2.4.14. In the rest of this section we present a few results concerning the special case of topolo-gies on a scheme. Hence, for any scheme X , we shall denote by Xet (resp. by XZar) the smalletale (resp. the small Zariski) site on X . It is clear that XZar is a small site, and it is not hard toshow that Xet is a U-site ([4, Exp.VII, §1.7]). The inclusion of underlying categories :

uX : XZar → Xet

is a continuous functor (see definition 2.1.39(i)) commuting with finite limits, whence a mor-phism of topoi :

uX := (u∗X , uX∗) : X∼et → X∼Zar.


XZaruX //

Xet

X∼Zar

u∗X // X∼et

commutes, where the vertical arrows are the Yoneda embeddings (lemma 2.1.49).The topoi X∼Zar and X∼et are locally ringed in a natural way, and by faithfully flat descent, we

see easily that uX∗OXet= OXZar

. By inspection, uX is a morphism of locally ringed topoi.

2.4.15. For any ring R (of our fixed universe U), denote by Sch/R the category of R-schemes,and by Sch/RZar (resp. Sch/Ret) the big Zariski (resp. etale) site on Sch/R. For R = Z,we shall usually just write SchZar and Schet for these sites. The morphisms uX of (2.4.13) areactually restrictions of a single morphism of sites :

u : SchZar → Schet

which, for every universe V such that U ∈ V, induces a morphism of V-topoi :

uV : (Schet)∼V → (SchZar)

∼V .

2.4.16. Let X be scheme; a geometric point of X is a morphism of schemes ξ : Specκ →X , where κ is an arbitrary separably closed field. Notice that both the Zariski and etaletopoi of Specκ are equivalent to the category Set, so ξ induces a topos-theoretic point ξ∼et :(Specκ)∼et → X∼et of X∼et (and likewise for X∼Zar). A basic feature of both the Zariski and etaletopologies, is that every point of X∼Zar and X∼et arise in this way.

More precisely, we say that two geometric points ξ and ξ′ of X are equivalent, if thereexists a third such point ξ′′ which factors through both ξ and ξ′. It is easily seen that this is anequivalence relation on the set of geometric points of X , and two topos-theoretic points ξ∼et andξ′∼et are isomorphic if and only if the same holds for the points ξ∼Zar and ξ′∼Zar, if and only if ξ isequivalent to ξ′.

Definition 2.4.17. Let X be a scheme, x a point of X , and x : Specκ→ X a geometric point.(i) We let κ(x) be the residue field of the local ring OX,x, and set

|x| := Specκ(x) κ(x) := κ |x| := Specκ(x)

If x ⊂ X is the image of x, we say that x is localized at x, and that x is the support of x.(ii) The localization of X at x is the local scheme

X(x) := Spec OX,x.


The strict henselization of X at x is the strictly local scheme

X(x) := Spec OX,x

where OX,x denotes the strict henselization of OX,x relative to the geometric point x ([33, Ch.IV,Def.18.8.7]) (recall that a local ring is called strictly local, if it is henselian with separablyclosed residue field; a scheme is called strictly local, if it is the spectrum of a strictly local ring:see [33, Ch.IV, Def.18.8.2]). By definition, the geometric point x lifts to a unique geometricpoint of X(x), which shall be denoted again by x.

(iii) Moreover, we shall denote by

ix : X(x)→ X ix : X(x)→ X(x)

the natural morphisms of schemes, and if F is any sheaf on XZar (resp. Xet), we let

F (x) := i∗xF F (x) := i∗xF (x)

and F (x) is a sheaf on X(x)Zar (resp. on X(x)et).(iv) If f : Y → X is any morphism of schemes, we let

f−1(x) := Y ×X |x| f−1(x) := Y ×X |x| Y (x) := Y ×X X(x) Y (x) := Y ×X X(x).

Also, if ξ is any geometric point of Y , we define f(ξ) as the geometric point f ξ of X .

2.4.18. Many discussions concerning the Zariski or etale site of a scheme, only make appeal togeneral properties of these two topologies, and therefore apply indifferently to either of them,with only minor verbal changes. For this reason, to avoid tiresome repetitions, the followingnotational device is often useful. Namely, instead of referring each time to XZar and Xet inthe course of an argument, we shall write just Xτ , with the convention that τ ∈ Zar, et hasbeen chosen arbitrarily at the beginning of the discussion. In the same manner, a τ -point ofX will mean a point of the topos X∼τ , and a τ -open subset of X will be any object of the siteXτ . With this convention, a Zar-point is a usual point of X , whereas an et-point shall be ageometric point. Likewise, if ξ is a given τ -point of X , the localization X(ξ) makes sensefor both topologies : if τ = et, then X(ξ) is the strict henselization as in definition 2.4.17(ii);if τ = Zar, then X(ξ) is the usual localization of X at the (Zariski) point ξ. If τ = et, thesupport of ξ is given by definition 2.4.17(i); if τ = Zar, then the support of ξ is just ξ itself (andcorrespondingly, in this case ξ is localized at ξ). Furthermore, OX,ξ is a local ring if τ = Zar,and it is a strictly local ring, in case τ = et.

2.4.19. Let f : X → Y be a morphism of schemes, x a geometric point ofX , and set y := f(x).The natural morphism fx : X(x) → Y (y) induces a unique local morphism of strictly localschemes

fx : X(x)→ Y (y)

([33, Ch.IV, Prop.18.8.8(ii)]) that fits in a commutative diagram :

|x| x // X(x)

fx

ix // X(x)

fx

ix // X

f

|y| y // Y (y)

iy // Y (y)iy // Y.

Let now F be any sheaf on Yet; there follows a natural isomorphism :

f ∗xF (y)∼→ (f ∗F )(x).

Notice also the natural bijections :

(2.4.20)Fy

∼→ Γ(Y (y),F (y))∼→ Γ(|y|, y∗F (y))

f ∗Fx∼→ Γ(X(x), f ∗F (x))

∼→ Γ(|x|, x∗f ∗F (x))


which induce a natural identification :

(2.4.21) Fy∼→ f ∗Fx : σ 7→ f ∗x(σ).

2.4.22. Let X be a scheme, x, x′ ∈ X any two points, such that x is a specialization of x′.Choose a geometric point x localized at x. The localization map OX,x → OX,x′ induces anatural specialization morphism of X-schemes :

X(x′)→ X(x).

Set W := X(x)×X(x)X(x′). The natural map g : W → X(x′) is faithfully flat, and is the limitof a cofiltered system of etale morphisms; hence we may find w ∈ W lying over the closedpoint of X(x′), and the induced map κ(x′) → κ(w) is algebraic and separable. Choose also ageometric point w of W localized at w, and set x′ := g(w). Then g induces an isomorphismgw : W (w)

∼→ X(x′), whence a unique morphism

(2.4.23) X(x′)→ X(x)

which makes commute the diagram :

W (w)gw //

iw

X(x′)

ix′ // X(x′)

W (w) // X(x)

ix // X(x)

where the left bottom arrow is the natural projection, and the right-most vertical arrow is thespecialization map. In this situation, we say that x is a specialization of x′ (and that x′ isa generization of x), and we call (2.4.23) a strict specialization morphism. Combining with(2.4.20), we obtain the strict specialization map induced by (2.4.23)

(2.4.24) Gx → Gx′

for every sheaf G on Xet.

Remark 2.4.25. (i) In the situation of (2.4.19), suppose that G = f ∗F for a sheaf F on Yet.Then (2.4.24) is a map Ff(x) → Ff(x′). By inspecting the definition, it is easily seen that thelatter agrees with the strict specialization map for F induced by a unique strict specializationmorphism Y (f(x′))→ Y (f(x)).

(ii) Notice that (2.4.23) and (2.4.24) depend not only on the choice of w (which may not beunique, when X(x) is not unibranch) but also on the geometric point w. Indeed, the groupof automorphisms of the X(x′)-scheme X(x′) is naturally isomorphic to the Galois groupGal(κ(x′)s/κ(x′)) ([33, Ch.IV, (18.8.8.1)]).

Lemma 2.4.26. Let X be a scheme, F a sheaf on Xet. We have :(i) The counit of the adjunction εF : u∗X uX∗F → F is a monomorphism.

(ii) Suppose there exists a sheaf G on XZar, and an epimorphism f : u∗G → F (resp. amonomorphism f : F → u∗G ). Then εF is an isomorphism.

(iii) The functor u∗X is fully faithful.(iv) (Hilbert 90) R1uX∗O

×Xet

= 1XZar.

Proof. (i): The assertion can be checked on the stalks. Hence, let ξ be any geometric point ofX; we have to show that the natural map (uX∗F )ξ → Fξ is injective. To this aim, say thats, s′ ∈ (uX∗F )ξ, and suppose that the image of s in Fξ agrees with the image of s′; we mayfind an open neighborhood U of ξ in XZar, such that s and s′ lie in the image of F (U), and byassumption, there exists an etale morphism f : V → U such that the images of s and s′ coincidein F (V ). However, f(V ) ⊂ U is an open subset ([31, Ch.IV, Th.2.4.6]), and the induced map


V → f(V ) is a covering morphism in Xet; it follows that the images of s and s′ agree alreadyin F (f(V )), therefore also in (uX∗F )ξ.

(iii): According to proposition 1.1.11(ii), it suffices to show that the unit of the adjunctionηG : G → uX∗ u∗XG is an isomorphism, for every G ∈ Ob(X∼Zar). However, we havemorphisms :

u∗XGu∗X(ηG )−−−−−→ u∗X uX∗ u∗XG

εu∗X

G

−−−→ u∗XG .

whose composition is the identity of u∗XG (see (1.1.8)); also, (i) says that εu∗XG is a monomor-phism, and then it follows formally that it is actually an isomorphism (e.g. from the dual of [10,Prop.1.9.3]). Hence the same holds for u∗X(ηG ), and by considering the stalks of the latter, weconclude that also ηG is an isomorphism, as required.

(ii): Suppose first that f : u∗G → F is an epimorphism. We have just seen that ηG is anisomorphism, therefore we have a morphism u∗ u∗f : u∗G → u∗ u∗F whose compositionwith εF is f ; especially, εF is an epimorphism, so the assertion follows from (i).

In the case of a monomorphism f : F → u∗G , set H := u∗G qF u∗G ; we may representf as the equalizer of the two natural maps j1, j2 : u∗G → H . However, the natural morphismu∗(G q G )→H is an epimorphism, hence the counit εH is an isomorphism, by the previouscase. Then (iii) implies that ji = u∗j′i for morphisms j′i : G → u∗H (i = 1, 2). Let F ′ bethe equalizer of j′1 and j′2; then u∗F ′ ' F , and since we have already seen that the unit ofadjunction is an isomorphism, the assertion follows from the triangular identitities of (1.1.8).

(iv): The assertion can be checked on the stalks. To ease notation, set F := R1uX∗O×Xet

.Let ξ be any geometric point of X , and say that s ∈ Fξ; pick a (Zariski) open neighborhoodU ⊂ X of ξ such that s lies in the image of F (U). We may then find a Zariski open covering(Uλ → U | λ ∈ Λ) of U , such that the image of s in F (Ui) is represented by a O×Uλ,et-torsor onUλ,et, for every λ ∈ Λ. After replacing U by any Uλ containing the support of ξ, we may assumethat s is the image of the isomorphism class of some O×Uet

-torsor Xet on Uet. By faithfully flatdescent, there exists a O×UZar

-torsor X on UZar, and an isomorphism of O×Uet-torsors :

Xet∼→ O×Uet

⊗u∗UO×UZar

u∗UX.

However, after replacing U by a smaller open neighborhood of ξ, we may suppose thatX(U) 6=∅, therefore Xet(U) 6= ∅ as well, i.e. s is the image of the trivial section of F (U).

3. MONOIDS AND POLYHEDRA

Unless explicitly stated otherwise, every monoid encountered in this chapter shall be com-mutative. For this reason, we shall usually economize adjectives, and write just “monoid” whenreferring to commutative monoids.

3.1. Monoids. If M is any monoid, we shall usually denote the composition law of M bymultiplicative notation: (x, y) 7→ x · y (so 1 is the neutral element). However, sometimes itis convenient to be able to switch to an additive notation; to allow for that, we shall denote by(logM,+) the monoid with additive composition law, whose underlying set is the same as forthe given monoid (M, ·), and such that the identity map is an isomorphism of monoids (then,the neutral element of logM is denoted by 0). For emphasis, we may sometimes denote bylog : M

∼→ logM the identity map, so that one has the tautological identities :

log 1 = 0 and log(x · y) = log x+ log y for every x, y ∈M.

Conversely, if (M,+) is a given monoid with additive composition law, we may switch to amultiplicative notation by writing (expM, ·), in the same way.


3.1.1. For any monoid M , and any two subsets S, S ′ ⊂M , we let :

S · S ′ := s · s′ | s ∈ S, s′ ∈ S ′and Sa is defined recursively for every a ∈ N, by the rule :

S0 := 1 and Sa := S · Sa−1 if a > 0.

Notice that the pair (P(M), ·) consisting of the set of all subsets of M , together with thecomposition law just defined, is itself a monoid : the neutral element is the subset 1. In thesame vein, the exponential notation for subsets of M becomes a multiplicative notation in themonoid (log P(M),+) = (P(logM),+), i.e. we have the tautological identity : logSa =a · logS, for every S ∈P(M) and every a ∈ N.

Furthermore, for any two monoids M and N , the set HomMnd(M,N) is naturally a monoid.The composition law assigns to any two morphisms ϕ, ψ : M → N their product ϕ · ψ, givenby the rule : ϕ · ψ(m) := ϕ(m) · ψ(m) for every m ∈M .

Basic examples of monoids are the set (N,+) of natural numbers, and the non-negative real(resp. rational) numbers (R+,+) (resp. (Q+,+)), with their standard addition laws.

3.1.2. Given a surjection X → Y of monoids, it may not be possible to express Y as a quotientof X – a problem relevant to the construction of presentations for given monoids, in termsof free monoids. For instance, consider the monoid (Z,) consisting of the set Z with thecomposition law such that :

x y :=

x+ y if either x, y ≥ 0 or x, y ≤ 0max(x, y) otherwise

for every x, y ∈ Z. Define a surjective map ϕ : N⊕2 → (Z,) by the rule (n,m) 7→ n −m,for every n ∈ N. Then one verifies easily that Kerϕ = 0, and nevertheless ϕ is not anisomorphism. The right way to proceed is indicated by the following :

Lemma 3.1.3. Every surjective map of monoids is an effective epimorphism (in the categoryMnd).

Proof. (See example 1.5.16 for the notion of effective epimorphism.) Let π : M → N be asurjection of monoids. For every monoid X , we have a natural diagram of sets :

HomMnd(N,X)j // HomMnd(M,X)

p∗2

//p∗1 // HomMnd(M ×N M,X)

where p1, p2 : M ×N M → M are the two natural projections, and we have to show that themap j identifies HomMnd(N,X) with the equalizer of p∗1 and p∗2. First of all, the surjectivity ofπ easily implies that j is injective. Hence, let ϕ : M → X be any map such that ϕp1 = ϕp2;we have to show that ϕ factors through π. To this aim, it suffices to show that the map of setsunderlying ϕ factors as a composition ϕ′ π, for some map of sets ϕ′ : N → X , since ϕ′ willthen be necessarily a morphism of monoids. However, the forgetful functor F : Mnd → Setcommutes with fibre products (lemma 2.3.29(iii)), and F (π) is an effective epimorphism, sincein the category Set all surjections are effective epimorphisms. The assertion follows.

3.1.4. Lemma 3.1.3 allows to construct presentations for an arbitrary monoid M , as follows.First, we choose a surjective map of monoids F := N(S) → M , for some set S. Then wechoose another set T and a surjection of monoids N(T ) → F ×M F . Composing with thenatural projections p1, p2 : F ×M F → F , we obtain a diagram :

(3.1.5) N(T )q1 //q2// N(S) // M

which, in view of lemma 3.1.3, identifies M to the coequalizer of q1 and q2.


Definition 3.1.6. Let M be a monoid, Σ ⊂M a subset.(i) Let (eσ | σ ∈ Σ) be the natural basis of the free monoid N(Σ). We say that Σ is a system

of generators for M , if the map of monoids N(Σ) → M such that eσ 7→ σ for everyσ ∈ Σ, is a surjection.

(ii) M is said to be finitely generated if it admits a finite system of generators.(iii) M is said to be fine if it is integral and finitely generated.(iv) A finite presentation for M is a diagram such as (3.1.5) that identifies M to the co-

equalizer of q1 and q2, and such that, moreover, S and T are finite sets.(v) We say that a morphism of monoids ϕ : M → N is finite, if N is a finitely generated

M -module, for the M -module structure induced by ϕ.

Lemma 3.1.7. (i) Every finitely generated monoid admits a finite presentation.(ii) Let M be a finitely generated monoid, and (Ni | i ∈ I) a filtered family of monoids.

Then the natural map :

colimi∈I

HomMnd(M,Ni)→ HomMnd(M, colimi∈I

Ni)

is a bijection.

Proof. (i): LetM be a finitely generated monoid, and choose a surjection π : N(S) →M with Sa finite set. We have seen thatM is the coequalizer of the two projections p1, p2 : P := N(S)×MN(S) → N(S). For every finitely generated submonoid N ⊂ P , let p1,N , p2,N : N → N(S) be therestrictions of p1 and p2, and denote by CN the coequalizer of p1,N and p2,N . By the universalproperty of CN , the map π factors through a map of monoids πN : CN → M , and since πis surjective, the same holds for πN . It remains to show that πN is an isomorphism, for Nlarge enough. We apply the functor M 7→ Z[M ] of (2.3.50), and we derive that Z[M ] is thecoequalizer of the two maps Z[p1],Z[p2] : Z[P ] → Z[S], i.e. Z[M ] ' Z[S]/I , where I isthe ideal generated by Im(Z[p1] − Z[p2]). Clearly I is the colimit of the filtered system ofanalogous ideals IN generated by Im(Z[p1,N ]−Z[p2,N ]), for N ranging over the filtered familyF of finitely generated submonoids of P . By noetherianity, there exists N ∈ F such thatI = IN , therefore Z[M ] is the coequalizer of Z[p1,N ] and Z[p2,N ]. But the latter coequalizer isalso the same as Z[CN ], whence the contention.

(ii): This is a standard consequence of (i). Indeed, say that f1, f2 : M → Ni are twomorphisms whose compositions with the natural map Ni → N := colim

i∈INi agree, and pick a

finite set of generators x1, . . . , xn for M . For any morphism ϕ : i→ j in the filtered category I ,denote by gϕ : Ni → Nj the corresponding morphism; then we may find such a morphism ϕ, sothat gϕf1(xk) = gϕf2(xk) for every k ≤ n, whence the injectivity of the map in (ii). Next, letf : M → N be a given morphism, and pick a finite presentation (3.1.5); we deduce a morphismg : N(S) → N , and since S is finite, it is clear that g factors through a morphism gi : N(S) → Ni

for some i ∈ I . Set g′i := gi q1 and g′′i := gi q2; by assumption, after composing g′i and ofg′′i with the natural map Ni → N , we obtain the same map, so by the foregoing there exists amorphism ϕ : i→ j in I such that gϕ g′i = gϕ g′′i . It follows that gϕ gi factors through M ,whence the surjectivity of the map in (ii).

Definition 3.1.8. Let M be a monoid, I ⊂M an ideal.(i) We say that I is principal, if it is cyclic, when regarded as an M -module.

(ii) The radical of I is the ideal rad(I) consisting of all x ∈M such that xn ∈ I for everysufficiently large n ∈ N. If I = rad(I), we also say that I is a radical ideal.

(iii) A face of M is a submonoid F ⊂ M with the following property. If x, y ∈ M are anytwo elements, and xy ∈ F , then x, y ∈ F .

(iv) Notice that the complement of a face is always an ideal. We say that I is a prime idealof M , if M \I is a face of M .


Proposition 3.1.9. LetM be a finitely generated monoid, and S a finitely generatedM -module.Then we have :

(i) Every submodule of S is finitely generated.(ii) Especially, every ideal of M is finitely generated.

Proof. Of course, (ii) is a special case of (i). To show (i), let S ′ ⊂ S be an M -submodule, Σ ⊂S ′ any system of generators. Let P ′(Σ) be the set of all finite subsets of Σ, and for every A ∈P ′(Σ), denote by S ′A ⊂ S ′ the submodule generated by A; clearly S ′ is the filtered union of thefamily (S ′A | A ∈ P ′(Σ)), hence Z[S ′] is the filtered union of the family of Z[M ]-submodules(Z[S ′] | S ∈P ′(Σ)). Since Z[M ] is noetherian and Z[S] is a finitely generated Z[M ]-module,it follows that Z[S ′A] = Z[S ′] for some finite subset A ⊂ Σ, whence the contention.

Corollary 3.1.10. Let M be any fine and sharp monoid. The set mM \m2M is finite, and is the

unique minimal system of generators of M .

Proof. It is easily seen that any system of generators of M must contain Σ := mM \m2M , hence

the latter must be a finite set. On the other hand, suppose that there exists an element x0 ∈ Mwhich is not contained in the submonoid M ′ generated by Σ. Then we may write x0 = x1y1

for some x0, y0 ∈ mM , with x1 /∈M ′, so x1 admits a similar decomposition. Proceeding in thisway, we obtain a sequence of elements (xn | n ∈ N) with the property that Mxn ⊂ Mxn+1 forevery n ∈ N. We claim that Mxn 6= Mxn+1 for every n ∈ N. Indeed, if the inequality fails forsome n ∈ N, we may write xn+1 = axn for some a ∈ N, and on the other hand, we have byconstruction xn = yxn+1 for some y ∈ mM ; summing up, we get xn = yaxn, whence ya = 1,since M is integral, therefore y ∈M×, a contradiction.

Thus, from the given x0, we have produced an infinite strictly ascending chain of ideals ofM , which is ruled out by virtue of proposition 3.1.9(ii). This means that x0 cannot exist, andthe corollary follows.

3.1.11. Let M be a monoid, (Iλ | λ ∈ Λ) any collection of ideals of M ; then it is easily seenthat both

⋃λ∈Λ Iλ and

⋂λ∈Λ Iλ are ideals of M . The spectrum of M is the set :

SpecM

consisting of all prime ideals of M . It has a natural partial ordering, given by inclusion of primeideals; the minimal element of SpecM is the empty ideal ∅ ⊂M , and the maximal element is:

mM := M \M×.

If (pλ | λ ∈ Λ) is any family of prime ideals of M , then⋃λ∈Λ pλ is a prime ideal of M .

Any morphism ϕ : M → N of monoids induces a natural map of partially ordered sets :

ϕ∗ : SpecN → SpecM p 7→ ϕ−1p.

We say that ϕ is local, if ϕ(mM) ⊂ mN .

Lemma 3.1.12. Let f1 : M → N1 and f2 : M → N2 be two local morphisms of monoids. IfN1 and N2 are sharp, then N1 qM N2 is sharp.

Proof. Let (a, b) ∈ N1 × N2, and suppose there exists c ∈ M , a′ ∈ N1, b′ ∈ N2 such that(a, b) = (a′f1(c), b′) and (1, 1) = (a′, f2(c)b′); since N2 is sharp, we deduce f2(c) = b′ = 1, sob = 1. Then, since f2 is local, we get c ∈ M×, hence f1(c) = 1 and a = a′ = 1. One arguessymmetrically in case (1, 1) = (a′f1(c), b′) and (a, b) = (a′, f2(c)b′). We conclude that (a, b)represents the unit class in N1 qM N2 if and only if a = b = 1. Now, suppose that the class of(a, b) is invertible in N1 qM N2; it follows that there exists (c, d) such that ac = 1 and bd = 1,which implies that a = 1 and b = 1, whence the contention.


Lemma 3.1.13. Let S ⊂ M be any submonoid. The localization j : M → S−1M inducesan injective map j∗ : SpecS−1M → SpecM which identifies SpecS−1M with the subset ofSpecM consisting of all prime ideals p such that p ∩ S = ∅.

Proof. For every p ∈ SpecM , denote by S−1p the ideal of S−1M generated by the image ofp. We claim that p = j∗(S−1p) for every p ∈ SpecM such that p ∩ S = ∅. Indeed, clearlyp ⊂ j∗(S−1p); next, if f ∈ j∗(S−1p), there exists s ∈ S and g ∈ p such that s−1g = fin S−1M ; therefore there exists t ∈ S such that tg = tsf in M , especially tsf ∈ p, hencef ∈ p, since t, s /∈ p. Likewise, one checks easily that S−1p is a prime ideal if p ∩ S = ∅, andq = S−1(j∗q) for every q ∈ SpecS−1M , whence the contention.

Remark 3.1.14. (i) If we take Sp := M\p, the complement of a prime ideal p of M , we obtainthe monoid

Mp := S−1p M

and SpecMp ⊂ SpecM is the subset consisting of all prime ideals q contained in p.(ii) Likewise, if p ⊂ M is any prime ideal, the spectrum Spec(M \p) is naturally identified

with the subset of SpecM consisting of all prime ideals q containing p (details left to the reader).(iii) Let S ⊂ M be any submonoid. Then there exists a smallest face F of M containing S

(namely, the intersection of all the faces that contain S). It is easily seen that F is the subset ofall x ∈M such that xM ∩ S 6= ∅. From this characterization, it is clear that S−1M = F−1M .In other words, every localization of M is of the type Mp for some p ∈ SpecM .

Lemma 3.1.15. Let M be a monoid, and I ⊂ M any ideal. Then rad(I) is the intersection ofall the prime ideals of M containing I .

Proof. It is easily seen that a prime ideal containing I also contains rad(I). Conversely, saythat f ∈ M \ rad(I); let ϕ : M → Mf be the localization map. Denote by m the maximalideal of Mf . We claim that I ⊂ ϕ−1m, Indeed, otherwise there exist g ∈ I , h ∈ M and n ∈ Nsuch that g−1 = f−nh in Mf ; this means that there exist m ∈ N such that fm+n = fmgh inM , hence fm+n ∈ I , which contradicts the assumption on f . On the other hand, obviouslyf /∈ ϕ−1m.

Lemma 3.1.16. (i) Let M be a monoid, and G ⊂M× a subgroup.(a) The map given by the rule : I 7→ I/G establishes a natural bijection from the set

of ideals of M onto the set of ideals of M/G.(b) Especially, the natural projection π : M →M/G induces a bijection :

π∗ : SpecM/G→ SpecM

(ii) Let (Mi | i ∈ I) be any finite family of monoids, and for each j ∈ I , denote byπj :

∏i∈IMi →Mj the natural projection. The induced map∏

i∈I

SpecMi → Spec∏i∈I

Mi : (pi | i ∈ I) 7→⋃i∈I

π∗i pi

is a bijection.(iii) Let (Mi | i ∈ I) be any filtered system of monoids. The natural map

Spec colimi∈I

Mi → limi∈I

SpecMi

is a bijection.

Proof. (i): By lemma 2.3.31(iii), M/G is the set-theoretic quotient of M by the translationaction of G. By definition, any ideal I of M is stable under the G-action, hence the quotientI/G is well defined, and one checks easily that it is an ideal of M/G. Moreover, if p ⊂ M


is a prime ideal, it is easily seen that p/G is a prime ideal of M/G. Assertions (a) and (b) arestraightforward consequences.

(ii): The assertion can be rephrased by saying that every face F of∏

i∈IMi is a product offaces Fi ⊂ Mi. However, if m := (mi | i ∈ I) ∈ F , then, for each i ∈ I we can writem = m(i) · n(i), where, for each j ∈ I , the j-th-component of m(i) (resp. of n(i)) equals 1(resp. mj), unless j = i, in which case it equals mi (resp. 1). Thus, m(i) ∈ F for every i ∈ I ,and the contention follows easily.

(iii): Denote by M the colimit of the system (Mi | i ∈ I), and ϕi : Mi → M the naturalmorphisms of monoids, as well as ϕf : Mi → Mj the transition maps, for every morphismf : i→ j in I . Recall that the set underlying M is the colimit of the system of sets (Mi | i ∈ I)(lemma 2.3.29(iii)). Let now p• := (pi | ∈ I) be a compatible system of prime ideals, i.e.such that pi ∈ SpecMi for every i ∈ I , and ϕ−1

f pj = pi for every f : i → j. We letβ(p•) :=

⋃i∈I ϕi(pi). We claim that β(p•) is a prime ideal ofM . Indeed, suppose that x, y ∈M

and xy ∈ β(p•); since I is filtered, we may find i ∈ I , xi, yi ∈ Mi, and zi ∈ pi, such thatx = ϕi(xi), y = ϕi(yi), and xy = ϕi(zi). Especially, ϕi(zi) = ϕi(xiyi), so there exists amorphism f : i → j such that ϕf (zi) = ϕf (xiyi). But ϕf (zi) ∈ pj , so either ϕf (xi) ∈ pj orϕf (yi) ∈ pj , and finally either x ∈ p or y ∈ p, as required.

Let p ⊂ M be any prime ideal; it is easily seen that β(ϕ−1i p | i ∈ I) = p. To conclude, it

suffices to show that pi = ϕ−1i β(p•), for every compatible system p• as above, and every i ∈ I .

Hence, fix i ∈ I and pick xi ∈Mi such that ϕi(xi) ∈ β(p•); then there exists j ∈ I and xj ∈ pjsuch that ϕi(xi) = ϕj(xj). Since I is filtered, we may find k ∈ I and morphisms f : i→ k andg : j → k such that ϕf (xi) = ϕg(xj), so ϕf (xi) ∈ pk, and finally xi ∈ pi, as sought.

Remark 3.1.17. In case (Mi | i ∈ I) is an infinite family of monoids, the natural map of lemma3.1.16(ii) is still injective, but it is not necessarily surjective. For instance, let I be any infiniteset, and let U ⊂ P(I) be a non-principal ultrafilter; denote by ∗N the quotient of NI underthe equivalence relation ∼U such that (ai | i ∈ I) ∼U (bi | i ∈ I) if and only if there existsU ∈ U such that ai = bi for every i ∈ U . It is clear that the composition law on NI descendsto ∗N; the resulting structure (∗N,+) is called the monoid of hypernatural numbers. Denote byπ : NI → ∗N the projection, and let 0 ∈ NI be the unit; then it is easily seen that π(0) is aface of ∗N, but π−1(π(0)) is not a product of faces Fi ⊂ N.

Definition 3.1.18. Let M be a monoid.(i) The dimension of M , denoted dimM ∈ N ∪ +∞, is defined as the supremum of all

r ∈ N such that there exists a chain of strict inclusions of prime ideals of M :

p0 ⊂ p1 ⊂ · · · ⊂ pr.

(ii) The height of a prime ideal p ∈ SpecM is defined as ht p := dimMp.(iii) A facet of M is the complement of a prime ideal of M of height one.

Remark 3.1.19. (i) Notice that not all epimorphisms in Mnd are surjections on the underlyingsets; for instance, every localization map M → S−1M is an epimorphism.

(ii) If ϕ : N → M is a map of fine monoids (see definition 3.1.6(vi)), then it will followfrom corollary 3.4.2 that N ×M N is also finitely generated. If M is not integral, then this failsin general : a counter-example is provided by the morphism ϕ constructed in (3.1.2).

(iii) Let Σ be any set; it is easily seen that a free monoid M ' N(Σ) admits a unique minimalsystem of generators, in natural bijection with Σ. Especially, the cardinality of Σ is determinedby the isomorphism class of M ; this invariant is called the rank of the free monoid M . This isthe same as the rank of M as an N-module (see example 1.2.27).

(iv) A submonoid of a finitely generated monoid is not necessarily finitely generated. Forinstance, consider the submonoid M ⊂ N⊕2, with M := (0, 0) ∪ (a, b) | a > 0. However,the following result shows that a face of a finitely generated monoid is again finitely generated.


Lemma 3.1.20. Let f : M → N be a map of monoids, F ⊂ N a face of N , and Σ ⊂ N asystem of generators for N . Then :

(i) N× is a face of N , and f−1F is a face of M .(ii) Σ ∩ F is a system of generators for F .

(iii) If N is finitely generated, SpecN is a finite set, and dimN is finite.(iv) If N is finitely generated (resp. fine) then the same holds for F−1N .

Proof. (i) and (ii) are left to the reader, and (iii) is an immediate consequence of (ii). To show(iv), notice that – in view of (ii) – the set Σ ∪ f−1 | f ∈ F ∩ Σ is a system of generators ofF−1N .

Definition 3.1.21. (i) If M is a (pointed or not pointed) monoid, and S is a pointed M -module,we say that S is integral, if for every x, y ∈ M and every s ∈ S such that xs = ys 6= 0, wehave x = y. The annihilator ideal of S is the ideal

AnnM(S) := m ∈M |ms = 0 for every s ∈ S.

If s ∈ S is any element, we also write AssM(s) := AssM(Ms). The support of S is the subset :

SuppS := p ∈ SpecM | Sp 6= 0.

(ii) A pointed monoid (M, 0M) is called integral, if it is integral when regarded as a pointedM -module; it is called fine, if it is finitely generated and integral in the above sense.

(iii) The forgetful functor Mnd → Set (notation of (2.3.28)) admits a left adjoint, whichassigns to any set Σ the free pointed monoid N(Σ)

:= (N(Σ)).(iv) A morphism ϕ : M → N of pointed monoids is local, if both M,N 6= 0, and ϕ is local

when regarded as a morphism of non-pointed monoids.

Remark 3.1.22. (i) Quite generally, a (non-pointed) monoid M is finitely generated (resp.free, resp. integral, resp. fine) if and only if M has the corresponding property for pointedmonoids. However, there exist integral pointed monoids which are not of the form M for anynon-pointed monoid M .

(ii) Let (M, 0M) be a pointed monoid. An ideal of (M, 0M) is a pointed submodule I ⊂M .Just as for non-pointed monoids, we say that I is a prime ideal, if M \ I is a (non-pointed)submonoid of M , and a non-pointed submonoid which is the complement of a prime ideal, iscalled a face of M . Hence the smallest ideal is 0. Notice though, that 0 is not necessarily aprime ideal, hence the spectrum Spec (M, 0M) does not always admit a least element. However,if M = N for some non-pointed monoid N , the natural morphism of monoids N → Ninduces a bijection :

Spec (N, 0N)→ SpecN.

(iii) Let I ⊂ M be any ideal; the inclusion map I → M can be regarded as a morphism ofpointed M -modules (if M is not pointed, this is achieved via the faithful imbedding (2.3.15)),whence a pointed M -module M/I , with a natural morphism M →M/I . The latter map is alsoa morphism of monoids, for the obvious monoid structure on M/I . One checks easily that, ifM is integral, M/I is an integral pointed monoid.

(iv) Let M be a (pointed or not pointed) monoid, p ⊂ M a prime ideal. Then the naturalmorphism of monoids M →M/p induces a bijection :

SpecM/p∼→ q ∈ SpecM | p ⊂ q = SpecM \p.

(v) Let M be a (pointed or not pointed) monoid, and S 6= 0 a pointed M -module. Thenthe support of S contains at least the maximal ideal of M . This trivial observation shows that apointed M -module is 0 if and only if its support is empty.


(vi) Let M be a pointed monoid, and Σ ⊂ M a non-pointed submonoid. The localizationΣ−1M (defined in the category of monoids, as in (2.3.33)) is actually a pointed monoid : itszero element 0Σ−1M is the image of 0M .

(vii) If M is a (pointed or not pointed) monoid, Σ ⊂ M any non-pointed submonoid, and Sa pointed M -module, we let as usual Σ−1S := Σ−1M ⊗M S (see remark 2.3.21(i), if M is notpointed). The resulting functor M -Mod → Σ−1M -Mod is exact. Indeed, it is right exact,since it is left adjoint to the restriction of scalars arising from the localization map M → Σ−1M(see (1.2.26)), and one verifies directly that it commutes with finite limits. Also, it is clear that

Supp Σ−1S = SuppS ∩ Spec Σ−1M.

(viii) Let M be a pointed monoid, and N ⊂ M a pointed submonoid. Since the final object1 of the category of pointed monoids is not isomorphic to the initial object 1, the push-out ofthe diagram 1 ← N → M is not an interesting object (it is always isomorphic to 1). Even ifwe form the quotient M/N in the category of non-pointed monoids, we still get always 1, since0M ∈ N , and therefore in the quotient M/N the images of 0M and of the unit of M coincide.

The only case that may give rise to a non-trivial quotient, is when N is non-pointed; in thissituation we may form M/N in the category of non-pointed monoids, and then remark that theimage of 0M yields a zero element 0M/N for M/N , so the latter is a pointed monoid.

Example 3.1.23. (i) Let M be a (pointed or not pointed) monoid, G ⊂ M× a subgroup, andS a pointed M -module. Then M/G ⊗M S = S/G is the set of orbits of S under the inducedG-action.

(ii) In the situation of (i), notice that the functor

M -Mod→M/G-Mod : S 7→ S/G

is exact, hence M/G is a flat M -module. (See definition 2.3.22(i).)(iii) Likewise, if Σ ⊂ M is a non-pointed submonoid, then the localization Σ−1M is a flat

M -module, due to remark 3.1.22(vii).(iv) Suppose that S is an integral pointed M -module (with M either pointed or not pointed),

and let Σ ⊂ M be a non-pointed submonoid. Then Σ−1S is also an integral pointed Σ−1M -module. Indeed, suppose that the identity

(3.1.24) (s−1a) · (s′−1b) = (s−1a) · (s′′−1c) 6= 0

holds for some a, b, c ∈ S and s, s′, s′′ ∈ Σ; we need to check that s′−1b = s′′−1c, or equiva-lently, that s′′b = s′c in Σ−1S. However, (3.1.24) is equivalent to s′′ab = s′ac in Σ−1S, andthe latter holds if and only if there exists t ∈ Σ such that ts′′ab = ts′ac in S. The two sides inthe latter identity are 6= 0, as the same holds for the two sides of the identity (3.1.24); therefores′′b = s′c holds already in S, and the contention follows.

(v) Likewise, in the situation of (iv), S/Σ := S⊗MM/Σ is an integral pointedM/Σ-module.Indeed, notice the natural identification S/Σ = Σ−1S ⊗Σ−1M (Σ−1M)/Σgp which – in view of(iv) – reduces the proof to the case where Σ is a subgroup of M×. Then the assertion is easilyverified, taking into account (i).

Especially, if M is an integral pointed monoid, and Σ ⊂ M is any non-pointed submonoid,then both Σ−1M and M/Σ are integral pointed monoids (this generalizes lemma 2.3.38).

(vi) Let G be any abelian group, ϕ : M → G a morphism of non-pointed monoids. ThenG is a flat M-module. For the proof, we may – in light of (iii) – replace M by Mgp, therebyreducing to the case where M is a group. Next, by (ii), we may assume that ϕ is injective, inwhich case G is a free M -module with basis G/M .

Remark 3.1.25. (i) Let M → N and M → N ′ be morphisms of pointed monoids; N andN ′ can be regarded as pointed M -modules in an obvious way, hence we may form the tensorproduct N ′′ := N ⊗M N ′; the latter is endowed with a unique monoid structure such that the


maps e : N → N ′′ and e′ : N ′ → N ′′ given by the rule n 7→ n ⊗ 1 for all n ∈ N (resp.n′ 7→ 1⊗n′ for all n′ ∈ N ′) are morphisms of monoids. Just as for usual ring homomorphisms,the monoid N ′′ is a coproduct of N and N ′ over M , i.e. there is a unique isomorphism ofpointed monoids:

(3.1.26) N ⊗M N ′∼→ N qM N ′

that identifies e and e′ to the natural morphisms N → N qM N ′ and N ′ → N qM N ′. As usualall this extends to non-pointed monoids. (Details left to the reader.)

(ii) Especially, if we take M = 1, the initial object in Mnd, we obtain an explicitdescription of the pointed monoid N ⊕ N ′ : it is the quotient (N × N ′)/∼, where ∼ denotesthe minimal equivalence relation such that (x, 0) ∼ (0, x′) for every x ∈ N , x′ ∈ N ′. Fromthis, a direct calculation shows that a direct sum of pointed integral monoids is again a pointedintegral monoid.

Remark 3.1.27. (i) Clearly every pointed M -module S is the colimit of the filtered familyof its finitely generated submodules. Moreover, S is the colimit of a filtered family of finitelypresented pointed M -modules. Recall the standard argument : pick a countable set I , and let Cbe the (small) full subcategory of the category M -Mod whose objects are the coequalizers ofevery pair of maps of pointed M -modules p, q : M (I1) → M (I2), for every finite sets I1, I2 ⊂ I(this means that, for every such pair p, q we pick one representative for this coequalizer). Thenthere is a natural isomorphism of pointed M -modules :

colimiC/S

ιS∼→ S

where i : C →M -Mod is the inclusion functor, and ιS is the functor as in (1.1.16).(ii) If S is finitely generated, we may find a finite filtration of S by submodules 0 = S0 ⊂

S1 ⊂ · · · ⊂ Sn = S such that Si+1/Si is a cyclic M -module, for every i = 1, . . . , n.(iii) Notice that, if S is integral and S ′ ⊂ S is any submodule, then S/S ′ is again integral.

Moreover, if S is integral and cyclic, we have a natural isomorphism of M -modules :

S∼→M/AnnM(S).

(Details left to the reader.)(iv) Suppose furthermore, that M ] is finitely generated, and S is any pointed M -module.

Lemma 3.1.16(i.a) and proposition 3.1.9(ii) easily imply that every ascending chain

I0 ⊂ I1 ⊂ I2 ⊂ · · ·of ideals of M is stationary; especially, the set AnnM(s) | s ∈ S \ 0 admits maximalelements. Let I be a maximal element in this set; a standard argument as in commutativealgebra shows that I is a prime ideal : indeed, say that xy ∈ I = AnnM(s) and x /∈ I; thenxs 6= 0, hence y ∈ AnnM(xs) = I , by the maximality of I . Now, if S is also finitely generated,it follows that we may find a finite filtration of S as in (ii) such that, additionally, each quotientSi+1/Si is of the form M/p, for some prime ideal p ⊂M .

These properties make the class of integral pointed modules especially well behaved : es-sentially, the full subcategory M -Int.Mod of M -Modo consisting of these modules mimicsclosely a category of A-modules for a ring A, familiar from standard linear algebra. This shallbe amply demonstrated henceforth. For instance, we point out the following combinatorialversion of Nakayama’s lemma :

Proposition 3.1.28. LetM be a (pointed or not pointed) monoid, S a finitely generated integralpointed M -module, and S ′ ⊂ S a pointed submodule, such that

S = S ′ ∪mM · S.Then S = S ′.


Proof. After replacing S by S/S ′, we may assume that mMS = S, in which case we haveto check that S = 0. Suppose then, that S 6= 0; from remark 3.1.27(ii,iii) it follows that Sadmits a (pointed) submodule T ⊂ S such that S/T ' M/mM . Especially, mM · (S/T ) = 0,i.e. mMS ⊂ T , and therefore S = T , which contradicts the choice of T . The contentionfollows.

Remark 3.1.29. (i) The integrality assumption cannot be omitted in proposition 3.1.28. Indeed,take M := N and S := 0, where 0 denotes the final N-module. Then S 6= 0, but mMS = S.

(ii) Let us say that an element of the M -module S is primitive, if it does not lie in mMS. Wededuce from proposition 3.1.28, the following :

Corollary 3.1.30. Let M be a sharp (pointed or not pointed) monoid, S a finitely generatedintegral pointed monoid. Then the set S \ mMS of primitive elements of S is finite, and is theunique minimal system of generators of S.

Proof. Indeed, it is easily seen that every system of generators of S must contain all the primitiveelements, so S \mMS must be finite. On the other hand, let S ′ ⊂ S be the submodule generatedby the primitive elements; clearly S ′ ∪mMS = S, hence S ′ = S, by proposition 3.1.28.

3.1.31. Let R be any ring, M a non-pointed monoid. Notice that the M -module underlying anyR[M ]-module is naturally pointed, whence a forgetful functor R[M ]-Mod → M -Mod. Thelatter admits a left adjoint

M -Mod → R[M ]-Mod : (S, 0S) 7→ R〈S〉 := CokerR[0S].

Likewise, the monoid (A, ·) underlying any (commutative unitary) R-algebra A is naturallypointed, whence a forgetful functor R-Alg→Mnd, which again admits a left adjoint

Mnd → R-Alg : (M, 0M) 7→ R〈M〉 := R[M ]/(0M)

where (0M) ⊂ R[M ] denotes the ideal generated by the image of 0M .If (M, 0M) is a pointed monoid, and S is a pointed (M, 0M)-module, then notice that R〈S〉

is actually a R〈M〉-module, so we have as well a natural functor

(M, 0M)-Mod → R〈M〉-Mod S 7→ R〈S〉

which is again left adjoint to the forgetful functor.For instance, let I ⊂ M be an ideal; from the foregoing, it follows that R〈M/I〉 is naturally

an R[M ]-algebra, and we have a natural isomorphism :

R〈M/I〉 ∼→ R[M ]/IR[M ].

Explicitly, for any x ∈ F := M \I , let x ∈ R〈M/I〉 be the image of x; then R〈M/I〉 is a freeR-module, with basis (x | x ∈ F ). The multiplication law of R〈M/I〉 is determined as follows.Given x, y ∈ F , then x · y = xy if xy ∈ F , and otherwise it equals zero.

Notice that, if I1 and I2 are two ideals of M , we have a natural identification :

R〈M/(I1 ∩ I2)〉 ∼→ R〈M/I1〉 ×R〈M/(I1∪I2)〉 R〈M/I2〉.

These algebras will play an important role in section 6.5. As a special case, suppose that p ⊂Mis a prime ideal; then the inclusion M \p ⊂M induces an isomorphism of R-algebras :

R[M \p]∼→ R〈M/p〉.

Furthermore, general nonsense yields a natural isomorphism of R-modules :

(3.1.32) R〈S ⊗M S ′〉 ∼→ R〈S〉 ⊗R〈M〉 R〈S ′〉 for all pointed M -modules S and S ′.


3.1.33. Let A be a commutative ring with unit, and f : M → (A, ·) a morphism of pointedmonoids. Then f induces (forgetful) functors :

A-Mod→M -Mod A-Alg→M/Mnd

(notation of (1.1.12)) which admit left adjoints :

M -Mod → A-Mod S 7→ S ⊗M A := Z〈S〉 ⊗Z〈M〉 A

M/Mnd → A-Alg N 7→ N ⊗M A := Z〈N〉 ⊗Z〈M〉 A.

Sometimes we may also use the notation :

S ⊗M N := Z〈S〉 ⊗Z〈M〉 N and SL⊗M K• := Z〈S〉

L⊗Z〈M〉 K•

for any pointed M -module S, any A-module N and any object K• of D−(A-Mod). The latterderived tensor product is obtained by tensoring with a flat Z〈M〉-flat resolution of Z〈S〉. (Suchresolutions can be constructed combinatorially, starting from a simplicial resolution of S.) Allthe verifications are standard, and shall be left to the reader.

Definition 3.1.34. LetM be a pointed monoid,A a commutative ring with unit, ϕ : M → (A, ·)a morphism of pointed monoids, N an A-module. We say that N is ϕ-flat (or just M -flat, if noambiguity is likely to arise), if the functor

M -Int.Mod → A-Mod : S 7→ S ⊗M N

is exact, in the sense that it sends exact sequences of pointed integral M -modules, to exactsequences of A-modules. We say that N is faithfully ϕ-flat if this functor is exact in the abovesense, and we have S ⊗M N = 0 if and only if S = 0.

Remark 3.1.35. (i) Notice that the functor S 7→ S ⊗M N of definition 3.1.34 is right exact inthe categorical sense (i.e. it commutes with finite colimits), since it is a right adjoint. However,even when N is faithfully flat, this functor is not always left exact in the categorical sense : itdoes not commute with finite products, nor with equalizers, in general.

(ii) Let M and S be as in definition 3.1.34, and let R be any non-zero commutative unitaryring. Denote by ϕ : M → (R〈M〉, ·) the natural morphism of pointed monoids. In light of(3.1.32), it is clear that if R〈S〉 is a flat R〈M〉-module, then S is a flat pointed M -module, andthe latter condition implies that R〈S〉 is ϕ-flat.

Lemma 3.1.36. Let M be a monoid, A a ring, ϕ : M → (A, ·) a morphism of monoids, andassume that ϕ is local and A is ϕ-flat. Then A is faithfully ϕ-flat.

Proof. Let S be an integral pointed M -module, and suppose that S ⊗M A = 0; we have toshow that S = 0. Say that s ∈ S, and let Ms ⊂ S be the M -submodule generated by s.Since A is ϕ-flat, it follows easily that Ms ⊗M A = 0, hence we are reduced to the casewhere S is cyclic. By remark 3.1.27(iii), we may then assume that S = M/I for some idealI ⊂ M . It follows that S ⊗M A = A/ϕ(I)A, so that ϕ(I) generates A. Since ϕ is local, thisimplies that I = M , whence the contention.

Lemma 3.1.37. Let M be an integral pointed monoid, I, J ⊂ M two ideals, A a ring, α :M → (A, ·) a morphism of monoids, N an α-flat A-module, and S a flat M -module. Then :

IS ∩ JS = (I ∩ J)S

α(I)N ∩ α(J)N = α(I ∩ J)N.


Proof. We consider the commutative ladder of pointed M -modules, with exact rows and injec-tive vertical arrows :

(3.1.38)

0 // I ∩ J //

I //

I/(I ∩ J) //

0

0 // J // M // M/(I ∪ J) // 0

By assumption, the ladder ofA-modules (3.1.38)⊗MN has still exact rows and injective verticalarrows. Then, the snake lemma gives the following short exact sequence involving the cokernelsof the vertical arrows :

0→ JN/(I ∩ J)N → N/INp−→ N/(IN + JN)→ 0

(where we have written JN instead of α(J)N , and likewise for the other terms). HoweverKer p = JN/(IN ∩ JN), whence the second stated identity. The first stated identity can bededuced from the second, by virtue of remark 3.1.35(ii).

Remark 3.1.39. By inspection of the proof, we see that the first identity of lemma 3.1.37 holds,more generally, whenever Z〈S〉 is ϕ-flat, where ϕ : M → Z〈M〉 is the natural morphism ofpointed monoids.

Proposition 3.1.40. LetM be a pointed integral monoid, A a ring, ϕ : M → (A, ·) a morphismof monoids, N an A-module. Then we have :

(i) The following conditions are equivalent :(a) N is ϕ-flat.(b) Tor

Z〈M〉i (Z〈T 〉, N) = 0 for every i > 0 and every pointed integral M -module T .

(c) TorZ〈M〉1 (Z〈M/I〉, N) = 0 for every ideal I ⊂M .

(d) The natural map I ⊗M N → N is injective for every ideal I ⊂M .(ii) If moreover, M ] is finitely generated, then the conditions (a)-(d) of (i) are equivalent to

either of the following two conditions :(e) Tor

Z〈M〉1 (Z〈M/p〉, N) = 0 for every prime ideal p ⊂M .

(f) The natural map p⊗M N → N is injective for every prime ideal p ⊂M .

Proof. Clearly (a)⇒(b)⇒(c)⇒(e). Next, by considering the short exact sequence of pointedintegral M -modules 0→ I →M →M/I → 0 we easily see that (c)⇔(d) and (e)⇔(f).

(c)⇒(a) : Let Σ := (0 → S ′ → S → S ′′ → 0) be a short exact sequence of pointedintegral M -modules; we need to show that the induced map S ′ ⊗M N → S ⊗M N is injective.Since the sequence Z〈Σ〉 is still exact, the long Tor-exact sequence reduces to showing thatTor

Z〈M〉1 (Z〈T 〉, N) = 0 for every pointed integral M -module T . In view of remark 3.1.27(i),

we are easily reduced to the case where T is finitely generated; next, using remark 3.1.27(ii,iii),the long exact Tor-sequence, and an easy induction on the number of generators of T , we mayassume that T = M/I , whence the contention.

Lastly, if M ] is finitely generated, then remark 3.1.27(iv) shows that, in the foregoing argu-ment, we may further reduce to the case where T = M/p for a prime ideal p ⊂ M ; this showsthat (e)⇒(a).

Lemma 3.1.41. Let M be a pointed monoid, S a pointed M -module, and suppose that thefollowing conditions hold for S :

(F1) If s ∈ S and a ∈ AnnM(s), then there exist b ∈ AnnM(a) such that s ∈ bS.(F2) If a1, a2 ∈ M and s1, s2 ∈ S satisfy the identity a1s1 = a2s2 6= 0, then there exist

b1, b2 ∈M and t ∈ S such that si = bit for i = 1, 2 and a1b1 = a2b2.Then the natural map I ⊗M S → S is injective for every ideal I ⊂M .


Proof. Let I be an ideal, and suppose that two elements a1⊗s1 and a2⊗s2 of I⊗MS are mappedto the same element of S. If aisi = 0 for i = 1, 2, then (F1) says that there exist b1, b2 ∈M andt1, t2 ∈ S such that aibi = 0 and si = biti for i = 1, 2; thus ai ⊗ si = ai ⊗ biti = aibi ⊗ si = 0in I ⊗M S. In case aisi 6= 0, pick b1, b2 ∈ M and t ∈ S as in (F2); we conclude thata1 ⊗ s1 = a1 ⊗ b1t = a1b1 ⊗ t = a2b2 ⊗ t = a2 ⊗ s2 in I ⊗M S, whence the contention.

Theorem 3.1.42. Let M be an integral pointed monoid, S a pointed M -module. The followingconditions are equivalent :

(a) S is M -flat.(b) For every morphism M → P of pointed monoids, P ⊗M S is P -flat.(c) For every short exact sequence Σ of integral pointedM -modules, the sequence Σ⊗M S

is again short exact.(d) Conditions (F1) and (F2) of lemma 3.1.41 hold for S.

Proof. Clearly (b)⇒(a)⇒(c).(c)⇒(d): To show (F1), set I := Ma, and denote by i : I →M the inclusion; (c) implies that

the induced map i⊗M S : I ⊗M S → S is injective. However, we have a natural isomorphismI∼→ M/AnnM(a) of M -modules (remark 3.1.27(iii)), whence an isomorphism : I ⊗M S

∼→S/AnnM(a)S, and under this identification, i ⊗M S is induced by the map S → S : s 7→ as.Thus, multiplication by a maps the subset S \AnnM(a)S injectively into itself, which is theclaim.

For (F2), notice that Z〈S〉 is ϕ-flat under condition (c), for ϕ : M → Z〈M〉 the naturalmorphism. Now, say that a1s1 = a2s2 6= 0 in S; set I := Ma1, J := Ma2; the assumptionmeans that a1s1 ∈ IS ∩ JS, in which case remark 3.1.39 shows that there exist t ∈ S andb1, b2 ∈ M such that a1b1 = a2b2, and a1s1 = a1b1t, hence a2s2 = a2b2t. Since we have seenthat multiplication by a1 maps S \AnnM(a1)S injectively into itself, we deduce that s1 = b1t,and likewise we get s2 = b2t.

To prove that (d)⇒(b), we observe :

Claim 3.1.43. Let P be a pointed monoid, Λ a small locally directed category (see definition1.1.37(iv)), and S• : Λ→ P -Mod a functor, such that Sλ fulfills conditions (F1) and (F2), forevery λ ∈ Ob(Λ). Then the colimit of S• also fulfills conditions (F1) and (F2).

Proof of the claim. In light of remark 1.1.38(ii), we may assume that Λ is either discrete orpath-connected. Suppose first that Λ is path-connected; then remark 2.3.17(ii) allows to checkdirectly that conditions (F1) and (F2) hold for the colimit of S•, since they hold for every Sλ.If Λ is discrete, the assertion is that conditions (F1) and (F2) are preserved by arbitrary (small)direct sums, which we leave as an exercise for the reader. ♦

To a given pointed M -module S, we attach the small category S∗, such that :

Ob(S∗) = S\0 and HomS∗(s′, s) = a ∈M | as = s′.

The composition of morphisms is induced by the composition law of M , in the obvious way.Notice that S∗ is locally directed if and only if S satisfies condition (F2).

We define a functor F : S∗ → M -Mod as follows. For every s ∈ Ob(S∗) we let F (s) :=M , and for every morphism a : s′ → s we let F (a) := a ·1M . We have a natural transformationτ : F ⇒ cS , where cS : S∗ → M -Mod is the constant functor associated to S; namely, forevery s ∈ Ob(S∗), we let τs : M → S be the map given by the rule a 7→ as for all a ∈ M .There follows a morphism of pointed M -modules :

(3.1.44) colimS∗

F → S

Claim 3.1.45. If S fulfills conditions (F1), the map (3.1.44) is an isomorphism.


Proof of the claim. Indeed, we have a natural decomposition of S∗ as coproduct of a family(S∗i | i ∈ I) of path-connected subcategories (for some small set I : see remark 1.1.38(ii));especially we have HomS∗(s, s

′) = ∅ if s ∈ Ob(S∗i ) and s′ ∈ Ob(S∗j ) for some i 6= j in I .For each i ∈ I , let Fi : S∗i → M -Mod be the restriction of F . There follows a natural

isomorphism : ⊕i∈I

colimS∗i

Fi∼→ colim

S∗F.

Since the colimit of Fi commutes with the forgetful functor to sets, an inspection of the defi-nitions yields the following explicit description of the colimit Ti of Fi. Every element of Ti isrepresented by some pair (s, a) where s ∈ Ob(S∗i ) ⊂ S\0 and a ∈ M ; such pair is mappedto as by (3.1.44), and two such pairs (s, a), (s′, a′) are identified in Ti if there exists b ∈ Msuch that bs = s′ and ba′ = a.

Hence, denote by Si the image under (3.1.44) of Ti; we deduce first, that Si ∩ Sj = 0 ifi 6= j. Indeed, say that t ∈ Si ∩ Sj; by the foregoing, there exist si ∈ Ob(S∗i ), sj ∈ Ob(S∗j ),and ai, aj ∈ M , such that aisi = t = ajsj . If t 6= 0, we get morphisms ai : t → si andaj : t → sj in S∗; say that t ∈ S∗k for some k ∈ I; it then follows that S∗i = S∗k = S∗j , acontradiction. Next, it is clear that (3.1.44) is surjective. It remains therefore only to show thateach Ti maps injectively onto Si. Hence, say that (s1, a1) and (s2, a2) represent two elements ofTi with t := a1s1 = a2s2. If t 6= 0, we get, as before, morphisms a1 : t→ s1 and a2 : t→ s2 inS∗i , and the two pairs are identified in Ti to the pair (t, 1). Lastly, if t = 0, condition (F1) yieldsb ∈ M and s′ ∈ S such that a1b = 0 and s1 = bs′, whence a morphism b : s1 → s′ in S∗i , andFi(b)(a1, s1) = (0, s′) which represents the zero element of Ti. The same argument applies aswell to (a2, s2), and the claim follows. ♦

Claim 3.1.46. Let M → P be any morphism of pointed monoids, and S a pointed M -modulefulfilling conditions (F1) and (F2). Then the natural map I ⊗M S → P ⊗M S is injective, forevery ideal I ⊂ P .

Proof of the claim. From claim 3.1.45 we deduce that P ⊗M S is the locally directed colimit ofthe functor P ⊗M F , and notice that the pointed P -module P fulfills conditions (F1) and (F2);by claim 3.1.43 we deduce that P ⊗M S also fulfills the same conditions, so the claim followsfrom lemma 3.1.41. ♦

After these preliminaries, suppose that conditions (F1) and (F2) hold for S, and let Σ :=(0→ T ′ → T → T ′′ → 0) be a short exact sequence of pointed M -modules. We wish to showthat Σ ⊗M S is still short exact. However, if U ′′ ⊂ T ′′ is any M -submodule, let U ⊂ T be thepreimage of U ′′, and notice that the induced sequence 0 → T ′ → U → U ′′ → 0 is still shortexact. Since a filtered colimit of short exact sequences is short exact, remark 3.1.27(i) allows toreduce to the case where T ′′ is finitely generated.

We shall argue by induction on the number n of generators of T ′′. Hence, suppose first thatT ′′ is cyclic, and let t ∈ T be any element whose image in T ′′ is a generator. Set C := Mt ⊂ T ,and let C ′ ⊂ T ′ be the preimage of C. We obtain a cocartesian (and cartesian) diagram ofpointed M -modules :

D :

C ′ //

C

T ′ // T.

The induced diagram D ⊗M S is still cocartesian, hence the same holds for the diagram of setsunderlying D ⊗M S (remark 2.3.17(ii)). Especially, if the induced map C ′ ⊗M S → C ⊗M Sis injective, the same will hold for the map T ′ ⊗M S → T ⊗M S. We may thus replace T ′ andT by respectively C ′ and C, which allows to assume that also T is cyclic. In this case, pick a


generator u ∈ T ; we claim that there exists a unique multiplication law µT on T , such that thesurjection p : M → T : a 7→ au is a morphism of pointed monoids. Indeed, for every t, t′ ∈ T ,write t = au for some a ∈ M , and set µT (t, t′) := at′. Using the linearity of p we easilycheck that µT (t, t′) does not depend on the choice of a, and the resulting composition law µTis commutative and associative. Then T ′ is an ideal of T , so claim 3.1.46 tells us that the mapT ′ ⊗M S → T ⊗M S is injective, as required.

Lastly, suppose that n > 1, and the assertion is already known whenever T ′′ is generated byat most n− 1 elements. Let U ′′ ⊂ T ′′ be a pointed M -submodule, such that U ′′ is generated byat most n− 1 elements, and T ′′/U ′′ is cyclic. Denote by U ⊂ T the preimage of U ′′; we deduceshort exact sequences Σ′ := (0→ T ′ → U → U ′′ → 0) and Σ′′ := (0→ U → T → T ′′/U ′′ →0), and by inductive assumption, both Σ′ ⊗M S and Σ′′ ⊗M S are short exact. Therefore thenatural map T ′⊗M S → T ⊗M S is the composition of two injective maps, hence it is injective,as stated.

Remark 3.1.47. In the situation of remark 3.1.35(ii), suppose that M is pointed integral. Thentheorem 3.1.42 implies that S is a flat pointed M -module if and only if R〈S〉 is ϕ-flat.

Corollary 3.1.48. Let M be an integral pointed monoid, S a pointed M -module. Then(i) The following conditions are equivalent :

(a) S is M -flat.(b) For every ideal I ⊂M , the induced map I ⊗M S → S is injective.

(ii) If moreover M ] is finitely generated, then these conditions are equivalent to :(c) For every prime ideal p ⊂M , the induced map p⊗M S → S is injective.

Proof. (i): Indeed, by remark 3.1.47 and proposition 3.1.40(i) (together with (3.1.32)), both (a)and (b) hold if and only if Z〈S〉 is ϕ-flat, for ϕ : M → Z〈M〉 the natural morphism.

(ii): This follows likewise from proposition 3.1.40(ii).

Corollary 3.1.49. Let ϕ : M → N be a morphism of pointed monoids, G ⊂ M×, H ⊂ N×

two subgroups such that ϕ(G) ⊂ H , and denote by ϕ : M/G → N/H the induced morphism.Let also S be any N -module. We have :

(i) If S(ϕ) is a flat M -module, then S/H(ϕ) is a flat M/G-module (notation of (1.2.26)).(ii) If moreover, M is a pointed integral monoid and S is a pointed integral H-module,

then also the converse of (i) holds.

Proof. (i): We have a natural isomorphism

(S/H)(ϕ)∼→ N/H ⊗N/ϕG S(ϕ)/ϕG.

However, by example 3.1.23(ii), N/H is a flat N/ϕG-module, and S(ϕ)/ϕG is a flat M/G-module, whence the contention.

(ii): By theorem 3.1.42, it suffices to check that conditions (F1) and (F2) of lemma 3.1.41hold in S(ϕ), and notice that, by the same token, both conditions hold for the M/G-moduleS/H(ϕ) = S(ϕ)/G, since M/G is pointed integral (example 3.1.23(v)).

Hence, say that ϕ(a)s = 0 for some a ∈ M and s ∈ S; we may then find b ∈ M , t′ ∈ S andg ∈ G such that ϕ(gb)t = s and ab = 0. Setting t := ϕ(g)t′, we deduce that (F1) holds.

Next, say that ϕ(a1)s1 = ϕ(a2)s2 6= 0 for some a1, a2 ∈M and s1, s2 ∈ S; then we may findg ∈ G, h1, h2 ∈ H , b1, b2 ∈M and t′ ∈ S such that ga1b1 = a2b2 and

(3.1.50) ϕ(bi)hit′ = si for i = 1, 2.

After replacing b1 by gb1 and h1 by h1ϕ(g−1), we reduce to the case where a1b1 = a2b2. From(3.1.50) we deduce that

ϕ(a1b1)h1t′ = ϕ(a1)s1 = ϕ(a2)s2 = ϕ(a2b2)h2t

′ = ϕ(a1b1)h2t′


whence h1 = h2, since S is a pointed integral H-module. Setting t := h1t′, we see that (F2)

holds.

Corollary 3.1.51. Let ϕ : M → N be a flat morphism of pointed monoids, with M pointedintegral, and let p ⊂ N be any prime ideal. Then the morphism M/ϕ−1p→ N/p induced by ϕis also flat.

Proof. Let F := N \p; by theorem 3.1.42, it suffices to check that conditions (F1) and (F2)of lemma 3.1.41 hold for the ϕ−1F -module F . However, since 0 /∈ F , condition (F1) holdstrivially. Moreover, by assumption these two conditions hold for the M -module N ; hence, saythat ϕ(a1) · s1 = ϕ(a2) · s2 in F , for some a1, a2 ∈ ϕ−1F and s1, s2 ∈ F . It follows that thereexist b1, b2 ∈M and t ∈ N such that a1b1 = a2b2 in M , and ϕ(bi) · t = si for i = 1, 2. Since Fis a face, this implies that ϕ(b1), ϕ(b2), t ∈ F , so (F2) holds for F , as stated.

Another corollary is the following analogue of a well known criterion due to Lazard.

Proposition 3.1.52. Let M be an integral pointed monoid, S an integral pointed M -module, Ra non-zero commutative ring with unit. The following conditions are equivalent :

(a) S is M -flat.(b) S is the colimit of a filtered system of free pointed M -modules (see remark 2.3.17(ii)).(c) R〈S〉 is a flat R〈M〉-module.

Proof. Obviously (b)⇒(a) and (b)⇒(c).(c)⇒(a) has already been observed in remark 3.1.35(ii).(a)⇒(b): It suffices to prove that if S is flat, the category S∗ attached to S as in the proof

of theorem 3.1.42, is pseudo-filtered. However, a simple inspection of the construction showsthat S∗ is pseudo-filtered if and only if S satisfies both condition (F2) of lemma 3.1.41, and thefollowing further condition. For every a, b ∈ M and s ∈ S such that as = bs 6= 0, there existst ∈ S and c ∈ M such that ac = bc and ct = s. This condition is satisfied by every integralpointed M -module, whence the contention.

3.1.53. Consider now, a cartesian diagram of integral pointed monoids :

D(P0, I, P1) :

P0//

P1

P2

// P3

where P2 (resp. P3) is a quotient P0/I (resp. P1/IP1) for some ideal I ⊂ P0, and the verticalarrows of D(P0, I, P1) are the natural surjections. In this situation, it is easily seen that theinduced map I → IP1 is bijective; especially, I is both a P0-module and a P1-module. Letϕi : Pi → Z〈Pi〉 be the units of adjunction, for i = 0, 1, 2. Let also M be any Z〈P0〉-module.

Lemma 3.1.54. In the situation of (3.1.53), we have :(i) Let J ⊂ P0 be any ideal. Then S := P0/J admits a three-step filtration

0 ⊂ Fil0S ⊂ Fil1S ⊂ Fil2S = S

such that Fil0S and gr2S are P2-modules, and gr1S is a P1-module.(ii) The following conditions are equivalent :

(a) M is ϕ0-flat.(b) M ⊗P0 Pi is ϕi-flat and Tor

Z〈P0〉1 (M,Z〈Pi〉) = 0, for i = 1, 2.

(iii) The following conditions are equivalent :(a) Tor

Z〈P0〉1 (M,Z〈Pi〉) = 0 for i = 1, 2, 3.

(b) TorZ〈Pi〉1 (M ⊗P0 Pi,Z〈P3〉) = 0 for i = 1, 2.


(iv) Suppose moreover, that P3 6= 0 is a free pointed P2-module. Then the Z〈P0〉-moduleM is ϕ0-flat if and only if the Z〈P1〉-module M ⊗P0 P1 is ϕ1-flat.

Proof. (i): Define Fil1S := Ker(S → P0/(I ∪ J)) = (I ∪ J)/J . Then it is already clear thatS/Fil1S is a P2-module. Next, let Fil0S := Ker(Fil1S → (I ∪ JP1)/JP1) = JP1/J . SinceIP1 = I , we see that Fil0S is a P2-module, and (I ∪ JP1)/JP1 is a P1-module.

(ii): Clearly (a)⇒(b), hence suppose that (b) holds, and let us prove (a). By proposition3.1.40(i), it suffices to show that Tor

Z〈P0〉1 (M,Z〈P0/I〉) = 0 for every ideal J ⊂ P0. In view of

(i), we are then reduced to showing that TorZ〈P0〉1 (M,Z〈S〉) = 0, whenever S is a Pi-module,

for i = 1, 2. However, for such Pi-module S, we have a base change spectral sequence

E2pq : TorZ〈Pi〉p (TorZ〈P0〉

q (M,Z〈Pi〉),Z〈S〉)⇒ TorZ〈P0〉p+q (M,Z〈S〉).

Under assumption (b), we deduce : TorZ〈P0〉1 (M,Z〈S〉) = Tor

Z〈Pi〉1 (M ⊗P0 Pi,Z〈S〉) = 0.

(iii): Notice that the induced diagram of rings Z〈D(P0, I, P1)〉 is still cartesian. Then, this isa special case of [36, Lemma 3.4.15].

(iv): Suppose that M ⊗P0 P1 is ϕ1-flat, and P3 is a free pointed P2-module, say P3 ' P(Λ)2 ,

for some set Λ 6= ∅; then the Z〈P2〉-module Z〈P3〉 is isomorphic to Z〈P2〉(Λ), especially it isfaithfully flat, and we deduce that Tor

Z〈P0〉1 (M,Z〈Pi〉) = 0 for i = 1, 2, by (iii). On the other

hand, M ⊗P0 P3 is ϕ3-flat, so it also follows that M ⊗P0 P2 is ϕ2-flat, by proposition 3.1.52.Summing up, this shows that M fulfills condition (ii.b), hence also (ii.a), as sought.

3.1.55. Let now P → Q be an injective morphism of integral pointed monoids, and supposethat P ] and Q] are finitely generated monoids, and Q is a finitely generated P -module. DenoteϕP : P → Z〈P 〉 and ϕQ : Q→ Z〈Q〉 the usual units of adjunction.

Theorem 3.1.56. In the situation of (3.1.55), let M be a Z〈P 〉-module, and suppose that theZ〈Q〉-module M ⊗P Q is ϕQ-flat. Then M is ϕP -flat.

Proof. Using lemma 3.1.54(iv), and an easy induction, it suffices to show that there exists afinite chain

(3.1.57) P = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn = Q

of inclusions of integral pointed monoids, and for every j = 0, . . . , n− 1 an ideal Ij ⊂ Qj , anda cartesian diagram of integral pointed monoids D(Qj, Ij, Qj+1) as in (3.1.53), and such thatQj+1/Ij 6= 0 is a free pointed Qj/Ij-module. (Notice that each Q]

i is a quotient of Qi/P×, and

the latter is a submodule of Q/P×, hence Q]i is still a finitely generated monoid, by proposition

3.1.9(i).) If P = Q, there is nothing to prove; so we may assume that P is strictly contained inQ, and – invoking again proposition 3.1.9(i) – we further reduce to showing that there exists amonoid Q1 ⊂ Q strictly containing P , and an ideal I ⊂ P , such that the diagram D(P, I,Q1)fulfills the above conditions.

Suppose first that the support of Q/P contains only the maximal ideal mP of P , and letx1, . . . , xr be a finite system of generators for mP (proposition 3.1.9(ii)). For each i = 1, . . . , r,the localization (Q/P )xi is a Pxi-module with empty support, hence (Q/P )xi = 0 (remark3.1.22(v)). It follows that every element of Q/P is annihilated by some power of xi, and sinceQ/P is finitely generated, we may find N ∈ N large enough, such that xNi Q/P = 0 fori = 1, . . . , r. After replacing N by some possibly larger integer, we get mN

P · Q/P = 0, andwe may assume that N is the least integer with this property. If N = 0, there is nothing toprove; hence suppose that N > 0, and set Q1 := P ∪ mN−1

P Q. Notice that Q1 is a monoid,and Q1/P 6= 0 is annihilated by mP . Especially, mPQ1 = mP . Moreover, the induced mapP/mP → Q1/mP is injective and Q1 is an integral pointed module, therefore the group P× acts


freely on Q1\mP , i.e. Q1/mP is a free pointed P/mP -module. It follows easily that, if we takeI := mP , we do obtain a diagram D(P,mP , Q1) with the sought properties, in this case.

For the general case, let p ⊂ P be a minimal element of SuppQ/P (for the ordering givenby inclusion). Then the induced morphism Pp → Qp still satisfies the conditions of (3.1.55)(lemma 3.1.20(iv)). Moreover, SuppQp/Pp = pPp by remark 3.1.22(vii). By the previouscase, we deduce that there exists a chain of inclusions of integral pointed monoids Pp ⊂ Q′1 ⊂Qp, such that the resulting diagram D(Pp, pPp, Q

′1) is cartesian, and Q′1/pPp 6= 0 is a free

pointed Pp/pPp-module. Let e1, . . . , ed be a basis of the latter Pp/pPp-module, with e1 = 1.Hence, ei ∈ Qp \ pPp for every i = 1, . . . , d, and after multiplying e2, . . . , ed by a suitableelement of P \ p, we may assume that each ei is the image in Qp of an element ei ∈ Q.Moreover, for every i, j ≤ d, either eiej = 0, or else there exists aij ∈ Pp and k(i, j) ≤ dsuch that eiej = aijek(i,j). Furthermore, fix a system of generators x1, . . . , xr for p; then, forevery i ≤ r and every j ≤ d we have xiej ∈ pPp. Again, after multiplying e2, . . . , ed by somec ∈ P \p, we may assume that aij ∈ P for every i, j ≤ d, and moreover that xiej lies in theimage of p for every i ≤ r and j ≤ d.

And if we multiply yet again e2, . . . , ed by a suitable element of P \p, we may finally reach asystem of elements e1, . . . , ed ∈ Q such that e1 = 1 and :

• For every i, j ≤ d, we have either eiej = 0 or else eiej = aijek(i,j).• xiej ∈ p for every i ≤ r and j ≤ d.

Clearly these elements span a P -module Q1 which is a monoid containing P and contained inQ; moreover, by construction we have pQ1 = p, hence the resulting diagram D(P, p, Q1) iscartesian. Notice also that (Q1/p)p ' Q′1/pPp, and that P/p = P ′ , where P ′ := P \p is anintegral (non-pointed) monoid. To conclude it suffices now to apply the following

Claim 3.1.58. Let P ′ be an integral non-pointed monoid, S a pointed P ′ -module, and e :=(e1, . . . , ed) a system of generators for S. Suppose that S ⊗P ′ P ′gp

is a free pointed P ′gp -

module, and the image of e is a basis for this module. Then S is a free pointed P ′ -module, withbasis e.

Proof of the claim. If e is not a basis, we have a relation in S of the type a1e1 = a2e2, forsome a1, a2 ∈ P ′. This relation must persist in S ⊗P ′ P ′gp

, and implies that a1 = a2 = 0 inP ′gp . However, under the stated assumptions the localization map P ′ → P ′gp

is injective, acontradiction.

3.2. Integral monoids. We begin presently the study of a special class of monoids, the integralnon-pointed monoids, and the subclass of saturated monoids (see definition 2.3.40(iii)). Later,we shall complement this section with further results on fine monoids (see sections 3.4 and 6.5).Throughout this section, all the monoids under consideration shall be non-pointed.

Definition 3.2.1. Let ϕ : M → N be a morphism of monoids.(i) ϕ is said to be integral if, for any integral monoid M ′, and any morphism M → M ′,

the push-out N ⊗M M ′ is integral.(ii) ϕ is said to be strongly flat (resp. strongly faithfully flat) if the induced morphism

Z[ϕ] : Z[M ]→ Z[N ] is flat (resp. faithfully flat).

Lemma 3.2.2. Let f : M → N and g : N → P be two morphisms of monoids.(i) If f and g are integral (resp. strongly flat), the same holds for g f .

(ii) If f is integral (resp. strongly flat), and M → M ′ is any other morphism, then themorphism 1M ′ ⊗M f : M ′ →M ′ ⊗M N is integral (resp. strongly flat).

(iii) If f is integral, and S ⊂ M and T ⊂ N are any two submonoids such that f(S) ⊂ T ,then the induced morphism S−1M → T−1N is integral.


(iv) If S ⊂M is any submonoid, the natural map M → S−1M is strongly flat.(v) If f is integral, then the same holds for f int, and the natural map M int ⊗M N → N int

is an isomorphism.(vi) The unit of adjunction M →M int is an integral morphism.

Proof. (i) is obvious. Assertion (ii) for integral maps is likewise clear, and (ii) for strongly flatmorphisms follows from (2.3.52). Assertion (iv) follows immediately from (2.3.53).

(iii): Let S−1M → M ′ be any map of integral monoids; in view of lemma 2.3.34, we haveP := M ′ ⊗S−1M T−1N ' T−1(M ′ ⊗M N), hence P is integral, which is the claim.

(v): The second assertion follows easily by comparing the universal properties (details left tothe reader); using this and and (ii), we deduce that f int is integral.

(vi) is left to the reader.

Theorem 3.2.3. Let ϕ : M → N be a morphism of integral monoids. Consider the followingconditions :

(a) ϕ is integral.(b) ϕ is flat (see remark 2.3.23(vi)).(c) ϕ is flat and injective.(d) ϕ is strongly flat.(e) For every field k, the induced map k[ϕ] : k[M ]→ k[N ] is flat.

Then : (e)⇔ (d)⇔ (c)⇒ (b)⇔ (a).

Proof. This result appears in [52, Prop.4.1(1)], with a different proof.(a)⇒(b): Let I ⊂M be any ideal; we consider the M -module :

R(M, I) :=⊕n∈N

In

where In denotes, for each n ∈ N, the n-th power of I in the monoid (P(M), ·) of (3.1.1).Then there exists an obvious multiplication law on R(M, I), such that the latter is a N-gradedintegral monoid, and the inclusion M → R(M, I) in degree zero is a morphism of monoids.We call R(M, I) the Rees monoid associated to M and I .

Denote also by j : R(M, I)→M ×N the natural inclusion map. By assumption, the monoidR(M, I) ⊗M N is integral. However, the natural map R(M, I) ⊗M N → (R(M, I) ⊗M N)gp

factors through j⊗MN . The latter means that, for every n ∈ N, the induced map In⊗MN → Nis injective. Then the assertion follows from proposition 3.1.40 and remark 2.3.21(ii).

(b)⇒(a): Let M → M ′ be any morphism of integral monoids; we need to show that thenatural map M ′ ⊗M N →M ′gp ⊗Mgp Ngp is injective (see remark 3.1.25(i)). The latter factorsthrough the morphism M ′ ⊗M N → M ′gp ⊗M N , which is injective by theorem 3.1.42 andremark 2.3.21(ii). We are thus reduced to proving the injectivity of the natural map :

M ′gp ⊗Mgp (Mgp ⊗M N)→M ′gp ⊗Mgp Ngp.

By comparing the respective universal properties, it is easily seen that Mgp ⊗M N is the lo-calization ϕ(M)−1N , which of course injects into Ngp. Then the contention follows from thefollowing general :

Claim 3.2.4. LetG be a group, T → T ′ an injective morphism of monoids, G→ P a morphismof monoids. Then the natural map P ⊗G T → P ⊗G T ′ is injective.

Proof of the claim. This follows easily from remark 3.1.25(i) and lemma 2.3.31(ii). ♦

(d)⇒(e) and (c)⇒(b) are trivial.(e)⇒(c): The flatness of ϕ has already been noticed in remark 3.1.35(ii). To show that ϕ is

injective, let a1, a2 ∈ M and let k be any field. Under assumption (e), the image in k[N ] of


the annihilator Annk[M ](a1 − a2) generates the ideal Annk[N ](ϕ(a1)− ϕ(a2)). Set b := a1a−12 ;

it follows that Annk[Mgp](1 − b) generates Annk[Ngp](1 − ϕ(b)). However, one checks easilythat the annihilator of 1 − b in k[Mgp] is either 0 if b is not a torsion element of Mgp, or elseis generated (as an ideal) by 1 + b + · · · + bn−1, where n is the order of b in the group Mgp.Now, if ϕ(a1) = ϕ(a2), we have ϕ(b) = 1, hence the annihilator of 1 − b cannot be 0, and infact k[Ngp] = Annk[Ngp](1 − ϕ(b)) = nk[Ngp]. Since k is arbitrary, it follows that n = 1, i.e.a1 = a2.

(c)⇒(d): since ϕ is injective, N is an integral pointed M-module, so the assertion is aspecial case of proposition 3.1.52.

Remark 3.2.5. (i) Let G be a group; then every morphism of monoids M → G is integral.Indeed, lemma 3.2.2(i,vi) reduces the assertion to the case where M is integral, in which case itis an immediate consequence of theorem 3.2.3 and example 3.1.23(vi).

(ii) Let M be an integral monoid, and S a flat pointed M-module. From theorem 3.1.42 wesee that T := S\0 is an M -submodule of S, hence S = T.

(iii) Let ϕ : M → N be a morphism of integral monoids, and set Γ := Cokerϕgp; thenthe natural map π : N → Γ defines a grading on N (see definition 2.3.8(i)), which we call theϕ-grading. As usual, we shall write Nγ instead of π−1(γ), for every γ ∈ Γ. We shall use theadditive notation for the composition law of Γ; especially, the neutral element shall be denotedby 0. Clearly ϕ factors through a morphism of monoids M → N0, and each graded summandNγ is naturally a M -module.

(iv) With the notation of (iii), we claim that the induced morphism ϕ : ϕ(M) → N is flat(hence strongly flat, by theorem 3.2.3), if and only ifNγ is a filtered union of cyclicM -modules,for every γ ∈ Γ. Indeed, notice that a cyclic M -submodule of Nγ is a free ϕ(M)-module ofrank one (since N is integral), hence the condition implies that Nγ is a flat ϕ(M)-module,hence ϕ flat. Conversely, suppose that ϕ is flat, and let n1, n2 ∈ Nγ (for some γ ∈ Γ); thismeans that there exist a1, a2 ∈ M such that ϕ(a1)n1 = ϕ(a2)n2 in N . Then, condition (F2) oftheorem 3.1.42 says that there exist n′ ∈ N and b1, b2 ∈ M such that ni = ϕ(bi)n

′ for i = 1, 2;especially, n′ ∈ Nγ , which shows that Nγ is a filtered union of cyclic M -modules.

(v) With the notation of (iii), notice as well, that a morphism ϕ : M → N of integral monoidsis exact (see definition 2.3.40(i)) if and only if Kerϕgp ⊂ M and ϕ induces an isomorphismM/Kerϕgp ∼→ N0. (Details left to the reader.)

Theorem 3.2.6. Let M → N be a finite, injective morphism of integral monoids, and S apointed M-module. Then S is M-flat if and only if N ⊗M S is N-flat.

Proof. In light of theorem 3.1.42, we may assume that N ⊗M S is N-flat, and we shall showthat S is M-flat. To this aim, let 1 denote the trivial monoid (the initial and final object inthe category of monoids); any pointed M-module X is a pointed 1-module by restriction ofscalars, and if X and Y are any two pointed M-modules, we define a M-module structure onX ⊗1 Y by the rule

a · (x⊗ y) := ax⊗ ay for every a ∈M, x ∈ X and y ∈ Y .

With this notation, we remark :

Claim 3.2.7. Let ϕ : M → G be a morphism of monoids, where G is a group and M isintegral. Then a pointed M-module S is M-flat if and only if the same holds for the M-module S ⊗1 G.

Proof of the claim. Suppose that S is M-flat; then S = T for some M -module T (remark3.2.5(ii)), and it is easily seen that S⊗1G = (T ×G). By theorem 3.1.42, it suffices to checkthat conditions (F1) and (F2) of lemma 3.1.41 hold for (T ×G).


Hence, suppose that a ∈ M and h ∈ (T × G) satisfy the identity ah = 0; in this case, asimple inspection shows that either a = 0 or h = 0; condition (F1) follows straightforwardly.To check (F2), say that a1 · (s1, g) = a2 · (s2, g2) 6= 0, for some ai ∈ M , si ∈ S, gi ∈ G(i = 1, 2); since (F2) holds for S, we may find b1, b2 ∈ M and t ∈ S such that a1b1 = a2b2 andbit = si (i = 1, 2). Notice that g := ϕ(b1)−1g1 = ϕ(b2)−1g2 in G, therefore bi · (t, g) = (si, gi)for i = 1, 2, whence the contention.

Conversely, suppose that S⊗1 G is M-flat; we wish to show that (F1) and (F2) hold for S.However, suppose that as = 0 for some a ∈ M and s ∈ S with s 6= 0; then a · (s ⊗ e) = 0(where e ∈ G is the neutral element); since (F1) holds for S ⊗1 G, we deduce that there existb ∈M and t⊗ g ∈ S⊗1G such that ba = 0 and s⊗ e = b · (t⊗ g) = bt⊗ϕ(b)g; this impliesthat bt = s, so (F1) holds for S, as sought.

Lastly, suppose that a1s1 = a2s2 6= 0 for some ai ∈ M and si ∈ S (i = 1, 2). It follows thata1 · (s1 ⊗ ϕ(a2)) = a2 · (s2 ⊗ ϕ(a1)). By applying condition (F2) to this identity in S ⊗1 G,we deduce that the same condition holds also for S. ♦

Next, we observe that there is a natural isomorphism of N-modules :

(3.2.8) N ⊗M (S ⊗1 Ngp )

∼→ (N ⊗M S)⊗1 Ngp n⊗ (s⊗ g) 7→ (n⊗ s)⊗ ϕ(n)g

whose inverse is given by the rule : (n⊗ s)⊗ g 7→ n⊗ (s⊗ ϕ(n)−1g) for every n ∈ N , s ∈ S,g ∈ Ngp

. We leave to the reader the verification that these maps are well defined, and they areinverse to each other. In view of (3.2.8) and claim 3.2.7, we may then replace S by S ⊗1 N

gp ,

which allows to assume that S is an integral pointed M-module and N ⊗M S is an integralpointed N-module. In this case, in view of proposition 3.1.52 we know that Z〈N ⊗M S〉 is aflat Z〈N〉-module, and it suffices to show that Z〈S〉 is a flat Z〈M〉-module.

However, under our assumptions, the ring homomorphism Z〈M〉 → Z〈N〉 is finite andinjective, so the assertion follows from [45, Part II, Th.1.2.4] and (3.1.32).

Lemma 3.2.9. Let M be an integral monoid, and S ⊂M a submonoid. We have :(i) The natural map S−1M sat → (S−1M)sat is an isomorphism.

(ii) If M is saturated, then M/S is saturated, and if S is a group, also the converse holds.(iii) The inclusion map M →M sat is a local morphism.(iv) The inclusion M ⊂M sat induces a natural bijection :

SpecM sat ∼→ SpecM.

Proof. (i) and (iii) are left to the reader. For (ii), the natural isomorphism S−1M/Sgp ∼→M/S,together with (i), reduces to the case where S is a group, in which case it suffices to remark that(M/S)sat = M sat/S. Lastly, to show (iv) it suffices to prove that, for any face F ⊂ M , thesubmonoid F sat ⊂ M sat is a face, and F sat ∩M = F , and that every face of M sat is of thisform. These assertions are easy exercises, which we leave as well to the reader.

Lemma 3.2.10. Let M be a saturated monoid such that M ] is fine. Then there exists an iso-morphism of monoids :

M∼→M ] ×M×

and if M is fine, M× is a finitely generated abelian group. Moreover, the projection M → M ]

induces a bijection :SpecM ] ∼→ SpecM : p 7→ p×M×.

Proof. Under the stated assumptions, G := Mgp/M× is a free abelian group of finite rank,hence the projection Mgp → G admits a splitting σ : G → Mgp. Set M0 := M ∩ σ(G); it iseasily seen that M = M0 ×M×, whence M0 ' M ]. If M is fine, Mgp is a finitely generatedabelian group, hence the same holds for its direct factor M×. The last assertion can be provendirectly, or can be regarded as a special case of lemma 3.1.16(i.b).


Definition 3.2.11. Let ϕ : M → N be a morphism of integral monoids.

(i) We say that ϕ is k-saturated (for some integer k > 0), if the push-out P ⊗M N isintegral and k-saturated, for every morphism M → P with P integral and k-saturated.

(ii) We say that ϕ is saturated, if the following holds. For every morphism of monoidsM → P such that P is integral and saturated, the monoid P ⊗M N is also integral andsaturated.

Clearly, if ϕ is k-saturated for every integer k > 0, then ϕ is saturated.

Lemma 3.2.12. Let ϕ : M → N be a morphism of integral monoids, and S ⊂ M , T ⊂ N twosubmonoids such that ϕ(S) ⊂ T . The following holds :

(i) The localization map M → S−1M is saturated.(ii) If ϕ is saturated, the same holds for the morphisms S−1M → T−1N andM/S → N/T

induced by ϕ.(iii) If S and T are two groups, then ϕ is saturated if and only if the same holds for the

induced morphism M/S → N/T .(iv) If ϕ is saturated, then the natural map N ⊗M M sat → N sat is an isomorphism.

Proof. (i) follows from the standard isomorphism : S−1M ⊗M N∼→ ϕ(S)−1N , together with

lemma 3.2.9(i). Next, let M/S → P and S−1M → Q be morphisms of monoids. Then

P ⊗M/S N/T ' (P ⊗M N)/T and Q⊗S−1M T−1N ' T−1(Q⊗M N)

(lemma 2.3.34) so assertions (ii) and (iii) follow from lemma 3.2.9(i,ii).(iv) follows by comparing the universal properties.

Example 3.2.13. (i) LetM be an integral monoid, I ⊂M an ideal, and consider again the Reesmonoid R(M, I) of the proof of theorem 3.2.3. Clearly R(M, I) is an integral monoid. However,easy examples show that, even when M is saturated, R(M, I) is not generally saturated. Moreprecisely, the following holds. For every ideal J ⊂M , set

J sat := a ∈Mgp | an ∈ Jn for some integer n > 0

where Jn denotes the n-th power of J in the monoid (P, ·) of (3.1.1). Then J sat is an ideal ofM sat. With this notation, we have the identity :

R(M, I)sat =⊕n∈N

(In)sat.

(Verification left to the reader.)(ii) For instance, take M := Q⊕2

+ , and let I ⊂ M be the ideal consisting of all pairs (x, y)such that x + y > 1. Then In = (x, y) ∈ Q⊕2

+ | x + y > n, and it is easily seen thatIn = (In)sat for every n ∈ N, hence in this case R(M, I) is saturated, in view of (ii). It is easilyseen that R(M, I) does not fulfill condition (F2) of lemma 3.1.41, hence the natural inclusionmap i : M → R(M, I) is not flat (theorem 3.1.42), hence it is not an integral morphism,according to theorem 3.2.3. On the other hand, we have :

Lemma 3.2.14. With the notation of example 3.2.13(ii), the morphism i is saturated.

Proof. We prefer to work with the multiplicative notation, so we shall argue with the monoid(expQ⊕2

+ , ·) (see (3.1)). Indeed, let ϕ : M → P be any morphism of monoids, with P saturated.Clearly R(M, I) is the direct sum of the P -modules P ⊗M In, for all n ∈ N.

Claim 3.2.15. The natural map P ⊗M In → InP is an isomorphism, for every n ∈ N


Proof of the claim. Indeed, the map is obviously surjective. Hence, suppose that a1x1 = a2x2

for some a1, a2 ∈ In, x1, x2 ∈ P such that x1⊗a1 = x2⊗a2. For every ϑ ∈ Q with 0 ≤ ϑ ≤ 1,set aϑ := aϑ1 · a1−ϑ

2 and notice that

xϑ := a1a−1ϑ x1 = a2a

−1ϑ x2 ∈ P gp

and if N ∈ N is large enough, so that Nϑ ∈ N, then xNϑ = xNϑ1 xNϑ2 ∈ P , hence xϑ ∈ P .Next, for any (a, b), (a′, b′) ∈ Q⊕2

+ , set (a, b) ∨ (a′, b′) := (min(a, a′),min(b, b′)). Choose anincreasing sequence 0 := ϑ0 < ϑ1 < · · · < ϑn := 1 of rational numbers, such that

bi := aϑi ∨ aϑi+1∈ In for every i = 0, . . . , n− 1.

Then there exist ci, di ∈ Q⊕2+ such that aϑi = bici and aϑi+1

= bidi for i = 0, . . . , n − 1. Wemay then compute in In ⊗M P :

aϑi ⊗ xϑi = bici ⊗ xϑi = bi ⊗ cixϑi and likewise aϑi+1⊗ xϑi+1

= bi ⊗ dixϑi+1.

By construction, we have cixϑi = dixϑi+1for i = 0, . . . , n− 1, whence the contention. ♦

In view of claim 3.2.15, we are reduced to showing that R(P, IP ) is saturated, and by example3.2.13, this comes down to proving that (InP )sat = InP for every n ∈ N. However, say thatx ∈ P gp, and xn = a1x1 · · · anxn for some ai ∈ I and xi ∈ P ; set a := (a1 · · · an)1/n,and notice that a ∈ I . Then xn = an · x1 · · ·xn, so that xa−1 ∈ P , and finally x ∈ IP , asrequired.

Lemma 3.2.14 shows that a saturated morphism is not necessarily integral. Notwithstanding,we shall see later that integrality holds for an important class of saturated morphisms (corollary3.4.4). Now we wish to globalize the class of saturated morphisms, to an arbitrary topos. Ofcourse, we could define the notion of saturated morphism of T -monoids, just by repeating wordby word definition 3.2.11(ii). However, it is not clear that the resulting condition would be ofa type which can be checked on stalks, in the sense of remark 2.2.14(ii). For this reason, weprefer to proceed as in Tsuji’s work [74].

Lemma 3.2.16. Let T be a topos, ϕ : M → N and ψ : N → P two morphisms of integralT -monoids. We have:

(i) If ϕ and ψ are exact, the same holds for ψ ϕ.(ii) If ψ ϕ is exact, the same holds for ϕ.

(iii) Consider a commutative diagram of integral T -monoids :

(3.2.17)

Mϕ //

ψ

N

ψ′

M ′ ϕ′ // N ′.

Then the following holds :(a) If (3.2.17) is a cartesian diagram and ϕ′ is exact, then ϕ is exact.(b) If (3.2.17) is cocartesian (in the category Int.MndT ) and ϕ is exact, then ϕ′ is

exact.

Proof. For all these assertions, remark 2.3.41(ii) easily reduces to the case where T = Set,which therefore we assume from start. Now, (i) and (ii) are left to the reader.

(iii.a): Let x ∈ Mgp such that ϕgp(x) ∈ N ; hence (ϕ′ ψ)gp(x) = ψ′(ϕgp(x)) ∈ N ′, andtherefore ψgp(x) ∈M ′, since ϕ′ is exact. It follows that x ∈M , so ϕ is exact.

(iii.b): Let x ∈ (M ′)gp such that (ϕ′)gp(x) ∈ N ′; then we may write (ϕ′)gp(x) = ϕ′(y) ·ψ′(z)for some y ∈ M ′ and z ∈ N . Therefore, (ϕ′)gp(xy−1) = ψ′(z); since the functor P 7→ P gp

commutes with colimits, it follows that we may find w ∈ Mgp such that ψgp(w) = xy−1 and


ϕgp(w) = z. Since ϕ is exact, we deduce that w ∈ M , therefore xy−1 ∈ M ′, and finallyx ∈M ′, whence the contention.

3.2.18. Let T be a topos; for two morphisms P ←M → N of integral T -monoids, we set

Nint

⊗M P := (N ⊗M P )int

which is the push-out of these morphisms, in the category Int.MndT . Notice that, for everymorphism f : T ′ → T of topoi, the natural morphism of T ′-monoids

(3.2.19) f ∗(Nint

⊗M P )→ f ∗Nint

⊗f∗M f ∗P

is an isomorphism (by lemmata 2.3.45(i) and 2.3.46(i)).Let now ϕ : M → N be a morphism of integral T -monoids. For any integer k > 0, let

kM and kN be the k-Frobenius maps of M and N (definition 2.3.40(ii)), and consider thecocartesian diagram :

(3.2.20)

MkM //

ϕ

M

ϕ′

N

kMint⊗M1N // P .

The endomorphism kN factors through kMint

⊗M 1N and a unique morphism β : P → N suchthat β ϕ′ = ϕ. A simple inspection shows that kP = (kM ⊗M 1N) β. Now, if kM is exact,then the same holds for kM

int

⊗M 1N , in view of lemma 3.2.16(iii.b). Hence, if P is k-saturated,then β is exact (lemma 3.2.16(ii)), and if M is k-saturated, also the converse holds. Likewise,if M is k-saturated and β is exact, then N is k-quasi-saturated.

These considerations motivate the following :

Definition 3.2.21. Let T be a topos, ϕ : M → N a morphism of integral T -monoids.

(i) A commutative diagram of integral T -monoids :

Mϕ //

N

M ′ ϕ′ // N ′

is called an exact square, if the induced morphism M ′ int

⊗M N → N ′ is exact.(ii) ϕ is said to be k-quasi-saturated if the commutative diagram :

(3.2.22)

Mϕ //

kM

N

kN

Mϕ // N

is an exact square (the vertical arrows are the k-Frobenius maps).(iii) ϕ is said to be quasi-saturated if it is k-quasi-saturated for every integer k > 0.

Proposition 3.2.23. Let T be a topos.

(i) If (3.2.17) is an exact square, and M → P is any morphism of integral T -monoids,the square P

int

⊗M (3.2.17) is exact.


(ii) Consider a commutative diagram of integral T -monoids :

(3.2.24)

Mϕ1 //

ψ

N

ϕ2 // P

ψ′

M ′ ϕ′1 // N ′

ϕ′2 // P ′

and suppose that the left and right square subdiagrams of (3.2.24) are exact. Then thesame holds for the square diagram :

Mϕ2ϕ1 //

ψ

P

ψ′

M ′ ϕ′2ϕ′1 // P ′.

Proof. (i): We have a commutative diagram :

(Pint

⊗M M ′)int

⊗M Nσ //

ω

Pint

⊗M (M ′ int

⊗M N)

1Pint⊗Mα

(Pint

⊗M M ′)int

⊗P (Pint

⊗M N)β // P

int

⊗M N ′

where ω, σ are the natural isomorphisms, and β and α : M ′ int

⊗M N → N ′ are the natural maps.By assumption, α is exact, hence the same holds for 1P

int

⊗M α, by lemma 3.2.16(iii.b). So β isexact, which is the claim.

(ii): Let α : M ′ int

⊗M N → N ′ and β : N ′int

⊗N P → P ′ be the natural maps; by assumptions,these are exact morphisms. However, we have a natural commutative diagram :

(M ′ int

⊗M N)int

⊗N Pω //

αint⊗N1P

M ′ int

⊗M P

γ

N ′int

⊗N Pβ // P ′

where ω is the natural isomorphism, and γ is the map deduced from ψ′ and ϕ′2 ϕ′1. Then theassertion follows from lemma 3.2.16(i,iii.b).

Corollary 3.2.25. Let T be a topos, ϕ : M → N , ψ : N → P two morphisms of integralT -monoids, and h, k > 0 any two integers. The following holds :

(i) If ϕ is both h-quasi-saturated and k-quasi-saturated, then ϕ is hk-quasi-saturated.(ii) If ϕ and ψ are k-quasi-saturated, the same holds for ψ ϕ.

(iii) If M → P is any map of integral T -monoids, and ϕ is k-quasi-saturated (resp. quasi-saturated), then the same holds for ϕ

int

⊗M 1P : P → Nint

⊗M P .(iv) Let S ⊂ ϕ−1(N×) be a T -submonoid. Then ϕ is k-quasi-saturated (resp. quasi-

saturated) if and only if the same holds for the induced map ϕS : S−1M → N .(v) Let G ⊂ Kerϕ be a subgroup. Then ϕ is k-quasi-saturated (resp. quasi-saturated) if

and only if the same holds for the induced map ϕ : M/G→ N .(vi) ϕ is quasi-saturated if and only if it is p-quasi-saturated for every prime number p.

(vii) M is k-saturated (resp. saturated) if and only if the unique morphism of T -monoids1 →M is k-quasi-saturated (resp. quasi-saturated).

(viii) If ϕ is k-quasi-saturated (resp. quasi-saturated), and M is k-saturated (resp. satu-rated), then N is k-saturated (resp. saturated).


(ix) If M is integral, and N is a T -group, ϕ is quasi-saturated.

Proof. (i) and (ii) are straightforward consequences of proposition 3.2.23(ii).To show (iii), set P ′ := N

int

⊗M P and let us remark that we have a commutative diagram

P

ϕint⊗M1P

// P 1//

P

ϕint⊗M1P

P ′ // P 2

// P ′

such that :

• the composition of the top (resp. bottom) arrows is the k-Frobenius map• the left square subdiagram is (3.2.22)

int

⊗M P• the right square subdiagram is cocartesian (hence exact).

then the assertion follows from proposition 3.2.23.(iv): Suppose that ϕ is k-quasi-saturated. Then the same holds for S−1ϕ : S−1M →

S−1Mint

⊗M N , according to (iii). However, we have a natural isomorphism S−1Mint

⊗M N∼→

ϕ(S)−1N = N , so ϕS is k-quasi-saturated.Conversely, suppose that ϕS is k-quasi-saturated. By (ii), in order to prove that the same

holds for ϕ, it suffices to show that the localisation map M → S−1M is k-quasi-saturated. Butthis is clear, since (3.2.22) becomes cocartesian if we take for ϕ the localisation map.

(v): To begin with, M/G is an integral monoid, by lemma 2.3.38. Suppose that ϕ is k-quasi-saturated. Arguing as in the proof of (iv) we see that the natural map N → (M/G)

int

⊗M N is anisomorphism, hence ϕ is k-quasi-saturated, by (iii).

Conversely, suppose that ϕ is k-quasi-saturated. By (ii), in order to prove that ϕ is saturated,it suffices to show that the same holds for the projection M → M/G. By (iii), we are furtherreduced to showing that the unique map G→ 1 is k-quasi-saturated, which is trivial.

(vi) is a straightforward consequence of (i). Assertion (vii) can be verified easily on thedefinitions. Next, suppose that ϕ is quasi-saturated and M is saturated. By (ii) and (vii) itfollows that the unique morphism 1 → N is quasi-saturated, hence (vii) implies that N issaturated, so (viii) holds.

(ix): In view of (iv), we are reduced to the case where M is also a group, in which case theassertion is obvious.

Proposition 3.2.26. Let ϕ : M → N be an integral morphism of integral monoids, k > 0 aninteger. The following conditions are equivalent :

(a) ϕ is k-quasi-saturated.(b) ϕ is k-saturated.(c) The push-out P of the cocartesian diagram (3.2.20), is k-saturated.

Proof. (a)⇒(b) by virtue of corollary 3.2.25(iii,viii), and trivially (b)⇒(c). Lastly, the implica-tion (c)⇒(a) was already remarked in (3.2.18).

Corollary 3.2.27. Let ϕ : M → N be an integral morphism of integral monoids. Then ϕ isquasi-saturated if and only if it is saturated.

Proposition 3.2.26 motivates the following :

Definition 3.2.28. Let T be a topos, ϕ : M → N a morphism of integral T -monoids, k > 0an integer. We say that ϕ is k-saturated (resp. saturated) if ϕ is integral and k-quasi-saturated(resp. and quasi-saturated).


In case T = Set, we see that an integral morphism of integral monoids is saturated in thesense of definition 3.2.28, if and only if it is saturated in the sense of the previous definition3.2.11. We may now state :

Corollary 3.2.29. Let P(T, ϕ) be the property “ϕ is an integral (resp. exact, resp. k-saturated,resp. saturated) morphism of integral T -monoids” (for a topos T ). Then P can be checked onstalks. (See remark 2.2.14(ii).)

Proof. The fact that integrality of a T -monoid can be checked on stalks, has already been es-tablished in lemma 2.3.46(ii). For the property “ϕ is an integral morphism” (between integralT -monoids), it suffices to apply theorem 3.2.3 and proposition 2.3.26.

For the property “ϕ is exact” one applies lemma 2.3.46(iii). From this, and from (3.2.19) onededuces that also the properties of k-saturation and saturation can be checked on stalks.

Lemma 3.2.30. Let f : M → N be a morphism of integral monoids. We have :(i) If f is exact, then f is local.

(ii) Conversely, if f is an integral and local morphism, then f is exact.

Proof. (i) is left to the reader as an exercise.(ii): Suppose x ∈ Mgp is an element such that b := fS(x) ∈ N . Write x = z−1y for certain

y, z ∈ M ; therefore b · f(z) = f(y) holds in N , and theorems 3.2.3, 3.1.42 imply that thereexist c ∈ N and a1, a2 ∈ M , such that 1 = c · f(a1), b = c · f(a2) and ya1 = za2. Since f islocal, we deduce that a1 ∈M×, hence x = a−1

1 a2 lies in M .

Proposition 3.2.31. Let f : M → N be an integral map of integral monoids, k > 0 an integer,and N =

⊕γ∈ΓNγ the f -grading of N (see remark 3.2.5(iii)). Then the following conditions

are equivalent :(a) f is k-quasi-saturated.(b) Nkγ = Nk

γ for every γ ∈ Γ. (Here the k-th power of subsets of N is taken in themonoid P(N), as in (3.1.1).)

Proof. (a)⇒(b): Let π : Ngp → Γ be the projection, suppose that y ∈ Nkγ for some γ ∈ Γ,and pick x ∈ Ngp such that π(x) = γ. This means that y = xk · f gp(z) for some z ∈ Mgp.By (a), it follows that we may find a pair (a, b) ∈ M × N and an element w ∈ Mgp, suchthat (aw−k, bw) = (z, x) in Mgp × Ngp. Especially, b, b · f(a) ∈ Nγ , and consequently y =bk−1 · (b · f(a)) ∈ Nkγ , as stated.

(b)⇒(a): Let S := f−1(N×); by lemmata 3.2.30(ii) and 3.2.2(iii), the induced map fS :S−1M → N is exact and integral. Moreover, by corollary 3.2.25(iv), f is k-quasi-saturated ifand only if the same holds for fS , and clearly the f -grading of N agrees with the fS-grading.Hence we may replace f by fS and assume from start that f is exact. In this case, G :=Ker f gp ⊂ M , and corollary 3.2.25(v) says that f is k-quasi-saturated if and only if the sameholds for the induced map f : M/G→ N ; moreover f is still integral, since it is deduced fromf by push-out along the map M → M/G. Hence we may replace f by f , thereby reducing tothe case where f is injective. Also, f is flat, by theorem 3.2.3. The assertion boils down to thefollowing. Suppose that (x, y) ∈Mgp ×Ngp is a pair such that a := f gp(x) · yk ∈ N ; we haveto show that there exists a pair (m,n) ∈M ×N whose class in the push-out P of the diagram

Nf←−M

kM−−−→M

agrees with the image of (x, y) in P gp. However, set γ := π(y), and notice that a ∈ Nkγ , hencewe may write a = n1 · · ·nk for certain n1, . . . , nk ∈ Nγ . Then, according to remark 3.2.5(iv),we may find n ∈ Nγ and x1, . . . , xk ∈ M such that ni = n · f(xi) for every i = 1, . . . , k. Itfollows that y · n−1 ∈ f gp(Mgp), say y = n · f(z) for some z ∈ Mgp. Set m := x1 · · ·xk; then


(x, y) = (x, n · f(z)) represents the same class as (x · zk, n) in P gp. Especially, f(x · zk) · nk =a = f(m) · nk, hence m = x · nk, since f is injective. The claim follows.

Corollary 3.2.32. Let f : M → N be an integral and local morphism of integral and sharpmonoids. Then :

(i) f is exact and injective.(ii) If furthermore, f is saturated, then Coker f gp is a torsion-free abelian group.

Proof. (i): It follows from lemma 3.2.31 that f is exact. Next, suppose that f(a1) = f(a2)for some a1, a2 ∈ M . By theorem 3.2.3 and 3.1.42, it follows that there exist b1, b2 ∈ M andt ∈ N such that 1 = f(b1)t = f(b2)t and a1b1 = a2b2. Since N is sharp, we deduce thatf(b1) = f(b2) = 1, and since f is local and M is sharp, we get b1 = b2 = 1, thus a1 = a2,whence x = 1, which is the contention.

(ii): Let us endow N with its f -grading. Suppose now that g ∈ G is a torsion element, andsay that gk = 1 in G for some integer k > 0; by propositions 3.2.31 and 3.2.26, we then haveN1 = Nk

g . Especially, there exist a1, . . . , ak ∈ Ng, such that a1 · · · ak = 1. Since N is sharp,we must have ai = 1 for i = 1, . . . , k, hence g = 1.

Corollary 3.2.33. Let ϕ : M → N be a morphism of integral monoids, F ⊂ N any face, andϕF : ϕ−1F → F the restriction of ϕ.

(i) If ϕ is flat, then the same holds for ϕF , and the induced map CokerϕgpF → Cokerϕgp

is injective.(ii) If ϕ is saturated, the same holds for ϕF .

Proof. (i): The fact that ϕF is flat, is a special case of corollary 3.1.51. The assertion aboutcokernels boils down to showing that the induced diagram of abelian groups :

(ϕ−1F )gpϕgpF //

F gp

Mgp

ϕgp

// Ngp

is cartesian. However, say that ϕgp(a1a−12 ) = f1f

−12 for some a1, a2 ∈ M and f1, f2 ∈ F . This

means that ϕ(a1)f2 = ϕ(a2)f1 in N ; by condition (F2) of theorem 3.1.42, we deduce that thereexist b1, b2 ∈ M and t ∈ N such that b2a1 = b1a2 and fi = t · ϕ(bi) (i = 1, 2). Since F is aface, it follows that b1, b2 ∈ ϕ−1F , hence a1a

−12 = b1b

−12 ∈ (ϕ−1F )gp, as required.

(ii): By (i), the morphism ϕF is integral, hence it suffices to show that ϕF quasi-saturated(proposition 3.2.26), and to this aim we shall apply the criterion of proposition 3.2.31. Indeed,let N =

⊕γ∈ΓNγ (resp. F =

⊕γ∈ΓF

Fγ) be the ϕ-grading (resp. the ϕF -grading); accordingto (i), the induced map j : ΓF → Γ is injective. This means that Fγ = F ∩ Nj(γ) for everyγ ∈ ΓF . Hence, for every integer k > 0 we may compute

Fkγ = F ∩Nkj(γ) = (F ∩Nj(γ))

k = F kγ

where the first identity follows by applying proposition 3.2.31 (and proposition 3.2.26) to thesaturated map ϕ, and the second identity holds because F is a face of N .

3.3. Polyhedral cones. Fine monoids can be studied by means of certain combinatorial ob-jects, which we wish to describe. Part of the material that follows is borrowed from [35].Again, all the monoids in this section are non-pointed.


3.3.1. Quite generally, a convex cone is a pair (V, σ), where V is a finite dimensional R-vectorspace, and σ ⊂ V is a non-empty subset such that :

R+ · σ = σ = σ + σ

where the addition is formed in the monoid (P(V ),+) as in (3.1.1), and scalar multiplicationby the set R+ is given by the rule :

R+ · S := r · s | r ∈ R+, s ∈ S for every S ∈P(V ).

A subset S ⊂ σ is called a ray of σ, if it is of the form R+ · s, for some s ∈ σ\0.We say that (V, σ) is a closed convex cone if σ is closed as a subset of V (of course, V is

here regarded as a topological space via any choice of isomorphism V ' Rn). We denote by〈σ〉 ⊂ V the R-vector space generated by σ. To a convex cone (V, σ) one assigns the dual cone(V ∨, σ∨), where V ∨ := HomR(V,R), the dual of V , and :

σ∨ := u ∈ V ∨ | u(v) ≥ 0 for every v ∈ σ.Also, the opposite cone of σ is the cone

−σ := −v ∈ V | v ∈ σ.Notice that σ∨ and−σ are always closed cones. Notice as well that the restriction of the additionlaw of V determines a monoid structure (σ,+) on the set σ. A map of cones

ϕ : (W,σW )→ (V, σV )

is an R-linear map ϕ : W → V such that ϕ(σW ) ⊂ σV . Clearly, the restriction of ϕ yields amap of monoids (σW ,+)→ (σV ,+). If S ⊂ V is any subset, we set :

S⊥ := u ∈ V ∨ | u(s) = 0 for every s ∈ S ⊂ V ∨.

Lemma 3.3.2. Let (V, σ) be a closed convex cone, Then, under the natural identification V ∼→(V ∨)∨, we have (σ∨)∨ = σ.

Proof. This follows from [19, Ch.II, §5, n.3, Cor.5].

3.3.3. A convex polyhedral cone is a cone (V, σ) such that σ is of the form :

σ := r1v1 + · · ·+ rsvs ∈ V | ri ≥ 0 for every i ≤ sfor a given set of vectors v1, . . . , vs ∈ V , called generators for the cone σ. One also says thatv1, . . . , vs is a generating set for σ, and that R+ · v1, . . . ,R+ · vs are generating rays for σ.We say that σ is a simplicial cone, if it is generated by a system of linearly independent vectors.

Lemma 3.3.4. Let (V, σ) be a convex polyhedral cone, S a finite generating set for σ. Then :(i) For every v ∈ σ there is a subset T ⊂ S consisting of linearly independent vectors,

such that v is contained in the convex polyhedral cone generated by T .(ii) (V, σ) is a closed convex cone.

Proof. (i): Let T ⊂ S be a subset such that v is contained in the cone generated by T ; up toreplacing T by a subset, we may assume that T is minimal, i.e. no proper subset of T generatesa cone containing v. We claim that T consists of linearly independent vectors. Otherwise, wemay find a linear relation of the form

∑w∈T aw · w = 0, for certain aw ∈ R, at least one of

which is non-zero; we may then assume that :

(3.3.5) aw > 0 for at least one vector w ∈ T .Say also that v =

∑w∈T bw · w with bw ∈ R+; by the minimality assumption on T , we must

actually have bw > 0 for every w ∈ T . We deduce the identity : v =∑

w∈T (bw − taw) · w forevery t ∈ R; let t0 be the supremum of the set of positive real numbers t such that bw− taw ≥ 0


for every w ∈ T . In view of (3.3.5) we have t0 ∈ R+; moreover bw− t0aw ≥ 0 for every w ∈ T ,and bw − t0aw = 0 for at least one vector w, which contradicts the minimality of T .

(ii): In view of (i), we are reduced to the case where σ is generated by finitely many linearlyindependent vectors, and for such cones the assertion is clear (details left to the reader).

3.3.6. A face of a convex cone σ is a subset of the form σ ∩ Keru, for some u ∈ σ∨. Thedimension (resp. codimension) of a face τ of σ is the dimension of the R-vector space 〈τ〉 (resp.of the R-vector space 〈σ〉/〈τ〉). A facet of σ is a face of codimension one. Notice that if σ isa polyhedral cone, and τ is a face of σ, then (V, τ) is also a convex polyhedral cone; indeed ifS ⊂ V is a generating set for σ, then S ∩ τ is a generating set for τ .

Lemma 3.3.7. Let (V, σ) be a convex polyhedral cone. Then the faces of (V, σ) are the same asthe faces of the monoid (σ,+).

Proof. Clearly we may assume that σ 6= 0. Let F be a face of the polyhedral cone σ, andpick u ∈ σ∨ such that F = σ ∩Keru. Then σ \ F = x ∈ σ | u(x) > 0, and this is clearly anideal of the monoid (σ,+); hence F is a face of (σ,+).

Conversely, suppose that F is a face of (σ,+). First, we wish to show that F is a cone in V .Indeed, let f ∈ F , and r > 0 any real number; we have to prove that r · f ∈ F . To this aim, itsuffices to show that (r/N) · f ∈ F for some integer N > 0, so that, after replacing r by r/Nfor N large enough, we may assume that 0 < r < 1. In this case, (1− r) · f ∈ σ, and we havef = r · f + (1− r) · f , so that r · f ∈ F , since F is a face of (σ,+).

Next, denote by W ⊂ V the R-vector space spanned by F . Suppose first that V = W , andconsider any m ∈ σ; then m = f1 − f2 for some f1, f2 ∈ F , hence f1 = f2 + m. Since F is aface of (σ,+), this implies that m lies in F , so F = σ in this case, especially F is a face of theconvex polyhedral cone σ.

So finally, we may assume thatW is a proper subspace of V . In this case, letN := σ+(−F ).We notice that N 6= V . Indeed, if m ∈ σ and −m ∈ N , we may write −m = m′ − f for somem′ ∈ σ and f ∈ F ; hence f = m + m′, and therefore m ∈ F ; in other words, N avoids thewhole of −(σ \ F ), which is not empty for σ 6= 0.

Thus,N is a proper convex cone of V . Now, let u1, . . . , uk ∈ N∨ be a system of generators ofthe R-subspace of V ∨ spanned by N∨, and set u := u1 + · · ·+ uk. Suppose that x ∈ σ ∩Keru;then −x ∈ Kerui for every i = 1, . . . , k, and therefore −x ∈ N∨∨ = N (lemma 3.3.2). Hence,we may write −x = m − f for some m ∈ σ and f ∈ F , or equivalently, f = m + x, whichshows that x ∈ F . Summarizing, we have proved that F = σ ∩ Keru, i.e. F is a face of theconvex cone σ.

Proposition 3.3.8. Let (V, σ) be a convex polyhedral cone. The following holds :(i) Any intersection of faces of σ is still a face of σ.

(ii) If τ is a face of σ, and γ is a face of (V, τ), then γ is a face of σ.(iii) Every proper face of σ is the intersection of the facets that contain it.

Proof. (i): Say that τi = σ ∩Kerui, where u1, . . . , un ∈ σ∨. Then⋂ni=1 τi = σ ∩Ker

∑ni=1 ui.

(ii): Say that τ = σ ∩ Keru and γ = τ ∩ Ker v, where u ∈ σ∨ and v ∈ τ∨. Then, for larger ∈ R+, the linear form v′ := v + ru is non-negative on any given finite generating set of σ,hence it lies in σ∨, and γ = σ ∩Ker v′.

(iii): To begin with, we prove the following :

Claim 3.3.9. (i) Every proper face of σ is contained in a facet.(ii) Every face of σ of codimension 2 is the intersection of exactly two facets.

Proof of the claim. We may assume that 〈σ〉 = V . Let τ be a face of σ of codimension at leasttwo, and denote by σ be the image of σ in the quotient V := V/〈τ〉; clearly (V , σ) is again a


polyhedral cone. Moreover, choose u ∈ σ∨ such that τ = σ∩Keru; the linear form u descendsto u ∈ σ∨, therefore σ ∩Keru = 0 is a face of σ.

(ii): If τ has codimension two, V has dimension two. Suppose that the assertion is known for(V , σ); then we find exactly two facets γ1, γ2 of σ whose intersection is 0. Their preimagesin V intersect σ in facets γ1, γ2 that satisfy γ1 ∩ γ2 = τ . Hence, we may assume from start thatτ = 0 and V has dimension two, in which case the verification is easy, and shall be left to thereader.

(i): Arguing by induction on the codimension, it suffices to show that τ is contained in aproper face spanning a larger subspace. To this aim, suppose that the claim is known for σ;since 0 is a face of σ of codimension at least two, it is contained in a proper face γ; thepreimage γ of the latter intersects σ in a proper face containing τ . Thus again, we are reducedto the case where τ = 0. Pick u0 ∈ σ∨ such that σ ∩ Keru0 = 0; choose also any otheru1 ∈ V ∨ such that σ ∩ Keru1 6= 0. Since dimR V

∨ ≥ 2, we may find a continuous mapf : [0, 1] → V ∨ \0 with f(0) = u0 and f(1) = u1. Let P+(V ) be the topological spaceof rays of V (i.e. the topological quotient V \0/∼ by the equivalence relation such thatv ∼ v′ if and only if v and v′ generate the same ray), and define likewise P+(V ∨); let alsoZ ⊂ P := P+(V ) × P+(V ∨) be the incidence correspondence, i.e. the subset of all pairs(v, u) such that u(v) = 0, for any representative u of the class u and v of the class v. Finally,let P+(σ) ⊂ P+(V ) be the image of σ \0. Then Z (resp. P+(σ)) is a closed subset of P(resp. of P+(V )), hence Y := Z ∩ (P+(σ) × P+(V ∨)) is a closed subset of P . Since theprojection π : P → P+(V ∨) is proper, π(Y ) is closed in P+(V ∨). Let f : [0, 1] → P+(V ∨)be the composition of f and the natural projection V ∨\0 → P+(V ∨); then f−1(π(Y )) is aclosed subset of [0, 1], hence it admits a smallest element, say a (notice that a > 0). Moreover,ua ∈ σ∨; indeed, otherwise we may find v ∈ σ\0 such that ua(v) < 0, and since u0(v) > 0,we would have ub(v) = 0 for some b ∈ (0, a). The latter means that f(b) ∈ π(Y ), whichcontradicts the definition of a. Since by construction, σ ∩Kerua 6= 0, the claim follows. ♦

Let τ be any face of σ; to show that (iii) holds for τ , we argue by induction on the codimensionof τ . The case of codimension 2 is covered by claim 3.3.9(ii). If τ has codimension > 2, weapply claim 3.3.9(i) to find a proper face γ containing τ ; by induction, τ is the intersection offacets of γ, and each of these is the intersection of two facets in σ (again by claim 3.3.9(ii)),whence the contention.

3.3.10. Suppose σ spans V (i.e. 〈σ〉 = V ) and let τ be a facet of σ; by definition there exists anelement uτ ∈ σ∨ such that τ = σ∩Keruτ , and one sees easily that the ray Rτ := R+ ·uτ ⊂ σ∨

is well-defined, independently of the choice of uτ . Hence the half-space :

Hτ := v ∈ V | uτ (v) ≥ 0depends only on τ . Recall that the interior (resp. the topological closure) of a subset E ⊂ V isthe largest open subset (resp. the smallest closed subset) of V contained in E (resp. containingE). The topological boundary of E is the intersection of the topological closures of E and ofits complement V \E.

Proposition 3.3.11. Let (V, σ) be a convex polyhedral cone, such that σ spans V . We have:(i) The topological boundary of σ is the union of its facets.

(ii) If σ 6= V , then σ =⋂τ⊂σHτ , where τ ranges over the facets of σ.

(iii) The rays Rτ , where τ ranges over the facets of σ, generate the cone σ∨.

Proof. (i): Notice that σ spans V if and only if the interior of σ is not empty. A proper faceτ is the intersection of σ with a hyperplane Keru ⊂ V with u ∈ σ∨ \0; therefore, everyneighborhood U ⊂ V of any point v ∈ τ intersects V \σ. This shows that τ lies in thetopological boundary of σ.


Conversely, if v is in the boundary of σ, choose a sequence (vi | i ∈ N) of points of V \σ,converging to the point v; by lemma 3.3.2, for every i ∈ N there exists ui ∈ σ∨ such thatui(vi) < 0. Up to multiplication by scalars, we may assume that the vectors ui lie on somesphere in V ∨ (choose any norm on V ∨); hence we may find a convergent subsequence, and wemay then assume that the sequence (ui | i ∈ N) converges to an element u ∈ V ∨. Necessarilyu ∈ σ∨, therefore v lies on the face σ ∩ Keru, and the assertion follows from proposition3.3.8(iii).

(ii): Suppose, by way of contradiction, that v lies in every half-space Hτ , but v /∈ σ. Chooseany point v′ in the interior of σ, and let t ∈ [0, 1] be the largest value such that w := tv +(1 − t)v′ ∈ σ. Clearly w lies on the boundary of σ, hence on some facet τ , by (i). Say thatτ = σ ∩Keru; then u(v′) > 0 and u(w) = 0, so u(v) < 0, a contradiction.

(iii): When σ = V , there is nothing to prove, hence we may assume that σ 6= V . In this case,suppose that u ∈ σ∨, and u is not in the cone C generated by the rays Rτ . Applying lemma3.3.2 to the cone (V ∨, C), we deduce that there exists a vector v ∈ V with v ∈ Hτ for all thefacets τ of σ, and u(v) < 0, which contradicts (ii).

Corollary 3.3.12. Let (V, σ) and (V, σ′) be two convex polyhedral cones. Then :

(i) (Farkas’ theorem) The dual (V ∨, σ∨) is also a convex polyhedral cone.(ii) If τ is a face of σ, then τ ∗ := σ∨∩τ⊥ is a face of σ∨ such that 〈τ ∗〉 = 〈τ〉⊥. Especially:

(3.3.13) dimR〈τ〉+ dimR〈τ ∗〉 = dimR V.

The rule τ 7→ τ ∗ is a bijection from the set of faces of σ to those of σ∨. The smallestface of σ is σ ∩ (−σ).

(iii) (V, σ ∩ σ′) is a convex polyhedral cone, and every face of σ ∩ σ′ is of the form τ ∩ τ ′,for some faces τ of σ and τ ′ of σ′.

Proof. (i): Set W := 〈σ〉 ⊂ V , and pick a basis u1, . . . , uk of W⊥; by proposition 3.3.11(iii),the assertion holds for the dual (W∨, σ∨) of the cone (W,σ). However, W∨ ' V ∨/W⊥, hencethe dual cone (V ∨, σ∨) is generated by lifts of generators of (W∨, σ∨), together with the vectorsui and −ui, for i = 1, . . . , k.

(ii): Notice first that the faces of σ∨ are exactly the cones σ∨∩u⊥, for u ∈ σ = (σ∨)∨. Fora given v ∈ σ, let τ be the smallest face of σ such that v ∈ τ ; this means that τ∨ ∩ v⊥ = τ⊥

(where (V ∨, τ∨) is the dual of (V, τ)). Hence σ∨∩v⊥ = σ∨∩(τ∨∩v⊥) = τ ∗, so every faceof σ∨ has the asserted form. The rule τ 7→ τ ∗ is clearly order-reversing, and from the obviousinclusion τ ⊂ (τ ∗)∗ it follows that τ ∗ = ((τ ∗)∗)∗, hence this map is a bijection. It follows fromthis, that the smallest face of σ is (σ∨)∗ = σ ∩ (σ∨)⊥ = (σ∨)⊥ = σ ∩ (−σ). In particular,we see that (σ ∩ (−σ))∗ = σ∨, and furthermore, (3.3.13) holds for τ := σ ∩ (−σ). Identity(3.3.13) for a general face τ can be deduced by inserting τ in a maximal chain of faces of σ,and comparing with the dual chain of faces of σ∨ (details left to the reader). Finally, it is clearthat 〈τ〉 ⊂ 〈τ ∗〉⊥; since these spaces have the same dimension, we deduce that 〈τ〉⊥ = 〈τ ∗〉.

(iii): Indeed, lemma 3.3.2 implies that σ∨ + σ′∨ is the dual of σ ∩ σ′, hence (i) impliesthat σ ∩ σ′ is polyhedral. It also follows that every face τ of σ ∩ σ′ is the intersection ofσ ∩ σ′ with the kernel of a linear form u + u′, for some u ∈ σ∨ and u′ ∈ σ′∨. Consequently,τ = (σ ∩Keru) ∩ (σ′ ∩Keru′).

Corollary 3.3.14. For a convex polyhedral cone (V, σ), the following conditions are equivalent:

(a) σ ∩ (−σ) = 0.(b) σ contains no non-zero linear subspaces.(c) There exists u ∈ σ∨ such that σ ∩Keru = 0.(d) σ∨ spans V ∨.


Proof. (a)⇔ (b) since σ ∩ (−σ) is the largest subspace contained in σ. Next, (a)⇔ (c) sinceσ ∩ (−σ) is the smallest face of σ. Finally, (a)⇔ (d) since dimR〈σ ∩ (−σ)〉 + dimR〈σ∨〉 = n(corollary 3.3.12(ii)).

3.3.15. A convex polyhedral cone fulfilling the equivalent conditions of corollary 3.3.14 is saidto be strongly convex. Suppose that (V, σ) is strongly convex; then proposition 3.3.11(iii) saysthat σ is generated by the rays Rτ , where τ ranges over the facets of σ∨. The rays Rτ areuniquely determined by σ, and are called the extremal rays of σ. Moreover, these Rτ form theunique minimal set of generating rays for σ. Indeed, concerning the minimality : for each facetτ of σ∨, pick vτ ∈ σ with R+ · vτ = Rτ ; suppose that vτ0 =

∑τ∈S tτ · vτ for some subset S of

the set of facets of σ∨, and appropriate tτ > 0, for every τ ∈ S. It follows easily that u(vτ ) = 0for every u ∈ τ0, and every τ ∈ S. But by definition of Rτ , this implies that S = τ0, whichis the claim. Concerning uniqueness : suppose that Σ is another system of generating rays;especially, for any facet τ ⊂ σ∨, the ray Rτ is in the convex cone generated by Σ; it followseasily that there exists ρ ∈ Σ such that u(ρ) = 0 for every u ∈ τ , in which case ρ = Rτ . Thisshows that Σ must contain all the extremal rays.

Example 3.3.16. (i) Suppose that dimR V = 2, and (V, σ) is a strongly convex polyhedralcone, and assume that σ generates V . Then the only face of codimension two of σ is 0, soit follows easily from claim 3.3.9(ii) that σ admits exactly two facets, and these are also theextremal rays of σ, especially σ is simplicial. Of course, these assertions are rather obvious; indimension > 2, the general situation is much more complicated.

(ii) Let (V, σ) be a convex polyhedral cone, and suppose that σ spans V . Let τ be a faceof σ. Notice that (σ,+)gp = (V,+), and (τ,+) is a face of the monoid (σ,+), by lemma3.3.7. Hence we may view the localization τ−1σ naturally as a submonoid of (V,+), and itis easily seen that τ−1σ is a convex cone. By proposition 3.3.11(iii), the polyhedral cone σ∨

is generated by its extremal rays Rl1, . . . ,Rln, and by proposition 3.3.8(iii), we may assumethat τ = σ ∩ Ker (l1 + · · · + lk) for some k ≤ n. Clearly li(v) ≥ 0 for every v ∈ τ−1σ andevery i ≤ k. Conversely, if l ∈ (τ−1σ)∨, we must have τ ⊂ Ker l and l ∈ σ∨; if we writel =

∑ni=1 aili for some ai ≥ 0, it follows easily that ai = 0 for every i > k. On the other

hand, suppose that v ∈ V satisfies the inequalities li(v) ≥ 0 for i = 1, . . . , k; then, for everyi = k + 1, . . . , n we may find ui ∈ τ such that li(v + ui) ≥ 0, hence v + uk+1 + · · ·+ un ∈ σ,and therefore v ∈ τ−1σ. This shows that τ−1σ is a closed convex cone, and its dual (τ−1σ)∨ isthe convex cone generated by l1, . . . , lk; especially, it is a convex polyhedral cone, and then thesame holds for τ−1σ, by virtue of lemma 3.3.2 and corollary 3.3.12(i).

(iii) In the situation of (ii), let v ∈ τ be any element that lies in the relative interior of τ , i.e.in the complement of the union of the facets of τ . Denote by Sv ⊂ τ the submonoid generatedby v. Then we claim that

τ−1σ = S−1v σ.

Indeed, let s ∈ σ and t ∈ τ be any two element; in view of proposition 3.3.11(i), it is easily seenthat there exists an integer N > 0 such that v −N−1t ∈ τ , hence t′ := Nv − t ∈ τ . Therefores− t = (s+ t′)−Nv ∈ S−1σ, and the assertion follows.

Lemma 3.3.17. Let f : V → W be a linear map of finite dimensional R-vector spaces, (V, σ)a convex polyhedral cone. The following holds :

(i) (W, f(σ)) is a convex polyhedral cone.(ii) Suppose moreover, that σ ∩ Ker f does not contain non-zero linear subspaces of V .

Then, for every face τ of f(σ) there exists a face τ ′ of σ such that f(τ ′) ⊂ τ , and frestricts to an isomorphism : 〈τ ′〉 ∼→ 〈τ〉.

Proof. (i) is obvious. To show (ii) we argue by induction on n := dimR Ker f . The assertion isobvious when n = 0, hence suppose that n > 0 and that the claim is already known whenever


Ker f has dimension < n. Let τ be a face of f(σ); then f−1τ is a face of f−1f(σ), henceσ ∩ f−1τ is a face of σ = σ ∩ f−1f(σ). In view of proposition 3.3.8(ii), we may then replaceσ by σ ∩ f−1τ , and therefore assume from start that τ = f(σ). We may as well assume thatV = 〈σ〉 and W = 〈τ〉. The assumption on σ implies especially that σ 6= V , hence σ is theintersection of the half-spacesHγ corresponding to its facets γ (proposition 3.3.11(ii)). For eachfacet γ of σ, let uγ be a chosen generator of the ray Rγ (notation of (3.3.10)). Since σ ∩ Ker fdoes not contain non-zero linear subspaces, we may find a facet γ such that V ′ := Keruγ doesnot contain Ker f . Then, the inductive assumption applies to the restriction f|V ′ : V ′ → Wand the convex polyhedral cone σ ∩ V ′, and yields a face τ ′ of the latter, such that f inducesan isomorphism 〈τ ′〉 ∼→ 〈f(σ ∩ V ′)〉. Finally, 〈f(σ ∩ V ′)〉 = W , since V ′ does not containKer f .

Lemma 3.3.18. Let f : V → W be a linear map of finite dimensional R-vector spaces, f∨ :W∨ → V ∨ the transpose of f , and (W,σ) a convex polyhedral cone. Then :

(i) (V, f−1σ) is a convex polyhedral cone and (f−1σ)∨ = f∨(σ∨).(ii) For every face δ of f−1σ, there exists a face τ of σ such that δ = f−1τ . If furthermore,〈σ〉+f(V ) = W , we may find such a τ so that additionally, f induces an isomorphism:

V/〈δ〉 ∼→ W/〈τ〉.

(iii) Conversely, for every face τ of σ, the cone f−1τ is a face of f−1σ, and (f−1τ)∗ is thesmallest face of (f−1σ)∨ containing f∨(τ ∗) (notation of corollary 3.3.12(ii)).

Proof. (i): By corollary 3.3.12(i), (W∨, σ∨) is a convex polyhedral cone, hence we may findu1, . . . , us ∈ σ∨ such that σ =

⋂si=1 u

−1i (R+). Therefore f−1σ =

⋂si=1(ui f)−1(R+). Let

γ ⊂ V ∨ be the cone generated by the set u1f, . . . , usf; then f−1σ = γ∨, and the assertionresults from lemma 3.3.2 and a second application of (i).

(iii): For every u ∈ σ∨, we have : f−1(σ ∩ Keru) = (f−1σ) ∩ Keru f . Since we alreadyknow that (f−1σ)∨ = f∨(σ∨), we see that the faces of f−1σ are exactly the subsets of the formf−1τ , where τ ranges over the faces of σ. Next, for any such τ , the set f∨(τ ∗) consists of allu ∈ V ∨ of the form u = w f for some w ∈ σ∨ such that w(τ) = 0. From this description itis clear that f∨(τ ∗) ⊂ (f−1τ)∗. To show that (f−1τ)∗ is the smallest face containing f∨(τ ∗), itthen suffices to prove that f∨(τ ∗)⊥ ∩ f−1σ ⊂ f−1τ . However, let v ∈ f∨(τ ∗)⊥ ∩ f−1σ; thenw f(v) = 0 for every w ∈ τ ∗, i.e. f(v) ∈ (τ ∗)∗ = τ , whence the contention.

(ii): The first assertion has already been shown; hence, suppose that 〈σ〉 + f(V ) = W . Wededuce that σ∨∩Ker f∨ does not contain non-zero linear subspaces of W∨; indeed, if u ∈ W∨,and both u and −u lie in σ∨, then u vanishes on 〈σ〉, and if u ∈ Ker f∨, then u vanishes as wellon f(V ), hence u = 0. We may then apply lemma 3.3.17(ii) to find a face γ of σ∨ such thatf∨(γ) ⊂ δ∗ and f∨ restricts to an isomorphism : 〈γ〉 ∼→ 〈δ∗〉. Especially, δ∗ is the smallest faceof (f−1σ)∨ containing f∨(γ), hence δ∗ = (f−1γ∗)∗ by (iii), i.e. δ = f−1γ∗. We also deducethat f induces an isomorphism : V/〈δ∗〉⊥ ∼→ W/〈γ〉⊥. Since 〈δ∗〉⊥ = 〈δ〉 and 〈γ〉⊥ = 〈γ∗〉, thesecond assertion holds with τ := γ∗.

Lemma 3.3.19. Let (V, σ) and (V ′, σ′) be two convex polyhedral cones. Then :

(i) (V ⊕ V ′, σ × σ′) is a convex polyhedral cone.(ii) Every face of σ × σ′ is of the form τ × τ ′, for some faces τ of σ and τ ′ of σ′.

Proof. Indeed, σ × σ′ = (p−11 σ) ∩ (p−1

2 σ′), where p1 and p2 are the natural projections ofV ⊕ V ′ onto V and V ′. Hence, assertions (i) and (ii) follow from corollary 3.3.12(iii) andlemma 3.3.18(i).


3.3.20. Let (L,+) be a free abelian group of finite rank, σ ⊂ LR := L⊗ZR a convex polyhedralcone. We say that σ is L-rational (or briefly : rational, when there is no danger of ambiguity)if it admits a generating set consisting of elements of L. Then it is clear that every face of arational convex polyhedral cone is again rational (see (3.3.6)). On the other hand, let

(M,+) ⊂ (L,+)

be a submonoid of L; we shall denote by (LR,MR) the convex cone generated by M (i.e. thesmallest convex cone in LR containing the image ofM ). IfM is fine,MR is a convex polyhedralcone. Later we shall also find useful to consider the subset :

MQ := m⊗ q |m ∈M, q ∈ Q+ ⊂ LQ := L⊗Z Qwhich is a submonoid of LQ.

Proposition 3.3.21. Let (L,+) be a free abelian group of finite rank, with dual

L∨ := HomZ(L,Z).

Let also (LR, σ) and (LR, σ′) be two L-rational convex polyhedral cones. We have :

(i) The dual (L∨R, σ∨) is an L∨-rational convex polyhedral cone.

(ii) (LR, σ ∩ σ′) is also an L-rational convex polyhedral cone.(iii) Let g : L′ → L (resp. h : L → L′) be a map of free abelian groups of finite rank,

and denote by gR : L′R → LR (resp. hR : LR → L′R) the induced R-linear map. Then,(L′R, g

−1R σ) and (L′R, hR(σ)) are L′-rational.

(iv) Let L′ be another free abelian group of finite rank, and (L′R, σ′) an L′-rational convex

polyhedral cone. Then (LR ⊕ L′R, σ × σ′) is L⊕ L′-rational.

Proof. (i) and (ii) follow easily, by inspecting the proof of corollary 3.3.12(i),(iii) : the detailsshall be left to the reader.

(iii): The assertion concerning hR(σ) is obvious. To show the assertion for g−1R σ, one ar-

gues as in the proof of lemma 3.3.18(i) : by (i), we may find u1, . . . , us ∈ L∨ such thatσ =

⋂si=1 u

−1i,R(R+). Therefore g−1

R σ =⋂si=1(ui g)−1

R (R+). Let γ ⊂ V ∨ be the cone gen-erated by the set (u1 g)R, . . . , (us g)R; then g−1

R σ = γ∨, and the assertion results fromlemma 3.3.2 and a second application of (i).

Lastly, arguing as in the proof of lemma 3.3.19(i), one derives (iii) from (ii) and (iii).

Parts (i) and (iii) of the following proposition provide the bridge connecting convex polyhe-dral cones to fine monoids.

Proposition 3.3.22. Let (L,+) be a free abelian group of finite rank, (LR, σ) an L-rationalconvex polyhedral cone, and set σL := L ∩ σ. Then :

(i) (Gordan’s lemma) σL is a fine and saturated submonoid of L, and L ∩ 〈σ〉 = σgpL .

(ii) For every v ∈ LR, the subset L ∩ (σ − v) is a finitely generated σL-module.(iii) For any submonoid M ⊂ L, we have : MQ = MR ∩ LQ and M sat = MR ∩ L.

Proof. (Here σ − v ⊂ LR denotes the translate of σ by the vector −v, i.e. the subset of allw ∈ LR such that w + v ∈ σ.) Choose v1, . . . , vs ∈ L that generate σ, and set

Cε :=

s∑i=1

tivi | ti ∈ [0, ε] for i = 1, . . . , s

for every ε > 0.

(i): Clearly L ∩ σ is saturated. Since C1 is compact and L is discrete, C1 ∩ L is a finite set.We claim that C1 ∩ L generates the monoid σL. Indeed, if v ∈ σL, write v =

∑si=1 rivi, with

ri ≥ 0 for every i = 1, . . . , s; hence ri = mi + ti for some mi ∈ N and ti ∈ [0, 1[, and thereforev = v′ +

∑si=1mivi, where v′, v1, . . . , vs ∈ C1 ∩ L. Next, it is clear that σgp

L ⊂ L ∩ 〈σ〉; forthe converse, say that w ∈ L ∩ 〈σ〉, and write w = w1 − w2, for some w1, w2 ∈ σ. Then


w1 =∑s

i=1 tivi for some ti ≥ 0; we pick t′i ∈ N such that t′i ≥ ti for every i ≤ s, and we setw′i :=

∑si=1 t

′ivi. It follows that w = w′1 − w′2, where w′2 := w2 + (w′1 − w1)), and notice that

w′1 ∈ σL and w′2 ∈ σ; then we must have w′2 ∈ σL as well, and therefore w ∈ σgpL .

(ii) is similar : from the compactness of C1 one sees that L ∩ (C1 − v) is a finite set; onthe other hand, arguing as in the proof of (i), one checks easily that the latter set generates theσL-module L ∩ (σ − v).

(iii): Let x ∈MR ∩ LQ; then we may write

(3.3.23) x =n∑i=1

ri ⊗mi where mi ∈M , ri > 0 for every i ≤ n.

Claim 3.3.24. In the situation of proposition 3.3.22(iii), let x ∈ MR ∩ LQ, and write x as in(3.3.23). Then, for every ε > 0 there exist q1, . . . , qn ∈ Q+ with |ri − qi| < ε for everyi = 1, . . . , n, and such that x =

∑ni=1 qi ⊗mi.

Proof of the claim. Up to a reordering, we may assume that m1, . . . ,mk form a basis of theQ-vector space generated by m1, . . . ,mn, therefore mk+i =

∑kj=1 qijmj for a matrix

A := (qij | i = 1, . . . , n− k; j = 1, . . . , k)

with entries in Q. Let r := (r1, . . . , rk) and r′ := (rk+1, . . . , rn); since x ∈ LQ, we deduce thatb := r+ r′ ·A ∈ Q⊕k. Moreover, if s := (s1, . . . , sk) ∈ Q⊕k and s′ := (sk+1, . . . , sn) ∈ Q⊕n−ksatisfy the identity b = s+ s′ ·A, then

∑ni=1 si ⊗mi = x. If we choose s′ very close to r′, then

s shall be very close to r; especially, we can achieve that both s and s′ are vectors with positivecoordinates. ♦

Claim 3.3.24 shows that x ∈MQ, whence the first stated identity; for the second identity, weare reduced to showing that M sat = MQ ∩ L, which is immediate.

For various algebraic and geometric applications of the theory of polyhedral cones, one is ledto study subdivisions of a given cone, in the sense of the following definition 3.3.25. Later weshall see a more abstract notion of subdivision, in the context of general fans, which howeverfinds its roots and motivation in the intuitive manipulations of polyhedra that we formalizehereafter.

Definition 3.3.25. Let V be a finite dimensional R-vector space.(i) A fan in V is a finite set ∆ consisting of convex polyhedral cones of V , such that :

• for every σ ∈ ∆, and every face τ of σ, also τ ∈ ∆;• for every σ, τ ∈ ∆, the intersection σ ∩ τ is also an element of ∆, and is a face of

both σ and τ .(ii) We say that ∆ is a simplicial fan if all the elements of ∆ are simplicial cones.

(iii) Suppose that V = L ⊗Z R for some free abelian group L; then we say that ∆ isL-rational if the same holds for every τ ∈ ∆.

(iv) A refinement of the fan ∆ is a fan ∆′ in V with⋃σ∈∆ σ =

⋃τ∈∆′ τ , and such that every

τ ∈ ∆ is the union of the γ ∈ ∆′ contained in τ .(v) A subdivision of a convex polyhedral cone (V, σ) is a refinement of the fan ∆σ consist-

ing of σ and its faces.

Lemma 3.3.26. Let (V, σ) be any convex polyhedral cone, ∆ a subdivision of (V, σ). We let

∆s := τ ∈ ∆ | 〈τ〉 = 〈σ〉.Then

⋃τ∈∆s τ = σ.

Proof. Let τ0 ∈ ∆ be any element. Then σ′ :=⋃τ 6=τ0 τ is a closed subset of σ. If 〈τ0〉 6= 〈σ〉,

then σ \ τ0 is a dense open subset of σ contained in σ′; it follows that σ′ = σ in this case.


Especially, τ0 =⋃τ 6=τ0(τ ∩ τ0); since each τ ∩ τ0 is a face of τ0, we see that there must exist

τ 6= τ0 such that τ0 is a face of τ . The lemma follows immediately.

Example 3.3.27. (i) Certain useful subdivisions of a polyhedral cone σ are produced by meansof auxiliary real-valued functions defined on σ. Namely, let us say that a continuous functionf : σ → R is a roof, if the following holds. There exist finitely many R-linear forms l1, . . . , lnon V , such that f(v) = min(l1(v), . . . , ln(v)) for every v ∈ σ. The concept of roof shall bereintroduced in section 3.6, in a more abstract and general guise; however, in order to grasp thelatter, it is useful to keep in mind its more concrete polyhedral incarnation. We attach to f asubdivision of σ, as follows. For every i, j = 1, . . . , n define lij := li − lj , and let τi ⊂ V ∨ bethe polyhedral cone σ∨ + Rli1 + · · · + Rlin. From the identity lik = lij + ljk we easily deducethat τ∨i ∩ τ∨j is a face of both τ∨i and τ∨j , for every i, j = 1, . . . , n. Denote by Θ the smallestsubdivision of σ containing all the τ∨i ; it is easily seen that

σ =n⋃i=1

τ∨i

and the restriction of f to each τ∨i agrees with li.(ii) Conversely, let f : σ → R a continuous function; suppose there exists a subdivision Θ

of σ, and a system (lτ | τ ∈ Θ) of R-linear forms on V such that• f(v) = lτ (v) for every τ ∈ Θ and every v ∈ τ .• f(u+ v) ≥ f(u) + f(v) for every u, v ∈ σ.

Then we claim that f is a roof on σ. Indeed, let Θs ⊂ Θ be the subset of all τ that span 〈σ〉.Notice fist that the system (lτ | τ ∈ Θs) already determines f uniquely, by virtue of lemma3.3.26. Next, let τ, τ ′ ∈ Θs be any two elements, and pick an element v of the interior of τ . Forany u ∈ τ ′ and any ε > 0 we have, by assumption : f(v + εu) ≥ f(v) + f(εu). If ε is smallenough, we have as well v + εu ∈ τ , in which case the foregoing inequality can be written as :

lτ (v) + ε · lτ (u) = lτ (v + εu) ≥ lτ (v) + ε · lτ ′(u)

whence lτ (u) ≥ lτ ′(u) = f(u) and the assertion follows.

Proposition 3.3.28. Let f : V → W be a linear map of finite dimensional R-vector spaces,(V, σ) a convex polyhedral cone, and h : σ → f(σ) the restriction of f . Then :

(i) There exists a subdivision ∆ of (W, f(σ)) such that :

h−1(a+ b) = h−1(a) + h−1(b) for every τ ∈ ∆ and every a, b ∈ τ .

(ii) Suppose moreover that V = L ⊗Z R, W = L′ ⊗Z R and f = g ⊗Z 1R for a mapg : L → L′ of free abelian groups. If σ is L-rational, then we may find an L-rationalsubdivision ∆ such that (i) holds.

Proof. Let V0 be the largest linear subspace contained in σ ∩ Ker f . Notice that, under theassumptions of (ii), we have : V0 = R ⊗Z Ker g. One verifies easily that the propositionholds for the given map f and the cone (V, σ), if and only if it holds for the induced mapf : V/V0 → W/f(V0) and the cone (V/V0, σ) (where σ is the image of σ in V/V0). Hence, wemay replace f by f , and assume from start that σ∩Ker f contains no non-zero linear subspaces.Moreover, we may assume that σ spans V and f(σ) spans W .

(i): Let S be the set of faces τ of σ such that f restricts to an isomorphism 〈τ〉 ∼→ W .

Claim 3.3.29. Let λ ⊂ f(σ) be any ray. Then :(i) λ′ := h−1λ is a strongly convex polyhedral cone. Especially, λ′ is generated by its

extremal rays (see (3.3.15)).


(ii) For every extremal ray ρ of λ′ with ρ * Ker f , there exists τ ∈ S such that ρ =τ ∩ f−1(λ).

Proof of the claim. λ′ is a convex polyhedral cone by lemma 3.3.18(i) and corollary 3.3.12(iii).To see that λ′ is strongly convex, notice that any subspace L ⊂ f−1(λ) lies already in Ker f ,and if L ⊂ σ, we must have L = 0 by assumption. Let ρ be an extremal ray of λ′ which is notcontained in Ker f ; notice that λ′ is the intersection of the polyhedral cones λ1 := h−1〈λ〉 andλ2 := f−1λ, hence we can find faces δi of λi (i = 1, 2) such that ρ = δ1∩δ2 (corollary 3.3.12(iii)and lemma 3.3.18(i)). However, the only proper face of λ2 is Ker f (lemma 3.3.18(ii)), henceδ2 = λ2. Likewise, f−1〈λ〉 has no proper faces, hence δ1 = γ ∩ f−1〈λ〉 for some face γ of σ(again by corollary 3.3.12(iii)). Since λ2 is a half-space in f−1〈λ〉, we deduce easily that eitherδ1 = ρ or δ1 = 〈ρ〉. Especially, dimR(f−1〈λ〉)/〈δ1〉 = dimR Ker f . We may then apply lemma3.3.18(ii) to the imbedding f−1〈λ〉 ⊂ V , to find a face τ of σ such that :

τ ∩ f−1〈λ〉 = δ1 〈τ〉 ∩ f−1〈λ〉 = 〈ρ〉 dimR V/〈τ〉 = dimR Ker f.

It follows that 〈τ〉 ∩Ker f = 0, and therefore τ ∈ S, as required. ♦

We construct as follows a subdivision of (W, f(σ)). For every τ ∈ S, let F (τ) be the setconsisting of the facets of the polyhedral cone f(τ); set also F :=

⋃τ∈S F (τ). Notice that, for

every γ ∈ F , the subspace 〈γ〉 is a hyperplane of W ; we let :

U := f(σ)\⋃γ∈F

〈γ〉.

Then U is an open subset of f(σ), and the topological closure C of every connected componentC of U is a convex polyhedral cone. Moreover, if C and D are any two such connected compo-nents, the intersection C ∩ D is a face of both C and D. We let ∆ be the subdivision of f(σ)consisting of the cones C – where C ranges over all the connected components of U – togetherwith all their faces.

Claim 3.3.30. For every δ ∈ ∆ and every τ ∈ S, the intersection δ ∩ f(τ) is a face of δ.

Proof of the claim. Due to proposition 3.3.8(iii), we may assume that δ is the topological closureof a connected component C of U . We may also assume that f(τ) 6= W , otherwise there isnothing to prove; in that case, we have f(τ) =

⋂γ∈F (τ) Hγ , where, for each γ ∈ F (τ), the

half-space Hγ is the unique one that contains both f(τ) and γ (proposition 3.3.11(ii)). It thensuffices to show that δ ∩ Hγ is a face of δ for each such Hγ . We may assume that δ * Hγ .Since C is connected and C ⊂ W \〈γ〉, it follows that δ ⊂ −Hγ , the topological closure of thecomplement of Hγ . Hence (−Hγ)

∨ ⊂ δ∨ (where (−Hγ)∨ is the dual of the polyhedral cone

(W,−Hγ)), and therefore δ ∩Hγ = δ ∩Hγ ∩ (−Hγ) = δ ∩ 〈γ〉 is indeed a face of δ. ♦

Next, for every w ∈ f(σ), let I(w) := τ ∈ S | w ∈ f(τ).

Claim 3.3.31. Let δ ∈ ∆, and w1, w2 ∈ δ. Then I(w1 + w2) ⊂ I(w1) ∩ I(w2).

Proof of the claim. Suppose first thatw1+w2 is contained in a face δ′ of δ; say that δ′ = δ∩Keru,for some u ∈ δ∨. This means that u(w1 +w2) = 0, hence u(w1) = u(w2) = 0, i.e. w1, w2 ∈ δ′.Hence, we may replace δ by δ′, and assume that δ is the smallest element of ∆ containingw1 +w2. Thus, suppose that τ ∈ I(w1 +w2); therefore w1 +w2 ∈ f(τ)∩ δ. From claim 3.3.30we deduce that δ ⊂ f(τ), hence τ ∈ I(w1) ∩ I(w2), as claimed. ♦

Finally, we are ready to prove assertion (i). Hence, let a, b ∈ f(σ) be any two vectors that liein the same element of ∆. Clearly :

h−1(a) + h−1(b) ⊂ h−1(a+ b)


hence it suffices to show the converse inclusion. However, directly from claim 3.3.29(ii) wederive the identity :

h−1(R+ · w) = (σ ∩Ker f) +∑τ∈I(w)

(τ ∩ f−1(R+ · w)) for every w ∈ f(σ).

Taking into account claim 3.3.31, we are then reduced to showing that :

τ ∩ f−1(R+ · (a+ b)) ⊂ (τ ∩ f−1(R+ · a)) + (τ ∩ f−1(R+ · b)) for every τ ∈ I(a+ b).

The latter assertion is obvious, since f restricts to an isomorphism 〈τ〉 ∼→ W .(ii): By inspecting the construction, one verifies easily that the subdivision ∆ thus exhibited

shall be L-rational, whenever σ is.

3.3.32. Later we shall also be interested in rational variants of the identities of proposition3.3.28(i). Namely, consider the following situation. Let g : L → L′ be a map of free abeliangroups of finite rank, gR : LR → L′R the induced R-linear map, and (LR, σ) anL-rational convexpolyhedral cone; set τ := gR(σ), and denote by hR : σ → τ (resp. hQ : σ ∩ LQ → τ ∩ L′Q) therestriction of gR. We point out, for later reference, the following observation :

Lemma 3.3.33. In the situation of (3.3.32), suppose that :

h−1R (x1) + h−1

R (x2) = h−1R (x1 + x2) for every x1, x2 ∈ τ

(where the sum is taken in the monoid (P(σ),+)). Then we have as well :

h−1Q (x1) + h−1

Q (x2) = h−1Q (x1 + x2) for every x1, x2 ∈ τ ∩ L′Q.

Proof. Let x1, x2 ∈ τ ∩ L′Q be any two elements, and v ∈ h−1Q (x1 + x2), so we may write

v = v1 + v2 for some vi ∈ h−1R (xi) (i = 1, 2). Let also u1, . . . , uk be a finite system of

generators for σ∨, and set

Ji := j ≤ k | uj(vi) = 0 Ei := g−1R (xi) ∩

⋂j∈Ji

Keruj (i = 1, 2).

Clearly LQ ∩ Ei is a dense subset of Ei for i = 1, 2, hence, in any neighborhood of (x1, x2) inL⊕2R we may find a solution (y1, y2) ∈ L⊕2

Q for the system of equations

gR(yi) = xi uj(yi) = 0 for i = 1, 2 and every j ∈ Ji.Since uj(xi) > 0 for every j /∈ Ji, we will also have uj(yi) > 0 for every j /∈ Ji, provided yi issufficiently close to xi. The lemma follows.

3.3.34. We conclude this section with some considerations that shall be useful later, in ourdiscussion of normalized lengths for model algebras (see (9.3.41)). Keep the notation of propo-sition 3.3.22, and for every subset U ⊂ LR, let

SL,σ(U) := L ∩ (σ − v) | v ∈ U and set SL,σ := SL,σ(LR).

There is a natural L-module structure on SL,σ; namely, notice that

(L ∩ (σ − v)) + l = L ∩ (σ − (v − l)) for every v ∈ LR and l ∈ Lhence the rule τl : S 7→ S+ l defines a bijection of SL,σ onto itself, for every l ∈ L, and clearlyτl τl′ = τl+l′ for every l, l′ ∈ L. Also, for every S ∈ SL,σ define

Ω(σ, S) := v ∈ LR | L ∩ (σ − v) = Sand denote by Ω(σ, S) the topological closure of Ω(σ, S) in LR. For given u ∈ L∨R and r ∈ R,set Hu,r := v ∈ LR | u(v) ≥ r. We shall say that a subset of LR is Q-linearly constructible,if it lies in the boolean subalgebra of P(LR) generated by the subsets Hu⊗Q1R,r, for u rangingover all the Q-linear forms LQ → Q, and r ranging over all rational numbers.


Proposition 3.3.35. With the notation of (3.3.34), the following holds :(i) SL,σ(U) is a finite set, for every bounded subset U ⊂ LR.

(ii) SL,σ is a finitely generated L-module.(iii) For every non-empty S ∈ SL,σ, the subset Ω(σ, S) is Q-linearly constructible.(iv) Suppose moreover, that σ spans LR. Then, for every S ∈ SL,σ, the subset Ω(σ, S) is

contained in the topological closure of its interior (see (3.3.10)).(v) For every S ∈ SL,σ, and every v ∈ Ω(σ, S), we have S ⊂ L ∩ (σ − v).

Proof. (i): Define Cε as in the proof of proposition 3.3.22; since U is bounded, it is containedin the union of finitely many subsets of LR of the form C1 + l, for l ranging over a finite subsetof L. On the other hand, τl induces a bijection

SL,σ(C1)∼→ LL,σ(C1 − l) for every l ∈ L.

Hence, it suffices to check the assertion for U = C1. However, the proof of proposition3.3.22(ii) shows that L ∩ (σ − v) is generated by L ∩ (C1 − v); if v ∈ C1, the latter subsetis contained in C ′ := C1∪ (−C1), which is a compact subset of LR. Therefore L∩C ′ is a finiteset, and the claim follows.

(ii): We have already observed that the L-module SL,σ is generated by SL,σ(C1), and this isa finite set, by (i).

(iii): Fix a minimal system S1, . . . , Sn of generators of the L-module SL,σ (i.e. the Si arechosen representatives for the orbits of the L-action on SL,σ). After replacing Si by sometranslates Si + l (for an appropriate l ∈ L) we may also assume that either Si = ∅, or else0 ∈ Si, and notice that this implies :

(3.3.36) Si ⊂ 〈σ〉 ∩ L = σgpL for every i = 1, . . . , n

(proposition 3.3.22(i)). Set

Aij := l ∈ L | Si ⊂ Sj − a for every i, j ≤ n

and notice that Aij is a σL-module, for every i, j ≤ n.

Claim 3.3.37. If Si, Sj 6= ∅, the σL-module Aij is finitely generated.

Proof of the claim. Fix l ∈ L such that σL + l ⊂ Si. Next, say that x1, . . . , xt is a finitesystem of generators for the σL-module Sj (proposition 3.3.22(ii)); by virtue of (3.3.36), forevery s = 1, . . . , t, we may write xs = as−bs for certain as, bs ∈ σL. Set l′ := b1 + · · ·+bt, andnotice that Sj ⊂ σL − l′. Now, if Si ⊂ Sj − a, we deduce that σL + l ⊂ σL − a− l′, especiallyl ∈ σL − a− l′, i.e. a ∈ σL − (l + l′). This shows that Aij is isomorphic to an ideal of σL, andthen the claim follows from proposition 3.1.9(ii). ♦

Now, let i, j ≤ n such that Si, Sj 6= ∅. Suppose first that i 6= j, and let A′ij ⊂ Aij be anyfinite generating system for the σL-module Aij . From the construction, it is clear that everyelement of LSj that contains Si, must contain Sj − l, for some l ∈ A′ij . To deal with the casewhere i = j, we remark, more generally :

Claim 3.3.38. Let P be any fine and saturated monoid,M ⊂ P gp a non-empty finitely generatedP -submodule, and a ∈ P gp an element such that aM ⊂M . Then a ∈ P .

Proof of the claim. Pick any m ∈ M , and denote by M ′ ⊂ M the submodule generated by(akm | k ∈ N). According to proposition 3.1.9(i), there exists N ≥ 0 such that M ′ is generatedby the finite system (akm | k = 0, . . . , N). Especially, aN+1m ∈M ′, and therefore there existsx ∈ P and i ≤ N such that aN+1m = aimx in M ; it follows that aN+1−i ∈ P , and finallya ∈ P , since P is saturated. ♦

From (3.3.36) we see that Aii ⊂ σgpL , if Si 6= ∅; combining with claim 3.3.38, we deduce

that Aii = σL. Moreover, notice as well that if Si = Si − a for some a ∈ σgpL , then both a


and −a ∈ Aii, so that a ∈ σ×L . Thus, let A′ii be any set of representatives of mσ \m2σ, where

mσ denotes the maximal ideal of σ]L. If a ∈ L, and Si − a contains strictly Si, then a is anon-invertible element of σL, and taking into account corollary 3.1.10, we see that A′ii is finite,and there exists l ∈ A′ii such that Si − l ⊂ Si − a. Next, for every i ≤ n such that Si 6= ∅, set

S i :=⋃j

Sj + l | l ∈ A′ij

where j ≤ n runs over the indices such that Sj 6= ∅. Summing up, we conclude that S i isa finite set for every i ≤ n with Si 6= ∅, and if an element of SL,σ contains strictly Si, thenit contains some element of S i. Lastly, in order to prove assertion (iii), we may assume thatS = Si for some i ≤ n, and notice that :

(3.3.39) Ω(σ, Si) = v ∈ LR | S ⊂ σ − v \⋃

S′∈S i

v ∈ LR | S ′ ⊂ σ − v.

Since S is finitely generated (proposition 3.3.22(ii)), we reduce to showing that, for every a ∈ L,the subset Ω(σ, a) := v ∈ LR | a ∈ σ − v is Q-linearly constructible, which follows easilyfrom proposition 3.3.21(i) and lemma 3.3.2.

(iv): We remark :

Claim 3.3.40. Let Cε be as in the proof of proposition 3.3.22; For every a ∈ Ω(σ, S) there existsε > 0 such that a+ Cε ⊂ Ω(σ, S).

Proof of the claim. Since σ is closed in LR, for every a ∈ LR and every b ∈ LR \ Ω(σ, a) thereexists ε > 0 such that (b+ Cε) ∩ Ω(σ, a) = ∅. Taking into account (3.3.39), the claim followseasily. ♦

If σ span LR, the subset Cε has non-empty interior Uε, for every ε > 0, and the topologicalclosure of Uε equals Cε. The assertion is then an immediate consequence of claim 3.3.40.

(v): The assertion follows easily from proposition 3.3.22(ii) : the details shall be left to thereader.

3.3.41. Let L be as in (3.3.34), and for all integers n,m > 0 set1

mL := v ∈ LQ |mv ∈ L

1

mL[1/n] :=

⋃k≥0

1

nkmL.

For future reference, let us also point out :

Lemma 3.3.42. With the notation of (3.3.41), let Ω ⊂ LR be a Q-linearly constructible subset.Then we have :

(i) The topological closure of Ω in LR is again Q-linearly constructible.(ii) There exists an integer m > 0 such that 1

mL[1/n] ∩ Ω is dense in Ω, for every n > 1.

Proof. (i): Ω is a finite union of non-empty subsets of the form H1 ∩ · · · ∩Hk, where each Hi

is either of the form Hu⊗1R,r for some non-zero Q-linear form u of LQ and some r ∈ Q (andthis is a closed subset of LR), or else is the complement in LR of a subset of this type (andthen its closure is a half-space H−u⊗1R,r). One verifies that the closure of H1 ∩ · · · ∩Hk is theintersection of the closures of H1, . . . , Hk, whence the assertion.

(ii): We may assume that Ω = Ω1 ∩ Ω2, where Ω1 is a finite intersection of rational hy-plerplanes, and Ω2 is a finite intersection of open half-spaces (i.e. of complements of closedhalf-spaces). Suppose that 1

mL[1/n] ∩ Ω1 is dense in Ω1; then clearly 1

mL[1/n] ∩ Ω is dense in

Ω. Hence, we may further assume that Ω is a non-empty intersection of rational hyperplanes.In this case, Ω is of the form VR + v0, where v0 ∈ LQ, and VR = V ⊗Z R for some subgroupV ⊂ L. Notice that L[1/n] ∩ VR is dense in VR for every integer n > 1. Then, any integerm > 0 such that v0 ∈ 1

mL will do.


3.4. Fine and saturated monoids. This section presents the more refined theory of fine andsaturated monoids. Again, all the monoids in this section are non-pointed. We begin with a fewcorollaries of proposition 3.3.22(i,iii).

Corollary 3.4.1. Let M be an integral monoid, such that M ] is fine. We have :

(i) The inclusion map M →M sat is a finite morphism of monoids.(ii) Especially, if M is fine, any monoid N with M ⊂ N ⊂M sat, is fine.

Proof. (i): From lemma 3.2.9(ii) we deduce that M sat is a finitely generated M -module if andonly if (M ])sat is a finitely generated M ]-module. Hence, we may replace M by M ], andassume that M is fine. Pick a surjective group homomorphism ϕ : Z⊕n → Mgp; it is easilyseen that :

ϕ−1(M sat) = (ϕ−1M)sat

and clearly it suffices to show that ϕ−1N is finitely generated, hence we may replace M byϕ−1M , and assume throughout that Mgp is a free abelian group of finite rank. In this case,proposition 3.3.22(i,iv) already implies that M sat is finitely generated. Let a1, . . . , ak ∈ M sat

be a finite system of generators, and pick integers n1, . . . , nk > 0 such that anii ∈ M fori = 1, . . . , k. For every i = 1, . . . , k let Σi := aji | j = 0, . . . , ni − 1; it is easily seenΣ1 · · ·Σk ⊂ M sat is a system of generators for the M -module M sat (where the product of thesets Σi is formed in the monoid P(M sat) of (3.1.1)).

(ii) follows from (i), in view of proposition 3.1.9(i).

Corollary 3.4.2. Let f : M1 → M and g : M2 → M be two morphisms of monoids, such thatM1 and M2 are finitely generated, and M is integral. Then the fibre product M1 ×M M2 is afinitely generated monoid, and if M1 and M2 are fine, the same holds for M1 ×M M2.

Proof. If the monoids M , M1 and M2 are integral, M1 ×M M2 injects in Mgp1 ×Mgp Mgp

2

(lemma 2.3.29(iii)), hence it is integral. To show that the fibre product is finitely generated,choose surjective morphisms N⊕a → M1 and N⊕b → M2, for some a, b ∈ N; by compositionwe get maps of monoids ϕ : N⊕a → M , ψ : N⊕b → M , such that the induced morphismP := N⊕a ×M N⊕b → M1 ×M M2 is surjective. Hence it suffices to show that P is finitelygenerated. To this aim, let L := Ker(ϕgp − ψgp : Z⊕a+b → Mgp); for every i = 1, . . . , a + b,denote also by πi : Z⊕a+b → Z the projection onto the i-th direct summand. The systemπi | i = 1, . . . , a+b generates a rational convex polyhedral cone σ ⊂ L∨⊗ZR, and one verifieseasily that P = L ∩ σ∨, so the assertion follows from propositions 3.3.21(i) and 3.3.22(i).

Corollary 3.4.3. Let (Γ,+, 0) be an integral monoid, M a finitely generated Γ-graded monoid.Then M0 is a finitely generated monoid, and Mγ is a finitely generated M0-module, for everyγ ∈ Γ.

Proof. We have M0 = M ×Γ 0, hence M0 is finitely generated, by corollary 3.4.2. Thegiven element γ ∈ Γ determines a unique morphism of monoids N → Γ such that 1 7→ γ.Let p1 : M ′ := M ×Γ N → N and p2 : M ′ → M be the two natural projections; by lemma2.3.29(iii), we have Mγ = p2(p−1

1 (1)). In light of corollary 3.4.2, M ′ is still finitely generated,hence we are reduced to the case where Γ = N and γ = 1. In this case, pick a finite set ofgenerators S for M . One checks easily that M1 ∩ S generates the M0-module M1.

Corollary 3.4.4. Let M be an integral monoid, such that M ] is fine, and ϕ : M → N asaturated morphism of monoids. Then ϕ is flat.

Proof. In view of corollary 3.4.1(i) and theorem 3.2.6, it suffices to show that the M sat-moduleM sat ⊗M N is flat. Hence we may replace M by M sat, and assume that M is saturated. Let


I ⊂ M be any ideal, and define R(M, I) as in the proof of theorem 3.2.3; by assumption,R(M, I)sat ⊗M N is a saturated – especially, integral – monoid, i.e. the natural map

R(M, I)⊗M N → R(M, I)gp ⊗Mgp Ngp

is injective. The latter factors through the morphism j⊗MN , where j : R(M, I)→M×N is theobvious inclusion. In light of example 3.2.13(i), we deduce that the induced map Isat⊗M N →N is injective. Now, if I is a prime ideal, then Isat = I , hence the contention follows fromcorollary 3.1.48(ii).

The following corollary generalizes lemma 3.2.10.

Corollary 3.4.5. Let f : M → N be a local, flat and saturated morphism of fine monoids, withM sharp. Then there exists an isomorphism of monoids

g : N∼→ N ] ×N×

that fits into a commutative diagram

Mf] //

f

N ]

N

g // N ] ×N×

whose right vertical arrow is the natural inclusion map.

Proof. From lemma 3.2.30(ii), we know that f is exact, and since M is sharp, we easily deducethat f(M)gp ∩ N× = 1. Hence, the induced group homomorphism Mgp ⊕ N× → Ngp isinjective. On the other hand, since f is flat, local and saturated, the same holds for f ] : M → N ]

(lemma 3.2.12(iii) and corollary 3.4.4); then corollary 3.2.32(ii) says that the cokernel of theinduced group homomorphism Mgp → (N ])gp = Ngp/N× is a free abelian group G (of finiterank). Summing up, we obtain an isomorphism of abelian groups :

h : Mgp ⊕N× ⊕G ∼→ Ngp

extending the map f gp. Set N0 := N ∩ h(Mgp ⊕ G); it follows easily that the natural mapN0×N× → N is an isomorphism; especially, the projection N → N ] maps N0 isomorphicallyonto N ], and the contention follows.

3.4.6. Let (M, ·) be a fine (non-pointed) monoid, so that Mgp is a finitely generated abeliangroup. We set Mgp

R := logMgp ⊗Z R, and we let MR be the convex polyhedral cone generatedby the image of logM . Then (MR,+) is a monoid, and we have a natural morphism of monoids

ϕ : logM → (MR,+).

Proposition 3.4.7. With the notation of (3.4.6), we have :(i) Every face of the polyhedral cone MR is of the form FR, for a unique face F of M .

(ii) The induced map :ϕ∗ : SpecMR → SpecM

is a bijection.

Proof. Clearly, we may assume that M 6= 1. Let p ⊂ M be a prime ideal; we denote bypR the ideal of (MR,+) generated by all elements of the form r · ϕ(x), where r is any strictlypositive real number, and x is any element of p. We also denote by (M\p)R the convex cone ofMgp

R generated by the image of M \p.

Claim 3.4.8. MR is the disjoint union of (M \p)R and pR.


Proof of the claim. To begin with, we show that MR = (M\p)R ∪ pR. Indeed, let x ∈MR; thenwe may write x =

∑hi=1mi ⊗ ai for certain a1, . . . , ah ∈ R+ and m1, . . . ,mh ∈ M . We may

assume that a1, . . . , ak ∈ p and ak+1, . . . , ah ∈M\p. Now, if k = 0 we have x ∈ (M\p)R, andotherwise x ∈ pR, which shows the assertion.

It remains to show that (M \p)R ∩ pR = ∅. To this aim, suppose by way of contradiction,that this intersection contains an element x; this means that we have finite subsets S0 ⊂ M \pand S1 ⊂M such that S1 ∩ p 6= ∅, and an identity of the form :

(3.4.9) x =∑σ∈S0

σ ⊗ aσ =∑σ∈S1

σ ⊗ bσ

where aσ > 0 for every σ ∈ S0 and bσ > 0 for every σ ∈ S1. For every σ ∈ S0, choosea rational number a′σ ≥ aσ; after adding the summand

∑σ∈S0

σ ⊗ (a′σ − aσ) to both sidesof (3.4.9), we may assume that aσ ∈ Q+ for every σ ∈ S0. Let N ⊂ M be the submonoidgenerated by S1; it follows that x ∈ NR ∩MQ = NQ (proposition 3.3.22(iii)), hence we mayassume that all the coefficients aσ and bσ are rational and strictly positive (see remark 3.3.24).We may further multiply both sides of (3.4.9) by a large integer, to obtain that these coefficientsare actually integers. Then, up to further multiplication by some integer, the identity of (3.4.9)lifts to an identity between elements of logM , of the form :

∑σ∈S0

aσ · σ =∑

σ∈S1bσ · σ. The

latter is absurd, since S1 ∩ p 6= ∅ and S0 ∩ p = ∅. ♦

Claim 3.4.8 implies that pR is a prime ideal of MR, and clearly p ⊂ ϕ∗(pR). Since we haveas well M \p ⊂ ϕ−1(M \p)R, we deduce that p = ϕ∗(pR). Hence the rule p 7→ pR yields aright inverse ϕ∗ : SpecM → SpecMR for the natural map ϕ∗. To show that ϕ∗ is also a leftinverse, let q ⊂MR be a prime ideal; by lemma 3.3.7 and proposition 3.3.21(i), the face MR\qis of the form MR ∩ Keru, for some u ∈ M∨

R ∩ (logMgp)∨. Then it is easily seen that MR\qis the convex cone generated by ϕ(M) ∩Keru, in other words, MR \ q = ϕ−1(Keru)R. Againby claim 3.4.8, it follows that q = (M \ϕ−1(Keru))R = (ϕ∗q)R, as stated. The argument alsoshows that every face of MR is of the from (M \p)R for a unique prime ideal p, which settlesassertion (i).

Corollary 3.4.10. Let M be a fine monoid. We have :(i) dimM = rkZ(Mgp/M×).

(ii) dim(M \p) + ht p = dimM for every p ∈ SpecM .(iii) If M 6= 1 is sharp (see (2.3.32)), there exists a local morphism M → N.(iv) IfM is sharp andMgp is a torsion-free abelian group of rank r, there exists an injective

morphism of monoids M → N⊕r.

Proof. (i): By proposition 3.4.7, the dimension of M can be computed as the length of thelongest chain F0 ⊂ F1 ⊂ · · · ⊂ Fd of strict inclusions of faces of MR. On the other hand, givensuch a maximal chain, denote by ri the dimension of the R-vector space spanned by Fi; in viewproposition 3.3.8(ii),(iii), it is easily seen that ri+1 − ri = 1 for every i = 0, . . . , d − 1. SinceMR ∩ (−MR) is the minimal face of MR, we deduce that

dimM = dimRMgpR − dimRMR ∩ (−MR).

Clearly dimRMgpR = rkZM

gp; moreover, by proposition 3.4.7, the faceMR∩(−MR) is spannedby the image of the face M× of M . whence the assertion.

(ii) is similar : again proposition 3.3.8(ii),(iii) implies that, every face F of MR fits into amaximal strictly ascending chain of faces of MR, and the length of any such maximal chain isdimM , by (i).

(iii): Notice that rkZMgp > 0, by (i). By proposition 3.4.7(i), MR is strongly convex, there-

fore, by proposition 3.3.21(i), we may find a non-zero linear map ϕ : Mgp ⊗Z Q → Q, suchthat MR ∩Kerϕ⊗Q R = 0 and ϕ(M) ⊂ Q+. A suitable positive integer of ϕ will do.


(iv): Under the stated assumption, we may regard M as a submonoid of MR, and the lattercontains no non-zero linear subspaces. By corollary 3.3.14 and proposition 3.3.21(i), we maythen find r linearly independent forms u1, . . . , ur : Mgp ⊗Z Q → Q which are positive onM . It follows that u1 ⊗Q R, . . . , ur ⊗Q R generate a polyhedral cone σ∨ ⊂ M∨

R , so its dualcone σ ⊂ Mgp

R contains MR. By construction, σ admits precisely r extremal rays, say therays generated by the vectors v1, . . . , vr, which we can pick in Mgp

Q , in which case they forma basis of the latter Q-vector space. Now, every x ∈ MR can be written uniquely in the formx =

∑ri=1 aivi for certain a1, . . . , ar ∈ Q+; since M is finitely generated, we may find an

integer N > 0 independent of x, such that Nai ∈ N for every i = 1, . . . , r. In other words, Mis contained in the monoid generated byN−1v1, . . . , N

−1vr; the latter is isomorphic to N⊕r.

3.4.11. For any monoid M , the dual of M is the monoid M∨ := HomMnd(M,N) (see (3.1.1)).As usual, there is a natural morphism

M →M∨∨ : m 7→ (ϕ 7→ ϕ(m)) for every m ∈M and ϕ ∈M∨.

We say that M is reflexive, if this morphism is an isomorphism.

Proposition 3.4.12. Let M be a monoid. We have :(i) M∨ is integral, saturated and sharp.

(ii) If M is finitely generated, M∨ is fine, and we have a natural identification :

(M∨)R∼→ (MR)∨.

Moreover, dimM = dimM∨.(iii) If M is finitely generated and sharp, we have a natural identification :

(M∨)gp ∼→ (Mgp)∨.

(iv) If M is fine, sharp and saturated, then M is reflexive.

Proof. (i): It is easily seen that the natural group homomorphism

(3.4.13) (M∨)gp → (Mgp)∨ := HomZ(Mgp,Z)

is injective. Now, say that ϕ ∈ (M∨)gp and Nϕ ∈ M∨ for some N ∈ N; we may view ϕ asgroup homomorphism ϕ : Mgp → Z, and the assumption implies that ϕ(M) ⊂ Z ∩ Q+ = N,whence the contention.

(ii): Indeed, let x1, . . . , xn be a system of generators of M . Define a group homomorphismf : (Mgp)∨ → Z⊕n by the rule : ϕ 7→ (ϕ(x1), . . . , ϕ(xn)) for every f : Mgp → Z. Then M∨ =ϕ−1(N⊕n), and since (Mgp)∨ is fine, corollary 3.4.2 implies that M∨ is fine as well. Next, theinjectivity of (3.4.13) implies especially that (M∨)gp is torsion-free, hence (3.4.13)⊗ZR is stillinjective; its restriction to (M∨)R factors therefore through an injective map f : (M∨)R →(MR)∨. The latter map is determined by the image of M∨, and by inspecting the definitions,we see that f(ϕ) := ϕgp ⊗ 1 for every ϕ ∈ M∨. To prove that f is an isomorphism, it sufficesto show that it has dense image. However, say that ϕ ∈ (MR)∨; then ϕ : Mgp → R is a grouphomomorphism such that ϕ(M) ⊂ R+. Since M is finitely generated, in any neighborhood ofϕ in Mgp

R we may find some ϕ′ : Mgp → Q+, and then Nϕ′ ∈ M∨ for some integer N ∈ Nlarge enough. It follows that ϕ′ is in the image of f , whence the contention.

The stated equality follows from the chain of identities :

dimM = dimMR = dim(MR)∨ = dim(M∨)R = dimM∨

where the first and the last follow from proposition 3.4.7(ii), and the second follows from corol-lary 3.3.12(ii).

(iii): Let us show first that, under these assumptions, (3.4.13) ⊗Z R is an isomorphism. In-deed, if M is sharp, (MR)∨ spans (Mgp

R )∨ (corollary 3.3.14 and proposition 3.4.7(i)); then the


assertion follows from (ii). We deduce that (M∨)gp and (Mgp)∨ are free abelian groups of thesame rank, hence we may find a basis ϕ1, . . . , ϕr of (M∨)gp (resp. ψ1, . . . , ψr of (Mgp)∨),and positive integers N1, . . . , Nr such that (3.4.13) is given by the rule : ϕi 7→ Niψi for everyi = 1, . . . , r. But then necessarily we have Ni = 1 for every i ≤ r, and (iii) follows.

(iv): It is easily seen that M∨ = (MR)∨ ∩ (Mgp)∨ (notation of (3.4.6)). After dualizing againwe find : M∨∨ = ((M∨)R)∨ ∩ (M∨gp)∨. From (ii) we deduce that ((M∨)R)∨ = (MR)∨∨ = MR(lemma 3.3.2), and from (iii) we get : (M∨gp)∨ = (Mgp)∨∨ = Mgp. Hence M∨∨ = MR ∩Mgp = M (proposition 3.3.22(iii)).

Remark 3.4.14. (i) Let M be a sharp and fine monoid. Proposition 3.4.12(iii) implies that thenatural map

HomMnd(M,Q+)gp → HomMnd(M,Q)

is an isomorphism. Indeed, it is easily seen that this map is injective. For the surjectivity, oneuses the identification HomMnd(M,Q)

∼→ (M∨)gp ⊗Z Q, which follows from loc.cit. (Detailsleft to the reader.)

(ii) For i = 1, 2, let Ni → N be two morphisms of monoids. By general nonsense, we havea natural isomorphism :

(N1 ⊗N N2)∨∼→ N∨1 ×N∨ N∨2 .

(iii) If fi : Mi → M (i = 1, 2) are morphisms of fine, saturated and sharp monoids, thereexists a natural surjection :

(3.4.15) M∨1 ⊗M∨ M∨

2 → (M1 ×M M2)∨

whose kernel is the subgroup of invertible elements. Indeed, set P := M∨1 ⊗M∨M∨

2 ; in view of(ii) and proposition 3.4.12(iv), we have a natural identification P∨∨ ∼→ (M1 ×M M2)∨, and thesought map is its composition with the double duality map P → P∨∨. Moreover, clearly P isfinitely generated, and it is also integral and saturated, since saturation commutes with colimits.Hence – again by proposition 3.4.12(iv) – the double duality map induces an isomorphismP/P×

∼→ P∨∨.(iv) In the situation of (iii), if fi : Mi → M (i = 1, 2) are epimorphisms, then (3.4.15) is an

isomorphism. Indeed, in this case the dual morphisms f∨i : M∨ → M∨i are injective, so that P

is sharp (lemma 3.1.12), whence the claim.

Theorem 3.4.16. Let M be a saturated monoid, such that M ] is fine. We have :

(i) M =⋂

ht p=1

Mp (where the intersection runs over the prime ideals of M of height one).

(ii) If moreover, dimM = 1, then there is an isomorphism of monoids :

M× × N ∼→M.

(iii) Suppose that Mgp is a torsion-free abelian group, and let R be any normal domain.Then the group algebra R[M ] is a normal domain as well.

Proof. (i): Pick a decompositionM = M ]×M× as in lemma 3.2.10, and notice thatM ] is fine,sharp and saturated. The prime ideals of M are of the form p = p0 ×M×, where p0 is a primeideal of M ]. Then it is easily seen that Mp = M ]

p0×M×. Therefore, the sought assertion holds

for M if and only if it holds for M ], and therefore we may replace M by M ], which reducesto the case where M is sharp, hence the natural morphism ϕ : logM → MR is injective. Insuch situation, we have M = MR ∩Mgp and Mp = Mp,R ∩Mgp for every prime ideal p ⊂ M(proposition 3.3.22(iii) and lemma 3.2.9(i)). Thus, we are reduced to showing that

MR =⋂

ht p=1

Mp,R.


However, set τ := (M \p)R; by inspecting the definitions, one sees that Mp,R = MR + (−τ),and proposition 3.4.7 shows that τ is a facet of MR, hence Mp,R is the half-space denoted Hτ in(3.3.10). Then the assertion is a rephrasing of proposition 3.3.11(ii).

(ii): Arguing as in the proof of (i), we may reduce again to the case where M is sharp, inwhich case M = MR ∩ Mgp. The foregoing shows that, in case dimM = 1, the cone MRis a half-space, whose boundary hyperplane is the only non-trivial face σ of MR. However, σis generated by the image of the unique non-trivial face of M , i.e. by M× = 1 (proposition3.4.7(i)), hence σ = 0, soMR is a half-line. Now, let u : Mgp

R → R be a non-zero linear form,such that u(M) ≥ 0, and x1, . . . , xn a system of non-zero generators for M ; say that u(x1) isthe least of the values u(xi), for i = 1, . . . , n. Since M is saturated, it follows easily that everyvalue u(xi) is an integer multiple of u(x1) (proposition 3.3.22(iii)), and then x1 is a generatorfor M , so M ' N.

(iii): To begin with, R[M ] ⊂ R[Mgp], and since Mgp is torsion-free, it is clear that [Mgp]is a domain, hence the same holds for R[M ]. Furthermore, from (i) we derive : R[M ] =⋂

htp=1R[Mp], hence it suffices to show that R[Mp] is normal whenever p has height one. How-ever, we have R[Mp] ' R[M×

p ] ⊗R R[N] in light of (ii), and since M×p is torsion free, it is a

filtered colimit of a family of free abelian groups of finite rank, so everything is clear.

Example 3.4.17. Let M be a fine, sharp and saturated monoid of dimension 2.(i) By corollary 3.4.10(i) and example 3.3.16, we see that M admits exactly two facets,

which are fine saturated monoids of dimension one; by theorem 3.4.16(ii) each of these facetsis generated by an element, say ei (for i = 1, 2). From proposition 3.3.22(iii) it follows thatQ+e1 ⊕Q+e2 = MQ. Especially, we may find an integer N > 0 large enough, such that :

Ne1 ⊕ Ne2 ⊂M ⊂ Ne1

N⊕ N

e2

N.

(ii) Moreover, clearly e1 and e2 are unimodular elements of Mgp (i.e. they generate directsummands of the latter free abelian group of rank 2). We may then find a basis f1, f2 of Mgp

with e1 = f1, and e2 = af1 + bf2, where a, b ∈ Z and (a, b) = 1. After replacing f2 by someelement of the form cf2 + df1 with c ∈ 1,−1 and d ∈ Z, we may assume that b > 0 and0 ≤ a < b. Clearly, such a normalized pair (a, b) determines the isomorphism class of M , sinceMR is the strictly convex cone of Mgp

R whose extremal rays are generated by e1 and e2, andM = Mgp ∩MR.

(iii) More precisely, suppose that M ′ is another fine, sharp and saturated monoid of dimen-sion 2, and ϕ : M → M ′ an isomorphism. Pick a basis f ′1, f

′2 of M ′gp and a normalized pair

(a′, b′) as in (ii), such that e′1 := f1 and e′2 := a′f ′1 + b′f ′2 generate the two facets of M ′. Clearly,ϕ must send a facet of M onto a facet of M ′; we distinguish two possibilities :

• either ϕ(e1) = e′1 and ϕ(e2) = e′2, in which case we get ϕ(f2) = b−1(a′−a)f ′1+b−1b′f ′2;especially, b′, a− a′ ∈ bZ. By considering ϕ−1, we get symmetrically that b ∈ b′Z, sob = b′ and therefore (a′, b) = 1 = (a, b) and 0 ≤ a, a′ < b, whence a = a′

• or else ϕ(e1) = e′2 and ϕ(e2) = e′1, in which case we get ϕ(f2) = b−1(1 − aa′)f1 +b−1b′af2. It follows again that b′ ∈ bZ, so b = b′, arguing as in the previous case.Moreover, 0 ≤ a′ < b, and 1 − aa′ ∈ bZ. In other words, the class of a′ in the group(Z/bZ)× is the inverse of the class of a.

Conversely, it is easily seen that, if M ′ is as above, f ′1, f′2 is a basis of M ′gp, and the two facets

of M ′ are generated by f ′1 and a′f ′1 + b′f ′2, for a pair (a′, b′) normalized as in (ii), and such thataa′ ≡ 1 (mod b), then there exists an isomorphism M

∼→ M ′ of monoids (details left to thereader). Hence, set (Z/bZ)† := (Z/bZ)×/∼, where∼ denotes the smallest equivalence relationsuch that [a] ∼ [a]−1 for every [a] ∈ (Z/bZ)×. We conclude that there exists a natural bijectionbetween the set of isomorphism classes of fine, sharp and saturated monoids of dimension 2,and the set of pairs (b, [a]), where b > 0 is an integer, and [a] ∈ (Z/bZ)†.


3.4.18. Let P be an integral monoid. A fractional ideal of P is a P -submodule I ⊂ P gp suchthat I 6= ∅ and x·I ⊂ P for some x ∈ P . Clearly the union and the intersection of finitely manyfractional ideals, are again fractional ideals. We may also define the product of two fractionalideals I1, I2 ⊂ P gp : namely, the subset

I1I2 := xy | x ∈ I1, y ∈ I2 ⊂ P gp

which is again a fractional ideal, by an easy inspection. If I is a fractional ideal of P , we saythat I is finitely generated, if it is such, when regarded as a P -module. For any two fractionalideals I1, I2, we let

(I1 : I2) := x ∈ P gp | x · I2 ⊂ I1.It is easily seen that (I1 : I2) is a fractional ideal of P (if x ∈ I2 and yI1 ⊂ P , then clearlyxy(I1 : I2) ⊂ P ). We set

I−1 := (P : I) and I∗ := (I−1)−1 for every fractional ideal I ⊂ P gp.

Clearly J−1 ⊂ I−1, whenever I ⊂ J , and I ⊂ I∗ for all fractional ideals I , J . We say that I isreflexive if I = I∗. We remark that I−1 is reflexive, for every fractional ideal I ⊂ P gp. Indeed,we have I−1 ⊂ (I−1)∗, and on the other hand (I−1)∗ = (I∗)−1 ⊂ I−1. It follows that I∗ isreflexive, for every fractional ideal I . Moreover, I∗ ⊂ J∗, whenever I ⊂ J ; especially, I∗ is thesmallest reflexive fractional ideal containing I . Notice furthermore, that aI−1 = (a−1I)−1 forevery a ∈ P gp; therefore, aI∗ = (aI)∗, for every fractional ideal I and a ∈ P gp.

Lemma 3.4.19. Let P be any integral monoid, I, J ⊂ P gp two fractional ideals. Then :(i) (IJ)∗ = (I∗J∗)∗.

(ii) I∗ is the intersection of the invertible fractional ideals of P that contain I (see definition2.3.6(iv)).

Proof. (i): Since IJ ⊂ I∗J∗, we have (IJ)∗ ⊂ (I∗J∗)∗. To show the converse inclusion,it suffices to check that I∗J∗ ⊂ (IJ)∗, since (IJ)∗ is reflexive, and (I∗J∗)∗ is the smallestreflaxive fractional ideal containing I∗J∗. Now, let a ∈ I be eny element; we get aJ∗ =(aJ)∗ ⊂ (IJ)∗, so IJ∗ ⊂ (IJ)∗, and therefore (IJ∗)∗ ⊂ (IJ)∗. Lastly, let b ∈ J∗ be anyelement; we get bI∗ = (bI)∗ ⊂ (IJ∗)∗, so I∗J∗ ⊂ (IJ∗)∗, whence the lemma.

(ii): It suffices to unwind the definitions. Indeed, a ∈ P gp lies in I∗ if and only if aI−1 ⊂ P ,if and only if ab ∈ P , for every b ∈ P gp such that bI ⊂ P . In other words, a ∈ I∗ if and only ifa ∈ b−1P for every b ∈ P gp such that I ⊂ b−1P , which is the contention.

3.4.20. Let P be any integral monoid. We denote by Div(P ) the set of all reflexive fractionalideals of P . We define a composition law on Div(P ) by the rule :

I J := (IJ)∗ for every I, J ∈ Div(P ).

It follows easily from lemma 3.4.19(i) that is an associative law; indeed we may compute :

(I J)K = ((IJ)∗K)∗ = (IJK)∗ = (I(JK)∗)∗ = I (J K)

for every I, J,K ∈ Div(P ). Clearly I J = J I and P I = I , for every I, J ∈ Div(P ),so (Div(P ),) is a commutative monoid. Notice as well that, if I ⊂ P , then also I∗ ⊂ P(lemma 3.4.19(ii)), so the subset of all reflexive fractional ideals contained in P is a submonoidDiv+(P ) ⊂ Div(P ).

Example 3.4.21. Let A be an integral domain, and K the field of fractions of A. Classically,one defines the notions of fractional ideal and of reflexive fractional ideal of A : see e.g. [61,p.80]. In our terminology, these are understood as follows. Set A′ := A ∩K×, and notice thatthe monoid (A, ·) is naturally isomorphic to the integral pointed monoid A′. Then a fractionalideal of A is an A′-submodule of K× = K of the form I, where I ⊂ K× is a fractional idealof A′. Likewise one may define the reflexive ideals of A. The set Div(A) of all reflexive ideals


of A is then endowed with the unique monoid structure, such that the map Div(A′)→ Div(A)given by the rule I 7→ I is an isomorphism of monoids.

Lemma 3.4.22. Let P be an integral monoid, and G ⊂ P× a subgroup. We have :(i) The rule I 7→ I/G induces a bijection from the set of fractional ideals of P to the set

of fractional ideals of P/G.(ii) A fractional ideal I of P is reflexive if and only if the same holds for I/G.

(iii) The rule I 7→ I/G defines an isomorphism of monoids

Div(P )∼→ Div(P/G).

(iv) If P ] is fine, every fractional ideal of P is finitely generated.

Proof. The first assertion is left to the reader. Next, we remark that I−1/G = (I/G)−1 and(IJ)/G = (I/G) · (J/G), for every fractional ideals I, J of P , which imply immediatelyassertions (ii) and (iii). Lastly, suppose that P ] is finitely generated, and let I be any fractionalideal of P ; pick x ∈ I−1; since P is integral, I is finitely generated if and only if the same holdsfor xI . Hence, in order to show (iv), we may assume that I ⊂ P , in which case the assertionfollows from proposition 3.1.9(ii) and lemma 3.1.16(i.a).

In order to characterize the monoids P such that Div(P ) is a group, we make the following :

Definition 3.4.23. Let P be an integral monoid, and a ∈ P gp any element.(a) We say that a is power-bounded, if there exists b ∈ P such that anb ∈ P for all n ∈ N.(b) We say that P is completely saturated, if all power-bounded elements of P gp lie in P .

Example 3.4.24. Let (Γ,≤) be an ordered abelian group, and set Γ+ := γ ∈ Γ | γ ≤ 1. ThenΓ+ is always a saturated monoid, but it is completely saturated if and only if the convex rank ofΓ is ≤ 1 (see [36, Def.6.1.20]). The proof shall be left as an exercise for the reader.

Proposition 3.4.25. Let P be an integral monoid. We have :(i) (Div(P ),) is an abelian group if and only if P is completely saturated.

(ii) If P is fine and saturated, then P is completely saturated.(iii) Let A be a Krull domain, and set A′ := A\0. Then (A′, ·) is a completely saturated

monoid.

Proof. (i): Suppose that I ∈ Div(I) admits an inverse J in the monoid (Div(P ),), and noticethat I I−1 ⊂ P ; it follows easily that I (J ∪ I−1)∗ = P , hence I−1 ⊂ J , by the uniquenessof the inverse. On the other hand, if J strictly contains I−1, then IJ strictly contains P , whichis absurd. Thus, we see that Div(P ) is a group if and only if II−1 = P for every I ∈ Div(P ).Now, suppose first that P is completely saturated. In view of lemma 3.4.19(ii), we are reducedto showing that P is contained in every invertible fractional ideal containing I−1I . Hence, saythat I−1I ⊂ aP for some a ∈ P gp; equivalently, we have a−1I−1I ⊂ P , i.e. a−1I−1 ⊂ I−1,and then a−kI−1 ⊂ I−1 for every integer k ∈ N. Say that b ∈ I−1 and c ∈ I∗; we conclude thata−kbc ∈ P for every k ∈ N, so a−1 ∈ P , by assumption, and finally P ⊂ aP , as required.

Conversely, suppose that Div(P ) is a group, and let a ∈ P gp be any power-bounded element.By definition, this means that the P -submodule I of P gp generated by (ak | k ∈ N) is a fractionalideal of P . Then I−1 is a reflexive fractional ideal, and by assumption I−1 admits an inverse,which must be I∗, by the foregoing. On the other hand, by construction we have aI ⊂ I , henceaI∗ = (aI)∗ ⊂ I∗. We deduce that aP = a(I∗ I−1) = aI∗ I−1 ⊂ I∗ I−1 = P , i.e.a ∈ P , as stated.

(ii) is a special case of claim 3.3.38.(iii): See [61, §12] for the basic generalities on Krull domains. One is immediately reduced

to the case where A is a valuation ring whose valuation group Γ has rank ≤ 1. Taking into


account (i) and lemma 3.4.22, it then suffices to show that the monoid A′/A× is completelysataurated. However, the latter is isomorphic to the submonoid Γ+ of elements ≤ 1 in Γ, so theassertion follows from example 3.4.24.

3.4.26. Let ϕ : P → Q be a morphism of integral monoids, and I any fractional ideal of P ;notice that IQ := ϕgp(I)Q ⊂ Qgp is a fractional ideal of Q. Moreover, the identities

(I1 ∪ I2)Q = I1Q∪ I2Q (I1I2)Q = (I1Q) · (I2Q) for all fractional ideals I1, I2 ⊂ P gp

are immediate from the definitions. Likewise, if A an integral domain and α : P → (A\0, ·)a morphism of monoids, then the A-submodule IA := αgp(I)A of the field of fractions of Ais a fractional ideal of the ring A (in the usual commutative algebraic meaning : see example3.4.21), and we have corresponding identities :

(I1∪ I2)A = I1A+ I2A (I1I2)A = (I1A) · (I2A) for all fractional ideals I1, I2 ⊂ P gp.

Lemma 3.4.27. In the situation of (3.4.26), suppose that ϕ is flat and A is α-flat, and letI, J, J ′ ⊂ P gp be three fractional ideals, with I finitely generated. Then we have :

(i) (J : I)Q = (JQ : IQ) and (J : I)A = (J : I)A.(ii) Especially, if I is reflexive, the same holds for IQ and IA (see (5.6)).

(iii) Suppose furthermore that A is local, and α is a local morphism. Then JA = J ′A ifand only if J = J ′.

(iv) If P is fine, the rule I 7→ IQ and I 7→ IA define morphisms of monoids

Div(ϕ) : Div(P )→ Div(Q) Div(α) : Div(P )→ Div(A)

(where Div(A) is defined as in example 3.4.21), and Div(α) is injective, if α is localand A is a local domain.

Proof. (i): Say that I = a1P ∪ · · · ∪ anP for elements a1, . . . , an ∈ P gp. Then

(J : I) = a−11 J ∩ · · · ∩ a−1

n J and (JQ : IQ) = a−11 JQ ∩ · · · ∩ a−1

n JQ

and likewise for (J : I)A, hence the assertion follows from an easy induction, and the following

Claim 3.4.28. For any two fractional ideals J1, J2 ⊂ P , we have (J1 ∩ J2)Q = J1Q ∩ J2Q and(J1 ∩ J2)A = J1A ∩ J2A.

Proof of the claim. Pick any x ∈ P such that xJ1, xJ2 ⊂ P ; since P is an integral monoid, andA is an integral domain, it suffices to show that x(J1 ∩ J2)Q = xJ1Q ∩ xJ2Q and likewise forx(J1 ∩ J2)A, and notice that x(J1 ∩ J2) = xJ1 ∩ xJ2. We may thus assume that J1 and J2 areideals of P , in which case the assertion is lemma 3.1.37. ♦

(ii): Suppose that I is reflexive; from (i) we deduce that ((IA)−1)−1 = IA. The assertionis an immediate consequence, once one remarks that, for any fractional ideal J ⊂ A, there isa natural isomorphism of A-modules : J−1 ∼→ J∨ := HomA(J,A). Indeed, the isomorphismassigns to any x ∈ J−1 the map µx : J → A : a 7→ xa for every a ∈ J (details left to thereader).

(iii): We may assume that JA = J ′A, and we prove that J = J ′, and by replacing J ′ byJ ∪ J ′, we may assume that J ⊂ J ′. Then the contention follows easily from lemma 3.1.36.

(iv): This is immediate from (i) and (iii).

Remark 3.4.29. In the situation of lemma 3.4.27(iv), obviously Div(ϕ) restricts to a morphismof submonoids :

Div+(ϕ) : Div+(P )→ Div+(Q).


3.4.30. Next, suppose that P is fine and saturated, and let I ⊂ P gp be any fractional ideal.Then theorem 3.4.16(i) easily implies that :

I−1 =⋂

ht p=1

(Ip)−1

where the intersection – running over the prime ideals of P of height one – is taken withinHomP (I, P gp), which naturally contains all the (Ip)

−1. The structure of the fractional idealsof Pp when ht p = 1 is very simple : quite generally, theorem 3.4.16(ii) easily implies that ifdimP = 1, then all fractional ideals are cyclic, and then clearly they are reflexive. On the otherhand, I is finitely generated, by lemma 3.4.22(iv). We deduce that I is reflexive if and only if :

(3.4.31) I =⋂

ht p=1

Ip.

Indeed, suppose that (3.4.31) holds; then we have I∗ =⋂

ht p=1(I−1p )−1 =

⋂ht p=1 Ip, since we

have just seen that Ip is a reflexive fractional ideal of Pp, for every prime ideal p of height one.

Proposition 3.4.32. Let P be a fine and saturated monoid, and denote by D ⊂ SpecP thesubset of all prime ideals of height one. Then the mapping :

(3.4.33) Z⊕D → Div(P ) :∑

ht p=1

np[p] 7→⋂

ht p=1

mnp

Pp

is an isomorphism of abelian groups.

Proof. Here mPp ⊂ Pp is the maximal ideal, and for n ≥ 0, the notation mnPp

means the usualn-th power operation in the monoid P(P gp), which we extend to all integers n, by lettingmnPp

:= m−nPpwhenever n < 0.

In order to show that (3.4.33) is well defined, set I :=⋂

ht p=1 mnpp . Pick, for every p such

that np < 0, an element xp ∈ p, and set yp := x−npp ; if np ≥ 0, set yp := 1. Then it is easy to

check (using theorem 3.4.16(i)) that∏

ht p=1 yp lies in I−1, hence I is a fractional ideal. Next,for given p, p′ ∈ D, notice that (Pp)p′ = P gp; it follows that

(3.4.34) Ip = mnpp for every p ∈ D

therefore I is reflexive. Furthermore, it is easily seen (from theorem 3.4.16(ii)), that everyreflexive ideal of Pp is of the form mn

P for some integer n, and moreover mnP = mm

P if andonly if n = m. Then (3.4.31) implies that the mapping (3.4.33) is surjective, and the injectivityfollows from (3.4.34). It remains to check that (3.4.33) is a group homomorphism, and to thisaim we may assume – in view of lemma 3.4.27(iv) – that dimP = 1, in which case the assertionis immediate.

3.4.35. A morphism ϕ : I → J of fractional ideals of P is, by definition, a morphism of P -modules. Let x, y ∈ I be any two elements; we may find a, b ∈ P such that ax = by in I , andtherefore aϕ(x) = ϕ(ax) = ϕ(by) = bϕ(y); thus, ϕ(y) = (b−1a) · ϕ(x) = (x−1y) · ϕ(x). Thisshows that, for every morphism ϕ : I → J of fractional ideals, there exists c ∈ P gp such thatϕ(x) = cx for every x ∈ I . Especially, I ' J if and only if there exists a ∈ P gp such thatI = aJ . Likewise one may characterize the morphisms and isomorphisms of fractional idealsof an integral domain. We denote

Div(P )

the set of isomorphism classes of reflexive fractional ideals of P . From the foregoing, it is clearthat, if I ' I ′, we have I J ' I ′ J for every J ∈ Div(P ); therefore the compositionlaw of Div(P ) descends to a composition law for Div(P ), which makes it into a (commutative)


monoid, and if P is completely saturated, then Div(P ) is an abelian group. We also deduce anexact sequence of monoids

(3.4.36) 1→ P× → P gp jP−−→ Div(P )→ Div(P )→ 1

where jP is given by the ruleA: a 7→ aP for every a ∈ P ; especially, jP restricts to a morphismof monoids

j+P : P → Div+(P ).

Likewise, we define Div(A), for any integral domain A : see example 3.4.21. Moreover, in thesituation of (3.4.26), we see from lemma 3.4.27(iv) that, if α is local, P is fine, A is α-flat, andϕ is flat, then Div(ϕ) and Div(α) descend to well defined morphisms of monoids

Div(ϕ) : Div(P )→ Div(Q) Div(α) : Div(P )→ Div(A).

Proposition 3.4.37. Let P be a fine and saturated monoid, I, J ⊂ P gp two fractional ideals, Aa local integral domain, and α : P → (A, ·) a local morphism of monoids. We have :

(i) (I : I) = P .(ii) Suppose that A is α-flat. Then IA ' JA if and only if I ' J . Especially, in this case

Div(α) is an injective map.

Proof. (i): Clearly it suffices to show that (I : I) ⊂ P . Hence, say that x ∈ (I : I), andpick any a ∈ I; it follows that xna ∈ P for every n > 0; in the additive group logP gp wehave therefore the identity n · log(x) + log(a) ∈ logP , so log(x) + n−1 log(a) ∈ (logP )Rfor every n > 0. Since (logP )R is a convex polyhedral cone in (logP gp)R, we deduce thatx ∈ (logP )R ∩ (logP gp) = logP (proposition 3.3.22(iii)), as claimed.

(ii): We may assume that IA is isomorphic to JA, and we show that I is isomorphic to J .Indeed, the assumption means that a(IA) = JA for some x ∈ Frac(A); therefore, a ∈ (JA :IA) and a−1 ∈ (IA : JA), so

A = (IA : JA) · (JA : IA) = ((I : J) · (J : I))A

by virtue of lemma 3.4.27(i). Since A is local, it follows that there exist a ∈ (I : J) andb ∈ (J : I) such that α(ab) ∈ A×, whence ab ∈ P×, since α is local. It follows easily thatI = aJ , as asserted.

Example 3.4.38. (i) Let P be a fine and saturated monoid, and D ⊂ SpecP the subset of allprime ideals of height one; for every p ∈ D, denote

vp : P → P ]p∼→ N

the composition of the localization map, and the natural isomorphism resulting from theorem3.4.16(ii). A simple inspection shows that the isomorphism (3.4.33) identifies the map jP of(3.4.36) with the morphism of monoids

vP : P gp → Z⊕D x 7→ (vgpp (x) | p ∈ D).

With this notation, the isomorphism (3.4.33) is the map given by the rule :

k• 7→ v−1P (k• + N⊕D) for every k• ∈ Z⊕D.

(ii) Suppose now that P is sharp and dimP = 2, in which caseD = p1, p2 contains exactlytwo elements. According to example 3.4.17(ii), we may find a basis f1, f2 of P gp, such that thetwo facets P \ p1 and P \ p2 of P are generated respectively by e1 := f1 and e2 := af1 + bf2,for some a, b ∈ N, with a < b and (a, b) = 1. It follows easily that P is a submonoid of the freemonoid

Q := Ne′1 ⊕ Ne′2 where e′1 := b−1e1 and e′2 := b−1e2


and Qgp/P gp ' Z/bZ (details left to the reader). The induced map SpecQ → SpecP is ahomeomorphism; especially Q admits two prime ideals q1, q2 of height one, so that qi∩P = pifor i = 1, 2, whence – by proposition 3.4.32 – a natural isomorphism

s∗ : Div(Q)∼→ Div(P )

and notice that jQ : Qgp → Div(Q) is the isomorphism given by the rule : e′i 7→ qi for i = 1, 2.Moreover, we have commutative diagrams of monoids :

Ps //

vpi

Q

vqi

Nti // N

(i = 1, 2).

Clearly, Q \ qi is the facet generated by e′i, so vqi is none else than the projection onto thedirect factor Ne′3−i, for i = 1, 2. In order to compute vpi , it then suffices to determine ti, orequivalently tgp

i . However, set τi := vgpqi sgp; clearly τ1(f2) = τ1(e′2 − ae′1) = 1, so τ1 is

surjective. Also, τ2(f1) = b and τ2(f2) = −a, so τ2 is surjective as well; therefore both t1 andt2 are the identity endomorphism of N. Summing up, we find that

jP = s∗ jQ sgp

and the morphism vP is naturally identified with sgp : P gp → Qgp. Especially, we have obtaineda natural isomorphism

Div(P )∼→ Z/bZ.

We may then rephrase in more intrinsic terms the classification of example 3.4.17(iii) : namelythe isomorphism class of P is completely determined by the datum of Div(P ) and the equiva-lence class of the height one prime ideals of P in the quotient set Div(P )† defined as in loc.cit.

(iii) In the situation of (ii), a simple inspection yields the following explicit description of allreflexive fractional ideals of P . Recall that such ideals are of the form

Ik1,k2 := mk11 ∩mk2

2 = x ∈ P gp | vp1(x) ≥ k1, vp2(x) ≥ k2where mi is the maximal ideal of Ppi , and ki ∈ Z, for i = 1, 2. Then

Ik1,k2 = x1e1 + x2e2 | x1, x2 ∈ b−1Z, x1 ≥ b−1k2, x2 ≥ b−1k1 ∩ P gp for all k1, k2 ∈ Z.With this notation, the cyclic reflexive ideals are then those of the form

(x1f1 + x2f2)P = Ix2b,x1b−x2a with x1, x2 ∈ Z.

Especially, we see that the classes of p1 = I0,1 and p2 = I1,0 are both of order b in Div(P ).

The following estimate, special to the two-dimensional case, will be applied – in section 9.6– to the proof of the almost purity theorem for towers of regular log schemes.

Lemma 3.4.39. Let P be a fine and saturated monoid of dimension 2, and denote by b the orderof the finite cyclic group Div(P ). We have :

m[b/2]P ⊂ I · I−1 for every I ∈ Div(P )

(where [b/2] denotes the largest integer ≤ b/2).

Proof. Notice first that the assertion holds for a given I ∈ Div(P ), if and only if it holds forxI , for any x ∈ P gp. If b = 1, then P = N⊕2, in which case Div(P ) = 0, so every reflexivefractional ideal of P is isomorphic to P , and the assertion is clear. Hence, assume that b > 1;let p1, p2 be the two prime ideals of height one of P , and define Q, e1, e2 and Ik1,k2 for everyk1, k2 ∈ Z, as in example 3.4.38(ii,iii). With this notation, notice that

mP \m2P = e1, e2 ∪ Σ where Σ ⊂ x1e1 + x2e2 | x1, x2 ∈ b−1Z, 0 ≤ x1, x2 < 1.


It follows easily that, for every i ∈ N, every element of miP is of the form x1e1 + x2e2 with

x1, x2 ∈ b−1N and max(x1, x2) ≥ b−1i. Hence, let I ∈ Div(P ) and x := x1e1 + x2e2 ∈ m[b/2]P ,

and say that bx1 ≥ [b/2]. According to example 3.4.38(iii), we may assume that I = pj2 = Ij,0for some j ∈ 0, . . . , b − 1. Moreover, notice that the assertion holds for I if and only if itholds for I−1, whose class in Div(P ) agrees with the class of pb−j2 . Clearly, either j ≤ [b/2] orb− j ≤ [b/2]; hence, we may assume that j ∈ 0, . . . , [b/2]. Thus, P ⊂ I−1, and I ⊂ I · I−1,and clearly x ∈ I , so we are done in this case. The case where bx2 ≥ [b/2] is dealt with inthe same way, by writing I = pj1 for some non-negative j ≤ [b/2] : the details are left to thereader.

If f : P → Q is a general morphism of integral monoids, and I a fractional ideal of Q, theP -module f gp−1(I) is not necessarily a fractional ideal of P (for instance, consider the naturalmap P → P gp). One may obtain some positive results, by restricting to the class of morphismsintroduced by the following :

Definition 3.4.40. Let f : P → Q be a morphism of monoids. We say that f is of Kummertype, if f is injective, and the induced map fQ : PQ → QQ is surjective (notation of (3.3.20)).

Lemma 3.4.41. Let f : P → Q be a morphism of monoids of Kummer type, SQ ⊂ Q asubmonoid, and set SP := f−1SQ. We have :

(i) The map Spec f : SpecQ→ SpecP is bijective; especially dimP = dimQ.(ii) If Q× is a torsion-free abelian group, P is the trivial monoid (resp. is sharp) if and

only if the same holds for Q.(iii) The induced morphism S−1

P P → S−1Q Q is of Kummer type.

(iv) If P is integral, the unit of adjunction P → P sat is of Kummer type.(v) Suppose that P is integral and saturated. Then f ] : P ] → Q] is of Kummer type.

(vi) If both P and Q are integral, and P is saturated, then f is exact.

Proof. (ii) and (iv) are trivial, and (iii) is an exercise for the reader.(i): Let F, F ′ ⊂ Q be two faces such that f−1F = f−1F ′, and say that x ∈ F . Then

xn ∈ f(P ) for some n > 0, so xn ∈ f(f−1F ′), whence x ∈ F ′, which implies that Spec f isinjective. Next, for a given face F of P , let F ′ ⊂ Q be the subset of all x ∈ Q such that thereexists n > 0 with xn ∈ f(F ). It is easily seen that F ′ is a face of Q, and moreover f−1F ′ = F ,which shows that Spec f is also surjective.

(v): Clearly the map (P ])Q → (Q])Q is surjective. Now, let x, y ∈ P such that the imagesof f(x) and f(y) agree in Q], i.e. there exists u ∈ Q× with u · f(x) = f(y); we may findn > 0 such that un, u−n ∈ f(P ). Say that un = f(v), u−n = f(w); since f(vw) = 1, we havevw = 1, and moreover f(vxn) = f(yn), so vxn = yn. Therefore xny−n, x−nyn ∈ P , and sinceP is saturated we deduce that xy−1, x−1y ∈ P , so the images of x and y agree in P ].

(vi): Notice first that f gp is injective, since the same holds for f . Suppose x ∈ P gp andf gp(x) ∈ Q; we may then find an integer k > 0 and y ∈ P such that f(y) = f(x)k. Since P issaturated, it follows that x ∈ P , so f is exact.

3.4.42. Suppose that ϕ : P → Q is a morphism of integral monoids of Kummer type, with Psaturated, and let I ⊂ Qgp be a fractional ideal. Then ϕ∗I := ϕgp−1(I) is a fractional ideal ofP . Indeed, by assumption there exists a ∈ Q such that aI ⊂ Q; we may find k > 0 and b ∈ Psuch that ak = ϕ(b), so ϕ(bx) ∈ ϕ(P )gp ∩ Q = ϕ(P ) for every x ∈ ϕ∗I , since ϕ is exact(lemma 3.4.41(v)); therefore b · ϕ∗(I) ⊂ P .

3.4.43. In the situation of (3.4.42), suppose that both P and Q are fine and saturated, and letgr•Q

gp be the ϕ-grading of Q, indexed by (Γ,+) := Qgp/P gp (see remark 3.2.5(iii)); for everyx ∈ Qgp, denote x ∈ Γ the image of x. Let also I ⊂ Qgp be any fractional ideal, and denote by


gr•I the Γ-grading on I deduced from the ϕ-grading of Qgp; arguing as in (3.4.42), it is easilyseen that, more generally, ϕ∗(x−1grxI) is a fractional ideal of P , for every x ∈ Q (details leftto the reader). For every prime ideal q of height one in Q, we have a commutative diagram ofmonoids :

(3.4.44)P

ϕ //

vϕ−1q

Q

vq

N

eq // Nwhere vq and vϕ−1q are defined as in example 3.4.38(i), and eq is the multiplication by a non-zero(positive) integer, which we call the ramification index of ϕ at q, and we denote also eq.

Lemma 3.4.45. In the situation of (3.4.43), suppose that I is a reflexive fractional ideal. Thenϕ∗(a−1 · graI) is a reflexive fractional ideal of P , for every a ∈ Qgp.

Proof. Clearly, we may replace I by a−1I , and reduce to the case where a = 1, in which case wehave to check that ϕ∗I is a reflexive fractional ideal. However, according to example 3.4.38(i),we may write I = v−1

Q (k• + N⊕D), where D ⊂ SpecQ is the subset of the height one primeideals, and k• ∈ Z⊕D. Set

k′• := ([e−1q kq] | q ∈ D)

(where, for a real number x, we let [x] be the smallest integer ≥ x). Since Specϕ is bijective(lemma 3.4.41(i)), the commutative diagrams (3.4.44) imply that ϕ∗I = v−1

P (k′•+N⊕D), whencethe contention.

Example 3.4.46. Let P be as in example 3.4.38(ii), set Q := Div+(P ), and take ϕ := j+P :

P → Q (notation of (3.4.35)). The discussion of loc.cit. shows that ϕ is a morphism ofKummer type, and notice that the ϕ-grading of Q is indexed by Qgp/P gp = Div(P ). Now,pick any x ∈ Q, and let x ∈ Div(P ) be the equivalence class of x; by lemma 3.4.45, the P -module grxQ is isomorphic to a reflexive fractional ideal of P . We claim that the isomorphismclass of grxQ is precisely x−1 (where the inverse is formed in the commutative group Div(P )).Indeed, let a ∈ P gp be any element; by definition, we have a ∈ ϕ∗(x−1grxQ) if and only ifϕgp(a) ∈ x−1grxQ, if and only if ax ∈ Q, if and only if a ∈ x−1, whence the claim. Thus, thefamily

(grγDiv+(P ) | γ ∈ Div(P ))

is a complete system of representatives for the isomorphism classes of the reflexive fractionalideals of a fine, sharp and saturated monoid P of dimension 2.

Remark 3.4.47. Further results on reflexive fractional ideals for monoids, and their divisorclass groups can be found in [23].

3.5. Fans. According to Kato ([53, §9]), a fan is to a monoid what a scheme is to a ring.More prosaically, the theory of fans is a reformulation of the older theory of rational polyhedraldecompositions, developed in [54].

Definition 3.5.1. (i) A monoidal space is a datum (T,OT ) consisting of a topological space Tand a sheaf of monoids OT on T .

(ii) A morphism of monoidal spaces is a datum

(f, log f) : (T,OT )→ (S,OS)

consisting of a continuous map f : T → S, and a morphism log f : f ∗OS → OT ofsheaves of monoids that is local, i.e. whose stalk (log f)t : OS,f(t) → OT,t is a localmorphism, for every t ∈ T . The strict locus of (f, log f) is the subset

Str(f, log f) ⊂ T


consisting of all t ∈ T such that (log f)t is an isomorphism.(iii) We say that a monoidal space (T,OT ) is sharp, (resp. integral, resp. saturated) if OT

is a sheaf of sharp (resp. integral, resp. integral and saturated) monoids.(iv) For any monoidal space (resp. integral monoidal space) (T,OT ), the sharpening (resp.

the saturation) of (T,OT ) is the sharp monoidal space (T,OT )] := (T,O]T ) (resp.

(T,OT )sat := (T,OsatT )).

It is easily seen that the rule (T,OT ) 7→ (T,OT )] extends to a functor from the category ofmonoidal spaces to the full subcategory of sharp monoidal spaces. This functor is right adjointto the corresponding fully faithful embedding of categories.

Likewise, the functor (T,OT ) 7→ (T,OT )sat is right adjoint to the fully faithful embeddingof the category of saturated monoidal spaces, into the category of integral monoidal spaces.

3.5.2. Let P be any monoid; for every f ∈ P , let us set

D(f) := p ∈ SpecP | f /∈ p.Notice that D(f) ∩ D(g) = D(fg) for every f, g ∈ P . We endow SpecP with the topologywhose basis of open subsets consists of the subsets D(f), for every f ∈ P . Notice that mP isthe only closed point of SpecP (especially, SpecP is trivially quasi-compact).

By lemma 3.1.13, the localization map jf : P → Pf induces an identification j∗f : SpecPf∼→

D(f). It is easily seen that (j∗f )−1D(fg) = D(j(g)) ⊂ SpecPf ; in other words, the topology

of SpecPf agrees with the topology induced from SpecP , via j∗.Next, for every f ∈ P we set :

OSpecP (D(f)) := Pf .

We claim that OSpecP (D(f)) depends only on the open subset D(f), up to natural isomorphism(and not on the choice of f ). More precisely, say that D(f) ⊂ D(g) for two given elementsf, g ∈ P ; it follows that the image of g in Pf lies outside the maximal ideal mPf , hence g ∈P×f , and therefore the localization map jf : P → Pf factors uniquely through a morphism ofmonoids :

jf,g : Pg → Pf .

Likewise, if D(g) ⊂ D(f) as well, the localization jg : P → Pg factors through a unique mapjg,f : Pf → Pg, whence the identities :

jf,g jg,f jf = jf jg,f jf,g jg = jg

and since jf and jg are epimorphisms, we see that jf,g and jg,f are mutually inverse isomor-phisms.

3.5.3. Say that D(f) ⊂ D(g) ⊂ D(h) for some f, g, h ∈ P ; by direct inspection, it is clear thatjf,gjg,h = jf,h, so the ruleD(f) 7→ Pf yields a well defined presheaf of monoids on the site CP

of open subsets of SpecP of the form D(f) for some f ∈ P . Then OSpecP is trivially a sheafon CP (notice that if D(f) =

⋃i∈I D(gi) is an open covering of D(f), then D(gi) = D(f)

for some i ∈ I). According to [26, Ch.0, §3.2.2] it follows that OSpecP extends uniquely toa well defined sheaf of monoids on SpecP , whence a monoidal space (SpecP,OSpecP ). Byinspecting the construction, we find natural identifications :

(3.5.4) (OSpecP )p∼→ Pp for every p ∈ SpecP

and moreover :P∼→ Γ(SpecP,OSpecP ).

It is also clear that the rule

(3.5.5) P 7→ (SpecP,OSpecP )


defines a functor from the category Mndo to the category of monoidal spaces.

Proposition 3.5.6. The functor (3.5.5) is right adjoint to the functor :

(T,OT ) 7→ Γ(T,OT )

from the category of monoidal spaces, to the category Mndo.

Proof. Let f : P → Γ(T,OT ) be a map of monoids. We define a morphism

ϕf := (ϕf , logϕf ) : (T,OT )→ (SpecP,OSpecP )

as follows. Given t ∈ T , let ft : P → OT,t be the morphism deduced from f , and denoteby mt ⊂ OT,t the maximal ideal. We set ϕf (t) := f−1

t (mt). In order to show that ϕf iscontinuous, it suffices to prove that Us := ϕ−1

f (D(s)) is open inMT , for every s ∈ P . However,Us = t ∈ T | ft(s) ∈ O×T,t, and it is easily seen that this condition defines an open subset(details left to the reader). Next, we define logϕf on the basic open subsets D(s). Indeed, letjs : Γ(T,OT )→ OT (Us) be the natural map; by construction, js f(s) is invertible in OT (Us),hence js f extends to a unique map of monoids :

Ps = OSpecP (D(s))→ ϕf∗OT (D(s)).

By [26, Ch.0, §3.2.5], the above rule extends to a unique morphism OSpecP → ϕf∗OT of sheavesof monoids, whence – by adjunction – a well defined morphism logϕf : ϕ∗fOSpecP → OT . Inorder to show that (ϕf , logϕf ) is the sought morphism of monoidal spaces, it remains to checkthat (logϕf )t : Pϕf (t) → OT,t is a local morphism, for every t ∈ T . However, let it : P → Pϕf (t)

be the localization map; by construction, we have (logϕf )t it = ft, and the contention is astraightforward consequence.

Conversely, say that (ϕ, logϕ) : (T,OT ) → (SpecP,OSpecP ) is a morphism of monoidalspaces; then logϕ corresponds to a unique morphism ψ : OSpecP → ϕ∗OT , and we set

fϕ := Γ(SpecP, ψ) : P → Γ(T,OT ).

By inspecting the definitions, it is easily seen that fϕf = f for every morphism of monoids f asabove. To conclude, it remains only to show that the rule (ϕ, logϕ) 7→ fϕ is injective. However,for a given morphism of monoidal spaces (ϕ, logϕ) as above, and every point t ∈ T , we have acommutative diagram of monoids :

Pfϕ //

Γ(T,OT )

jt

Pϕ(t)logϕt // OT,t

.

Since logϕt is local, it follows that ϕ(t) = (fϕ jt)−1mt, especially fϕ determines ϕ : T →SpecP . Finally, since the map P → Pϕ(t) is an epimorphism, we see that logϕt is determinedby fϕ as well, and the proposition follows.

Definition 3.5.7. Let (T,OT ) be a sharp monoidal space.(i) We say that (T,OT ) is an affine fan, if there exists a monoid P and an isomorphism of

sharp monoidal spaces (SpecP,OSpecP )]∼→ (T,OT ).

(ii) In the situation of (i), if P can be chosen to be finitely generated (resp. fine), we saythat (T,OT ) is a finite (resp. fine) affine fan.

(iii) We say that (T,OT ) is a fan, if there exists an open covering T =⋃i∈I Ui, such that the

induced sharp monoidal space (Ui,OT |Ui) is an affine fan, for every i ∈ I . We denoteby Fan the full subcategory of the category of monoidal spaces, whose objects are thefans.


(iv) In the situation of (iii), if the covering (Ui | i ∈ I) can be chosen, so that (Ui,OT |Ui) isa finite (resp. fine) affine fan for every i ∈ I , we say that (T,OT ) is locally finite (resp.locally fine).

(v) We say that the fan (T,OT ) is finite (resp. fine) if it is locally finite (resp. locally fine)and quasi-compact.

(vi) Let (T,OT ) be a fan. The simplicial locus Tsim ⊂ T is the subset of all t ∈ T such thatOT,t is a free monoid of finite rank.

Remark 3.5.8. (i) For every monoid P , let TP denote the affine fan (SpecP )]. In light ofproposition 3.5.6, it is easily seen that the functor P 7→ TP is an equivalence from the oppositeof the full subcategory of sharp monoids, to the category of affine fans.

(ii) Since the saturation functor commutes with localizations (lemma 3.2.9(i)), it is easilyseen that the saturation of a fan is a fan, and more precisely, the saturation of an affine fan TP ,is naturally isomorphic to TP sat .

(iii) Let Q1 and Q2 be two monoids; since the product P ×Q is also the coproduct of P andQ in the category Mnd (see example 2.3.30(i)), we have a natural isomorphism in the categoryof fans :

TP×Q∼→ TP × TQ.

More generally, suppose that P → Qi, for i = 1, 2, are two morphisms of monoids. Then wehave a natural isomorphism of fans :

TQ1⊗PQ2

∼→ TQ1 ×TP TQ2 .

From this, a standard argument shows that fibre products are representable in the category offans.

(iv) Furthermore, lemma 3.1.16(ii) implies that the natural map :

π : Spec (P ×Q)→ SpecP × SpecQ

is a homeomorphism (where the product of SpecP and SpecQ is taken in the category oftopological spaces and continuous maps).

(v) Moreover, we have natural isomorphisms of monoids :

OTP×Q,π−1(s,t)∼→ OTP ,s × OTQ,t for every s ∈ SpecP and t ∈ SpecQ.

(vi) For any fan T := (T,OT ), and any monoid M , we shall use the standard notation :

T (M) := HomFan((SpecM)], T ).

Especially, if T is an affine fan, say T = (SpecP )], then T (M) = HomMnd(P,M ]); forinstance, if T is affine, T (N) is a monoid, and T (Q+)gp is a Q-vector space. Furthermore, bystandard general nonsense we have natural identifications of sets :

(T1 ×T T2)(M)∼→ T1(M)×T (M) T2(M)

for any pair of T -fans T1 and T2, and every monoid M . If T , T1 and T2 are affine, this identifi-cation is also an isomorphism of monoids.

Example 3.5.9. (i) The topological space underlying the affine fan (SpecN,OSpecN)] consistsof two points : SpecN = ∅,m, where m := N\0 is the closed point. The structure sheafO := OSpecN is determined as follows. The two stalks are O∅ = 1 (the trivial monoid) andOm = N; the global sections are Γ(SpecN,O) = N.

(ii) Let (T,OT ) be any fan, P any monoid, with maximal ideal mP , and ϕ : TP :=(SpecP,OSpecP )] → (T,OT ) a morphism of fans. Say that ϕ(mP ) ∈ U for some affineopen subset U ⊂ F ; then ϕ(SpecP ) ⊂ U , hence ϕ factors through a morphism of fansTP → (U,OT |U). In view of proposition 3.5.6, such a morphism corresponds to a unique


morphism of monoids ϕ\ : OT (U) → P ], and then ϕ(mP ) = ϕ\−1(mP ) ∈ Spec OT (U) = U .The map on stalks determined by ϕ is the local morphism

OT,ϕ(mP )∼→ OT (U)ϕ(mP )/OT (U)×ϕ(mP ) → P ]

obtained from ϕ\ after localization at the prime ideal ϕ\−1(mP ).(iii) For any two monoids M and N , denote by loc.HomMnd(M,N) the set of local mor-

phisms of monoids M → N . The discussion in (ii) leads to a natural identification :

T (P )∼→∐t∈T

loc.HomMnd(OT,t, P])

For any monoid P . The support of a P -point ϕ ∈ T (P ) is the unique point t ∈ T such that ϕcorresponds to a local morphism OT,t → P ].

Example 3.5.10. (i) Let P be any monoid, k > 0 any integer, set TP := (SpecP )], and letkP : P → P be the k-Frobenius map of P (definition 2.3.40(ii)). Then

kTP := SpeckP : TP → TP

is a well defined endomorphism inducing the identity on the underlying topological space.(ii) More generally, let F be any fan; for every integer k > 0 we have the k-Frobenius

endomorphismkF : F → F

which induces the identity on the underlying topological space, and whose restriction to anyaffine open subfan U ⊂ F is the endomorphism kU defined as in (i).

3.5.11. Let P be a monoid, M a P -module, and set TP := (SpecP )]. We define a presheafM∼ on the site of basic affine open subsets D(f) ⊂ SpecP (for all f ∈ P ), by the rule :

U 7→M∼(U) := M ⊗P OTP (U)

(and for an inclusion U ′ ⊂ U of basic open subsets, the corresponding morphism M∼(U) →M∼(U ′) is deduced from the restriction map OTP (U) → OTP (U ′)). It is easily seen thatM∼ is a sheaf, hence it extends to a well defined sheaf of OTP -modules on TP ([26, Ch.0,§3.2.5]). Clearly Γ(TP ,M

∼) = M , and the rule M 7→ M∼ yields a well defined functorP -Mod → OTP -Mod, which is left adjoint to the global section functor on OTP -modules :M 7→ Γ(TP ,M ) (verification left to the reader).

Definition 3.5.12. Let (T,OT ) be a fan, M a OT -module. We say that M is quasi-coherent,if there exists an open covering T =

⋃i∈I Ui of T by affine open subsets, and for each i ∈ I a

OT (Ui)-module Mi with an isomorphism of OT |Ui-modules M|Ui∼→M∼

i .

Remark 3.5.13. (i) Let (T,OT ) be a fan, M a quasi-coherent OT -module, and U ⊂ T be anyopen subset, such that (U,OT |U) is an affine fan (briefly : an affine open subfan of T ). Then,since U admits a unique closed point t ∈ U , it is easily seen that M|U is naturally isomorphicto M∼

t as a OT |U -module.(ii) In the same vein, if M is an invertible OT -module (see definition 2.3.6(iv)), then the

restriction M|U of M to any affine open subset, is isomorphic to OT |U .(iii) For any fan (T,OT ), the sheaf of abelian groups Ogp

T is quasi-coherent (exercise for thereader). Suppose that T is integral; then an OT -submodule I ⊂ Ogp

T is called a fractional ideal(resp. a reflexive fractional ideal) of OT if I is quasi-coherent, and I (U) is a fractional ideal(resp. a reflexive fractional ideal) of OT (U), for every affine open subset U ⊂ T .

(iv) Let P be any integral monoid, I ⊂ P gp a fractional ideal of P , and set TP := (SpecP )].It follows easily from lemma 3.4.22(i) that I∼ ⊂ Ogp

TPis a fractional ideal of OTP , and I∼ is

reflexive if and only if I is a reflexive fractional ideal of P (lemma 3.4.22(ii)).


(v) Suppose that T is locally finite; in this case, it follows easily from proposition 3.1.9(ii)that every quasi-coherent ideal of OT is coherent. Likewise, if T is also integral, and I ⊂ Ogp

T

is a (quasi-coherent) fractional ideal of OT , then I is coherent, provided the stalks It arefinitely generated OT,t-modules, for every t ∈ T . (Details left to the reader.)

Remark 3.5.14. (i) Let T be any integral fan. We define a sheaf DivT on T , by letting DivT (U)be the set of all reflexive fraction ideals of OU , for every open subset U ⊂ T .

(ii) Now, suppose that T is locally fine; in this case, we can endow DivT with a naturalstructure of T -monoid, as follows. First, we define a presheaf of monoids on the site CT ofaffine open subsets of T by the rule :

U 7→ DivT (U) := (Div(OT (U)),)

(notation of (3.4.20)) and for an inclusion U ′ ⊂ U of affine open subset, the correspondingmorphism of monoids DivT (U)→ DivT (U ′) is deduced from the flat map OT (U)→ OT (U ′),by virtue of lemma 3.4.27(iv). Arguing as in (3.5.3), we see that DivT is a sheaf on CT , andthen [26, Ch.0, §3.2.2] implies that DivT extends uniquely to a sheaf of monoids on T . It isthen clear that the sheaf of sets underlying this T -monoid is (naturally isomorphic to) the sheafdefined in (i).

(iii) In the situation of (ii), we have likewise a T -submonoid Div+T ⊂ DivT (remark 3.4.29),

and we may also define a T -monoid DivT (see (3.4.35)). Moreover, we have the global versionof (3.4.36) : namely, the sequence of T -monoids

1→ OgpT

jT−−→ DivT → DivT → 1

is exact (recall that O×T = 1T , the initial T -monoid), and jT restricts to a map of T -monoids

OT → Div+T .

Indeed, the assertion can be checked on the stalks over each t ∈ T , where it reduces to the exactsequence (3.4.36) for P := OT,t. Lastly, we remark that, if T is locally fine and saturated, thenDivT and DivT are abelian T -groups (proposition 3.4.25(i,ii)).

3.5.15. If f : T ′ → T is a morphism of fans, and M is any OT -module, then we define as usualthe OT ′-module :

f ∗M := f−1M ⊗f−1OT OT ′

where f−1M denotes the usual sheaf-theoretic inverse image of M (so f−1OT means herewhat was denoted f ∗OT in definition 3.5.1(ii)). The rule M 7→ f ∗M yields a left adjoint to thefunctor

OT ′-Mod→ OT -Mod N 7→ f∗N

(verification left to the reader). Notice that, if M is quasi-coherent, then f ∗M is a quasi-coherent OT ′-module. Indeed, the assertion is local on T ′, hence we are reduced to the casewhere T ′ = (SpecP ′) and T = (SpecP ) for some monoids P and P ′. In this case, the functorM 7→ f ∗(M∼) : P -Mod → OT ′-Mod is left adjoint to the functor M 7→ Γ(T ′,M ) on OT ′-modules. The latter functor also admits the left adjoint given by the rule : M 7→ (M ⊗P P ′)∼,whence a natural isomorphism of OT ′-modules :

f ∗(M∼)∼→ (M ⊗P P ′)∼.

3.5.16. Let T := (T,OT ) be a fan, t ∈ T any point. The height of t is :

htT (t) := dim OT,t ∈ N ∪ +∞(see definition 3.1.18) and the dimension of T is dimT := sup(htT (t) | t ∈ T ). (If T = ∅ isthe empty fan, we let dimT := −∞.)


Suppose that T is locally finite; then it follows from (3.5.4) and lemma 3.1.20(iii),(iv) thatthe height of any point of T is an integer. Moreover, let U(t) ⊂ T denote the subset of allpoints x ∈ T which specialize to t (i.e. such that the topological closure of x in T containst); clearly U(t) is the intersection of all the open neighborhoods of t in T , and we have a naturalhomeomorphism :

(3.5.17) Spec OT,t∼→ U(t).

Especially, if T is locally finite, U(t) is a finite set, and moreover U(t) is an open subset :indeed, if U ⊂ T is any finite affine open neighborhood of t, we have U(t) ⊂ U , hence U(t)can be realized as the intersection of the finitely many open neighborhoods of t in U . In thiscase, (3.5.17) induces an isomorphism of fans :

(3.5.18) (Spec OT,t)] ∼→ (U(t),OT |U(t)).

Therefore, for every h ∈ N, let Th ⊂ T be the subset of all points of T of height ≤ h; clearlyU(t) ⊂ Th whenever t ∈ Th, hence the foregoing shows that – if T is locally finite – Th is anopen subset of T for every h ∈ N, and T =

⋃h∈N Th.

Notice also that the simplicial locus of a fan T is closed under generizations. Therefore, Tsim

is an open subset of T , whenever T is locally finite.

3.5.19. In the situation of remark 3.5.8(v), suppose additionally that P and Q are finitely gen-erated. The natural projection P ×Q→ P induces a morphism j : TP → TP×Q of affine fans,and it is easily seen that j(t) = π−1(t,∅) for every t ∈ TP . It follows that the restriction of j isa homeomorphism U(t)

∼→ U(j(t)) for every t ∈ TP , and moreover log j : j∗OTP×Q → OTP isan isomorphism. We conclude that j is an open immersion.

3.5.20. Let P be a fine, sharp and saturated monoid, and set Q := P∨ (notation of (3.4.11)).By proposition 3.4.12(iv), we have a natural identification P ∼→ Q∨. By proposition 3.4.7(ii)and corollary 3.3.12(ii), the rule

(3.5.21) F 7→ F ∗ := F ∗R ∩ Pestablishes a natural bijection from the faces of Q to those of P . For every face F of Q, setpF := P \F ∗; there follows a natural bijection F 7→ pF between the set of all faces of Q andSpecP , such that

F ⊂ F ′ ⇔ pF ⊂ pF ′ .

Moreover, set TP := (SpecP )]; we have natural identifications :

F∨∼→ OTP ,pF for every face F of Q

under which, the specialization maps OTP ,pF ′→ OTP ,pF correspond to the restriction maps

(F ′)∨ → F∨ : ϕ 7→ ϕ|F .

Definition 3.5.22. Let T := (T,OT ) be a fan.(i) An integral (resp. a rational) partial subdivision of T is a morphism f : (T ′,OT ′)→ T

of fans such that, for every t ∈ T ′, the group homomorphism

(log f)gpt : Ogp

T,f(t) → OgpT ′,t (resp. the Q-linear map (log f)gp

t ⊗Z 1Q)

is surjective.(ii) If f : T ′ → T is an integral (resp. rational) partial subdivision, and the induced map

T ′(N)→ T (N) (resp. T ′(Q+)→ T (Q+)) : ϕ 7→ f ϕis bijective, we say that f is an integral (resp. a rational) subdivision of T .

(iii) A morphism of fans f : T ′ → T is finite (resp. proper), if the fibre f−1(t) is a finite(resp. and non-empty) set, for every t ∈ T .


(iv) A subdivision T ′ → T of T is simplicial, if T ′sim = T ′.

Remark 3.5.23. (i) Let T be any integral fan. Then the counit of adjunction

T sat → T

is an integral subdivision. This morphism is also a homeomorphism on the underlying topolog-ical spaces, in light of 3.2.9(iv).

(ii) Let f : T ′ → T be any integral subdivision. Then f restricts to a bijection T ′0∼→ T0 on

the sets of points of height zero. Indeed, notice that, if t′ ∈ T ′ is a point of height zero, thenf(t′) ∈ T0 since the map (log f)gp

t′ must be local; moreover loc.HomMnd(OT,t′ ,N) consists ofprecisely one element, namely the unique map σt : N → 1, and if t′1, t

′2 ∈ T ′0 have the same

image in T0, the sections σt′1 and σt′2 have the same image in T (N), hence they must coincide,so that t′1 = t′2, as claimed.

(iii) Let f : T ′ → T be an integral subdivision of locally fine and saturated fans. In general,the image of a point t′ ∈ T ′ of height one may have height strictly greater than one. On theother hand, for any t ∈ T of height one, and any t′ ∈ f−1(t), the map Z ∼→ Ogp

T,t → OgpT ′,t′

must be surjective (theorem 3.4.16(ii)), therefore OgpT ′,t′ is a cyclic group; however OT ′,t′ is also

sharp and saturated, so it must be either the trivial monoid 1 or N. The first case is excludedby (ii), so ht(t′) = 1, and moreover (log f)t′ is an isomorphism (and there exists a unique suchisomorphism). Since the induced map T ′(N) → T (N) is bijective, it follows easily that f−1(t)

consists of exactly one point, and therefore f restricts to an isomorphism f−1(T1)∼→ T1.

Proposition 3.5.24. Let f : T ′ → T be a morphism of fans, with T ′ locally finite, and considerthe following conditions :

(a) The induced map T ′(N)→ T (N) is injective.(b) For every integral saturated monoid P , the induced map T ′(P )→ T (P ) is injective.(c) f is a partial rational subdivision.

Then we have : (a)⇔ (b)⇒ (c).

Proof. Obviously (b) ⇒ (a). Conversely, assume that (a) holds, let P be a saturated monoid,and suppose we have two sections in T ′(P ) whose images in T (P ) agree. In light of example3.5.9(iii), this means that we may find two points t′1, t

′2 ∈ T ′, such that f(t′1) = f(t′2) = t,

and two local morphisms of monoids σi : OT ′,t′1→ P/P× whose compositions with log ft′i

(i = 1, 2) yield the same morphism OT,t → P/P×, and we have to show that these mapsare equal. In view of lemma 3.2.9(ii), we may then replace P by P/P×, and assume thatP is sharp. Since the stalks of OT ′ are finitely generated, the morphisms σi factor through afinitely generated submonoid M ⊂ P . We may then replace P by its submonoid M sat, whichallows to assume additionally that P is finitely generated (corollary 3.4.1(ii)). In this case, wemay find an injective map j : P → N⊕r (corollary 3.4.10(iv); notice that j is trivially a localmorphism), hence we may replace σi by j σi (for i = 1, 2), after which we may assumethat P = N⊕r for some r ∈ N. Let δ : P → N be the local morphism given by the rule :(x1, . . . , xr) 7→ x1 + · · ·+xr for every x1, . . . , xr ∈ N; the compositions δ σi (for i = 1, 2) aretwo elements of T ′(N) whose images agree in T (N), hence they must coincide by assumption.This implies already that t′1 = t′2. Next, let πk : P → N (for k = 1, . . . , r) be the naturalprojections, and fix k ≤ r; the morphisms πk σi for i = 1, 2 are not necessarily local, but theydetermine elements of T ′(N) whose images agree again in T (N), hence they must coincide.Since k is arbitrary, we deduce that σ1 = σ2, as stated.

Next, we suppose that (b) holds, and we wish to show assertion (c); the latter is local on F ′,hence we may assume that both F and F ′ are affine, say F = (SpecQ)] and F ′ = (SpecQ′)],with Q′ finitely generated and sharp, and then we are reduced to checking that the map Qgp ⊗Z


Q→ Q′gp ⊗Z Q induced by f is surjective, or equivalently, that the dual map :

HomMnd(Q′,Q)→ HomMnd(Q,Q)

is injective. To this aim, we may further assume that Q′ is integral, in which case, by remark3.4.14(i) we have HomMnd(Q′,Q) = HomMnd(Q′,Q+)gp; the contention is an easy conse-quence.

3.5.25. In light of proposition 3.5.24, we may ask whether the surjectivity of the map on P -points induced by a morphism f of fans can be similarly characterized. This turns out to bethe case, but more assumptions must be made on the morphism f , and also some additionalrestrictions must be imposed on the type of monoid P . Namely, we shall consider monoids ofthe form Γ+, where (Γ,≤) is any totally ordered abelian group, and Γ+ ⊂ Γ is the subgroup ofall elements ≤ 1 (where 1 ∈ Γ denotes the neutral element). With this notation, we have thefollowing :

Proposition 3.5.26. Let f : T ′ → T be a finite partial integral subdivision, with T locally finite.The following conditions are equivalent :

(a) The induced map T ′(N)→ T (N) is surjective.(b) For every totally ordered abelian group (Γ,≥), the induced map T ′(Γ+) → T (Γ+) is

surjective.

Proof. Obviously, we need only to show that (a) ⇒ (b). Thus suppose, by way of contradic-tion, that (a) holds, but nevertheless there exists a totally ordered abelian group (Γ,≤), and anelement of T (Γ+) which is not in the image of T ′(Γ+). Such element corresponds to a localmorphism of monoids ϕ : OT,t → Γ+, for some t ∈ T , and the assumption means that ϕ doesnot factor through the monoid OT ′,s, for any s ∈ f−1(t). Set

P := O intT,t Qs := P gp ×Ogp

T ′,sO intT ′,s for every s ∈ f−1(t)

Notice that, since the map P gp → OgpT ′,s is surjective, we have

Qs/Q×s ' O int

T ′,s/(OintT ′,s)

×

and there is a natural injective morphism of monoids gs : P → Qs, determined by the pair(i, (log f)int

s ), where i : P int → P gp is the natural morphism; moreover, P gp = Qgps for every

s ∈ f−1(t). Clearly ϕ factors through a morphism ϕ : P → Γ+; since the unit of adjunctionOT,t → P is surjective, it follows that P is sharp and ϕ is local. Moreover, the group homomor-phism ϕgp : P gp → Γ factors uniquely through each Qgp

s . Our assumption then states that wemay find, for each s ∈ f−1(t), an element xs ∈ Qs whose image in Γ lies in the complement ofΓ+, i.e. the image of x−1

s lies in the maximal ideal m ⊂ Γ+. Let P ′ ⊂ P gp be the submonoidgenerated by P and by (x−1

s | s ∈ f−1(t)). By construction, P ′ is finitely generated, and themorphism ϕ extends uniquely to a morphism P ′ → Γ+, which maps each x−1

s into m. It followsthat all the x−1

s lie in the maximal ideal of P ′. Let us now pick any local morphism ψ′ : P ′ → N(corollary 3.4.10(iii)); by restriction, ψ′ induces a local morphism ψ : P → N, which – accord-ing to (a) – must factor through a local morphism ψs : Qs → N, for at least one s ∈ f−1(t).However, on the one hand we have ψ′gp = ψgp = ψgp

s ; on the other hand ψ′(x−1s ) 6= 0, hence

ψs(xs) = ψ′gp(xs) /∈ N, a contradiction.

Example 3.5.27. Let T be a locally fine fan, ϕ : F → T an integral subdivision, with F locallyfine and saturated, and k > 0 an integer. Suppose we have a commutative diagram of fans :

(3.5.28)F

ϕ //

g

T

kT

Fϕ // T.


where kT is the k-Frobenius endomorphism (example 3.5.10(ii)). Then we claim that necessar-ily g = kF . Indeed, suppose that this fails; then we may find a point t ∈ F such that the compo-sition of g and the open immersion jt : U(t)→ F is not equal to jt kU(t). Set P := OF,t; theng jt 6= jt kU(t) in F (P ). However, an easy computation shows that ϕ(g jt) = ϕ(jt kU(t)),which contradicts proposition 3.5.24.

3.5.29. Let P :=∐

n∈N Pn be a N-graded monoid (see definition 2.3.8); then P0 is a submonoidof P , every Pn is a P0-module, and P+ :=

∐n>0 Pn is an ideal of P . For every a ∈ P , the

localization Pa is Z-graded in an obvious way, and we denote by P(a) ⊂ Pa the submonoid ofelements of degree 0. Notice that there is a natural identification of P0-monoids

(3.5.30) P(an)∼→ P(a) for every integer n > 0.

Set as well :D+(a) := (SpecP(a))

]

and notice that the natural map P → P(a) induces a morphism of fans πa : D+(a) → TP :=(SpecP0)]. If b ∈ P is any other element, in order to determine the fibre product D+(a) ×TPD+(b) we may assume – in light of (3.5.30) – that a, b ∈ Pn for the same integer n, in whichcase we have natural isomorphisms

P(a) ⊗P0 P(b)∼→ P(ab)

∼←− P(a)[b−1a]

(see remark 3.1.25(i)) onto the localization of P(a) obtained by inverting its element a−1b; thisis of course the same as P(b)[a

−1b]. In other words D+(a) ×TP D+(b) is naturally isomorphicto D+(ab), identified to an open subfan in both D+(a) and D+(b). We may then glue the fansD+(a) for a ranging over all the elements of P , to obtain a new fan, denoted :

ProjP

called the projective fan associated to P . By inspecting the construction, we see that the mor-phisms πa assemble to a well defined morphism of fans πP : ProjP → TP . Each elementa ∈ P yields an open immersion ja : D+(a) → ProjP , and if b ∈ P is any other element, jabfactors through an open immersion D+(ab)→ D+(a).

3.5.31. Let ϕ : P → P ′ be a morphism of N-graded monoids (so ϕPn ⊂ P ′n for every n ∈ N).Set :

G(ϕ) :=⋃a∈P

D+(ϕ(a)) ⊂ ProjP ′.

Notice that, for every a ∈ P , ϕ induces a morphism ϕ(a) : P(a) → P ′(a), whence a morphism ofaffine fans (Projϕ)a : D+(ϕ(a))→ D+(a) ⊂ ProjP . Moreover, if b ∈ P is any other element,it is easily seen that (Projϕ)a and (Projϕ)b agree on D+(ϕ(a)) ∩ D+(ϕ(b)). Therefore, themorphisms (Projϕ)a glue to a well defined morphism :

Projϕ : G(ϕ)→ ProjP.

Notice that G(ϕ) = ProjP ′, whenever ϕP generates the ideal P ′+. Moreover, we have

(3.5.32) (Projϕ)−1D+(a) = D+(ϕ(a)) for every a ∈ P .

Indeed, say that D+(b) ⊂ G(ϕ) for some b ∈ P ′, and (Projϕ)D+(b) ⊂ D+(a). In order toshow that D+(b) ⊂ D(ϕ(a)), it suffices to check that D+(bϕ(c)) ⊂ D+(ϕ(a)) for every c ∈ P .However, the assumption means that the natural map

P(c) → P ′(ϕ(c)) → P ′(bϕ(c)) → P ′(bϕ(c))/P′×(bϕ(c))

factors through the localization P(c) → P(ac). This is equivalent to saying that ϕ(c−1a) is invert-ible in P ′(bϕ(c)), in which case the localization P ′(ϕ(c)) → P ′(bϕ(c)) factors through the localization


P ′(ϕ(c)) → P ′(ϕ(ac)). The latter means that the open immersion D+(bϕ(c)) ⊂ D+(ϕ(c)) factorsthrough the open immersion D+(ϕ(ac)) ⊂ D+(ϕ(c)), as claimed.

3.5.33. In the situation of (3.5.29), set Y := ProjP to ease notation. Let M be a Z-gradedP -module; for every a ∈ P , let M(a) ⊂Ma := M ⊗P Pa be the P(a)-submodule of degree zeroelements (for the natural grading on Ma). We deduce a quasi-coherent OD+(a)-module M∼

(a)

(see definition 3.5.12). Moreover, if b ∈ P is any other element, we have a natural identification

ωa,b : M∼(a)|D+(a)∩D+(b)

∼→M∼(b)|D+(a)∩D+(b).

This can be verified as follows. First, in view of (3.5.30), we may assume that a, b ∈ Pn, forsome n ∈ N, in which case we consider the P -linear morphism :

M(a) →M(b) ⊗P(b)P(b)[a

−1b] :x

am7→ x

bm⊗ bm

amfor every x ∈Mnm.

It is easily seen that this map is actually P(a)-linear, hence it extends to a OT (D+(a) ∩D+(b))-linear morphism :

ωa,b : M(a) ⊗P(a)P(a)[b

−1a]∼→M(b) ⊗P(b)

P(b)[a−1b].

Moreover, ωa,b ωb,a is the identity map, hence ωa,b induces the sought isomorphism ωa,b.Furthermore, for any a, b, c ∈ P , set D+(a, b, c) := D+(a) ∩ D+(b) ∩ D+(c); we have theidentity :

ωa,c|D+(a,b,c) = ωb,c|D+(a,b,c) ωa,b|D+(a,b,c)

which shows that the locally defined sheaves M∼(a) glue to a well defined OY -module, which we

shall denote M∼. Especially, for every n ∈ Z, let P (n) be the Z-graded P -module such thatP (n)k := Pn+k for every k ∈ Z (with the convention that Pn := ∅ if n < 0); we set :

OY (n) := P (n)∼.

Every element a ∈ Pn induces a natural isomorphism :

OY (n)|D+(a)∼→ OD+(a) : x 7→ f−kx for every local section x.

Hence on the open subset :

Un(P ) :=⋃a∈Pn

D+(a)

the sheaf OY (n) restricts to an invertible OUn(P )-module (see definition 2.3.6(iv)). Especially,if P1 generates P+, the OY -modules OY (n) are invertible, for every n ∈ Z.

3.5.34. In the situation of (3.5.31), let M be a Z-graded P -module. Then M ′ := M ⊗P P ′ is aZ-graded P ′-module, with the grading defined by the rule :

(3.5.35) M ′n :=

⋃j+k=n

Im(Mj ⊗P0 P′k →M ′).

There follows a P(a)-linear morphism :

(3.5.36) M(a) →M ′(ϕ(a)) :

x

ak7→ x⊗ 1

ϕ(a)kfor every a ∈ P

and since both localization and tensor product commute with arbitrary colimits, it is easily seenthat (3.5.36) extends an injective P ′(ϕ(a))-linear map

M(a) ⊗P(a)P ′(ϕ(a)) →M ′

(ϕ(a))


whence a map of OD+(ϕ(a))-modules (Projϕ)∗M∼|D+(ϕ(a)) → (M ′)∼|D+(ϕ(a))

, and the systemof such maps, for a ranging over the elements of P , is compatible with all open immersionsD+(ϕ(ab)) ⊂ D+(a), whence a well defined monomorphism of OG(ϕ)-modules

(3.5.37) (Projϕ)∗M∼ → (M ′)∼|G(ϕ).

Moreover, if a ∈ P1, then for every m ∈Mj and x ∈ P ′k, we may write

m⊗ xϕ(a)j+k

=m

aj⊗ x

ϕ(a)j

so the above map is an isomorphism on D+(a). Thus, (3.5.37) restricts to an isomorphism onthe open subset

G1(ϕ) :=⋃a∈P1

D+(ϕ(a)).

Especially (3.5.37) is an isomorphism whenever P1 generates P+. Notice as well that G1(ϕ) ⊂U1(P ′) ∩G(ϕ), and actually G1(ϕ) = U1(P ′) if ϕ(P ) generates P ′+.

3.5.38. Let P be as in (3.5.29), and f : P0 → Q a given morphism of monoids. Then P ′ :=P ⊗P0 Q is naturally N-graded, so that the natural map fP : P → P ′ is a morphism of gradedmonoids. Every element of P ′ is of the form a⊗ b = (a⊗ 1) · (1⊗ b), where a ∈ P and b ∈ Q.Then lemma 2.3.34 yields a natural isomorphism of Q-monoids :

P(a) ⊗P0 Q[b−1]∼→ P ′(a⊗b)

whence an isomorphism of affine fans :

βa⊗b : D+(a⊗ b) ∼→ D+(a)×TP D(b)

such that (πa ×TP j∗b ) βa⊗b = πa⊗b, where j∗b : D(b) → TQ := (SpecQ)] is the natural openimmersion. Especially, it is easily seen that the isomorphisms βa⊗1 assemble to a well definedisomorphism of fans :

(Proj fP , πP ′) : ProjP ′∼→ ProjP ×TP TQ

such that (πP ×TP 1TQ) (Proj fP , πP ′) = πP ′ . Lastly, if g : Q → R is another morphism ofmonoids, TR := (SpecR)], and P ′′ := P ′ ⊗Q R, then we have the identity :

(3.5.39) ((Proj fP , πP ′)×TQ 1TR) (Proj gP ′ , πP ′′) = (Proj (g f)P , πP ′′).

Moreover, for every Z-graded module M , the map (Proj fP )∗M∼ → (M ⊗P0 Q)∼|G(fP ) of(3.5.37) is an isomorphism, regardless of whether or not P1 generates P+ (verification left tothe reader). Especially, we get a natural identification :

(Proj fP )∗OY (n)∼→ OY ′(n) for every n ∈ Z

where Y := ProjP and Y ′ := ProjP ′.

3.5.40. In the situation of (3.5.38), let ϕ : R→ P be a morphism of N-graded monoids. Therefollow morphisms of fans :

Projϕ : G(ϕ)→ ProjR Proj(fP ϕ) : G(fP ϕ)→ ProjR

and in view of (3.5.32), it is easily seen that :

(3.5.41) G(fP ϕ) = (Proj fP )−1(G(ϕ)).


3.5.42. Let now (T,OT ) be any fan. A N-graded OT -monoid is a N-graded T -monoid P , witha morphism OT → P of T -monoids. We say that such a OT -monoid is quasi-coherent, if it issuch, when regarded as a OT -module. To a quasi-coherent N-graded OT -monoid P , we attacha morphism of fans :

πP : Proj P → T

constructed as follows. First, for every affine open subfan U ⊂ T , the monoid P(U) is N-graded, so we have the projective fan Proj P(U), and the morphism of monoids OT (U) →P(U) induces a morphism of fans Proj P(U)→ U . Next, say that U1, U2 ⊂ T are two affineopen subsets; for any affine open subset V ⊂ U1∩U2 we have restriction maps ρV,i : P(Ui)→P(V ) inducing isomorphisms of graded OT (V )-monoids :

P(Ui)⊗OT (Ui) OT (V )∼→P(V ).

whence isomorphisms of V -fans :

Proj P(V )∼→ Proj P(Ui)⊗OT (Ui) OT (V )

Proj ρV,i−−−−−→ Proj P(Ui)×Ui V

which in turn yield natural identifications :

ϑV : Proj P(U1)×U1 V∼→ Proj P(U2)×U2 V.

If W ⊂ V is a smaller affine open subset, (3.5.39) implies that ϑV ×V 1W = ϑW , and thereforethe isomorphisms ϑV glue to a single isomorphism of U1 ∩ U2-fans :

Proj P(U1)×U1 (U1 ∩ U2)∼→ Proj P(U2)×U2 (U1 ∩ U2)

which is furthermore compatible with base change to any triple intersectionU1∩U2∩U3 of affineopen subsets (details left to the reader). In such situation, we may glue the fans Proj P(U) –with U ⊂ T ranging over all the open affine subsets – along the above isomorphisms, to obtainthe sought fan Proj P; the construction also comes with a well defined morphism to T , asrequired. Then, for every such open affine U , the induced morphism Proj P(U)→ Proj P isan open immersion; finally a direct inspection shows that, for every smaller affine open subsetV ⊂ U we have :

Un(P(U)) ∩ Proj P(V ) = Un(P(V )) for every n ∈ N

(where the intersection is taken in Proj P). Hence the union of all the open subsets Un(P(U))is an open subset Un(P) ⊂ ProjP , intersecting each Proj P(U) in its subset Un(P(U)).

3.5.43. To ease notation, set Y := Proj P , and let π : Y → T be the projection. Let Mbe a Z-graded P-module, quasi-coherent as a OT -module; for every affine open subset U ⊂T , the graded P(U)-module M (U) yields a quasi-coherent Oπ−1U -module M∼

U , and everyinclusion of affine open subset U ′ ⊂ U induces a natural isomorphism M∼

U |U ′∼→ M∼

U ′ ofOπ−1U ′-modules. Therefore the modules M∼

U glue to a well defined OY -module M∼.For every n ∈ Z, denote by M (n) the Z-graded P-module such that M (n)k := Mn+k for

every k ∈ Z (especially, with the convention that Pk := 0 whenever k < 0, we obtain in thisway the P-module P(n)). We set :

OY (n) := P(n)∼ and M∼(n) := M∼ ⊗OY OY (n).

Clearly the restriction of OY (n) to Un(P) is invertible, for every n ∈ Z.Moreover, for every n ∈ Z, the scalar multiplication P(n) ⊗OT M →M (n) determines a

well defined morphism of OY -modules :

M∼(n)→M (n)∼


and arguing as in (3.5.34) we see that the restriction of this map is an isomorphism on U1(P).Especially, we have natural morphisms of OY -modules :

OY (n)⊗OY OY (m)→ OY (n+m) for every n,m ∈ Zwhose restrictions to U1(P) are isomorphisms.

Example 3.5.44. Let T be a fan, L an invertible OT -module, and set P(L ) := Sym•OTL(see example 2.3.10). Then the morphism

πP(L ) : P(L ) := Proj P(L )→ T

is an isomorphism. Indeed, the assertion can be checked locally on every affine open subsetU ⊂ T , hence say that U = (SpecP )] for some monoid P , and L ' OT |U , in which case theP -monoid P(L )(U) is isomorphic to P × N (with its natural morphism P → P × N : x 7→(x, 0) for every x ∈ P ), and the sought isomorphism corresponds to the natural identification :

(3.5.45) P = (P × N)(1,1)

where (1, 1) ∈ P × 1 = (P × N)1. Likewise, OP(L )(n) is the OP(L )-module associated tothe graded (P × N)-module P × N(n) = L ⊗n(U) ⊗P (P × N), so (3.5.45) induces a naturalisomorphism

π∗P(L )L⊗n ∼→ OP(L )(n) for every n ∈ N.

3.5.46. In the situation of (3.5.42), let ϕ : P → P ′ be a morphism of quasi-coherent N-graded OT -monoids (defined in the obvious way). By the foregoing, for every affine opensubset U ⊂ T , we have an induced morphism Projϕ(U) : G(ϕ(U)) → ProjP(U) of U -fans,where G(ϕ(U)) ⊂ Proj P(U) is an open subset of Proj P ′. Let V ⊂ U be a smaller affineopen subset; in light of (3.5.41), we have

G(ϕ(V )) = G(ϕ(U)) ∩ Proj P(V ).

It follows that the union of all the open subsets G(ϕ(U)) is an open subset G(ϕ) such that

G(ϕ) ∩ Proj P(U) = G(ϕ(U)) for every affine open subset U ⊂ T

and the morphisms Projϕ(U) assemble to a well defined morphism

Projϕ : G(ϕ)→ Proj P.

Moreover, if M is a Z-graded quasi-coherent P-module, the morphisms (3.5.37) assemble toa well defined morphism of OG(ϕ)-modules :

(3.5.47) (Projϕ)∗M∼ → (M ′)∼|G(ϕ)

where the grading of M ′ := M ⊗P P ′ is defined as in (3.5.35). Likewise, the union of allopen subsets G1(ϕ(U)) is an open subset G1(ϕ) ⊂ U1(P) ∩ G(ϕ), such that the restrictionof (3.5.47) to G1(ϕ) is an isomorphism. Especially, set Y := Proj P and Y ′ := Proj P ′; wehave a natural morphism :

(Projϕ)∗OY (n)∼→ OY ′(n)|G(ϕ)

which is an isomorphism, if P1 generates P+ :=∐

n>0 Pn locally on T .

3.5.48. On the other hand, let f : T ′ → T be a morphism of fans. The discussion in (3.5.38)implies that f induces a natural isomorphism of T ′-fans :

(3.5.49) Proj f ∗P∼→ Proj P ×T T ′.

Moreover, set Y := Proj P , Y ′ := Proj f ∗P , and let πY : Y ′ → Y be the projection deducedfrom (3.5.49); then there follows a natural identification :

OY ′(n)∼→ π∗Y OY (n) for every n ∈ Z.


3.5.50. Keep the notation of (3.5.42), and to ease notation, set Y := Proj P . Let C be thecategory whose objects are all the pairs (ψ : X → T,L ), where ψ is a morphism of fans,and L is an invertible OX-module; the morphisms (ψ : X → T,L ) → (ψ′ : X ′ → T,L ′)

are the pairs (β, h), where β : X → X ′ is a morphism of T -fans, and h : β∗L ′ ∼→ L isan isomorphism of OX′-modules (with composition of morphisms defined in the obvious way).Consider the functor

FP : C o → Set

which assigns to any object (ψ,L ) of C , the set consisting of all morphisms of graded OX-monoids

g : ψ∗P → Sym•OXL

which are epimorphisms on the underlying OX-modules (notation of example 2.3.10). On amorphism (β, h) as in the foregoing, and an element g′ ∈ FP(ψ′,L ′), the functor acts by therule :

FP(β, h) := (Sym•OXh) β∗g′.

Lemma 3.5.51. In the situation of (3.5.50), the following holds :(i) The object (πP|U1(P) : U1(P)→ T,OY (1)|U1(P)) represents the functor FP .

(ii) If P is an integral T -monoid, the OT -monoid Psat admits a unique grading such thatthe unit of adjunction P → Psat is a N-graded morphism, and there is a naturalisomorphism of Proj P-fans :

Proj (Psat)∼→ (Proj P)sat.

Proof. (i): The proof is mutatis mutandis, the same as that of lemma 6.4.26 (with some minorsimplifications). We leave it as an exercise for the reader.

(ii): The first assertion shall be left to the reader. The second assertion is local on Proj P ,hence we may assume that T = (SpecP0), and P = P∼ for some N-graded integral P0-monoid P . Let a ∈ P sat be any element; by definition we have an ∈ P for some n > 0, and weknow that the open subsets D+(a) et D+(an) coincide in Proj(Psat); hence we come down toshowing that (P(a))

sat = (P sat)(a) for every a ∈ P , which can be left to reader.

Definition 3.5.52. Let (T,OT ) be a fan (resp. an integral fan), I ⊂ OT an ideal (resp. afractional ideal) of OT .

(i) Let f : X → T be a morphism of fans (resp. of integral fans); then f−1I is an ideal(resp. a fractional ideal) of f−1OT , and we let :

IOX := log f(f−1I ) · OXwhich is the smallest ideal (res. fractional ideal) OX containing the image of f−1I .

(ii) A blow up of the ideal I is a morphism of fans (resp. of integral fans) ϕ : T ′ → Twhich enjoys the following universal property. The ideal (resp. fractional ideal) I OT ′

is invertible, and every morphism of fans (resp. of integral fans) X → T such thatI OX is invertible, factors uniquely through ϕ.

3.5.53. Let T be a fan (resp. an integral fan), I ⊂ OT a quasi-coherent ideal (resp. fractionalideal), and consider the N-graded OT -monoid :

B(J ) :=∐n∈N

I n

where I n ⊂ OT is the ideal (resp. fractional ideal) associated to the presheaf U 7→ I (U)n

for every open subset U ⊂ T (notation of (3.1.1), with the convention that I 0 := OT ) and themultiplication law of B(I ) is defined in the obvious way.


Proposition 3.5.54. The natural projection

Proj B(I )→ T

is a blow up of the ideal I .

Proof. We shall consider the case where T is not necessarily integral, and I ⊂ OT ; the caseof a fractional ideal of an integral fan is proven in the same way. Set Y := Proj B(I ); tobegin with, let us show that IOY is invertible. The assertion is local on T , hence we mayassume that T = (SpecP )], and I = I∼ for some ideal I ⊂ P , so Y = ProjB(I), whereB(I) =

∐n∈N I

n. Let a ∈ B(I)1 = I be any element; then the restriction of IOY to D+(a) isgenerated by 1 = a/a ∈ B(I)(a), so clearly I|D+(a) ' OY ; since U1(B(I)) = Y (notation of(3.5.33)), the contention follows.

Next, let ϕ : X → T be a morphism of fans, such that IOX is an invertible ideal. It followseasily that I nOX is invertible for every n ∈ N, so the natural map of N-graded OX-monoids

ϕ∗Sym•OT (I )∼→ Sym•OX (IOX)→ B(IOX)

is an isomorphism. On the other hand, the projection Proj B(IOX) → X is an isomorphism(example (3.5.44)), whence – in view of (3.5.49) – a natural morphism of T -fans :

(3.5.55) X → Proj B(I ).

To conclude, it remains to show that (3.5.55) is the only morphism of T -fans from X toProj B(I ). The latter assertion can be checked again locally on T , so we are reduced asabove to the case where T is the spectrum of P , and I is associated to I . We may also as-sume that X = (SpecQ)], and ϕ is given by a morphism of monoids f : P → Q. Then thehypothesis means that the ideal f(I)Q is isomorphic to Q (see remark 3.5.13(ii)), hence it isgenerated by an element of the form f(a), for some a ∈ I , and the endomorphism x 7→ f(a)xof f(I)Q, is an isomorphism. In such situation, it is clear that f factors uniquely through amorphism of monoids P → B(I)(a); namely, one defines g : B(I)(a) → Q by the rule :a−kx 7→ f(a)−kf(x) (for every x ∈ Ik), which is well defined, by the foregoing observations.The morphism (Spec g)] : X → D+(a) must then agree with (3.5.55).

Example 3.5.56. (i) Let P be a monoid, I ⊂ P any finitely generated ideal, a1, . . . , an afinite system of generators of I; set T := (SpecP )], and let ϕ : T ′ → T be the blow up ofthe ideal I∼ ⊂ OT . Then T ′ admits an open covering consisting of the affine fans D+(fi).The latter are the spectra of the monoids Qi consisting of all fractions of the form a · f−ti , forevery a ∈ In; we have a · f−ti = b · f−si in Qi if and only if there exists k ∈ N such thata · f s+ki = b · f t+ki , if and only if the two fractions are equal in Pfi , in other word, Qi is thesubmonoid of Pfi generated by P and fj · f−1

i | j ≤ n, for every i = 1, . . . , n.(ii) Consider the special case where P is fine, and the ideal I ⊂ P is generated by two

elements f, g ∈ P . Let t ∈ T ′ be any point; up to swapping f and g, we may assume that tcorresponds to a prime ideal p ⊂ P [f/g], hence ϕ(t) corresponds to q := j−1p ⊂ P , wherej : P → P [f/g] is the natural map. We have the following two possibilities :

• Either f/g ∈ p, in which case let y ∈ F ′ := P [f/g] \ p be any element; writingx = y·(f/g)n for some n ≥ 0 and y ∈ P , we deduce that n = 0, so x = y ∈ F := P\q,therefore F ′ = j(F ). Notice as well that in this case f/g is not invertible in P [f/g],hence P [f/g]× = P×, whence dimP [f/g] = dimP , and , by corollary 3.4.10(i).• Or else f/g /∈ p, in which case the same argument yields F ′ = j(F )[f/g]. In this

case, f/g is invertible in P [f/g] if and only if it is invertible in the face F ′, henceht p = dimP − rkZF

′ ≥ dimP − dimF − 1, by corollary 3.4.10(i),(ii).In either event, corollary 3.4.10(i),(ii) implies the inequality :

1 ≥ ht(ϕ(t))− ht(t) ≥ 0 for every t ∈ T ′.


3.6. Special subdivisions. In this section we explain how to construct – either by geometricalor combinatorial means – useful subdivisions of given fans.

3.6.1. Let T be any locally fine and saturated fan, and t ∈ T any point. By reflexivity(proposition 3.4.12(iv)), the elements s ∈ Ogp

T,t ⊗Z Q correspond bijectively to Q-linear formsρs : U(t)(Q+)gp → Q, and s ∈ Ogp

T,t if and only if ρs restricts to a morphism of monoidsU(t)(N)→ Z. Moreover, this bijection is compatible with specialization maps : if t′ is a gener-ization of t in T , then the form U(t′)(Q+)gp → Q induced by the image of s in Ogp

T,t′ ⊗Z Q isthe restriction of ρs (see (3.5.20)).

Hence, any global section λ ∈ Γ(T,OgpT ) yields a well defined function

ρλ : T (N)→ Zwhose restriction to U(t)(N) is the restriction of a Z-linear form on U(t)(N)gp, for every t ∈ T ;conversely, any such function arises from a unique global section of Ogp

T . Likewise, we havea natural isomorphism between the Q-vector space of global sections λ of Ogp

T ⊗Z Q, and thespace of functions ρλ : T (Q+)→ Q with a corresponding linearity property.

Let now ρ : T (Q+) → Q be any function; we may attach to ρ a sheaf of fractional ideals ofOT,Q (notation of (3.3.20)), by the rule :

Iρ,Q(U) := s ∈ OT,Q(U) | ρs ≥ ρ|U for every open subset U ⊂ T

In this generality, not much can be said concerning Iρ,Q; to advance, we restrict our attentionto a special class of functions, singled out by the following :

Definition 3.6.2. Let T be a locally fine and saturated fan.(a) A roof on T is a function :

ρ : T (Q+)→ Qsuch that, for every t ∈ T , there exist k := k(t) ∈ N and Q-linear forms

λ1, . . . , λk : U(t)(Q+)gp → Qwith ρ(s) = min(λi(s) | i = 1, . . . , k) for every s ∈ U(t)(Q+).

(b) An integral roof on T is a roof ρ on T such that ρ(s) ∈ Z for every s ∈ T (N).

3.6.3. The interest of the notion of roof on a fan T , is that it encodes in a geometrical way,an integral subdivision of T , together with a coherent sheaf of fractional ideals of OT (seedefinition 2.3.6(iii)). This shall be seen in several steps. To begin with, let T and ρ be as indefinition 3.6.2(a). For any t ∈ T , pick a system λ := λ1, . . . , λk of Q-linear forms fulfillingcondition (b) of the definition; then for every i = 1, . . . , k let us set :

U(t, i)(N) := x ∈ U(t)(N) | ρ(x) = λi(x).Notice that U(t)(N) = OT,t, by proposition 3.4.12(iv). Moreover, say that λ is irredundant fort if no proper subsystem of λ fulfills condition (b) of definition 3.6.2 relative to U(t).

Lemma 3.6.4. With the notation of (3.6.3), the following holds :(i) U(t, i)(N) is a saturated fine monoid for every i ≤ k.

(ii) There is a unique system of Q-linear forms which is irredundant for t.(iii) If λ is irredundant for t, then dimU(t, i)(N) = htT (t) for every i = 1, . . . , k.

Proof. (i): We leave to the reader the verification that U(t, i) is a saturated monoid. Next, letσi ⊂ U(t)(R+)gp∨ be the convex polyhedral cone spanned by the linear forms

((λj − λi)⊗Q R | j = 1, . . . , k).

Then σi is Ogp∨T,t -rational, so σ∨i is Ogp

T,t-rational, and σ∨i ∩ OgpT,t is a fine monoid (propositions

3.3.21(i), and 3.3.22(i)), therefore the same holds for U(t, i)(N) = σ∨i ∩ OT,t (corollary 3.4.2).


(iii): Notice that htT (t) = dimU(t)(N), by proposition 3.4.12(ii) and (3.5.18). In view ofcorollary 3.4.10(i), it follows already that dimU(t, i) ≤ htT (t). Now, let λ1, . . . , λk be anirredundant system, and suppose, by contradiction, that dim U(t, i)(N) < dim U(t)(N) forsome i ≤ k. Especially, σ∨i ∩ U(t)(R+) does not span the R-vector space U(t)(R+)gp, andtherefore the dual σi +U(t)(R)∨ is not strongly convex (corollary 3.3.14). After relabeling, wemay assume that i = 1. Hence there exist aj, bj ∈ R+ and ϕ, ϕ′ ∈ U(t)(R)∨, and an identity :

k∑j=2

aj(λj − λi) + ϕ = −k∑j=2

bj(λj − λi)− ϕ′.

Moreover,∑k

j=2(aj + bj) > 0. It follows that there exist ψ ∈ U(t)(R)∨, and non-negative realnumbers (cj | j = 2, . . . k) such that

λi =k∑j=2

cjλj + ψ andk∑j=2

cj = 1.

On the other hand, the irredundancy condition means that there exists x ∈ U(t, 1)(N) such thatλj(x) > λ1(x) for every j > 1. Since ψ(x) ≥ 0, we get a contradiction.

(ii): The assertion is clear, if htT (t) ≤ 1. Hence suppose that the height of t is ≥ 2, and letλ := λ1, . . . , λk and µ := µ1, . . . , µr be two irredundant systems for t. Fix i ≤ k, and pickx ∈ U(t, i)(N) which does not lie on any proper face of U(t, i)(N) (the existence of x is ensuredby (iii) and proposition 3.4.7(i)); say that µ1(x) = ρ(x). Since i is arbitrary, the assertion shallfollow, once we have shown that µ1 agrees with λi on the whole of U(t, i)(N).

However, by definition we have µ1(y) ≥ λi(y) for every y ∈ U(t, i)(N), and then it is easilyseen that Ker(µ1 − λi)∩U(t, i)(N) is a face of U(t, i)(N); since x ∈ Ker(µ1 − λi), we deducethat µ1 − λi vanishes identically on U(t, i)(N).

3.6.5. Henceforth, we denote by λ(t) := λ1, . . . , λk the irredundant system of Q-linear formsfor t. Let 1 ≤ i, j ≤ k; then we claim that U(t, i, j)(N) := U(t, i)(N) ∩ U(t, j)(N) is a face ofboth U(t, i)(N) and U(t, j)(N). Indeed say that x, x′ ∈ U(t, i)(N) and x + x′ ∈ U(t, i, j)(N);these conditions translate the identities :

λi(x) + λi(x′) = λj(x) + λj(x

′) λi(x) ≤ λj(x) λi(x′) ≤ λj(x

′)

whence x, x′ ∈ U(t, i, j)(N). Define :

U(t, i) := (SpecU(t, i)(N)∨)] U(t, i, j) := (SpecU(t, i, j)(N)∨)] for every i, j ≤ k.

According to (3.5.20) and (3.5.18), the inclusion maps U(t, i, j)(N)→ U(t, l)(N) (for l = i, j)are dual to open immersions

(3.6.6) U(t, i)← U(t, i, j)→ U(t, j).

We may then attach to t and ρ|U(t) the fan U(t, ρ) obtained by gluing the affine fans U(t, i)along their common intersections U(t, i, j). The duals of the inclusions U(t, i)(N)→ U(t)(N)determine a well defined morphism of locally fine and saturated fans :

(3.6.7) U(t, ρ)→ U(t)

which, by construction, induces a bijection on N-points : U(t, ρ)(N)∼→ U(t)(N), so it is a

rational subdivision, according to proposition 3.5.24.


3.6.8. Let now t′ ∈ T be a generization of t; clearly the system λ′ := λ′1, . . . , λ′k consisting ofthe restrictions λ′i of the linear forms λi to the Q-vector subspace U(t′)(Q+)gp, fulfills condition(b) of definition 3.6.2, relative to t′. However, λ′ may fail to be irredundant; after relabeling,we may assume that the subsystem λ′1, . . . , λ′l is irredundant for t′, for some l ≤ k. With theforegoing notation, we have obvious identities :

U(t′, i)(N) = U(t, i)(N) ∩ U(t′)(N) U(t′, i, j)(N) = U(t, i, j)(N) ∩ U(t′)(N)

for every i, j ≤ l; whence, in light of remark 3.4.14(iii), a commutative diagram of fans :

U(t′, i)

U(t′, i, j)oo //

U(t′, j)

U(t, i)×U(t) U(t′) U(t, i, j)×U(t) U(t′)oo // U(t, i)×U(t) U(t′)

whose top horizontal arrows are the open immersions (3.6.6) (with t replaced by t′), whosebottom horizontal arrows are the open immersions (3.6.6)×U(t)U(t′), and whose vertical arrowsare natural isomorphisms. Since U(t′) is an open subset of U(t), we deduce an open immersion

jt,t′ : U(t′, ρ)→ U(t, ρ).

If t′′ is a generization of t′, it is clear that jt′,t′′ jt,t′ = jt,t′′ , hence we may glue the fans U(t, ρ)along these open immersions, to obtain a locally fine and saturated fan T (ρ). Furthermore, themorphisms (3.6.7) glue to a single rational subdivision :

(3.6.9) T (ρ)→ T.

Remark 3.6.10. (i) In the language of definition 3.3.25 the foregoing lengthy procedure trans-lates as the following simple geometric operation. Given a fan ∆ (consisting of a collectionof convex polyhedral cones of a R-vector space V ), a roof on ∆ is a piecewise linear functionF :=

⋃σ∈∆ σ → R, which is concave on each σ ∈ ∆ (and hence it is a roof on each such

σ, in the sense of example 3.3.27). Then, such a roof determines a natural refinement ∆′ of∆; namely, ∆′ is the coarsest refinement such that, for each σ′ ∈ ∆′, the function ρ|σ′ is therestriction of a R-linear form on V . This refinement ∆′ corresponds to the present T (ρ).

(ii) Moreover, let P be a fine, sharp and saturated monoid of dimension d, set TP :=(SpecP )], and suppose that f : T → TP is any integral, fine, proper and saturated subdivi-sion. Then f corresponds to a geometrical subdivision ∆ of the strictly convex polyhedral coneTP (R+) = P∨R , and we claim that ∆ can be refined by the subdivision associated to a roofon TP . Namely, let ∆d−1 be the subset of ∆ consisting of all σ of dimension d − 1; everyσ ∈ ∆d−1 is the intersection of a d-dimensional element of ∆ and a hyperplane Hσ ⊂ P gp∨

R ;such hyperplane is the kernel of a linear form λσ on P gp∨

R . Let us define

ρ(x) :=∑

σ∈∆d−1

min(0, λσ(x)) for every x ∈ TP (R+).

Then it is easily seen that the subdivision of TP (R+) associated to the roof ρ as in (i), refinesthe subdivision ∆. In the language of fans, this construction translates as follows. For everypoint σ ∈ T of height d − 1, let Hσ ⊂ P gp be the kernel of the surjection P gp → Ogp

T,σ

induced by log f ; notice that Hσ is a free abelian group of rank one, and pick a generatorsσ of Hσ, which – as in (3.6.1) – corresponds to a function λσ : TP (N) → Z, so we mayagain consider the integral roof ρ on TP defined as in the foregoing. Then it is easily seenthat the rational subdivision T (ρ) → TP associated to ρ, factors as the composition of f anda (necessarily unique) integral subdivision g : T (ρ) → T . More precisely, for every mapping


ε : σ ∈ T | ht(σ) = d− 1 → 0, 1, let us set

λε :=∑

ht(σ)=d−1

ε(σ) · λσ and U(ε)(N) := x ∈ TP (N) | ρ(x) = λε(x).

Whenever U(ε)(N) has dimension d, let tε ∈ T (ρ) be the unique point such that U(ε)(N)∨ =OT (ρ),tε . As the reader may check, there exists a unique closed point τ ∈ T , such that ε(σ) ·λσ(x) ≤ 0 for every x ∈ U(τ)(N) and every σ ∈ U(τ) of height d−1. Then we have g(tε) = τ ,and the restriction U(tε) → U(τ) of g is deduced from the inclusion U(ε)(N) ⊂ U(τ)(N) ofsubmonoids of TP (N).

(iii) Furthermore, in the situation of (ii), the roof ρ on TP can also be viewed as a roof on T ,and then it is clear from the construction that the morphism g : T (ρ)→ T is also the subdivisionof T attached to the roof ρ.

We wish now to establish some basic properties of the sheaf of fractional ideals

Iρ := Iρ,Q ∩ OgpT

attached to a given roof on T . First we remark :

Lemma 3.6.11. Keep the notation of (3.6.5), and let s ∈ OgpT,t ⊗Z Q be any element such that

ρs ≥ ρ|U(t). Then we have :(i) There exist ϕ ∈ (OT,t)Q (notation of (3.3.20)) and c1, . . . , ck ∈ R+ such that :

ρs = ρϕ +k∑i=1

ciλi

k∑i=1

ci = 1.

(ii) The stalk Iρ,t is a finitely generated OT,t-module.

Proof. (i): Let σ ⊂ U(t)(R+)gp be the convex polyhedral cone spanned by the linear forms((λi − ρs)⊗Q R | i = 1, . . . , k). Then the assumption on s means that

U(t)(R+) ∩ σ∨ = U(t)(R+) ∩Ker ρs ⊗Q R.Especially σ∨ ∩ U(t)(R+) does not span U(t)(R+)gp. Then one can repeat the proof of lemma3.6.4(iii) to derive the assertion.

(ii): By remark 3.4.14(i), we may write λi = ρsi − ρs′i , where si, s′i ∈ (OT,t)Q for each i ≤ k;pick N ∈ N large enough, so that Ns′i ∈ OT,t for every i ≤ k, and set τ := N

∑ki=1 s

′i. Then

τ + Iρ,t ⊂ OgpT,t ∩ (OT,t)Q = OT,t. By proposition 3.1.9(ii), we deduce that τ + Iρ,t is a finitely

generated ideal, whence the contention.

Proposition 3.6.12. Let T be a locally fine and saturated fan, ρ a roof on T . Then the associatedfractional ideal Iρ of OT is coherent.

Proof. In view of lemma 3.6.11(ii) and remark 3.5.13(v), it suffices to show that Iρ is quasi-coherent, i.e. for every generization t′ of t, the image of Iρ,t in OT,t′ generates the OT,t′-module Iρ,t′ . Fix such t′; by propositions 3.3.21(i) and 3.4.7(i), there exists λ ∈ OT,t =U(t)(N)∨ such that U(t′)(N) = Kerλ; especially, we see that U(t′)(N)gp is a direct summandof U(t)(N)gp. Now, let s′ ∈ Iρ,t′ be any local section; it follows that we may find s ∈ Ogp

T,t

such that ρs : U(t)(N)gp → Z is a Z-linear extension of the corresponding Z-linear formρs′ : U(t′)(N)gp → Z. Let also λ1, . . . , λk be the irredundant system of Q-linear forms for t(relative to the roof ρ). For every i ≤ k we have the following situation :

λR := λ⊗Q R ∈ U(t, i)(R+)∨ (ρs − λi)⊗Q R ∈ U(t′, i)(R+)∨.

However, U(t′, i)(R+) = U(t, i)(R+) ∩ KerλR, hence U(t′, i)(R+)∨ = U(t, i)(R+)∨ + RλR.Especially, there exists ri ∈ R+ and ϕ ∈ U(t)(R+)∨ such that (ρs − λi)⊗Q R = ϕ− riλR. Let


N be an integer greater than max(r1, . . . , rk); it follows that s + Nλ ∈ Iρ,t and its image inIρ,t′ equals s′.

Proposition 3.6.13. In the situation of (3.6.3) suppose that ρ is an integral roof on T . Then themorphism (3.6.9) is the saturation of a blow up of the fractional ideal Iρ.

Proof. We have to exhibit an isomorphism f : T (ρ) → X := Proj B(I )sat of T -fans. Webegin with :

Claim 3.6.14. (i) The fractional ideal IρOT (ρ) is invertible.(ii) For every n ∈ N, denote by nρ : T (Q+) → Q+ the function given by the rule x 7→

n · ρ(x) for every x ∈ T (Q+). Then :

B(I )sat = B′ :=∐n∈N

Inρ

Proof of the claim. (ii): The assertion is local on T , hence we may assume that T = U(t)for some t ∈ T , in which case, denote by λ := λ1, . . . , λk the irredundant system of Q-linear forms for t. Let n ∈ N, and s ∈ Inρ(U(t)); it is easily seen that the T -monoid B′ issaturated, hence it suffices to show that there exists an integer k > 0 such that kρs = ρs′ forsome s′ ∈ I k(U(t)) (notation of (3.6.1)). However, lemma 3.6.11(ii) implies more preciselythat we may find such k, so that the corresponding s′ lies in the ideal generated by λ.

(i): The assertion is local on T (ρ), hence we consider again t ∈ T and the corresponding λas in the foregoing. It suffices to show that J := IρOU(t,i) is invertible for every i = 1, . . . , k(notation of (3.6.5)). However, by inspecting the constructions it is easily seen that J (U(t, i))consists of all s ∈ (U(t, i)(N)∨)gp such that ρs(x) ≥ ρ(x) for every x ∈ U(t, i)(Q+), i.e.ρs(x) ≥ λi(x) for every x ∈ U(t, i)(Q+). However, since ρ is integral, we have λi ∈ Ogp

T,t;if we apply lemma 3.6.11(i) with T replaced by U(t, i), we conclude that J (U(t, i)) is thefractional ideal generated by λi, whence the contention. ♦

In view of claim 3.6.14(i) we see that there exists a unique morphism f of T -fans from T (ρ)to X . It remains to check that f is an isomorphism; the latter assertion is local on X , hence wemay assume that T = U(t) for some t ∈ T , and then we let again λ be the irredundant systemfor t. A direct inspection yields a natural identification of OT,t-monoids :

B′(U(t, i))(λi)∼→ U(t, i)(N)∨ for every i = 1, . . . , k

whence an isomorphism U(t, i)∼→ D+(λi) ⊂ X , which – by uniqueness – must coincide

with the restriction of f . On the other hand, the proof of claim 3.6.14(ii) also shows thatX = D+(λ1) ∪ · · · ∪D+(λk), and the proposition follows.

Example 3.6.15. Let P be a fine, sharp and saturated monoid.(i) The simplest non-trivial roofs on TP := (SpecP )] are the functions ρλ such that

ρλ(x) := min(0, λ(x)) for every x ∈ TP (Q+).

where λ is a given element of HomQ(TP (Q+)gp,Q) ' P gp ⊗Z Q. Such a ρλ is an integralroof, provided λ ∈ P gp. In the latter case, we may write λ = ρs1 − ρs2 , for some s1, s2 ∈ P .Set ρ′ := ρλ + ρs2 , i.e. ρ′ = min(ρs1 , ρs2)mplest non-trivial roofs on TP := (SpecP )] are thefunctions ρλ such that

ρλ(x) := min(0, λ(x)) for every x ∈ TP (Q+).

where λ is a given element of HomQ(TP (Q+)gp,Q) ' P gp⊗Z Q. Such a ρλ is an integral roof,provided λ ∈ P gp. In the latter case, we may write λ = ρs1 − ρs2 , for some s1, s2 ∈ P . Setρ′ := ρλ + ρs2 , i.e. ρ′ = min(ρs1 , ρs2); clearly T (ρλ) = T (ρ′), and on the other hand lemma3.6.11(i) implies that the ideal Iρ′ is the saturation of the ideal generated by s1 and s2.


(ii) More generally, any system λ1, . . . , λn ∈ HomQ(TP (Q+)gp,Q) of Q-linear forms yieldsa roof ρ on TP , such that ρ(x) :=

∑ni=1 min(0, λi(x)) for every x ∈ TP (Q+). A simple inspec-

tion shows that the corresponding subdivision T (ρ) → T can be factored as the compositionof n subdivisions gi : Ti → Ti−1, where T0 := TP , Tn := T (ρ), and each gi (for i ≤ n) is thesubdivision of Ti−1 corresponding to the roof ρλi as defined in (i).

(iii) These subdivisions of TP ”by hyperplanes” are precisely the ones that occur in remark3.6.10(ii),(iii). Summing up, we conclude that every proper integral and saturated subdivisiong : T → TP of TP can be dominated by another subdivision f : T (ρ) → TP of the typeconsidered in (ii), so that f factors as the composition of g and a subdivision h : T (ρ) → Twhich is also of the type (ii). Especially, both f and h can be realized as the composition offinitely many saturated blow up of ideals generated by at most two elements of P .

3.6.16. Let P be a fine, sharp and saturated monoid. A proper, integral, fine and saturatedsubdivision of

TP := (SpecP )]

is essentially equivalent to a (P gp)∨-rational subdivision of the polyhedral cone σ := P∨R (see(3.4.6) and definition 3.3.25). A standard way to subdivide a polyhedron σ consists in choosinga point x0 ∈ σ \0, and forming all the polyhedra x0 ∗F , where F is any proper face of σ, andx0 ∗ F denotes the convex span of x0 and F . We wish to describe the same operation in termsof the topological language of affine fans.

Namely, pick any non-zero ϕ ∈ TP (Q+) (ϕ corresponds to the point x0 in the foregoing).Let U(ϕ) ⊂ SpecP be the set of all prime ideals p such that ϕ(P \p) 6= 0; in other words,the complement of U(ϕ) is the topological closure of the support of ϕ in TP , especially, U(ϕ)is an open subset of TP . Denote by j : P = Γ(TP ,OTP ) → Γ(U(ϕ),OTP ) the restriction map.The morphism of monoids :

P → Γ(U(ϕ),OTP )×Q+ x 7→ (j(x), ϕ(x))

determines a cocartesian diagram of fans

U(ϕ)× (SpecQ+)]

ψ′

β′ // U(ϕ)× (SpecN)]

ψ

TPβ // Tϕ−1N := (Specϕ−1N)]

Lemma 3.6.17. With the notation of (3.6.16), the morphisms ψ and ψ′ are proper rationalsubdivisions, which we call the subdivisions centered at ϕ.

Proof. Notice that both β and β′ are homeomorphisms on the underlying topological spaces,and moreover both log βgp ⊗Z 1Q and log β′gp ⊗Z 1Q are isomorphisms. Thus, it suffices toshow that ψ is a proper rational subdivision, hence we may replace P by ϕ−1N, which allowsto assume that ϕ is a morphism of monoids P → N, and ψ is a morphism of fans

T ′ := U(ϕ)× (SpecN)] → TP .

In this situation, by inspecting the construction, we find that ψ restricts to an isomorphism :

ψ−1U(ϕ)∼→ U(ϕ)

and the preimage of the closed subset TP \U(ϕ) is the preimage of the closed point m ∈(SpecN)] under the natural projection T ′ → (SpecN)]; in view of the discussion of (3.5.19),this is naturally identified with U(ϕ)×m. Moreover, the restriction U(ϕ)×m → TP \U(ϕ)of ψ is the map (t,m) 7→ t ∪ ϕ−1m (recall that t ⊂ P is a prime ideal which does not containϕ−1m). Thus, the assertion will follow from :


Claim 3.6.18. The Q-linear map

logψgp(t,m) ⊗Z 1Q : P gp ⊗Z Q→ (Ogp

TP ,t× Z)⊗Z Q

is surjective for every t ∈ U(ϕ), and the induced map :

(3.6.19) T ′(Q+)→ Tϕ−1N(Q+) = TP (Q+)

is bijective.

Proof of the claim. Indeed, let Ft ⊂ P∨ be the face of P∨ corresponding to the point t ∈ U(ϕ),under the bijection (3.5.21); then ϕ /∈ Ft, whence a natural isomorphism of monoids :

(Ft + Nϕ)∨∼→ OTP ,t × N : λ 7→ (λ|F , λ(ϕ))

whose inverse, composed with logψ(t,m), yields the restriction map P → (Ft + Nϕ)∨. Thisinterpretation makes evident the surjectivity of logψgp

(t,m) ⊗Z 1Q. In view of example 3.5.9(iii),the bijectivity of (3.6.19) is also clear, if one remarks that :

P∨R =⋃

t∈U(ϕ)

(Ft + Nϕ)R.

The latter identity is obvious from the geometric interpretation in terms of polyhedral cones. Aformal argument runs as follows. Let ϕ′ ∈ P∨R ; since P spans P gp

R , the cone (PR)∨ is stronglyconvex (corollary 3.3.14), hence the line ϕ′+Rϕ ⊂ P gp

R is not contained in P∨R , therefore thereexists a largest r ∈ R such that ϕ′ − rϕ ∈ P∨R , and necessarily r ≥ 0. If ϕ′ − rϕ = 0, theassertion is clear; otherwise, let F be the minimal face of P∨ such that ϕ′ − rϕ ∈ FR, so thatϕ′ = (ϕ′ − rϕ) + rϕ ∈ (F + Nϕ)R. Thus, we are reduced to showing that ϕ /∈ F . But noticethat ϕ′ − rϕ lies in the relative interior of F ; therefore, if ϕ ∈ F , we may find ε > 0 such thatϕ′ − (r + ε)ϕ still lies in FR, contradicting the definition of r.

3.6.20. Lemma 3.6.17 is frequently used to construct subdivisions centered at an interior pointof TP , i.e. a point ϕ ∈ TP (N) which does not lie on any proper face of TP (N) (equivalently,the support of ϕ is the closed point mP of TP ). In this case U(ϕ) = TP \mP = (TP )d−1,where d := dim P . By lemma 3.6.17, the N-point ϕ lifts to a unique Q+-point ϕ of (TP )d−1 ×(SpecN)], and by inspecting the definitions, it is easily seen that – under the identification ofremark 3.5.8(iii) – the support of ϕ is the point (∅,mN), where ∅ ∈ TP is the generic point.More precisely, we may identify (SpecN)](Q+) with Q+, and (TP )d−1(Q+) with a cone in theQ-vector space TP (Q+)gp, and then ϕ corresponds to the point (0, 1) ∈ TP (Q+)gp ×Q+.

Suppose now that we have an integral roof

ρ : (TP )d−1(Q+)→ Q

and let π : T (ρ)→ (TP )d−1 be the associated subdivision. Let also ψ : (TP )d−1× (SpecN)] →TP be the subdivision centered at ϕ; we deduce a new subdivision :

(3.6.21) T ∗ := T (ρ)× (SpecN)]π×1

(Spec N)]−−−−−−−→ (TP )d−1 × (SpecN)]ψ−→ TP

whose restriction to the preimage of (TP )d−1 is TP -isomorphic to π.

Lemma 3.6.22. In the situation of (3.6.20), there exist an integral roof

ρ : TP (Q+)→ Q

whose restriction to (TP )d−1(Q+) agrees with ρ, and a morphism T ∗ → T (ρ) of TP -monoids,whose underlying continuous map is a homeomorphism.


Proof. According to remark 3.5.8(vi), we have a natural identification :

TP (Q+) = T (ρ)(Q+)× (SpecN)](Q+) = T (ρ)(Q+)×Q+ ⊂ TP (Q+)gp ×Q.mapping the point ϕ to (0, 1). For a given c ∈ R, denote by ρc : TP (Q+) → Q the functiongiven by the rule : (x, y) 7→ ρ(x) + cy for every x ∈ T (ρ)(Q+) and every y ∈ Q+.

Let t ∈ T (ρ) be any point of height d − 1; by assumption, there exists a Q-linear form λt :U(π(t))(Q+)gp → Q whose restriction to U(t)(Q+) agrees with the restriction of ρ. Therefore,the restriction of ρc to U(t,mN)(Q+) = U(t)(Q+) × Q+ agrees with the restriction of theQ-linear form

λt,c : U(π(t))(Q+)gp ×Q = TP (Q+)gp → Q (x, y) 7→ λt(x) + cy.

For any two points t, t′ ∈ T (ρ) of height d − 1, with π(t) = π(t′), pick a finite system ofgenerators x1, . . . , xn for U(t′)(N), and let y1, . . . , yn ∈ U(t)(Q)gp, a1, . . . , an ∈ Q such that

xi = yi + aiϕ in the vector space U(π(t))(Q+)gp.

In case xi ∈ U(t)(N), we shall have ai = 0, and otherwise we remark that ai > 0. In-deed, if ai < 0 we would have xi − aiϕ ∈ U(t)(Q+)gp ∩ TP (Q+) ⊂ U(π(t))(Q+); however,U(π(t))(Q+) is a proper face of TP (Q+), hence ϕ ∈ U(π(t))(Q+), a contradiction.

We may then find c > 0 large enough, so that λt′(xi) < λt,c(xi) for every xi /∈ U(t)(N). Itfollows easily that

(3.6.23) λt′,c(x) < λt,c(x) for all x ∈ U(t′,mN)(Q+)\U(t,mN)(Q+).

Clearly we may choose c large enough, so that (3.6.23) holds for every pair t, t′ as above,and then it is clear that ρc will be a roof on TP . Notice that the points of height d of T ∗ areprecisely those of the form (t,mN), for t ∈ TP of height d− 1, so the points of T (ρc) of heightd are in natural bijection with those of T ∗, and if τ ∈ T (ρc) corresponds to τ ∗ ∈ T ∗ underthis bijection, we have an injective map U(τ ∗)(N) → U(τ)(N), commuting with the inducedprojections to TP (N). There follows a morphism of TP -fans U(τ ∗)→ U(τ) inducing a bijectionU(τ ∗)(Q+)

∼→ U(τ)(Q+). Since T (ρc) (resp. T ∗) is the union of all such U(τ) (resp. U(τ ∗)),we deduce a morphism T ∗ → T (ρc) inducing a homeomorphism on underlying topologicalspaces. Lastly, say that τ ∗ = (t,m); then U(τ ∗)(N)gp = U(t)(N)gp ⊕ Zϕ, and on the otherhand U(t)(N)gp is a direct factor of U(τ)(N)gp (since the specialization map Ogp

T (ρc),τ→ Ogp

T,t issurjective). It follows easily that we may choose for c a suitable positive integer, in such a waythat the resulting roof ρ := ρc will also be integral.

3.6.24. Let P be a fine, sharp and saturated monoid, and set as usual TP := (SpecP )]. There isa canonical choice of a point in TP (N) which does not lie on any proper face of TP (N); namely,one may take the N-point ϕP defined as the sum of the generators of the one-dimensional facesof TP (N) (such faces are isomorphic to N, by theorem 3.4.16(ii)).

Set d := dim P ; if ρ is a given integral roof for (TP )d−1, and c ∈ N is a sufficiently large,we may then attach to the datum (ϕP , ρ, c) an integral roof ρ of TP extending ρ as in lemma3.6.22, and such that ρ(ϕP ) = c. More generally, let T be a fine and saturated fan of dimensiond, and suppose we have a given integral roof ρd−1 on Td−1; for every point t ∈ T of height dwe have the corresponding canonical point ϕt in the “interior” of U(t)(N), and we may thenpick an integer cd large enough, so that ρd−1 extends to an integral roof ρd : T (Q+) → Qwith ρd(ϕt) = cd for every t ∈ T of height d, and such that the associated subdivision isTP -homeomorphic to T (ρd−1)× (SpecN).

This is the basis for the inductive construction of an integral roof on TP which is canonical in acertain restricted sense. Indeed, fix an increasing sequence of positive integers c := (c2, . . . , cd);first we define ρ1 : T1(Q+) → Q+ to be the identically zero map. This roof is extendedrecursively to a function ρh on Th, for each h = 2, . . . , d, by the rule given above, in such a way


that ρh(ϕt) = ci for every point t of height i ≤ h. By the foregoing we see that the sequence ccan be chosen so that ρd shall again be an integral roof.

3.6.25. More generally, let S := P1, . . . , Pk be any finite set of fine, sharp and saturatedmonoids. We let S -Fan be the full subcategory of Fan whose objects are the fans T suchthat, for every t ∈ T , there exists P ∈ S and an open immersion U(t) ⊂ (SpecP )]. Thenthe foregoing shows that we may find a sequence of integers c(S ) := (c2, . . . , cd), withd := max(dim Pi | i = 1, . . . , k) such that the following holds. Every object T of S -Fanis endowed with an integral roof ρT : T (Q+)→ Q+ such that :

• ρT (ϕt) = ci whenever ht(t) = i ≥ 2, and ρT vanishes on T1(Q+).• Every open immersion g : T ′ → T in S -Fan determines an open immersion g :T ′(ρT ′)→ T (ρT ) such that the diagram

T ′(ρT ′)g //

πT ′

T (ρT )

πT

T ′

g // T

commutes (where πT and πT ′ are the subdivisions associated to ρT and ρT ′).• If dim T = d, there exists a natural rational subdivision

(3.6.26) Td−1(ρTd−1)× (SpecN)] → T (ρT )

which is a homeomorphism on the underlying topological spaces.

3.6.27. Notice as well that, by construction, T1(ρT1) = T1; hence, by composing the morphisms(3.6.26), we obtain a rational subdivision of T (ρT ) :

T1×(SpecN⊕d−1)] → · · · → Td−2(ρTd−2)×(SpecN⊕2)] → Td−1(ρTd−1

)×(SpecN)] → T (ρT ).

In view of theorem 3.4.16(ii), it is easily seen that every affine open subsect of T1 is isomorphicto (SpecN)], and any two such open subsets have either empty intersection, or else intersect intheir generic points. In any case, we deduce a natural epimorphism

(Q+)T1 → OT1,Q

(notation of (3.3.20)) from the constant T1-monoid arising from Q+; whence an epimorphism :

ϑT : (Q⊕d+ )T (ρT ) → OT (ρT ),Q

which is compatible with open immersions g : T ′ → T in S -Fan, in the following sense. Setd′ := dim T ′, and notice that d′ ≤ d; denote by πdd′ : Q⊕d+ → Q⊕d′+ the projection on the first d′

direct summands; then the diagram of T ′-monoids :

(3.6.28)

(Q⊕d+ )T ′(ρT ′ )g∗ϑT //

(πdd′ )T ′(ρT ′ )

g∗OT (ρT ),Q

(log g)Q

(Q⊕d′+ )T ′(ρT ′ )ϑT ′ // OT ′(ρT ′ ),Q

commutes.


3.6.29. Let f : T ′ → T be a proper morphism, with T ′ locally fine, such that the induced mapT (Q+)→ T ′(Q+) is injective, and let s ∈ T be any element. For every t ∈ f−1(s), we set

Gt := U(t)(Q+)gp ∩ U(s)(N)gp Ht := U(t)(N)gp δt := (Gt : Ht)

and define δ(f, s) := max(δt | t ∈ f−1(s)).

Lemma 3.6.30. For every s ∈ T we have :(i) δ(f, s) ∈ N.

(ii) If t, t′ ∈ f−1(s), and t is a specialization of t′ in T ′, then δt′ ≤ δt.(iii) If f is a rational subdivision, the following conditions are equivalent :

(a) δ(f, s) = 1.(b) U(s)(N) =

⋃t∈f−1(s) U(t)(N).

Proof. (i): Since f−1(s) is a finite set, the assertion means that δt < +∞ for every t ∈ f−1(s).However, for such a t, let P := OT,s and Q := OT ′,t; then

U(t)(Q+)gp = HomZ(Qgp,Q) and U(t)(N)gp = HomZ(Qgp,Z)

(remark 3.4.14(i) and proposition 3.4.12(iii)). By proposition 3.5.24 we know that the map(log f)gp

t ⊗Z Q : P gp ⊗Z Q→ Qgp ⊗Z Q is surjective. Since Q is finitely generated, it followsthat the image Q′ of P gp in Qgp is a subgroup of finite index, and then it is easily seen thatδt = (Qgp : Q′).

(ii): Under the stated assumptions we have :

Gt′ = Gt ∩ U(t′)(Q+)gp Ht′ = Ht ∩ U(t′)(Q+)gp

whence the contention.(iii): Assume that (b) holds, and let ϕ ∈ U(t)(Q+)gp ∩U(s)(N)gp for some t ∈ f−1(s). Pick

any element ϕ0 ∈ U(t)(N) which does not lie on any proper face of O∨T ′,t; we may then find aninteger a > 0 large enough, so that ϕ + aϕ0 ∈ U(t)(Q+). Set ϕ1 := ϕ + aϕ0 and ϕ2 := aϕ0;then ϕ1, ϕ2 ∈ U(t)(Q+) ∩ U(s)(N) = U(t)(N), hence ϕ = ϕ1 − ϕ2 ∈ U(t)(N)gp. Since ϕ isarbitrary, we see that δt = 1, whence (a).

Conversely, suppose that (a) holds, and let ϕ ∈ U(s)(N); then there exists t ∈ f−1(s) suchthat ϕ ∈ U(t)(Q+). Thus, ϕ ∈ U(t)(N)gp ∩ U(t)(Q+) = U(t)(N), whence (b).

Theorem 3.6.31. Every locally fine and saturated fan T admits an integral, proper, simplicialsubdivision f : T ′ → T , whose restriction f−1Tsim → Tsim is an isomorphism of fans.

Proof. Let T be such a fan. By induction on h ∈ N, we shall construct a system of integral,proper simplicial subdivisions fh : S(h) → Th of the open subsets Th (notation of (3.5.16)),such that, for every h ∈ N, the restriction f−1

h+1(Th)→ Th of fh+1 is isomorphic to fh, and suchthat f−1

h Th,sim → Th,sim is an isomorphism. Then, the colimit of the morphisms fh will be thesought subdivision of T .

For h ≤ 1, we may take S(h) := Th.Next, suppose that h > 1, and that fh−1 : S(h − 1) → Th−1 has already been given; as

a first step, we shall exhibit a rational, proper simplicial subdivision of Th. Indeed, for everyt ∈ Th \ Th−1, choose a N-point ϕt ∈ U(t)(N) in the following way. If t ∈ Tsim, then let ϕt bethe (unique) generator of an arbitrarily chosen one-dimensional face of U(t)(N); and otherwisetake any point ϕt which does not lie on any proper face of U(t)(N).

With these choices, notice that U(ϕt) × (SpecN)] = U(t) in case t ∈ Tsim, and otherwiseU(ϕt) = U(t)h−1 (notation of (3.6.16)). By lemma 3.6.17, we obtain corresponding rationalsubdivisions U(ϕt)× (SpecN)] → U(t) of U(t) centered at ϕt. Notice that if t lies in the sim-plicial locus, this subdivision is an isomorphism, and in any case, it restricts to an isomorphismon the preimage of U(t)h−1.


By composing with the restriction of fh−1 × (SpecN)], we get a rational subdivision:

gt : T ′t := f−1h−1U(ϕt)× (SpecN)] → U(t)

whose restriction to the preimage of U(t)h−1 is an isomorphism. Moreover, gt is an isomor-phism if t lies in Tsim. Also notice that gt is simplicial, since the same holds for fh−1.

If t, t′ are any two distinct points of T of height h, we deduce an isomorphism

g−1t (U(t) ∩ U(t′))

∼→ g−1t′ (U(t) ∩ U(t′))

hence we may glue the fans T ′t and the morphisms gt along these isomorphism, to obtain thesought simplicial rational subdivision g : T ′ → Th.

For the next step, we shall refine g locally at every point s of height h; i.e. for such s, we shallfind an integral, proper simplicial subdivision fs : T ′′s → U(s), whose restriction to U(s)h−1

agrees with g, hence with fh−1. Once this is accomplished, we shall be able to build the soughtsubdivision fh by gluing the morphisms fs and fh−1 along the open subsets U(s)h−1.

Of course, if s lies in the simplicial locus of T , we will just take for fs the restriction of g,which by construction is already an isomorphism.

Henceforth, we may assume that T = U(s) is an affine fan of dimension h with s /∈ Tsim,and g : T ′ → T is a given proper rational simplicial subdivision, whose restriction to g−1Th−1

is an integral subdivision. We wish to apply the criterion of lemma 3.6.30(iii), which shows thatg is an integral subdivision if and only if δ(g, s) = 1. Thus, let t1, . . . , tk be the points of T ′

such that δti = δ(g, s) for every i = 1, . . . , k. Since δ(g, s) is anyway a positive integer (lemma3.6.30(i)), a simple descending induction reduces to the following :

Claim 3.6.32. Given g as above, we may find a proper rational simplicial subdivision g′ : T ′′ →T such that the following holds :

(i) The restriction of g′ to g′−1Th−1 is isomorphic to the restriction of g.(ii) Let t′1, . . . , t

′k′ ∈ T ′′ be the points such that δt′i = δ(g′, s) for every i = 1, . . . , k′. We

have δ(g′, s) ≤ δ(g, s), and if δ(g′, s) = δ(g, s) then k′ < k.

Proof of the claim. Set t := t1; by definition, there exists ϕ′ ∈ U(s)(N) which lies inU(t)(Q+) \ U(t)(N); this means that there exists a morphism ϕ fitting into a commutativediagram :

OT,s(log g)t //

ϕ′

OT ′,t

ϕ

N // Q+.

Say that OT ′,t ' N⊕r, and let (π1, . . . , πr) be the (essentially unique) basis of O∨T ′(h),t; then ϕ =a1π1 + · · · arπr, for some a1, . . . , ar ≥ 0, and after subtracting some positive integer multipleof π, we may assume that 0 ≤ ai < 1 for every i = 1, . . . , r. Moreover, the coefficients areall strictly positive if and only if ϕ is a local morphism; more generally, we let t′ be the uniquegenerization of t such that ϕ factors through a local morphism OT ′,t′ → N. Set e := ht(t′), anddenote by π1, . . . , πe the basis of O∨T ′,t′ , so that :

(3.6.33) ϕ = b1π1 + · · ·+ beπe

for unique rational coefficients b1, . . . , be such that 0 < bi < 1 for every i = 1, . . . , e.Denote by Z ⊂ T ′ the topological closure of t′; for every u ∈ Z ∩ T ′, the morphism ϕ

factors through a morphism ϕu : OT,u → Q+, and we may therefore consider the subdivision


of U(u) centered at ϕu as in (3.6.16), which fits into a commutative diagram of fans :

(U(u)\Z)× (SpecQ+)] //

(U(u)\Z)× (SpecN)]

ψu

U(u) // U ′(u) := (Specϕ−1u N)].

We complete the family (ψu | u ∈ Z), by letting U ′(u) := U(u) and ψu := 1U(u) for everyu ∈ T ′ \Z. Notice then, that the topological spaces underlying U(u) and U ′(u) agree for everyu ∈ T ′, and for every u1, u2 ∈ T ′, the restrictions of ψu1 and ψu2 :

ψ−1ui

(U(u1) ∩ U(u2))→ U(u1) ∩ U(u2) (i = 1, 2)

are isomorphic. Furthermore, by construction, each restriction U(u)→ T of g factors uniquelythrough a morphism βu : U ′(u) → T , hence the family (βu ψu | u ∈ T ′) glues to a welldefined morphism of fans g′ : T ′′ → T . By a direct inspection, it is easily seen that g′ is aproper rational simplicial subdivision which fulfills condition (i) of the claim.

Moreover, the map of topological spaces underlying g′ factors naturally through a continuousmap p : T ′′ → T ′, so that g′ = g p. The restriction V := p−1(T ′ \Z)→ T of g′ is isomorphicto the restriction T ′ \Z → T of g, hence :

δt = δp(t) for every t ∈ p−1(T ′ \Z).

It follows that, if k > 1 and T ′ \Z contains at least one of the points t2, . . . , tk, then δ(g′, s) ≥δ(g, s). On the other hand, p−1(T ′ \Z) contains at most k − 1 points u of T ′′ such that δu =δ(g, s), and for the remaining points u′ ∈ p−1(T ′ \Z) we have δu′ < δ(g, s). Since obviously

δ(g′, s) = max(δ(g′|p−1(T ′\Z), s), δ(g′|p−1Z , s))

we see that condition (ii) holds provided we show :

(3.6.34) δ(g′|p−1Z , s) = max(b1, . . . , be) · δ(g, s).

Hence, let us fix u ∈ Z, and let (v, x) ∈ (U(u)\Z) × (SpecN)] be any point (see (3.5.19)); ifx = ∅, then (v, x) /∈ p−1Z, so it suffices to consider the points of the form (v,m) (where m ⊂ Nis the maximal ideal). Moreover, say that ht(u) = d; in view of lemma 3.6.30(ii), it suffices toconsider the points (v,m) such that ht(v) = d− 1. There are exactly e such points, namely theprime ideals vi := (πi σ)−1m, where π1, . . . , πe are as in (3.6.33), and σ : OT ′,u → OT ′,t′ isthe specialization map.

In order to estimate δ(v,m) for some v := vi, we look at the transpose of the map

(log g′)gp(v,m) : Ogp

T,s → OgpT ′,v × Z : z 7→ ((log g)gp

v (z), ϕgp(z)).

Let (p1, . . . , pd) be the basis of O∨T,u, ordered in such a way that pi = πiσ for every i = 1, . . . , e.By a little abuse of notation, we may then denote (pj | j 6= i) the basis of O∨T ′,v, so that the dualgroup (Ogp

T ′,v × Z)∨ admits the basis pgpj | j 6= i ∪ q, where q is the natural projection onto

Z. Set as well p′i := pi (log g)gpu for every i = 1, . . . , d. With this notation, the above transpose

is the group homomorphism given by the rule :

pj 7→ p′i for j 6= i and q 7→ b1p′1 + · · ·+ bep

′e

from which we deduce easily that δ(v,m) = bi · δu, whence (3.6.34).

Proposition 3.6.35. Let (Γ,+, 0) be a fine monoid, M a fine Γ-graded monoid. Then :(i) There exists a finite set of generators C := γ1, . . . , γk of Γ, with the following prop-

erty. For every γ ∈ Γ, we may find a1, . . . , ak ∈ N such that:

γ = a1γ1 + · · ·+ akγk and Mγ = Ma1γ1· · ·Mak

γk.


(ii) There exists a subgroup H ⊂ Γgp of finite index, such that :

Maγ = Maγ for every γ ∈ H ∩ Γ and every integer a > 0.

(In (i) and (ii) we use the multiplication law of P(M), as in (3.1.1)).

Proof. Obviously we may assume that M maps surjectively onto Γ, in which case G := Γgp isfinitely generated, and its image G′ into GR := G ⊗Z R is a free abelian group of finite rank.The same holds as well for the image L of logMgp in Mgp

R := logMgp ⊗Z R. Let p : G→ G′

be the natural projection. According to proposition 3.3.22(iii), we have :

(3.6.36) MQ = MR ∩MgpQ

(notation of (3.3.20)). Let f gpR : Mgp

R → GR be the induced R-linear map, and denote byfR : MR → GR the restriction of f gp

R . By proposition 3.3.28(ii), we may find a G′-rationalsubdivision ∆ of fR(MR) such that :

(3.6.37) f−1R (a+ b) = f−1

R (a) + f−1R (b)

for every σ ∈ ∆ and every a, b ∈ σ. After choosing a refinement, we may assume that ∆ is asimplicial fan (theorem 3.6.31). Let τ ∈ ∆ be any cone; by proposition 3.3.22(i), the monoidN := τ ∩ G′ is finitely generated, and then the same holds for M ×G′ N , by corollary 3.4.2.However, the latter is just M ′ :=

⊕γ∈p−1N Mγ (lemma 2.3.29(iii)). Set Γ′ := Γ ∩ p−1N .

Claim 3.6.38. p(Γ′) generates τ .

Proof of the claim. Clearly the Γ-grading of M induces a surjection MQ → ΓQ (notation of(3.3.20)). By proposition 3.3.22(iii), we have ΓQ = fR(MR) ∩ GQ, hence N ⊂ τ ∩ GQ ⊂ ΓQ.Thus, for every n ∈ N we may find a ∈ N such that a · n ∈ p(Γ), hence a · n ∈ p(Γ′). On theother hand, N generates τ , since the latter is G′-rational. The claim follows. ♦

In view of claim 3.6.38, we may replace M by M ′, and Γ by Γ′, which allows to assumethat fR(MR) is a simplicial cone, and (3.6.37) holds for every a, b ∈ fR(MR). Next, let S :=e1, . . . , en ⊂ p(Γ) be a set of generators of the cone fR(MR); from the discussion in (3.3.15)we see that, up to replacing S by a subset, the rays R+ · ei (with i = 1, . . . , n) are precisely theextremal rays of fR(MR), especially, the vectors e1, . . . , en are R-linearly independent. Chooseg1, . . . , gn ∈ Γ such that p(gi) = ei for every i = 1, . . . , n. According to corollary 3.4.3, thereexist finite subsets Σ1 . . . ,Σn ⊂M , such that :

Mgi = M0 · Σi for every i = 1, . . . , n.

Claim 3.6.39. (M0)Q = f−1R (0) ∩Mgp

Q .

Proof of the claim. To begin with, we may write f−1R (0) = (f gp

R )−1(0) ∩MR, hence f−1R (0) ∩

MgpQ = (f gp

Q )−1(0) ∩ MQ, by (3.6.36). Now, suppose x ∈ (f gpQ )−1(0) ∩ MQ; then we may

find an integer a > 0 such that ax = m ⊗ 1 for some m ∈ logM . Say that m ∈ logMγ;then f gp

Q (γ) = 0, therefore γ is a torsion element of G′, and consequently bm ∈ logM0 for aninteger b > 0 large enough. We conclude that x = (bm)⊗ (ba)−1 ∈ (M0)Q, as claimed. ♦

On the other hand, since R+ei is a G′-rational polyhedral cone, f−1R (R+ei) is an L-rational

polyhedral cone (proposition 3.3.21(ii,iii)), hence it admits a finite set of generators Si ⊂ L. Upto replacing the elements of Si by some positive rational multiples, we may assume that fR(s)is either 0 or ei, for every s ∈ Si. In this case, it follows easily that

(3.6.40) f−1R (ei) = f−1

R (0) + Ti

where :

Ti :=

∑s∈Si

ts · s | ts ∈ R+,∑s∈Si

ts = 1


is the convex hull of Si (and as usual, the addition of sets in (3.6.40) refers to the addition lawof P(Mgp

R ), see (3.1.1)). Next, notice that Si ⊂ MQ, by (3.6.36); thus, we may find an integera > 0 such that a · s lies in the image of logM , for every s ∈ Si. After replacing ei by a · ei andSi by a · s | s ∈ Si, we may then achieve that (3.6.40) holds, and furthermore Si lies in theimage of logM , therefore in the image of logMgi . It follows easily that (3.6.40) still holds withSi replaced by the set Σi⊗1 := m⊗1 |m ∈ Σi. Let Σ0 ⊂ logM be a finite set of generatorsfor the monoid M0; claim 3.6.39 implies that Σ0 is also a set of generators for the L-rationalpolyhedral cone f−1

R (0). Let P ⊂M be the submonoid generated by Σ := Σ0 ∪Σ1 ∪ · · · ∪Σn,and ∆ ⊂ Γ the submonoid generated by g1, . . . , gn; clearly the Γ-grading of M restricts to a∆-grading on P . Notice that g1 ⊗ 1, . . . , gn ⊗ 1 are linearly independent in GQ, since the sameholds for e1, . . . , en; especially, ∆ ' N⊕n.

Claim 3.6.41. (i) The set Σ⊗ 1 := s⊗ 1 | s ∈ Σ generates the cone MR.(ii) Pa+b = Pa · Pb for every a, b ∈ ∆.

(iii) There exists a finite set A ⊂M such that M = A · P .

Proof of the claim. By (3.6.40) and the foregoing discussion, we know that (Σ0∪Σi)⊗1 generatef−1R (R+ei), for every i = 1, . . . , n. Since the additivity property (3.6.37) holds for every a, b ∈fR(MR), assertion (i) follows. (ii) is a straightforward consequence of the definitions. Next,from (i) and proposition 3.3.22(iii), we deduce that MQ = PQ. Thus, let m1, . . . ,mr be asystem of generators for the monoid M ; it follows that there are integers k1, . . . , kr > 0, suchthat mk1

1 , . . . ,mkrr ∈ P , and therefore the subset A :=

∏ri=1m

tii | 0 ≤ ti < ki for every i ≤ r

fulfills the condition of (iii). ♦

We introduce a partial ordering on G, by declaring that a ≤ b for two elements a, b ∈ G, ifand only if b− a ∈ ∆. Now, let a ∈ G be any element; we set

G(a) := g ∈ G | g ≤ a.

The subset G(a) inherits a partial ordering from G, and a is the maximum of the elements ofG(a); moreover, notice that every finite subset S ⊂ G(a) admits a supremum supS ∈ G(a).Indeed, it suffices to show the assertion for a set of two elements S = b1, b2; we may thenwrite a−bi =

∑nj=1 kijgj for certain kij ∈ N, and then sup(b1, b2) = a−

∑nj=1 min(k1j, k2j)·gj .

Let A be as in claim 3.6.41(iii), and denote by B ⊂ Γ the image of A; then

M = M0 ·MB where MB :=⊕b∈B

logMb.

For every a ∈ G, let also B(a) := B ∩G(a); invoking several times claim 3.6.41(ii), we get :

Ma =⋃

b∈B(a)

Mb · Pa−b

=⋃

b∈B(a)

Mb · PsupB(a)−b · Pa−supB(a)

⊂MsupB(a) · Pa−supB(a).

Finally, say that a−supB(a) =∑n

i=1 tigi for certain t1, . . . , tn ∈ N; applying once more claim3.6.41(ii), we conclude that :

Ma ⊂MsupB(a) ·n∏i=1

M tigi.

The converse inclusion is clear, and therefore the set C := g1, . . . , gn ∪ supB(a) | a ∈ Gfulfills condition (i) of the proposition.


(ii): For h ∈ ∆gp, say h =∑n

i=1 aigi, with integers a1, . . . , an, we let |h| :=∑n

i=1 |ai|gi ∈ ∆.Choose any positive integer α such that :

(3.6.42) |b| ≤ α ·n∑i=1

gi for every b ∈ B ∩∆gp

and let H ⊂ ∆gp be the subgroup generated by αg1, . . . , αgn.

Claim 3.6.43. B(h) = B(kh) for every h ∈ H and every integer k > 0.

Proof of the claim. Let h :=∑n

i=1 αigi ∈ H , and suppose that b ∈ B(kh) for some k > 0;therefore kh− b ∈ ∆, hence b ∈ ∆gp, and we can write b =

∑ni=1 βigi for integers β1, . . . , βn,

such that kαi − βi ≥ 0 for every i = 1, . . . , n. In this case, (3.6.42) implies that αi ≥ 0 forevery i ≤ n, and αi ≥ α ≥ βi whenever βi > 0. It follows easily that k′h − b ∈ ∆ for everyinteger k′ > 0, whence the claim. ♦

Using claims 3.6.41(ii) and 3.6.43, and arguing as in the foregoing, we may compute :

Mah = MsupB(ah) · Pah−supB(ah)

= MsupB(h) · Pah−supB(h)

= MsupB(h) · Ph−supB(h) · P a−1h

⊂Mah .

The converse inclusion is clear, so (ii) holds.

3.6.44. Let M be an integral monoid, and w ∈ logMgp any element. For ε ∈ 1,−1 we havea natural inclusion

jε : logM →M(ε) := logM + εNw(i.e. M(ε) is the submonoid of Mgp generated by M and wε). Let us write w := b−1a forsome a, b ∈ M ; then the induced morphisms of affine schemes ιε := SpecZ[jε] have a naturalgeometric interpretation. Namely, let f : X → SpecZ[M ] be the blow up of the ideal I ⊂ Z[M ]generated by a and b; we have X = U1 ∪ U−1, where Uε, for ε = ±1, is the largest opensubscheme of X such that wε ∈ OX(Uε). Then ιε is naturally identified with the restrictionUε → SpecZ[M ] of the blow up f . More generally, by adding to M any finite number ofelements of Mgp, we may construct in a combinatorial fashion, the standard affine charts of ablow up of an ideal of Z[M ] generated by finitely many elements of M . These considerationsexplain the significance of the following flattening theorem :

Theorem 3.6.45. Let j : M → N be an inclusion of fine monoids. Then there exists a finite setΣ ⊂ logMgp, and an integer k > 0 such that the following holds :

(i) For every mapping ε : Σ→ ±1, the induced inclusion :

M(ε) := logM +∑σ∈Σ

ε(σ)Nσ → N(ε) := logN +∑σ∈Σ

ε(σ)Nσ

is a flat morphism of fine monoids.(ii) Suppose that j is a flat morphism, and let ι : M → M sat be the natural inclusion.

Denote by Q the push-out of the diagram Nj←− M

ιkM−−−→ M sat (where kM is thek-Frobenius). Then the natural map M sat → Qsat is flat and saturated.

Proof. (i): (Notice that logM , logN and P (ε) :=∑

σ∈Σ ε(σ)Nσ may be regarded as sub-monoids of logNgp, and then the above sum is taken in the monoid (P(logNgp),+) definedas in (3.1.1).) Set G := Ngp/Mgp, and let Ngp =

⊕γ∈GN

gpγ be the j-grading of Ngp (re-

mark 3.2.5(iii)); notice that this grading restricts to j-gradings for N and N(ε). Let alsoΓ := γ ∈ G | Nγ 6= ∅, and choose a finite generating set γ1, . . . , γr of Γ with the


properties of proposition 3.6.35(i). According to corollary 3.4.3, for every i ≤ r there ex-ists a finite subset Σi := ti1, . . . , tini ⊂ M such that Nγi = N0 · Σi. Moreover, N0 is afinitely generated monoid, by corollary 3.4.2. Let Σ0 be a finite system of generators for N0,and for every i ≤ r define Σ′i := tij − til | 1 ≤ j < l ≤ ni. We claim that the subsetΣ := Σ0 ∪ Σ′1 ∪ · · · ∪ Σ′r will do. Indeed, first of all notice that Σ ⊂ logMgp. We shall applythe flatness criterion of remark 3.2.5(iv). Thus, we have to show that, for every g ∈ Γ, theM(ε)-module N(ε)g is a filtered union of cosets m+M(ε) (for certain m ∈ Ng). Hence, letx1, x2 ∈ N(ε)g; we may write xi = mi + pi for some mi ∈ logN and pi ∈ P (ε) (i = 1, 2);since xi+M(ε) ⊂ mi+M(ε), we may then assume that p1 = p2 = 0, hence x1, x2 ∈ N .Thus, it suffices to show that Ng is contained in a filtered union of cosets as above. However,by assumption there exist a1, . . . , ar ∈ N such that g =

∑ri=1 aiγi and Ng = Na1

γ1· · ·Nar

γr ; weare then easily reduced to the case where g = γi for some i ≤ r. Therefore, xj = σj + bj ,where σj ∈ Σi and bj ∈ N0, for j = 1, 2. Notice that P (ε) contains either σ1 − σ2, or σ2 − σ1

(or both); in the first occurrence, set σ := σ2, and otherwise, let σ := σ1. Likewise, say thatΣ0 = y1, . . . , yn, so that bj =

∑ns=1 ajsys for certain ajs ∈ N (j = 1, 2); for every s ≤ n, we

set a∗s := min(a1s, a2s) if ys ∈ P (ε), and otherwise we set a∗s := max(a1s, a2s). One sees easilythat x1, x2 ∈ σ +

∑ns=1 a

∗sys+M(ε), whence the contention.

(ii): By proposition 3.6.35(ii), there exists a subgroup H ⊂ G of finite index, such that :

π−1(hn) = π−1(h)n

for every integer n > 0 and every h ∈ H . Let k := (G : H), and define N ′ as the fibre productin the cartesian diagram :

(3.6.46)N ′

π′ //

µ

G

kG

Nπ // G

The trivial morphism 0M : M → G (i.e. the unique one that factors through 1) and theinclusion j satisfy the identity : kG 0M = π j, hence they determine a well-defined mapϕ : M → N ′.

Claim 3.6.47. ϕ is flat and saturated.

Proof of the claim. For the flatness, we shall apply the criterion of remark 3.2.5(iv). First, ϕ isinjective, since µ ϕ = j. Next, notice that the sequence of abelian groups :

0→Mgp ϕgp

−−→ (N ′)gp (π′)gp

−−−→ G→ 0

is short exact; indeed, this is none else than the pullback E ∗kgpG along the morphism kG, of the

short exact sequence

(3.6.48) E := (0→Mgp jgp

−−→ Ngp πgp

−−→ G→ 0).

It follows that ϕ is flat if and only if, for every x ∈ π′(N ′), the preimage (π′)−1(x) is a filteredunion of cosets of the form n · ϕ(M ′). However, the induced map (π′)−1(g)→ π−1(gk) is abijection for every g ∈ G, and the flatness of j implies that π−1(xk) is a filtered union of cosets,whence the contention. Notice also that ImkG ⊂ H; hence, by the same token, we derive that(π′)−1(gn) = (π′)−1(g)n for every g ∈ G, therefore ϕ is quasi-saturated, by proposition 3.2.31.Since we know already that j is integral (theorem 3.2.3), the claim follows. ♦


Next, we wish to consider the commutative diagram of monoids :

(3.6.49)M

j //

kM

N

ψ

Mϕ // N ′

such that ψ is the map determined by the pair of morphisms (f,kN). Let P be the push-out ofthe maps j and kM ; the maps ϕ and ψ determine a morphism τ : P → N ′.

Claim 3.6.50. (i) The diagram (3.6.46)gp of associated abelian groups, is cartesian.(ii) The diagram of abelian groups (3.6.49)gp is cocartesian (i.e. τ gp is an isomorphism).

(iii) There exists a morphism λ : N ′ → P such that λ τ = kP and τ λ = kN ′ .

Proof of the claim. (i): Suppose that x ∈ (N ′)gp is any element such that (π′)gp(x) = 1 andµgp(x) = 1; we may write x = b−1a for some a, b ∈ N ′, and it follows that π′(a) = π′(b) andµ(a) = µ(b), hence a = b in N ′, since the forgetful functor Mnd→ Set commutes with fibreproducts. Thus x = 1. On the other hand, suppose that πgp(b−1a) = xk for some a, b ∈ Nand x ∈ G; therefore, π(bk−1a) = (bx)k, so there exists c ∈ N ′ such that µ(c) = bk−1a andπ′(c) = bx. Likewise, there exists d ∈ N ′ with µ(d) = bk and π′(d) = b. Consequently,(π′)gp(d−1c) = x and µ(d−1c) = b−1a. The assertion follows.

(ii): Quite generally, let E := (0→ A→ B → C → 0) be a short exact sequence of objectsin an abelian category C , and for every object X of C , let kX := k · 1X : X → X . Then onehas a natural map of complexes αE : E ∗kC → E (resp. βE : E → kA ∗E) from E to the pull-back of E along kC (resp. from the push-out of E along kA to E), and a natural isomorphismωE : kA∗E

∼→ E ∗kC in the category ExtC (C,A) of extensions of C byA (this is the categorywhose objects are all short exact sequences in C of the form 0 → A → X → C → 0, andwhose morphisms are the maps of complexes which are the identity on A and C). These mapsare related by the identities :

(3.6.51) βE αE ωE = k · 1kA∗E ωE βE αE = k · 1E∗kC .We leave to the reader the construction of ωE . In the case at hand, we obtain a natural isomor-phism ωE : kgp

M ∗ E∼→ E ∗ kG, where E is the short exact sequence of (3.6.48). An inspection

of the construction shows that the map τ gp is precisely the isomorphism defined by ωE .(iii): Let µ′ : N → P be the natural map, and set λ := µ′µ. By inspecting the constructions,

one checks easily that λgp is the map defined by βE αE . Then the assertion follows from(3.6.51). ♦

Let P ′ be the push-out of the diagram N ′ϕ←− M

ι−→ M sat; from claim 3.6.47, lemma 3.2.2(i)and corollary 3.2.25(iii), we deduce that the natural map M sat → P ′ is flat and saturated, henceP ′ is saturated. On the other hand, directly from the definitions we get a cocartesian diagram

(3.6.52)

P //

τ

Q

τ ′

N ′ // P ′.

The induced diagram (3.6.52)sat of saturated monoids is still cocartesian (remark 2.3.41(v));however, claim 3.6.50(iii) implies easily that τ sat is an isomorphism, therefore the same holdsfor (τ ′)sat, and assertion (ii) follows.

4. COMPLEMENTS OF COMMUTATIVE AND HOMOLOGICAL ALGEBRA

This chapter is a miscellanea of results of commutative algebra that shall be needed in therest of the treatise.


4.1. Complexes in an abelian category. Let A be any abelian category. We denote by C(A )the category of (cochain) complexes of objects of A , and by D(A ) the derived category of A .Hence, an object of C(A ) or D(A ) is a pair

K• := (K•, d•K)

consisting of a system of objects (Kn | n ∈ Z) and morphisms (dnK : Kn → Kn+1 | n ∈ Z) ofA , called the differentials of the complex K•, such that

dn+1K dnK = 0 for every n ∈ Z.

The morphisms ϕ• : K• → L• in C(A ) are the systems of morphisms (ϕn : Kn → Ln | n ∈ Z)of A such that

ϕn+1 dnK = dn+1L ϕn for every n ∈ Z.

We will usually omit the subscript when referring to the differentials of a complex, unless thereis a danger of confusion. Let i ∈ Z be any integer; the cohomology of K• in degree i is theobject of A :

H iK• := Ker di/Im di−1.

Clearly, for every i ∈ Z, the rule K• 7→ H iK• extends to a functor

H i : C(A )→ A .

Also, let I ⊂ Z be any (bounded or unbounded) interval, i.e. I is either of the form Z∩ [a,+∞[,or Z∩]−∞, b] (for some a, b ∈ N), or the intersection of any of these two. We shall denote by

CI(C ) ( resp. DI(C ) )

the full subcategory of C(C ) (resp. of D(C )) whose objects are the complexes K• such thatKi = 0 whenever i /∈ I (resp. such that H iK• = 0 whenever i ∈/∈ Z). For instance, ifI = Z ∩ [a,+∞[, then CI(C ) (resp. DI(C )) is also denoted C≥a(C ) (resp. D≥a(C)), andlikewise for the case of an upper bounded interval. Moreover, we set

C−(A ) :=⋃n∈Z

C≤n(A ) C+(A ) :=⋃n∈Z

C≥n(A )

so C−(A ) (resp. C+(A )) is the full subcategory of C(A ) whose objects are the bounded above(resp. bounded below) complexes of A . Likewise we define the full subcategories D−(A )and D+(A ) of D(A ). Recall that, for any interval I , the category DI(C ) is also naturallyequivalent to the localization of CI(C ) by the multiplicative set of all morphisms in CI(C ) thatare quasi-isomorphisms, and likewise for D−(A ) and D+(A ) (verification left to the reader).

For every a ∈ Z, the inclusion functor

C≥a(A )→ C(A ) (resp. C≤a(A )→ C(A ) )

admits a right (resp. left) adjoint

t≥a : C(A )→ C≥a(A ) (resp. t≤a : C(A )→ C≤a(A ) )

called the brutal truncation functor; namely, for any complex K•, we let t≥a(K•) be the uniqueobject of C≥a(A ) that agrees with K• in all degrees≥ a, and with the same differentials as K•,in this range of degrees (and likewise for t≤a(K•)).


4.1.1. There is an obvious functor

A → C(A ) : A 7→ A[0]

that sends any object A of A to the complex with A placed in degree zero, i.e. such that A[0]i

equals A if i = 0, and equals 0 otherwise (clearly, there is a unique such complex). On the otherhand, the shift operator is the functor

C(A )→ C(A ) : K• → K•[1]

given by the rule :

K•[1]n := Kn+1 dnK[1] := −dn+1K for every n ∈ Z.

Clearly the shift operator is an automorphism of C(A ), and one defines the operator K• 7→K•[n], for every n ∈ Z, as the n-th power of the shift operator (in the automorphism group ofC(A )). Then, we can combine the two previous operators, to define the complex

A[n] := (A[0])[n] for every n ∈ Z and every A ∈ Ob(A ).

Definition 4.1.2. Let A be an abelian category, ϕ•, ψ• : K• → L• two morphisms in C(A ).(i) A (chain) homotopy from ϕ• to ψ• is the datum (sn : Kn → Ln−1 | n ∈ Z) of a system

of morphisms in A such that

ϕn − ψn = sn+1 dnK + dn+1L sn for every n ∈ Z.

(ii) We say that ϕ• and ψ• as in (i) are chain homotopic if there is a chain homotopybetween them. It is easily seen that this defines an equivalence relation ∼ on the setHomC(A )(K

•, L•), which is preserved by composition of morphisms : if ϕ• ∼ ψ• andα• : K ′• → K•, β• : L• → L′• are any two morphisms, then ϕ• α• ∼ ψ• α• andβ• ϕ• ∼ β• ψ•. It follows that there exists a well defined homotopy category

Hot(A )

whose objects are the same as those of C(A ), and whose morphisms are the homotopyclasses of morphisms of complexes, and a natural functor

C(A )→ Hot(A )

which is the identity on objects, and the quotient map on Hom-sets.(iii) We also say that a morphism ϕ : K• → L• is a homotopic equivalence, if the class of

ϕ• is an isomorphism in Hot(A ), i.e. if there exists a morphism ψ• : L• → K• suchthat ψ• ϕ• ∼ 1K• and ϕ ψ ∼ 1L• . We say that a complex K• is homotopicallytrivial, if the zero endomorphism 0 · 1K• is a homotopy equivalence.

Remark 4.1.3. (i) If F : A → A ′ is any additive functors of abelian categories, we getinduced functors

C(F ) : C(A )→ C(A ′) Hot(F ) : Hot(A )→ Hot(A ′)

by the rule :

F (K•)n := F (Kn) and dnF (K) := F (dnK) for every n ∈ Z and every K• ∈ Ob(C(A )).

(ii) Furthermore, if ϕ•, ψ• : K• → L• are chain homotopic morphisms in C(A ), then it iseasily seen that, for every i ∈ Z, the induced morphisms in cohomology

H iϕ•, H iψ• : H iK• → H iL•

coincide. Hence, the cohomology functor H i factors (uniquely) through a functor

H i : Hot(A )→ A for every i ∈ Z.


Remark 4.1.4. The indexing notation that makes use of superscripts to denote the degrees in acomplex, is known traditionally as cohomological degree notation. Sometimes it is more naturalto switch to the homological degree notation, that makes use of subscript indexing; namely, oneassociates to any cochain complex K•, the chain complex K• given by the rule :

Kn := K−n and dn := d−n : Kn → Kn−1 for every n ∈ Z.Likewise, one sets HnK• := H−nKn for every n ∈ Z, and calls this object of A the homologyof K• in degree n.

4.1.5. A double complex of A is an object of C(C(A )) (and likewise for a morphism of doublecomplexes). In other words, a double complex is a triple

K•• := (K••, d••h , d••v )

consisting of a system (Kpq | p, q ∈ Z) of objects of A , and morphisms dpqh , dpqv called respec-tively the horizontal and vertical differentials, fitting into a commutative diagram

Kpqdpqh //

dpqv

Kp+1,q

dp+1,qv

Kp,q+dpqh // Kp+1,q+1

for every p, q ∈ Z

and such that

dp+1,qh dpqh = 0 dp,q+1

v dpqv = 0 for every p, q ∈ Z.

4.1.6. There are natural functors

C(C(A ))→ C(C(A )) : K•• 7→ fl(K••) C(C(A ))→ C(A ) : K•• 7→ (K••)∆

where :• The flip fl(K••) of K•• is the double complex F •• such that F pq := Kqp for everyp, q ∈ Z, with differentials deduced from those of K••, in the obvious way• The diagonal (K••)∆ is the complex D• such that Dp := Kpp for every p ∈ Z and

with differentials

dp+1,qv dpqh : Dp → Dp+1 for every p ∈ Z.

Suppose that all coproducts (resp. all products) are representable in A . Then there are twoother natural functors

Tot⊕ : C(C(A ))→ C(A ) ( resp. TotΠ : C(C(A ))→ C(A ) )

defined as follows. The total complex Tot⊕(K••) (resp. TotΠ(K••)) is the complex T • suchthat

T n :=⊕p+q=n

Kpq ( resp. T n :=∏

p+q=n

Kpq )

and with differentials T n → T n+1 given by the sum (resp. the product) of the morphisms

dpqh + (−1)p · dpqv : Kpq → Kp,q+1 ⊕Kp+1,q for all p, q ∈ Z such that p+ q = n.

We often omit the superscript ⊕ when dealing with the total complex functor; to avoid confu-sion, we stipulate that the notation Tot shall always refer to the functor Tot⊕, so if we need touse the other total complex functor, we shall denote it explicitly by TotΠ. Notice that we havenatural isomorphisms

Tot(K••)∼→ Tot(fl(K••)) (resp. TotΠ(K••)

∼→ TotΠ(fl(K••)) )

given, in each degree n ∈ Z, by the direct sum (resp. the direct product) of the morphisms(−1)pq · 1Kpq , for every p, q ∈ Z such that p+ q = n.


Example 4.1.7. (i) To any two objects K• and L• of C(A ), we may attach the double complexof abelian groups Hom••A (K•, L•), given by the rule :

Homp,qA (K•, L•) := HomA (K−p, Lq) for every p, q ∈ Z

with differentials

dp,qh := HomA (d−pK ,1Lq) dp,qv := (−1)q+1 · HomA (1K−p , dqL) for every p, q ∈ Z.

Then set(Hom•A (K•, L•), d•K,L) := TotΠHom••A (K•, L•).

Let n ∈ Z be any integer; with this notation, a simple inspection shows that :• HomC(A )(K

•, L•[n]) = Ker dnK,L.• Let ϕ•, ψ• : K• → L•[n] be any two morphisms; then the set of homotopies from ϕ•

to ψ• is naturally identified with the subset

s• ∈ Hom−1−nA (K•, L•) | d−1−n

K,L (s•) = ψ• − ϕ•.• Consequently, we have a natural identification :

HomHot(A )(K•, L•[n])

∼→ HnHom•A (K•, L•).

(ii) The constructions of (i) can be used to endow C(A ) with a natural 2-category structure,whose 2-cells are given by homotopies of complexes. Indeed, let us write

s• : ϕ• ⇒ ψ•

if s• := (sn | n ∈ N) is a homotopy from ϕ• to ψ•. Then, if λ• : K• → L• is another morphism,and t• : ψ• ⇒ λ• another homotopy, we define a composition law by setting

s• t• := (sn + tn | n ∈ N)

and it is immediate that s• t• is a homotopy ϕ• ⇒ λ•. Moreover, if β•, γ• : L• → P •

are two other morphisms in C(A ), and u : β• ⇒ γ• another homotopy, we have a Godementcomposition law by the rule

u• ∗ s• := (βn+1 sn + un ψn | n ∈ N) : β• ϕ• ⇒ γ• ψ•.The associativity of the laws ∗ and ⊕ thus defined are easily checked by direct computation.

Now, suppose that δ• : L• → P • is yet another morphism, and v : γ ⇒ δ another homotopy.A direct calcultation yields the identity

(u• ∗ s•) (v• ∗ t•) = (u• v•) ∗ (s• t•) + c•

where cn := dn−2P un−1 tn − un tn+1 dnK for every n ∈ Z, which can be rewritten as

c• = d−2K,L((u t)•) where (u t)n := un−1 tn for every n ∈ Z.

Summing up, we conclude that C(A ) carries a 2-category structure, whose 2-cells ϕ⇒ ψ (forany two 1-cells ϕ, ψ : K• → L•) are the classes

s ∈ Hom−1A (K•, L•)/Im(d−2

K,L) such that d−1K,L(s) = ψ• − ϕ•.

(iii) Moreover, if F is any additive functor as in remark 4.1.3(i), the induced functor C(F )extends to a pseudo-functor for the 2-category structures given by (ii); indeed, if s• : K• ⇒ L•

is a homotopy, obviously the system (Fsn | n ∈ N) is a homotopy Fs• : F (K•)⇒ F (L•).

Example 4.1.8. (i) Suppose that (A ,⊗,Φ,Ψ) is a tensor abelian category. Then there exists anatural functor

−− : C(A )× C(A )→ C(C(A ))

defined as follows. Given two complexes K• and L•, we let

(K• A L•)p,q := Kp ⊗ Lq for every p, q ∈ Z


with differentials

dp,qh := dpK ⊗ 1Lq dp,qv := 1Kp ⊗ dqL for every p, q ∈ Z.If all coproducts are representables in A , we may set as well :

K• ⊗ L• := Tot(K• L•).

The commutativity constraints for (A ,⊗) yield natural isomorphisms K•L• ∼→ fl(L•K•),as well as

(4.1.9) Ψ•K,L : K• ⊗ L• ∼→ L• ⊗K•.Namely, one takes the direct sum of the maps (−1)pq ·ΨKp,Lq , for every p, q ∈ Z.

(ii) Likewise, if P • is another complex of A , the double complexes (K• ⊗ L•) P • andK• (L• ⊗ P •) are not isomorphic, but the associativity constraints of A induce naturalisomorphisms in C(A ) :

Φ•K,L,P : K• ⊗ (L• ⊗ P •) ∼→ (K• ⊗ L•)⊗ P •.Namely, one takes the direct sum of the morphisms ΦKi,Lj ,Pk (for every i, j, k ∈ Z). With thesenatural isomorphisms, C(A ) is then naturally a tensor abelian category as well.

(iii) In the situation of (i), notice that the natural morphismKi⊗Lj → (K•⊗L•)i+j inducesmorphisms

Ker (diK)⊗Ker (djL)→ Ker (di+jK⊗L)

(Ker (diK)⊗ Im (dj−1L ))⊕ (Im (di−1

K )⊗Ker (djL))→ Im (di+jK⊗L)

so, the induced map Ker (diK)⊗Ker (djL)→ H i+j(K•⊗L•) factors through a natural pairing :

H i(K•)⊗Hj(L•)→ H i+j(K• ⊗ L•) for every i, j ∈ Z.(iv) Moreover, if P • is a third complex, in view of (ii) we get a commutative diagram

H iK• ⊗ (HjL• ⊗HkP •)

γ

α // (H iK• ⊗HjL•)⊗HkP •β // H i+j(K• ⊗ L•)⊗HkP •

δ

H iK• ⊗Hj+k(L• ⊗ P •) λ // H i+j+k(K• ⊗ (L• ⊗ P •)) σ // H i+j+k((K• ⊗ L•)⊗ P •)

where α is the associativity constraint, β, γ, δ and λ are given by the above pairing, and σ isdeduced from Φ•K,L,P .

(v) If A also admits an internal Hom functor, we may define as well a functor

Hom•• : C(A )o × C(A )→ C(C(A ))

following the trace of example 4.1.7(i); namely, we set

Homp,q(K•, L•) := Hom(K−p, Lq) for every p, q ∈ Zwith differentials

dp,qh := Hom(d−pK ,1Lq) dp,qv := (−1)q+1 ·Hom(1K−p , dqL) for every p, q ∈ Z.

If all products are representable in A , we may then define

Hom•(K•, L•) := TotΠHom••(K•, L•).

(vi) Suppose that A is both complete and cocomplete, and P • is any other complex; to easenotation, set H• := Hom•(K•, L•) and N• := P • ⊗K•; we have natural isomorphisms

HomC(A )(P•, H•)

∼→ Equal(Hom0A (P •, H•)

dv //

dh

// Hom1A (P •, H•))


wheredh :=

∏n∈Z

HomA (P n, dnH) and dv :=∏n∈Z

HomA (dnP , Hn+1)

(see example 4.1.7(i)). However, notice that

HomaA (P •, H•) =

∏n∈Z

HomA (P n,∏

p+q=n+a

Hom(K−p, Lq))

=∏n∈Z

∏p+q=n+a

HomA (P n,Hom(K−p, Lq))

=∏n∈Z

∏p+q=n+a

HomA (P n ⊗K−p, Lq))

=∏q∈Z

HomA (⊕

n−p=q−a

P n ⊗K−p, Lq))

for every n, a ∈ Z, whence

HomC(A )(P•, H•)

∼→ Equal(Hom0A (N•, L•)

d′v //

d′h

// Hom1A (N•, L•))

whered′h :=

∏n∈Z

HomA (dqN , Lq) and d′v :=

∏n∈Z

HomA (N q, dqL)

so finally :HomC(A )(P

•, H•)∼→ HomC(A )(N

•, L•)

which says that Hom• is an internal Hom functor for C(A ).(vii) In the situation of (vi), set

HomC(A )(K•, L•) := Ker (d0 : H0 → H1)

and take P • := Z[0], where Z is any object of A ; it is easily seen that the natural map

HomA (Z,HomC(A )(K•, L•))→ HomC(A )(P

•, H•)→ HomC(A )(Z[0]⊗K•, L•)

is an isomorphism : details left to the reader.

4.1.10. Suppose that A is a small abelian category, and let (G•, d•G) be any complex of finitelygenerated abelian groups; with the notation of (1.2.43), we obtain an object (G•A , d

•A ) of C(A †);

on the other hand, if (K•, d•K) is any object of C(A ), we may also consider the object h†K :=

(h†Kn , h†dn | n ∈ N) of C(A †), and since A † is an abelian tensor category, we may form the

tensor productG• Z K

• := G•A h†K

according to example 4.1.8(i). Arguing as in (1.2.43), we see that this object of C(C(A †)) isisomorphic to an object of C(C(A )), and after choosing representing objects, we get a functor

C(Z-Modfg)× C(A )→ C(C(A )) (G•, K•) 7→ G• Z K•

which is additive in both arguments. Arguing as in remark 1.2.45(ii,iii), we may also definemore generally this functor in case A is an arbitrary abelian category, and if A is cocomplete,we can extend the functor to the whole of C(Z-Mod). It is then natural to define

G• ⊗Z K• := TotG• Z K

• for every G• and K• as above.

Likewise, we set K• Z G• := fl(G• Z K

•) and K• ⊗Z G• := Tot(K• Z G

•).


Remark 4.1.11. (i) With the notation of (4.1.10), notice the natural isomorphism

K•[1]∼→ Z[1]⊗Z K

• for every K• ∈ Ob(C(A ))

which explains the sign convention in the definition of the shift operator in (4.1.1).(ii) Moreover, denote by K〈1〉• ∈ Ob(C[−1,0](Z-Mod)) the object such that K〈1〉−1 := Z,

K〈1〉0 := Z ⊕ Z, and with differential d−1 given by the rule : n 7→ (n,−n) for every n ∈ Z.Let e0 := (0, 1) and e1 := (1, 0) be the canonical basis of K〈1〉0; we have two morphisms

ιi : Z[0]→ K〈1〉• for i = 0, 1

given, in degree zero, by the rule : n 7→ n · ei for every n ∈ N. For any pair of morphismsϕ•, ψ• : L• → M• in C(A ), and any homotopy s• := (sn | n ∈ Z) from ϕ• to ψ•, we obtain amorphism of complexes

σ• : K〈1〉• ⊗Z L• →M•

as follows. For every n ∈ Z, the morphism σn : Ln ⊕ Ln ⊕ Ln+1 → Mn restricts to ϕn (resp.ψn, resp. sn) on the first (resp. second, resp. third) summand. Conversely, the datum of amorphism σ• : K〈1〉•⊗Z L

• →M• yields a homotopy from σ• (ι•1⊗Z L•) to σ• (ι•0⊗Z L

•).

4.1.12. In the situation of example 4.1.8, suppose now that A is a category with enough pro-jective objects. Then, for every bounded above complex K• we may find a quasi-isomorphismρ•K : P •K → K• with P •K a bounded above complex of projective objects, and if K• → L• isany morphism of complexes, there exists a commutative diagram in C−(A )

P •KP •ϕ //

ρ•K

P •L

ρ•L

K•ϕ• // L•

where P •ϕ is unique up to homotopy, so the rules K• 7→ P •K and ϕ• 7→ P •ϕ yield a well definedfunctor

C−(A )→ Hot(A )

and moreover notice that, if ϕ• as above is a quasi-isomorphism, then P •ϕ is a homotopic equiv-alence. Hence, let K• and L• be any two objects of C−(A ); we may define two functors

K•L⊗− ( resp. −

L⊗ L•) : C(A )→ Hot(A )

by the rules :M• 7→ K• ⊗ P •M ( resp. M 7→ P •M ⊗ L• )

and in light of example 4.1.7(iii), we see that both functors transform quasi-isomorphisms intohomotopic equivalences, so they induce well defined functors on the derived categories, and

moreover it follows that the notation K•L⊗ L• is unambiguous : we may compute this object

by applying the functor K•L⊗ − to L•, or by applying the functor −

L⊗ L• to K•, and the two

resulting complexes are naturally isomorphic in D(A ) to the same complex P •K⊗P •L. The latter

assertion also implies that K•L⊗ L• is independent, up to unique isomorphism in D(A ), of the

choices of P •M and P •L. Hence we have obtained a natural functor

−L⊗− : D−(A )× D−(A )→ D−(A )

called the derived tensor product. We also let

TorAi (K•, L•) := Hi(K

• L⊗ L•) for every i ∈ Z.


In case A = A-Mod for some ring A, it is customary to denote this functor by −L⊗A −, and

then one also writes TorAi instead of TorA-Modi .

Remark 4.1.13. (i) Using the commutativity and associativity constraints for the tensor productin A , we deduce – in light of example 4.1.8(i,ii) – natural associativity isomorphisms

K•L⊗ (L•

L⊗Q•) ∼→ (K•

L⊗ L•)

L⊗Q• in D−(A )

as well as commutativity isomorphisms

K•L⊗ L• ∼→ L•

L⊗K• in D−(A )

for any bounded above complexes K•, L•, and Q•.

(ii) Also, it is easily seen that, ifK• ∈ Ob(D≤a(A )) and L• ∈ Ob(D≤b(A )), thenK•L⊗L• ∈

Ob(D≤a+b(A )), for every a, b ∈ Z.(iii) In the situation of (4.1.12), take A = A-Mod for some ringA, and suppose furthermore

that ϕ : A→ B is a ring homomorphism, K• a bounded above complex of A-modules, and L•

a complex of B-modules. Notice that P • ⊗A L• is naturally a complex of B-modules; also, ifL• → Q• is any morphism of complexes of B-modules, then the induced map P • ⊗A L• →P • ⊗A Q• is B-linear. It follows easily that the derived tensor product yields a functor

D−(A-Mod)× D−(B-Mod)→ D−(B-Mod) (K•, L•) 7→ K•L⊗A L•

such that, denoting ϕ∗ : D−(B-Mod) → D−(A-Mod) the “forgetful” functor, we have anatural isomorphism

K•L⊗A ϕ∗L•

∼→ ϕ∗(K•L⊗A L•) in D−(A-Mod).

4.1.14. Let now M•1 , M•

2 , N•1 , N•2 be any four objects of C−(A ), and to ease notation, set

M•12 := M•

1 ⊗M•2 N•12 := N•1 ⊗N•2 P •12 := P •M12

ρ•12 := ρ•M12

as well as P •i := P •Miand ρ•i := ρ•Mi

for i = 1, 2. There is a commutative diagram in C−(A )

P •1 ⊗ P •2ϕ•12 //

ρ•1⊗ρ•2 $$IIIIIIIIIP •12

ρ•12zzzzzzzz

M•12

where ϕ•12 is uniquely determined up to homotopy, whence a map

(M•1

L⊗N•1 )⊗ (M•

2

L⊗N•2 )

∼→ (P •1 ⊗ P •2 )⊗N•12

ϕ•12⊗N•12−−−−−−→ P •12 ⊗N•12∼→M•

12

L⊗N•12.

Taking into account example 4.1.8(iii), we deduce a bilinear pairing

TorAi (M•

1 , N•1 )⊗ TorA

j (M•2 , N

•2 )→ TorA

i+j(M•12, N

•12) for every i, j ∈ Z.


Moreover, suppose that M•3 and N•3 are two other bounded above complexes of A ; by inspect-

ing the constructions, we find a commutative diagram

P •1 ⊗ (P •2 ⊗ P •3 )P •1⊗ϕ•23 //

ρ•1⊗(ρ•2⊗ρ•3)++VVVVVVVVVVVVVVVVVVVVVV

Φ•P

P •1 ⊗ P •23

ϕ•1,23 //

ρ•1⊗ρ•23

P •1,23

ρ•1,23uujjjjjjjjjjjjjjjjjjjjj

P •ΦM

M•1,23

Φ•M

M•12,3

(P •1 ⊗ P •2 )⊗ P •3ϕ•12⊗P •3 //

(ρ•1⊗ρ•2)⊗ρ•333hhhhhhhhhhhhhhhhhhhhhhP •12 ⊗ P •3

ϕ•12,3 //

ρ•12⊗ρ•3

OO

P •12,3

ρ•12,3

jjTTTTTTTTTTTTTTTTTTTTT

where M•1,23 := M•

12 ⊗M•3 , M•

23 := M•2 ⊗M•

3 , M•1,23 := M•

1 ⊗M•23, and likewise for P •1,23,

P •23, and P •12,3 and the morphism ϕ•23, ϕ•1,23, ϕ•12,3, ρ•23, ρ•1,23, ρ•12,3. Here Φ•M and Φ•P are theassociativity constraints.

Therefore, set T ji := TorAi (M•

j , N•j ) for every i ∈ Z and j = 1, 2, 3, and also

T jki := TorAi (M•

jk, N•jk) T 1,23

i := TorAi (M•

1,23, N•1,23) T 12,3

i := TorAi (M•

12,3, N•12,3)

for every i ∈ Z, with j = 1, 2 and k = j + 1; in light of example 4.1.8(iv), we deduce acommutative diagram in A :

(4.1.15)

T 1i ⊗ (T 2

j ⊗ T 3k ) //

T 1i ⊗ T 23

j+k// T 1,23i+j+k

(T 1i ⊗ T 2

j )⊗ T 3k

// T 12i+j ⊗ T 3

k// T 12,3i+j+k

whose horizontal arrows are given by the above bilinear pairing, and whose left (resp. right)vertical arrow is the associativity constraint (resp. is induced by the associativity constraintΦ•M ).

4.1.16. Koszul complex and regular sequences. Let f := (fi | i = 1, . . . , r) be a finite system ofelements of a ring A, and (f) ⊂ A the ideal generated by this sequence; we recall the definitionof the Koszul complex K•(f) (see [28, Ch.III, §1.1]). First, suppose that r = 1, so f = (f) for asingle element f ∈ A; in this case :

K•(f) := (0→ Af−→ A→ 0)

concentrated in cohomological degrees 0 and −1. In the general case one lets :

K•(f) := K•(f1)⊗A · · · ⊗A K•(fr).

For every complex of A-modules M• one sets :

K•(f ,M•) := M• ⊗A K•(f) K•(f ,M•) := Tot•(Hom•A(K•(f),M•))

and denotes by H•(f ,M•) (resp. H•(f ,M•)) the homology of K•(f ,M•) (resp. the cohomol-

ogy of K•(f ,M•)). Especially, if M is any A-module :

H0(f ,M) = M/(f)M H0(f ,M) = HomA(A/(f),M)

(where, as usual, we regard M as a complex placed in degree 0).


4.1.17. Let r > 0 and f be as in (4.1.16), and set f ′ := (f1, . . . , fr−1). We have a short exactsequence of complexes : 0 → A[0] → K•(fr) → A[1] → 0, and after tensoring with K•(f

′)we derive a distinguished triangle:

K•(f′)→ K•(f)→ K•(f

′)[1]∂−→ K•(f

′)[1].

By inspecting the definitions one checks easily that the boundary map ∂ is induced by multipli-cation by fr. There follow exact sequences :

(4.1.18) 0→ H0(fr, Hp(f′,M))→ Hp(f ,M)→ H0(fr, Hp−1(f ′,M))→ 0

for every A-module M and for every p ∈ N, whence the following :

Lemma 4.1.19. With the notation of (4.1.17), the following conditions are equivalent :

(a) Hi(f ,M) = 0 for every i > 0.(b) The scalar multiplication by fr is a bijection on Hi(f

′,M) for every i > 0, and is aninjection on M/(f ′)M .

Definition 4.1.20. The sequence f is said to be completely secant on the A-module M , if wehave Hi(f ,M) = 0 for every i > 0.

The interest of definition 4.1.20 is due to its relation to the notion of regular sequence ofelements of A (see e.g. [15, Ch.X, §9, n.6]). Namely, we have the following criterion :

Proposition 4.1.21. With the notation of (4.1.16), the following conditions are equivalent :

(a) The sequence f is M -regular.(b) For every j ≤ r, the sequence (f1, . . . , fj) is completely secant on M .

Proof. Lemma 4.1.19 shows that (b) implies (a). Conversely, suppose that (a) holds; we showthat (b) holds, by induction on r. If r = 0, there is nothing to prove. Assume that the assertionis already known for all j < r. Since f is M -regular by assumption, the same holds for thesubsequence f ′ := (f1, . . . , fr−1), and fr is regular on M/(f ′)M . Hence Hp(f

′,M) = 0 forevery p > 0, by inductive assumption. Then lemma 4.1.19 shows that Hp(f ,M) = 0 for everyp > 0, as claimed.

Notice that any permutation of a completely secant sequence is again completely secant,whereas a permutation of a regular sequence is not always regular. As an application of theforegoing, we point out the following :

Corollary 4.1.22. If a sequence (f, g) of elements of A is M -regular, and M is f -adicallyseparated, then (g, f) is M -regular.

Proof. According to proposition 4.1.21, we only need to show that the sequence (g) is com-pletely secant, i.e. that g is regular on M . Hence, suppose that gm = 0 for some m ∈ M ; itsuffices to show that m ∈ fnM for every n ∈ N. We argue by induction on n. By assump-tion g is regular on M/fM , hence m ∈ fM , which shows the claim for n = 1. Let n > 1,and suppose we already know that m = fn−1m′ for some m′ ∈ M . Hence 0 = gfn−1m′, sogm′ = 0 and the foregoing case shows that m′ = fm′′ for some m′′ ∈ M , thus m = fnm′′, asrequired.

4.1.23. In the situation of (4.1.16), suppose that g := (gi | i = 1, . . . , r) is another sequence ofelements of A; we set fg := (figi | i = 1, . . . , r) and define a map of complexes

ϕg : K•(fg)→ K•(f)


as follows. First, suppose that r = 1; then f = (f), g = (g) and the sought map ϕg is thecommutative diagram :

0 // Afg //

g

A // 0

0 // Af // A // 0.

For the general case we let :ϕg := ϕg1

⊗A · · · ⊗A ϕgr .Especially, for every m,n ≥ 0 we have maps ϕfn : K•(f

n+m)→ K•(fm), whence maps

ϕ•fn : K•(fm,M)→ K•(fm+n,M)

and clearly ϕ•fp+q = ϕ•fp ϕ•fq for every m, p, q ≥ 0.

4.1.24. Let A be any ring, I ⊂ A a finitely generated ideal, f := (fi | i = 1, . . . , r) a finitesystem of generators for I . Moreover, for every n > 0 let I(n) ⊂ A denote the ideal generatedby fn := (fni | i = 1, . . . , r). For all m ≥ n > 0 we deduce natural commutative diagrams ofcomplexes :

K•(fm) //

ϕfm−n

A/I(m)[0]

πmn

K•(fn) // A/I(n)[0]

(notation of (4.1.23)) where πmn is the natural surjection, whence a compatible system of maps:

(4.1.25) HomD(A-Mod)(A/I(n)[0], C•)→ HomD(A-Mod)(K•(f

n), C•).

for every n ≥ 0 and every complex C• in D+(A-Mod). Especially, let us take C• of the formM [−i], for someA-moduleM and integer i ∈ N; since K•(f

n) is a complex of freeA-modules,(4.1.25) translates as a direct system of maps :

(4.1.26) ExtiA(A/I(n),M)→ H i(fn,M) for all n ∈ N and every i ∈ N.

Notice that Inr−r+1 ⊂ I(n) ⊂ In for every n > 0, hence the colimit of the system (4.1.26) isequivalent to a natural map :

(4.1.27) colimn∈N

ExtiA(A/In,M)→ colimn∈N

H i(fn,M) for every i ∈ N.

Lemma 4.1.28. With the notation of (4.1.24), the following conditions are equivalent :(a) The map (4.1.27) is an isomorphism for every A-module M and every i ∈ N.(b) colim

n∈NH i(fn, J) = 0 for every i > 0 and every injective A-module J .

(c) The inverse system (HiK•(fn) | n ∈ N) is essentially zero whenever i > 0, i.e. for

every p ∈ N there exists q ≥ p such that HiK•(fq)→ HiK•(f

p) is the zero map.

Proof. (a)⇒ (b) is obvious. Next, if J is an injective A-module, we have natural isomorphisms

(4.1.29) H i(fn, J) ' HomA(HiK•(fn), J) for all n ∈ N.

which easily implies that (c)⇒ (b).(b)⇒ (c) : Indeed, for any p ∈ N let us choose an injection ϕ : HiK•(f

n) → J into aninjective A-module J . By (4.1.29) we can regard ϕ as an element of H i(fn, J); by (b) theimage of ϕ in HomA(HiK•(f

q), J) must vanish if q > p is large enough. This can happen onlyif HiK•(f

q)→ HiK•(fp) is the zero map.


(b)⇒ (a) : Let M → J• be an injective resolution of the A-module M . The double complexcolimn∈N

Hom•A(K•(fn), J•) determines two spectral sequences :

Epq1 := colim

n∈NHomA(Kp(f

n), HqJ•)⇒ colimn∈N

Extp+qA (K•(fn),M)

F pq1 := colim

n∈NHp(fn, Jq) ' colim

n∈NHomA(HpK•(f

n), Jq)⇒ colimn∈N

Extp+qA (K•(fn),M).

Clearly Epq1 = 0 whenever q > 0, and (b) says that F pq

1 = 0 for p > 0. Hence these two spectralsequences degenerate and we deduce natural isomorphisms :

colimn∈N

ExtqA(A/I(n),M) ' F 0q2∼→ Eq0

2 ' colimn∈N

Hq(fn,M).

By inspection, one sees easily that these isomorphisms are the same as the maps (4.1.27).

Lemma 4.1.30. In the situation of (4.1.24), suppose that the following holds. For every finitelypresented quotient B of A, and every b ∈ B, there exists p ∈ N such that

AnnB(bq) = AnnB(bp) for every q ≥ p.

Then the inverse system (HiK•(fn) | n ∈ N) is essentially zero for every i > 0.

Proof. We shall argue by induction on r. If r = 1, then f = (f) for a single element f ∈ A. Inthis case, our assumption ensures that there exists p ∈ N such that AnnA(f q) = AnnA(fp) forevery q ≥ p. It follows easily that H1(ϕfp) : H1K•(f

p+k) → H1K•(fk) is the zero map for

every k ≥ 0 (notation of (4.1.23)), whence the claim.Next, suppose that r > 1 and that the claim is known for all sequences of less than r elements.

Set g := (f1, . . . , fr−1) and f := fr. Specializing (4.1.18) to our current situation, we deriveshort exact sequences :

0→ H0(fn, HpK•(gn))→ HpK•(f

n)→ H0(fn, Hp−1K•(gn))→ 0

for every p > 0 and n ≥ 0; for a fixed p, this is an inverse system of exact sequences, wherethe transition maps on the rightmost term are given by fm−n. By induction, the inverse system(HiK•(g

n) | n ∈ N) is essentially zero for i > 0, so we deduce already that the inverse system(HiK•(f

n) | n ∈ N) is essentially zero for all i > 1. To conclude, we are thus reduced to show-ing that the inverse system (Tn := H0(fn, H0K•(g

n)) | n ∈ N) is essentially zero. HoweverAn := H0K•(g

n) = A/(gn1 , . . . , gnr−1) is a finitely presented quotient of A for any fixed n ∈ N,

hence the foregoing case r = 1 shows that the inverse system (Tmn := AnnAn(fm) |m ∈ N) isessentially zero. Letm ≥ n be chosen so that Tmn → Tnn is the zero map; then the compositionTm = Tmm → Tmn → Tnn = Tn is zero as well.

Remark 4.1.31. Notice that the condition of lemma 4.1.30 is verified when A is noetherian.

4.1.32. Minimal resolutions. Let A be a local ring, k its residue field, M an A-module of finitetype, and :

· · · d3−→ L2d2−→ L1

d1−→ L0ε−→M

a resolution of M by free A-modules. We say that (L•, d•, ε) is a finite-free resolution if eachLi has finite rank. We say that (L•, d•, ε) is a minimal free resolution of M if it is a finite-freeresolution, and moreover the induced maps k ⊗A Li → k ⊗A Im di are isomorphisms for alli ∈ N (where we let d0 := ε). One verifies easily that if A is a coherent ring, then every finitelypresented A-module admits a minimal resolution.


4.1.33. Let L := (L•, d•, ε) and L′ := (L′•, d′•, ε′) be two free resolutions of M . A morphism

of resolutions L → L′ is a map of complexes ϕ• : (L•, d•) → (L′•, d′•) that extends to a

commutative diagram

L•ε //

ϕ•

M

L′•ε′ // M

Lemma 4.1.34. Let L := (L•, d•, ε) be a minimal free resolution of an A-module M of finitetype, L′ := (L′•, d

′•, ε′) any other finite-free resolution, ϕ• : L′ → L a morphism of resolutions.

Then ϕ• is an epimorphism (in the category of complexes of A-modules), Kerϕ• is a nullhomotopic complex of free A-modules, and there is an isomorphism of complexes :

L′•∼→ L• ⊕Kerϕ•.

Proof. Suppose first that L = L′. We set d0 := ε, L−1 := M , ϕ−1 := 1M and we showby induction on n that ϕn is an isomorphism. Indeed, this holds for n = −1 by definition.Suppose that n ≥ 0 and that the assertion is known for all j < n; by a little diagram chasing(or the five lemma) we deduce that ϕn−1 induces an automorphism Im dn

∼→ Im dn, thereforeϕn ⊗A 1k : k ⊗A Ln → k ⊗A Ln is an automorphism (by minimality of L), so the same holdsfor ϕn (e.g. by looking at the determinant of ϕn).

For the general case, by standard arguments we construct a morphism of resolutions: ψ• :L → L′. By the foregoing case, ϕ• ψ• is an automorphism of L, so ϕ• is necessarily anepimorphism, and L′ decomposes as claimed. Finally, it is also clear that Kerϕ• is an acyclicbounded above complex of free A-modules, hence it is null homotopic.

Remark 4.1.35. (i) Suppose that A is a coherent local ring, and let L := (L•, d•, ε) and L′ :=(L′•, d

′•, ε′) be two minimal resolutions of the finitely presented A-module M . It follows easily

from lemma 4.1.34 that L and L′ are isomorphic as resolutions of M .(ii) Moreover, any two isomorphisms L → L′ are homotopic, hence the rule: M 7→ L•

extends to a functorA-Modcoh → Hot(A-Mod)

from the category of finitely presented A-modules to the homotopy category of complexes ofA-modules.

(iii) The sequence of A-modules (SyziAM := Im di | i > 0) is determined uniquely by M(up to non-unique isomorphism). The graded module Syz•AM is sometimes called the syzygy ofthe module M . Moreover, if L′′ is any other finite free resolution of M , then we can choose amorphism of resolutions L′′ → L, which will be a split epimorphism by lemma 4.1.34, and thesubmodule d•(L′′•) ⊂ L• decomposes as a direct sum of Syz•AM and a free A-module of finiterank.

Lemma 4.1.36. LetA→ B a faithfully flat homomorphism of coherent local rings,M a finitelypresented A-module. Then there exists an isomorphism of graded B-modules :

B ⊗A Syz•AM → Syz•B(B ⊗AM).

Proof. Left to the reader.

Proposition 4.1.37. Let A→ B be a flat and essentially finitely presented local ring homomor-phism of local rings, M an A-flat finitely presented B-module. Then the B-module M admits aminimal free resolution

Σ• : · · · d3−−→ L2d2−−→ L1

d1−−→ L0d−1−−−→ L−1 := M.


Moreover, Σ is universally A-exact, i.e. for every A-module N , the complex Σ• ⊗A N is stillexact.

Proof. To start out, let us notice :

Claim 4.1.38. In order to prove the proposition, it suffices to show that, for every A-flat finitelypresented B-module N , and every B-linear surjective map d : L → N , from a free B-moduleL of finite rank, Ker d is also an A-flat finitely presented B-module.

Proof of the claim. Indeed, in that case, we can build inductively a minimal resolution Σ• ofM , such that Ni := Ker(di : Li → Li−1) is an A-flat finitely presented B-module for everyi ∈ N. Namely, suppose that a complex Σ

(i)• with these properties has already been constructed,

up to degree i, and let κB be the residue field of B; by Nakayama’s lemma, we may find asurjection di+1 : Li+1 → Ni, where Li+1 is a free B-module of rank dimκB(Ni ⊗B κB). Underthe assumption of the claim, the resulting complex Σ

(i+1)• : (Li+1 → Li → Li−1 → · · · → M)

fulfills the sought conditions, up to degree i+ 1.It is easily seen that the complex Σ• thus obtained shall be universally A-exact. ♦

Let us write A as the union of the filtered family (Aλ | λ ∈ Λ) of its noetherian subalgebras.Say that B = Cp, for some finitely presented A-algebra C, and a prime ideal p ⊂ C, andM = Np for some finitely presented C-module N . We may find λ ∈ Λ, a finitely generatedAλ-algebra Cλ and a Cλ-module Nλ such that C = Cλ ⊗Aλ A, and N = Nλ ⊗Aλ A; for everyµ ≥ λ, let Cµ := Aµ ⊗Aλ Cλ and Nµ := Aµ ⊗Aλ Nλ; also, denote by pµ the preimage of p inCµ, and set Bµ := (Cµ)pµ , Mµ := (Nµ)pµ . According to [32, Ch.IV, Cor.11.2.6.1(ii)], we mayassume that Mµ is a flat Aµ-module, for every µ ≥ λ. Moreover, suppose that d : L → Mis a B-linear surjection from a free B-module L of rank r; then we may find µ ∈ Λ such thatd descends to a Bµ-linear surjection dµ : Lµ → Mµ from a free Bµ-module Lµ of rank r. Itfollows easily that Kµ := Ker dµ is a flat Aµ-module, for every µ ≥ λ, and the induced mapKµ ⊗Bµ B → K := Ker d is a surjection, whose kernel is a quotient of Tor

Aµ1 (A,Mµ)p; the

latter vanishes, since Mµ is Aµ-flat. Hence, K is A-flat; furthermore, Kµ is clearly a finitelygenerated Bµ-module, hence K is a finitely presented B-module. Then the proposition followsfrom claim 4.1.38.

4.2. Simplicial objects. In this section, we introduce the simplicial formalism, which providesthe language for the homotopical algebra of section 4.5.

Definition 4.2.1. Let C be any category, and k ∈ N any integer.

(i) We denote by ∆ the simplicial category, whose objects are the finite ordered sets :

[n] := 0 < 1 < · · · < n for every n ∈ N

and whose morphisms are the non-decreasing functions.(ii) ∆ is a full subcategory of the augmented simplicial category ∆∧, whose set of objects is

Ob(∆)∪∅, with Hom∆∧(∅, [n]) consisting of the unique mapping of sets ∅→ [n],for every n ∈ N. It is convenient to set [−1] := ∅.

(iii) The augmented k-truncated simplicial category ∆∧k, is the full subcategory of ∆∧

whose objects are the elements of Ob(∆∧) of cardinality≤ k+1. The k-truncated sim-plicial category is the full subcategory ∆k of ∆∧k whose set of objects is Ob(∆∧k )\∅.

(iv) A simplicial object (resp. an augmented simplicial object, resp. a k-truncated simpli-cial object, resp. a k-truncated augmented simplicial object) of C is a functor ∆o → C(resp. (∆∧)o → C , resp. ∆o

k → C , resp. (∆∧k )o). The morphisms of simplicial objectsof C are just the natural transformations (and likewise for the truncated or augmented


variants). Clearly, these objects form a category, and we use the notations.C := Fun(∆o,C ) s.C := Fun((∆∧)o,C )

sk.C := Fun(∆ok,C ) sk.C := Fun((∆∧k )o,C ).

(v) Dually, a cosimplicial object F • of C is a functor F : ∆ → C , or – which is thesame – a simplicial object in C o. Likewise one defines the truncated or augmentedcosimplicial variants, and we set

c.C := Fun(∆,C ) c.C := Fun(∆∧,C )

ck.C := Fun(∆k,C ) ck.C := Fun(∆∧k ,C ).

4.2.2. Notice the front-to-back involution :

(4.2.3) ∆→ ∆ : (α : [n]→ [m]) 7→ (α∨ : [n]→ [m]) for every n,m ∈ Ndefined as the endofunctor which induces the identity on Ob(∆), and such that :

α∨(i) := m− α(n− i) for every α ∈ Hom∆([n], [m]) and every i ∈ [n].

Another construction of interest is the endofunctor

γ : ∆→ ∆

given by the rule : [n] 7→ [n+ 1] for every n ∈ N, and which takes any morphism α : [n]→ [m]of ∆, to the morphism γ(α) : [n + 1] → [m + 1] which is the unique extension of α such thatγ(α)(n+ 1) := m+ 1. Notice that γ restricts to functors γk : ∆k+1 → ∆k for every k ∈ N.• Given a simplicial object F of C , one gets a cosimplicial object F o of C , by the (obvious)

rule : (F o)[n] := F [n] for every n ∈ N, and F o(α) := F (α)o for every morphism α in ∆.• Moreover, by composing a simplicial (resp. cosimplicial) object F (resp. G) with the

involution (4.2.3), one obtains a simplicial (resp. cosimplicial) object F∨ (resp. G∨). Likewise,given a morphism α : F1 → F2, the Godement product α∨ := α ∗ (4.2.3) is a morphismF∨1 → F∨2 .• For F and G as above, we may also consider the simplicial (resp. cosimplicial) object

γF := F γo (resp. γG := G γ), and this definition extends again to morphisms, by takingGodement products. The object γF (resp. γG) is called the path space of F (resp. of G). IfF is a (k + 1)-truncated simplicial object, then we can consider γkF := F γok, which is ak-truncated simplicial object (and likewise for truncated cosimplicial objects).

4.2.4. There is an obvious fully faithful functor :

C → s.C : A 7→ s.A ( resp. C → sk.C : A 7→ sk.A )

that assigns to each object A of C the constant simplicial object s.A (resp. constant truncatedsimplicial object sk.A) such that s.A[n] := A for every n ∈ N (resp. for every n ≤ k), ands.A(α) := 1A for every morphism α of ∆ (resp. of ∆k). Of course, we have as well augmentedvariants s.A and sk.A, and cosimplicial versions c.A, ck.A, c.A, ck.A.

Moreover, we have, for every integer k ∈ N, the k-truncation functor

s.trunck : s.C → sk.C ( resp. s.trunck : s.C → sk.C )

that assigns to any simplicial (resp. augmented simplicial) object F : ∆o → C (resp. F :(∆∧)o → C ) its composition with the inclusion functor ∆o

k → ∆o (resp. (∆∧k )o → (∆∧)o).Again, we have as well the corresponding cosimplicial versions c.trunck and c.trunck. Also,for every n ∈ N, we have the functor

•[n] : s.A → A A 7→ A[n].

Lastly, any functor ϕ : B → C induces functors

s.ϕ : s.B → s.C sk.ϕ : sk.B → sk.C : F 7→ ϕ F


and there are of course augmented variants s.ϕ and sk.ϕ, as well as the corresponding cosim-plicial versions.

4.2.5. For given n ∈ N, and every i = 0, . . . , n, let

εi : [n− 1]→ [n] (resp. ηi : [n+ 1]→ [n])

be the unique injective map in ∆∧ whose image misses i (resp. the unique surjective map in∆ with two elements mapping to i). The morphisms εi (resp. ηi) are called face maps (resp.degeneracy maps). By direct inspection, one checks that they fulfill the identities :

εj εi = εi εj−1 if i < j

ηj ηi = ηi ηj+1 if i ≤ j

ηj εi =

εi ηj−1 if i < j1 if i = j or i = j + 1εi−1 ηj if i > j + 1.

Example 4.2.6. (i) For instance, notice the identities :

ε∨i = εn−i η∨i = ηn−i for every n ∈ N and every i = 0, . . . , n.

(ii) For every r, s ∈ N, we set

εsr,0 := ε0 · · · ε0 : [r]→ [r + s].

This is the injective mapping whose image is s, . . . , r + s; therefore, its front-to-back dual

εs∨r,0 := εr+s · · · εr+1 : [r]→ [r + s]

is just the natural inclusion map. We shall also use the notation

εs−1,0 : [−1]→ [s] for every s ∈ Nfor the unique morphism ∅→ [s] in ∆∧; of course, we have εs∨−1,0 = εs−1,0 for every s ∈ N.

4.2.7. It is easily seen that every morphism α : [n] → [m] in ∆ admits a unique factorizationα = ε η, where the monomorphism ε is uniquely a composition of faces :

ε = εi1 · · · εis with 0 ≤ is ≤ · · · ≤ i1 ≤ m

and the epimorphism η is uniquely a composition of degeneracy maps :

η = ηj1 · · · ηjt with 0 ≤ j1 < · · · < jt ≤ m

(see [75, Lemma 8.1.2]). It follows that, to give a simplicial object A[•] of a category C , itsuffices to give a sequence of objects (A[n] | n ∈ N) of C , together with face operators

∂i := A[εi] : A[n]→ A[n− 1] i = 0, . . . , n

for every integer n > 0 and degeneracy operators

σi := A[ηi] : A[n]→ A[n+ 1] i = 0, . . . , n

for every n ∈ N, satisfying the following simplicial identities :

(4.2.8)

∂i ∂j = ∂j−1 ∂i if i < j

σi σj = σj+1 σi if i ≤ j

∂i σj =

σj−1 ∂i if i < j1 if i = j or i = j + 1σj ∂i−1 if i > j + 1.

Under this correspondence we have ∂i = A(εi) and σi = A(ηi) ([75, Prop.8.1.3]). Likewise, ak-truncated simplicial object of C is the same as the datum of a sequence (A[n] | n = 0, . . . , k)


of objects of C , and of a system of face and degeneracy operators restricted to this sequence ofobjects, and fulfilling the same identities (4.2.8).

Dually, a cosimplicial objectA[•] of C is the same as the datum of a sequence (A[n] | n ∈ N)of objects of C , together with coface operators

∂i : A[n− 1]→ A[n] i = 0, . . . , n

and codegeneracy operators

σi : A[n+ 1]→ A[n] i = 0, . . . , n

which satisfy the cosimplicial identities :

(4.2.9)

∂j ∂i = ∂i ∂j−1 if i < j

σj σi = σi σj+1 if i ≤ j

σj ∂i =

∂i σj−1 if i < j1 if i = j or i = j + 1∂i−1 σj if i > j + 1

and likewise for k-truncated cosimplicial objects.

4.2.10. An augmented simplicial object of a category C can be viewed as the datum of a sim-plicial object A[•] of C , together with an object A[−1] ∈ Ob(C ), and a morphism ε : A[0] →A[−1], which is an augmentation, i.e. such that :

ε ∂0 = ε ∂1.

Dually, an augmented cosimplicial object of C can be viewed as a cosimplicial object A[•],together with a morphism η : A[−1] → A[0] in C , such that ηo is an augmentation for Ao[•].We say that η is an augmentation for A[•].

Remark 4.2.11. Let A be a simplicial object of the category C .(i) For every n ∈ N we have γA[n] := A[n+ 1] (notation of (4.2.2)), and the face operators

γA[εi] : γA[n + 1] → γA[n] for i ≤ n + 1 (resp. degeneracy operators γA[ηi] : γA[n] →γA[n+ 1] for i ≤ n) of γA are ∂i : A[n+ 2]→ A[n+ 1] (resp. σi : A[n+ 1]→ A[n+ 2]); i.e.we drop ∂n+2 and σn+1. Likewise for the truncated variants.

(ii) The discarded faces ∂n+2 and degeneracies σn+1 can be used to produce natural mor-phisms

s.A[0]fA−−→ γA

gA−−→ A.

Namely, we set

fA[n] := σn · · · σ1 gA[n] := ∂n+1 for every n ∈ N.

For every k ∈ N, the same operation on an object A ∈ Ob(sk+1.C ) yields natural morphisms

s.trunckAfA−−→ γkA

gA−−→ s.trunckA.

(iii) Since there is a unique morphism σn,0 : [n]→ [0] in ∆ for every n ∈ N, it is easily seenthat the system (A[σn,0] : A[0]→ A[n]) defines a natural morphism

s.A[0]→ A in s.C .

(iv) Likewise, suppose ε : A[0] → A[−1] is an augmentation for A; since there is exactlyone morphism εn+1

−1,0 : ∅→ [n] in ∆∧ for every n ∈ N, we see that the system (A[εn+1−1,0] | n ∈ N)

defines a natural morphismA→ s.A[−1] in s.C .


Definition 4.2.12. Let C be any category. Denote by

ei : ∆o → [1]/∆o i = 0, 1

the functor that assigns to each [n] ∈ Ob(∆) the unique morphism [n]→ [1] of ∆ whose imageis i. Let also t : [1]/∆o → ∆o be the target functor (see (1.1.12)). Let A and B be twosimplicial objects of C , and f, g : A→ B two morphisms.

(i) A homotopy from f to g is the datum of a natural transformation

u : A t⇒ B t

such that u ∗ e0 = f and u ∗ e1 = g.(ii) IfA[•] is an augmented simplicial object of C , with augmentation given by a morphism

ϕ : A → s.A[−1] in s.C , then we say that A is homotopically trivial, if there existsa morphism ψ : s.A[−1] → A and homotopies u and v, respectively from 1s.A[−1] toϕ ψ, and from 1A to ψ ϕ.

(iii) Dually, ifC andD are cosimplicial objects of C , and p, q : C → D any two morphismsin c.C , then a homotopy from p to q is a homotopy from po to qo in s.C o. Likewise wedefine homotopically trivial cosimplicial objects.

Remark 4.2.13. (i) In the situation of definition 4.2.12, suppose that p : A′ → A and q : B →B′ are any two morphisms in s.C ; then the natural transformation

(q ∗ t) u (p ∗ t) : A′ t⇒ B′ t

is a homotopy from f p to q g. Moreover, if F : C → D is any functor, then

F ∗ u : FA ∗ t⇒ FB ∗ t

is a homotopy from Ff to Fg.(ii) However, unlike the case for chain homotopies, simplicial homotopies cannot be com-

posed in this generality; hence, the simplicial category s.C cannot be made into a 2-category,by taking the homotopies as 2-cells.

(iii) In the same vein, for a general category C , the relation “there exists a homotopy fromf to g” on morphisms of s.C is neither symmetric nor transitive (though it will follow fromtheorem 4.2.58 that this is an equivalence relation, in case C is abelian).

4.2.14. Notice that there are exactly n + 1 morphisms ϕ : [n] → [1] for every [n] ∈ Ob(∆),and they can be labeled by the cardinality of ϕ−1(0) : for every n ∈ N and every k ≤ n+ 1, weshall write ϕn,k : [n] → [1] for the unique morphism such that ϕ−1

n,k(0) has cardinality k. Withthis notation, notice that

ϕn,k εi =

ϕn−1,k if i ≥ kϕn−1,k−1 if i < k

and ϕn,k ηi =

ϕn+1,k if i ≥ kϕn+1,k+1 if i < k.

Hence, a homotopy u from f to g as in definition 4.2.12, is the same as a system of morphisms

un,k : A[n]→ B[n] for every n ∈ N and every k ≤ n+ 1

such that un,n+1 = f [n] and un,0 = g[n] for every n ∈ N, and the diagrams

A[n]un,k //

∂i

B[n]

∂i

A[n]un,k //

σi

B[n]

σi

A[n− 1]un−1,k−a // B[n− 1] A[n+ 1]

un+1,k+a // B[n+ 1]

commute for every n ∈ N and every k ≤ n+ 1, where a := 0 if i ≥ k, and a := 1 if i < k.


4.2.15. A bisimplicial object in a category C is an object of the category s.(s.C ). The lattercan also be regarded as the category of all functors ∆o×∆o → C ; it follows that a bisimplicialobject of C is the same as a system (A[p, q] | (p, q) ∈ N × N) of objects of C , together withmorphisms

A[α, β] : A[p, q]→ A[p′, q′] for all morphisms α : [p′]→ [p], β : [q′]→ [q] of ∆

compatible with compositions of morphisms in ∆, in the obvious way. More generally, wemay define inductively the category of n-simplicial objects sn.A , for every n ∈ N, by lettings0.A := A , and sn.A := s.(sn−1.A ), for every n > 0. The diagonal functor

∆→ ∆×∆ [n] 7→ ([n], [n]) α 7→ (α, α) for all n ∈ N and all morphisms α of ∆

induces a functor∆C : s2.C → s.C A 7→ A∆.

Especially, we have A∆[n] := A[n, n] for every n ∈ N, and the face operators ∂i on A∆[n] areof the form A[εi, εi], for every i = 0, . . . , n (and likewise for the degeneracies). Also, the flipfunctor

∆×∆→ ∆×∆ ([m], [n]) 7→ ([n], [m])

induces an endofunctorfl : s2.A → s2.A

in the obvious way. Furthermore, the endofunctors

∆×∆γ×1∆−−−−→ ∆×∆

1∆×γ←−−−− ∆×∆

induce functorss2.A

γ1←−− s2.Aγ2−−→ s2.A

that admit descriptions as in remark 4.2.11(i). Correspondingly, we get natural morphisms

g(i)A : γiA→ A for i = 1, 2 and every A ∈ Ob(s2.A )

as in remark 4.2.11(ii).

Remark 4.2.16. (i) Let C be a category whose finite coproducts are representable. Let alsof .Set be the category of finite sets. To every object S of s.f .Set and every X ∈ Ob(s.C ), weattach a bisimplicial object S X of C as follows. For every n,m ∈ N, we let S X[n,m]be the coproduct of finitely many copies of X[m], indexed by the elements of S[n]; hence, forevery a ∈ S[n] we have a natural morphism ia : X[m] → S X[n,m]. If ϕ : [n] → [n′] andψ : [m]→ [m′] are any two morphisms in ∆o, we let SX[ϕ, ψ] : SX[n,m]→ SX[n′,m′]be the unique morphism such that SX[ϕ, ψ] ia = iS[ϕ](a) X[ψ] for every a ∈ S[n]. Clearly,this rules extends to a well defined functor

s.f .Set× s.C → s2.C (S,X) 7→ S X.

Likewise, we define X S := fl(S X) (notation of (4.2.15)). If all coproducts of C arerepresentable, we may even extend the above construction to arbitrary simplicial sets.

(ii) In the same vein, let A be any abelian category, and M any object of s.Z-Modfg (nota-tion of (1.2.43)). For any A ∈ Ob(s.A ), we may define a bisimplicial object M Z A of A ,by the rule : [n,m] 7→ M [n] ⊗Z A[n] for every n,m ∈ N and [ϕ, ψ] 7→ M [ϕ] ⊗Z A[ψ] for allmorphisms ϕ, ψ of ∆ (where these mixed tensor products are as defined in (1.2.43)). Clearlythese rules yield a well defined functor

s.Z-Modfg × s.A → s2.A (M,A) 7→M Z A.

Likewise, we set AZ M := fl(M Z A).


(iii) Furthermore, if (C ,⊗) is any tensor category, and X, Y any two simplicial objects ofC , we may define a bisimplicial object X Y , by the same rule as in (ii). This yields a functor

s.C × s.C → s2.C (X, Y ) 7→ X Y.

In this situation (resp. in the situation of (i), resp. of (ii)), we shall let also

X ⊗ Y := (X Y )∆ (resp. S ⊗X := (S X)∆, resp. M ⊗Z A := (M Z A)∆ )

which we shall call the tensor product of X and Y (resp. of S and X , resp. of M and A).Notice the natural identifications

S A∼→ (S ⊗ s.Z)Z A for every S ∈ Ob(s.f .Set) and every A ∈ Ob(s.A ).

Likewise, if U is any unit object for (C ,⊗), we get natural identifications

S X∼→ (S ⊗ s.U)X for every S ∈ Ob(s.f .Set) and every X ∈ Ob(s.C )

via the isomorphisms uX : X∼→ U ⊗X provided by proposition 1.2.6.

(iv) Notice also that, if f, g : S → T are any two morphisms of simplicial finite sets, then– in light of remark 4.2.13(i) – any homotopy u from f to g induces a homotopy u ⊗ X fromf ⊗X to g⊗X . Moreover, for every k ∈ N, let ∆k be the simplicial finite set given by the rule

∆k[n] := Hom∆([n], [k]) and ∆k[ϕ] := Hom∆(ϕ, [k])

for every n ∈ N and every morphism ϕ in ∆. Especially, ∆0 is the constant simplicial setassociated to the set with one element, and therefore ∆0 ⊗ X = X for every X ∈ Ob(C ).Also, the rules : [n] 7→ (ei[n] : [n] → [1]) (for i = 0, 1; here ei is the functor introduced indefinition 4.2.12) define morphisms

e∗i : ∆0 →∆1 i = 0, 1

and we notice that the datum of a homotopy u from f to g as in definition 4.2.12(i), is the sameas that of a morphism

u : ∆1 ⊗ A→ B such that u (e∗0 ⊗ A) = f and u (e∗1 ⊗ A) = g.

Indeed, given u, we construct u as follows. For every n ∈ N and every ϕ ∈ ∆1[n], let u[n]be the unique morphism such that u[n] iϕ = uϕ. The naturality of u easily implies thatthis rule amounts to a morphism u as sought. Conversely, given u, we can construct a naturaltransformation u, by reversing the foregoing rule.

Remark 4.2.17. (i) Let (C ,⊗) be a tensor category with internal Hom functor Hom, and unitobject U . To every two objects X, Y of s.C , we may attach an object

Hom(X, Y ) of c.s.C

as follows. For every n,m ∈ N, we let Hom(X, Y )[n,m] := Hom(X[n], Y [m]), and for ev-ery two morphisms ϕ, ψ of ∆, we define Hom(X, Y )[ϕ, ψ] := Hom(X[ϕ], Y [ψ]). Since thisis a mixed simplicial-cosimplicial object, we cannot extract a diagonal object from it; however,if C is complete, we can at least define

Homs.C (X, Y ) := Equal(∏

n∈N Hom(X, Y )[n, n]d1 //

d0

//∏

ϕ:[n]→[m] Hom(X, Y )[m,n])

where the second product ranges over the morphisms ϕ of ∆, and where

d0 :=∏

ϕ:[n]→[m]

Hom(X[ϕ], Y [n]) and d1 :=∏

ϕ:[n]→[m]

Hom(X[m], Y [ϕ]).

Arguing as in example 4.1.8(vi), it is easily seen that there are natural isomorphisms

HomC (Z,Homs.C (X, Y ))∼→ Homs.C (s.Z ⊗X, Y )


for every Z ∈ Ob(C ) and every X, Y ∈ Ob(s.C ).(ii) Suppose additionally, that all finite coproducts of C are representable. For every Z ∈

Ob(C ), and every X ∈ Ob(s.C ), consider the simplicial set

HomC (Z,X)

such that HomC (Z,X)[n] := HomC (Z,X[n]) for every n ∈ N, with face and degeneraciesdeduced from those of X , in the obvious way. For any k ∈ N, let also ∆k be the simplicial setdefined in remark 4.2.16(iv); we have natural isomorphisms

HomC (Z,Homs.C (∆k ⊗ s.U,X))∼→Homs.C (∆k ⊗ s.Z,X)∼→Homs.Set(∆k,HomC (Z,X))∼→HomC (Z,X)[k]

where the last isomorphism follows from Yoneda’s lemma (proposition 1.1.20(ii) : details leftto the reader). Applying again Yoneda’s lemma, we deduce a natural isomorphism

Homs.C (∆k ⊗ s.U,X)∼→ X[k] for every k ∈ N.

(iii) Notice that every morphism ϕ : [k]→ [k′] in ∆ induces a morphism

∆ϕ := Hom∆(−, ϕ) : ∆k →∆k′

and clearly ∆ψ ∆ϕ = ∆ψϕ, if ψ : [k′]→ [k′′] is any other morphism of ∆. Hence, the system(∆i | i ∈ N) amounts to an object of c.s.Set. Moreover, since the Yoneda isomorphisms arenatural in both X and ∆k, we get a commutative diagram

Homs.C (∆k′ ⊗ s.U,X)∼ //

Homs.C (∆ϕ⊗s.U,X)

X[k′]

X[ϕ]

Homs.C (∆k ⊗ s.U,X)

∼ // X[k]

for every morphism ϕ as above.(iv) Notice as well that the considerations of (ii) and (iii) can be repeated, mutatis mutandi,

for truncated simplicial objects : if X is an object of sn.C , then HomC (Z,X) shall be an objectof sk.Set, and we shall have natural isomorphisms

Homsn.C (s.truncn(∆k ⊗ s.U), X)∼→ X[k] for every k ≤ n

and similarly for the commutative diagrams of (iii) (details left to the reader).

4.2.18. Let C be a category with small Hom-sets, and suppose that all finite colimits of C arerepresentable. Then proposition 1.1.34 and remark 1.1.38(v) say that, for every integer k ∈ N,the k-truncation functor on s.C admits a left adjoint

skk : sk.C → s.C

which is called the k-th skeleton functor. By inspecting the proof of loc.cit. we see that, forevery k-truncated simplicial object F , this adjoint is calculated by the rule :

(4.2.19) skkA[n] := colimϕ:[i]→[n]

F [i]

where i ranges over all the integers ≤ k, and ϕ over all the morphisms [i]→ [n] in ∆o, and thetransition maps F [i] → F [j] in the colimit are the morphisms F [ψ] given by all commutative


triangles

(4.2.20)

[i]ψ //

ϕ @@@@@@@@[j]

ϕ~~

[n].

A morphism α : [n]→ [m] in ∆o induces a morphism skkF [α] : skkA[n]→ skkA[m]; namely,for every ϕ : [i]→ [m] one has a natural morphism jϕ : F [i]→ coskkF [m], and skkF [α] is thecolimit of the system of morphisms

jαϕ : F [i]→ skkA[m] for all ϕ : [i]→ [n].

It is clear that, for every n,m ≤ k and every α : [n] → [m], the colimit (4.2.19) is realized byF [n], and under this identification, skkA[α] agrees with F [α], so the unit of adjunction

F → s.trunck skkF

is an isomorphism. Dually, if all finite limits are representable in C , the truncation functoradmits a right adjoint

coskk : sk.C → s.C

called the k-th coskeleton functor, and a simple inspection of the proof of loc.cit. yields therule:

coskkF [n] := limϕ:[n]→[i]

F [i]

where i ranges over the integers ≤ k, and ϕ : [n] → [i] over the morphisms in ∆o, and thetransition maps are as in the foregoing (except that the downwards arrows in the commutativetriangles (4.2.20) are reversed). Especially, we easily deduce that the counit of adjunction

s.trunck coskkF → F

is an isomorphism. Moreover, if B is another category with small Hom-sets, whose finitecolimits (resp. finite limits) are all representable, and ϕ : B → C is any functor, then for anyF ∈ Ob(sk.B) there is a natural transformation

skk(sk.ϕF )→ s.ϕ(skkF ) ( resp. s.ϕ(coskkF )→ coskk(sk.ϕF ) )

which is an isomorphism, if ϕ is right exact (resp. if ϕ is left exact).

4.2.21. Let A be an abelian category, and A any object of s.A . For every n > 0, set

dn :=n∑i=0

(−1)i · ∂i A[n]→ A[n− 1].

Directly from the simplicial identities (4.2.8) we may compute

dn dn+1 =n∑i=0

n+1∑j=0

(−1)i+j · ∂i ∂j

=n∑i=0

n+1∑j>i

(−1)i+j · ∂j−1 ∂i +n∑i=0

∂i ∂i +n∑i=0

n∑j<i

(−1)i+j · ∂i ∂j

=n∑i=0

n+1∑j−1>i

(−1)i+j · ∂j−1 ∂i +n∑i=0

n∑j<i

(−1)i+j · ∂i ∂j

= 0

for every n ∈ N, so we are led to the following :


Definition 4.2.22. Let A be an abelian category, and k ∈ N any integer.(i) If A[•] is any simplicial object of A , with face operators ∂i, the unnormalized complex

associated to A[•] is the complex (A•, d•) ∈ Ob(C≤0(A )) such that An := A[n] forevery n ∈ N, and dn is defined as in (4.2.21), for every n > 0.

(ii) If A[•] is a k-truncated simplicial object of A , the unnormalized complex associatedto A[•] as the complex (A•, d•) where An and dn are defined as in (i) for every n ≤ k,and An := 0, dn := 0 for every n > k.

(iii) If A[•] is as in (i) (resp. as in (ii)), the normalized complex associated to A[•] is thesubcomplex N•A of A• such that

N0A := A[0] and NnA :=n⋂i=1

Ker ∂i for every n > 0 (resp. for 0 < n ≤ k).

So, the differential NnA→ Nn−1A equals ∂0, for every n ∈ N (resp. for every n ≤ k).(iv) If A[•] is as in (i) or (ii), the homology of A in degree n is

HnA := HnA• for every n ∈ N.(v) Let ε : A→ A−1 be an augmentation for A[•]. One says that the augmented simplicial

object (A, ε) is aspherical, if HnA = 0 for every n > 0, and ε induces an isomorphismH0A

∼→ A−1.

4.2.23. Let A be an abelian category, and recall that sn.A and sk.A are both abelian categoriesas well, for every n, k ∈ N (remark 1.2.36(ii)); also, clearly the rule of definition 4.2.22(iii)yields natural additive functors

NA : s.A → C≤0(A ) NA ,k : sk.A → C[−k,0](A ) A[•] 7→ N•A for every k ∈ Nand the rules of definition 4.2.22(i,ii) yield additive functors

UA : s.A → C≤0(A ) UA ,k : sk.A → C[−k,0](A ) A[•] 7→ A• for every k ∈ N.Remark 4.2.24. Let A and A be as in (4.2.21); directly from the simplicial identities (4.2.8)we may compute

dn σk =k−1∑i=0

(−1)i · σk−1 ∂i +n∑

i=k+2

(−1)i · σk ∂i−1.

Especially, if we let

D0 := 0 and DnA :=n−1∑i=0

Im (σi : A[n− 1]→ A[n]) for every n > 0

we see that dn restricts to a morphism d′n : DnA → Dn−1A for every n > 0, hence (D•A, d′•)

is a subcomplex of A•, called the degenerate subcomplex, and clearly we obtain an additivefunctor

DA : s.A → C(A ) A 7→ D•A.

Proposition 4.2.25. The natural injections induce a decomposition

A• = N•A⊕D•A in C≤0(A ).

Proof. First, we notice that NnA ∩DnA = 0 for every n ∈ N. Indeed, it suffices to check thatNnA ∩ Im(σi) = 0 for every i = 0, . . . , n − 1, but the latter follows easily from the identity∂i+1 σi = 1A[n−1]. To conclude the proof, it suffices to show that NnA + DnA = An forevery n ∈ N. Indeed, set K0 := An and define inductively Ki := Ki−1 ∩ Ker ∂i for everyi = 1, . . . , n; to prove the latter assertion, it suffices to check that

(4.2.26) (1An − σi · ∂i)(Ki) ⊂ Ki+1 for every i = 0, . . . , n.


However, we have :

∂j (1An − σi · ∂i) = ∂j − σi−1 ∂i−1 ∂j whenever j < i

and ∂i (1An − σi · ∂i) = ∂i − ∂i = 0, whence (4.2.26).

Example 4.2.27. (i) Let S be any simplicial set, and set ZS := S ⊗ s.Z (notation of remark4.2.16(iii)); we wish to give an explicit description of the complexN•ZS . To begin with, noticethat ZS[n] is the free abelian group with basis indexed by S[n], for every n ∈ N, and for everyx ∈ S[n] denote by ex ∈ ZS[n] the corresponding basis element. Next, set

DnS :=n−1⋃i=0

Im (σi : S[n− 1]→ S[n]) and NnS := S[n] \DnS for every n ∈ N.

From proposition 4.2.25, we see that NnZS can be naturally identified with the direct summandof ZS[n] generated by the system (ex | x ∈ NnS), for every n ∈ N. In order to describe thedifferential dn, let us define ex := ex if x ∈ NnS, and ex := 0 if x ∈ DnS. Then we may write

dn(ex) =n∑i=0

(−1)i · e∂ix for every n ∈ N and every x ∈ NnS.

The verifications are straightforward, and shall be left to the reader.(ii) For instance, for any i ∈ N consider the simplicial set ∆i as in remark 4.2.16(iv), and

defineK〈i〉• := N•Z∆i

where Z∆i is defined as in (i). Notice that this notation agrees with that of remark 4.1.11(ii). Itis easily seen that a morphism [n]→ [i] of ∆ lies inDn∆i if and only if it is not injective; henceNn∆i is the set of all injective maps ϕ : [n]→ [i] of ∆. We deduce a natural isomorphism

K〈i〉n∼→ Λn+1

Z Z⊕i+1 for every i, n ∈ N.Namely, to any map ϕ as above, we assign the exterior product eϕ(0) ∧ · · · ∧ eϕ(n), wheree0, . . . , ei denotes the canonical basis of Z⊕i+1. Under this isomorphism, the differential ofK〈i〉• gets identified with the differential of the Koszul complex attached to the sequence1i+1 := (1, . . . , 1) ∈ Z⊕i+1 (see example 4.4.48). Summing up, we obtain natural short ex-act sequences

0→ K〈i〉•[1]→ K•(1i+1)→ Z[0]→ 0 for every i ∈ Nwhere [1] denotes the shift operator, and Z[0] is the complex with Z placed in degree zero : see(4.1.1).

(iii) Let A be any abelian category, A any object of s.A and Z any object of A . Noticethat the simplicial set HomA (Z,A) defined in remark 4.2.17(ii) is actually a simplicial abeliangroup, and a direct inspection of the arguments of loc.cit. yields a natural isomorphism ofabelian groups

Homs.Z-Mod(Z∆i ,HomA (Z,A))∼→ HomA (Z,A[i]) for every i ∈ N.

4.2.28. Keep the notation of remark 4.2.24, and let

jA• : N•A→ A• qA• : A• → N•A

be respectively the injection and the projection with kernel D•A. Then the rules : A 7→ jA• andA 7→ qA• define natural transformations

j• : NA ⇒ UA q• : UA ⇒ NA .

With this notation, we may state :

Theorem 4.2.29. With the notation of remark (4.2.28), we have :


(i) The injection jA• is a homotopy equivalence, and D•A is homotopically trivial.(ii) More precisely, there exist natural modifications

j• q• 1UAq• j• 1NA

on the 2-category C(A ) as in example 4.1.7(ii) (see definition 1.3.7).(iii) Especially, the natural map

HiN•A→ HiA

is an isomorphism, for every i ∈ N.

Proof. Clearly (ii)⇒(i), and (i)⇒(iii), by virtue of remark 4.1.3(ii). To show (ii), set ϕA0 := 1A0

and ϕAn := 1A[n] − σn−1 ∂n : An → An for every n > 0. We notice :

Claim 4.2.30. The system (ϕAn | n ∈ N) defines an endomorphism of A•, which is homotopi-cally equivalent to 1A• .

Proof of the claim. Indeed, let sn := (−1)n · σn : An → An+1 for every n ∈ N, and definesn := 0 for every n < 0. Using the simplicial identities (4.2.8) we compute :

sn−1 dn + dn+1 sn = (−1)n−1 ·n∑j=0

(−1)j · (σn−1 ∂j − ∂j σn)− 1A[n]

=ϕAn + (−1)n−1 · (n−1∑j=0

(−1)j · (σn−1 ∂j − σn−1 ∂j)− 1A[n]

=ϕAn − 1[n]

for every n > 0, and for n = 0 we have as well s−1 d0 + d1 s0 = 0, whence the claim. ♦Notice that the homotopy exhibited in the proof of claim 4.2.30 is natural in A, so it already

yields the first sought modification. Next, for every simplicial object B of A , let γB andgB : γB → B be as in remark 4.2.11(i,ii), and define inductively

β0A := A and βn+1A := Ker gβnA for every n ∈ N.A simple inspection shows that

Nn+iA ⊂ βnA[i] and βnA[0] = NnA for every i, n ∈ N.

Hence, we may define a subcomplex B(n)• of A• for every n ∈ N, as follows. For i = 0, . . . , n

we let B(n)i := NiA, and for i > n we let B(n)

i := (βnA)[n + i] (notation of (4.1.1)). Thedifferential of B(n)

• is of course just the restriction of that of A•. By construction, N•A is asubcomplex of B(n)

• , for all n ∈ N. Next, we define an endomorphism f(n)• of B(n)

• , by setting

f(n)i := 1

B(n)i

for every i ≤ n, and f(n)i := ϕβ

nAi for i > n.

Notice that the restriction of f (n)• to the subcomplex N•A is just the inclusion map N• → B

(n)• .

Clearly B(n+1) ⊂ B(n) for every n ∈ N, and we remark that f (n)• factors through the inclusion

map B(n+1) ⊂ B(n). Indeed, since B(n)i = B

(n+1)i for every i ≤ n, the assertion is obvious

for this range of degrees; so we have only to check that ϕβnA• factors through βn+1A• for everyn ∈ N, and an easy induction reduces to checking that ϕA• factors through B(1)

• . But the latterassertion comes down to the identity ∂i ϕAi = 0 for every i > 0, which follows easily from thesimplicial identities (4.2.8).

If (si | i ∈ N) is the homotopy between 1βnA• and ϕβnA• supplied by claim 4.2.30, then weobtain a homotopy (t

(n)i | i ∈ N) between 1

B(n)•

and f (n)• , by setting

t(n)i := 0 for every i < n, and t

(n)i := si−n for i ≥ n.


Notice that Im t(n)i ⊂ Di+1A for every i, n ∈ N. Thus, for every n ∈ N, let h(n)

• : B(n)• →

B(n+1)• be the morphism of complexes deduced from f

(n)• and j(n)

• : B(n) → A• the inclusionmap, and set q(n)

• := h(n)• · · · h(0)

• ; it follows easily that the composition

p(n)• := j(n+1)

• q(n)• : A• → A•

is homotopically equivalent to 1A• , for every n ∈ N. More precisely, a direct inspection showsthat the system of morphisms

τ(n)i :=

n−1∑k=0

j(k+1)• t(k+1)

• q(k)• for every i ∈ N

provides a homotopy between 1A• and p(n)• . Furthermore, it is clear that

p(n)i = p

(m)i and τ

(n)i = τ

(m)i for every m ≥ n and every i ≤ n

so, we finally get an endomorphism p• : A• → A• by setting pi := p(i)i for every i ∈ N, and a

homotopy τ• between p• and 1A• , with τi := τ(i)i for every i ∈ N. By construction, p• factors

through jA• , and moreover, the restriction of p• to the subcomplex N•A is just jA• , so jA• is ahomotopy equivalence, via natural homotopies, as stated.

Lastly, notice that Im τi ⊂ Di+1A for every i ∈ N, from which it follows easily thatpi(DiA) ⊂ DiA for every i ∈ N, and then the foregoing implies that Ker p• = D•A. Weconclude that D•A is homotopically trivial, as stated.

4.2.31. By iterating U, we get a functor from bisimplicial objects to double complexes

U2A : s2.A

s.UA−−−−→ s.C(A )UC(A )−−−−→ C(C(A )) A[•, •] 7→ A••

and notice that this functor is naturally isomorphic to the functor

s2.AUs.A−−−−→ C(s.A )

C(UA )−−−−−→ C(C(A )).

In the same vein, theorem 4.2.29 yields natural decompositions of additive functors :

(4.2.32) U2A = (C(NA )⊕ C(DA )) (Ns.A ⊕ Ds.A )

and if we letN2

A := C(NA ) Ns.A : s2.A → C(C(A )) A 7→ N••A

the natural morphismiA• : TotN••A→ TotA••

is a homotopy equivalence.

4.2.33. We wish next to exhibit two natural transformations

A∆•

AWA•−−−→ TotA••

ShA•−−−→ A∆• for every A ∈ Ob(s2.A ).

Namely, for every n ∈ N, define :• ShAn as the sum, for all p, q ∈ N such that p+ q = n, of the shuffle maps

ShAp,q :=∑µν

εµν · A[ην1 · · · ηνq , ηµ1 · · · ηµp ] : Ap,q → An,n

where the sum ranges over the shuffle permutations (µ, ν) of type (p, q) of the set0, . . . , p + q − 1 (these are the permutations described in [36, §4.3.15]), and εµν isthe sign of the permutation (µ, ν)• AWA

n as the sum, for all p, q ∈ N such that p+ q = n, of the Alexander-Whitney maps

AWAp,q := A[εq∨p,0, ε

pq,0] : An,n → Ap,q


(notation of example 4.2.6(ii)). Clearly, for every p, q ∈ N the rule A 7→ Ap,q defines a functor

•[p, q] : s2.A → A

and the maps ShAp,q and AWAp,q yield natural transformations

Shp,q : •[p, q]⇒ •[p+ q, p+ q] AWp,q : •[p+ q, p+ q]⇒ •[p, q].

Proposition 4.2.34. With the notation of (4.2.23), the sequence (ShAn | n ∈ N) defines a mor-phism of chain complexes

ShA• : TotA•• → A∆• for every A ∈ Ob(s2.A ).

Proof. For any bisimplicial object A of A , set A−1,q = Ap,−1 := 0 for every p, q ∈ Z, and let

A[ε0,1[q]] : A0,q → A−1,q A[1[p], ε0] : Ap,0 → Ap,−1 A[ε0, ε0] : A0,0 → A−1,−1

be the zero maps; likewise, let ShA−1,q : A−1,q → Aq−1,q−1 and ShAp,−1 : Ap,−1 → Ap−1,p−1 bethe zero maps, and notice that also ShA0,0 is the zero map. We define

dA,hp,q :=

p∑i=0

(−1)i ·A[εi,1[q]] : Ap,q → Ap−1,q dA,vp,q :=

q∑i=0

(−1)j ·A[1[p], εi] : Ap,q → Ap,q−1

so, for every n ∈ N the differential dn in degree n of TotA•• is the sum of the maps

dA,hp,q + (−1)p · dA,vp,q : Ap,q → Ap−1,q ⊕ Ap,q−1 for all p, q ∈ N such that p+ q = n

whereas the differential of A∆• is the morphism

dAn :=n∑i=1

A[εi, εi] : An,n → An−1,n−1

and we have to check the identity

(4.2.35) dAp+q ShAp,q = ShAp−1,q dA,hp,q + (−1)p · ShAp,q−1 dA,vp,q for all p, q ∈ N.

Now, we set B := γ1A, C := γ2A, D := γ2B (notation of (4.2.15)), but for the purpose ofthis proof, we shall modify the differentials of the double complexes B•• and C•• in certain lowdegrees : namely, we define

dB,h0,q := A[ε0,1[q]] dC,vp,0 := A[1[p], ε0] for every p, q ∈ N.

Claim 4.2.36. With the foregoing notation, the following holds :

(i) ShAp,q = (−1)q · ShDp−1,q A[1[p], ηq] + ShDp,q−1 A[ηp,1[q]] for every p, q ∈ N.(ii) A[εp+q−1, εp+q−1] ShDp−1,q−1 = ShAp−1,q−1 A[εp, εq] for every p, q > 0.

(iii) A[εp+q, εp+q] ShAp,q = (−1)q · ShAp−1,q A[εp,1[q]] + ShAp,q−1 A[1[p], εq] for everyp, q ∈ N.

Proof of the claim. (i): First, we notice the identities :

(4.2.37)A[1[p+q], ηp+q] ShAp,q = ShCp,q A[1[p], ηq]

A[ηp+q,1[p+q]] ShAp,q = ShBp,q A[ηp,1[q]].for every p, q ∈ N

that are deduced from the identitiesηµ1 · · · ηµp ηp+q = ηq ηµ1 · · · ηµpην1 · · · ηνq ηp+q = ηp ην1 · · · ηνq


which in turn follow from the simplicial identities for degeneracy maps. By applying the first(resp. second) identity (4.2.37) with A replaced by B (resp. by C) and p replaced by p − 1(resp. with q replaced by q − 1) we get :

(4.2.38)A[1[p+q], ηp+q−1] ShBp−1,q = ShDp−1,q A[1[p], ηq]

A[ηp+q−1,1[p+q]] ShCp,q−1 = ShDp,q−1 A[ηp,1[q]].for every p, q ∈ N.

Now, suppose first that p, q > 0, and let (µ, ν) be any (p, q)-shuffle of 0, . . . , p+ q − 1; theneither µp = p+ q−1 or νq = p+ q−1. In the first (resp. second) case, after removing µp (resp.νq) we get a (p− 1, q)-shuffle (resp. a (p, q − 1)-shuffle) (µ, ν) with

εµν = (−1)q · εµν (resp. εµν = εµν).

However, the first (resp. second) left-hand side of (4.2.38) contains precisely all the terms ofthe first (resp. second) type occurring in the definition of ShAp,q, so we get (i) in this case. Thecases where either p = 0 or q = 0 can be dealt with by a similar, but simpler, argument.

(ii) follows by naturality of Shp−1,q−1, applied to the morphism g(1)A g

(2)B : D → A from

(4.2.15).(iii): The case where p = q = 0 is obvious, and the other cases follow by composing both

sides of (i) with A[εp+q, εp+q], applying (ii), and recalling that ηq εq = 1[q] : details left to thereader. ♦

Now, a simple inspection shows that (4.2.35) follows from claim 4.2.36(iii) and the following

Claim 4.2.39. dDp+q−1 ShAp,q = ShAp−1,q dB,hp−1,q + (−1)p · ShAp,q−1 d

C,vp,q−1 for every p, q ∈ N.

Proof of the claim. Consider first the case where p = 0. If q ≤ 1, it is easily seen that bothsides of the stated identity vanish; if q ≥ 2, the left-hand side is

dDq−1 A[η0 · · · ηq−1,1[q]] =

q−1∑i=0

(−1)i · A[η0 · · · ηq−1 εi, εi]

and the right-hand side is

A[η0 · · · ηq−2,1[q]] dC,v0,q−1 =

q−1∑i=0

(−1)i · A[η0 · · · ηq−2, εi]

so in this case the assertion comes down to the obvious identity

η0 · · · ηq−1 εi = η0 · · · ηq−2 : [q − 2]→ [0] for every i = 0, . . . , q − 1.

Likewise we deal with the case where q = 0. It follows already that (4.2.35) holds for thesevalues of (p, q), and for every bisimplicial object A of A ; especially, we get

(4.2.40) dDq ShD0,q = ShD0,q−1 dD,v0,q for every q ∈ N.

Next, for p = q = 1, a direct computation shows that both sides equal

A[ε0 η0,1[1]]− A[1[1], ε0 η0] : A1,1 → A1,1.

We prove now, by induction on q, that the assertion holds for p = 1. This is already known forq ≤ 1, so suppose that r > 1, and that the assertion holds for p = 1 and every q < r. The latterimplies that also (4.2.35) holds for these values of (p, q), and for every bisimplicial object A ofA ; especially, we get

(4.2.41) dDr ShD1,r−1 = ShD0,r−1 dD,h1,r−1 − ShD1,r−2 d

D,v1,r−1.

On the other hand, claim 4.2.36(i) says that

dDr ShA1,r = (−1)r · dDr ShD0,r A[1[1], ηr] + dDr ShD1,r−1 A[η1,1[r]]


Combining with (4.2.40) and (4.2.41) we obtain

dDr ShA1,r = (−1)r ·ShD0,r−1 dD,v0,r A[1[1], ηr]+(ShD0,r−1 d

D,h1,r−1−ShD1,r−2 d

D,v1,r−1)A[η1,1[r]].

However, we have

dD,h1,r−1 A[η1,1[r]] = A[η1 ε0,1[r]]− A[1[1],1[r]] = A[ε0 η0,1[r]]− 1A[1,r]

so we can rewrite dDr ShA1,r = ϕ1 + ϕ2 − ϕ3, where

ϕ1 := ShD0,r−1 ((−1)r · dD,v0,r A[1[1], ηr]− 1A[1,r]) = (−1)r · ShD0,r−1 A[1[1], ηr−1] dC,v1,r−1

ϕ2 := ShD0,r−1 A[ε0 η0,1[r]]

ϕ3 := ShD1,r−2 dD,v1,r−1 A[η1,1[r]].

On the other hand, using claim 4.2.36(i), we can compute

ShA0,r dB,h0,r−1 = ShD0,r−1 A[η0,1[r]] dB,h0,r−1 = ϕ2

ShA1,r−1 dC,v1,r−1 = ((−1)r−1 · ShD0,r−1 A[1[1], ηr−1] + ShD1,r−2 A[η1,1[r−1]]) dC,v1,r−1

=ϕ3 − ϕ1

which shows that the claim holds for p = 1 and q = r, and concludes the induction.Lastly, we prove the claim for every p, q ∈ N, by induction on p+q; notice that the foregoing

already shows that the assertion holds whenever p + q ≤ 2. Thus, let r > 2, and suppose thatthe claim is already known for every pair (p, q) such that p+ q < r; then also (4.2.35) holds forsuch values of p and q, and especially we get

(4.2.42) dDp+q ShDp,q = ShDp−1,q dD,hp,q + (−1)p · ShDp,q−1 dD,vp,q whenever p+ q < r.

Let (p′, q′) be a pair such that p′ + q′ = r; combining (4.2.42) with claim 4.2.36(i), we get

dDp′+q′−1 ShAp′,q′ = (−1)q′ · (ShDp′−2,q′ d

D,hp′−1,q′ − (−1)p

′ · ShDp′−1,q′−1 dD,vp′−1,q′) A[1[p′], ηq′ ]

+ (ShDp′−1,q′−1 dD,hp′,q′−1 + (−1)p

′ · ShDp′,q′−2 dD,vp′,q′−1) A[ηp′ ,1[q′]].

On the other hand, after noticing that

dD,hp′,q′−1 A[ηp′ ,1[q′]]− (−1)p′ · 1A[p′,q′] =A[ηp′−1,1[q′]] dB,hp′−1,q′

dD,vp′−1,q′ A[1[p′], ηq′ ]− (−1)q′ · 1A[p′,q′] =A[1[p′], ηq′−1] dC,vp′,q′−1

we may apply claim 4.2.36(i), to compute

ShAp′−1,q′ dB,hp′−1,q′ = ((−1)q

′ · ShDp′−2,q′ A[1[p′−1], ηq′ ] + ShDp′−1,q′−1 A[ηp′−1,1[q′]]) dB,hp′−1,q′

= (−1)q′ · ShDp′−2,q′ d

D,hp′−1,q′ A[1[p′], ηq′ ]− (−1)p

′ · ShDp′−1,q′−1

+ ShDp′−1,q′−1 dD,hp′,q′−1 A[ηp′ ,1[q′]]

and

ShAp′,q′−1 dC,vp′,q′−1 = ((−1)q

′−1 · ShDp′−1,q′−1 A[1[p′], ηq′−1]

+ ShDp′,q′−2 A[ηp′ ,1[q′−1]]) dC,vp′,q′−1

= (−1)q′−1 · ShDp′−1,q′−1 d

D,vp′−1,q′ A[1[p′], ηq′ ] + ShDp′−1,q′−1

+ ShDp′,q′−2 dD,vp′,q′−1 A[ηp′ ,1[q′]].

Finally, the sought identity for the pair (p′, q′) follows by comparing the last two identities withthe previous one for dDp′+q′−1 ShAp′,q′ , and this concludes the proof of the inductive step.


Proposition 4.2.43. With the notation of (4.2.23), the sequence (AWAn | n ∈ N) defines a

morphism of chain complexes

AWA• : A∆

• → TotA•• for every A ∈ Ob(s2.A ).

Proof. Denote by dT• the differential of TotA••, and keep the notation of the proof of proposition4.2.34; we have to check the identity

(4.2.44) AWn−1 dAn = dTn AWn for every n ∈ N.However, say that p, q ∈ N and p + q = n; the projection of the right-hand side of (4.2.44) onthe direct summand Ap−1,q of (TotA••)n−1 equals

(4.2.45) dA,hp,q AWp,q + (−1)p−1 · dA,vp−1,q+1 AWp−1,q+1.

Unwinding the definition, (4.2.45) is found to bep∑i=0

(−1)i · A[εq∨p,0 εi, εpq,0]− (−1)p ·

q+1∑i=0

(−1)i · A[εq+1∨p−1,0, ε

p−1q+1,0 εi]

which we may rewrite as

(4.2.46)p−1∑i=0

(−1)i · A[εq∨p,0 εi, εpq,0] +

p+q∑i=p

(−1)i · A[εq+1∨p−1,0, ε

p−1q+1,0 εi−p+1].

On the other hand, the projection of the left-hand side of (4.2.44) onto Ap−1,q equalsn∑i=0

(−1)i · A[εi εq∨p−1,0, εi εp−1q,r ].

To compare the latter with (4.2.46), it suffices to remark that

εi εq∨p−1,0 =

εq∨p,0 εi if i < p

εq+1∨p−1,0 if i ≥ p

and εi εp−1q,r =

εpq,0 if i < p

εp−1q+1,0 εi if i ≥ p

which are all deduced from the simplicial identities for the εi. The proposition follows.

4.2.47. Propositions 4.2.34 and 4.2.43 yield natural transformations of functors

Tot U2A

Sh• // UA ∆A .AW•

oo

Now, denote by hA : C(A ) → Hot(A ) the natural functor (this induces the identity on theobjects, and the projection on the group of morphisms); there follow natural transformations

hA Tot U2A

hA ∗Sh• // hA UA ∆A .hA ∗AW•

oo between functors s2.A → Hot(A ).

Theorem 4.2.48 (Eilenberg-Zilber-Cartier). With the notation of (4.2.47), we have :(i) hA ∗ AW• and hA ∗ Sh• are mutually inverse isomorphisms of functors.

(ii) More precisely, there exist natural modifications (see definition 1.3.7)

AW• Sh• 1TotU2A

Sh• AW• 1UA ∆A.

Proof. Let pA• : TotA•• → TotN••A be the projection whose kernel is the sum of the re-maining three direct summands in the decomposition Tot (4.2.32); explicitly, in each degreen ∈ N, this kernel is the sum of the subobjects ImA[ηi,1[q]] and ImA[1[p], ηj] of Ap,q, for everyi = 0, . . . , p − 1, j = 0, . . . , q − 1, and every p, q ∈ N such that p + q = n. Likewise, letjA• : N•A

∆ → A∆• and qA• : A∆

• → N•A∆ be as in (4.2.28); theorem 4.2.29 and the discussion


of (4.2.31) show that the pairs (pA• , iA• ) and (jA• , q

A• ) induce mutually inverse isomorphisms in

Hot(A ), so it suffices to check that the same holds for the compositions

ShA• := qA• ShA• iA• : TotN••A→ N•A∆ AWA

• := pA• AWA• jA• : N•A

∆ → TotN••A

for every bisimplicial object A of A . However, we have

Claim 4.2.49. pA• AWA• ShA• iA• = 1TotN••A.

Proof of the claim. More precisely, say that p+ q = n, consider the composition

fp,q : Ap,qShAp,q−−−→ Ap+q,p+q

AWAn−−−→ (TotA••)n

and let gp,q : Ap,q → (TotA••)n be the inclusion map; we shall show that fp,q − gp,q ⊂ Ker pAn .Indeed, we have

fp,q =n∑i=0

∑(µ,ν)

εµν · A[ην1 · · · ηνq εn−i∨i,0 , ηµ1 · · · ηµp εin−i,0]

(notation of (4.2.33), and εin−i,0 is defined as in the proof of proposition 4.2.43). However, ifi < p, then ηµ1 · · · ηµp εin−i,0 is of the form τ ηµp for some τ : [p+ q− i− 1]→ [q], so thecorresponding term does lie in Ker pAn . If i > p, then ην1 · · · ηνq εn−i∨i,0 is of the form τ ηkfor some k ≤ νq and some τ : [i − 1] → [p], so this term likewise lies in Ker pAn . It remains toconsider the terms with i = p; however, a simple inspection shows that ην1 · · · ηνq ε

n−p∨p,0 is

of the same form as above, unless ν1 is either p− 1 or p (details left to the reader); furthermore,if ν1 = p− 1, then µp ≥ p, in which case ηµ1 · · · ηµp εin−i,0 is of the form described above.In all these cases, the corresponding term again lies in Ker pAn . So, it remains only to considerthe single case where (µ, ν) is the identity permutation, whose sign equals 1; in this case, thecorresponding term is none else than A[1[p],1[q]], whence the claim. ♦

Claim 4.2.49 and theorem 4.2.29(ii) already yield the existence of the sought modificationAW• Sh• 1TotU2

A. Next we define, for every n ∈ N, a natural transformation

sn : •[n] ∆A ⇒ •[n+ 1] ∆A

(notation of (4.2.4); so sAn is a morphismA[n, n]→ A[n+1, n+1] for every objectA of s2.A ).The construction is by induction on n : for n = 0 we let sA0 : A[0, 0] → A[1, 1] be the zeromorphism; for n > 0 we set

sAn := ShA′

n AWA′

n A[η0, η0]− sA′n−1 where A′ := (γ2 γ1(A∨))∨

(notation of (4.2.15) and (4.2.2) : explicitly, we have A′[p, q] := A[p + 1, q + 1] for everyp, q ∈ N, and A′[εi, εj] := A[εi+1, εj+1], and likewise for A′[εi, ηj], A′[ηi, εj] and A′[ηi, ηj], forevery face and degeneracy map of ∆). We have

Claim 4.2.50. The system (qAn+1 sAn jAn | n ∈ N) is a homotopy qA• ShA• AWA• jA• ⇒ 1N•A∆ .

Proof of the claim. Denote by dA• the differential ofA∆• , and let dA0 : A∆

0 → 0 and sA−1 : 0→ A∆0

be the zero morphisms. We check, more precisely, that

qAn (dAn+1 sAn + sAn−1 dAn ) = qAn ShAn AWAn − qAn for every n ∈ N.

We argue by induction on n, and the assertion is clear for n = 0. For n = 1, notice that

ShA1 AWA1 = A[ε1 η0,1[1]] + A[1[1], ε0 η0] : A1,1 → A1,1.

It follows that

sA1 = ShA′

1 AWA′

1 A[η0, η0] = (A[ε1 η0 η0, η0] + A[η0, η1])


whencedA2 sA1 = ShA1 AWA

1 − 1A[1,1] + A[ε1 ε1 η0, ε1 η0]

(details left to the reader). Since ImA[η0, η0] ⊂ Ker qA2 , the assertion follows in this case. Next,suppose that r > 1, and that the sought identity is already known for every n < r, and everybisimplicial object A; especially, it holds for A′, so if we let dA′• be the differential of A′•, we get

(4.2.51) qA′

r−1 (dA′

r sA′

r−1 + sA′

r−2 dA′

r−1) = qA′

r−1 ShA′

r−1 AWA′

r−1 − qA′

r−1.

On the other hand, after noticing that

dA′

r A[η0, η0] = 1A[r,r] − A[η0, η0] dA′r−1

we may compute

dA′

r sAr = dA′

r ShA′

r AWA′

r A[η0, η0]− dA′r sA′

r−1

= ShA′

r AWA′

r dA′

r A[η0, η0]− dA′r sA′

r−1

= ShA′

r AWA′

r − ShA′

r AWA′

r A[η0, η0] dA′r−1 − dA′

r sA′

r−1

which, combined with (4.2.51), implies

(4.2.52) qA′

r−1 (dA′

r sAr + sAr−1 dA′

r−1) = −qA′

r−1.

Furthermore, notice the natural morphism in s2.A

(g(1)A∨ g

(2)γ1A∨

)∨ : A′ → A

supplied by (4.2.15); explicitly, for every p, q ∈ N, this morphism is given by the discarded faceoperator A[ε0, ε0] : A[p+ 1, q + 1]→ A[p, q]. Then, the naturality of s•, Sh• and AW• implies

A[ε0, ε0] sAr =A[ε0, ε0] ShA′

r AWA′

r A[η0, η0]− A[ε0, ε0] sA′r−1

= ShAr AWAr A[η0 ε0, η0 ε0]− sAr−1 A[ε0, ε0]

= ShAr AWAr − sAr−1 A[ε0, ε0].

Lastly, recalling that dAr = A[ε0, ε0]−dA′r−1, the latter identity can be added to (4.2.52), to deduce

qA′

r−1 (dAr+1 sAr + sAr−1 dAr ) = qA′

r−1 ShAr AWAr − qA

′

r−1.

Now, to prove the assertion in degree r, it suffices to observe that qAr factors through qA′

r−1. ♦

Claim 4.2.50 and theorem 4.2.29(ii) supply the second sought modification, and conclude theproof of the theorem.

4.2.53. Let (A ,⊗) be an abelian tensor category, A[•] and B[•] two objects of s.A , and definethe bisimplicial objectsAB andBA, as well as the simplicial objectsA⊗B andB⊗A of Aas in remark 4.2.16(iii). Notice that the system of commutativity constraints (ΨA[n],B[n] | n ∈ N)amounts to an isomorphism

ΨA⊗B : A⊗B ∼→ B ⊗ A in s.A

whence an isomorphism Ψ(A⊗B)• : (A ⊗ B)•∼→ (B ⊗ A)• on the respective unnormalized

complexes.

Proposition 4.2.54. With the notation of (4.2.53), the diagram of chain complexes

Tot(AB)••Ψ•A•,B• //

ShAB•

Tot(B A)••

ShBA•

(A⊗B)•

Ψ(A⊗B)• // (B ⊗ A)•


commutes, where Ψ•A•,B• is the commutativity constraint for the unnormalized chain complexesA• and B•, as in (4.1.9).

Proof. The assertion boils down to the identity

(4.2.55) (−1)pq · ShBAq,p ΨA[p],B[q] = ΨA[n],B[n] ShABp,q

for every p, q ∈ N with p+ q = n. For the latter, suppose first that p = 0, in which case

ShABp,q = A[η0 · · · ηq−1]⊗ 1B[q] and ShBAq,p = 1B[q] ⊗ A[η0 · · · ηq−1]

from which we derive (4.2.55), using the naturality of Ψ (details left to the reader). Likewisewe argue for the case where q = 0. For the general case, we proceed by induction on n. Thecases n = 0, 1 have already been dealt with, so suppose r ≥ 2, and that the sought identity isalready known for every pair of integers whose sum is < n, and every objects A,B of s.A . Bythe foregoing, we may also assume that both p, q > 0, and then claim 4.2.36(i) implies that

ShABp,q = (−1)q · ShγAγBp−1,q (1A[p] ⊗B[ηq]) + ShγAγBp,q−1 (A[ηp]⊗ 1B[q]).

However, it follows easily from remark 1.2.34(iii) that Ψ is an additive functor in both of itsarguments; combining with the inductive assumption, we deduce that

ΨA[n],B[n] ShABp,q = (−1)pq · ShγBγAq,p−1 ΨA[p],B[q+1] (1A[p] ⊗B[ηq])

+ (−1)p(q−1) · ShγBγAq−1,p ΨA[p+1],B[q] (A[ηp]⊗ 1B[q])

= (−1)pq · ShγBγAq,p−1 (B[ηq]⊗ 1A[p]) ΨA[p],B[q]

+ (−1)p(q−1) · ShγBγAq−1,p (1B[q] ⊗ A[ηp]) ΨA[p],B[q]

= (−1)pq · ShBAq,p ΨA[p],B[q]

where the last equality follows again from claim 4.2.36(i) and the additivity of Ψ.

4.2.56. The shuffle map is also associative, in the following sense. Let

A := (A[p, q, r] | p, q, r ∈ N)

be any triple simplicial object of the abelian category A , and denote by A(1,2) (resp. A(2,3)) thediagonal bisimplicial object of A extracted from A by the rule

A(1,2)[p, q] := A[p, p, q] ( resp. A(2,3)[p, q] := A[p, q, q] ) for every p, q ∈ N.

Let also A••• be the triple chain complex associated to A, and A∆••• the diagonal chain complex

extracted from A•••. Moreover, denote by A′ (resp. A′′) the bisimplicial object of s.A given bythe rule :

[p, q] 7→ A[p, q, •] ( resp. [p, q] 7→ A[•, p, q] ) for every p, q ∈ N.

Proposition 4.2.57. With the notation of (4.2.56), we have a commutative diagram in C(A )

Tot(A•••) //

Tot(A(1,2)•• )

ShA(1,2)

•

Tot(A(2,3)•• )

ShA(2,3)

• // A∆•••

whose top horizontal (resp. left vertical) arrow is obtained as the composition of ShA′

• (resp.ShA

′′

• ) with the functor Tot C(UA ) : C(s.A )→ C(C(A ))→ C(A ).


Proof. For every p, q, r ∈ N, denote by

ShA′

p,q[r] :A[p, q, r]→ A[p+ q, p+ q, r] = A(1,2)[p+ q, r]

ShA′′

q,r [p] :A[p, q, r]→ A(2,3)[p, q + r]

respectively the [r]-component of ShA′

p,q and the [p]-component of ShA′′

q,r . The assertion boilsdown to the identity :

ShA(1,2)

p+q,r ShA′

p,q[r] = ShA(2,3)

p,q+r ShA′′

q,r [p] for every p, q, r ∈ N.To check the latter, set

D′ := γ2 γ1A′ D(1,2) := γ2 γ1(A(1,2))

and define likewise D′′ and D(2,3) (notation of (4.2.15)). Also, let A[p, q, r] := 0 whenever oneof the indices p, q, r is < 0, and define Shp,q to be the zero map, when either p or q is strictlynegative (cp. the proof of proposition 4.2.34). We argue by induction on n := p + q + r;the case n = 0 is trivial, so suppose that n > 0, and that the assertion is already known forall indices whose sum is < n, and all triple simplicial objects of A . Notice as well that theassertion trivially holds as well if any of the indices p, q, r is strictly negative, since in this caseboth sides are the zero map. Hence, we may assume that p, q, r ∈ N; in this case, by applyingclaim 4.2.36(i), first to A(1,2) and then to A′, we compute

ShA(1,2)

p+q,r ShA′

p,q[r] = (−1)r · ShD(1,2)

p+q−1,r A[1[p+q],1[p+q], ηr] ShA′

p,q[r]

+ ShD(1,2)

p+q,r−1 A[ηp+q, ηp+q,1[r]] ShA′

p,q[r]

= (−1)r · ShD(1,2)

p+q−1,r ShA′

p,q[r + 1] A[1[p],1[q], ηr]

+ ShD(1,2)

p+q,r−1 ShD′

p,q[r] A[ηp, ηq,1[r]] (by (4.2.37))

= (−1)q+r · ShD(1,2)

p+q−1,r ShD′

p−1,q[r + 1] A[1[p], ηq, ηr]

+ (−1)r · ShD(1,2)

p+q−1,r ShD′

p,q−1[r + 1] A[ηp,1[q], ηr]

+ ShD(1,2)

p+q,r−1 ShD′

p,q[r] A[ηp, ηq,1[r]]

= (−1)q+r · ShD(2,3)

p−1,q+r ShD′′

q,r [p] A[1[p], ηq, ηr]

+ (−1)r · ShD(2,3)

p,q+r−1 ShD′′

q−1,r[p+ 1] A[ηp,1[q], ηr]

+ ShD(2,3)

p,q+r−1 ShD′′

q,r−1[p+ 1] A[ηp, ηq,1[r]]

where the last identity holds by inductive assumption. On the other hand, by applying claim4.2.36(i) to A(2,3) we get

ShA(2,3)

p,q+r ShA′′

q,r [p] = (−1)q+r · ShD(2,3)

p−1,q+r A[1[p], ηq+r, ηq+r] ShA′′

q,r [p]

+ ShD(2,3)

p,q+r−1 A[ηp,1[q+r],1[q+r]] ShA′′

q,r [p]

= (−1)q+r · ShD(2,3)

p−1,q+r ShD′′

q,r [p] A[1[p], ηq, ηr]

+ ShD(2,3)

p,q+r−1 ShA′′

q,r [p+ 1] A[ηp,1[q],1[r]] (by (4.2.37))

and after applying again claim 4.2.36(i) to A′′ and comparing with the foregoing expression forShA

(1,2)

p+q,r ShA′

p,q[r], we obtain the sought identity.

Theorem 4.2.58 (Dold-Puppe-Kan). For any abelian category A , and any k ∈ N, we have :(i) The functors NA and NA ,k are equivalences.

(ii) If f, g are any two morphisms in s.A , then there exists a simplicial homotopy from fto g if and only if there exists a chain homotopy from N•f to N•g.


Proof. We easily reduce to the case where A is small, and then there exists a fully faithfulimbedding A → B, where B is a complete and cocomplete abelian tensor category, withinternal Hom functor (lemma 1.2.42).

(i): We first construct an explicit quasi-inverse for the functors NB and NB,k, as follows. Forevery i ∈ N, consider the cochain complex K〈i〉• defined in example 4.2.27(ii); notice thatevery morphism ϕ : [i]→ [i′] in ∆ induces a morphism

K〈ϕ〉• := N•Z∆ϕ : K〈i〉• → K〈i′〉•(notation of remark 4.2.17(iii)). Hence, the system (K〈i〉• | i ∈ N) amounts to a cosimplicialobject of C≤0(Z-Mod). Now, let U be a unit of the tensor category B; for any object C• ofC≤0(B) (resp. of C[−k,0](B)), we set

KC [i] := HomC(B)(K〈i〉• ⊗Z U [0], C•) for every i ∈ N (resp. for every i ≤ k)

where HomC(B) is the functor constructed in example 4.1.8(vii), and the mixed tensor productis defined as in (4.1.10). By the foregoing, it is clear that the system (KC [i] | i ∈ N) amounts toan object of s.B (resp. of sk.B). In order to compute N•KC , let us set

Z∆i+ :=

i∑n=1

Im (Z∆εn : Z∆i−1 → Z∆i) Z∆i0 := Z∆i/Z∆i

+ K〈i〉• := N•Z∆i0

for every i > 0, as well as Z∆00 := Z∆0 and K〈0〉• := K〈0〉•; thus, K〈i〉• is also the quotient of

K〈i〉• by the sum of the images of the morphisms K〈εn〉, for n = 1, . . . , i. With this notation,a simple inspection of the definition shows that

NiKC = HomC(B)(K〈i〉• ⊗Z U [0], C•) for every i ∈ N (resp. for every i ≤ k).

On the other hand, using the explicit description of example 4.2.27(ii), it is easily seen that

K〈i〉n∼→

Z if n = i or n = i− 1 ≥ 00 otherwise.

More precisely, K〈i〉i (resp. K〈i〉i−1) is generated by the basis element e1 of K〈i〉i (resp. eε0of K〈i〉i−1) corresponding to the identity map 1[i] (resp. corresponding to ε0 : [i − 1] → [i]).Furthermore, example 4.2.27(i) shows that the differential K〈i〉i → K〈i〉i−1 maps e1 to eε0 ,so it corresponds to the identity map Z → Z, under the foregoing identification. Taking intoaccount remark 1.2.12(iv), we deduce that NiKC is the kernel of the morphism

(4.2.59) Ci ⊕ Ci−1 → Ci−1 ⊕ Ci−2

given by the matrix ((−1)i · dCi 1Ci−1

0 (−1)i+1 · dCi−1

)and since dCi−1 dCi = 0, the latter is just Ci; more precisely, Ci is identified with this kernel,via the monomorphism

(4.2.60) (1Ci , (−1)i+1 · dCi ) : Ci → Ci ⊕ Ci−1.

This identification NiKC∼→ Ci can also be described as follows. For every Z ∈ Ob(B), let

HomB(Z,C•)

be the cochain complex such that HomB(Z,C•)n := HomB(Z,Cn) for every n ∈ Z, withdifferentials induced by those of C•, in the obvious way; then we have natural identifications

HomC(Z-Mod)(K〈i〉•,HomB(Z,C•))∼→ HomC(B)(K〈i〉• ⊗Z Z[0], C•)

∼→ HomB(Z,NiKC)

(see example 4.1.8(vii)) whose composition with the induced isomorphism

HomB(Z,NiKC)∼→ HomB(Z,Ci)


is given by the rule :

(4.2.61) (ϕ• : K〈i〉• → HomB(Z,C•)) 7→ (ϕi(ε1) : Z → Ci).

It remains to determine the differential dNi+1 : Ni+1KC → NiKC ; by definition, the latter isinduced by the morphism K〈ε0〉• : K〈i〉• → K〈i+ 1〉•. In turn, the foregoing description saysthat K〈ε0〉• is naturally identified with the morphism of C(Z-Mod)

K〈i〉•

0 // 0 //

Z1Z

// Z

// 0

K〈i+ 1〉• 0 // Z // Z // 0 // 0

(where the two copies of Z on the top horizontal row are placed in homological degrees i andi− 1), so dNi+1 is deduced from the morphism in C(B)

0 // Ci+1 ⊕ Ci //

Ci ⊕ Ci−1

1

// 0 //

0

0 // 0 // Ci ⊕ Ci−1// Ci−1 ⊕ Ci−2

// 0

whose rows are given by the morphisms (4.2.59). Therefore, via the identification (4.2.60),the morphism dNi+1 becomes none else than (−1)i+1 · dCi+1, i.e. we have obtained a naturalisomorphism

ωC• : N•KC∼→ C•

for every complex in C≤0(B) (resp. in C[−k,0](B)).Conversely, let B[•] be any object of s.B, and Z any object of B; clearly

N•(∆k ⊗ s.Z) = K〈i〉• ⊗Z Z[0] for every i ∈ N.

In view of example 4.1.8(vii) and remark 4.2.17(ii), we deduce a natural transformation

(4.2.62)

HomB(Z,B[i])a // Homs.B(∆i ⊗ s.Z,B)

b

HomB(Z,KN•B[i]) HomC(B)(K〈i〉• ⊗Z Z[0], N•B)coo

which, by Yoneda’s lemma, comes from a unique morphism in B

ψBi : B[i]→ KN•B[i] for every i ∈ N.

The same construction applies, in case B is an object of sk.B : we need only replace thegroup Homs.B(∆i ⊗ s.Z,B) by Homsk.B(s.trunck(∆i ⊗ s.Z), B) in (4.2.62) : see remark4.2.17(iv). Moreover, remark 4.2.17(iii) implies that the system ψB := (ψBi | i ∈ N) amounts toa morphism B → KN•B in s.B (resp. in sk.B). By the same token, the Z-linear isomorphisma maps the abelian subgroup HomB(Z,NiB) isomorphically onto the subgroup

Homs.B(Z∆i0 ⊗Z s.Z,B)

∼→ Homs.Z-Mod(Z∆i0 ,HomB(Z,B))

for every Z ∈ Ob(B) and every i ∈ N, where HomB(Z,B) is the simplicial abelian group asin example 4.2.27(iii). Explicitly, if β : Z → NiB is any morphism, then a(β) is the uniquemorphism Z∆i

0 → HomB(Z,B) of simplicial abelian groups such that a(β)(e1) = β, wheree1 ∈ K〈i〉i ⊂ Z∆i

0 [i] is the basis element described in the foregoing. Likewise, we have naturalidentifications

HomC(Z-Mod)(K〈i〉•,HomB(Z,N•B))∼→ HomC(B)(K〈i〉• ⊗Z Z[0], N•B)


and b restricts to a map

Homs.Z-Mod(Z∆i0 ,HomB(Z,B))→ HomC(Z-Mod)(K〈i〉•,HomB(Z,N•B)) ϕ 7→ N•ϕ.

Again, the same applies to an object of sk.B, by taking suitable truncated variants of the aboveconstructions. Taking into account (4.2.61), we conclude that ψB induces an isomorphism

N•ψB : N•B

∼→ N•KN•B

which is inverse to ωN•B. To finish the proof of assertion (i) for the category B, it then sufficesto remark :

Claim 4.2.63. The functors NB and NB,k are conservative.

Proof of the claim. We show, by induction on n, that if a morphism h in s.B or sk.B (for anyk ∈ N) induces an isomorphism N•h, then h[n] is an isomorphism for every n ∈ N (resp. forevery n ≤ k). The assertion is obvious for n = 0, hence suppose that n > 0, and that h[n−1] isknown to be an isomorphism whenever h is a morphism in s.B or in sk.B (for arbitrary k ∈ N)such that N•h is an isomorphism. Let h : A → B be any such morphism in s.B or in sk.B.If h is a morphism in s0.B, we are done, hence we may suppose that either h is a morphism ins.B or k > 0. Set

A′ := Ker(gA : γA→ A) ( resp. A′ := Ker(gA : γk−1A→ s.trunck−1A) )

as well as B′ := Ker(gB) (notation of remark 4.2.11(ii)). Notice that gA is an epimorphism,since ∂n+2 admits the section σn+1, for every n ∈ N (resp. for every n ≤ k). Therefore, wehave a commutative diagram in s.B with exact rows

0 // A′ //

h′

γA //

γh

A //

h

0

0 // A′ // γA // A // 0

(resp. a corresponding diagram in sk−1.B). By inspecting the definitions, it is easily seenthat N•A′ = (N•A)[−1], N•A′ = (N•A)[−1], and N•h′ = (N•h)[−1]; especially, N•h′ is anisomorphism, so h′[n − 1] is an isomorphism, by inductive assumption. The same holds alsofor h[n − 1], and we conclude that γh[n] is an isomorphism. But γh[n] = h[n + 1], so we aredone.

Lastly, in order to prove assertion (i) for the original category A , it suffices to notice :

Claim 4.2.64. Let C• be any object of C(A ) (resp. of C[−k,0](A )), and regard C• as an object ofC(B) (resp. of C[−k,0](B)), via the fully faithful imbedding A → B. Then KC is isomorphicto an object of s.A (resp. sk.A ), regarded as a full subcategory of s.B (resp. of sk.B), via thesame imbedding.

Proof of the claim. This follows easily, by remarking that K〈i〉• lies in C[−i,0](Z-Mod) forevery i ∈ N, and K〈i〉j is a finitely generated abelian group for every i, j ∈ N : details left tothe reader. ♦

(ii): First, let f, g : A → B be two morphisms in s.A , and u : ∆1 ⊗ A → B a homotopyfrom f to g (see remark 4.2.16(iv)); especially,

u (∆ε1 ⊗ A) = f and u (∆ε0 ⊗ A) = g.

Notice thatZ∆1• ⊗Z A• = Tot(∆1 A)••

(notation of remark 4.2.16(iii) and example 4.2.27(i)); there follows a morphism in C(A )

u• : Z∆1• ⊗Z A•

Sh•−−−→ (∆1 ⊗ A)•q•−−→ N•(∆1 ⊗ A)

N•u−−−→ N•B


where Sh• denotes the shuffle map for the bisimplicial object A∆1, and q• is the projectiondefined in (4.2.28). Moreover, the maps ∆εi [0] : ∆0[0]→∆1[0] (i = 0, 1) induce morphisms

ei,n := Z∆εi [0] ⊗Z An : An → Z∆1[0] ⊗Z An ⊂ (Z∆1• ⊗Z A•)n

that amount to morphisms of cochain complexes ei,• : A• → Z∆1• ⊗ZA• (i = 0, 1), and a simple

inspection of the definitions shows that

Sh• ei,• = (∆εi ⊗ A)• for i = 0, 1

whenceu• e1,• = f• and u• e0,• = g•.

The construction makes it clear that ei restricts to a morphism N•A→ K〈1〉• ⊗Z N•A, and thelatter is none else than the map ιi ⊗Z N•A, with the notation of remark 4.1.11(ii). We concludethat the morphism

u• : K〈1〉• ⊗Z N•A → Z∆1• ⊗Z A•

u•−−→ N•B

is a homotopy from N•f to N•g (notation of (4.2.28)).Conversely, suppose that v• : K〈1〉•⊗ZN•A→ N•B is a homotopy fromN•f toN•g. Since

K〈1〉• is a direct summand of Z∆1• and N•A is a direct summand of A•, we may extend v• to a

morphismv• : Z∆1

• ⊗Z A• → N•B

such that v• is the zero morphism on the direct summands other than K〈1〉• ⊗Z N•A. Next,consider the composition

v• : N•(∆1 ⊗ A)j•−−→ (∆1 ⊗ A)•

AW•−−−→ A• ⊗Z Z∆1•

Ψ•−−→ Z∆1• ⊗Z A•

v•−−→ N•B

where j• is the natural injection (see (4.2.28)), AW• is the Alexander-Whitney map for thebisimplicial object A ∆1 (notice that ∆1 ⊗ A = (A ∆1)∆), and Ψ• is the commutativityconstraint (see example 4.1.8(i)). By (i), the morphism v• comes from a unique morphism

v : ∆1 ⊗ A→ B in s.A .

On the other hand, since Np+qA is contained in the kernel of A[εq∨p,0] for every p, q ∈ N, it iseasily seen that the diagram

N•Aιi⊗ZN•A //

N•(∆εi⊗A)

K〈1〉• ⊗Z N•A

N•(∆1 ⊗ A)

Ψ•AW•j• // Z∆1• ⊗Z A•

commutes for i = 0, 1. We conclude that v is a homotopy from f to g.

Corollary 4.2.65. Let A be any abelian category. We have :(i) If k ∈ N is any integer, and A any k-truncated simplicial object of A , then

Hi(coskkA) = 0 for every i ≥ k.

(ii) Every homotopically trivial augmented simplicial object of A is aspherical.

Proof. (i): Denote by t≤k : C≤0(A ) → C[−k,0](A ) the brutal truncation functor (see (4.1)). Inlight of theorem 4.2.58, we see that t≤k admits a right adjoint vk : C[−k,0](A ) → C≤0(A ), andclearly there are natural isomorphisms

N•coskkA∼→ vkN•A for every A ∈ Ob(sk.A ).

Taking into account theorem 4.2.29(iii), we are then reduced to showing

Claim 4.2.66. Hi(vkK•) = 0 for every (K•, d•) ∈ Ob(C[−k,0]) and every i ≥ k.


Proof of the claim. Indeed it is easily seen that :

(vkK•)i =

Ki for ≤ kKer dk for i = k + 10 for i > k + 1

and the differential of vkK• in degree ≤ k agrees with that of K•, whereas in degree k + 1 it isthe natural inclusion map (details left to the reader). The claim follows immediately. ♦

(ii) follows directly from theorems 4.2.58(ii) and 4.2.29(iii), and remark 4.1.3(ii).

4.3. Injective modules, flat modules and indecomposable modules.

4.3.1. Indecomposable modules. Recall that a unitary (not necessarily commutative) ring R issaid to be local ifR 6= 0 and, for every x ∈ R either x or 1−x is invertible. IfR is commutative,this definition is equivalent to the usual one.

4.3.2. Let C be any abelian category andM an object of C . One says thatM is indecomposableif it is non-zero and cannot be presented in the form M = N1 ⊕ N2 with non-zero objects N1

and N2. If such a decomposition exists, then EndC (N1)× EndC (N2) ⊂ EndC (M), especiallythe unitary ring EndC (M) contains an idempotent element e 6= 1, 0 and therefore it is not alocal ring.

However, if M is indecomposable, it does not necessarily follow that EndC (M) is a localring. Nevertheless, one has the following:

Theorem 4.3.3 (Krull-Remak-Schmidt). Let (Ai)i∈I and (Bj)j∈J be two finite families of ob-jects of C , such that:

(a) ⊕i∈IAi ' ⊕j∈JBj .(b) EndC (Ai) is a local ring for every i ∈ I , and Bj 6= 0 for every j ∈ J .

Then we have :(i) There is a surjection ϕ : I → J , and isomorphismsBj

∼→ ⊕i∈ϕ−1(j)Ai, for every j ∈ J .(ii) Especially, if Bj is indecomposable for every j ∈ J , then I and J have the same

cardinality, and ϕ is a bijection.

Proof. Clearly, (i)⇒(ii). To show (i), let us begin with the following :

Claim 4.3.4. LetM1,M2 be two objects of C , and setM := M1⊕M2. Denote by ei : Mi →M(resp. pi : M → Mi) the natural injection (resp. projection) for i = 1, 2. Suppose thatα : P →M is a subobject of M , such that p1α : P →M1 is an isomorphism. Then the naturalmorphism β : P ⊕M2 →M is an isomorphism.

Proof of the claim. Denote by πP : P⊕M2 → P and π2 : P⊕M2 →M2 the natural projections;then β := απP + e2π2. Of course, the assertion follows easily by applying the 5-lemma (whichholds in any abelian category) to the commutative ladder with exact rows :

0 // M2// P ⊕M2

πP //

β

P

p1α

// 0

0 // M2e2 // M

p1 // M1// 0.

Equivalently, one can argue directly as follows. By definition, Coker β represents the functor

C → Z-Mod : X 7→ Ker HomC (β,X)

and likewise for Coker p1α; however, it is easily seen that these functors are naturally isomor-phic, hence the natural morphism Coker β → Coker p1α is an isomorphism, so β is an epimor-phism. Next, let t : Ker β → P ⊕M2 be the natural morphism; since p1α is an isomorphism,


πP t = 0, so t factors through a morphism t : Ker β →M2. It follows that e2 t = β t = 0,therefore t = 0, and finally Ker β = 0, since t is a monomorphism. ♦

Set A := ⊕i∈IAi, and denote ei : Ai → A (resp. pi : A → Ai) the natural injection (resp.projection), for every i ∈ I . Also, endow I ′ := I ∪ ∞ with any total ordering (I ′, <) forwhich∞ is the maximal element.

Claim 4.3.5. Let (aj : A → A)j∈J be a finite system of endomorphisms such that∑

j∈J aj =1A. Then there exists a mapping ϕ : I → J such that the following holds :

(i) aϕ(i)ei : Ai → A is a monomorphism, for every i ∈ I , and let Pi := Im(aϕ(i)ei).(ii) For every l ∈ I ′, the natural morphism βl :

⊕i<l Pi⊕

⊕i≥lAi → A is an isomorphism.

Proof of the claim. We both define ϕ(l) and prove (ii), by induction on l. If l is the smallestelement of I , then βl = 1A, so (ii) is obvious. Hence, let l ∈ I be any element, l′ ∈ I ′ thesuccessor of l, and assume that βl is an isomorphism. Set a′j := β−1

l ajβl for every j ∈ J , andMl :=

⊕i<l Pi ⊕

⊕i≥lAi; also, let e′l : Al → Ml and p′l : Ml → Al be the natural morphisms.

Clearly∑

j∈J a′j = 1Ml

, so∑

j∈J p′la′je′l = 1Al; since EndC (Al) is a local ring, it follows that

there exists jl ∈ J such that p′la′jle′l is an automorphism of Al. However,

(4.3.6) a′jle′l = β−1

l ajlβle′l = β−1

l ajlel

so assertion (i) follows for the index l, by setting ϕ(l) := jl.Next, set Ml,1 := Al and Ml,2 :=

⊕i<l Pi ⊕

⊕i>lAi. The foregoing argument provides

a subobject α : P ′l := Im(a′ϕ(l)e′l) → Ml = Ml,1 ⊕ Ml,2 such that p′lα : P ′l → Ml,1 is an

isomorphism. By claim 4.3.4, it follows that the natural map β′l′ : P ′l ⊕ Ml,2 → Ml is anisomorphism. On the other hand, (4.3.6) implies that βl restricts to an isomorphism γl : P ′l

∼→Pl. We deduce a commutative diagram

P ′l ⊕Ml,2

β′l′ //

γl⊕1Ml,2

Ml

βl

Ml′

βl′ // A

whose vertical arrows and top arrow are isomorphisms. Thus, the bottom arrow is an isomor-phism as well, as required. ♦

For every j ∈ J , let e′j : Bj → A (resp. p′j : A → Bj) be the injection (resp. projec-tion) deduced from a given isomorphism as in (a), and set aj := e′jp

′j for every j ∈ J . Then∑

j∈J aj = 1A, so claim 4.3.5 yields a mapping ϕ : I → J such that the following holds. SetPj := ⊕i∈ϕ−1(j)Im(ajei) for every j ∈ J ; then the natural morphism β : ⊕j∈JPj

∼→ A is anisomorphism. However, clearly β(Pj) is a subobject of Im(e′j), for every j ∈ J . Hence, ϕmust be a surjection, and β restricts to isomorphisms Pj

∼→ Im(e′j) for every j ∈ J , whenceisomorphisms Pj

∼→ Bj , from which (ii) follows immediately.

Remark 4.3.7. There are variants of theorem 4.3.3, that hold under different sets of assump-tions. For instance, in [37, Ch.I, §6, Th.1] it is stated that the theorem still holds when I andJ are no longer finite sets, provided that the category C admits generators and that all filteredcolimits are representable and exact in C .

4.3.8. Let us now specialize to the case of the category A-Mod, where A is a (commutative)local ring, say with residue field k. Let M be any finitely generated A-module; arguing byinduction on the dimension of M ⊗A k, one shows easily that M admits a finite decompositionM = ⊕ri=1Mi, where Mi is indecomposable for every i ≤ r.


Lemma 4.3.9. Let A be a henselian local ring, and M an A-module of finite type. Then M isindecomposable if and only if EndA(M) is a local ring.

Proof. In view of (4.3.2), we can assume that M is indecomposable, and we wish to prove thatEndA(M) is local. Thus, let ϕ ∈ EndA(M).

Claim 4.3.10. The subalgebra A[ϕ] ⊂ EndA(M) is integral over A.

Proof of the claim. This is standard: choose a finite system of generators (fi)1≤i≤r for M , thenwe can find a matrix a := (aij)1≤i,j≤r of elements of A, such that ϕ(fi) =

∑rj=1 aijfj for every

i ≤ r. The matrix a yields an endomorphism ψ of the free A-module A⊕r; on the other hand,let e1, . . . , er be the standard basis of A⊕r and define an A-linear surjection π : A⊕r → M bythe rule: ei 7→ fi for every i ≤ r. Then ϕ π = π ψ; by Cayley-Hamilton, ψ is annihilated bythe characteristic polynomial χ(T ) of the matrix a, whence χ(ϕ) = 0. ♦

Since A is henselian, claim 4.3.10 implies that A[ϕ] decomposes as a finite product of localrings. If there were more than one non-zero factor in this decomposition, the ring A[ϕ] wouldcontain an idempotent e 6= 1, 0, whence the decomposition M = eM ⊕ (1− e)M , where bothsummands would be non-zero, which contradicts the assumption. Thus, A[ϕ] is a local ring, sothat either ϕ or 1M − ϕ is invertible. Since ϕ was chosen arbitrarily in EndA(M), the claimfollows.

Corollary 4.3.11. Let A be a henselian local ring. Then:(i) If (Mi)i∈I and (Nj)j∈J are two finite families of indecomposable A-modules of finite

type such that ⊕ri=1Mi ' ⊕sj=1Nj , then there is a bijection β : I∼→ J such that

Mi ' Nβ(i) for every i ∈ I .(ii) If M and N are two finitely generated A-modules such that M⊕k ' N⊕k for some

integer k > 0, then M ' N .(iii) If M , N and X are three finitely generated A-modules such that X ⊕M ' X ⊕ N ,

then M ' N .

Proof. It follows easily from theorem 4.3.3 and lemma 4.3.9; the details are left to the reader.

Proposition 4.3.12. Let A → B be a faithfully flat map of local rings, M and N two finitelypresented A-modules with a B-linear isomorphism ω : B ⊗AM

∼→ B ⊗A N . Then M ' N .

Proof. Under the standing assumptions, the natural B-linear map

B ⊗A HomA(M,N)→ HomB(B ⊗AM,B ⊗A N)

is an isomorphism ([36, Lemma 2.4.29(i.a)]). Hence we can write ω =∑r

i=1 bi ⊗ ϕi for somebi ∈ B and ϕi : M → N (1 ≤ i ≤ r). Denote by k and K the residue fields of A and Brespectively; we set ϕi := 1k ⊗A ϕi : k ⊗A M → k ⊗A N . From the existence of ω wededuce easily that n := dimk k ⊗A M = dimk k ⊗A N . Hence, after choosing bases, we canview ϕ1, . . . , ϕr as endomorphisms of the k-vector space k⊕n. We consider the polynomialp(T1, . . . , Tr) := det(

∑ri=1 Ti · ϕi) ∈ k[T1, . . . , Tr]. Let b1, . . . , br be the images of b1, . . . , br

in K; it follows that p(b1, . . . , br) 6= 0, especially p(T1, . . . , Tr) 6= 0.

Claim 4.3.13. The proposition holds if k is an infinite field or if k = K.

Proof of the claim. Indeed, in either of these cases we can find a1, . . . , ar ∈ k such thatp(a1, . . . , ar) 6= 0. For every i ≤ r choose an arbitrary representative ai ∈ A of ai, and setϕ :=

∑ri=1 aiϕi. By construction, 1k ⊗A ϕ : k ⊗A M → k ⊗A N is an isomorphism. By

Nakayama’s lemma we deduce that ϕ is surjective. Exchanging the roles of M and N , the same


argument yields an A-linear surjection ψ : N →M . Finally, [61, Ch.1, Th.2.4] shows that bothψ ϕ and ϕ ψ are isomorphisms, whence the claim. ♦

Let Ah be the henselization of A, and Ash, Bsh the strict henselizations of A and B respec-tively. Now, the induced mapAsh → Bsh is faithfully flat, the residue field ofAsh is infinite, andω induces a Bsh-linear isomorphism Bsh ⊗A M

∼→ Bsh ⊗A N . Hence, claim 4.3.13 yields anAsh-linear isomorphism β : Ash ⊗A M

∼→ Ash ⊗A N . However, Ash is the colimit of a filteredfamily (Aλ | λ ∈ Λ) of finite etale Ah-algebras, so β descends to an Aλ-linear isomorphismβλ : Aλ ⊗A M ' Aλ ⊗A N for some λ ∈ Λ. The Ah-module Aλ is free, say of finite rank n,hence βλ can be regarded as an Ah-linear isomorphism (Ah ⊗AM)⊕n

∼→ (Ah ⊗A N)⊕n. ThenAh ⊗AM ' Ah ⊗A N , by corollary 4.3.11(ii). Since the residue field of Ah is k, we concludeby another application of claim 4.3.13.

4.3.14. Injective hulls. The notion of injective hull plays a central role in the theory of localduality : for a noetherian local ring, one constructs a dualizing module as the injective hull ofthe residue field (see [44, Exp.IV, Th.4.7]), and in section 5.9, injective hulls of the residue fieldsof a monoid algebra will also enable us to perform a certain computation of local cohomology,which is a crucial step in the proof of Hochster’s theorem. We present here the basic results oninjective hulls, in the context of arbitrary abelian categories.

Definition 4.3.15. Let A be any abelian category, and f : N →M a monomorphism in A .(i) We say that M is an essential extension of N if the following holds. For any subobject

P ⊂ M we have either P = 0 or P ∩ Im f 6= 0 (here 0 denotes the zero object of A :see remark 1.2.29(i)).

(ii) We say that M is a proper essential extension of N if it is an essential extension of N ,and f is not an isomorphism.

(iii) We say that M is an injective hull of N , if M is both an essential extension of N , andan injective object of A .

Lemma 4.3.16. Let A be an abelian category, and I an object of A . Suppose that :(a) A is cocomplete.(b) All colimits of A are universal (see example 1.1.24(v)).

Then we have :(i) I is injective if and only if it does not admit any proper essential extensions.

(ii) If N → M is any monomorphism, the set of all essential extensions of N contained inM admits maximal elements.

Proof. (i): Suppose that I is injective, and let f : I → M be any monomorphism which isnot an isomorphism. Then f admits a left inverse, so I is a direct summand of M , hence Mis not an essential extension of I . Conversely, suppose that I does not admit proper essentialextensions; let f : N →M be a monomorphism in A and g : N → I any morphism in A . Weconsider the cocartesian diagram in A

Nf //

g

M

g′

I

f ′ // P

and notice that f ′ is a monomorphism, since the same holds for f . By Zorn’s lemma – and dueto conditions (a) and (b) – we may find a maximal subobject Q ⊂ P such that Q ∩ I = 0, andclearly P/Q is an essential extension of I (via h); therefore P/Q = I , so P = I ⊕Q, and if welet p : P → I be the resulting projection, we see that p g′ : M → I is an extension of f . Thisshows that I is injective.


(ii): By Zorn’s lemma, it suffices to check the following assertion. Suppose that (Pj | j ∈ J)is a totally ordered family of essential extensions of N contained in N (for some totally orderedsmall indexing set J). Then P :=

⋃j∈J Pj is again an essential extension of N (notice that this

union (i.e. colimit) is a subobject of M , due to condition (b)). However, say that Q ⊂ P andQ ∩N = 0; due to condition (b), we have Q =

⋃j∈J(Q ∩ Pj), and clearly (Q ∩ Pj) ∩N = 0,

so Q ∩ Pj = 0 for every j ∈ J , and finally Q = 0.

Proposition 4.3.17. Let A be an abelian category fulfilling condition (a) and (b) of lemma4.3.16, and M any object of A . We suppose moreover that :

(c) A has enough injective objects.Then we have :

(i) M admits an injective hull. More precisely, if M → I is any monomorphism into aninjective object, then a maximal essential extension of M in I is an injective hull of M .

(ii) Let f : M → E be an injective hull of M , and g : M → I any monomorphism into aninjective object. Then g factors through f and a monomorphism h : E → I .

(iii) If f : M → E and g : M → E ′ are two injective hulls of M , there exists an isomor-phism h : E

∼→ E ′ such that h f = f ′.

Proof. (i): Let E ⊂ I be such a maximal essential extension of M (which exists by lemma4.3.16(ii)); by virtue of lemma 4.3.16(i), it suffices to check that E does not admit properessential extensions. However, suppose that E → E ′ is a proper essential extension; since I isinjective, the inclusion morphism E → I extends to a morphism f : E ′ → I . By maximality ofE, we must then have Ker f 6= 0; on the other hand, obviously E ∩Ker f = 0, which is absurd,since E ′ is an essential extension of E.

(ii): Since I is injective, g extends to a morphism h : E → I , and clearly M ∩ Kerh = 0.Since E is an essential extension of M , it follows that Kerh = 0.

(iii): By (ii) there exists a monomorphism h : E → E ′ such that h f = f ′. Since E isinjective, it follows that Imh is a direct summand of E ′; but E ′ is an essential extension of M ,so Imh = E ′, i.e. h is also an epimorphism, hence an isomorphism.

4.3.18. We specialize now to the case where A is the category A-Mod of A-modules, withA an arbitrary noetherian ring. Recall that, if M is any A-module, then the set AssM of allassociated primes of M consists of the prime ideals p ⊂ A such that there exists m ∈ Mwith AnnA(m) = p. (This is the correct definition only for noetherian rings : we shall see indefinition 5.5.1 a more general notion that is well behaved for arbitrary rings.) By proposition4.3.17(i,iii) the injective hull of M exists and is well defined up to (in general, non-unique)isomorphism, and we shall denote by EA(M) a choice of such hull.

Lemma 4.3.19. Let A be a noetherian ring. The following holds :(i) If (Iλ | λ ∈ Λ) is a (small) family of injective A-modules, then I :=

⊕λ∈Λ Iλ is an

injective A-module.(ii) If S ⊂ A is any multiplicative subset, and M any injective A-module, then MS is an

injective AS-module.

Proof. (i): Let us first recall :

Claim 4.3.20. Let R be any ring, M an R-module. The following conditions are equivalent :(a) M is an injective R-module.(b) For every ideal J ⊂ R, every R-linear map J → M extends to an R-linear map

R→M .(c) Ext1

R(R/J,M) = 0 for every ideal J ⊂ R.


Proof of the claim. Of course, (a)⇒(b)⇒(c). To check that (c)⇒(a), let N ⊂ P be an inclusionof R-modules, and f : N →M an A-linear map. We let S be the set of all pairs (N ′, f ′), whereN ′ ⊂ P is an R-submodule containing N , and f ′ : N ′ →M an R-linear map extending f . Theset S is partially ordered, by declaring that (N ′, f ′) ≥ (N ′′, f ′′) if N ′′ ⊂ N ′ and f ′|N ′′ = f ′′, forany two pairs (N ′, f ′), (N ′′, f ′′) ∈ S. By Zorn’s lemma, S admits a maximal element (Q, g).Suppose Q 6= P , and let x ∈ P \Q. Set Q′ := Q + Rx and J := AnnR(Q′/Q); there followsan exact sequence of R-modules

0→ Q→ Q′ → R/J → 0

and then (c) implies that the restriction map HomR(Q′,M) → HomR(Q,M) is surjective;especially, g extetnds to an R-linear map Q′ → M , contradicting the maximality of Q. HenceQ = P , which shows (a). ♦

In view of claim 4.3.20, it suffices to show that every A-linear map f : J → I from any idealJ ⊂ A, extends to an A-linear map A→ I . However, since J is finitely generated, there existsa finite subset Λ′ ⊂ Λ such that the image of f is contained in I ′ :=

⊕λ∈Λ′ Iλ. Clearly I ′ is an

injective A-module, so f extends to an A-linear map A→ I ′, and the assertion follows.(ii): Let J ⊂ AS be any ideal, and write J = IS for some ideal I ⊂ A; since A is noetherian,

we haveExt1

AS(AS/J,MS) = AS ⊗A Ext1

A(A,M)

so the assertion follows from claim 4.3.20.

We may now state :

Proposition 4.3.21. Let A be any noetherian ring, M any A-module. We have :(i) For every p ∈ SpecA, the A-module EA(A/p) is indecomposable (see (4.3.2)).

(ii) Let I be any non-zero injective A-module, and p ∈ Ass I any associated prime. ThenEA(A/p) is a direct summand of I . Especially, if I is indecomposable, then I is iso-morphic to EA(A/p).

(iii) AssAM = AssAEA(M).(iv) If p, q ∈ SpecA are any two prime ideals, then the A-modules EA(A/p) and EA(A/q)

are isomorphic if and only if p = q.(v) EAS(MS) ' AS ⊗A EA(M) for any multiplicative subset S ⊂ A.

Proof. (i): Set B := A/p, and suppose that EA(B) is decomposable; especially, there existnon-zero submodules M1,M2 ⊂ EA(B) with M1 ∩M2 = 0. Thus, (M1 ∩B)∩ (M2 ∩B) = 0,and since B is a domain, we deduce that Mi ∩ B = 0 for i = 1, 2. But since EA(B) is anessential extension of B, this is absurd.

(ii): By assumption, there exists x ∈ I such that AnnA(x) = p, so the submodule Ax ⊂I is isomorphic to A/p; then, by proposition 4.3.17(ii) there exists a monomorphism f :EA(A/p)→ I , and since EA(A/p) in injective, the image of f is a direct summand of I .

(iii): Clearly AssA(M) ⊂ AssAEA(M). Conversely, suppose p ∈ AssAEA(M); then thereexists anA-submoduleN ⊂ EA(M) isomorphic toA/p. SinceEA(M) is an essential extensionof M , we have N ∩M 6= 0, so p ∈ AssA(M).

(iv) follows directly from (iii).(v): We know already that E ′ := AS ⊗A EA(M) contains MS and is injective (lemma

4.3.19(iii)), so it remains only to check that E ′ is an essential extension of M . Thus, letx ∈ E ′ \ MS; we have to check that N := ASx ∩ MS 6= 0, and clearly we may assumethat x ∈ EA(M). Set F := AnnA(tx) | t ∈ S; since A is noetherian, F admits maximalelements, and notice that ASx = AStx for any t ∈ S. Hence, we may replace x by tx for somet ∈ S, after which we may assume that AnnA(x) is maximal in F . We may writeAx∩M = Ixfor some ideal I ⊂ A, so N = ISx; let a1, . . . , ak ∈ A be a system of generators for I , and


notice that Ix 6= 0, since EA(M) is an essential extension of M . Suppose that N = 0; thenthere exists t ∈ S such that the identity taix = 0 holds in A for i = 1, . . . , k. However,AnnA(x) = AnnA(tx) by construction, so Ix = 0, a contradiction.

Theorem 4.3.22. Let A be a noetherian ring, I an injective A-module. We have :(i) I decomposes as a direct sum of the form

(4.3.23) I '⊕

p∈SpecA

EA(A/p)(Rp)

for a system of (small) sets (Rp | p ∈ SpecA).(ii) Moreover, the cardinality of Rp equals dimκ(p) HomAp(κ(p), Ip), for every p ∈ SpecA

(where κ(p) := Ap/pp). Especially, this cardinality is independent of the decomposi-tion (4.3.23).

Proof. (i): Denote by F the set of all indecomposable injective submodules of I , and by Sthe set of all subsets G ⊂ F such that the natural map IG :=

⊕G∈G G → I is injective. The

set S is partially ordered by inclusion, and by Zorn’s lemma, S admits a maximal elementM ; in light of proposition 4.3.21(ii), it suffices to check that IM = I . However, IM is injective(lemma 4.3.19(i)), hence I = IM ⊕ J for some A-module J , and it is easily seen that J isinjective as well. We are thus reduced to showing that J = 0. Now, if J 6= 0, let p ∈ AssAJbe any associated prime ([61, Th.6.1(i)]); then EA(A/p) is an indecomposable injective directsummand of J (proposition 4.3.21(i,ii)), and therefore IM ⊕ EA(A/p) is a submodule of I ,contradicting the maximality of M .

(ii): In light of (i) and proposition 4.3.21(iii,v) we have

HomAp(κ(p), Ip) = HomAp(κ(p), EA(A/p)(Rp)p ) = κ(p)(Rp) ⊗κ(p) HomAp(κ(p), EAp(κ(p)))

so we are reduced to showing the following :

Claim 4.3.24. Let m ⊂ A be any maximal ideal, and set κ := A/m; then

d := dimκ HomA(κ,EA(κ)) = 1.

Proof of the claim. Since κ ⊂ EA(κ), obviously d ≥ 1. On the other hand, we have a naturalidentification

HomA(κ,EA(κ)) = F := x ∈ EA(κ) | mx = 0.If d > 1, we may find x ∈ F such that Ax ∩ κ = 0; but this is absurd, since EA(κ) is anessential extension of κ.

4.3.25. When dealing with the more general coherent rings that appear in sections 5.7 and 5.8,the injective hull is no longer suitable for the study of local cohomology, but it turns out thatone can use instead a “coh-injective hull”, which works just as well.

Definition 4.3.26. Let A be a ring; we denote by A-Modcoh the full subcategory of A-Modconsisting of all coherent A-modules.

(i) An A-module J is said to be coh-injective if the functor

A-Modcoh → A-Modo : M 7→ HomA(M,J)

is exact.(ii) An A-module M is said to be ω-coherent if it is countably generated, and every finitely

generated submodule of M is finitely presented.

Lemma 4.3.27. Let A be a ring.


(i) Let M , J be two A-modules, N ⊂ M a submodule. Suppose that J is coh-injectiveand both M and M/N are ω-coherent. Then the natural map:

HomA(M,J)→ HomA(N, J)

is surjective.(ii) Let (Jλ | λ ∈ Λ) be a filtered family of coh-injective A-modules. Then colim

λ∈ΛJλ is

coh-injective.(iii) Assume that A is coherent, let M• be an object of D−(A-Modcoh) and I• a bounded

below complex of coh-injective A-modules. Then the natural map

Hom•A(M•, I•)→ RHom•A(M•, I

•)

is an isomorphism in D(A-Mod).

Proof. (i): Since M is ω-coherent, we may write it as an increasing union⋃n∈NMn of finitely

generated, hence finitely presented submodules. For each n ∈ N, the image of Mn in M/N isa finitely generated, hence finitely presented submodule; therefore Nn := N ∩Mn is finitelygenerated, hence it is a coherent A-module, and clearly N =

⋃n∈NNn. Suppose ϕ : N → J is

any A-linear map; for every n ∈ N, set Pn+1 := Mn +Nn+1 ⊂M , and denote by ϕn : Nn → Jthe restriction of ϕ. We wish to extend ϕ to a linear map ϕ′ : M → J , and to this aim weconstruct inductively a compatible system of extensions ϕ′n : Pn → J , for every n > 0. Forn = 1, one can choose arbitrarily an extension ϕ′1 of ϕ1 to P1. Next, suppose n > 0, andthat ϕ′n is already given; since Pn ⊂ Mn, we may extend ϕ′n to a map ϕ′′n : Mn → J . SinceMn ∩Nn+1 = Nn, and since the restrictions of ϕ′′n and ϕn+1 agree on Nn, there exists a uniqueA-linear map ϕ′n+1 : Pn+1 → J that extends both ϕ′′n and ϕn+1.

(ii): Recall that a coherent A-module is finitely presented. However, it is well known thatan A-module M is finitely presented if and only if the functor Q 7→ HomA(M,Q) on A-modules, commutes with filtered colimits (see e.g. [36, Prop.2.3.16(ii)]). The assertion is aneasy consequence.

(iii): Since A is coherent, one can find a resolution ϕ : P• → M• consisting of free A-modules of finite rank (cp. [36, §7.1.20]). One looks at the spectral sequences

Epq1 : HomA(Pp, I

q)⇒Rp+qHom•A(M•, I•)

F pq1 : HomA(Mp, I

q)⇒Hp+qHom•A(M•, I•).

The resolution ϕ induces a morphism of spectral sequences F ••1 → E••1 ; on the other hand,since Iq is coh-injective, we have : Epq

2 ' HomA(HpF•, Iq) ' HomA(HpM•, I

q) ' F pq2 , so

the induced map F pq2 → Epq

2 is an isomorphism for every p, q ∈ N, whence the claim.

4.3.28. Let A be a coherent ring, Z ⊂ SpecA a constructible closed subset; we denote by

A-Modcoh,Z

the full subcategory of A-Modcoh whose objects are the (coherent) A-modules with supportcontained in Z. Let I ⊂ A be any finitely generated ideal such that Z = SpecA/I; it is easilyseen that

Ob(A-Modcoh,Z) =⋃n∈N

Ob(A/In-Modcoh).

Now, consider a functorT : A-Modocoh,Z → Z-Mod.

Notice that TM is naturally an A-module, for every coherent A-module M : indeed, if a ∈ Ais any element, we may define the scalar multiplication by a on TM as the Z-linear endomor-phism T (a · 1M). In other words, T factors through the forgetful functor A-Mod→ Z-Mod;


especially, if we setHT := colim

n∈NT (A/In)

we get a natural A-linear map

M∼→ colim

n∈NHomA(A/In,M)→ colim

n∈NHomA(TM, T (A/In))→ HomA(TM,HT )

whence a bilinear pairingM × TM → HT

which in turns yields a natural transformation

ωM : TM → HomA(M,HT ) for every M ∈ Ob(A-Modcoh,Z).

Lemma 4.3.29. In the situation of (4.3.28), the following conditions are equivalent :(a) ω is an isomorphism of functors T ∼→ HomA(−, HT ).(b) T is left exact.

Proof. Clearly (a)⇒(b). For the converse, suppose first that Z = SpecA, in which case, noticethat HT = TA and ωA is the natural isomorphism TA

∼→ HomA(A, TA); then, if M is anycoherent A-module, pick a finite presentation

Σ : A⊕n → A⊕m →M → 0

and apply the 5-lemma to the resulting ladder of A-modules ωΣ with left exact rows, to deducethat ωM is an isomorphism. Next, for a general ideal I and M an A-module with SuppM ⊂ Z,we may find n ∈ N such that M is an A/In-module; since A/Ik is coherent, the foregoing casethen shows that the induced map

TM → HomA(M,T (A/Ik))

is an isomorphism, for every k ≥ n. To conclude, it suffices to remark that the natural map

colimk∈N

HomA(M,T (A/Ik))→ HomA(M,HT )

is an isomorphism, since M is finitely presented ([36, Prop.2.3.16(ii)]).

We wish next to present a criterion that allows to detect, among the functors T as in (4.3.28),those that are exact. A complete characterization shall be given only for a restricted class ofcoherent ring; namely, we make the following :

Definition 4.3.30. LetA be a coherent ring. We say thatA is an Artin-Rees ring, if the followingholds. For every finitely generated ideal I ⊂ A, every coherent A-module M , and every finitelygenerated A-submodule N ⊂M , the I-adic topology of M induces the I-adic topology on N .

We can then state :

Proposition 4.3.31. In the situation of (4.3.28) suppose furthermore that A is an Artin-Reesring. Then the following conditions are equivalent :

(a) HT is a coh-injective A-module.(b) T is exact.

Proof. Clearly (a)⇒(b). For the converse, let M be any coherent A-module, and N ⊂ M anyfinitely generated A-submodule; it suffices to check that the induced map

HomA(M,HT )→ HomA(N,HT )

is surjective. However, let f : N → HT be any A-linear map; since N is finitely presented,there exists n ∈ N such that f factors through an A-linear map fn : N → T (A/In) ([36,Prop.2.3.16(ii)]), and clearly InN ⊂ Ker fn, so InN ⊂ Ker f as well. Since the I-adic topol-ogy ofN agrees with the topology induced by the I-adic topology ofM , there exists k ∈ N such


that IkM ∩ N ⊂ InN . Set N := N/(IkM ∩ N); then N is a finitely generated A-submoduleof M/IkM , and especially it is a coherent A-module. By construction, f factors through an A-linear map f : N → HT ; assumption (b) and lemma 4.3.29 imply that f extends to an A-linearmap M/IkM → HT , and the resulting map M → HT extends f , whence (a).

Remark 4.3.32. (i) In the terminology of definition 4.3.30, the standard Artin-Rees lemmaimplies that every noetherian ring is an Artin-Rees ring. We shall see later that, if V is any val-uation ring, then every essentially finitely presented V -algebra is an Artin-Rees ring (corollary5.7.2 and theorem 5.7.29).

(ii) On the other hand, if A is noetherian, claim 4.3.20 easily implies that an A-module iscoh-injective if and only if it is injective.

(iii) Combining (i) and (ii) with proposition 4.3.31, we recover [44, Exp.IV, Prop.2.1].

Example 4.3.33. Let κ0 be a field, A a noetherian κ0-algebra, m ⊂ A a maximal ideal suchthat κ := A/m is a finite extension of κ0, and set Z := m ⊂ SpecA. Then every object ofA-Modcoh,Z is a finite dimensional κ0-vector space, and therefore the functor

T : A-Modcoh,Z → κ0-Mod M 7→ Homκ0(M,κ0)

is exact. Proposition 4.3.31 and remark 4.3.32(ii) then say that

HT := colimn∈N

Homκ0(A/mn, κ0)

is an injectiveA-module. More precisely, notice that HomA(κ,HT ) = AnnHT (m) = T (κ) ' κ,therefore HT is the injective hull of the residue field κ.

4.3.34. Flatness criteria. The following generalization of the local flatness criterion answersaffirmatively a question raised in [32, Ch.IV, Rem.11.3.12].

Lemma 4.3.35. Let A be a ring, I ⊂ A an ideal, B a finitely presented A-algebra, p ⊂ Ba prime ideal containing IB, and M a finitely presented B-module. Then the following twoconditions are equivalent:

(a) Mp is a flat A-module.(b) Mp/IMp is a flat A/I-module and TorA1 (Mp, A/I) = 0.

Proof. Clearly, it suffices to show that (b)⇒(a), hence we assume that (b) holds. We write Aas the union of the filtered family (Aλ | λ ∈ Λ) of its local noetherian subalgebras, and setA′ := A/I , Iλ := I ∩ Aλ, A′λ := Aλ/Iλ for every λ ∈ Λ. Then, for some λ ∈ Λ, the A-algebra B descends to an Aλ-algebra Bλ of finite type, and M descends to a finitely presentedBλ-module Mλ. For every µ ≥ λ we set Bµ := Aµ ⊗Aλ Bλ and Mµ := Bµ ⊗Bλ Mλ. Up toreplacing Λ by a cofinal family, we can assume that Bµ and Mµ are defined for every µ ∈ Λ.Then, for every λ ∈ Λ let gλ : Bλ → B be the natural map, and set pλ := g−1

λ p.

Claim 4.3.36. There exists λ ∈ Λ such that Mλ,pλ/IλMλ,pλ is a flat A′λ-module.

Proof of the claim. Set B′λ := Bλ ⊗Aλ A′λ and M ′λ := Mλ ⊗Aλ A′λ for every λ ∈ Λ; clearly the

natural maps

colimλ∈Λ

A′λ → A/IA colimλ∈Λ

B′λ → B/IB colimλ∈Λ

M ′λ →M/IM

are isomorphisms. Then the claim follows from (b) and [32, Ch.IV, Cor.11.2.6.1(i)]. ♦

In view of claim 4.3.36, we can replace Λ by a cofinal subset, and thereby assume thatMλ,pλ/IλMλ,pλ is a flat A′λ-module for every λ ∈ Λ.

Claim 4.3.37. (i) The natural map: colimλ∈Λ

TorAλ1 (Mλ, A′λ)→ TorA1 (M,A′) is an isomorphism.


(ii) For every λ, µ ∈ Λ with µ ≥ λ, the natural map:

fλµ : Aµ ⊗Aλ TorAλ1 (Mλ, A′λ)→ Tor

Aµ1 (Mµ, A

′µ)

is surjective.

Proof of the claim. For every λ ∈ Λ, let L•(Mλ) denote the canonical free resolution of theAλ-module Mλ ([17, Ch.X, §3, n.3]). Similarly, denote by L•(M) the canonical free resolutionof the A-module M . It follows from [18, Ch.II, §6, n.6, Cor.] and the exactness properties offiltered colimits, that the natural map: colim

λ∈ΛL•(Mλ)→ L•(M) is an isomorphism. Hence:

colimλ∈Λ

H•(A′λ

L⊗Aλ Mλ) ' colim

λ∈ΛH•(A

′λ ⊗Aλ L•(Mλ))

'H•(A′ ⊗A colimλ∈Λ

L•(Mλ))

'H•(A′ ⊗A L•(M))

'H•(A′L⊗AM)

which proves (i). To show (ii) we use the base change spectral sequences for Tor ([75, Th.5.6.6])

E2pq : TorAµp (TorAλq (Mλ, Aµ), A′µ)⇒ TorAλp+q(Mλ, A

′µ)

F 2pq : Tor

A′λp (TorAλq (Mλ, A

′λ), A

′µ)⇒ TorAλp+q(Mλ, A

′µ).

Since F 210 = 0, the natural map

F 201 := A′µ ⊗A′λ TorAλq (Mλ, A

′λ)→ TorAλ1 (Mλ, A

′µ)

is an isomorphism. On the other hand, we have a surjection:

TorAλ1 (Mλ, A′µ)→ E2

10 := TorAµ1 (Mµ, A

′µ)

whence the claim. ♦

We deduce from claim 4.3.37(i) that the natural map

colimλ∈Λ

TorAλ1 (Mλ,pλ , A′λ)→ TorA1 (Mp, A

′) = 0

is an isomorphism. However, TorAλ1 (Mλ,pλ , A′λ) is a finitely generated Bλ,pλ-module by [17,

Ch.X, §6, n.4, Cor.]. We deduce that fλµ,pλ = 0 for some µ ≥ λ; therefore TorAµ1 (Mµ,pµ , A

′µ) =

0, in view of claim 4.3.37(ii), and then the local flatness criterion of [28, Ch.0, §10.2.2] saysthat Mµ,pµ is a flat Aµ-module, so finally Mp is a flat A-module, as stated.

Lemma 4.3.38. Let A be a ring, I ⊂ A an ideal. Then :(i) The following are equivalent :

(a) The map A→ A/I is flat.(b) The map A→ A/I is a localization.(c) For every prime ideal p ⊂ A containing I , we have IAp = 0, especially, V (I) is

closed under generizations in SpecA.(ii) Suppose I fulfills the conditions (a)-(c) of (i). Then the following are equivalent :

(a) I is finitely generated.(b) I is generated by an idempotent.(c) V (I) ⊂ SpecA is open.

Proof. (i): Clearly (b)⇒(a). Also (a)⇒(c), since every flat local homomorphism is faithfullyflat. Suppose that (c) holds; we show that the natural map B := (1 + I)−1A → A/I isan isomorphism. Indeed, notice that IB is contained in the Jacobson radical of B, hence itvanishes, since it vanishes locally at every maximal ideal of B.


(ii): Clearly (c)⇒(b)⇒(a). By condition (i.c), we see that every element of I vanishes on anopen subset of SpecA containing V (I), hence (a)⇒(c) as well.

4.3.39. For every ring A, we let I (A) be the set of all ideals I ⊂ A fulfilling the equivalentconditions (i.a)-(i.c) of lemma 4.3.38, and Z (A) the set of all closed subsetZ ⊂ SpecA that areclosed under generizations in SpecA. In light of lemma 4.3.38(i.c), we have a natural mapping:

(4.3.40) I (A)→ Z (A) : I 7→ V (I).

Lemma 4.3.41. The mapping (4.3.40) is a bijection, whose inverse assigns to any Z ∈ Z (A)the ideal I[Z] consisting of all the elements f ∈ A such that fAp = 0 for every p ∈ Z.

Proof. Notice first that I[V (I)] = I for every I ∈ I (A); indeed, clearly I ⊂ I[V (I)], and iff ∈ I[V (I)], then fAp = 0 for every prime ideal p containing I , hence the image of f in A/Ivanishes, so f ∈ I . To conclude the proof, it remains to show that, if Z is closed and closedunder generizations, then V (I[Z]) = Z. However, say that Z = V (J), and suppose that f ∈ J ;then SpecAp ⊂ Z for every p ∈ Z, hence f is nilpotent in Ap, so there exists n ∈ N and anopen neighborhood U ⊂ SpecA of p such that fn = 0 in U . Since Z is quasi-compact, finitelymany such U suffice to cover Z, hence we may find n ∈ N large enough such that fn ∈ I[Z],whence the contention.

Proposition 4.3.42. Let f : X ′ → X a quasi-compact and faithfully flat morphism of schemes.Then the topological space underlying X is the quotient of X ′ under the equivalence relationinduced by f .

Proof. The assertion means that a subset Z ⊂ X is open (resp. closed) if and only if the sameholds for the subset f−1Z of X ′. For any subset Z of X (resp. of Y ′), we denote by Z thetopological closure of Z inX (resp. in Y ′). The proposition will result from the following moregeneral :

Claim 4.3.43. Let g : Y ′ → X be a flat morphism of schemes, h : Y ′′ → X a quasi-compactmorphism of schemes, and set Z := h(Y ′′). Then

g−1Z = g−1Z.

Proof of the claim. Quite generally, if T is a topological space, U ⊂ T an open subset, Z ⊂ Tany subset, and Z the topological closure of Z in X , then Z ∩ U is the topological closureof Z ∩ U in U (where the latter is endowed with the topology induced by X) : indeed, setZ ′ := Z ∩ U , W := X \ U , and notice that

Z ∪W = Z ∪W = Z ′ ∪W = Z ′ ∪Whence we may assume that Z ⊂ U , in which case the assertion is obvious.

Now, let (Ui | i ∈ I) be any affine open covering of X , and set U ′i := g−1Ui, U ′′ := h−1Uifor every i ∈ I; by the foregoing, we are reduced to showing the claim with g and h replacedby the morphisms g|U ′i : U ′i → Ui and h|U ′′i : U ′′i → Ui, hence we may assume that X is affine.Likewise, by considering an affine open covering of Y ′ and arguing similarly, we reduce to thecase where Y ′ is affine as well. In this situation, since h is quasi-compact, Y ′′ is a finite union ofaffine open subsets U ′′1 , . . . , U

′′n , hence we may replace Y ′′ by the disjoint union of U ′′1 , . . . , U

′′n ,

and assume that Y ′′ is also affine. Say that X = SpecA, Y ′ = SpecB′ and Y ′′ = SpecB′′,and denote by I ⊂ A the kernel of the ring homomorphism A → B′′ corresponding to h; thenZ = SpecA/I . Let h′ : Y ′×X Y ′′ → Y ′ be the morphism obtained from h by base change, andnotice that g−1Z is the image of h′; therefore, if we let I ′ ⊂ B′ be the kernel of the induced mapB′ → B′⊗AB′′, we have g−1Z = SpecB′/I ′. However, since B′ is a flat A-algebra, I ′ = IB′,whence the claim. ♦


Let now Z ⊂ X be a subset, such that Z ′ := f−1Z is closed in X ′, and endow Z ′ with thereduced subscheme structure. The restriction h : Z ′ → X of f is then quasi-compact, andclearly f−1(hZ ′) = Z ′. Thus, claim 4.3.43 says that f−1Z = Z ′, which means that Z = Z, i.e.Z is closed, as stated.

Corollary 4.3.44. LetA be a ring whose set MinA (resp. MaxA) of minimal prime (resp. max-imal) ideals is finite. If Z ⊂ SpecA is a subset closed under specializations and generizations,then Z is open and closed.

Proof. When MinA is finite, any such Z is a finite union of irreducible components, hence it isclosed. But the same applies to the complement of Z, whence the contention. When A is semi-local, consider the faithfully flat map A→ B, where B is the product of the localizations of Aat its maximal ideals. Clearly the assertion holds for B, hence for A, by proposition 4.3.42.

The following result is borrowed from [65, Cor.2.6].

Proposition 4.3.45. Let R be a ring, R′ any finitely generated R-algebra, and suppose thatSpecR is a noetherian topological space. Then the same holds for SpecR′.

Proof. Let us first remark :

Claim 4.3.46. Let T be a quasi-compact and quasi-separated topological space. We have :(i) T is noetherian if and only if every closed subset of T is constructible.

(ii) Denote by C the set of all non-constructible closed subsets of T , partially ordered byinclusion. If C is not empty, it admits minimal elements, and every minimal elementof C is an irreducible subset of T .

Proof of the claim. (i) shall be left to the reader.(ii): Let Zi | i ∈ I be a totally ordered subset of C , and set Z :=

⋂i∈I Zi; we claim

that Z ∈ C . Indeed, if this fails, Z is constructible, hence its complement is a quasi-compactopen subset of T , and therefore it equals T \Zi for some i ∈ I; but then Z = Zi, so Zi isconstructible, a contradiction. By Zorn’s lemma, we deduce that C admits minimal elements.Let Z ∈ C be such a minimal element, and suppose that Z is not irreducible; then Z = Z1∪Z2

for some closed subsets Z1, Z2 strictly contained in Z. By minimality of Z, both Z1 and Z2 areconstructible, and therefore the same must hold for Z, a contradiction. ♦

Let R′ be a finitely generated R-algebra. In order to show that SpecR′ is noetherian, wemay assume that R′ is a free polynomial R-algebra of finite type, and then an easy inductionreduces to the case where R′ = R[X]. Now, suppose Y := SpecR[X] is not noetherian, andlet f : Y → X := SpecR; by claim 4.3.46 we may find an irreducible non-contructible closedsubset Z ⊂ Y . By assumption, the topological closure W of f(Z) in X is an irreducibleconstructible closed subset, and therefore f−1W is a constructible closed subset of Y (recallthat every morphism of schemes is continuous for the constructible topology). It follows thatevery constructible subset of f−1W is also constructible as a subset of Y ; especially, Z is notconstructible in f−1W . We may then replace X by W and Y by f−1W , and assume from startthat f(Z) is a dense subset of X and R is a domain, in which case we set K := FracR. Letp ∈ Y be the generic point of Z; so p is a prime ideal of R[X], and pK[X] is a principal idealgenerated by some polynomial p(X) ∈ p. Let c ∈ R be the leading coefficient of p(X), setU := SpecR[c−1] and notice that f(Z) ∩ U 6= ∅. Hence, Z \ f−1U is a closed subset of Ystrictly contained in Z, so it is constructible. It follows that Z ∩ f−1U is a non-constructiblesubset of Y ; since f−1U is a constructible subset of Y , we conclude that Z ∩ f−1U is a non-constructible subset of f−1U . Hence, we may replace R by R[c−1] and assume from start thatpK[X] is generated by a monic polynomial p(X) ∈ p. However, we notice :


Claim 4.3.47. Let A be a domain, K the field of fractions of A, p ⊂ A[X] any ideal, andp(X) ∈ p a monic polynomial that generates pK[X]. Then p(X) generates p.

Proof of the claim. This is elementary : consider any g(X) ∈ p, and let g(X) = q(X) · p(X) +r(X) be the euclidean division of g by p in K[X]. Since p generates pK[X], we have r = 0,and since p is monic, it is easily seen that q ∈ A[X], whence the claim. ♦

Claim 4.3.47 says that p is a principal ideal ofR[X]; but in that case, Z must be constructible,a contradiction.

4.4. Graded rings, filtered rings and differential graded algebras.

4.4.1. Graded rings. To motivate the results of this paragraph, let us review briefly the wellknown correspondance between Γ-gradings on a module M (for a given commutative group Γ),and actions of the diagonalizable group D(Γ) on M .

Let S be a scheme; on the category Sch/S of S-schemes we have the presheaf of rings :

OSch/S : Sch/S → Z-Alg (X → S) 7→ Γ(X,OX).

Also, let M be an OS-module; following [24, Exp.I, Def.4.6.1], we attach to M the OSch/S-module WM given by the rule : (f : X → S) 7→ Γ(X, f ∗M). If M and N are two quasi-coherent OS-modules, and f : S ′ → S an affine morphism, it is easily seen that

(4.4.2) Γ(S ′,HomOSch/S(WM ,WN)) = HomOS(M, f∗OS′ ⊗OS N)

(see [24, Exp.I, Prop.4.6.4] for the details).Let f : G→ S be a group S-scheme, i.e. a group object in the category Sch/S. If f is affine,

we say that G is an affine group S-scheme; in that case, the mutiplication law G×SG→ G andthe unit section S → G correspond respectively to morphisms of OS-algebras

∆G : f∗OG → f∗OG ⊗OS f∗OG εG : OS → f∗OG

which make commute the diagram :

f∗OG∆G //

∆G

f∗OG ⊗OS f∗OG

1f∗OS⊗∆G

f∗OG ⊗OS f∗OG

∆G⊗1f∗OS // f∗OG ⊗OS f∗OG ⊗OS f∗OG

as well as a similar diagram, which expresses the unit property of εG : see [24, Exp.I, §4.2].

Example 4.4.3. Let G be any commutative group. The presheaf of groups

DS(G) : Sch/S → Z-Mod (X → S) 7→ HomZ-Mod(G,O×X (X))

is representable by an affine group S-scheme DS(G), called the diagonalizable group schemeattached to G. Explicitly, if S = SpecR is an affine scheme, the underlying S-scheme ofDS(G) is SpecR[G], and the group law is given by the map of R-algebras

∆G : R[G]→ R[G]⊗R R[G]∼→ R[G×G] g 7→ (g, g) for every g ∈ G

with unit εG : R[G] → R given by the standard augmentation (see [24, Exp.I, §4.4]). For ageneral scheme S, we have DS(G) = DSpecZ(G) ×SpecZ S (with the induced group law andunit section).

Definition 4.4.4. Let M be an OS-module, G a group S-scheme. A G-module structure on Mis the datum of a morphism of presheaves of groups on Sch/S :

hG → AutOSch/S(WM)

(where hG denotes the Yoneda imbedding : see (1.1.19)).


4.4.5. Suppose now that f : G → S is an affine group S-scheme, and M a quasi-coherentOS-module; in view of (4.4.2), a G-module structure on M is then the same as a map of OS-modules

µM : M → f∗OG ⊗OS M

which makes commute the diagrams :

MµM //

µM

f∗OG ⊗OS M

∆G⊗1M

MµM //

1M((QQQQQQQQQQQQQQQQQQ f∗OG ⊗OS M

εG⊗1M

f∗OG ⊗OS M

1f∗OG⊗µM

// f∗OG ⊗OS f∗OG ⊗OS M M.

Example 4.4.6. Let Γ be a commutative group; a DS(Γ)-module structure on a quasi-coherentOS-module M is the datum of a morphism of OS-modules

µM : M → OS[Γ]⊗OS M = Z[Γ]⊗Z M

which makes commute the diagrams of (4.4.5). If S = SpecR is affine, M is associated to anR-module which we denote also by M ; in this case, µM is the same as a system (µ

(γ)M | γ ∈ Γ)

of OS-linear endomorphisms of M , such that :

• for every x ∈M , the subset γ ∈ Γ | µ(γ)M (x) 6= 0 is finite.

• µ(γ)M µ

(τ)M = δγ,τ · 1M and

∑γ∈Γ µM = 1M .

In other words, the µ(γ)M form an orthogonal system of projectors of M , summing up to the

identity 1M . This is the same as the datum of a Γ-grading on M : namely, for a given DS(Γ)-module structure µM , one lets

grγM := µ(γ)M (M) for every γ ∈ Γ

and conversely, given a Γ-grading gr•M on M , one defines µM as the R-linear map given bythe rule : x 7→ γ ⊗ x for every γ ∈ Γ and every x ∈ grγM .

4.4.7. Suppose now that g : X → S is an affine S-scheme, and f : G → S an affine groupS-scheme. A G-action on X is a morphism of presheaves of groups :

hG → AutSch/S∧(hX).

(notation of (1.1.19)); the latter is the same as a morphism of S-schemes

(4.4.8) G×S X → X

inducing a G-module structure on g∗OX :

g∗OX → f∗OG ⊗OS g∗OX

which is also a morphism of OS-algebras. For instance, if S = SpecR is affine, andG = DS(Γ)for an abelian group Γ, we may write X = SpecA for some R-algebra B, and in view ofexample 4.4.6, the G-action on X is the same as the datum of a Γ-graded R-algebra structureon B, in the sense of the following :

Definition 4.4.9. Let (Γ,+) be a commutative monoid, R a ring.(i) A Γ-graded R-algebra is a pair B := (B, gr•B) consisting of an R-algebra B and a

Γ-grading B =⊕

γ∈Γ grγB of the R-module B, such that

grγB · grγ′B ⊂ grγ+γ′B for every γ, γ′ ∈ Γ.

A morphism of Γ-graded R-algebras is a map of R-algebras which is compatible withthe gradings, in the obvious way.


(ii) Let B := (B, gr•B) be a Γ-graded R-algebra. A Γ-graded B-module is a datumM := (M, gr•M) consisting of a B-module M and a Γ-grading M =

⊕γ∈Γ grγM of

the R-module underlying M , such that

grγB · grγ′M ⊂ grγ+γ′M for every γ, γ′ ∈ Γ.

A morphism of Γ-graded B-module is a map f : M → N of B-modules, such thatf(grγM) ⊂ grγN , for every γ ∈ Γ.

(iii) If f : Γ′ → Γ is any morphism of commutative monoids, and M is any Γ-gradedR-module, we define the Γ′-graded R-module Γ′ ×Γ M by setting

grγ(Γ′ ×Γ M) := grf(γ)M for every γ ∈ Γ′.

Notice that if B := (B, gr•B) is a Γ-graded R-algebra, then Γ′ ×Γ B with its gradinggives a Γ′-gradedR-algebra Γ′×ΓB, with the multiplication and addition laws deducedfrom those of B, in the obvious way. Likewise, if (M, gr•M) is a Γ-graded B-module,then Γ′ ×Γ M is naturally a Γ′-graded Γ′ ×Γ B-module.

(iv) Furthermore, if N is any Γ′-graded R-module, we define the Γ-graded R-module N/Γ

whose underlying R-module is the same as N , and whose grading is given by the rule

grγ(N/Γ) :=⊕

γ′∈f−1(γ)

grγN.

Just as in (iii), if C := (C, gr•C) is a Γ′-graded R-algebra, then we get a Γ-gradedR-algebra C/Γ, whose underlying R-algebra is the same as C. Lastly, if (N, gr•N) isa Γ′-graded C-module, then N/Γ is a Γ-graded C/Γ-module.

Example 4.4.10. (i) For instance, the R-algebra R[Γ] is naturally a Γ-graded R-algebra, whenendowed with the Γ-grading such that grγR[Γ] := γR for every γ ∈ Γ.

(ii) Suppose that Γ is an integral monoid. Then, to a Γ-graded R-algebra B, the correspon-dance described in (4.4.7) associates a DS(Γgp)-action on SpecB (where S := SpecR), givenby the map of R-algebras

ϑB : B → B[Γ] ⊂ B[Γgp] : b 7→ b · γ for every γ ∈ Γ, and every b ∈ grγB.

Remark 4.4.11. (i) Let R be a ring; consider a cartesian diagram of monoids

Γ3//

Γ1

Γ2

// Γ0

and let B be any Γ1-graded R-algebra. A simple inspection of the definitions yields an identityof Γ2-graded R-algebras :

Γ2 ×Γ0 B/Γ0 = (Γ3 ×Γ1 B)/Γ2 .

(ii) Suppose that Γ is a finite abelian group, whose order is invertible in OS . Then DS(Γ) isan etale S-scheme. Indeed, in light of (2.3.52), the assertion is reduced to the case where Γ =Z/nZ for some integer n > 0 which is invertible in OS . However, R[Z/nZ] ' R[T ]/(T n − 1),which is an etale R-algebra, if n ∈ R×.

(iii) More generally, suppose that Γ is a finitely generated abelian group, such that the orderof its torsion subgroup is invertible in OS . Then we may write Γ = L ⊕ Γtor, where L is a freeabelian group of finite rank, and Γtor is a finite abelian group as in (ii). In view of (2.3.52) and(ii), we conclude that DS(Γ) is a smooth S-scheme in this case.

(iv) Let Γ be as in (iii), and suppose that X is an S-scheme with an action of G := DS(Γ).Then the corresponding morphism (4.4.8) and the projection pG : G×SX → G induce an auto-morphism of theG-schemeG×SX , whose composition with the projection pX : G×SX → X


equals (4.4.8). We then deduce that both (4.4.8) and pX are smooth morphisms. This observa-tion, together with the above correspondance between Γ-graded algebras and DS(Γ)-actions, isthe basis for a general method, that allows to prove properties of graded rings, provided they canbe translated as properties of the corresponding schemes which are well behaved under smoothbase changes. We shall present hereafter a few applications of this method.

Proposition 4.4.12. Let Γ be an integral monoid, B a Γ-graded (commutative, unitary) ring,and suppose that the order of any torsion element of Γgp is invertible in B. Then :

(i) nil(B[Γ]) = nil(B) ·B[Γ].(ii) nil(B) is a Γ-graded ideal of B.

Proof. (i): Clearly nil(B) · B[Γ] ⊂ nil(B[Γ]). To show the converse inclusion, it suffices toprove that nil(B[Γ]) ⊂ pB[Γ] for every prime ideal p ⊂ B, or equivalently that B/p[Γ] is areduced ring for every such p. Since the natural map Γ → Γgp is injective, we may furtherreplace Γ by Γgp, and assume that Γ is an abelian group. In this case, B/p[Γ] is the filteredunion of the rings B/p[H], where H runs over the finitely generated subgroups of Γ; it sufficestherefore to prove that each B/p[H] is reduced, so we may assume that Γ is finitely generated,and the order of its torsion subgroup is invertible in B. In this case, B/p[Γ] is a smooth B/p-algebra (remark 4.4.11(iii)), and the assertion follows from [33, Ch.IV, Prop.17.5.7].

(ii): The assertion to prove is that nil(B) =⊕

γ∈Γ nil(B) ∩ grγB. However, let ϑB : B →B[Γ] be the map defined as in example 4.4.10(ii); now, (i) implies that ϑB restricts to a mapnil(B)→ nil(B) ·B[Γ], whence the contention.

For a ring homomorphism A → B, let us denote by (A,B)ν the integral closure of A in B.We have the following :

Proposition 4.4.13. Let Γ be an integral monoid, f : A → B a morphism of Γ-graded rings,and suppose that the order of any torsion element of Γgp is invertible in A. Then :

(i) (A,B)ν [Γgp] = (A[Γgp], B[Γgp])ν .(ii) The grading of B restricts to a Γ-grading on the subring (A,B)ν .

Proof. (i): We easily reduce to the case where Γ is finitely generated, in which case A[Γgp] is asmooth A-algebra (remark 4.4.11(iii)), and the assertion follows from [31, Ch.IV, Prop.6.14.4].

(ii): We consider the commutative diagram

AϑA //

f

A[Γ]

f [Γ]

BϑB // B[Γ]

where ϑA and ϑB are defined as in example 4.4.10(ii). Say that b ∈ (A,B)ν ; then ϑB(b) ∈(A[Γ], B[Γ])ν . In light of (i), it follows that ϑB(b) ∈ (A,B)ν [Γgp] ∩ B[Γ] = (A,B)ν [Γ]. Theclaim follows easily.

Corollary 4.4.14. In the situation of proposition 4.4.12, suppose moreover, that B is a domain.Then we have :

(i) The integral closure Bν of B in its field of fractions is Γgp-graded, and the inclusionmap B → Bν is a morphism of Γgp-graded rings.

(ii) Suppose furthermore, that Γ is saturated. Then Bν is a Γ-graded ring.

Proof. (Notice that, since Γ is integral, the grading of B extends trivially to a Γgp-grading, bysetting grγB := 0 for every γ ∈ Γgp \ Γ.) Clearly B is the filtered union of its subalgebras∆ ×Γ B, where ∆ ranges over the finitely generated submonoids of Γ. Hence, we are easily


reduced to the case where Γ is finitely generated. Let S be the multiplicative system of all non-zero homogeneous elements of B. It is easily seen that A := S−1B is a Γgp-graded algebra,K := gr0A is a field, and dimK grγA = 1 for every γ ∈ Γgp. Moreover, we have

grγA · grγ′A = grγ+γ′A for every γ, γ′ ∈ Γgp.

Claim 4.4.15. A is a normal domain, and if Γ is saturated, Γ×Γgp A is normal as well.

Proof of the claim. Pick a decomposition Γgp = L⊕ T , where L is a free abelian group, and Tis the torsion subgroup of Γgp. It follows easily that the induced map of K-algebras

(L×Γgp A)⊗K (T ×Γgp A)→ A

is an isomorphism. Moreover, E := T ×Γgp A is a finite field extension of K, and L×Γgp A 'K[L]. Summing up, A is isomorphic to E[L], whence the first assertion.

Next, suppose that Γ is saturated; then Γ ' Γ]×Γ× (lemma 3.2.10), and we have a decompo-sition Γ× = H⊕T , whereH is a free abelian group. In this case, we may take L := (Γ])gp⊕H ,and the foregoing isomorphism induces an identification

Γ×Γgp A ' E[Γ] ⊕H]

so the second assertion follows, taking into account theorem 3.4.16(iii). ♦

The corollary now follows straightforwardly from claim 4.4.15 and proposition 4.4.13.

Proposition 4.4.16. Suppose that (Γ′,+)→ (Γ,+) is a morphism of fine monoids, B a finitelygenerated (resp. finitely presented) Γ-graded R-algebra, and M a finitely generated (resp.finitely presented) Γ-graded B-module. We have :

(i) Γ′ ×Γ B is a finitely generated (resp. finitely presented) R-algebra.(ii) Γ′ ×Γ M is a finitely generated (resp. finitely presented) Γ′ ×Γ B-module.

(iii) grγM is a finitely generated (resp. finitely presented) gr0B-module, for every γ ∈ Γ.

Proof. Let BM denote the direct sum B ⊕M , endowed with the R-algebra structure given bythe rule :

(b1,m1) · (b2,m2) = (b1b2, b1m2 + b2m1) for every b1, b2 ∈ B and m1,m2 ∈M.

Notice that BM is characterized as the unique R-algebra structure for which M is an ideal withM2 = 0, the natural projection BM → B is a map of R-algebras, and the B-module structureon M induced via π agrees with the given B-module structure on M .

Claim 4.4.17. The following conditions are equivalent :(a) The R-algebra BM is finitely generated (resp. finitely presented).(b) B is a finitely generated (resp. finitely presented) R-algebra and M is a finitely gener-

ated (resp. finitely presented) B-module.

Proof of the claim. (b)⇒(a): Suppose first that B is a finitely generated R-algebra, and Mis a finitely generated B-module. Pick a system of generators ΣB := b1, . . . , bs for B andΣM := m1, . . . ,mk for M . Then it is easily seen that ΣB ∪ΣM generates the R-algebra BM .

For the finitely presented case, pick a surjection of R-algebras ϕ : R[T1, . . . , Ts]→ B and ofB-module ψ : B⊕k →M . Let Σ′B be a finite system of generators of the ideal Kerϕ. Pick alsoa finite system b1, . . . , br of generators of the B-module Kerψ; we may write bi =

∑kj=1 bijej

for certain bij ∈ B (where e1, . . . , ek is the standard basis of B⊕k). For every i ≤ r and j ≤ k,pick Pij ∈ ϕ−1(bij). It is easily seen thatBM is isomorphic to theR-algebraR[T1, . . . , Ts+k]/I ,where I is generated by Σ′B ∪

∑kj=1 PijTj+s | i = 1, . . . , r ∪ Ti+sTj+s | 0 ≤ i, j ≤ k.

(a)⇒(b): Suppose that BM is a finitely generated R-algebra, and let c1, . . . , cn be a system ofgenerators. For every i = 1, . . . , n, we may write ci = bi +mi for unique bi ∈ B and mi ∈M .


Since M2 = 0, it is easily seen that m1, . . . ,mn is a system of generators for the B-module M ,and clearly b1, . . . , bn is a system of generators for the R-algebra B.

Next, suppose that BM is finitely presented over R. We may find a system of generators ofthe type b1, . . . , bs,m1, . . . ,mk for certain bi ∈ B and mj ∈ M . We deduce a surjection ofR-algebras

ϕ : R[T1, . . . , Ts+k]/(Ts+iTs+j | 0 ≤ i, j ≤ k)→ BM

such that Ti 7→ bi for every i ≤ s and Tj+s 7→ mj for every j ≤ k. It is easily seen that Kerϕis generated by the classes of finitely many polynomials P1, . . . , Pr, where

Pi = Qi(T1, . . . , Ts) +k∑j=1

Ts+jQij(T1, . . . , Ts) i = 1, . . . , r

for certain polynomials Qi, Qij ∈ R[T1, . . . , Ts]. It follows easily that B = R[T1, . . . , Ts]/I ,where I is the ideal generated by Q1, . . . , Qr, and M is isomorphic to the B-module B⊕k/N ,where N is the submodule generated by the system

∑kj=1 Qij(b1, . . . , bs)ej | i = 1, . . . , r ♦

Suppose now that B is a finitely generated R-algebra; then B is generated by finitely manyhomogeneous elements, say b1, . . . , bs of degrees respectively γ1, . . . , γs. Thus, we may definesurjections of monoids

(4.4.18) N⊕s → Γ : ei 7→ γi for i = 1, . . . , s

(where e1, . . . , es is the standard basis of N⊕s) and of R-algebras ϕ : C → B, where C :=R[N⊕s] is a free polynomial R-algebra. Notice that C is a N⊕s-graded R-algebra, and via(4.4.18) we may regard ϕ as a morphism of Γ-graded R-algebras C/Γ → B. Then I := ker ϕis a Γ-graded ideal of C, and if we set P := N⊕s×Γ Γ′ we deduce an isomorphism of Γ′-gradedR-algebras

B′ := Γ′ ×Γ B∼→ (P ×N⊕s C)/Γ′/(Γ

′ ×Γ I)

(see remark 4.4.11(i)).

Claim 4.4.19. P ×N⊕s C is a finitely presented R-algebra.

Proof of the claim. Indeed, this R-algebra is none else than R[P ], hence the assertion followsfrom corollary 3.4.2 and lemma 3.1.7(i). ♦

From claim 4.4.19 it follows already that B′ is a finitely generated R-algebra. Now, supposethat M is a finitely generated B-module, and set M ′ := Γ′ ×Γ M ; notice that

(4.4.20) Γ′ ×Γ (BM) = B′M ′ .

In view of claim 4.4.17, we deduce that M ′ is a finitely generated B′-module. Next, in case Bis a finitely presented R-algebra, I is a finitely generated ideal of C/Γ; as we have just seen, thisimplies that Γ′ ×Γ I is a finitely generated (Γ′ ×Γ C/Γ)-module, and then claim 4.4.19 showsthat B′ is a finitely presented R-algebra. This concludes the proof of (i).

Lastly, if moreover M is a finitely presented B-module, assertion (i), together with (4.4.20)and claim 4.4.17 say thatM ′ is a finitely presentedB′-module; thus, also assertion (ii) is proven.

(iii): For any given γ ∈ Γ, let us consider the morphism f : N→ Γ such that 1 7→ γ, and setB′ := N×Γ B. By (i), the R-algebra B′ is finitely generated (resp. finitely presented), and theB′-module M ′ := N×Γ M is finitely generated (resp. finitely presented). After replacing B byB′ and M by M ′, we may then assume from start that Γ = N, and we are reduced to showingthat gr1M is a finitely generated (resp. finitely presented) gr0B-module.


Let m1, . . . ,mt be a system of generators of M consisting of homogeneous elements ofdegrees respectively j1, . . . , jt. We endow B⊕t with the N-grading such that

grkB⊕t :=

t⊕i=1

grk−jiBei

(where e1, . . . , et is the standard basis of B⊕t); then the B-linear map B⊕t → M given bythe rule ei 7→ mi for every i = 1, . . . , t is a morphism of N-graded B-modules, and if M isfinitely presented, its kernel is generated by finitely many homogeneous elements b1, . . . , bs.In the latter case, endow again B⊕s with the unique N-grading such that the B-linear mapϕ : B⊕s → B⊕t given by the rule ei 7→ bi for every i = 1, . . . , s is a morphism of N-gradedB-modules. Now, in order to check that gr1M is a finitely generated gr0B-module, it sufficesto show that the same holds for gr1B

⊕t. The latter is a direct sum of gr0B-modules isomorphicto either gr0B or gr1B. Likewise, if M is finitely presented, gr1M = Coker gr1ϕ, and again,gr1B

⊕s is a direct sum of gr0B-modules isomorphic to either gr0B or gr1B; hence in order tocheck that gr1M is a finitely presented gr0B-module, it suffices to show that gr1B is a finitelypresented gr0B-module. In either event, we are reduced to the case where Γ = N and M = B.

However, from (ii) we deduce especially that gr0B = 0 ×N B is a finitely generated (resp.finitely presented) R-algebra, hence B is a finitely generated (resp. finitely presented) B0-algebra as well; we may then assume that R = gr0B. Let Σ be a system of homogeneousgenerators for the R-algebra B; we may then also assume that

(4.4.21) Σ ∩ gr0B = ∅.Then it is easily seen that the R-module gr1B is generated by Σ∩gr1B. Lastly, if B is a finitelypresentedR-algebra, we consider the natural surjection ψ : R[Σ]→ B from the free polynomialR-algebra generated by the set Σ, and endowR[Σ] with the unique grading for which ψ is a mapof N-graded R-algebras; then I := Kerψ is a finitely generated N-graded ideal with gr0I = 0.As usual, we pick a finite system Σ′ of homogeneous generators for I; clearly B1 is isomorphicto gr1B0[Σ]/gr1I . On the other hand, (4.4.21) easily implies that gr1R[Σ] is a free R-module offinite rank, and moreover gr1I is generated by Σ′ ∩ gr1I; especially, gr1B is a finitely presentedR-module in this case, and the proof is complete.

4.4.22. Let (Γ,+) be a monoid, R a ring, B := (B, gr•B) a Γ-graded R-algebra, and M aΓ-graded B-module. We denote by M [γ] the Γ-graded B-module whose underlying B-moduleis M , and whose grading is given by the rule :

grβM [γ] := grβ+γM for every γ ∈ Γ.

Remark 4.4.23. (i) In the situation of (4.4.22), pick any system x := (xi | i ∈ I) of homo-geneous generators of M , and say that xi ∈ grγiM for every i ∈ I . Then we may define asurjective map of Γ-graded B-modules

L :=⊕i∈I

B[−γi]→M : ei 7→ xi for every i ∈ I

where (ei | i ∈ I) denotes the canonical basis of the free B-module L (notice that ei ∈ grγiLfor every i ∈ I).

(ii) In case M is a finitely generated B-module, we may pick a finite system x as above, andthen L shall be a free B-module of finite rank.

(iii) Especially, suppose that B is a coherent ring and M is finitely presented as a B-module;then, in the situation of (ii), the kernel of the surjection L → M shall be again a finitelypresented Γ-graded B-module, so we can repeat the above construction, and find inductively aresolution

Σ : · · · → Lndn−−→ Ln−1 → · · · → L0

d0−−→M


such that Ln is a free B-module of finite rank, and the map dn is a morphism of Γ-gradedB-modules, for every n ∈ N.

(iv) In the situation of (iii), suppose furthermore that B is a flat R-algebra, in which casegrγB is a flat R-module, for every γ ∈ Γ. Then it is clear that the resolution Σ yields, in eachdegree γ ∈ Γ a flat resolution Σγ of the R-module grγM .

4.4.24. Filtered rings and Rees algebras. Some of the following material is borrowed from [9,Appendix III], where much more can be found.

Definition 4.4.25. Let R be a ring, A an R-algebra.(i) An R-algebra filtration on A is an increasing exhaustive filtration Fil•A indexed by Z

and consisting of R-submodules of A, such that :

1 ∈ Fil0A and FiliA · FiljA ⊂ Fili+jA for every i, j ∈ Z.The pair A := (A,Fil•A) is called a filtered R-algebra.

(ii) Let M be an A-module. An A-filtration on M is an increasing exhaustive filtrationFil•M consisting of R-submodules, and such that :

FiliA · FiljM ⊂ Fili+jM for every i, j ∈ Z.The pair M := (M,Fil•M) is called a filtered A-module.

(iii) Let U be an indeterminate. The Rees algebra ofA is the Z-graded subring ofA[U,U−1]

R(A)• :=⊕i∈Z

U i · FiliA.

(iv) Let M := (M,Fil•M) be a filtered A-module. The Rees module of M is the gradedR(A)•-module :

R(M)• :=⊕i∈Z

U i · FiliM.

Lemma 4.4.26. Let R be a ring, A := (A,Fil•A) a filtered R-algebra, gr•A the associatedgraded R-algebra, R(A)• ⊂ A[U,U−1] the Rees algebra of A. Then there are natural isomor-phisms of graded R-algebras :

R(A)•/UR(A)• ' gr•A R(A)•[U−1] ' A[U,U−1]

and of R-algebras :R(A)•/(1− U)R(A)• ' A.

Proof. The isomorphisms with gr•A and with A[U,U−1] follow directly from the definitions.For the third isomorphism, it suffices to remark that A[U,U−1]/(1− U) ' A.

Definition 4.4.27. Let R be a ring, A := (A,Fil•A) a filtered R-algebra.(i) Suppose that A is of finite type over R, let x := (x1, . . . , xn) be a finite set of gener-

ators for A as an R-algebra, and k := (k1, . . . , kn) a sequence of n integers; the goodfiltration Fil•A attached to the pair (x,k) is the R-algebra filtration such that FiliA isthe R-submodule generated by all the elements of the form

n∏j=1

xajj where :

n∑j=1

ajkj ≤ i and a1, . . . , an ≥ 0

for every i ∈ Z. A filtration Fil•A on A is said to be good if it is the good filtrationattached to some system of generators x and some sequence of integers k.

(ii) The filtration Fil•A is said to be positive if it is the good filtration associated to a pair(x,k) as in (i), such that moreover ki > 0 for every i = 1, . . . , n.


(iii) Let M be a finitely generated A-module. An A-filtration Fil•M is called a good filtra-tion if R(M,Fil•M)• is a finitely generated R(A)•-module.

Example 4.4.28. Let A := R[t1, . . . , tn] be the free R-algebra in n indeterminates. Choose anysequence k := (k1, . . . , kn) of integers, and denote by Fil•A the good filtration associated tot := (t1, . . . , tn) and k. Then R(A,Fil•A)• is isomorphic, as a graded R-algebra, to the freepolynomial algebra A[U ] = R[U, t1, . . . , tn], endowed with the grading such that U ∈ gr1A[U ]and tj ∈ grkjA[U ] for every j ≤ n. Indeed, a graded isomorphism can be defined by the rule :U 7→ U and tj 7→ Ukj · tj for every j ≤ n. The easy verification shall be left to the reader.

4.4.29. Let A and M be as in definition 4.4.27(iii); suppose that m := (m1, . . . ,mn) is a finitesystem of generators of M , and let k := (k1, . . . , kn) be an arbitrary sequence of n integers. Tothe pair (m,k) we associate a filtered A-module M := (M,Fil•M), by declaring that :

(4.4.30) FiliM := m1 · Fili−k1A+ · · ·+mn · Fili−knA for every i ∈ Z.

Notice that the homogeneous elements m1 ·Uk1 , . . . ,mn ·Ukn generate the graded Rees moduleR(M)•, hence Fil•M is a good A-filtration. Conversely :

Lemma 4.4.31. For every good filtration Fil•M on M , there exists a sequence m of generatorsof M and a sequence of integers k, such that Fil•M is of the form (4.4.30).

Proof. Suppose that Fil•M is a good filtration; then R(M)• is generated by finitely many ho-mogeneous elements m1 · Uk1 , . . . ,mn · Ukn . Thus,

R(M)i := U i · FiliM = m1 · Uk1 · Ai−k1 + · · ·+mn · Ukn · Ai−kn for every i ∈ Z

which means that the sequences m := (m1, . . . ,mn) and k := (k1, . . . , kn) will do.

4.4.32. Let A → B be a map of noetherian rings, I ⊂ A an ideal, M a finitely generatedA-module, and N a finitely generated B-module. It is easily seen that

Ti := TorAi (M,N)

is a finitely generated B-module, for every i ∈ N. We endow A (resp. B) with its I-adic (resp.IB-adic) filtration, extended to negative integers, by the rule Ik := A for every k ≤ 0, anddenote by A (resp. B) the resulting filtered ring. Also, we define a B-filtration on Ti, by therule :

FilnTi := Im (TorAi (InM,N)→ Ti) for every i ∈ N and n ∈ Z.

Proposition 4.4.33. The B-filtration Fil•Ti is good, for every i ∈ N.

Proof. Pick a finite set of generators f1, . . . , fr for I , and consider the surjective map of gradedA-algebras

(4.4.34) A[U, t1, . . . , tr]→ R(A)• : U 7→ 1 ∈ R(A)−1 ti 7→ fi for i = 1, . . . , r

(for the grading of A[U, t1, . . . , tr] that places the indeterminates t1, . . . , tr in degree 1, and Uin degree −1). The B-module

(4.4.35) Pi,• :=⊕n∈Z

TorAi (InM,N)

carries a natural structure of graded B ⊗A R(A)•-module, whence a graded B[U, t1, . . . , tr]-module structure as well, via (4.4.34), and it suffices to show :

Claim 4.4.36. Pi,• is a finitely generated B[U, t1, . . . , tr]-module, for every i ∈ N.


Proof of the claim. Let Fil•M be the I-adic filtration on M , and notice that R(M,Fil•M)• isa finitely generated R(M)•-module; a fortiori, it is a finitely generated graded A[U, t1, . . . , tr]-module. By remark 4.4.23(iv), we may then find a resolution

· · · → Lndn−−→ Ln−1 → · · · → L0

d0−−→ R(M,Fil•M)•

where each Ln is a free A[U, t1, . . . , tr]-module of finite rank, each dn is a morphism of gradedA[U, t1, . . . , tr]-modules, and the restriction of the resolution to the summands of degree i isa flat resolution of I iM , for every i ∈ N. There follows a natural isomorphism of gradedB-modules

(4.4.37) Pi,•∼→ Hi(L• ⊗A B) for every i ∈ N

and a simple inspection shows that the B[U, t1, . . . , tr]-module structure on Pi,• deduced via(4.4.37) agrees with the foregoing one, so the assertion follows.

4.4.38. In the situation of (4.4.32), set An := A/In+1 for every n ∈ N. We deduce, for everyi ∈ N, a morphism of projective systems of B-modules

X i• := (TorAi (M,N)⊗A An | n ∈ N)

ϕi•−−→ Y i• := (TorAi (M ⊗A An, N) | n ∈ N)

where the transition maps of X i• and Y i

• are induced by the projections An+1 → An, for everyn ∈ N, and the morphisms ϕin are induced by the projections M →M ⊗A An.

Corollary 4.4.39. With the notation of (4.4.38), we have :(i) The morphism ϕi• is an isomorphism of pro-B-modules, for every i ∈ N.

(ii) The natural map

limn∈N

TorAi (M,N)⊗A An → limn∈N

TorAi (M/InM,N)

is an isomorphism, for every i ∈ N.

Proof. (i): The assertion means that the systems (Kerϕin | n ∈ N) and (Cokerϕin | n ∈ N) areessentially zero, for every i ∈ N. However, set U := 1 ∈ R(A)−1, and notice that the long exactTorA• (−, N)-sequence arising from the short exact sequence

0→ InM →M →M/InM → 0

yields a natural identification

Cokerϕn = Ker (Un : Pi,n → Pi,0)

where Pi,• is defined as in (4.4.35), and Un denotes the scalar multiplication by the same ele-ment, for the natural R(A)•-module structure of Pi,•. Moreover, under this identification, thesystem (Cokerϕi• | n ∈ N) becomes a direct summand of the system of B-modules

(4.4.40) (KerUn : Pi,• → Pi,• | n ∈ N)

whose transition maps are given by multiplication by U . By the same token, the projectivesystem (Kerϕin | n ∈ N) is a quotient of the projective system Zi

• := (FilnTi ⊗A An | n ∈ N),for every i ∈ N. Now, it follows easily from proposition 4.4.33 and lemma 4.4.31, that, forevery i ∈ N, there exists c ∈ N such that

Filn+k+cTi ⊂ In+kTi ⊂ IkFilnTi for every k, n ∈ N.Taking k = n, we conclude that the system Zi

• is essentially zero, so the same holds for(Kerϕin | n ∈ N), for every i ∈ N. Lastly, since B[U, t1, . . . , tr] is noetherian, claim 4.4.36 im-plies that there exists N ∈ N such that KerUN+k = KerUN for every k ∈ N. It follows easilythat the system (4.4.40) is essentially zero, and then the same holds for (Cokerϕin | n ∈ N), forevery i ∈ N.


(ii) is a standard consequence of (i) : see [75, Prop.3.5.7].

4.4.41. Differential graded algebras. The material of this paragraph shall be applied in section4.5, in order to study certain strictly anti-commutative graded algebras constructed via homo-topical algebra. Especially, the graded algebras appearing in this paragraph are usually notcommutative, unless it is explicitly said otherwise.

Definition 4.4.42. Let A be any ring.(i) A differential graded A-algebra is the datum of• a complex (B•, d•B) of A-modules• an A-linear map µpq : Bp ⊗A Bq → Bp+q for every p, q ∈ Z

such that the following holds :(a) Set B :=

⊕p∈ZB

p; then the system of maps µ•• adds up to a map µ : B ⊗A B → B,and one requires that the resulting pair (B, µ) is an associative unitary Z-graded A-algebra. Then, one sets a · b := µ(a⊗ b), for every a, b ∈ B.

(b) We have the identities

dp+qB (a · b) = dpB(a) · b+ (−1)p · a · dqB(b) for every p, q ∈ Z and every a ∈ Bp, b ∈ Bq.

We callB the associated gradedA-algebra ofB•. A morphismB• → C• of differential gradedA-algebras is a map of complexes ofA-modules such that the induced map of associated gradedA-modules is a map of A-algebras. We denote by

A-dga

the resulting category of differential graded A-algebras.(ii) We say that the differential graded A-algebra B• is strictly anti-commutative, if we have

(i) a · b = (−1)pq · b · a for every p, q ∈ Z and every a ∈ Bp, b ∈ Bq

(ii) a · a = 0 for every p ∈ Z and every a ∈ B2p+1.If only condition (a) holds, we say that B• is anti-commutative.

(iii) Let (B•, d•B) be a differential graded A-algebra, B its associated graded A-algebra, and(M•, d•M) a complex of A-modules.

(a) We say that M• is a left B•-module if the A-module M :=⊕

p∈ZMp is a graded left

B-module (for the natural Z-grading on M ), and we have

dp+qM (b ·m) = (dpBb) ·m+ (−1)p · b · dqM(m)

for every p, q ∈ Z, every b ∈ Bp, and every m ∈M q.(b) We say that M• is a right B•-module if M is a graded right B-module, and we have

dp+qM (m · a) = dqM(m) · a+ (−1)q ·m · dpM(a)

for every p, q ∈ Z, every b ∈ Bp, and every m ∈M q.(c) We say that M• is a B-bimodule if it is both a left and right B•-module, and with

these B•-modules structures, the A-module M becomes a B-bimodule (i.e. the leftmultiplication commutes with the right multiplication).

We call M the associated graded B-module of M . A morphism M• → N• of left (resp. right,resp. bi-) B•-modules is a map of complexes of A-modules, such that the induced map ofassociated graded A-modules is a map of left (resp. right, resp. bi-) B-modules.

(iv) Let C• be any Z-graded A-algebra, and M• any Z-graded C•-bimodule. An A-lineargraded derivation from C• to M• is a morphism of graded A-modules ∂ : C• →M• such that

∂(xy) = ∂(x) · y + x · ∂(y) for every x, y ∈ C.


Remark 4.4.43. Let B• be any differential graded A-algebra.(i) Notice that condition (b) of definition 4.4.42(i) is the same as saying that µ•• induces a

map of complexesµ• : B• ⊗A B• → B•.

Moreover, in light of example 4.1.8(i), we see that B• is anti-commutative if and only if thediagram

B• ⊗A B•

µ• %%KKKKKKKKKK∼ // B• ⊗A B•

µ•yyssssssssss

B•

commutes, where the horizontal arrow is the isomorphism (4.1.9) that swaps the two tensorfactors. Hence, in some sense this is actually a commutativity condition.

(ii) Likewise, ifM• is a complex ofA-modules with a graded left (right)B-module structureon the associated graded A-module M , then M• is a left (resp. right) B•-module if and only ifthe scalar multiplication of the B-module M induces a map of complexes

B• ⊗AM• →M• ( resp. M• ⊗A B• →M• ).

(iii) It is easily seen that the multiplication maps µpq induce A-linear maps

(HpB•)⊗A (HqB•)→ Hp+qB• for every p, q ∈ Zanf if we let H•B• :=

⊕p∈ZH

pB•, then the resulting map

(H•B•)⊗A (H•B•)→ H•B•

endowsH•B• with a structure of Z-graded associative unitaryA-algebra, which shall be strictlyanti-commutative whenever the same holds forB•. Likewise, ifM• is any left (resp. right, resp.bi) B•-module, then H•M• is naturally a Z-graded left (resp. right, resp. bi) H•B•-module.

(iv) Let M• be a left B•-module, and denote by µpqM : Bp ⊗AM q →Mp+q the (p, q)-gradedcomponent of the scalar multiplication of M•, for every p, q ∈ Z. Then M•[1] is also naturallya left B•-module, with scalar multiplication µ••M [1] given by the rule :

µpqM [1] := (−1)p · µp,q+1M for every p, q ∈ Z.

Likewise, if N• is a right B•-module, with scalar multiplication µ••N , then N•[1] is naturally aright B•-module, with multiplication µ••N [1] given by the rule :

µpqN [1] := µp,q+1N for every p, q ∈ Z.

Lastly, if P • is a B•-bimodule, then the left and right B•-module structure defined above onP •[1], amount to a natural B-bimodule structure on P •[1].

(v) Let C• be any Z-graded A-algebra, N• any Z-graded left (resp. right, resp. bi-) B-module. We let N [1]• be the graded A-module given by the rule N [1]p := Np+1 for everyp ∈ Z. We shall always view N [1]• as a left (resp. right, resp. bi-) B-module, via the scalarmultiplications obtained from those of N•, following the rules spelled out in (iv). This ensuresthat the functor from B•-modules to H•B•-modules that assigns to any B•-module M• itshomology H•M•, is compatible with shift operators.

4.4.44. Let A be ring, (B•, d•B) any differential graded A-algebra, and I• ⊂ B• a (graded)two-sided ideal of B• (i.e. a bi-submodule of the B•-bimodule B•). Let

∂ : H•(B•/I•)→ H•I•[1]

denote the natural map induced by the short exact sequence of complexes

0→ I• → B• → B•/I• → 0.


We have :

Lemma 4.4.45. In the situation of (4.4.44), the map ∂ is an A-linear graded derivation of thegraded A-algebra H•(B•/I•).

Proof. Indeed, let a and b be any two cycles of the complex B•/I• in degree p and q, and a,b the respective classes; lift a and b to some elements a ∈ Bp and b ∈ Bq, so that ∂(a · b) isthe class in Hp+q+1I• of dB(a · b) = dB(a) · b + (−1)p · a · dB (b). Since dB(a) (resp. dB (b))represents the class of ∂(a) (resp. of ∂(b)), the assertion follows from the explicit descriptionof the bimodule structure on I•[1] provided by remark 4.4.43(iv).

4.4.46. Let B• and C• be two differential graded A-algebras. We may endow the complexB• ⊗A C• with a structure of differential graded A-algebra, by the following rule :

(x1⊗y1) ·(x2⊗y2) := (−1)j1i2 ·(x1x2)⊗(y1y2) for every xl ∈ Bil , yl ∈ Cjl , with l = 1, 2.

It is easily seen that, if both B• and C• are anti-commutative, the same holds for B• ⊗A C•. Interms of morphisms of complexes, this multiplication law corresponds to the composition

(B• ⊗A C•)⊗A (B• ⊗A C•)∼→ (B• ⊗A B•)⊗A (C• ⊗A C•)

µ•B⊗Aµ•C−−−−−−→ B• ⊗A C•

where the first isomorphism is obtained by composing the associativity isomorphisms of exam-ple 4.1.8(ii) and the isomorphisms (4.1.9) that swap the tensor factors. Here µ•B and µ•C are themultiplication maps of B• and C•.

Suppose now that both B• and C• are bounded above complexes; presumably, in this casethere is a canonical way to define a differential graded algebra structure as well on (suitablerepresentatives for)

D• := B•L⊗A C•

in such a way that this structure is well defined as an object of the derived category of A-dga(the latter should be, as usual, the localization of A-dga, by the multiplicative system of quasi-isomorphisms). More modestly, we shall endow the graded A-module H•D• with a naturalstructure of associative graded A-algebra, in such a way that the natural map

(4.4.47) H•D• :=

⊕i∈Z

TorAi (B•, C•)→ H•(B• ⊗A C•)

is a morphism of graded A-algebras. The multiplication of H•D• is defined as the composition

HiD• ⊗A HjD

• α−→ TorAi+j(B• ⊗A B•, C• ⊗A C•)

µ−→ Hi+jD• for every i, j ∈ Z

where α is the bilinear pairing provided by (4.1.14), and with µ := TorAi+j(µ•B, µ

•C).

Let us check first that the foregoing rule does define an associative multiplication on H•D•.To this aim, set B•2 := B• ⊗A B•, B•3 := B• ⊗A B•2 and define likewise C•2 and C•3 ; a littlediagram chase, together with (4.1.15), reduces to verifying the commutativity of the diagram

TorAi (B•2 , C•2)⊗A HjD

•µ⊗A1HiD• //

HiD• ⊗A HjD

•

HiD• ⊗A TorAj (B•2 , C

•2)

1HjD•⊗Aµoo

TorAi+j(B•3 , C

•3)

γ // TorAi+j(B•2 , C

•2) TorAi+j(B

•3 , C

•3)

δoo

whose vertical arrows are the bilinear pairings of (4.1.14), and with

γ := TorAi+j(µ•B ⊗A 1B• , µ

•C ⊗A 1C•) δ := TorAi+j(1B• ⊗A µ•B,1C• ⊗A µ•C).


We show the commutativity of the left subdiagram; the same argument applies to the right one.Unwinding the definitions, we come down to checking the commutativity, in D(A-Mod), ofthe diagram of complexes

(P •B2⊗A C•2)⊗A (P •B ⊗A C•)

∼ //

ϑ•

(P •B2⊗A P •B)⊗A C•3

ϕ•12,3⊗A1C•3 // P •B3⊗A C•3η•

(P •B ⊗A C•)⊗A (P •B ⊗A C•)

∼ // (P •B ⊗A P •B)⊗A C•2ϕ•1,23⊗A1C•3 // P •B2

⊗A C•2

(notation of (4.1.12)), with ϑ• := (P •µB⊗Aµ•C)⊗A1P •B⊗AC• and η• := P •µB⊗A1B

⊗A(µ•C⊗A1C•),and where ϕ•12,3 and ϕ•1,23 are as in (4.1.14) (and with the isomorphisms given by compositionsof associativity and swapping isomorphisms). A direct inspection shows that this diagram com-mutes up to homotopy, as required.

Next, we check that (4.4.47) is a map of graded A-algebras. Unwinding the definitions, wesee that – with the notation of (4.1.14) – the multiplication of H•D• is the map on homologyinduced by the composition

(P •B ⊗A C•)⊗A (P •B ⊗A C•)∼→ (P •B ⊗A P •B)⊗A C•2 → P •B2

⊗A C• → P •B ⊗A C

where the first isomorphism is again a composition of associativity isomorphisms and isomor-phisms that swap the factors; the second map is ϕ•12 ⊗A µ•C , where ϕ•12 : P •B ⊗A P •B → P •B2

isdefined as in (4.1.14). The last map is P •µB ⊗A 1C• , where P •µB is given by (4.1.12). Since theassociativity and swapping isomorphisms are obviously natural in all their arguments, we comedown to checking the commutativity, in D(A-Mod), of the diagram of complexes

(P •B ⊗A P •B)⊗A C•2ϕ•12⊗Aµ•C //

(ρ•B⊗Aρ•B)⊗A1C•2

P •B2⊗A C•

P •µB⊗A1C•

B•2 ⊗A C•2

µ•B⊗Aµ•C // B• ⊗A C• P •B ⊗A C•.

ρ•B⊗A1C•oo

The latter is further reduced to the commutativity of

P •B ⊗A P •Bϕ•12 //

ρ•B⊗Aρ•B

P •B2

P •µB

B•2µ•B // B• P •B.

ρ•Boo

But a simple inspection shows that this diagram indeed commutes up to homotopy.Lastly, we claim that if B• and C• are both anti-commutative, then the same holds for H•D•.

Indeed, consider the diagram

(P •B ⊗A P •B)⊗A C•2∼ //

(ρ•B⊗Aρ•B)⊗A1C•2 ((QQQQQQQQQQQQ

ϕ•12⊗Aµ•C

(P •B ⊗A P •B)⊗A C•2

(ρ•B⊗Aρ•B)⊗A1C•2vvmmmmmmmmmmmm

ϕ•12⊗Aµ•C

B•2 ⊗A C•2∼ //

µ•B⊗Aµ•C &&MMMMMMMMMM

B•2 ⊗A C•2

µ•B⊗Aµ•Cxxqqqqqqqqqq

B• ⊗A C•

P •B2⊗A C•

P •µB⊗A1C•// P •B ⊗A C•

ρ•B⊗A1C•

OO

P •B2⊗A C•.

P •µB⊗A1C•oo


(The upper isomorphism is obtained from the automorphism of P •B ⊗A P •B that swaps the twofactors, and from the automorphism of C•2 of the same type; likewise for the lower isomor-phism.) The assumption on B• and C• says that the inner triangular subdiagram commutes,and the same holds – up to homotopy – for the upper and the left and right subdiagrams, by asimple inspection. Then also the outer rectangular subdiagram commutes up to homotopy, andthe contention follows easily.

Example 4.4.48. An important example of differential graded algebra arises from the Koszulcomplex of (4.1.16). Indeed, let f := (f1, . . . , fr) be a finite sequence of elements of a ring A.We can endow K•(f) with a natural structure of strictly anti-commutative differential gradedA-algebra, as follows. First, notice that there are natural isomorphisms

Ki(f)∼→ Λi

A(A⊕r) for every i = 0, . . . , r.

This is clear if r = 1, and for r > 1, such an isomorphism is established inductively, by meansof the natural decomposition

ΛiA(L⊕ A)

∼→ Λi−1A (L)⊕ Λi

A(L) for every i ∈ N and every A-module L

(see [36, (4.3.18)]) and the corresponding decomposition

Ki(f)∼→ (Ki−1(f1, . . . , fr−1)⊗A K1(fr))⊕ (Ki(f1, . . . , fr−1)⊗A K0(fr)).

Let e1, . . . , er denote the canonical basis of A⊕r; under this isomorphism, the differentials ofthe Koszul complex are identified with the operators df ,• defined inductively as follows. We letdf ,1 : A⊕r → A be the A-linear map given by the rule : ei 7→ fi for i = 1, . . . , r. If i > 1, set

df ,i(ej1 ∧ · · · ∧ eji) := fj1ej2 ∧ · · · ∧ eji − ej1 ∧ df ,i−1(ej2 ∧ · · · ∧ ej−i)

for every sequence (j1, . . . , ji) ∈ 1, . . . , ri. It is easily seen that the complex (Λ•A(A⊕r), df ,•)underlies a differential graded A-algebra, whose multiplication is given by usual the exteriorproduct :

x · y := x ∧ y for every x, y ∈ Λ•A(A⊕r).

Notice as well that the isomorphism

κf : K•(f)∼→ (Λ•A(A⊕r), df ,•)

thus obtained, is compatible with extensions of sequences : if g := (g1, . . . , gs) is any othersequence of elements of A, we have a commutative diagram

K•(f)⊗A K•(g)κf⊗Aκg //

(Λ•A(A⊕r), df ,•)⊗A (Λ•A(A⊕s), dg,•)

K•(f ,g)

κf ,g // (Λ•A(A⊕r+s), df ,g,•)

whose left vertical arrow comes directly from the definition of Kf ,g, and whose right verticalarrow is given by [36, (4.3.18)]. Moreover, the right vertical arrow is even an isomorphism ofdifferential graded A-algebras, if we endow the tensor product with the multiplication law as in(4.4.46).

4.4.49. Let A be a ring, f := (f1, . . . , fr) a regular sequence of elements of A; denote by J theideal generated by f , and set A0 := A/J . We may regard the complex A0[0] as an (especiallysimple) differential graded A-algebra, with multiplication µ• deduced from that of A0, in theobvious way. We may then state :


Proposition 4.4.50. In the situation of (4.4.49), theA0-module J/J2 is free of rank r, and thereexists a unique isomorphism of strictly anti-commutative graded A-algebras

(4.4.51) Λ•A0(J/J2)

∼→ H•(A0[0]L⊗A A0[0])

which restricts, in degree 1, to the natural identification

Λ1A0

(J/J2) = J/J2 ∼→ TorA1 (A0, A0).

Proof. In this situation, proposition 4.1.21 says that the Koszul complex, with its natural aug-mentation, yields a resolution

ε• : K•(f)→ A0[0]

by free A-modules. Hence, there is a unique isomorphism ω• : P •A0

∼→ K•(f) in D(A-Mod),whose composition with ε• agrees with ρ•A0

(notation of (4.1.12)). Let us endow K•(f) withthe differential graded A-algebra structure inherited via the isomorphism κf of example 4.4.48.Then ε• is a map of differential graded A-algebras, we easily deduce a commutative diagram inD(A-Mod)

P •A0⊗A P •A0

P •µ //

ω•⊗Aω•

P •A0

ω•

K•(f)⊗A K•(f) // K•(f)

whose bottom horizontal arrow is the multiplication map of K•(f), and where P •µ is defined as in(4.1.12). We conclude that ω• induces an isomorphism of anti-commutative graded A-algebras

(4.4.52) H•(A0[0]L⊗A A0[0])

∼→ K•(f , A0[0])∼→ H•((Λ

•A(A⊕r), df ,•)⊗A A0).

By simple inspection, we see that df ,i ⊗A 1A0 = 0 for every i ∈ Z, whence an isomorphism ofstrictly anti-commutative A-graded algebras :

H•((Λ•A(A⊕r), df ,•)⊗A A0)

∼→ Λ•A0(A⊕r0 ).

Combining with (4.4.52), we obtain in degree one a natural isomorphism

Λ1A0

(A⊕r0 ) = A⊕r0∼→ TorA1 (A0, A0)

∼→ J/J2

which, finally, delivers the sought isomorphism of differential graded A-algebras. The unique-ness of (4.4.51) is clear, since the exterior algebra is generated by its degree one summand.

Remark 4.4.53. (i) Simplicial A-algebras are another important source of differential gradedalgebras, thanks to the following construction. Let R be any simplicial A-algebra, R• the asso-ciated chain complex of A-modules, and denote by µR : R ⊗A R → R the multiplication mapof R. By considering the shuffle map for the bisimplicial A-module R A R (notation as in(4.2.53)), we deduce a natural map of complexes

µR• : R• ⊗A R•ShRAR•−−−−−→ (R⊗A R)•

(µR)•−−−−→ R•

and taking into account propositions 4.2.54 and 4.2.57, it is easily seen that (R•, µR•) is astrictly anti-commutative differential graded algebra. By remark 4.4.43(iii), we deduce that thegraded A-module H•R :=

⊕p∈NHpR is naturally an N-graded associative unitary and strictly

anti-commutative A-algebra.(ii) Likewise, if M is any R-module, then we obtain on the associated chain complex M• a

natural structure of R•-bimodule, so that H•M is naturally a graded H•R-bimodule.(iii) Clearly, a morphism ϕ : R → S of simplicial A-algebras induces a morphism ϕ• :

R• → S• of differential graded A-algebras, and a morphism f : M → N of R-modulesinduces a morphism ϕ• : M• → N• of R•-bimodules.


4.5. Some homotopical algebra. The methods of homotopical algebra allow to construct de-rived functors of non-additive functors, in a variety of situations. In this section, we presentthe basics of this theory, beginning with its cornerstone, the standard resolution associated to atriple, as explained in the following paragraph.

4.5.1. A triple (>, η, µ) on a category C is the datum of a functor > : C → C together withnatural transformations :

η : 1C ⇒ > µ : > > ⇒ >such that the following diagrams commute :

(> >) >µ∗>

> (> >)>∗µ +3 > >

µ

>>∗η +3

1> OOOOOOOOOOOOOOO

OOOOOOOOOOOOOOO > >µ

>η∗>ks

1>ooooooooooooooo

ooooooooooooooo

> >µ +3 > >.

Dually, a cotriple (⊥, ε, δ) in a category C is a functor ⊥: C → C together with naturaltransformations :

ε :⊥⇒ 1C δ :⊥⇒⊥ ⊥such that the following diagrams commute :

⊥ δ +3

δ

⊥ ⊥δ∗⊥

⊥1⊥

PPPPPPPPPPPPPPP

PPPPPPPPPPPPPPP1⊥

ooooooooooooooo

ooooooooooooooo

δ

⊥ ⊥ ⊥∗δ +3 ⊥ (⊥ ⊥) (⊥ ⊥) ⊥ ⊥ ⊥ ⊥⊥∗εks ε∗⊥ +3 ⊥ .

Notice that a cotriple in C is the same as a triple in C o.

4.5.2. A cotriple ⊥ on C and an object A of C determine a simplicial object ⊥A[•] in C ;namely, for every n ∈ N set ⊥ A[n] :=⊥n+1 A, and define face and degeneracy operators :

∂i := (⊥i ∗ε∗ ⊥n−i)A : ⊥ A[n]→⊥ A[n− 1]

σi := (⊥i ∗δ∗ ⊥n−i)A : ⊥ A[n]→⊥ A[n+ 1].

Using the foregoing commutative diagrams, one verifies easily that the simplicial identities(4.2.8) hold. Moreover, the morphism εA :⊥A→ A defines an augmentation ⊥A[•]→ A.

Dually, a triple > and an object A of C determine a cosimplicial object >A[•] := >•+1A,such that

∂i := (>i ∗ η ∗ >n−i)A : >nA→ >n+1A

σi := (>i ∗ µ ∗ >n−i)A : >n+2A→ >n+1A

which is augmented by the morphism ηA : A → >A. Clearly, the rule : A 7→⊥A[•] (resp.A 7→ >A[•]) defines a functor

C → s.C (resp. C → c.C ).

4.5.3. An adjoint pair (G,F ) as in (1.1.8), with its unit η and counit ε, determines a triple(>,η,µ), where :

> := F G : B → B µ := F ∗ ε ∗G : > > ⇒ >as well as a cotriple (⊥, ε, δ), where :

⊥ := G F : A → A δ := G ∗ η ∗ F :⊥⇒⊥ ⊥ .

Indeed, the naturality of ε and η easily implies the commutativity of the diagrams of (4.5.1).The following proposition explains how to use these triples and cotriples to construct canonicalresolutions.


Proposition 4.5.4. In the situation of (4.5.3), the following holds :(i) For every A ∈ Ob(A ) and B ∈ Ob(B), the augmented simplicial objects :

⊥GB[•] εGB−−−→ GB F ⊥A[•] F∗εA−−−−→ FA

are homotopically trivial (notation of (4.5.2)).(ii) Dually, the same holds for the augmented cosimplicial objects :

GBG∗ηB−−−−→ G>B[•] FA

ηFA−−−→ >FA[•].Proof. Notice that

F ⊥A[n] = >FA[n] for every [n] ∈ Ob(∆∧).

Therefore, for every morphism ϕ : [n]→ [m] in ∆∧, we have two morphisms

F ⊥A[ϕ] : F ⊥A[m]→ F ⊥A[n] >FA[ϕ] : F ⊥A[n]→ F ⊥A[m].

Now, recall that the augmentation εA defines a natural morphism

⊥A[ε•+1−1,0] :⊥A[•]→ s.A [n] 7→⊥A[εn+1

−1,0]

(notation of remark 4.2.11(iv)), and one sees that the system (>FA[εn+1−1,0] | [n] ∈ Ob(∆))

defines a morphism>FA[ε•+1

−1,0] : s.FA→ F ⊥A[•]which is right inverse to F ⊥A[ε•+1

−1,0]. For every n, k ∈ N such that k ≤ n+ 1, set

un,k := (>FA[εkn−k,0]) (F ⊥A[εkn−k,0])

(notation of example 4.2.6(ii)).

Claim 4.5.5. The system u•• defines a homotopy from 1F⊥A[•] to (>FA[ε•+1−1,0]) (F ⊥A[ε•+1

−1,0])(see (4.2.14)).

Proof of the claim. By unwinding the definitions, we see that F ⊥A[εkn−k,0] is the composition

F (ε⊥n+1−kA · · · ε⊥n−1A ε⊥nA) : F ⊥n+1A→ F ⊥n+1−kA for k > 0

and >FA[εkn−k,0] is the composition

ηF⊥nA ηF⊥n−1A · · · ηF⊥n+1−kA : F ⊥n+1−kA→ F ⊥n+1A for k > 0

and both equal 1F⊥n+1A for k = 0. Now, since the face operator ∂i of F ⊥A[n] is F ⊥i (ε⊥n−iA)for every i ≤ n, the naturality of η yields a commutative diagram

F ⊥nA∂i−1 //

ηF⊥nA

F ⊥n−1A

ηF⊥n−1A

F ⊥n+1A

∂i // F ⊥nA

for every i > 0 and every n ∈ N

whereas ∂0 ηF⊥nA = 1F⊥nA for every n ∈ N; whence the identities :

∂i >FA[εkn−k,0] =

>FA[εk−1

n−k,0] for i < k>FA[εkn−k−1,0] ∂i−k for i ≥ k.

On the other hand, we have the simplicial identities :

εi εkn−k,0 =

εk+1n−k for i < kεkn+1−k εi−k for i ≥ k.

Summing up, we conclude that

∂i un,k =

un−1,k−1 ∂i for i < kun−1,k ∂i for i ≥ k.


Arguing likewise, one deduces as well the corresponding commutation rule – spelled out in(4.2.14) – for the degeneracies σi, and the claim follows. ♦

Likewise, notice that

G>B[n] =⊥GB[n] for every [n] ∈ ∆∧.

Therefore, for every morphism ϕ : [n]→ [m] in ∆∧, we have as well two morphisms

⊥GB[ϕ] :⊥GB[m]→⊥GB[n] G>B[ϕ] :⊥GB[n]→⊥GB[m]

and the system (G>B[εn+1−1,0] | [n] ∈ Ob(∆∧)) defines a morphism

G>B[ε•+1−1,0] : s.GB →⊥GB[•]

which is right inverse to the morphism ⊥GB[ε•+1−1,0] deduced from the augmentation. Arguing

as in the proof of claim 4.5.5, one checks that the rule

vn,k := (G>B[εk∨n−k,0]) (⊥GB[εk∨n−k,0]) for every n, k ∈ N such that k ≤ n+ 1

yields a homotopy v from 1⊥GB[•] to (G>B[ε•+1−1,0]) (⊥GB[ε•+1

−1,0]).The dual statements admit the dual proof.

4.5.6. Now, denote by Set the category of pointed sets, whose objects are all the pairs (S, s)where S is a small set, and s ∈ S is any element; the morphisms f : (S, s) → (S ′, s′) are themappings f : S → S ′ such that f(s) = s′. Next, let A be any ring, and consider the pointedforgetful functor

F : A-Mod→ Set M 7→ (M, 0M)

(where 0M ∈M is the zero element of M ). The functor F admits the left adjoint

L : Set → A-Mod (S, s) 7→ A(S)/As

(where A(S) denotes the free A-module with basis given by S, so As ⊂ A(S) is the directsummand generated by the basis element s ∈ S). According to proposition 4.5.4, the adjointpair (L,F) yields a functor

⊥A• : A-Mod→ s.A-Mod

into the category of augmented simplicial A-modules (notation of (4.2.10)), such that

F ⊥A• M → FM

is a homotopically trivial augmented pointed simplicial set, for everyA-moduleM . This functoris obtained by iterating the functor⊥A:= LF, so in each degree it consists of a free A-module.We call ⊥A• M the standard free resolution of M .

Remark 4.5.7. Notice that ⊥A• 0 = s.0, the constant simplicial A-module associated to thetrivial A-module (see (4.2.4)). This is the reason why we prefer the adjoint pair (L,F), ratherthan the analogous pair considered in [56, I.1.5.5], arising from the forgetful functor from A-modules to non-pointed sets.

4.5.8. Let R be a simplicial A-algebra, and define the category R-Mod of R-modules, as in[36, §8.1]. Notice that any morphism S → R of simplicial rings induces a base change functor

S-Mod→ R-Mod (M [n] | n ∈ N) 7→ (R[n]⊗S[n] M [n] | n ∈ N)

which is left adjoint to the forgetful functor (details left to the reader).Now, by applying the functors ⊥R[n]

• to the terms M [n] of a given R-module M , we obtainan augmented simplicial R-module

(4.5.9) ⊥R• M →M


which amounts to a bisimplicialA-module, whose columns⊥R[n]• M [n]→M [n] are augmented

simplicial R[n]-modules, for every n ∈ N. Likewise, the row ⊥Rn M is an R-module, for everyn ∈ N.

Lemma 4.5.10. In the situation of (4.5.8), let ϕ : M → N be any quasi-isomorphism ofR-modules. Then the induced morphism

⊥Rn ϕ :⊥RnM →⊥Rn Nis a homotopy equivalence, for every n ∈ N.

Proof. Set ⊥R−1M := M , and notice the natural isomorphisms

(4.5.11) ⊥RnM∼→ R⊗s.Z ⊥s.Z ⊥Rn−1M for every n ∈ N

(where s.Z is the constant simplicial ring arising from Z (see (4.2.4)), which is the initial objectin the category of simplicial rings). The assumption means that ⊥R−1 ϕ is a quasi-isomorphism.However, (4.5.11) and Whitehead’s theorem ([56, I.2.2.3]) imply that, if – for a given n ∈ N –the map ⊥Rn−1 ϕ is a quasi-isomorphism, then ⊥Rn ϕ is a homotopy equivalence. We may thusconclude by a simple induction on n.

Example 4.5.12. Let M and N two R-modules. We may define two functors

M`⊗R − ( resp. −

`⊗R N ) : R-Mod→ R-Mod

by the rules :

P 7→M ⊗R (⊥R• P )∆ ( resp. P 7→ (⊥R• P )∆ ⊗R N ) for every R-module P

where ∆ is the diagonal functor, from C -bisimplicial to C -simplicial objects (see (4.2.15)).(i) We claim that these functors descend to the derived category. That is, suppose that

ϕ : P → P ′ is a quasi-isomorphism; then M`⊗R ϕ and ϕ

`⊗R N are both quasi-isomorphisms.

Indeed, lemma 4.5.10 implies that the induced morphisms

M⊗R ⊥Rn ϕ : M⊗R ⊥Rn P →M⊗R ⊥Rn P ′

are quasi-isomorphisms, for every n ∈ N, and likewise for ϕ⊗R ⊥Rn N , so the assertion followsfrom Eilenberg-Zilber’s theorem 4.2.48(i).

(ii) Also, we claim that the notation M`⊗RN is unambiguous. That is, we may compute this

object by applying the functor M`⊗R− to N , or by applying the functor−

`⊗RN to M , and the

resulting two R-modules are naturally isomorphic in D(R-Mod). Indeed, it suffices to checkthat the natural morphisms

M ⊗R (⊥R• N)∆ α←− (⊥R• M)∆ ⊗R (⊥R• N)∆ β−→ (⊥R• M)∆ ⊗R Ninduced by the augmentation (4.5.9), are both quasi-isomorphisms. We check this for α; thesame argument shall apply to β. To ease notation, set L := (⊥R• N)∆. Then α is deduced byextracting the diagonal from the augmented simplicial R-module

(4.5.13) ⊥R• M ⊗R L→M ⊗R L(so, the n-th column of s.L is a constant and flat simplicial R[n]-module, for every n ∈ N).Thus, we are reduced to checking that the columns of (4.5.13) are aspherical. However, forevery n ∈ N, the n-th column is the augmented simplicialR[n]-module (⊥R[n]

• M [n])⊗R[n]L[n];since L[n] is flat, it then suffices to recall that the standard free resolution is aspherical.

(iii) The foregoing discussion yields a well defined functor

−`⊗R − : D(R-Mod)× D(R-Mod)→ D(R-Mod)


called the left derived tensor product. The same method shall be applied hereafter to constructderived functors of certain non-additive functors.

Remark 4.5.14. Let M and N be as in example 4.5.12 and set P := (⊥R• M)⊗R s.N (this is asimplicial R-module; especially, it is a bisimplicial A-module); by Eilenberg-Zilber’s theorem

4.2.48, the cochain complex associated to M`⊗R N is naturally isomorphic, in D(A-Mod) to

the complex Tot(P ••), where P •• denotes the double complex associated to P . There followsa spectral sequence :

E1pq := TorR[p]

q (M [p], N [p])⇒ Hp+q(M`⊗R N)

whose differentials d1pq : E1

pq → E1p−1,q are obtained as follows. For every integer q ∈ N, let

Tq be the R-module given by the rule : Tq[p] := E1pq, with face and degeneracy maps deduced

from those of M , N and R, in the obvious way. Then d1pq is the differential in degree p of the

chain complex Tq• associated to Tq (details left to the reader).

Example 4.5.15. (i) Another useful triple arises from the forgetful functor A-Alg → Setand its left adjoint, that attaches to any set S the free A-algebra A[S]. There results, for everyA-algebra B, a standard simplicial resolution by free A-algebras

FA(B)→ B.

Now, if R is again a simplicial A-algebra, and S any R-algebra, we can proceed as in theforegoing, to obtain a bisimplicial resolution FR(S) → S whose columns FR[n](S[n]) → S[n]are augmented simplicial R[n]-algebras, for every n ∈ N. A simple inspection shows that theproof of lemma 4.5.10 carries over – mutatis mutandi – to the present situation, hence a quasi-isomorphism S → S ′ of R-algebra induces a morphism FR(S)[n]→ FR(S ′)[n] of R-algebras,which is a homotopy equivalence on the underlying R-modules, for every n ∈ N.

(ii) We may then define a derived tensor product for R-algebras, proceeding as in example4.5.12. Namely, if S and S ′ are any two R-algebras, we define two functors

S`⊗R − ( resp. −

`⊗R S ′ ) : R-Alg→ R-Alg

by the rules :

T 7→ S ⊗R FR(T )∆ ( resp. T 7→ FR(T )∆ ⊗R N ) for every R-algebra T .

Arguing like in loc.cit. we see that both functors transform quasi-isomorphisms into quasi-isomorphisms, hence they descend to the derived category D(R-Alg), and moreover, the two

functors are naturally isomorphic, so the notation S`⊗R S ′ is unambiguous : the detailed verifi-

cation is left as an exercise for the reader.

4.5.16. Let A-Alg.Mod be the category of all pairs (B,M), where B is any A-algebra, andM any B-module. The morphisms (B,M)→ (B′,M ′) are the pairs (f, ϕ), where f : B → B′

is a morphism of A-algebras, and ϕ : M → f ∗M ′ a B-linear map (here f ∗M ′ denotes the B-module obtained from the B′-module M ′, by restriction of scalars along the map f ). Supposenow that we have a commutative diagram of categories :

A-Alg.ModT //

F

C

f

A-Alg

g // B

such that :


• For every X ∈ Ob(B), the fibre f−1X is an abelian category whose filtered colimitsare representable and exact.• F is given by the rule (B,M) 7→ B for every (B,M) ∈ Ob(A-Alg.Mod).• For every A-algebra B, the restriction

TB : F−1B → f−1(gB)

of T commutes with filtered colimits, i.e., if ((B,Mi) | i ∈ I) is any filtered system ofobjects of A-Alg.Mod (over the same A-algebra B), then the induced morphism

colimi∈I

T (B,Mi)→ T (B, colimi∈I

Mi)

is an isomorphism.The functors f and g extend to functors s.f : s.C → s.B, respectively s.g : s.A-Alg → s.B,and notice that – for R any simplicial A-algebra – T induces a functor

TR : R-Mod→ s.C/R := s.f−1(s.gR) (M [n] | n ∈ N) 7→ (T (R[n],M [n]) | n ∈ N).

For every R-module M , we set

LTR(M) := TR(⊥R• M)∆.

The augmentation TR(⊥R• M)→ TRM can be regarded as a morphism of bisimplicial objects

TR(⊥R• M)→ s.TRM

(the columns of s.TRM are constant C -simplicial objects), whence a morphism in s.C/R :

(4.5.17) LTR(M)→ s.TRM∆ = TRM for every R-module M.

In many applications, s.C/R will be also an abelian category, but anyhow we can state thefollowing :

Proposition 4.5.18. With the notation of (4.5.16), suppose that C is an abelian category. Thenthe following holds :

(i) If M is a flat R-module, then (4.5.17) is a quasi-isomorphism.(ii) If ϕ : M → N is a quasi-isomorphism of R-modules, then the induced map

LTR(M)→ LTR(N)

is a quasi-isomorphism.

Proof. (i): The assumption means that M [n] is a flat R[n]-module for every n ∈ N. Denoteby C the unnormalized double complex associated to TR(⊥R• M), and by C∆ (resp. by D) theunnormalized complex associated to LTR(M) (resp. to TRM ). By Eilenberg-Zilber’s theorem4.2.48, we have a natural quasi-isomorphism

(4.5.19) C∆ → Tot(C)

given by the Alexander-Whitney map of (4.2.33), and a simple inspection shows that the mapC∆ → D induced by (4.5.17) factors through (4.5.19). Hence, it suffices to show that the map

Tot(C)→ D

induced by the augmentation of TR(⊥R• M), is a quasi-isomorphism, under the stated condition.To this aim, it suffices to check that the augmented C -simplicial object

(4.5.20) TR(⊥R[n]• M [n])→ TRM [n]

is aspherical for every n ∈ N. However, since M [n] is a flat R[n]-module, it can be writtenas a filtered colimit of a system of free R[n]-modules ([57, Ch.I, Th.1.2]); on the other hand,the functor ⊥R[n]

• commutes with all filtered colimits, and the same holds for TR, by assump-tion. Since the filtered colimits of the fibres of the functor f are exact, we are then reduced


to checking the assertion in case M [n] is a free R[n]-module; but in this case, (4.5.20) is evenhomotopically trivial, by virtue of proposition 4.5.4.

(ii): In light of Eilenberg-Zilber’s theorem 4.2.48, it suffices to show that the induced map

TR(⊥RnM)→ TR(⊥Rn N)

is a quasi-isomorphism for every n ∈ N. In turns, this follows readily from lemma 4.5.10.

Remark 4.5.21. (i) The proof of proposition 4.5.18(i) applies as well to the derived tensorproduct; namely, for any two R-modules M and N there is a natural morphism of R-modules

M`⊗R N →M ⊗R N

which is a quasi-isomorphism, if either M or N is a flat R-module (details left to the reader).(ii) Especially, let S and S ′ be any two R-algebras. Then, since the standard free resolution

FR(S) of example 4.5.15 is a flat simplicialR-module, (i) implies that theR-module underlying

the R-algebra S`⊗R S ′, is naturally isomorphic, in D(R-Mod), to the derived tensor product

of the R-modules underlying S and S ′. In other words, the notation −`⊗R − is unambiguous,

whether one refers to derived tensor products of algebras, or of their underlying modules.(iii) Denote by σ, ω : R-Mod → R-Mod respectively the suspension and loop functors

([56, I.3.2.1]), and recall that σ is left adjoint to ω. A simple inspection of the definitions yieldsnatural identifications

(σM)⊗R N∼→ σ(M ⊗R N) (ωM)⊗R N

∼→ ω(M ⊗R N)

for every R-modules M and N . By the same token, it is clear that σ and ω transform flatR-modules into flat R-modules. In view of (i), we deduce natural isomorphisms

(σM)`⊗R N

∼→ σ(M`⊗R N) (ωM)

`⊗R N

∼→ ω(M`⊗R N) in D(R-Mod)

for every R-modules M and N .(iv) Let f : N → N ′ be any morphism of R-modules; in the same vein, we get a natural

identification :

Cone(M ⊗R f)∼→M ⊗R Cone f for every R-module M

(see [56, I.3.2.2] for the definition of the cone of a morphism ofR-modules). It follows immedi-

ately that the functor M`⊗R − : D(R-Mod)→ D(R-Mod) transforms distinguished triangles

into distinguished triangles (see [56, I.3.2.2.4] for the definition of distinguished triangle inD(R-Mod)). For future reference, let us also point out :

Lemma 4.5.22. Let R be a simplicial A-algebra, X and Y two R-modules, n,m ∈ N twointegers, and suppose that HiX = 0 = HjY for every i < n and j < m. Then

Hi(X`⊗R Y ) = 0 for every i < n+m.

Proof. Set σ0 := 1R-Mod, and define inductively σk : R-Mod → R-Mod by the rule : σk :=σ σk−1 for every k > 0. Define likewise ωk, and notice that σk is left adjoint to ωk, for everyk ∈ N (remark 4.5.21(iii)). Let f : σn ωnX → X denote the counit of adjunction, and noticethat Cone f = 0 in D(R-Mod). In view of remark 4.5.21(iv,v), we deduce that the naturalmorphisms

σn(ωnX`⊗R Y )→ (σn ωnX)

`⊗R Y → X

`⊗R Y

are isomorphisms in D(R-Mod). Likewise, the natural morphism

σm(ωnX`⊗R ωmY )→ ωnX

`⊗R Y


is an isomorphism in D(R-Mod), so finally the same holds for the morphism

σn+m(ωnX`⊗R ωmY )→ X

`⊗R Y

whence the claim.

4.5.23. In view of proposition 4.5.18, it is clear that, if C is an abelian category, LTR yields awell defined functor on derived categories

LTR : D(R-Mod)→ D(s.C/R) M 7→ LTR(M)

(where D(s.C/R) is the localization of s.C/R with respect to the class of quasi-isomorphisms,and likewise for D(R-Mod) : see [36, Def.8.1.3]). More generally, suppose that C is endowedwith a functor

Φ : C → A

to an abelian category A (usually, this will be a forgetful functor of some sort), and denote byDΦ(s.C/R) the localization of s.C/R with respect to the system of its morphisms whose imageunder s.Φ are quasi-isomorphisms in s.A . Then, since clearly

s.Φ LTR = L(s.Φ TR)

we see that LTR descends to a well defined functor

LTR : D(R-Mod)→ DΦ(s.C/R).

We will need also the following slight refinement :

Corollary 4.5.24. In the situation of proposition 4.5.18, the following holds :(i) Let ϕ : M → N be a morphism of R-modules, and n ∈ N an integer such that

Hiϕ : HiM → HiN

is an isomorphism, for every i < n. Then the same holds for the induced map

Hi(LTRϕ) : Hi(LTRM)→ Hi(LTRN).

(ii) Suppose that T (B, 0) = 0gB (the initial and final object of f−1(gB)) for every A-algebra B. Let n ∈ N be an integer, and M an R-module such that HiM = 0 for everyi < n. Then Hi(LTRM) = 0gRi for every i < n.

Proof. (i): Denote by

ϕX : X → cosknX for every X ∈ Ob(R-Mod)

the unit of adjunction (see (4.2.18)). Taking into account corollary 4.2.65(i), we have the fol-lowing properties :

• X[i] = cosknX[i] and ϕ[i] is the identity map of X[i], for every i ≤ n.• Hi(cosknX) = 0 for every i ≥ n.

In view of proposition 4.5.18, we are then easily reduced to checking the assertion for thespecial where N := cosknM , and ϕ := ϕM . However, since in this case N [i] = M [i] for everyi ≤ n, it is clear that ⊥R[i]

• ϕ[i] is the identity automorphism of ⊥R[i]• M [i], for every i ≤ n.

After applying the functor TR, and extracting the diagonal, we see that the induced morphismLTRM → LTRN in s.C is given by the identity automorphism of (LTRM)[i], in every degreei ≤ n, whence the contention.

(ii): In light of (i), we may assume that M = s.0 (the trivial R-module). In this case, theassertion follows easily from remark 4.5.7.


4.5.25. In the situation of (4.5.16), take C := V -Alg.Mod, f := F , and g the identity au-tomorphism of A-Alg. If B → B′ is a map of A-algebras, and (B,M) ∈ Ob(C ), we defineB′ ⊗B (B,M) := (B′, B′ ⊗B M).

Corollary 4.5.26. In the situation of (4.5.25), suppose moreover that, for every flat morphismB → B′ of A-algebras, the natural map

B′ ⊗B T (B,M)→ T (B′ ⊗B (B,M))

is an isomorphism, for every (B,M) ∈ Ob(C ). Then, for every flat morphism ϕ : R → S ofsimplicial V -algebras, the natural morphism

(4.5.27) S ⊗R LTRM → LTS(S ⊗RM)

is an isomorphism in D(S-Mod), for every R-module M .

Proof. (Recall that ϕ is flat if and only if S[n] is a flat R[n]-algebra, for every n ∈ N. Themap of the proposition is deduced from the natural morphism ⊥R• M →⊥S• (S⊗RM), given byfunctoriality of the standard free resolution.)

Let π :⊥R• M → M be the standard free resolution of M ; by the flatness assumption on S,the morphism S ⊗R π : S⊗R ⊥R• M → S ⊗RM is a free resolution of the S-module S ⊗RM ,whence a natural isomorphism

LTS(S ⊗RM)∼→ TS((S⊗R ⊥R• M)∆) in D(S-Mod)

by proposition 4.5.18(i) and Eilenberg-Zilber’s theorem 4.2.48. On the other hand, the assump-tions on T yield a natural isomorphism

S ⊗R LTR(M) = S ⊗R TR((⊥R• M)∆)∼→ TS((S⊗R ⊥R• M)∆) in D(S-Mod).

By combining these isomorphisms, we get an isomorphism S ⊗R LTRM → LTS(S ⊗R M),and a simple inspection shows that the latter is realized by the natural map (4.5.27).

4.5.28. Let now R be any simplicial A-algebra, and I ⊂ R any ideal, and set R0 := R/I . TheRees algebra of I is the graded R-algebra

R(R, I)• :=⊕p∈N

Ip.

Notice the natural isomorphism of graded R0-algebras

R(R, I)⊗R R0∼→ gr•IR :=

⊕p∈N

Ip/Ip+1

as well as the exact sequence of R(R, I)-modules

0→ gr•+1I R→ R(R, I)⊗R R/I2 → gr•IR→ 0

deduced from the natural projection R/I2 → R0. Next, pick any quasi-isomorphism of R-algebras P → R0, with P a flat R-algebra; there follows an exact sequence of P ⊗R R(R, I)-modules

0→ P ⊗R gr•+1I R→ P ⊗R R(R, I)⊗R R/I2 β−→ P ⊗R gr•IR→ 0

and notice that β is actually a morphism of P ⊗R R(R, I)-algebras. According to remark4.4.53(i), after forming the associated cochain complexes, we obtain an epimorphism β• ofdifferential graded (P ⊗R R(R, I))•-algebras, whose kernel (P ⊗R gr•+1

I R)• is a two-sided

ideal of (P ⊗R R(R, I)⊗R R/I2)•. In this situation, we deduce a map of H•(R0

`⊗R R(P, I))-

modules

(4.5.29) δ : H•(R0

`⊗R gr•IR)→ H•(R0

`⊗R gr•+1

I R)[1]


which is a graded derivation, according to lemma 4.4.45 (recall that here the shift [1] refers tothe homological grading; the notation gr•+1 denotes also a shift in degrees, which however doesnot alter the signs of the scalar multiplication in the way prescribed by remark 4.4.43(v)).

4.5.30. Keep the situation of (4.5.28), and suppose furthermore that R is a constant A-algebraand I a constant ideal, i.e. R = s.B, and I = s.J for some A-algebra B and some ideal J ⊂ B.We set B0 := B/J and

G• := R0

`⊗R grIR

• Λ• := Λ•B(J/J2) S• := Sym•B(J/J2)

where the derived tensor product is formed in D(R-Alg), as in example 4.5.15, so G• is rep-resented by P ⊗R gr•IR, and H•G• is a bigraded A-algebra, strictly anti-commutative for the(homological) grading H•, and commutative for the (cohomological) grading deduced from thegrading of G• (see remark 4.4.53(i)). Also, Λ• (resp. S•) is a strictly anti-commutative (resp.a commutative) graded B0-algebra, and we use the homological (resp. cohomological) con-ventions for grading, so Λp (resp. Sp) is placed in degree −p (resp. p). Moreover, a standardcalculation yields a natural isomorphism of B0-modules

H0G0 ∼→ B0 H1G

0 ∼→ TorB1 (B0, B0)∼→ J/J2

whence a natural map Λ• → H•G0 of strictly anti-commutative graded B0-algebras, restricting

to an isomorphism in degrees≤ 1. On the other hand, we have a surjective map of commutativegraded B0-algebras

S• → gr•JB :=⊕p∈N

Jp/Jp+1

restricting as well to an isomorphism in degrees ≤ 1; since P ⊗R gr•IR is naturally a simplicialgr•JB-algebra, we obtain a natural map of bigraded S•-algebras

ω•• : Λ• ⊗B S• → H•G•.

Now, H•G• has been endowed with a derivation δ : H•G• → H•G

•+1[1] in (4.5.29); on theother hand, there exists on Λ• ⊗B S• a natural S•-linear graded derivation

∂ : Λ• ⊗B S• → Λ• ⊗B S•+1[1]

as explained in [56, I.4.3.1.2]. We may then consider the diagram of S•-linear maps

(4.5.31)

Λ• ⊗B S•ω•• //

∂

H•G•

δ

Λ• ⊗B S•+1[1]ω•+1• [1]

// H•G•+1[1].

Definition 4.5.32. (i) In the situation of (4.5.30), we say that the ideal J is quasi-regular, if thefollowing conditions hold :

• The map ω•• restricts to an isomorphism ω0•∼→ H•G

0.• The B0-module J/J2 is flat.

(ii) Let R be any simplicial A-algebra, and I ⊂ R any ideal. We say that I is quasi-regular,if I[n] is a quasi-regular ideal of R[n], for every n ∈ N.

Remark 4.5.33. (i) Suppose that B := (Bi | i ∈ I) is any filtered system of rings, andJ := (Ji ⊂ Bi | i ∈ I) a filtered system of ideals, such that Ji is quasi-regular in Bi, for everyi ∈ I . Denote by B (resp. J) the colimit of B (resp. of J); then it is easily seen that J isquasi-regular in B.

(ii) Let B be any ring, and C := B[Xi | i ∈ I] a free B-algebra (for any set I). Then theideal J := (Xi | i ∈ I) ⊂ C is quasi-regular. Indeed, (i) reduces to the case where I is a finiteset, for which the assertion is a special case of the following :


Lemma 4.5.34. In the situation of (4.5.30), suppose that J is generated by a (finite) regularsequence of elements of B. Then J is quasi-regular.

Proof. The assertion becomes a paraphrase of proposition 4.4.50, once we have identified the

graded B-algebra H•G• with the graded B-algebra H•(B0[0]L⊗B B0[0]). However, denote

by P • the cochain complex associated to the flat resolution P → s.B0; according to remark4.4.53(i), P • is naturally a differential graded B-algebra, and the induced map ε• : P • → B0[0]is a morphism of differential gradedB-algebras (for the multiplication law µ• onB0[0] inheritedfrom that ofB0). Then there exists in D(B-Mod) a unique isomorphism ω• : P •B0

→ P • whosecomposition with ε• agrees with ρ•B0

(notation of (4.1.12)), and moreover we get a commutativediagram in D(B-Mod) :

P •B0⊗B P •B0

P •µ //

ω•⊗Bω•

P •B0

ω•

P • ⊗B P • // P •

whose bottom horizontal arrow is the multiplication map of P •, and where P •µ is defined as in(4.1.12). The assertion follows immediately.

Proposition 4.5.35. In the situation of (4.5.30), we have :(i) (4.5.31) is a commutative diagram.

(ii) Suppose that J is a quasi-regular ideal. Then(a) ω•• is an isomorphism.(b) There exists a natural isomorphism of B0-modules :

TorBi (B0, B/Jn)∼→ Coker(∂n−2

i+1 : Λi+1 ⊗B Sn−2 → Λi ⊗B Sn−1) for every n, i > 0.

Proof. (i): Since both derivations are S•-linear, it suffices to check that (4.5.31) commutes in(upper) degree 0. Morever, since the B-algebra Λ• is generated by Λ1, it suffices to check thecommutativity of the diagram

(4.5.36)Λ1

ω01 //

∂01

H1G0

δ01

S1ω1

0 // H0G1.

However, according to [56, I.4.3.1.2], the map ∂01 is just the identity of J/J2. The map δ0

1 is theboundary map arising – by the snake lemma – from the exact sequence of complexes

0→ JP •/J2P • → P •/J2P • → P •/JP • → 0

(where P • denotes the cochain complex associated to the flat resolution P → s.B0). The sameboundary map is used to define the isomorphism ω0

1 , and ω10 is the natural isomorphism J/J2 →

H0(P •/JP •). Summing up, the commutativity of (4.5.36) follows by simple inspection.(ii.a): We prove, by induction on n ∈ N, that every ωn• is an isomorphism. For n = 0, 1 there

is nothing to show, so suppose that n > 1, and the assertion is already known for every degree< n. In order to prove the assertion in degree n, it suffices to check that ωn0 : Sn → grnJBis an isomorphism; indeed, in this case grnJB is a flat B0-module, and then a standard spectralsequence argument allows to conclude that ωnj is also an isomorphism, for every j ∈ N.

Let P → R0 be a resolution as in (4.5.28). The I-adic filtration has finite length on P/IkP ,for every k ∈ N, and therefore yields a convergent spectral sequence

E1pq ⇒ TorBp+q(B0, B/J

k)


with

E1pq :=

TorBp+q(B0, grJB

−p) for p > −k0 otherwise

whose differentials E1pq → E1

p−1,q agree – by direct inspection – with the derivation δ−pp+q, when-ever p > 1− k. Especially, for k = n+ 1, we get a complex

Σ : H2Gn−2 δn−2

2−−−→ H1Gn−1 δn−1

1−−−→ grnJB → 0.

On the other hand, since J/J2 is a flat B0-module, the corresponding sequence

Σ′ : Λ2 ⊗B Sn−2 ∂−→ Λ1 ⊗B Sn−1 ∂−→ Sn → 0

is an exact complex (this is a segment of the Koszul complex of [56, I.4.3.1.7]). In light of(i), the maps ωn−2

2 and ωn−11 yield a morphism of complexes Σ′ → Σ, so we are reduced to

checking that Σ is exact, and the inductive assumption already says that Σ is exact at the middleterm H1G

n−1. Now notice that, since the Koszul complex is exact, the inductive assumptionimplies that E2

pq = 0 for all (p, q) such that p > 1− n and q > 0; moreover, obviously we haveE1pq = 0 for p+ q < 0 or p > 0, therefore

E∞00 = E100 and E∞−n,n = E2

−n,n = Coker δn−11 .

But a simple inspection shows that the natural map H0P → E∞00 is an isomorphism, so E∞−n,n =

0, therefore δn−11 is surjective, and Σ is exact, as required.

(ii.b): We use the previous spectral sequence, for k = n. Taking into account (ii.a), and theexactness of the Koszul complex ([56, I.4.3.1.7]) we get

E1pq = 0 for p+ q < 0 or p > 0 or p ≤ −n

E2p,q = 0 for every (p, q) with p > 1− n and q > 0

E21−n,i+n−1

∼→ Coker ∂n−2i+1 for every i ∈ N.

It follows that Er1−n+r,i+n−r = 0 for every r ≥ 2 and every i ∈ N; indeed, this is clear if

1 − n + r > 0, and if the latter fails, we get 1 − n + r > 1 − n and i + n − r > 0, so thestated vanishing holds nevertheless. We deduce that E∞1−n,i+n−1 = E2

1−n,i+n−1 for every i ∈ N.Next, suppose that i > 0. Clearly we have E∞p,i−p = 0 both for p > 0 and p ≤ −n; and if0 ≥ p > 1 − n, we get i − p ≥ i > 0, so again E∞p,i−p = 0. In conclusion, E∞p,i−p = 0 exceptpossibly for p = 1− n, and the contention follows.

The following is a well known theorem of Quillen, which is found in his typewritten notes“On the homology of commutative rings” that have been circulated since the late 60s, but havenever been published.

Theorem 4.5.37. Let R be any simplicial A-algebra, and I ⊂ R a quasi-regular ideal suchthat H0I = 0. Then we have :

HiIn = 0 for every n ∈ N and every i < n.

Proof. We argue by induction on n ∈ N. For n ≤ 1 there is nothing to prove, so suppose thatn > 1, and that the assertion is already known for In−1. Set Rk := R/Ik+1 for every k ∈ N,and let

Λ• ⊗R S• := Λ•R(I/I2)⊗R Sym•R(I/I2)

which is a bigraded R0-module, endowed with the bigraded derivation ∂•• such that ∂•• [i] is theKoszul derivation of Λ• ⊗R S•[i], defined as in [56, I.4.3.1.2]. Set as well

Ci := Coker (∂n−3i : Λi ⊗R Sn−3 → Λi−1 ⊗R Sn−2) for every i ∈ N.


From the inductive assumption and lemma 4.5.22, we already know that

(4.5.38) Hi(I`⊗R In−1) = 0 for every i < n.

However, according to remark 4.5.14, there is a spectral sequence

E1pq := TorR[p]

q (I[p], I[p]n−1)⇒ Hp+q(I`⊗R In−1)

and proposition 4.5.35(ii.b) yields natural isomorphisms

E1pq∼→ Tor

R[p]q+1(R0[p], I[p]n−1)

∼→ TorR[p]q+2(R0[p], Rn−2[p])

∼→ Cq+3[p] for every q > 0

under which, the differentials E1pq → E1

p−1,q are identified with those of the chain complexassociated to the R0-module Cq+3. Now, since I/I2 is a flat R0-module, [56, I.4.3.1.7] says thatthe sequence of R0-modules

Σi : 0→ Λn+i−3∂−→ Λn+i−4 ⊗R S1 → · · · → Λi ⊗R Sn−3 ∂−→ Λi−1 ⊗R Sn−2 → Ci → 0

is an exact complex, for every i > 0. On the other hand, since H0I/I2 = 0, (and again, since

I/I2 is a flat R0-module), combining [56, I.4.3.2.1] and lemma 4.5.22, we see that

Hi(Λj ⊗R Sq−j) = 0 for every q ∈ N, every j ≤ q, and every i < q.

We recall the following general

Claim 4.5.39. Let A be any abelian category, d ∈ N any integer, and consider an exact sequence

0→ Kdfd−−→ Kd−1 → · · · → K1

f1−−→ K0 → 0

of objects of C(A ), such that Ki = 0 in D(A ), for every i = 1, . . . , d − 1. Then there is anatural isomorphism

K0∼→ Kd[d− 1] in D(A ).

Proof of the claim. For every i = 0, . . . , d, set Zi := Ker fi. We have short exact sequences

0→ Zi → Ki → Zi−1 → 0 for i = 1, . . . , d.

Since Ki = 0 in D(A ), there follows natural isomorphisms Zi−1∼→ Zi[1] for every i =

1, . . . , d− 1. However, Z0 = K0 and Zd−1 = Kd, whence the claim. ♦

By applying claim 4.5.39 to the exact sequence coskn+i−3Σi, we easily deduce that

HjCi = 0 for every i > 0 and every j < n+ i− 3

so finally E2pq = 0 for every q > 0 and every p < n + q, and clearly E1

pq = 0 for every q < 0.Consequently, for every i < n the termE∞i−q,q = 0 vanishes for q > 0, and equalsE2

i,0 for q = 0.Lastly, a simple calculation shows that

E2i,0 = Hi(I ⊗R In−1) for every i ∈ N.

Summing up, and taking into account 4.5.38, we conclude that Hi(I ⊗R In−1) = 0 for everyi < n, so we are reduced to checking :

Claim 4.5.40. The natural map Hi(I ⊗R In−1)→ Hi(In) is surjective, for every i ≤ n.

Proof of the claim. Indeed, denote by K the kernel of the epimorphism I ⊗R In−1 → In; as inthe foregoing, we get natural isomorphisms

K[i]∼→ Tor

R[i]1 (R0[i], I[i]n−1)

∼→ TorR[i]2 (R0[i], R[i]/I[i]n−1)

∼→ C3[i] for every i ∈ N

that amount to an isomorphism K∼→ C3 of R0-modules. We have already seen that HiC3 = 0

for every i < n, whence the claim.


4.6. Witt vectors, Fontaine rings and divided power algebras. We begin with a quick reviewof the ring of Witt vectors associated to an arbitrary ring; for the complete details we refer to[14, Ch.IX, §1]. Henceforth we let p be a fixed prime number. To start out, following [14,Ch.IX, §1, n.1] one defines, for every n ∈ N, the n-th Witt polynomial

ωn(X0, . . . , Xn) :=n∑i=0

piXpn−i

i ∈ Z[X0, . . . , Xn].

Notice the inductive relations :

(4.6.1)ωn+1(X0, . . . , Xn+1) = ωn(Xp

0 , . . . , Xpn) + pn+1Xn+1

=Xpn+1

0 + p · ωn(X1, . . . , Xn+1).

Then one shows ([14, Ch.IX, §, n.3]) that there exist polynomials

Sn, Pn ∈ Z[X0, . . . , Xn, Y0, . . . , Yn] In ∈ Z[X0, . . . , Xn] Fn ∈ Z[X0, . . . , Xn+1]

characterized by the identities :

ωn(S0, . . . , Sn) = ωn(X0, . . . , Xn) + ωn(Y0, . . . , Yn)

ωn(P0, . . . , Pn) = ωn(X0, . . . , Xn) · ωn(Y0, . . . , Yn)

ωn(I0, . . . , In) = − ωn(X0, . . . , Xn)

ωn(F0, . . . , Fn) = ωn+1(X0, . . . , Xn+1)

for every n ∈ N. A simple induction shows that :

(4.6.2) Fn ≡ Xpn (mod pZ[X0, . . . , Xn+1]) for every n ∈ N.

(See [14, Ch.IX, §1, n.3, Exemple 4].)

4.6.3. Let now A be an arbitrary (commutative, unitary) ring. For every a := (an | n ∈ N),b := (bn | n ∈ N) in AN, one sets :

SA(a, b) := (Sn(a0, . . . , an, b0, . . . , bn) | n ∈ N)

PA(a, b) := (Pn(a0, . . . , an, b0, . . . , bn) | n ∈ N)

IA(a) := (In(a0, . . . , an) | n ∈ N)

ωA(a) := (ωn(a0, . . . , an) | n ∈ N).

There follow identities :ωA(SA(a, b)) = ωA(a) + ωA(b)

ωA(PA(a, b)) = ωA(a) · ωA(b)

ωA(IA(a)) = − ωA(a)

for every a, b ∈ AN. With this notation one shows that the set AN, endowed with the additionlaw SA : AN × AN → AN and product law PA : AN × AN → AN is a commutative ring, whosezero element is the identically zero sequence 0A := (0, 0, . . . ), and whose unit is the sequence1A := (1, 0, 0, . . . ) ([14, Ch.IX, §1, n.4, Th.1]). The resulting ring (AN, SA, PA, 0A, 1A) isdenoted W (A), and called the ring of Witt vectors associated to A. Furthermore, let a ∈ W (A)be any element; then :

• The opposite −a (that is, with respect to the addition law SA) is the element IA(a).• For every n ∈ N, the element ωn(a) := ωn(a0, . . . , an) ∈ A is called the n-th ghost

component of a, and the map ωn : W (A)→ A : b 7→ ωn(b) is a ring homomorphism.


Hence, the map W (A) → AN : a 7→ ωA(a) is also a ring homomorphism (where AN is en-dowed with termwise addition and multiplication). Henceforth, the addition and multiplicationof elements a, b ∈ W (A) shall be denoted simply in the usual way : a + b and a · b, and theneutral element of addition and multiplication shall be denoted respectively 0 and 1. The ruleA 7→ W (A) is a functor from the category of rings to itself; indeed, if ϕ : A → B is anyring homomorphism, one obtains a ring homomorphism W (ϕ) : W (A) → W (B) by settinga 7→ (ϕ(an) | n ∈ N). In other words, W (ϕ) is induced by the map of sets ϕN : AN → BN, andmoreover one has the identity :

ϕN ωA = ωB W (ϕ).

4.6.4. Next, one defines two mappings on elements of W (A), by the rule :FA(a) := (Fn(a0, . . . , an+1) | n ∈ N)

VA(a) := (0, a0, a1, . . . ).

FA and VA are often called, respectively, the Frobenius and Verschiebung maps ([14, Ch.IX, §1,n.5]). Let also fA : AN → AN be the ring endomorphism given by the rule a 7→ (a1, a2, . . . ), andvA : AN → AN the endomorphism of the additive group ofAN defined by a 7→ (0, pa0, pa1, . . . ).Then the basic identity characterizing (Fn | n ∈ N) can be written in the form :

(4.6.5) ωA FA = fA ωA.On the other hand, (4.6.1) yields the identity :

ωA VA = vA ωA.Let ϕ : A→ B be a ring homomorphism; directly from the definitions we obtain :

(4.6.6) W (ϕ) FA = FB W (ϕ) W (ϕ) VA = VB W (ϕ).

The map FA is an endomorphism of the ring W (A), and VA is an endomorphism of the additivegroup underlying W (A) ([14, Ch.IX, §1, n.5, Prop.3]). Moreover one has the identities :

(4.6.7)FA VA = p · 1W (A)

VA(a · FA(b)) = VA(a) · b for every a, b ∈ W (A)

and FA is a lifting of the Frobenius endomorphism of W (A)/pW (A), i.e. :

FA(a) ≡ ap (mod pW (A)) for every a ∈ W (A).

It follows easily from (4.6.7) that Vm(A) := ImV mA is an ideal of W (A) for every m ∈ N.

The filtration (Vm(A) | m ∈ N) defines a linear topology T on W (A). This topology can bedescribed explicitly, due to the identity :

(4.6.8) a = (a0, . . . , am−1, 0, . . . ) + V mA fmA (a) for every a ∈ W (A).

(See [14, Ch.IX, §1, n.6, Lemme 4] : here the addition is taken in the additive group underlyingW (A).) Indeed, (4.6.8) implies that T agrees with the product topology on AN, regarded as theinfinite countable power of the discrete spaceA. Hence (W (A),T ) is a complete and separatedtopological ring.

4.6.9. The projection ω0 admits a useful set-theoretic splitting, the Teichmuller mapping :

τA : A→ W (A) a 7→ (a, 0, 0, . . . ).

For every a ∈ A, the element τA(a) is known as the Teichmuller representative of a. Noticethat:

ωn τA(a) = apn

for every n ∈ N.Moreover, τA is a multiplicative map, i.e. the following identity holds :

τA(a) · τA(b) = τA(a · b) for every a, b ∈ A


(see [14, Ch.IX, §1, n.6, Prop.4(a)]). Furthermore, (4.6.8) implies the identity :

(4.6.10) a =∞∑n=0

V nA (τA(an)) for every a ∈ W (A)

(where the convergence of the series is relative to the topology T : see [14, Ch.IX, §1, n.6,Prop.4(b)]).

4.6.11. For every n ∈ N we set :

Wn(A) := W (A)/Vn(A)

(the n-truncated Witt vectors of A). This defines an inverse system of rings (Wn(A) | n ∈ N),whose limit is W (A). Clearly the ghost components descend to well-defined maps :

ωm : Wn(A)→ A for every m < n.

Especially, the map ω0 : W1(A)→ A is an isomorphism.

4.6.12. Let now A be a ring of characteristic p. In this case (4.6.2) yields the identity :

(4.6.13) FA(a) = (apn | n ∈ N) for every a := (an | n ∈ N) ∈ W (A).

As an immediate consequence we get :

(4.6.14) p · a = VA FA(a) = FA VA(a) = (0, ap0, ap1, . . . ) for every a ∈ W (A).

For such a ring A, the p-adic topology on W (A) agrees with the V1(A)-adic topology, andboth are finer than the topology T . Moreover, W (A) is complete for the p-adic topology ([14,Ch.IX, §1, n.8, Prop.6(b)]). Furthermore, let ΦA : A → A be the Frobenius endomorphisma 7→ ap; (4.6.13) also implies the identity :

(4.6.15) FA τA = τA ΦA.

4.6.16. Suppose additionally that A is perfect, i.e. that ΦA is an automorphism on A. It followsfrom (4.6.13) that FA is an automorphism on W (A). Hence, in view of (4.6.7) and (4.6.14) :

pn ·W (A) = Vn(A) for every n ∈ N.

So, the topology T agrees with the p-adic topology; more precisely, one sees that the p-adicfiltration coincides with the filtration (Vn(A) | n ∈ N) and with the V1(A)-adic filtration. Espe-cially, the 0-th ghost component descends to an isomorphism

ω0 : W (A)/pW (A)∼→ A.

Also, (4.6.10) can be written in the form :

(4.6.17) a =∞∑n=0

pn · τA(ap−n

n ) for every a ∈ W (A).

The next few results are proposed as a set of exercises in [14, Ch.IX, pp.70-71]. For the sakeof completeness, we sketch the proofs.


4.6.18. Let A be a ring, and (Jn | n ∈ N) a decreasing sequence of ideals of A, such thatpJn + Jpn ⊂ Jn+1 for every n ∈ N. Let πn : A→ A/Jn and π′n : A/Jn → A/J0 be the naturalprojections. Let also R be a ring of characteristic p, and ϕ : R→ A/J0 a ring homomorphism.We say that a map of sets ϕn : R→ A/Jn is a lifting of ϕ if ϕn(xp) = ϕn(x)p for every x ∈ R,and π′n ϕn = ϕ.

Lemma 4.6.19. In the situation of (4.6.18), the following holds :(i) If ϕn and ϕ′n are two liftings of ϕ, then ϕn and ϕ′n agree on Rpn .

(ii) If R is a perfect ring, then there exists a unique lifting ϕn : R → A/Jn of ϕ. We haveϕn(1) = 1 and ϕn(xy) = ϕn(x) · ϕn(y) for every x, y ∈ R.

(iii) IfR is perfect andA is complete and separated for the topology defined by the filtration(Jn | n ∈ N), then there exists a unique map of sets ϕ : R → A such that π0 ϕ = ϕand ϕ(xp) = ϕ(x)p for every x ∈ R. Moreover, ϕ(1) = 1 and ϕ(xy) = ϕ(x) · ϕ(y)for every x, y ∈ R. If furthermore, A is a ring of characteristic p, then ϕ is a ringhomomorphism.

Proof. (i) is an easy consequence of the following :

Claim 4.6.20. Let x, y ∈ R; if x ≡ y (mod J0), then xpn ≡ ypn

(mod Jn) for every n ∈ N.

Proof of the claim. It is shown by an easy induction on n ∈ N, which we leave to the reader towork out. ♦

(ii): The uniqueness follows from (i). For the existence, pick any map of sets ψ : R → Asuch that π0 ψ = ϕ. We let :

ϕn(x) := πn ψ(xp−n

)pn

for every x ∈ R.Using claim 4.6.20 one verifies easily that ϕn does not depend on the choice of ψ. Especially,define ψ′ : R → A by the rule : x 7→ ψ(xp

−1)p for every x ∈ R. Clearly π0 ψ′ = ϕ as well,

hence :

ϕn(xp) = πn ψ′(xp1−n

)pn

= πn ψ(xp−n

)pn+1

= ϕn(x)p for every x ∈ Ras claimed. If we choose ψ so that ψ(1) = 1, we obtain ϕn(1) = 1. Finally, since ϕ is a ringhomomorphism we have ψ(x) · ψ(y) ≡ ψ(xy) (mod J0) for every x, y ∈ R. Hence

ψ(x)pn · ψ(y)p

n ≡ ψ(xy)pn

(mod Jn)

(again by claim 4.6.20) and finally ϕn(xpn) ·ϕn(yp

n) = ψ((xy)p

n), which implies the last stated

identity, since R is perfect.(iii): The existence and uniqueness of ϕ follow from (ii). It remains only to show that ϕ(x)+

ϕ(y) = ϕ(x + y) in case A is a ring of characteristic p. It suffices to check the latter identityon the projections onto A/Jn, for every n ∈ N, in which case one argues analogously to theforegoing proof of the multiplicative property for ϕn : the details shall be left to the reader.

Proposition 4.6.21. In the situation of (4.6.18), suppose that R is a perfect ring and A iscomplete and separated for the topology defined by the filtration (Jn | n ∈ N). Then thefollowing holds :

(i) For every n ∈ N there exists a unique ring homomorphism vn : Wn+1(A/J0)→ A/Jnsuch that the following diagram commutes :

Wn+1(A)ωn //

Wn+1(π0)

A

πn

Wn+1(A/J0)vn // A/Jn.


(ii) Let un := vn Wn+1(ϕ) F−nR , where FR : Wn+1(R) → Wn+1(R) is induced bythe Frobenius automorphism of W (R). Then, for every n ∈ N the following diagramcommutes :

Wn+2(R)un+1 //

A/Jn+1

ϑ

Wn+1(R)un // A/Jn

where the vertical maps are the natural projections.(iii) There exists a unique ring homomorphism u such that the following diagram commutes:

W (R)u //

ω0

A

π0

Rϕ // A/J0.

Furthermore, u is continuous for the p-adic topology on W (R), and for every a :=(an | n ∈ N) ∈ W (R) we have :

(4.6.22) u(a) =∞∑n=0

pn · ϕ(ap−n

n )

where ϕ : R→ A is the unique map of sets characterized as in lemma 4.6.19(iii).

Proof. (i): We have to check that ωn(a0, . . . , an) ∈ Jn whenever a0, . . . , an ∈ J0, which isclear from the definition of the Witt polynomial ωn(X0, . . . , Xn).

(ii): Since FR is an isomorphism, it boils down to verifying :

Claim 4.6.23. ϑ vn+1(ϕ(a0), . . . , ϕ(an+1)) = vn(ϕ(F0(a0, a1)), . . . , ϕ(Fn(a0, . . . , an+1))) forevery a := (a0, . . . , an+1) ∈ Wn+1(A).

Proof of the claim. Directly from (4.6.6) we derive :

(ϕ(F0(a0, a1)), . . . , ϕ(Fn(a0, . . . , an+1))) = (F0(ϕ(a0), ϕ(a1)), . . . , Fn(ϕ(a0), . . . , ϕ(an)))

for every a ∈ Wn+1(A). Hence, it suffices to show that

ϑ vn+1(b0, . . . , bn+1) = vn(F0(b0, b1), . . . , Fn(b0, . . . , bn+1))

for every (b0, . . . , bn+1) ∈ Wn+1(A/J0). The latter identity follows from (4.6.5). ♦

(iii): The existence of u follows from (ii); indeed, it suffices to take u := limn∈N un. Withthis definition, it is also clear that u is continuous for the topology T of W (R); however, sinceR is perfect, the latter coincides with the p-adic topology (see (4.6.12)). Next we remark thatπ0 u τR = ϕ ω0 τR = ϕ, hence :

(4.6.24) u τR = ϕ

by lemma 4.6.19(iii). Identity (4.6.22) follows from (4.6.17), (4.6.24) and the continuity ofu.

4.6.25. Let now R be any ring; we let (An;ϕn : An+1 → An | n ∈ N) be the inverse system ofrings such that An := R/pR and ϕn is the Frobenius endomorphism ΦR : R/pR → R/pR forevery n ∈ N. We set :

E(R)+ := limn∈N

An.

So E(R)+ is the set of all sequences (an | n ∈ N) of elements an ∈ R/pR such that an = apn+1

for every n ∈ N. We denote by TE the pro-discrete topology on E(R)+ obtained as the inverse


limit of the discrete topologies on the rings An. As an immediate consequence of the definition,E(R)+ is a perfect ring. Let ΦE(R)+ be the Frobenius automorphism; notice that :

Φ−1E(R)+(an | n ∈ N) = (an+1 | n ∈ N) for every (an | n ∈ N) ∈ E(R)+.

The projection to A0 defines a natural ring homomorphism

uR : E(R)+ → R/pR

and a basis of open neighborhoods of 0 ∈ E(R)+ is given by the descending family of ideals :

Ker(E(R)+ → An) = Ker(uR Φ−nE(R)+) for every n ∈ N.

4.6.26. Suppose now that (R, |·|R) is a valuation ring with value group ΓR, residue characteristicp and generic characteristic 0. Set Γp

∞

R :=⋂n∈N Γp

n

R , and let ΓE(R) ⊂ Γp∞

R be the convexsubgroup consisting of all γ ∈ Γp

∞

R such that |p|−nR > γ > |p|nR for every sufficiently largen ∈ N. We define a mapping

| · |E(R) : E(R)+ → ΓE(R) ∪ 0

as follows. Let a := (an | n ∈ N) ∈ E(R)+ be any element, and for every n ∈ N choose arepresentative an ∈ R for the class an ∈ R/pR. If |an|R ≤ |p|R for every n ∈ N, then we set|a|E(R) := 0; if there exists n ∈ N such that |an| > |p|R, then we set |a|E(R) := |an|p

n

R . Oneverifies easily that the definition is independent of all the choices : the details shall be left to thereader.

Lemma 4.6.27. (i) In the situation of (4.6.26), the pair (E(R)+, | · |E(R)) is a valuation ring.(ii) Especially, if R is deeply ramified of rank one, then ΓR = ΓE(R) is the value group of

E(R)+.

Proof. (i): One checks easily that |a · b|E(R) = |a|E(R) · |b|E(R) for every a, b ∈ E(R)+, as wellas the inequality : |a + b|E(R) ≤ max(|a|E(R), |b|E(R)). It is also clear from the definition that|a|E(R) = 0 if and only if a = 0. Next, let us show that E(R)+ is a domain. Indeed, supposethat a := (an | n ∈ N) and b := (bn | n ∈ N) are two non-zero elements such that a · b = 0;we may assume that 0 < |a|E(R) ≤ |b|E(R), in which case |an|p

n

R ≤ |bn|pn

R for every sufficientlylarge n ∈ N, and every choice of representative an ∈ R (resp. bn ∈ R) for an (resp. for bn).By assumption we have |an · bn|R ≤ |p|R for every n ∈ N, whence |an|2R ≤ |p|R for sufficientlylarge n ∈ N, hence |apn|R ≤ |p|R, consequently ap = 0, and since E(R)+ is perfect, we deducethat a = 0, a contradiction. To conclude the proof of (i), it then suffices to show :

Claim 4.6.28. |a|E(R) ≤ |b|E(R) if and only if a · E(R)+ ⊂ b · E(R)+.

Proof of the claim. We may assume that 0 < |a|E(R) ≤ |b|E(R), and we need to show that bdivides a in E(R)+. Choose as usual representatives an and bn for an and bn, for every n ∈ N;we may assume that |bn| ≥ |p| for every n ∈ N, and then directly from the definitions, it is clearthat |an|R ≤ |bn|R, hence there exist n0 ∈ N, xn ∈ R such that 0 6= an = xn · bn in R/pR forevery n ≥ n0. We compute : an = apn+1 = xpn+1 · b

pn+1 = xpn+1 · bn, hence (xpn+1 − xn) · bn = 0

for every n ≥ n0. For every n ∈ N, let yn := xpn+1 − xn; notice that b2n 6= 0 and yn · bn = 0

in R/pR whenever n ≥ n0, which implies especially that yn does not divide bn in R/pR. Itfollows that bn divides yn when n ≥ n0, and therefore y2

n = 0 for all n ≥ n0. In turn, this showsthat xp

2

n+1 = xpn for every n ≥ n0, so that, up to replacing finitely many terms, the sequencex := (xpn+1 | n ∈ N) is a well-defined element of E(R)+. Clearly a · x = b, which is the soughtassertion. ♦


(ii): If R is deeply ramified of rank one, then the Frobenius endomorphism ΦR is surjective([36, Prop.6.6.6]), so that ΓR = Γp

∞

R = ΓE, and moreover the map uR : E(R)+ → R/pR isobviously surjective, so that |E(R)+|E(R) = Γ+

R ∪ 0.

4.6.29. We resume the general situation of (4.6.25), and we define :

A(R)+ := W (E(R)+)

where W (−) denotes the ring of Witt vectors as in (4.6.3). According to (4.6.16), A(R)+

is complete and separated for the p-adic topology; moreover we see from (4.6.14) that p is aregular element of A(R)+, and the ghost component ω0 descends to an isomorphism

ω0 : A(R)+/pA(R)+ ∼→ E(R)+.

The map ω0 also admits a set-theoretic multiplicative splitting, the Teichmuller mapping of(4.6.9), which will be here denoted :

τR : E(R)+ → A(R)+.

Furthermore, let R∧ denote the p-adic completion of R; by proposition 4.6.21(iii), the compo-sition uR ω0 lifts to a unique map

uR : A(R)+ → R∧

which is continuous for the p-adic topologies. Finally, A(R)+ is endowed with an automor-phism, which we shall denote by σR : A(R)+ → A(R)+, which lifts the Frobenius map ofE(R)+, i.e. such that the horizontal arrows of the diagram :

(4.6.30)

A(R)+ σR //

ω0

A(R)+

ω0

E(R)+

ΦE(R)+ //

τR

OO

E(R)+

τR

OO

commute with the downward arrows (see (4.6.16)). The horizontal arrows commute also withthe upward ones, due to (4.6.15).

4.6.31. Let a := (an | n ∈ N) be any element of E(R)+, and for every n ∈ N choose arepresentative an ∈ R for the class an. In view of (4.6.22) and by inspecting the proof oflemma 4.6.19 we deduce easily the following identity

(4.6.32) uR τR(a) = limn→∞

apn

n

where the convergence is relative to the p-adic topology of R∧.

Proposition 4.6.33. With the notation of (4.6.29), suppose moreover that there exist π, a ∈ Rsuch that π is regular, a is invertible, p = πp · a, and ΦR induces an isomorphism :

R/πR∼→ R/pR : xmod πR 7→ xp mod pR.

Then the following holds :(i) uR and uR are surjections.

(ii) There exists a regular element ϑ ∈ A(R)+ such that :

KeruR = ϑ ·A(R)+.

Proof. (i): Indeed, if ΦR is surjective, the same holds for uR, and then the claim follows easilyfrom [61, Th.8.4].

(ii): The surjectivity of uR says especially that we may find π := (πn | n ∈ N) ∈ E(R)+

such that π0 is the image of π in R/pR.

Claim 4.6.34. (i) πpn generates Ker (ΦnR : R/pR→ R/pR) for every n ∈ N.


(ii) Let fn ∈ R be any lifting of πn; then there exists an invertible element an ∈ R∧ suchthat fpn+1

n = an · p.(iii) πp is a regular element of E(R)+, and generates KeruR.

Proof of the claim. (i): We argue by induction on n ∈ N. We have πp0 = 0, so the claimis clear for n = 0. For n = 1, the assertion is equivalent to our assumption on π. Next,suppose that the assertion is known for some n ≥ 1, and let x ∈ R/pR such that Φn+1

R (x) = 0;hence ΦR(x) = y · πpn for some y ∈ R/pR. Since ΦR is surjective, we may write y = zp forsome z ∈ R/pR, therefore (x − z · πn)p = 0, and consequently x − z · πn ∈ π0(R/pR), sox ∈ πn(R/pR) = πpn+1(R/pR), as stated.

(ii): By assumption, fpnn − π ∈ pR, hence fpn+1

n ≡ a−1p (mod πpR), in other words,fp

n+1

n = p(a−1 + πb) for some b ∈ R. Since R∧ is complete and separated for the π-adictopology, the element an := a−1 + πb is invertible in R∧; whence the contention.

(iii): In view of (i), for every n ∈ N we have a ladder of short exact sequences :

0 // (πpn+1) //

ΦR

R/pRΦn+1R //

ΦR

R/pR // 0

0 // (πpn) // R/pRΦnR // R/pR // 0

and since all the vertical arrows are surjections, we deduce a short exact sequence ([75, Lemma3.5.3]) :

0→ limn∈N

(πpn)j−→ E(R)+ uR−→ R/pR→ 0.

Next, choose fn as in (ii); since p is regular in R, it is also regular in R∧, and then the sameholds for fn. It follows easily that the kernel of the multiplication map µn : R/pR → R/pR :x 7→ πpnx is the ideal In := (πp

n+1−pn ), whence a ladder of short exact sequences :

0 // In //

ΦR

R/pRµn //

ΦR

(πpn)

ΦR

// 0

0 // In−1// R/pR

µn−1 // (πpn−1) // 0

for every n > 0. Notice that pn+1 − p ≥ pn, hence πpn+1−pn−1 = 0 for every n > 0, so the

inverse system (In | n ∈ N) is essentially zero, and the system of maps (µn | n ∈ N) defines anisomorphism ([75, Prop.3.5.7]) :

µ : E(R)+ ∼→ limn∈N

(πpn).

By inspecting the construction, one sees that the composition j µ : E(R)+ → E(R)+ is noneelse than the map : x 7→ x · πp. Both assertions follow immediately. ♦

Claim 4.6.35. ω0 restricts to a surjection KeruR → KeruR.

Proof of the claim. By the snake lemma, we are reduced to showing that uR restricts to asurjection Kerω0 → pR∧, which is clear since Kerω0 = pA(R)+ and uR is surjective. ♦

From [61, Th.8.4] and claims 4.6.35, 4.6.34(iii) we deduce that every ϑ ∈ KeruR ∩ω−10 (πp)

generates KeruR. Furthermore, claim 4.6.34(iii) also says that the sequence (p, ϑ) is regular onA(R)+. Since A(R)+ is p-adically separated, the sequence (ϑ, p) is regular as well (corollary4.1.22); especially ϑ is regular, as stated.


Example 4.6.36. Let R be a ring such that p is regular in R, and ΦR is surjective. Supposealso that we have a compatible system of elements un := p1/pn ∈ R∧ (so that upn+1 = un forevery n ∈ N). In this case, an element ϑ as provided by proposition 4.6.33 can be chosenexplicitly. Indeed, the sequence a := (p, p1/p, p1/p2

, . . . ) defines an element of E(R)+ such thatuR τR(a) = p (see (4.6.32)), and by inspecting the proof of the proposition, we deduce thatthe element

ϑ := p− τR(p, p1/p, p1/p2

, . . . )

is a generator of KeruR.

Definition 4.6.37. Let m,n : Z→ N be any two maps of sets.(i) The support of n is the subset :

Supp n := i ∈ Z | n(i) 6= 0.

(ii) For every t ∈ Z, we let n[t] : Z → N be the mapping defined by the rule: n[t](k) :=n(k + t) for every k ∈ Z.

(iii) We define a partial ordering on the set of all mappings Z→ N by declaring that n ≥mif and only if n(k) ≥m(k) for every k ∈ Z.

4.6.38. Keep the assumptions of proposition 4.6.33, and let n : Z → N be any mapping withfinite support. We set :

ϑn :=∏

i∈Supp n

σiR(ϑn(i))

where ϑ ∈ A(R)+ is – as in proposition 4.6.33(ii) – any generator of KeruR, and σR is theautomorphism introduced in (4.6.29). For every such n, the element ϑn is regular in A(R)+,and if n ≥m, then ϑm divides ϑn. We endow A(R)+ with the linear topology TA which admitsthe family of ideals (ϑn ·A(R)+ | n : Z→ N) as a fundamental system of open neighborhoodsof 0 ∈ A(R)+.

Theorem 4.6.39. In the situation of (4.6.38), the following holds :(i) The topological ring (A(R)+,TA) is complete and separated.

(ii) The homomorphism of topological rings ω0 : (A(R)+,TA) → (E(R)+,TE) is aquotient map.

Proof. We resume the notation of the proof of proposition 4.6.33; especially, the image of ϑ inE(R)+ is the element πp, and π lifts the image of π in R/pR.

Claim 4.6.40. (i) The topology TE on E(R)+ agrees with the π-adic topology.(ii) The sequences (p, ϑn) and (ϑn, p) are regular in A(R)+ for every n : Z → N of finite

support.

Proof of the claim. Clearly :ω0(ϑn) = π

∑i∈Z n(i)·pi+1

and then the regularity of (p, ϑn) follows easily from claim 4.6.34(iii), which also implies that :

πpn+1 · E(R)+ = Ker (uR Φ−nE(R)+) for every n ∈ N

which, in turn, yields (i). To show that also the sequence (ϑn, p) is regular, it suffices to invokecorollary 4.1.22. ♦

Claim 4.6.40(i) already implies assertion (ii). For every n ∈ N, let A+∧n denote the comple-

tion of A+n := A(R)+/pn+1A(R)+ for the topology TAn , defined as the unique topology such


that the surjection (A(R)+,TA) → (A+n ,TAn) is a quotient map. For every n > 0 we have a

natural commutative diagram :

(4.6.41)

0 // A+0

//

A+n

//

A+n−1

//

0

0 // A+∧0

// A+∧n

// A+∧n−1

// 0

whose vertical arrows are the completion maps. Since p is regular in A(R)+, the top row of(4.6.41) is short exact.

Claim 4.6.42. The bottom row of (4.6.41) is short exact as well.

Proof of the claim. It suffices to show that the topology TA0 on A+0 agrees with the subspace

topology induced from A+n ([61, Th.8.1]). This boils down to verifying the equality of ideals of

A(R)+ :

(4.6.43) (ϑn) ∩ (pk) = (ϑn · pk)

for every k ∈ N and every n : Z → N as in (4.6.38). The latter follows easily from claim4.6.40(ii) and [61, Th.16.1]. ♦

Claim 4.6.44. The topological ring (A+n ,TAn) is separated and complete for every n ∈ N.

Proof of the claim. We shall argue by induction on n ∈ N. For n = 0 the assertion reducesto claim 4.6.40(i) (and the fact that TE is obviously a complete and separated topology onE(R)+). Suppose that n > 0 and that the assertion is known for every integer < n. Thismeans that the right-most and left-most vertical arrows of (4.6.41) are isomorphisms. Thenclaim 4.6.42 implies that the middle arrow is bijective as well, as required. ♦

Claim 4.6.45. For every n : Z → N as in (4.6.38), the ring A(R)+/ϑnA(R)+ is complete andseparated for the p-adic topology.

Proof of the claim. Since A(R)+ is complete for the p-adic topology, [61, Th.8.1(ii)] reducesto showing that the p-adic topology on A(R)+ induces the p-adic topology on its submoduleϑnA(R)+. The latter assertion follows again from (4.6.43). ♦

To conclude the proof of (i) we may now compute :

limn:Z→N

A(R)+/ϑnA(R)+ ' limn:Z→N

limk∈N

A+k /ϑnA+

k (by claim 4.6.45)

' limk∈N

limn:Z→N

A+k /ϑnA+

k

' limk∈N

A+k (by claim 4.6.44)

' A(R)+

which is the contention.

We conclude this section with a review of the theory of divided power modules and algebras.

Definition 4.6.46. Let A be a ring, k ∈ N an integer, and M := (M1, . . . ,Mk) an object of theabelian category (A-Mod)k, i.e. any sequence ofA-modules. Let alsoQ be anotherA-module,and d := (d1, . . . , dk) ∈ N⊕k any sequence of k integers.

(i) We denote by FM , GQ : A-Alg→ Set the functors given respectively by the rules :

B 7→ (M1 × · · · ×Mk)⊗A B and B 7→ Q⊗A B for every A-algebra B.


Also, let M∨i := HomA(Mi, A) for every i = 1, . . . , k, and set :

Sym•A(M∨) :=k⊗i=1

Sym•A(M∨i )

which we view as a N⊕k-graded A-algebra, with the grading given by the rule :

SymnA(M∨) :=

k⊗i=1

SymniA (M∨

i ) for every n := (n1, . . . , nk) ∈ N⊕k.

(ii) A homogeneous multipolynomial law of degree d on the sequence of A-modules M ,with values in Q, is a natural transformation λ : FM → GQ, which we write also

λ : M Q

such that the following holds. For every A-algebra B, every (x1, . . . , xk) ∈ FM(B)and every (b1, . . . , bk) ∈ Bk, we have

λB(b1x1, . . . , bkxk) = bd11 · · · b

dkk · λB(x1, . . . , xk).

We denote by PoldA(M,Q) the set of all homogeneous multipolynomial laws M Qof degree d.

Remark 4.6.47. (i) Let f := (fi : M ′i → Mi | i = 1, . . . , k) be a sequence of A-linear maps,

and set M ′ := (M ′1, . . . ,M

′k); let also g : Q → Q′ be another A-linear map. Suppose that

λ : M Q is a homogeneous multipolynomial law of degree d; then we obtain a law

g λ f : M ′ Q′

of the same type, by the rule :

B 7→ (g ⊗A B) λB ((f1 × · · · × fk)⊗A B) : (M ′1 × · · · ×M ′

k)⊗A B → Q′ ⊗A B

for every A-algebra B. This shows that the rule (M,Q) 7→ PoldA(M,Q) is functorial in botharguments. However, it is not a priori obvious that this functor takes values in (essentially)small sets. The latter assertion will nevertheless be proven shortly.

(ii) Suppose that M ′ := (M ′1, . . . ,M

′k′) is another sequence of A-modules, Q′ another A-

module, and d′ := (d′1, . . . , d′k′) ∈ N⊕k′ another sequence of integers. If we have two homoge-

neous polynomial laws λ : M Q and λ′ : M ′ Q′ of degrees respectively d and d′, we canform the tensor product

λ⊗ λ′ : (M,M ′) Q⊗A Q′

which is the homogeneous multipolynomial law of degree (d, d′) given by the rule :

(λ⊗ λ′)B(x1, . . . , xk, x′1, . . . , x

′k′) := λB(x1, . . . , xk)⊗ λ′B(x′1, . . . , x

′k′)

for every A-algebra B and every (x1, . . . , x′k′) ∈ (M1 × · · · ×M ′

k′)⊗A B.(iii) Let B be any A-algebra; if λ : M Q is a law as in (i), then the restriction of λ to

B-algebras yields a law λ/B : M ⊗A B → Q⊗B. In this way, we obtain a natural mapping

(4.6.48) PoldA(M,Q)→ PoldB(M ⊗A B,Q⊗A B).

Lemma 4.6.49. In the situation of definition 4.6.46, suppose that the A-module Mi is free offinite rank, for every i = 1, . . . , k. Then, there exists a natural bijection

PoldA(M,Q)∼→ Q⊗A Symd

A(M∨).


Proof. Indeed, fix bases (ei1, . . . , eiri) for each A-module Mi, and denote by (e∗i1, . . . , e∗iri

) thedual basis of M∨

i . For every i = 1, . . . , k there is a natural (injective) map of A-algebras

Sym•A(M∨i )→ Sym•A(M∨) : s 7→ 1⊗ · · · ⊗ s⊗ · · · ⊗ 1

which allows to view naturally the e∗ij as elements of Sym•A(M∨). Suppose that λ : M Q isa given homogeneous multipolynomial law of degree d, and set

Pλ(e∗ij) := λSym•A(M∨)

(r1∑j=1

e1j ⊗ e∗1j, . . . ,rk∑j=1

ekj ⊗ e∗kj

)∈ Q⊗A Sym•A(M∨).

Notice that the terms in parenthesis are elements of Mi ⊗A M∨i ⊂ Mi ⊗A Sym•A(M∨

i ) that donot depend on the chosen bases, since they correspond to the identity automorphisms of Mi,under the natural identification Mi ⊗A M∨

i∼→ EndA(Mi). Now, let B be any A-algebra, and

(bij | i = 1, . . . , k; j = 1, . . . , ri) any sequence of elements of B. By considering the uniquemap of A-algebras Sym•A(M) → B given by the rule : e∗ij 7→ bij for every i = 1, . . . , k andevery j = 1, . . . , ri, it is easily seen that

λB

(r1∑j=1

b1je1j, . . . ,

rk∑j=1

bkjekj

)= Pλ(bij) ∈ Q⊗A B.

In other words, λ is completely determined by the polynomial Pλ(e∗ij) with coefficients in Q.Next, set B := Sym•A(M∨)[Y1, . . . , Yk]; the homogeneity condition on λ implies that

Pλ(Yie∗ij) = λB

(Y1 ·

r1∑j=1

e1j ⊗ e∗1j, . . . , Yk ·rk∑j=1

ekj ⊗ e∗kj

)= Y d1

1 · · ·Ydkk · P (e∗ij)

which means that Pλ ∈ Q ⊗A SymdA(M∨). Conversely, it is clear that any element of Q ⊗A

SymdA(M∨) yields a homogeneous law of degree d, whence the lemma.

Remark 4.6.50. (i) Lemma 4.6.49 shows especially that, if M1, . . . ,Mk are free A-modules offinite rank, the rule Q 7→ PoldA(M,Q) yields a functor with values in essentially small sets, andit is clear that the sum of two homogeneous multipolynomial laws of degree d is still a law of thesame type; likewise, if we multiply such a law by an element ofA, we get a new law of the sametype. Up to isomorphism, we may therefore assume that this is a functor A-Mod→ A-Mod.

(ii) Next, set

ΓdA(M) := (SymdAM

∨)∨.

The lemma can be rephrased by saying that, under the stated assumptions, theA-module ΓdA(M)represents the functor

A-Mod→ A-Mod : Q 7→ PoldA(M,Q).

(iii) The rule (A,M) 7→ ΓdA(M) is functorial for sequences M of free A-modules of finiterank. Indeed, let ϕ : M → N be a morphism of such sequences. For every degree d of lengthk, the induced map PoldA(N,Q)→ PoldA(M,Q) corresponds to a map

(4.6.51) SymdA(N∨)⊗A Q→ Symd

A(M∨)⊗A Q

that can be worked out as follows. Pick bases (eij | j = 1, . . . , ri) forNi and (eij | j = 1, . . . , si)for Mi, for every i = 1, . . . , k, and denote as usual by (e∗ij | j = 1, . . . , ri), (f ∗ij | j = 1, . . . , si)the respective dual bases. Say that λ : N Q is a given homogeneous law of degree d, and let


Pλ(e∗ij) ∈ Symd

A(N∨)⊗A Q be the corresponding homogeneous polynomial. By construction,the polynomial Pλϕ ∈ Symd

A(M∨)⊗A Q corresponding to λ ϕ is calculated as

Pλϕ := (λ ϕ)Sym•A(M∨)

(si∑j=1

fij ⊗ f ∗ij | i = 1, . . . , k

)

=λSym•A(M∨)

(si∑j=1

ϕi(fij)⊗ f ∗ij | i = 1, . . . , k

)

=λSym•A(M∨)

(si∑j=1

eij ⊗ ϕ∨i (e∗ij) | i = 1, . . . , k

)=Pλ(ϕ

∨i (e∗ij)).

In other words, (4.6.51) equals SymdA(ϕ∨)⊗A 1Q, so if we let

ΓdA(ϕ) := SymdA(ϕ∨)∨

we obtain :• a functor ΓdA(−) from the full subcategory (A-Modo)kff of (A-Modo)k consisting of

all sequences of free A-modules of finite rank, into the category of A-modules• for every A-module Q, a well defined isomorphism of functors on this subcategory

(Q,M) 7→ (PoldA(M,Q)∼→ HomA(ΓdAM,Q) : A-Mod× (A-Modo)kff → A-Mod

The following result extends the above observations to arbitrary sequences M of A-modules.

Theorem 4.6.52. Let M be any sequence as in definition 4.6.46(i). We have :

(i) PoldA(M,Q) is an essentially small set, for every A-module Q.(ii) There exists a functor

(4.6.53) ΓdA : (A-Mod)k → A-Mod

with a natural isomorphism of A-module valued functors

PoldA(−, Q)∼→ HomA(ΓdA(−), Q) for every A-module Q.

Proof. Suppose first that all the A-modules M1, . . . ,Mk are finitely presented. Then, for everyi = 1, . . . , k, pick a surjective A-linear map fi : Li → Mi, with Li free of finite rank. Next, letgi : Li ⊕ Li →Mi be the map given by the rule : (l, l′) 7→ fi(l − l′) for every (l, l′) ∈ Li ⊕ Li;since Mi is finitely presented, Ker gi is finitely generated, for every i = 1, . . . , k, so we mayfind a surjective A-linear map L′i → Ker gi. By composing with the inclusion Ker gi → Li⊕Liand the two projections Li ⊕ Li → Li, we deduce a presentation of Mi :

(4.6.54) Coker (h1i , h

2i : L′i

//// Li)∼→Mi

such that the induced mapL′i → Li ×Mi

Li

is surjective. Notice that the latter condition is stable under arbitrary base changes A→ B. SetL := (L1, . . . , Lk), L′ := (L′1, . . . , L

′k) and hj := (hj1, . . . , h

jk) for j = 1, 2.

Claim 4.6.55. The presentation (4.6.54) induces a presentation of sets :

PoldA(M,Q)∼→ Ker (PoldA(L,Q)

//// PoldA(L′, Q)) for every A-module Q.


Proof of the claim. Indeed, suppose that λ : L Q is a given homogeneous multipolynomiallaw of degree d, such that λ h1 = λ h2 (notation of remark 4.6.47(i)). Since the induced mapL′i ⊗A B → (Li ⊗A B) ×M⊗AB (Li ⊗A B) is surjective for every i = 1, . . . , k, it follows thatλB descends (uniquely) to a map µB : (M1 × · · · ×Mk)⊗A B → Q⊗A B. An easy inspectionshows that the resulting rule : B 7→ µB is a homogeneous multipolynomial law of degree d. ♦

From claim 4.6.55 and lemma 4.6.49, it follows easily that the theorem holds for M , with

ΓdA(M) := Coker (ΓdA(h1),ΓdA(h2) : ΓdA(L′)//// ΓdA(L))

(details left to the reader). Lastly, let M be an arbitrary sequence. We may then find a filteredsystem of sequences (Mσ | σ ∈ Σ) consisting of finitely presented A-modules and with A-linear transition maps, whose colimit is isomorphic to M . Since such colimits are preserved byarbitrary base changes, and commute with the forgetful functor to sets, it follows easily that thesystem induces an isomorphism of sets :

PoldA(M,Q)∼→ lim

σ∈ΣPoldA(Mσ, Q) for every A-module Q

(details left to the reader). In view of the foregoing, it then follows that the theorem holds forM , with

ΓdA(M) := colimσ∈Σ

ΓdA(Mσ).

Again, we invite the reader to spell out the details.

4.6.56. The identity map of ΓdAM corresponds to a homogeneous multipolynomial law

λdM : M ΓdAM

such that every other law M Q homogeneous of the same degree, factors uniquely throughλdM and an A-linear map ΓdAM → Q (in the sense explained in remark 4.6.47(i)). Especially, ifM ′ ∈ Ob((A-Mod)k

′) is another sequence, and d′ ∈ N⊕k′ any sequence of integers, the tensor

product λdM ⊗ λd′

M ′(remark 4.6.47(ii)) factors uniquely through λd,d

′

M,M ′and an A-linear map

(4.6.57) Γd,d′

A (M,M ′)→ ΓdA(M)⊗A Γd′

A(M ′).

Furthermore, notice that, in the situation of remark 4.6.47(iii), the mapping (4.6.48) is clearlyA-linear; hence, we get an induced B-linear map

(4.6.58) B ⊗A ΓdA(M)→ ΓdB(B ⊗AM)

for every M and d as above, and every A-algebra B.

Corollary 4.6.59. With the notation of (4.6.56), we have :(i) The functor (4.6.53) commutes with all filtered colimits.

(ii) The map (4.6.57) is an isomorphism of A-modules, for every M , M ′, d and d′.(iii) For every A-algebra B, the map (4.6.58) is an isomorphism of B-modules.(iv) If the A-modules M1, . . . ,Mk are all free (resp. finitely presented, resp. finitely gener-

ated, resp. flat), then the same holds for ΓdA(M).

Proof. (i): It suffices to check that the natural map

PoldA(colimσ∈Σ

Mσ, Q)→ limσ∈Σ

PoldA(Mσ, Q)

is an isomorphism, for every filtered system of A-modules (Mσ | σ ∈ Σ) and every A-moduleQ; the latter assertion is obvious (details left to the reader).

(iv): The assertion concerning free modules of finite rank, and finitely generated or finitelypresented modules follows directly from the explicit construction in lemma 4.6.49 and theorem4.6.52. The assertion for flat modules follows from the assertion for free modules of finite


rank and from (i), by means of Lazard’s theorem [57, Ch.I, Th.1.2]. Moreover, from remark4.6.50(iii), we see that, if ϕ : M → N is a morphism of sequences of free A-modules of finiterank, such that ϕi is a split injection for every i = 1, . . . , n, then ΓdA(ϕ) is also a split injectivemap of free A-modules; from this, and from (i), it follows easily that ΓdA transforms sequencesof free A-modules (of possibly infinite rank) into free A-modules : details left to the reader.

(iii): In light of (i), it is easily seen that the natural transformation (4.6.57) commutes with ar-bitrary filtered colimits of A-modules, hence we are reduced to the case where M is a sequenceof finitely presented A-modules. Next, claim 4.6.55 shows that a presentation (4.6.54) yields acommutative diagram

ΓdA(L′)// //

ΓdA(L) //

ΓdA(M) //

0

ΓdB(B ⊗A L′)//// ΓdB(B ⊗A L) // ΓdB(B ⊗AM) // 0

with exact rows. Therefore, we are further reduced to the case where M is a sequence of freeA-modules of finite rank, in which case the assertion follows by a simple inspection of the proofof lemma 4.6.49.

(ii): Let t : Γd,d′

A (M,M ′)→ Q be an A-linear map. To t we associate an A-linear map

t∗ : ΓdA(M)⊗A Γd′

A(M ′)→ Q

as follows. Set τ := t λd,d′

M,M ′: (M,M ′) Q (notation of remark 4.6.47(i)). To every

A-algebra B, the law τ associates the mapping

B ⊗AM → Pold′

B(B ⊗AM ′, B ⊗A Q) x 7→ τB,x

where (τB,x)C(y) := τC(1 ⊗ x, y) for every x ∈ B ⊗A M , every B-algebra C and everyy ∈ C ⊗AM ′. In light of (iii), the latter is the same as a mapping

B ⊗AM → HomB(B ⊗A Γd′

A(M ′), B ⊗A Q) x 7→ tB,x

and notice that, for every x ∈ B ⊗AM , the B-linear map tB,x is characterized by the identity

(4.6.60) tB,x(λd′

M ′,B(y)) = (τB,x)B(y) for every y ∈ B ⊗AM ′.

We deduce a compatible system of mappings

B ⊗A Γd′

A(M ′)→ PoldB(B ⊗AM,B ⊗A Q) : y 7→ τ ∗B,y for every A-algebra B

where (τ ∗B,y)C(x) := tC,x(1 ⊗ y) for every B-algebra C, every x ∈ C ⊗A M , and every y ∈B⊗A Γd

′

A(M). The C-linearity of tC,x implies that these latter mappings are B-linear (for everyA-algebra B), and then they are the same as a compatible system of B-linear maps

(4.6.61) B ⊗A Γd′

A(M ′)→ HomB(B ⊗A ΓdA(M), B ⊗A Q) : y 7→ t∗B,y

for every A-algebra B. Notice that, for every y ∈ B ⊗A Γd′

A(M ′), the B-linear map t∗B,y ischaracterized by the identity :

(4.6.62) t∗B,y(λdM,B(x)) = (τ ∗B,y)B(x) for every x ∈ B ⊗AM.

Obviously, the datum of a compatible system (4.6.61) is the same as that of a map t∗ as sought.

Claim 4.6.63. t∗ (4.6.57) = t.


Proof of the claim. It suffices to show that t∗ (4.6.57) λd,d′

M,M ′= τ . However, recall that

(4.6.57)λd,d′

M,M ′= λdM⊗λ

d′

M ′, so we are reduced to checking that t∗ (λdM⊗λ

d′

M ′) = τ . Hence,

let B be any A-algebra, and m ∈ B⊗AM , m′ ∈ B⊗AM ′ any two elements. To ease notation,set x := λdM(m) and y := λd

′

M ′(m′). We compute :

(t∗ (λdM ⊗ λd′

M ′))B(m,m′) = (B ⊗A t∗)(x⊗ y)

= t∗B,y(x)

= (τ ∗B,y)B(m) (by (4.6.62))

= tB,m(y)

= (τB,m)B(m′) (by (4.6.60))

= τB(m,m′)

as claimed. ♦

Especially, if we let t be the identity map of Γd,d′

A (M,M ′), claim 4.6.63 shows that the result-ing t∗ is a left inverse for (4.6.57). Now, if all the modules M1, . . . ,M

′k′ are free of finite rank,

then lemma 4.6.49 shows that both of the A-modules appearing in (4.6.57) are free and havethe same rank, so the assertion follows, in this case.

Next, suppose that the modules of the two sequences are finitely presented; in this case,we argue by descending induction on the number f of free A-modules appearing in the twosequences. If f = k + k′, the assertion has just been shown. Suppose that f < k + k′, andthe assertion is already known for all pairs of sequences containing at least f + 1 free modules.After permutation of the sequences, we may assume that Mi is not free, for some index i ≤ k.Then, pick a presentation of Mi as in (4.6.54), and let N ′ (resp. N ′′) be the sequences obtainedfrom M by replacing Mi with Li (resp. with L′i); by naturality of (4.6.57), the maps h1

i and h2i

induce a commutative diagram with exact rows :

ΓdA(N ′′,M ′)////

ΓdA(N ′,M ′) //

ΓdA(M,M ′) //

0

ΓdA(N ′′)⊗A ΓdA(M ′)// // ΓdA(N ′)⊗A ΓdA(M ′) // ΓdA(M)⊗A ΓdA(M ′) // 0

so it suffices to prove the assertion for the pair of sequences (N ′,M ′) and (N ′′,M ′); but thelatter contain f + 1 free A-modules, so we conclude by induction.

Lastly, if M and N are arbitrary sequences, we may present them as filtered colimits ofsequences of finitely presented A-modules; since tensor products commute with such colimits,we are done, by the foregoing case (details left to the reader).

Remark 4.6.64. (i) Let M and d be as in (4.6.56), suppose that all the A-modules Mi arefree of finite rank, and set S(M) := Sym•A(M∨). Let Q be another A-module, and λ : M Q a homogeneous multipolinomial law of degree d. From lemma 4.6.49 we may extract thefollowing calculation for the A-linear map λ† : ΓdA(M)→ Q corresponding to λ. Set

Pλ := λS(M)(1M1 , . . . ,1Mk) ∈ ΓdA(M)∨ ⊗A Q.

Then λ† is the image of Pλ under the natural identification ΓdA(M)∨⊗AQ∼→ HomA(ΓdA(M), Q).

(ii) Let M , M ′, d and d′ be as in (4.6.56), suppose that all the A-modules Mi and M ′i are free

of finite rank, and to ease notation let

λ := λdM λ′ := λd′

M ′λ := λ⊗ λ′.


By (i), we compute

Pλ = λS(M,M ′)(1M1 , . . . ,1Mk)⊗ λ′S(M,M ′)(1M ′1 , . . . ,1M ′k′ ).

However, λS(M,M ′)(1M1 , . . . ,1Mk) is the image of Pλ under the A-linear induced by the A-

algebra homomorphism S(M)→ S(M,M ′) = S(M)⊗A S(M ′) given by the rule s 7→ s⊗ 1,for every s ∈ S(M). Likewise we determine λ′S(M,M ′)(1M ′1 , . . . ,1M ′k′ ), and we conclude thatPλ = Pλ ⊗ Pλ′ , whence λ† = λ† ⊗ λ†, under the natural identification

EndA(Γd,d′

A (M,M ′))∼→ EndA(ΓdA(M))⊗A EndA(Γd

′

A(M ′)).

But by definition, (λdM)† is the identity automorphism of ΓdA(M), for every sequence M , andevery degree d. We see therefore that :

(λdM ⊗ λd′

M ′)† = (λd,d

′

M,M ′)†

which translates as saying that the A-linear map (4.6.57) is none else than the transpose of thenatural isomorphism

SymdA(M∨)⊗A Symd′

A(M ′∨)∼→ Symd,d′

A (M∨,M ′∨).

This gives an alternative way to prove corollary 4.6.59(ii), in the case of free A-modules.

4.6.65. Let M and M ′ be two sequences of A-modules, of the same length k, and e, e′ twodegrees of length k. If λ : (M,M ′) Q is any homogeneous law of degree (e, e′), then we canalso regard λ as a homogeneous multipolynomial law of degree e+ e′ on the sequence M ⊕M ′

of length k. In this way, we obtain a natural A-linear mapping

(4.6.66) Pole,e′

A ((M,M ′), Q)→ Pole+e′

A (M ⊕M ′, Q) for every A-module Q

which, by transposition, corresponds to a unique A-linear map

(4.6.67) Γe+e′

A (M ⊕M ′)→ Γe,e′

A (M,M ′).

Fix d ∈ N⊕k; in view of corollary 4.6.59(ii), we see that the sum of the maps (4.6.67) for allpairs (e, e′) with e+ e′ = d, is a well defined A-linear map

∆d

M,M ′:=

⊕e+e′=d

∆e,e′

M,M ′: ΓdA(M ⊕M ′)→

⊕e+e′=d

ΓeA(M)⊗A Γe′

A(M ′).

Proposition 4.6.68. The map ∆d

M,M ′is an isomorphism, for every M , M ′, and every degree d.

Proof. Arguing as in the proof of corollary 4.6.59(ii), we are easily reduced to the case whereMand M ′ are sequences of free A-modules of finite rank. In this case, (4.6.67) can be computedas in remark 4.6.64(i); namely, it corresponds to the element

P ∈ Syme+e′

A ((M ⊕M ′)∨)⊗A Syme,e′

A (M∨,M ′∨)∨

determined as follows. Take Q := Γe,e′

A (M,M ′), and let λ : M ⊕M ′ Γe,e′

A (M,M ′) be theimage of λe,e

′

M,M ′under (4.6.66); then

P = λS(M⊕M ′)(1M1⊕M ′1 , . . . ,1Mk⊕M ′k).

Thus, we come down to evaluating the image of 1Mi⊕M ′i under the natural identification

(Mi ⊕M ′i)⊗A S(Mi ⊕M ′

i)∼→ (Mi ⊗A S(Mi ⊕M ′

i))× (M ′i ⊗A S(Mi ⊕M ′

i)).

for every i = 1, . . . , k. We easily find that 1Mi⊕M ′i 7→ (pi, p′i) under this identification, where

pi : Mi ⊕M ′i →Mi and p′i : Mi ⊕M ′

i →M ′i are the natural projections; so finally

(4.6.69) P = (λe,e′

M,M ′)S(M⊕M ′)(p1, . . . , pk, p

′1, . . . , p

′k).


Pick a basis (ei1, . . . , eiri) (resp. (e′i1, . . . , e′ir′i

)) for each Mi (resp. M ′i), let (e∗i1, . . . , e

∗iri

) (resp.(e′∗i1, . . . , e

′∗ir′i

))) be the dual basis, and for every = 1, . . . , k and every j = 1, . . . , ri (resp.j = 1, . . . , r′i) denote by εij (resp. ε′ij) the element of (Mi ⊕M ′

i)∨ that agrees with e∗ij on Mi

and that vanishes on M ′i (resp. that agrees with e′∗ij on M ′

i and vanishes on Mi). Notice that

pi =

ri∑j=1

ε∗ij ⊗ eij p′i =

r′i∑j=1

ε′∗ij ⊗ e′ij for every i = 1, . . . , k.

With this notation, it follows that the right hand-side of (4.6.69) is the image of

(λe,e′

M,M ′)S(M,M ′)(1M1 , . . . ,1M ′k) = 1

Syme,e′A (M,M ′)

under the natural map

(4.6.70) S(M,M ′)→ S(M ⊕M ′) : e∗ij 7→ ε∗ij e′∗ij 7→ ε′∗ij.

In other words, the sought P is none else than the restriction of (4.6.70) to the direct factorSyme,e′

A (M∨,M ′∨), and this restriction is an isomorphism onto a direct factor of the A-moduleSyme+e′

A (M ⊕M ′). By transposition, we see that (4.6.67) is identified with the projection ontoa natural direct factor of Γe+e

′

A (M⊕M ′), and a simple inspection shows that the sum of all theseprojections is the identity map of the latter A-module, whence the proposition.

Remark 4.6.71. (i) Let M,M ′, Q and λ be as in (4.6.65). The image of λ under (4.6.66)is characterized as the unique law ϕ : M ⊕M ′ Q of degree e + e′ such that ϕ(m,m′) =λB(m,m′) for everyA-algebraB and every (m,m′) ∈ B⊗A(M⊕M ′). In turns, ϕ correspondsto the unique A-linear map f : Γe+e

′

A (M ⊕M ′)→ Q such that (B ⊗A f)(λe+e′

M⊕M ′)B(m,m′) =

λB(m,m′) for every B and (m,m′) as above. Especially, if we take λ := λe,e′

M,M ′, we see that

(4.6.67) is characterized as the unique A-linear map such that

(λe+e′

M⊕M ′)B(m,m′) 7→ (λe,e′

M,M ′)B(m,m′)

for every B and (m,m′) as above. Lastly, it follows that ∆e,e′

M,M ′is characterized as the unique

A-linear map such that

(λe+e′

M⊕M ′)B(m,m′) 7→ (λeM)B(m)⊗ (λe′

M ′)B(m′)

for every A-algebra B and every (m,m′) ∈ B ⊗A (M ⊕M ′).(ii) In the same vein, let f : M → N be any morphism in (A-Mod)a, and d ∈ N⊕k any

degree. Then we see that the induced map

ΓdA(f) : ΓdA(M)→ ΓdA(N)

is characterized as the unique A-linear map such that

(λdM)B(m) 7→ (λdN)B(f(m))

for every A-algebra B, and every m ∈ B ⊗AM (details left to the reader).

4.6.72. Let M , M ′, M ′′ be three sequences of A-modules of length k, and d, d′, d′′ threedegrees, also of lengths k. We obtain a diagram of A-linear maps :

Γe+e′+e′′

A (M ⊕M ′ ⊕M ′′)∆e,e′+e′′

M,M′⊕M′′ //

∆e+e′,e′′

M⊕M′,M′′

ΓeA(M)⊗A Γe′+e′′

A (M ′ ⊕M ′′)

ΓeA(M)⊗A∆

e′,e′′

M′,M′′

Γe+e′

A (M ⊕M ′)⊗A Γe′′

A (M ′′)∆e,e′

M,M′⊗AΓe′′A (M ′′)

// ΓeA(M)⊗A Γe′

A(M ′)⊗A Γe′′

A (M ′′).


Proposition 4.6.73. The diagram of (4.6.72) commutes.

Proof. By remark 4.6.71(i), the map ∆e,e′+e′′

M,M ′⊕M ′′ is characterized as the unique one such that

(λe+e′+e′′

M⊕M ′ )B(m,m′,m′′) 7→ (λeM)B(m)⊗ (λe′+e′′

M⊕M ′)B(m′,m′′)

for every A-algebra B and every (m,m′,m′′) ∈ B ⊗A (M ⊕M ′ ⊕M ′′), and ∆e′,e′′

M ′,M ′′is the

unique A-linear map such that

(λe′+e′′

M ′⊕M ′′)B(m′,m′′) 7→ (λe′

M ′)B(m′)⊗ (λe

′′

M ′′)B(m′′)

for every A-algebra B and every (m′,m′′) ∈ B ⊗A (M ′ ⊕M ′′). Therefore, the composition ofthe top horizontal and right vertical arrows in the diagram is the unique A-linear map such that

(λe+e′+e′′

M⊕M ′ )B(m,m′,m′′) 7→ (λeM)B(m)⊗ (λe′

M ′)B(m′)⊗ (λe

′′

M ′′)B(m′′)

for every B and (m,m′,m′′) as above. The same calculation can be carried out for the compo-sition of the other two arrows, and clearly one obtains the same result.

4.6.74. Let M be any sequence of A-modules of length k, and denote by s : M ⊕M →M theaddition map : (m,m′) 7→ m+m′ for every m,m′ ∈M . The map s induces A-linear maps

µe,e′

M : ΓeA(M)⊗A Γe′(M)

δe,e′

−−−→ Γe+e′

A (M ⊕M)Γe+e′A (s)−−−−−→ ΓdA(M) for every e, e′ ∈ N⊕k

where δe,e′ is the natural inclusion map obtained by restricting the inverse of ∆e+e′

M,M to the direct

factor ΓeA(M) ⊗A Γe′(M). In light of remark 4.6.71(i,ii), the map µe,e

′

M is characterized as theunique A-linear map such that

(4.6.75) (λeM)B(m)⊗ (λe′

M)B(m′) 7→ (λe+e′

M )B(m+m′)

for every A-algebra B and every (m,m′) ∈ B ⊗A (M ⊕M). Using this characterization, it iseasily seen that the diagram

ΓeA(M)⊗A Γe′

A(M)⊗A Γe′′

A (M)µe,e′M ⊗AΓ

e′′A (M)

//

ΓeA(M)⊗Aµ

e′,e′′M

Γe+e′

A (M)⊗A Γe′′

A (M)

µe+e′,e′′M

ΓeA(M)⊗A Γe′+e′′

A (M)µe,e′+e′′M // Γe+e

′+e′′

A (M)

commutes, for every e, e′, e′′ ∈ N⊕k. The foregoing shows that the maps µ••M define on

Γ•A(M) :=⊕d∈N⊕k

ΓdA(M).

a natural structure of associative N⊕k-graded A-algebra. Moreover, (4.6.75) also easily impliesthat this algebra is commutative. We call Γ•A(M) the divided power envelope of M . It iscustomary to use the notation

m[d] := (λdM)B(m) ∈ B ⊗AM

for every A-algebra B, every m ∈ B ⊗AM , and every d ∈ N⊕k. Then, proposition 4.6.68 anda simple inspection of the definitions yield the identity

(4.6.76) (a+ b)[d] =∑e+e′=d

a[e] · b[e′]


for every A-algebra B and every a, b ∈ B ⊗AM . Clearly, the above construction yields a welldefined functor

(4.6.77) (A-Mod)k → A-Alg : M 7→ Γ•A(M)

and notice that, for every morphism f : M → N of sequences, we have

(Γ•Af)(a[d]) = (fa)[d] = (Γ•Af(a))[d]

for every degree d, every A-algebra B, and every a ∈ B ⊗AM .

Theorem 4.6.78. With the notation of (4.6.74), we have :(i) For every A-algebra B, there exists a natural isomorphism of N⊕k-graded algebras :

B ⊗A Γ•A(M)∼→ Γ•B(B ⊗AM).

(ii) The functor (4.6.77) commutes with filtered colimits.(iii) For any two sequences of A-modules M and M ′ of the same length, there exists a

natural isomorphism of graded A-algebras :

Γ•A(M ⊕M ′)∼→ Γ•A(M)⊗A Γ•A(M ′)

(with the N⊕k-grading of the tensor product defined in the obvious way).

Proof. (i) and (ii) follow immediately from corollary 4.6.59(i,iii), and (iii) follows directly fromproposition 4.6.68.

4.6.79. Consider the special case of a sequence M consisting of a single A-module M . From(4.6.76), by evaluating the expression (2m)[d] = 2d ·m[d], a simple induction on d shows that

(4.6.80) d! ·m[d] = md in Γ•A(M), for every m ∈M.

Corollary 4.6.81. In the situation of (4.6.79), we have

m[i] ·m[j] =

(i+ j

i

)·m[i+j] in Γ•A(M), for every m ∈M and every i, j ∈ N.

Proof. By considering the A-linear map f : A→M such that f(1) = m, we reduce to the casewhere M = A. By considering the unique ring homomorphism Z → A we reduce further –in light of theorem 4.6.78(i) – to the case where A = Z and M = Z. In this case, ΓZ(Z) is atorsion-free Z-module, by lemma 4.6.49, so the identity follows easily from (4.6.80).

4.7. Regular rings. In section 9.6, we shall need a criterion for the regularity of suitable ex-tensions of regular local rings. Such a criterion is stated incorrectly in [30, Ch.0, Th.22.5.4];our first task is to supply a corrected version of loc.cit.

4.7.1. Consider a local ring A, with maximal ideal mA, and residue field kA := A/mA ofcharacteristic p > 0. From [30, Ch.0, Th.20.5.12(i)] we get a complex of kA-vector spaces

(4.7.2) 0→ mA/(m2A + pA)

dA−−→ Ω1A/Z ⊗A kA → Ω1

kA/Z → 0.

Lemma 4.7.3. The complex (4.7.2) is exact.

Proof. Set (A0,mA0) := (A/pA,mA/pA) and Fp := Z/pZ; the induced sequence of ringhomomorphisms Fp → A0 → kA yields a distinguished triangle ([56, Ch.II, §2.1.2.1]) :

LA0/FpL⊗A0 kA → LkA/Fp → LkA/A0 → LA0/Fp

L⊗A0 kA[1].

However, [36, Th.6.5.12(ii)] says that HiLkA/Fp = 0 for every i > 0 (one applies loc.cit. to thevalued field (kA, | · |), where | · | is the trivial valuation; more directly, one may observe that kA


is the colimit of a filtered system of smooth Fp-algebras, since Fp is perfect, and then recall thatthe functor L commutes with filtered colimits). We deduce a short exact sequence

0→ H1LkA/A0 → Ω1A0/Fp ⊗A0 kA → Ω1

kA/Fp → 0

which, under the natural identifications

Ω1kA/Z

∼→ Ω1kA/Fp Ω1

A/Z ⊗A kA∼→ Ω1

A0/Fp ⊗A0 kA H1LkA/A0

∼→ mA0/m2A0

([56, Ch.III, Cor.1.2.8.1]) becomes (4.7.2), up to replacing dA by −dA ([56, Ch.III, Prop.1.2.9];details left to the reader).

4.7.4. Keep the situation of (4.7.1), and define

A2 := A×kA W2(kA)

where W2(kA) is the ring of 2-truncated Witt vectors of kA, as in (4.6.11), which is augmentedover kA, via the ghost component map ω0 : W2(kA) → kA. Recall that the addition (resp.multiplication) law of W2(kA) are given by polynomials S0(X0, Y0) and S1(X0, X1, Y0, Y1)(resp. P0(X0, Y0) and P1(X0, X1, Y0, Y1)) uniquely determined by the identities detailed in(4.6); after a simple calculation, we find :

S0 =X0 + Y0 S1 =X1 + Y1 −p−1∑i=1

1

p

(p

i

)X i

0Yp−i

0

P0 =X0Y0 P1 =Xp0Y1 +X1Y

p0 + pX1Y1.

Notice that V1(kA) := Kerω0 is an ideal of W2(kA) such that V1(kA)2 = 0; especially, itis naturally a kA-vector space, with addition (resp. scalar multiplication) given by the rule :(0, a) + (0, b) := (0, a+ b) (resp. x · (0, a) := (x, 0) · (0, a) = (0, xp0 · a)) for every a, b, x ∈ kA.In other words, the map

(4.7.5) V1(kA)→ k1/pA (0, a) 7→ a1/p

is an isomorphism of kA-vector spaces, and we also see that V1(kA) is a one-dimensional k1/pA -

vector space, with scalar multiplication given by the rule : a·(0, b) := (0, apb) for every a ∈ k1/pA

and b ∈ kA. By construction, we have a natural exact sequence of A2-modules :

(4.7.6) 0→ V1(kA)→ A2π−→ A→ 0.

Especially, A2 is a local ring, and π is a local ring homomorphism inducing an isomorphism onresidue fields.

4.7.7. It is easily seen that the rule (A,mA) 7→ A2 defines a functor on the category Localof local rings and local ring homomorphisms, to the category of (commutative, unitary) rings.Hence, let us set

(A,mA) 7→ ΩA := Ω1A2/Z ⊗A2 A.

It follows that the rule (A,mA) 7→ (A,ΩA) yields a functor Local→ Alg.Mod to the categorywhose objects are the pairs (B,M) where B is a ring, and M is a B-module (the morphisms(B,M) → (B′,M ′) are the pairs (ϕ, ψ) where ϕ : B → B′ is a ring homomorphism, andψ : B′ ⊗B M →M is a B′-linear map : cp. [36, Def.2.5.22(ii)]).

In view of (4.7.6) and [30, Ch.0, Th.20.5.12(i)] we get an exact sequence of A-modules

V1(kA)→ ΩA → Ω1A/Z → 0

whence, after tensoring with kA, a sequence of kA-vector spaces

(4.7.8) 0→ V1(kA)j−→ ΩA ⊗A kA

ρ−→ Ω1A/Z ⊗A kA → 0

which is right exact by construction.


Proposition 4.7.9. With the notation of (4.7.7), we have :(i) If p /∈ m2

A, then the sequence (4.7.8) is exact.(ii) If p ∈ m2

A, then Ker j = (0, ap) | a ∈ kA. Especially, the isomorphism (4.7.5)identifies Ker j with the subfield kA of k1/p

A .

Proof. In view of lemma 4.7.3, the kernel of ρ is naturally identified with the kernel of thenatural map

mA2/(m2A2

+ pA2)→ mA/(m2A + pA).

On the other hand, it is easily seen that mA2 = mA ⊕ V1(kA), and under this identification, themultiplication law of A2 restricts on mA2 to the mapping given by the rule : (a, b) · (a′, b′) :=(aa′, 0) for every a, a′ ∈ mA and b, b′ ∈ V1(kA). There follows a natural isomorphism

(4.7.10) mA2/m2A2

∼→ (mA/m2A)⊕ V1(kA)

which identifies the map mA2/m2A2→ mA/m

2A deduced from π (notation of (4.7.6)), with the

projection on the first factor. Moreover, by inspecting the definitions, we easily get a commuta-tive diagram

V1(kA)j //

ΩA ⊗A kA

mA2/m2A2

// mA2/(m2A2

+ pA2)

dA2

OO

whose bottom horizotal arrow is the quotient map, and whose left vertical arrow is the inclusionmap of the direct summand V1(kA) resulting from (4.7.10).

The map A2 → mA2/m2A2

given by the rule : a 7→ pa := pa (mod m2A2

) factors (uniquely)through a kA-linear map

tA2 : kA → mA2/m2A2

a (mod mA) 7→ pa (mod m2A2

)

and likewise we may define a map tA : kA → mA/m2A. The snake lemma then gives an induced

map ∂ : Ker tA → V1(kA), and in view of the foregoing, the proposition follows from :

Claim 4.7.11. (i) If p /∈ m2A, then tA is injective.

(ii) If p ∈ m2A, then Im ∂ = (0, ap) | a ∈ kA.

Proof of the claim. (i) is obvious.(ii): By virtue of (4.6.7), we have p = (p, (0, 1)) in A2, and if (a, y) ∈ A2 is any element,

then p · (a, y) = (pa, (0, ap)), so in case p ∈ m2A, the map tA2 is given by the rule :

a 7→ (pa, (0, ap)) = (0, (0, ap)) ∈ (mA/m2A)⊕ V1(kA) for every a ∈ A.

By the same token, in this case tA is the zero map. The claim follows straightforwardly.

4.7.12. Keep the notation of (4.7.7), and assume that p /∈ m2A, so (4.7.8) is a kA-extension of

Ω1A/Z⊗AkA by V1(kA), by virtue of proposition 4.7.9; hence, (4.7.8)⊗kA k

1/pA is a k1/p

A -extension

of the corresponding k1/pA -vector spaces. Recall now that V1(kA) is naturally a k1/p

A -vector spaceof dimension one, and let ψ : V1(kA)⊗kA k

1/pA → V1(kA) be the scalar multiplication. By push

out along ψ, we obtain therefore an extension ψ ∗ (4.7.8) ⊗kA k1/pA fitting into a commutative

ladder with exact rows :

0 // V1(kA)⊗kA k1/pA

//

ψ

ΩA ⊗A k1/pA

//

ψ

Ω1A/Z ⊗A k

1/pA

//

0

0 // V1(kA)j // ΩA

// Ω1A/Z ⊗A k

1/pA

// 0


(see [36, §2.5.5]). We consider now the mapping :

d : A→ ΩA a 7→ ψ(d(a, τkA(a))⊗ 1) for every a ∈ Awhere :

• a ∈ kA is the image of a in kA• τkA is the Teichmuller mapping (see (4.6.9)), so τkA(a) = (a, 0) ∈ W2(kA)• d : A2 → ΩA2/Z is the universal derivation of A2.

Since τkA is multiplicative, the map d satisfies Leibniz’s rule, i.e. we have

d(ab) = a · d(b) + b · d(a) for every a, b ∈ A.However, d is not quite a derivation, since additivity fails for τkA , hence also for d. Instead,recalling that (p− 1)! = −1 in Fp, we get the identity :

τkA(a+ b) = τkA(a) + τkA(b)−p−1∑i=1

(0,ai

i!· bp−i

(p− i)!

)

= τkA(a) + τkA(b)−p−1∑i=1

ai/p

i!· b

1−i/p

(p− i)!· p

for every a, b ∈ kA. On the other hand, notice thatψ(d(0, x · p)⊗ 1) = j ψ(x · p⊗ 1)

= x · j ψ(p⊗ 1)

= x ·ψ(d(0, p)⊗ 1)

= x ·ψ(d(p− (p, 0))⊗ 1)

= − x · d(p)

for every x ∈ k1/pA , whence :

d(a+ b) = d(a) + d(b) +

p−1∑i=1

ai/p

i!· b

1−i/p

(p− i)!· d(p) for every a, b ∈ A.

4.7.13. Especially, notice we do have d(a + b) = d(a) + d(b) in case either a or b lies inmA. Hence, d restricts to an additive map mA → ΩA, and Leibniz’s rule implies that the latterdescends to a kA-linear map

d : mA/m2A → ΩA.

Proposition 4.7.14. In the situation of (4.7.1), suppose that p /∈ m2A. Then there exists a natural

exact sequence of k1/pA -vector spaces :

0→ mA/m2A ⊗kA k

1/pA

d⊗k1/pA−−−−−→ ΩA → Ω1

kA/Z ⊗kA k1/pA → 0.

Proof. By inspecting the constructions in (4.7.12), we obtain a commutative ladder of kA-vectorspaces, with exact rows :

0 // pA/(m2A ∩ pA) //

i

mA/m2A

//

d

mA/(m2A + pA) //

dA

0

0 // V1(kA)j // ΩA

// Ω1A/Z ⊗A k

1/pA

// 0

such that dA ⊗kA kB is injective with cokernel naturally isomorphic to Ω1kA/Z ⊗kA k

1/pA (lemma

4.7.3). By the snake lemma, it then suffices to show that i⊗kA k1/pA is an isomorphism; but since


both the source and target of the latter map are one-dimensional k1/pA -vector spaces, we come

down to checking that i is not the zero map. But a simple inspection yields i(p) = (0, 1), asrequired.

4.7.15. Keep the notation of (4.7.1), and let now f1, . . . , fn be a finite sequence of elements ofA, and e1, . . . , en a sequence of integers such that ei > 1 for every i = 1, . . . , n. Set

C := A[T1, . . . , Tn]/(T e11 − f1, . . . , Tenn − fn).

Fix a prime ideal n ⊂ C such that n ∩ A = mA, and let B := Cn. So the induced map A → Bis a local ring homomorphism; we denote by mB the maximal ideal of B, and set kB := B/mB.Also, define the integer ν as follows :

• If p ∈ m2A, then ν := dimkA E, where E ⊂ Ω1

A/Z⊗A kA is the kA-vector space spannedby df1, . . . , dfn.• If p /∈ m2

A, then ν := dimk

1/pAE, where E ⊂ ΩA is the k1/p

A -vector space spanned byd(f1), . . . ,d(fn).

Theorem 4.7.16. In the situation of (4.7.15), suppose moreover that :(a) fi ∈ mA, for every i ≤ n such that p does not divide ei.(b) mA/m

2A is a finite dimensional kA-vector space.

Then mB/m2B is a finite dimensional kB-vector space, and we have :

dimkB mB/m2B = n+ dimkA mA/m

2A − ν.

Proof. Let us begin with the following general result :

Claim 4.7.17. Let K be any field, and L a field extension of K of finite type. Then

dimL Ω1L/K − dimLH1LL/K = tr. deg[L : K].

Proof of the claim. This is known as Cartier’s identity, and a proof is given in [30, Ch.0,Th.21.7.1]. We present a proof via the cotangent complex formalism. Suppose first that K isof finite type over its prime field K0 (so K0 is either Q or a finite field of prime order). In thiscase, notice that H1LL/K0 = 0 (cp. the proof of lemma 4.7.3); then the transitivity triangle forthe sequence of maps K0 → K → L ([56, Ch.II, §2.1.2.1]) yields an exact sequence

0→ H1LL/K → Ω1K/K0

⊗K L→ Ω1L/K0

→ Ω1L/K → 0

and the claim follows easily, after one remarks that dimK Ω1K/K0

= tr. deg[K : K0], and like-wise for Ω1

L/K0. Next, let K be an arbitrary field, and write K as the union of the filtered

family (Kλ | λ ∈ Λ) of its subfields that are finitely generated over K0. Choose elementsx1, . . . , xt ∈ L such that K(x1, . . . , xt) = L. Let ϕ : K[X1, . . . , Xt] → L be the map of K-algebras given by the rule : Xi 7→ xi for i = 1, . . . , t. Then I := Kerϕ is a finitely generatedideal, so we may find λ ∈ Λ and an ideal Iλ ⊂ Kλ[X1, . . . , Xt] such that I = Iλ⊗KλK (detailsleft to the reader). Denote by Lλ the field of fractions of Kλ[T1, . . . , Tt]/Iλ; it is easily seen thatLλ ⊗Kλ K is an integral domain, and its field of fractions is a K-algebra naturally isomorphicto L. Especially, we have tr. deg[Lλ : Kλ] = tr. deg[L : K], and on the other hand, there is anatural isomorphism in D−(L-Mod) :

LLλ/Kλ ⊗Lλ L∼→ LL/K

([56, Ch.II, Prop.2.2.1, Cor.2.3.1.1]). Then the sought identity for the extension K ⊂ L followsfrom the same identity for the extension Kλ ⊂ Lλ. The latter is already known, by the previouscase. ♦

Let q ⊂ A[T1, . . . , Tn] be the preimage of n, set R := A[T1, . . . , Tn]q, and denote by p themaximal ideal of R. We prove first the following special case :


Claim 4.7.18. (i) dimkB p/p2 = n+ dimkA mA/m2A.

(ii) The natural map γ : mA/m2A → p/p2 is injective.

Proof of the claim. (i): According to [36, Th.6.5.12(i)], we have HiLkB/kA = 0 for everyi > 1 (we apply loc.cit. to the extension (kA, | · |kA) ⊂ (kB, | · |kB) of valued fields withtrivial valuations); then the transitivity triangle for the cotangent complex relative to the mapsA→ kA → kB yields a short exact sequence of kB-vector spaces :

0→ H1LkA/A ⊗kA kBα−→ H1LkB/A → H1LkB/kA → 0

([56, Ch.II, §2.1.2.1]). Likewise, since HiLR/A = 0 for every i > 1 ([56, Ch.II, Cor.1.2.6.3]),the sequence of maps A→ R→ kB yields an exact sequence of kB-vector spaces :

0→ H1LkB/Aβ−→ H1LkB/R → Ω1

R/A ⊗R kB → ΩkB/A → 0.

However, we have natural isomorphisms

H1LkA/A∼→ mA/m

2A H1LkB/R

∼→ p/p2

([56, Ch.III, Cor.1.2.8.1]), and clearly dimkB ΩR/A ⊗R kB = n. Thus :

dimkA mA/m2A = dimkB H1LkB/A − dimkB H1LkB/kA

dimkB p/p2 = dimkB H1LkB/A + n− dimkB ΩkB/kA .

Taking into account the identity

dimkB H1LkB/kA = dimkB ΩkB/kA

provided by claim 4.7.17, the assertion follows.(ii): Notice that the composition of α and β yields an injective map mA/m

2A → p/p2 so it

suffices to check that this composition equals γ. However, let

Σ : H1LkB/Rd−→ H0kB ⊗R LR/A

be the boundary map of the transitivity triangle

(4.7.19) kB ⊗R LR/A → LkB/A → LkB/R → kB ⊗R σLR/Aarising from the sequence A → R → kB; we regard Σ as a complex placed in degrees [−1, 0],and then it is clear that the triangle τ≥−1(4.7.19) is naturally isomorphic in D(kB-Mod) to thetriangle

(4.7.20) kB ⊗R H0LR/A[0]→ Σ→ H1LkB/R[1]→ kB ⊗R H0LR/A[1].

According to [56, III.1.2.9.1], the complex Σ is naturally isomorphic to the complex

Θ : p/p2−dR/A−−−−→ kB ⊗R Ω1

R/A

(placed in degrees [−1, 0]) where dR/A is induced by the universal derivation R → Ω1R/A.

Likewise, we have natural isomorphisms

(4.7.21) τ≥−1LkB/R∼→ p/p2[1] τ≥−1LR/A

∼→ Ω1R/A[0]

and under these identifications, 4.7.20 is the obvious triangle deduced from Θ. Especially, themap β is naturally identified with the inclusion map Ker dR/A → p/p2.

Likewise, there exists a natural identification

(4.7.22) τ≥−1LkA/A∼→ mA/m

2A[1] in D(kA-Mod)

as well as a natural map of complexes

(4.7.23) kB ⊗kA (mA/m2A)[1]→ Θ


deduced from γ. To conclude, it suffices to check that the map

(4.7.24) τ≥−1kB ⊗A LkA/A → τ≥−1LkB/Acoming from the transitivity triangle for the sequence A → kA → kB, corresponds to the mor-phism (4.7.23), under the identification (4.7.22) and the previous identification of τ≥−1LkB/Awith Θ. To this aim, it suffices to compare the maps obtained by applying to these two mor-phisms the functor Ext1

kB(−,M [0]), for arbitrary kB-modules M . Now, recall that there exists

a natural isomorphism

Ext1kB

(τ≥−1LkB/R,M [0])∼→ ExalR(kB,M) for every kB-module M.

As explained in [56, III.1.2.8], under the identification (4.7.21), this becomes the followingkB-linear isomorphism

(4.7.25) HomkB(p/p2,M)∼→ ExalR(kB,M) ϕ 7→ ϕ ∗ U

(notation of [36, §2.5.5]), where

U : 0→ p/p2 → R/p2 → kB → 0

is the natural extension. There is a natural surjection

HomkB(p/p2,M)→ Ext1kB

(Θ,M [0])

and the foregoing implies that the natural isomorphism

Ext1kB

(τ≥−1LkB/A,M [0])∼→ ExalA(kB,M)

is also realized as in (4.7.25) : given a class c in Ext1kB

(Θ,M |0]), take an arbitrary representativeϕ : p/p2 →M , and the correspondence associates to c the class of the push out ϕ ∗ U .

By the same token, we have a natural isomorphism

Ext1kB

(τ≥−1kB ⊗kA LkA/A,M(0])∼→ ExalA(kA,M) for every kB-module M

and on the one hand, the map Ext1kB

((4.7.24),M [0]) is identified naturally with the map

ExalA(kB,M)→ ExalA(kA,M)

given by pull back along the inclusion map j : kA → kB. On the other hand, by [56, III.1.2.8],the identification (4.7.21) induces the isomorphism

HomkA(mA/m2A,M)

∼→ ExalA(kA,M) ψ 7→ ψ ∗ U ′

whereU ′ : 0→ mA/m

2A → A/m2

A → kA → 0

is the natural extension. So finally, the assertion boils down to the identity

(ϕ γ) ∗ U ′ = ϕ ∗ U ∗ j in ExalA(kA,M)

for every kB-module M and every ϕ : p/p2 → M . We leave the verification as an exercise forthe reader. ♦

Now, let F ⊂ p/p2 be the kB-vector space spanned by T e11 − f1, . . . , Tenn − fn. Clearly, we

have a short exact sequence

0→ F → p/p2 → mB/m2B → 0.

Suppose first that p /∈ m2A, in which case p /∈ p2, by claim 4.7.18(ii), hence ΩR is well defined.

On the other hand, notice that d(T eii ) = d(T eii − fi) + d(fi) in ΩR, since T eii − fi ∈ p. Now, ifei is a multiple of p, Leibniz’s rule yields d(T eii ) = ei ·T ei−1

i d(Ti) = 0. If ei is not a multiple ofp, we have fi ∈ mA by assumption; hence Ti ∈ p and therefore d(T eii ) = 0 again, since ei > 1.In either case, we find

d(fi) = −d(T eii − fi) in ΩR, for every i = 1, . . . , n.


In view of proposition 4.7.14, it follows that dimkB mB/m2B = dimkB p/p2−dim

k1/pBE ′, where

E ′ ⊂ ΩR is the k1/pB -vector space spanned by d(f1), . . . ,d(fn). Taking into account claim

4.7.18(i), it then suffices to remark :

Claim 4.7.26. With the foregoing notation, we have :(i) The natural map Ω1

A/Z⊗AR→ Ω1R/Z is injective, and its image is a direct summand of

Ω1R/Z.

(ii) If p /∈ mA, the natural map ΩA ⊗k1/pAk

1/pB → ΩR is injective.

Proof of the claim. (i) is a standard calculation (more precisely, the complement of Ω1A/Z ⊗A R

in Ω1R/Z is the free R-module generated by dT1, . . . , dTn).

(ii): By inspecting the constructions, we get a natural commutative ladder of k1/pB -vector

spaces

0 // V1(kA)⊗k

1/pAk

1/pB

//

ΩA ⊗k1/pAk

1/pB

//

Ω1A/Z ⊗A k

1/pB

//

0

0 // V1(kB) // ΩR// Ω1

R/Z ⊗R k1/pB

// 0

whose central vertical arrow is the map of the claim. However, it is easily seen that the leftvertical arrow is an isomorphism, so the assertion follows from (i). ♦

Lastly, suppose p ∈ m2A, so that p ∈ p2 as well. Arguing as in the foregoing case, we

see that dimkB F equals the dimension of the kB-vector subspace of Ω1R/Z ⊗R kB spanned by

df1, . . . , dfn, and in view of claim 4.7.26(i), the latter equals ν, whence the contention.

Corollary 4.7.27. In the situation of theorem 4.7.16, the following conditions are equivalent :(a) A is a regular local ring, and ν = n.(b) B is a regular local ring.

Proof. Suppose first that (b) holds. Since the map A → B is faithfully flat, it is easily seenthat A is noetherian, and then [30, Ch.0, Prop.17.3.3(i)] shows already that A is a regular localring. Moreover, C is clearly a finite A-algebra, therefore dimkB mB/m

2B = dimB ≤ dimA =

dimkA mA/m2A. Theorem 4.7.16 then implies that ν = n, so (a) holds.

Next, suppose that (a) holds. We apply theorem 4.7.16 as in the foregoing, to deduce thatdimB = dimA = dimkB mB/m

2B, whence (b) : details left to the reader.

Remark 4.7.28. (i) Keep the notation of corollary 4.7.27. In [30, Ch.0, Th.22.5.4] it is assertedthat condition (b) is equivalent to the following :

(a’) A is regular and the space E ⊂ Ω1A/Z ⊗A kA spanned by df1, . . . , dfn has dimension n.

When p ∈ m2A, this condition (a’) agrees with our condition (a), so in this case of course we

do have (a’)⇔(b). However, the latter equivalence fails in general, in case p /∈ mA : the mistakeis found in [30, Ch.0, Rem.22.4.8], which is false. The implication (a’)⇒(b) does remain true inall cases : this is easily deduced from theorem 4.7.16, since the image of d(f) in Ω1

A/Z⊗kA k1/pA

agrees with df , for every f ∈ A. The proof of loc.cit. is correct for p ∈ m2A, which is the only

case that is used in the proof of corollary 4.7.27.(ii) Moreover, in the situation of corollary 4.7.27, suppose that B is a regular local ring.

Then we have :• If p /∈ m2

B, then the image of the sequence df 1/e11 , . . . , df

1/enn in Ω1

B/Z ⊗B kB spans akB-vector space of dimension n.


• If p ∈ m2B, then the image of the sequence d(f

1/e11 ), . . . ,d(f

1/enn ) in ΩB spans a k1/p

B -vector space of dimension n.

Indeed, consider the ring C := B[T1, . . . , Tn]/(T p1 − f1/e11 , . . . , T pn − f

1/enn ). Then, corollary

4.7.27 applies to the extension A ⊂ C, the sequence (f1, . . . , fn), and the sequence of integers(pe1, . . . , pen), so C is a regular local ring. But the same corollary applies as well to the ex-tension B ⊂ C, the sequence (f

1/e11 , . . . , f

1/enn ), and the sequence of integers (p, . . . , p), and

yields the assertion.(iii) Let us say that the sequence (f1, . . . , fn) is maximal in A if the following holds :• If p ∈ m2

A, then (df1, . . . , dfr) is a basis of the kA-vector space Ω1A/Z ⊗A kA.

• If p /∈ m2A, then (d(f1), . . . ,d(fr)) is a basis of the k1/p

A -vector space ΩA.Then, in the situation of (ii), we claim that the sequence (f1, . . . , fn) is maximal inA if and onlyif the sequence (f

1/e11 , . . . , f

1/enn ) is maximal in B. Indeed, under the current assumptions we

have dimkA Ω1kA/Z = dimkB Ω1

kB/Z, since these integers are equal to the transcendence degreeof kA (and kB) over Fp, and on the other hand dimkA mA/m

2A = dimkB mB/m

2B, since A and B

are regular local rings of the same dimension; then the assertion follows from (ii), lemma 4.7.3and proposition 4.7.14 (details left to the reader).

4.7.29. Let p > 0 be a prime integer, and A an Fp-algebra. Denote by ΦA : A → A theFrobenius endomorphism of A, given by the rule : a 7→ ap for every a ∈ A. For every A-module M , we let M(Φ) be the A-module obtained from M via restriction of scalars along themap ΦA (that is, a ·m := apm for every a ∈ A and m ∈M ). Notice that ΦA is an A-linear mapA→ A(Φ). Theorem 4.7.30, and part (i) of the following theorem 4.8.42 are due to E.Kunz.

Theorem 4.7.30. Let A be a noetherian local Fp-algebra. Then the following conditions areequivalent :

(i) A is regular.(ii) ΦA is a flat ring homomorphism.

(iii) There exists n > 0 such that ΦnA is a flat ring homomorphism.

Proof. (i)⇒(ii): Let A∧ be the completion of A, and f : A→ A∧ the natural map. Clearly

f ΦA = ΦA∧ f.

Since f is faithfully flat, it follows that ΦA is flat if and only if the same holds for ΦA∧ , so wemay replace A by A∧, and assume from start that A is complete, hence A = k[[T1, . . . , Td]], fora field k of characteristic p, and d = dimA ([30, Ch.0, Th.19.6.4]). Then, it is easily seen that

ΦA(A) = Ap = kp[[T p1 , . . . , Tpd ]].

Set B := k[[T p1 , . . . , Tpd ]]; the ring A is a free B-module (of rank pd), hence it suffices to check

that the inclusion map Ap → B is flat. However, denote by m the maximal ideal of Ap; clearlyB is an m-adically ideal-separated A-module (see [61, p.174, Def.]), hence it suffices to checkthat B/mkB is a flat Ap/mk-module for every k > 0 ([61, Th.22.3]). The latter is clear, sincek[T p1 , . . . , T

pd ] is a flat kp[T p1 , . . . , T

pd ]-module.

(ii)⇒(iii) is obvious.(iii)⇒(i): Notice first that Spec Φn

A is the identity map on the topological space underlyingSpecA; especially, Φn

A is flat if and only if it is faithfully flat, and the latter condition impliesthat Φn

A is injective. We easily deduce that if (iii) holds, then A is reduced. Now, consider quitegenerally, any finite system x• := (x1, . . . , xt) of elements of A, and let I ⊂ A be the idealgenerated by x•; we shall say that x• is a system of independent elements, if I/I2 is a freeA/I-module of rank n. We show first the following :


Claim 4.7.31. Let y, z, x2, . . . , xt be a family of elements of A, such that x• := (yz, x2, . . . , xt)is a system of independent elements, and denote by J ⊂ A the ideal generated by x•. We have :

(i) (y, x2, . . . , xt) is a system of independent elements of A.(ii) If lengthAA/J is finite, then

lengthAA/J = lengthAA/(y, x2, . . . , xt) + lengthAA/(z, x2, . . . , xt).

Proof of the claim. (i): Suppose a1y+a2x2 + · · ·+atxt = 0 is a linear relation with a1, . . . , at ∈A, and let I ⊂ A be the ideal generated by y, x2, . . . , xt. We have to show that a1, . . . , at ∈ I .However, as a1yz + a2zx2 + · · · + atzxt = 0, it follows by assumption, that a1 lies in J ⊂ I .Write a1 = b1yz + b2x2 + · · ·+ btxt; then

b1y2z + (a2 + b2y)x2 + · · ·+ (at + bty)xt = 0.

Therefore ai + biy ∈ J for i = 2, . . . , t, and therefore a2, . . . , at ∈ I , as required.(ii): It suffices to show that the natural map of A-modules :

A/(z, x2, · · · , xt)→ I/J a 7→ ay + J

is an isomorphism. However, the surjectivity is immediate. To show the injectivity, supposethat ay ∈ J , i.e. ay = b1yz + b2x2 + · · · + btxt for some b1, . . . , bt ∈ A; we deduce that(b1z − a)yz + b2zx2 + · · ·+ btzxt = 0, hence a− b1z ∈ J by assumption, so a lies in the idealgenerated by z, x2, . . . , xt, as required. ♦

Now, set q := pn, and Aν := Aqν ⊂ A for every integer ν > 0; pick a minimal system

x• := (x1, . . . , xt) of generators of the maximal ideal mA of A, and notice that, since A isreduced, Φnν

A induces an isomorphism A → Aν , hence x(ν)• := (xq

ν

1 , . . . , xqν

t ) is a minimalsystem of generators for the maximal ideal mν ofAν . Set as well Iν := mνA; since the inclusionmap Aν → A is flat by assumption for every ν > 0, we have a natural isomorphism of A-modules

(mν/m2ν)⊗Aν A

∼→ Iν/I2ν .

On the other hand, set kν := Aν/mν ; by Nakayama’s lemma, dimkν mν/m2ν = t, so Iν/I2

ν isa free A-module of rank t, i.e. x(ν)

• is an independent system of elements of A. From claim4.7.31(ii) and a simple induction, we deduce that

(4.7.32) lengthAA/Iν = lengthA∧A∧/IνA

∧ = qνt for every ν > 0

(where the first equality holds, since Iν is an open ideal in the mA-adic topology of A). Accord-ing to [30, Ch.0, Th.19.9.8] (and its proof), A∧ contains a field isomorphic to k0 := A/mA, andthe inclusion map k0 → A∧ extends to a surjective ring homomorphism k0[[X1, . . . , Xt]]→ A∧,such thatXi 7→ xi for i = 1, . . . , t. Denote by J the kernel of this surjection; in view of (4.7.32),we have

lengthA∧k0[[X1, . . . , Xt]]/(J,Xqν

1 , . . . , Xqν

t ) = qνt

which means that J ⊂ (Xqν

1 , . . . , Xqν

t ) for every ν > 0. We conclude that J = 0, and A∧ =k0[[X1, . . . , Xt]] is regular, so the same holds for A.

The last result of this section is a characterization of regular local rings via the cotangentcomplex, borrowed from [1], which shall be used in the following section on excellent rings.

Lemma 4.7.33. Let A be a ring, (a1, . . . , an) a regular sequence of elements of A, that gener-ates an ideal I ⊂ A, and set A0 := A/I . Then there is a natural isomorphism

LA0/A∼→ I/I2[1] in D(A0-Mod)

and I/I2 is a free A0-module of rank n.


Proof. Notice that H0LA0/A = 0, and there is a natural isomorphism H1LA0/A∼→ I/I2 ([56,

Ch.III, Cor.1.2.8.1]). There follows a natural morphism LA0/A → I/I2[1], and we shall showmore precisely, that this morphism is an isomorphism. We proceed by induction on n. Hence,suppose first that n = 1, set a := a1, and B := A[T ], the free polynomial A-algebra in onevariable; define a map of A-algebras B → A by the rule T 7→ a. Set also B0 := B/TB (so B0

is isomorphic to A). Since a is regular, it is easily seen that the natural morphism

B0

L⊗B A→ B0 ⊗B A = A0

is an isomorphism in D(B-Mod). It follows that the induced morphism LB0/B⊗B0A0 → LA0/A

is an isomorphism in D(A0-Mod) ([56, Ch.II, Prop.2.2.1]), so it suffices to check the assertionfor the ring B and its regular element T . However, the sequence of ring homomorphismsA→ B → B0 induces a distinguished triangle ([56, Ch.II, Prop.2.1.2])

LB/A ⊗B B0 → LB0/A → LB0/B → LB/A ⊗B B0[1]

and since clearly LB0/A ' 0 in D(B0-Mod), and LB/A ' Ω1B/A[0] ' B[0] ([56, Ch.II,

Prop.1.2.4.4]), the assertion follows (details left to the reader). Next, suppose that n > 1,and that the assertion is already known for regular sequences of length < n. Denote by I ′ ⊂ Athe ideal generated by a1, . . . , an−1, and set A′ := A/I ′. There follows a sequence of ring ho-momorphisms A→ A′ → A0, and the inductive assumption implies that the natural morphisms

LA′/A → I ′/I ′2[1] LA0/A′ → I/(I2 + I ′)[1]

are isomorphisms, and I ′/I ′2 (resp. I/(I2 + I ′)) is a free A′-module (resp. A0-module) ofrank n − 1 (resp. of rank 1). Then the assertion follows easily, by inspecting the distinguishedtriangle

LA′/A ⊗A′ A0 → LA0/A → LA0/A′ → LA′/A ⊗A′ A0[1]

given again by [56, Ch.II, Prop.2.1.2] (details left to the reader).

Proposition 4.7.34. Let A be a local noetherian ring, a ∈ A a non-invertible element, and setA0 := A/aA. The following conditions are equivalent :

(a) a is a regular element of A.(b) H2LA0/A = 0, and aA/a2A is a free A0-module of rank one.

Proof. For any ring R, any non-invertible element x ∈ R, and any n ∈ N, set Rn := R/xn+1R,and consider the R0-linear map

βx,n : R0 → xnR/xn+1R

induced by multiplication by xn. We remark :

Claim 4.7.35. Suppose that R is a noetherian local ring, denote by κR the residue field of R,and let n > 0 be any given integer. The following conditions are equivalent :

(c) βx,n is an isomorphism.(d) xn 6= 0 and TorR0

1 (xnR/xn+1R, κR) = 0.(e) xn 6= 0 and the surjection R0 → κR induces a surjective map

H2(LRn−1/R ⊗Rn−1 R0)→ H2(LRn−1/R ⊗Rn−1 κR).

Proof of the claim. It is easily seen that (c)⇒(d).Conversely, if (d) holds, notice that xnR/xn+1R 6= 0, since

⋂n∈N x

nR = 0. Hence κR⊗Rβx,nis an isomorphism of one-dimensional κR-vector spaces. On the other hand, under assumption(d), the natural map κR ⊗R Ker βx,n → Ker(κR ⊗R βx,n) is an isomorphism. By Nakayama’slemma, we conclude that Ker βx,n = 0, i.e. (c) holds.


Next, recall the natural isomorphism of R0-modules

H1(LRn−1/R ⊗Rn−1 M)∼→ xnR/x2nR⊗Rn−1 M

∼→ xnR/xn+1R⊗R0 M

for every R0-module M ([56, Ch.III, Cor.1.2.8.1]). Denote by m0 the kernel of the surjectionR0 → κR; there follows a left exact sequence

0→ TorR01 (xnR/xn+1R, κR)→ H1(LRn−1/R ⊗Rn−1 m0)→ H1(LRn−1/R ⊗Rn−1 R0)

which shows that (d)⇔(e) (details left to the reader). ♦

Claim 4.7.36. In the situation of claim 4.7.35, the following conditions are equivalent :(f) x is a regular element of R.(g) βx,n is an isomorphism, for every n ∈ N.

Proof of the claim. If (f) holds, xn is a regular element of R for every n > 0, and then (g)follows easily. Conversely, assume (g), and suppose that yx = 0 for some y ∈ R; we claimthat y ∈ xnR for every n ∈ N. We argue by induction on n : for n = 0, there is nothing toprove. Suppose that we have already obtained a factorization y = xnz for some z ∈ R. Thenxn+1z = 0, so the class of z in R0 lies in Ker βn+1, hence this class must vanish, i.e. z ∈ xR,and therefore y ∈ xn+1R. Since

⋂n∈N x

nR = 0, we deduce that y = 0, whence (f). ♦

Claim 4.7.37. Let f : R → R′ be any ring homomorphism, and set x′ := f(x). Suppose thatβx,n and βx′,n are both isomorphisms, for some integer n ∈ N, and set R′n := R′/x′n+1R′. Thenthe induced morphism

LR0/Rn ⊗R0 R′0 → LR′0/R′n

is an isomorphism in D(R′0-Mod).

Proof of the claim. If n = 0, there is nothing to prove, hence assume that n > 0. We remarkthat, for every i = 0, . . . , n, the complex

Rnxi+1

−−−→ Rnxn−i−−−→ Rn

is exact. Indeed, for i = 0, this results immediately from the assumption that Ker βn,x = 0.Suppose that i > 0, and that the assertion is already known for i−1; then, if yxn−i ∈ xn+1R forsome y ∈ R, the inductive hypothesis yields y ∈ xiR, so say that y = xiu and yxn−i = zxn+1

for some u, z ∈ R. It follows that xn(u − zx) = 0, and then u − zx ∈ xR, again sinceKer βx,n = 0; thus, u ∈ xR, and y ∈ xi+1R, as asserted.

We deduce that the Rn-module R0 admits a free resolution

Σ : · · · → Rnx−→ Rn

xn−−→ Rnx−→ Rn → R0.

The same argument applies to R′ and its element x′, and yields a corresponding free resolutionΣ′ of the R′n-module R′0. A simple inspection shows that Σ′ ⊗Rn R′n = Σ, i.e. the natural

morphism R0

L⊗Rn R′n → R′0 is an isomorphism in D(R′0-Mod). The claim then follows from

[56, Ch.II, Prop.2.2.1]. ♦

With these preliminaries, we may now return to the situation of the proposition : first, lemma4.7.33 says that (a)⇒(b). For the converse, we shall apply the criterion of claim 4.7.36 : namely,we shall show, by induction on n, that βa,n : A0 → anA/an+1A is bijective for every n ∈ N.

For n = 0, there is nothing to prove. Assume that n > 0, and that the assertion is alreadyknown for n− 1. Let m ⊂ A[T ] be the (unique) maximal ideal containing T , and set B := Am.We let f : B → A be the map of A-algebras given by the rule : T 7→ a. Define as usualBn := B/T n+1B and An := A/an+1A for every n ∈ N. Clearly βT,n−1 : B0 → T n−1B/T nBis bijective, and the same holds for βa,n−1, by inductive assumption. Then, claim 4.7.37 saysthat the induced morphism LB0/Bn−1 ⊗B0 A0 → LA0/An−1 is an isomorphism in D(A0-Mod).


Denote by κ the residue field of A and B, and notice as well that H2(LA0/A ⊗A0 κ) = 0, byvirtue of (b). Consequently, the commutative diagram of ring homomorphisms

B //

Bn−1//

B0

A // An−1

// A0

induces a commutative ladder with exact rows ([56, Ch.II, Prop.2.1.2]) :

H3(LB0/Bn−1 ⊗B0 κ) //

H2(LBn−1/B ⊗Bn−1 κ)

// H2(LB0/B ⊗B0 κ)

H3(LA0/An−1 ⊗A0 κ) // H2(LAn−1/A ⊗An−1 κ) // 0

whose left vertical arrow is an isomorphism. It follows that the central vertical arrow is surjec-tive. Consider now the commutative diagram

(4.7.38)

H2(LBn−1/B ⊗Bn−1 B0) //

H2(LBn−1/B ⊗Bn−1 κ)

H2(LAn−1/A ⊗An−1 A0) // H2(LAn−1/A ⊗An−1 κ)

induced by the mapsB0 → A0 → κ. We have just seen that the right vertical arrow of (4.7.38) issurjective, and the same holds for its top horizontal arrow, in light of claim 4.7.35. Thus, finally,the bottom horizontal arrow is surjective as well, so βa,n is an isomorphism (claim 4.7.35), andthe proposition is proved.

Theorem 4.7.39. Let A be a noetherian local ring, I ⊂ A an ideal, and set A0 := A/I . Thefollowing conditions are equivalent :

(a) Every minimal system of generators of I is a regular sequence of elements of A.(b) I is generated by a regular sequence of A.(c) The natural morphism LA0/A → I/I2[1] is an isomorphism in D(A0-Mod), and I/I2

is a flat A0-module.(d) H2LA0/A = 0, and I/I2 is a flat A0-module.

Proof. Clearly (a)⇒(b) and (c)⇒(d); also, lemma 4.7.33 shows that (b)⇒(c).(d)⇒(a): Let (a1, . . . , an) be a minimal system of generators for I; recall that the length n

of the sequence equals dimκ I ⊗A κ, where κ denotes the residue field of A. We shall argue byinduction on n. For n = 0, there is nothing to show, and the case n = 1 is covered by proposition4.7.34. Set B := A/a1A and J := IB. Assumption (d) implies that H2(LA0/A ⊗A0 κ) = 0,therefore the sequence of ring homomorphisms A→ B → A0 induces an exact sequence

0→ H2(LA0/B ⊗A0 κ)→ H1(LB/A ⊗B κ)→ I ⊗A κ→ J ⊗B κ→ 0

([56, Ch.II, Prop.2.1.2 and Ch.III, Cor.1.2.8.1]). However, clearly J admits a generating systemof length n − 1, hence n′ := dimκ J ⊗B κ < n. On the other hand, dimκH1(LB/A ⊗B κ) =1, so we have necessarily n′ = n − 1 and H2(LA0/B ⊗A0 κ) = 0. The latter means thatH2LA0/B = 0 and J/J2 is a flat B-module. By inductive assumption, we deduce that thesequence (a2, . . . , an) of the images in B of (a2, . . . , an), is regular. By virtue of lemma 4.7.33,it follows that H3(LA0/B ⊗A0 κ) = 0, whence a left exact sequence

0→ H2(LB/A ⊗B κ)→ H2(LA0/A ⊗A0 κ) = 0


obtained by applying again [56, Ch.II, Prop.2.1.2 and Ch.III, Cor.1.2.81] to the sequence A →B → A0. Thus, H2LB/A = 0 and a1A/a

21A is a flat B-module, so a1 is a regular element

(proposition 4.7.34) and finally, (a1, . . . , an) is a regular sequence, as required.

Corollary 4.7.40. Let A be a local noetherian ring, with maximal ideal m, and residue field κ.Then the following conditions are equivalent :

(a) A is regular.(b) The natural morphism Lκ/A → m/m2[1] is an isomorphism.(c) H2Lκ/A = 0.

Proof. It follows immediately, by invoking theorem 4.7.39 with I := m, and lemma 4.7.33.

4.8. Excellent rings. Recall that a morphism of schemes f : X → Y is called regular, ifit is flat, and for every y ∈ Y , the fibre f−1(y) is locally noetherian and regular ([31, Ch.IV,Def.6.8.1]).

Lemma 4.8.1. Let f : X → Y and g : Y → Z be two morphisms of locally noetherianschemes. We have :

(i) If f and g are regular, then the same holds for g f .(ii) If g f is regular, and f is faithfully flat, then g is regular.

Proof. (i): Clearly h := g f is flat. Let z ∈ Z be any point, and K any finite extension ofκ(z). Set

X ′ := h−1(z)×κ(z) K and Y ′ := g−1(z)×κ(z) K.

It is easily seen that the induced morphism f ′ : X ′ → Y ′ is regular. Moreover, for every y ∈ Y ′,the local ring OY ′,y′ is regular, since g is regular. Then the assertion follows from :

Claim 4.8.2. Let A → B be a flat and local ring homomorphism of local noetherian rings.Denote by mA ⊂ A the maximal ideal, and suppose that both A and B0 := B/mAB are regular.Then B is regular.

Proof of the claim. On the one hand, dimB = dimA + dimB0 ([30, Ch.IV, Cor.6.1.2]). Onthe other hand, let a1, . . . , an be a minimal generating system for mA, and b1, . . . , bm a systemof elements of the maximal ideal mB of B, whose images in B0 is a minimal generating systemfor mB/mAB. By Nakayama’s lemma, it is easily seen that the system a1, . . . , an, b1, . . . , bmgenerates the ideal mB. Since A and B0 are regular, n = dimA and m = dimB, so n + m =dimB, and the claim follows. ♦

(ii): Clearly g is flat. Then the assertion follows easily from [30, Ch.0, Prop.17.3.3(i)] :details left to the reader.

Definition 4.8.3. Let A be a noetherian ring.(i) We say that A is a G-ring, if the formal fibres of SpecA are geometrically regular, i.e.

for every p ∈ SpecA, the natural morphism SpecA∧p → SpecAp from the spectrum ofthe p-adic completion of A, is regular : see [31, Ch.IV, §7.3.13].

(ii) We say that A is quasi-excellent, if A is a G-ring, and moreover the following holds.For every prime ideal p ⊂ A, and every finite radicial extension K ′ of the field offractions K of B := A/p, there exists a finite B-subalgebra B′ of K ′ such that the fieldof fractions of B′ is K ′, and the regular locus of SpecB′ is an open subset (the latteris the set of all prime ideals q ⊂ B′ such that B′q is a regular ring).

(iii) We say that A is a Nagata ring, if the following holds. For every p ∈ SpecA and everyfinite field extension κ(p) ⊂ L, the integral closure of A/p in L is a finite A-module.(The rings enjoying this latter property are called universally japanese in [30, Ch.0,Def.23.1.1].)


(iv) We say thatA is universally catenarian if everyA-algebraB of finite type is catenarian,i.e. any two saturated chains (p0 ⊂ · · · ⊂ pn), (q0 ⊂ · · · ⊂ qm) of prime ideals of B,with p0 = q0 and pn = qm, have the same length (so n = m) ([31, Ch.IV, Def.5.6.2]).

(v) We say thatA is excellent, if it is quasi-excellent and universally catenarian ([31, Ch.IV,Def.7.8.2]).

Lemma 4.8.4. Let A be a noetherian ring.

(i) If A is quasi-excellent, then A is a Nagata ring.(ii) If A is a G-ring, then every quotient and every localization of A is a G-ring.

(iii) Suppose that the natural morphism SpecA∧m → SpecAm is regular for every maximalideal m ⊂ A. Then A is a G-ring.

(iv) If A is a local G-ring, then A is quasi-excellent.(v) If A is a complete local ring, then A is excellent.

Proof. (i): This is [31, Ch.IV, Cor.7.7.3].(ii): The assertion for localizations is obvious. Next, if I ⊂ A is any ideal, and p ⊂ A any

prime ideal containing I , then (A/I)∧p = A∧p /IA∧p , from which it is immediate that A/I is a

G-ring, if the same holds for A.(iv) follows from [31, Ch.IV, Th.6.12.7, Prop.7.3.18, Th.7.4.4(ii)].(v): In light of (iv), it suffices to remark that every complete noetherian local ring is univer-

sally catenarian ([31, Ch.IV, Prop.5.6.4] and [30, Ch.0, Th.19.8.8(i)]) and is a G-ring ([30, Ch.0,Th.22.3.3, Th.22.5.8, and Prop.19.3.5(iii)]).

(iii): Let us remark, more generally :

Claim 4.8.5. Let ϕ : A → B be a faithfully flat ring homomorphism of noetherian rings, suchthat f := Specϕ is regular. If B is a G-ring, the same holds for A.

Proof of the claim. In light of (ii), we easily reduce to the case where bothA andB are local, ϕ isa local ring homomorphism, and it suffices to show that the natural morphism πA : SpecA∧ →SpecA is regular (where A∧ is the completion of A). Consider the commutative diagram :

(4.8.6)SpecB∧

f∧ //

πB

SpecA∧

πA

SpecBf // SpecA.

By assumption, πB is a regular morphism; Then the same holds for f πB = πA f∧ (lemma4.8.1(i)). However, it is easily seen that the induced map ϕ∧ : A∧ → B∧ is still a local ringhomomorphism, hence f∧ is faithfully flat, so the claim follows from lemma 4.8.1(ii). ♦

Now, in order to prove (iii), it suffices to check that Am is a G-ring for every maximal idealm ⊂ A. In view of our assumption, the latter assertion follows from claim 4.8.5 and (v).

Definition 4.8.7. Let A be any topological ring, whose topology is linear; we shall consider :

• For any A-algebra C, the category ExalA(C) whose objects are all short exact se-quences of A-modules

(4.8.8) Σ : 0→M → Eψ−→ C → 0

such that E is an A-algebra (with A-module structure given by the structure map A→E), and ψ is a map of A-algebras. The morphisms in ExalA(C) are the commutative


ladders of A-modules :

(4.8.9)

0 // M ′ //

E ′ //

g

C // 0

0 // M ′′ // E ′′ // C // 0

where g is a map of A-algebras.• For any topological A-algebra C, whose topology is linear, the category ExaltopA(C)

whose objects are the short exact sequences of A-modules (4.8.8) such that E is atopological A-algebra whose topology is linear (and again, with A-module structuregiven by the structure mapA→ E), and ψ is a continuous and open map of topologicalA-algebras, whose kernel M is a discrete topological space, for the topology inducedfrom E. The morphisms in ExaltopA(C) are the commutative ladders (4.8.9) such thatg is a continuous map of topological A-algebras.

4.8.10. Now, let A be as in definition 4.8.7, C any topological A-algebra whose topology islinear, ϕ : A → C the structure morphism, and (Iλ | λ ∈ Λ) (resp. (Jλ′ | λ′ ∈ Λ′)) a filteredsystem of open ideals of A (resp. of C), which is a fundamental system of open neighborhoodsof 0. Let Λ′′ ⊂ Λ × Λ′ be the subset of all (λ, λ′) such that ϕ(Iλ) ⊂ Jλ′; the set Λ′′ ispartially ordered, by declaring that (λ, λ′) ≤ (µ, µ′) if and only if Iµ ⊂ Iλ and Jµ′ ⊂ Jλ′ . SetAλ := A/Iλ for every λ ∈ Λ (resp. Cλ′ := C/Jλ′ for every λ′ ∈ Λ′); for (λ, λ′), (µ, µ′) ∈ Λ′′

with (λ, λ′) ≤ (µ, µ′), the surjection πµ′λ′ : Cµ′ → Cλ′ induces a functor

ExalAλ(Cλ′)→ ExalAµ(Cµ′) Σ 7→ Σ ∗ πµ′λ′(see [36, §2.5.5]) and clearly the rule (λ, λ′) 7→ ExalAλ(Cλ′) yields a pseudo-functor

E : (Λ′′,≤)→ Cat.

Moreover, we have a pseudo-cocone :

(4.8.11) E⇒ ExaltopA(C)

defined as follows. To any (λ, λ′) ∈ Λ′′ and any object Σλ,λ′ : 0 → M → Eλ′ → Cλ′ →0 of ExalAλ(Cλ′), one assigns the extension (4.8.8) obtained by pulling back Σλ,λ′ along theprojection πλ′ : C → Cλ′ . Let β : E → Eλ′ be the induced projection; we endow E with thelinear topology defined by the fundamental system of all open ideals of the form ψ−1J ∩ β−1J ′

where J (resp. J ′) ranges over the set of open ideals of C (resp. over the set of all ideals ofEλ′). With this topology, it is easily seen that both ψ and the induced structure map A→ E arecontinuous ring homomorphisms. Moreover, if J ⊂ Jλ′ , then ψ−1J ∩ β−1J ′ = J × (J ′ ∩M),from which it follows that ψ is an open map. Furthermore, M ∩ β−10 = 0, which shows thatM is discrete, for the topology induced by the inclusion map M → E. Summing up, we haveassociated to Σλ,λ′ a well defined object Σλ,λ′ ∗πλ′ of ExaltopA(C), and it is easily seen that therule Σλ,λ′ 7→ Σλ,λ′ ∗ πλ′ is functorial in Σλ,λ′ , and for (λ, λ) ≤ (µ, µ′) in Λ′′, there is a naturalisomorphism in ExaltopA(C) :

Σλ,λ′ ∗ πλ′∼→ (Σλ,λ′ ∗ πµ′λ′) ∗ πµ′

(details left to the reader).

Lemma 4.8.12. The pseudo-cocone (4.8.11) induces an equivalence of categories :

β : 2-colimΛ′′

E∼→ ExaltopA(C).

Proof. Let Σ as in (4.8.8) be any object of ExaltopA(C). By assumption, there exists an openideal J ⊂ E such that M ∩ J = 0. Since ψ is an open map, there exists λ′ ∈ Λ′ such thatJλ′ ⊂ ψ(J), and after replacing J by J ∩ψ−1Jλ′ , we may assume that ψ(J) = Jλ′ . Likewise, if


ϕE : A→ E is the structure morphism, there exists λ ∈ Λ such that Iλ ⊂ ϕ−1E J , and it follows

that the induced extension

Σλ,λ′ : 0→M → E/J → Cλ′ → 0

is an object of ExalAλ(Cλ′). We notice :

Claim 4.8.13. There exists a natural isomorphism Σ∼→ Σλ,λ′ ∗ πλ′ in ExaltopA(C).

Proof of the claim. (i): By construction, ψ and the projection πJ : E → E/J define a uniquemorphism γ : E → (E/J)×Cλ′ C of A-algebras, restricting to the identity map on M (which isan ideal in both of these A-algebras). It is clear that γ is an isomorphism, and therefore it yieldsa natural isomorphism Σ

∼→ Σλ,λ′ ∗ πλ′ in ExalA(C). It remains to check that γ is continuousand open. For the continuity, it suffices to remark that, for every ideal I ⊂ E/J and everyµ′ ∈ Λ′, the ideals γ−1(I ×Cλ′ C) = π−1

J I and γ−1(E/J ×Cλ′ Jµ′) = ψ−1Jµ′ are open in E,which is obvious, since J is an open ideal and ψ is continuous. Lastly, let I ⊂ E be any openideal such that I ⊂ J ∩ ψ−1Jλ′; since ψ is an open map, it is easily seen that γ(I) = 0× ψ(I)is an open ideal of (E/J)×Cλ′ C, so γ is open. ♦

From claim 4.8.13 we see already that β is essentially surjective. It also follows easily thatβ is full. Indeed, consider any morphism s : Σ′ → Σ′′ of ExaltopA(C) as in (4.8.9), and pickan open ideal J ′′ ⊂ E ′′ with J ∩ M ′′ = 0; set J ′ := g−1J ′′, and notice that J ′ ∩ M ′ = 0.Moreover, if the image of J ′′ in C equals Jλ′ for some λ′ ∈ Λ′, then clearly the same holdsfor the image of J ′ in C. Therefore, in this case the foregoing construction yields objects Σ′λ,λ′and Σ′′λ,λ′ of ExalAλ(Cλ′) (for a suitable λ ∈ Λ), whose middle terms are respectively E ′/J ′

and E ′′/J ′′, and s descends to a morphism sλ′ : Σ′λ,λ′ → Σ′′λ,λ′ , whose middle term is the mapgλ′ : E ′/J ′ → E ′′/J ′′ induced by g. By inspecting the proof of claim 4.8.13, we deduce acommutative diagram

E ′∼ //

g

(E ′/J ′)×Cλ′ Cgλ′×Cλ′C

E ′′∼ // (E ′′/J ′′)×Cλ′ C

whose horizontal arrows are the maps that define the isomorphisms Σ′∼→ Σ′λ,λ′ ∗ πλ′ and

Σ′′∼→ Σ′′λ,λ′ ∗ πλ′ in ExaltopA(C). It follows easily that sλ ∗ πλ′ = s, whence the assertion.

Lastly, the faithfulness of β is immediate, since the projections πλ′ are surjective maps.

4.8.14. Let A be as in definition 4.8.7, and C any A-algebra (resp. any topological A-algebra);we denote by

nilExalA(C) ( resp. nilExaltopA(C) )

the full subcategory of ExalA(C) (resp. of ExaltopA(C)) whose objects are the nilpotent exten-sions of C, i.e. those extensions (4.8.8), where M is a nilpotent ideal of E. Moreover, in thesituation of (4.8.10), clearly E restricts to a pseudo-functor

nilE : (Λ′′,≤)→ Cat (λ, λ′) 7→ nilExalAλ(Cλ′).

Also, (4.8.11) restricts to a pseudo-cocone on nilE, and lemma 4.8.12 immediately implies anequivalence of categories

(4.8.15) 2-colimΛ′′

nilE∼→ nilExaltopA(C).

Proposition 4.8.16. Let A and C be as in (4.8.10). The following holds :


(i) Suppose that J2λ′ is open in C, for every λ′ ∈ Λ′. Then the forgetful functor

(4.8.17) nilExaltopA(C)→ nilExalA(C)

is fully faithful.(ii) Suppose additionally, that :

(a) C is a noetherian ring, and I ⊂ C is an ideal such that the topology of C agreeswith the I-preadic topology.

(b) I2λ is open in A, for every λ ∈ Λ.

Then the essential image of (4.8.17) is the (full) subcategory of all nilpotent extensions(4.8.8) such that the C-module M/M2 is annihilated by a power of I .

Proof. (i): The functor is obviously faithful, and in light of (4.8.15), we come down to thefollowing situation. We have a commutative ladder of extensions of A-algebras

0 // M //

Eψ //

g

C //

πλ′

0

0 // N // E ′ // Cλ′ // 0

for some λ′ ∈ Λ′, whose top (resp. bottom) row is an object of nilExaltopA(C) (resp. ofnilExalAλ(Cλ′), for some λ ∈ Λ), and we need to show that the kernel of g contains an openideal. To this aim, we remark :

Claim 4.8.18. For any open ideal J ⊂ E, the ideal J2 is open as well.

Proof of the claim. We may assume that J ∩M = 0. We have I := ψ(J2) = ψ(J)2, so theassumption in (i) say that I is open in C, and therefore ψ−1I = J2⊕M is open in E, so finallyJ ∩ ψ−1I = J2 is open in E, as stated. ♦

Now, set J := ψ−1Jλ′; then J is an open ideal of E, and clearly g(I) ⊂ N . Say that Nk = 0;then g(Ik) = 0, and Ik is an open ideal of E, by claim 4.8.18, whence the contention.

(ii): It is easily seen that, for every extension (4.8.8) in the essential image of (4.8.17), wemust have Ik(M/M2) = 0 for every sufficiently large k ≥ 0 (details left to the reader). Con-versely, consider a nilpotent extension Σ as in (4.8.8), with Ik(M/M2) = 0 and M t = 0 forsome k, t ∈ N. Pick a finite system f := (f1, . . . , fr) of elements of E whose images in B forma system of generators for Ik; notice that the ideal (fn)B is open inB, and (fn)M = 0 for everyn ≥ t. There follows an inverse system of exact sequences

H1(fn, B)→Mβn−−→ E/(fn)E → B/(fn)B → 0 for every n ≥ t

(notation of (4.1.16)) with transition maps induced by the morphisms ϕf of (4.1.23). As Bis noetherian, lemma 4.1.30 and remark 4.1.31 imply that the system (H1(fn, B) | n ∈ N) isessentially zero. Therefore, the same holds for the inverse system (Ker βn | n > 0). However,the transition map Ker βn+1 → Ker βn is obviously injective for every n ≥ t, so we concludethat βn is injective for some sufficiently large integer n. For such n, we obtain a nilpotentextension

(4.8.19) 0→M → E/(fn)E → B/(fn)B → 0.

Lastly, let ϕ : A → B be the structure map, and pick λ ∈ Λ such that Iλ ⊂ ϕ−1Ik; it iseasily seen that I tλM = 0 and ϕ(Isλ) ⊂ (fn)B for s ∈ N large enough. Therefore, somesufficiently large power of Iλ annihilates E/(fn)E; under our assumption (b), such power of Iλcontains another open ideal Iµ, so (4.8.19) is an object of nilExalAµ(B/(fn)B) whose image innilExaltopA(B) agrees with Σ.


Proposition 4.8.20. Let A be a topological ring (whose topology is linear), B a noetherianA-algebra, I ⊂ B an ideal, and suppose that :

(i) The structure map A→ B is continuous for the I-adic topology on B.(ii) For every open ideal J ⊂ A, the ideal J2 is also open.

Then the following conditions are equivalent :(a) B (with its I-adic topology) is a formally smooth A-algebra.(b) Ω1

B/A ⊗B B/I is a projective B/I-module, and H1(LB/A ⊗B B/I) = 0.

Proof. More generally, let A and C be as in (4.8.10), and M a discrete C-module, i.e. a C-module annihilated by an open ideal; we denote by ExaltopA(C,M) the C-module of squarezero topological A-algebra extensions of C by M . Likewise, for every (λ, λ′) ∈ Λ′′, and everyCλ′-module M , let ExalAλ(Cλ′ ,M) be the Cλ′-module of isomorphism classes of square zeroAλ-algebra extensions of Cλ′ by M . For (λ, λ′) ≤ (µ, µ′), we get a natural map of Cµ′-modules

(4.8.21) ExalAλ(Cλ′ ,M)→ ExalAµ(Cµ′ ,M) for every Cλ′-module M

and (4.8.15) implies a natural isomorphism :

colim(λ,λ′)∈Λ′′

ExalAλ(Cλ′ ,M)∼→ ExaltopA(C,M)

which is well defined for every discrete C-module M . By inspecting the definitions, and takinginto account [30, Ch.0, Prop.19.4.3], it is easily seen that C is a formally smooth A-algebra ifand only if ExaltopA(C,M) = 0 for every such discrete C-module M . We also have a naturalC-linear map :

(4.8.22) ExaltopA(C,M)→ ExalA(C,M) for every discrete C-module M

and proposition 4.8.16(i) shows that (4.8.22) is an injective map, provided J2λ′ is an open ideal

of C, for every λ′ ∈ Λ′. Moreover, for C := B (with its I-adic topology), the map (4.8.22) isan isomorphism, by virtue of proposition 4.8.16(ii).

Now, recall the natural isomorphism of Cλ-modules ([56, Ch.III, Th.1.2.3])

ExalA(B,M)∼→ Ext1

B(LB/A,M) for every B-module M.

Combining with the foregoing, we deduce a natural B-linear isomorphism

ExaltopA(B,M)∼→ Ext1

B(LB/A,M)

for every B-module M annihilated by some ideal Ik. Summing up, B is a formally smooth A-algebra if and only if Ext1

B(LB/A,M) vanishes for every discreteB-moduleM . SetB0 := B/I;by considering the I-adic filtration on a given M , a standard argument shows that the lattercondition holds if and only if it holds for every B0-module M , and in turns, this is equivalentto the vanishing of Ext1

B0(LB/A ⊗B B0,M) for every B0-module M . This last condition is

equivalent to (b), as stated.

Proposition 4.8.23. Let A be a ring, B a noetherian A-algebra of finite (Krull) dimension,n ∈ N an integer, and suppose that Hk(LB/A ⊗B κ(p)) = 0 for every prime ideal p ⊂ B andevery k = n, . . . , n+ dimB. Then Hn(LB/A ⊗B M) = 0 for every B-module M .

Proof. Let us start out with the following more general :

Claim 4.8.24. Let A be a ring, B a noetherian A-algebra, p ∈ SpecB a prime ideal, n ∈ N aninteger, and suppose that the following two conditions hold :

(a) Hn(LB/A ⊗B κ(p)) = 0.(b) Hn+1(LB/A ⊗B B/q) = 0 for every proper specialization q of p in SpecB.

Then, Hn(LB/A ⊗B B/p) = 0.


Proof of the claim. Let b ∈ B \ p be any element, and set M := B/(p + bB). Since Bis noetherian, M admits a finite filtration M0 ⊂ M1 ⊂ · · · ⊂ Mn := M such that, for everyi = 0, . . . , n−1, the subquotientMi+1/Mi is isomorphic toB/q, for some proper specializationq of p ([61, Th.6.4]). From (b), and a simple induction, we deduce that Hn+1(LB/A⊗BM) = 0.Whence, by considering the short exact sequence of B-modules

0→ B/pb−→ B/p→M → 0

we see that scalar multiplication by b is an injective map on the B-module Hn(LB/A ⊗B B/p).Since this holds for every b ∈ B \ p, we conclude that the natural map

Hn(LB/A ⊗B B/p)→ Hn(LB/A ⊗B B/p)⊗B Bp = Hn(LB/A ⊗B κ(p))

is injective. Then the assertion follows from (a). ♦

Now, let p ⊂ B be any prime ideal; the assumption, together with claim 4.8.24 and a simpleinduction on d := dimB/p, shows that

Hk(LB/A ⊗B B/p) = 0 for every k = n, . . . , n+ dimB − d.

For anyB-moduleM , setH(M) := Hn(LB/A⊗BM). Especially, we getH(B/p) = 0, for anyprime ideal p ⊂ B. Since B is noetherian, it follows easily that H(M) = 0 for any B-moduleM of finite type (details left to the reader). Next, if M is arbitrary, we may write it as the unionof the filtered family (Mi | i ∈ I) of its submodules of finite type; since H(M) is the colimit ofthe induced system (H(Mi) | i ∈ I), we see that H(M) = 0, as sought.

Corollary 4.8.25. Let A → B be a homomorphism of noetherian rings. Then the followingconditions are equivalent :

(a) Ω1B/A is a flat B-module, and HiLB/A = 0 for every i > 0.

(b) Ω1B/A is a flat B-module, and H1LB/A = 0.

(c) The induced morphism of schemes f : SpecB → SpecA is regular.(d) H1(LB/A ⊗B κ(x)) = 0 for every x ∈ SpecB.

Proof. Let x ∈ X := SpecB be any point; according to [30, Ch.0, Th.19.7.1], the followingconditions are equivalent :

(e) the map on stalks OY,f(x) → OX,x is formally smooth for the preadic topologies definedby the maximal ideals.

(f) f is flat at the point x, and the κ(f(x))-algebra Of−1f(x),x is geometrically regular.On the other hand, by virtue of proposition 4.8.20, condition (e) is equivalent to the vanishing ofH1(LB/A⊗B κ(x)), whence (b)⇒(c)⇔(d). It remains to check that (d)⇒(a). However, assume(d); taking into account proposition 4.8.23, and arguing by induction on i, we easily show that(a) will follow, provided

Hi(LB/A ⊗B κ(x)) = 0 for every x ∈ X and every i > 1.

For every x ∈ X , set Bx := OX,x ⊗A κ(f(x)); since B is A-flat, the latter holds if and only if

Hi(LBx/κ(f(x)) ⊗B κ(x)) = 0 for every x ∈ X and every i > 1

([56, Ch.II, Prop.2.2.1 and Ch.III, Cor.2.3.1.1]). Hence, we may replace A by κ(f(x)), and Bby Bx, and assume from start that A is a field and B is a local geometrically regular A-algebrawith residue field κB, and it remains to check that Hi(LB/A ⊗B κB) = 0 for every i > 1.However, in view of corollary 4.7.40, the sequence of ring homomorphisms A → B → κByields an isomorphism

Hi(LB/A ⊗B κB)∼→ HiLκB/A for every i > 1

([56, Ch.II, Prop.2.1.2]) so we conclude by the following general :


Claim 4.8.26. Let K ⊂ E be any extension of fields. Then HiLE/K = 0 for every i > 1.

Proof of the claim. By [56, Ch.II, (1.2.3.4)], we may reduce to the case where E is a finitelygenerated extension of K, say E = K(a1, . . . , an). We proceed by induction on n. If n = 1,then E is either an algebraic extension or a purely transcendental extension of K. In the lattercase, the assertion is immediate ([56, Ch.II, Prop.1.2.4.4 and Ch.III, Cor.2.3.1.1]). For the caseof an algebraic extension, the assertion is a special case of [36, Th.6.3.32(i)] (we apply loc.cit.to the valued field (K, |·|) with trivial valuation |·|). Lastly, if n > 1, set L := K(a1, . . . , an−1).By inductive assumption we have HiLL/K = HiLE/L = 0 for i > 1; on the other hand, there isa distinguished triangle ([56, Ch.II, Prop.2.1.2])

LL/K ⊗L E → LE/K → LE/L → LL/K ⊗L E[1] in D(E-Mod).

The sought vanishing follows immediately,

Corollary 4.8.27. LetA be a local noetherian ring, A∧ the completion ofA. Then the followingconditions are equivalent :

(a) A is quasi-excellent.(b) Ω1

A∧/A is a flat A∧-module, and H1LA∧/A = 0.(c) Ω1

A∧/A is a flat A∧-module, and HiLA∧/A = 0 for every i > 0.

Proof. Taking into account lemma 4.8.4(iii), this is a special case of corollary 4.8.25.

Proposition 4.8.28. Let p > 0 be a prime integer, A a regular local and excellent Fp-algebra,B a local noetherian A-algebra, mB the maximal ideal of B, and M a B-module of finite type.Then the B-module Ω1

A/Fp ⊗AM is separated for the mB-preadic topology.

Proof. Let ϕ : A → B be the structure morphism, and set p := ϕ−1mB; the localization Ap isstill excellent (lemma 4.8.4(ii,iv)) and regular, and clearly Ω1

Ap/Fp⊗ApM = Ω1A/Fp⊗AM , hence

we may replace A by Ap, and assume that ϕ is local. Let A∧ and B∧ be the completions of Aand B, set M∧ := B∧ ⊗B M , and notice that both of the natural maps Fp → A and A → A∧

are regular. In view of corollary 4.8.25, it follows that both of the natural B-linear maps

Ω1A/Fp ⊗AM → Ω1

A/Fp ⊗AM∧ Ω1

A/Fp ⊗AM → Ω1A∧/Fp ⊗A∧ M

∧

are injective (to see the injectivity of the second map, one applies the transitivity triangle arisingfrom the sequence of ring homomorphisms Fp → A → A∧ : details left to the reader). Thus,we may assume that A is complete. Now, for every ring R, and every integer m ∈ N, set

(4.8.29) R(m) := R[[T1, . . . , Tm]] R〈m〉 := R[[T p1 , . . . , Tpm]] ⊂ R(m).

With this notation, we have an isomorphism A∼→ κ(d) of Fp-algebras, where κ is the residue

field of A, and d := dimA ([30, Ch.0, Th.19.6.4]).

Claim 4.8.30. Let K be any field of characteristic p, and m ∈ N any integer. We have :(i) There exists a cofiltered system (Kλ | λ ∈ Λ) of subfields of K such that [K : Kλ] is

finite for every λ ∈ Λ, and⋂λ∈ΛK

λ = Kp.(ii) For every system (Kλ | λ ∈ Λ) fulfilling the condition of (i), the following holds :

(a) Ω1K(m)/K

λ〈m〉

is a free K(m)-module of finite rank, for every λ ∈ Λ, and the rule

M 7→ Ωm(M) := limλ∈Λ

(Ω1K(m)/K

λ〈m〉⊗K(m)

M)

defines an exact functor K(m)-Mod→ K(m)-Mod.(b) The natural map

ηm(M) : Ω1K(m)/Fp ⊗K(m)

M → Ωm(M)

is injective for every K(m)-module M .


(c) Let F (resp. F λ) denote the field of fractions of K(m) (resp. of Kλ〈m〉, for every

λ ∈ Λ); then⋂λ∈Λ F

λ = F p.

Proof of the claim. (i) and (ii.c) follow from [30, Ch.0, Prop.21.8.8] (and its proof).(ii.a): For given λ ∈ Λ, say that x1, . . . , xr is a p-basis ofK overKλ; then it is easily seen that

x1, . . . , xr, T1, . . . , Tm is a p-basis of K(m) over Kλ〈m〉 (see [30, Ch.0, Def.21.1.9]). According

to [30, Ch.0, Cor.21.2.5], it follows that Ω1K(m)/K

λ〈m〉

is the free K(m)-module of finite type with

basis dx1, . . . , dxr, dT1, . . . , dTm. Moreover, say that Kµ ⊂ Kλ, and let xr+1, . . . , xs be a p-basis of Kλ over Kµ; then x1, . . . , xs is a p-basis of K over Kµ ([30, Ch.0, Lemme 21.1.10]),so the induced map

Ω1K(m)/K

µ〈m〉→ Ω1

K(m)/Kλ〈m〉

is a projection onto a direct factor, and the assertion follows easily.(ii.b): We are easily reduced to the case whereM is aK(m)-module of finite type, and in light

of (ii.a), we may further assume that M is a cyclic K(m)-module. Next, we remark that, due to(ii.c), the natural map

Ω1F/Fp → lim

λ∈ΛΩ1F/Fλ

is injective ([30, Ch.0, Th.21.8.3]); in other words, ηm(F ) is injective. In order to show theinjectivity of ηm(M), it then suffices to check that the functor M 7→ Ω1

K(m)/Fp ⊗K(m)M is

exact. The latter holds by virtue of corollary 4.8.25, since the (unique) morphism of schemesSpecK(m) → SpecFp is obviously regular. This completes the proof for m = 0. Suppose nowthat m > 0, and that the injectivity of ηn(M) is already known for every n < m and everyK(n)-module M . By the foregoing, it remains to check that ηm(K(m)/I) is injective, for everynon-zero ideal I ⊂ K(m). Pick any non-zero f ∈ I , and set R := K(m−1); according to [14,Ch.VII, n.7, Lemme 3] and [14, Ch.VII, n.8, Prop.6], there exist an automorphism σ of the ringK(m), and elements g ∈ R[Tm], u ∈ K×(m) such that σ(f) = u·g, and g = T dm+a1T

d−1m +· · ·+ad

for some d ≥ 0 and certain elements a1, . . . , ad of the maximal ideal of R. However, setM ′ := K(m)/σ(I); in view of the commutative diagram of Fp-modules :

Ω1K(m)/Fp ⊗K(m)

Mηm(M)

//

Ωm(M)

Ω1K(m)/Fp ⊗K(m)

M ′ ηm(M ′)// Ωm(M ′)

(whose vertical arrows are induced by σ) we see that ηm(M) is injective, if and only if the sameholds for ηm(M ′). Hence, we may replace I by σ(I), and assume that g ∈ I . In this case, setalso Rλ := Kλ

〈m−1〉 for every λ ∈ Λ, and notice that the natural maps

R[Tm]/gpR[Tm]→ K(m)/gpK(m) Rλ[T pm]/gpRλ[T pm]→ Kλ

〈m〉/gpKλ〈m〉

are bijective; there follows a commutative diagram of K(m)-modules :

Ω1R[Tm]/Fp ⊗R[Tm] M //

αλ⊗R[Tm]M

Ω1K(m)/Fp ⊗K(m)

M

ηλm⊗K(m)M

Ω1R[Tm]/Rλ

⊗R[Tm] M // Ω1K(m)/K

λ〈m〉⊗K(m)

M


whose horizontal arrows are isomorphisms, and ηm(M) = limλ∈Λ ηλm ⊗K(m)

M . On the otherhand, for every λ ∈ Λ we have a commutative ladder of R[Tm]-modules with exact rows :

Σλ :

0 // Ω1R/Fp ⊗R R[Tm] //

ηλm−1⊗RR[Tm]

Ω1R[Tm]/Fp

//

αλ

Ω1R[Tm]/R

// 0

0 // Ω1R/Rλ

⊗R R[Tm] // Ω1R[Tm]/Rλ

// Ω1R[Tm]/R

// 0

and notice that the rows of Σλ ⊗R[Tm] M are still short exact, for every λ ∈ Λ. By induc-tive assumption, limλ∈Λ η

λm−1 ⊗R M is an injective map; we deduce that the same holds for

limλ∈Λ αλ ⊗R[T ] M , and the claim follows. ♦

Take K := κ, and pick any cofiltered system (Kλ | λ ∈ Λ) as provided by claim 4.8.30(i);in light of claim 4.8.30(ii.b), it now suffices to show that the resulting Ωd(M) is a separated B-module, for every noetherian κ(d)-algebra B and every B-module M of finite type. However,Ωd(M) is a submodule of

∏λ∈Λ(Ω1

K(m)/Kλ〈m〉⊗K(m)

M), hence we are reduced to checking that

each direct factor of the latter B-module is separated. But in view of claim 4.8.30(ii.a), wesee that each such factor is a finite direct sum of copies of M , so finally we come down to theassertion that M is separated for the mB-adic topology, which is well known.

Theorem 4.8.31. Let ϕ : A → B be a local ring homomorphism of local noetherian rings.Suppose that A is quasi-excellent, and ϕ is formally smooth for the preadic topologies definedby the maximal ideals. Then Specϕ is regular.

Proof. This is the main result of [2]. We begin with the following general remark :

Claim 4.8.32. Let R be a local noetherian ring, mR ⊂ R the maximal ideal, H : R-Mod →R-Mod an additive functor, and suppose that

(a) H is R-linear, i.e. H(t · 1M) = t ·H(1M) for every R-module M , and every t ∈ R.(b) H is semi-exact, i.e. for any short exact sequence 0 → M ′ → M → M ′′ → 0 of

R-modules, the induced sequence H(M ′)→ H(M)→ H(M ′′) is exact.(c) H commutes with filtered colimits.(d) H(M) is separated for the mR-adic topology, for every R-module M of finite type.(e) H(R/mR) = 0.

Then H(M) = 0 for every R-module M .

Proof of the claim. Since H commutes with filtered colimits, it suffices to show that H(M) = 0for every finitely generatedR-moduleM , and sinceH is semi-exact, a simple induction reducesfurther to the case where M is a cyclic R-module. Let now F be the family of all ideals I ofR such that H(R/I) 6= 0, and suppose, by way of contradiction, that F 6= ∅; pick a maximalelement J of F , and set M := R/J . By assumption, J 6= mR; thus, let t ∈ R be a non-invertible element with t /∈ J ; we get an exact sequence

H(M)t−→ H(M)→ H(B(m+n)/(J + tB(m+n)))

whose third term vanishes, by the maximality of J . On the other hand,⋂n∈N t

nH(M) = 0,since H(M) is separated. Thus H(M) = 0, contradicting the choice of J , and the claimfollows. ♦

Denote by κ the residue field of A, and for every m,n ∈ N, let ϕm,n : A(m) → B(m+n) be thecomposition of ϕ(m) : A(m) → B(m) (the T -adic completion of ϕ ⊗A A[T1, . . . , Tm]) with thenatural inclusion map B(m) → B(m+n) (notation of (4.8.29)).


Claim 4.8.33. In the situation of the theorem, suppose furthermore that A is either a field ora complete discrete valuation ring of mixed characteristic. Then, the morphism Specϕm,n isregular for every m,n ∈ N.

Proof of the claim. Set

H(M) := H1(LB(m+n)/A(m)⊗B(m+n)

M).

We shall consider separately three different cases :• Suppose first that A is a field of characteristic p > 0. According to corollary 4.8.25, it

suffices to show that H(M) vanishes for every B(m+n)-module M . Notice that the natural mapFp → B(m+n) is regular; from the distinguished triangle ([56, Ch.II, Prop.2.1.2])

LA(m)/Fp ⊗A(m)B(m+n) → LB(m+n)/Fp → LB(m+n)/A(m)

→ LA(m)/Fp ⊗A(m)B(m+n)[1]

and corollary 4.8.25, we deduce an injective B(m+n)-linear map

H(M)→ Ω1A(m)/Fp ⊗A(m)

M

from which it follows that, ifM is aB(m+n)-module of finite type,H(M) is a separatedB(m+n)-module, for the preadic topology defined by the maximal ideal of B(m+n) (proposition 4.8.28).Since ϕ is formally smooth, ϕm,n is also formally smooth for the preadic topologies defined bythe maximal ideals of A(m) and B(m+n); from proposition 4.8.20, we see that H(M) = 0, if Mis the residue field of B(m+n). Then the assertion follows from claim 4.8.32.• Next, suppose thatA is either a field of characteristic zero, or a complete discrete valuation

ring of mixed characteristic (so either κ = A, or else κ is a field of positive characteristic).According to corollary 4.8.25, it suffices to show that H(M) vanishes for M = κ(q), whereq ⊂ B(m+n) is any prime ideal. Fix such q, and set p := q ∩ A(m); if p = 0, then M is aK-algebra, where K is the field of fractions of A(m); now, K is a field of characteristic zero,and B′ := B(m+n) ⊗A(m)

K is a regular local K-algebra, so the induced morphism SpecB′ →SpecK is regular; since H(M) = H1(LB′/K ⊗B′ M), the assertion follows from corollary4.8.25. Notice that this argument applies especially to the case where m = 0; for the generalcase, we argue by induction on m. Hence, suppose that n ∈ N, m > 0, and that the assertion isalready known for ϕn,m−1.

Consider first the case where A is a discrete valuation ring, and p contains the maximal idealof A, and set B := B ⊗A κ. Since ϕm,n is flat, and since M is a B(m+n)-module, we have anatural isomorphism

H(M)∼→ H1(LB(m+n)/κ(m)

⊗B(m+n)M).

Then the sought vanishing follows from the foregoing, since B is a formally smooth κ-algebra(for the preadic topology of its maximal ideal).

Lastly, suppose that either A is a field, or p does not contain the maximal ideal of A (and p 6=0). In either of these two cases, we may find f ∈ p whose image in κ(m) is not zero. Accordingto [14, Ch.VII, n.7, Lemme 3] and [14, Ch.VII, n.8, Prop.6], we may find an automorphism σof the A-algebra A(m) and elements g ∈ A(m−1)[Tm], u ∈ A×(m) such that σ(f) = u · g, andg = T dm + a1T

d−1m + · · · + ad for some d ≥ 0 and certain elements a1, . . . , ad of the maximal

ideal of A(m−1). Denote by σ′ : B(m)∼→ B(m) the T -adic completion of σ ⊗A B, and let

σB : B(m+n)∼→ B(m+n) be the T -adically continuous automorphism that restricts to σ′ on B(m),

and such that σB(Ti) = Ti for i = m+1, . . . ,m+n. SetM ′ := B(m+n)/σB(q); by construction,


we have a commutative diagram of A-algebras

A(m)ϕm,n //

σ

B(m+n)

σB

A(m)ϕm,n // B(m+n)

inducing an isomorphism

LB(m+n)/A(n)⊗B(m+n)

M∼→ LB(m+n)/A(n)

⊗B(m+n)M ′ in D(A-Mod).

Thus, we may replace M by M ′, and assume from start that g ∈ p. In this case, set B′ :=B[[Tm+1, . . . , Tm+n]], and notice that both of the natural maps

A(m−1)[Tm]/gA(m−1)[Tm]→ A(m)/gA(m) B′(m−1)[Tm]/gB′(m−1)[Tm]→ B(m+n)/gB(m+n)

are isomorphisms. Since both ϕm,n and the map A(m−1)[Tm] → B′(m−1)[Tm] induced by ϕ areflat ring homomorphisms, there follows a natural isomorphism of B(m+n)-modules :

LB(m+n)/A(n)⊗B(m+n)

M∼→ LB′

(m−1)[Tm]/A(m−1)[Tm] ⊗B′

(m−1)[Tm] M

∼→ LB′(m−1)

/A(m−1)⊗B′

(m−1)M

([56, Ch.II, Prop.2.2.1]). However, the resulting map A(m−1) → B′(m−1) is none else thanϕm−1,n, up to a relabeling of the variables; the vanishing of H(M) then follows from the induc-tive assumption (and from corollary 4.8.25). ♦

Claim 4.8.34. In the situation of the theorem, suppose furthermore that A and B are complete,and let M be a B-module of finite type. Then H1(LB/A ⊗B M) is a B-module of finite type.

Proof of the claim. Let f : A0 → κ be a surjective ring homomorphism, with A0 a Cohen ring([30, Ch.0, Th.19.8.6(ii)]), and set B := B ⊗A κ; in light of [30, Ch.0, Lemme 19.7.1.3], thereexists a flat local, complete and noetherian A0-algebra B0 fitting into a cocartesian diagram :

A0

ψ //

f

B0

fB

κ // B.

Denote by κB the residue field of B; there follow natural isomorphisms of B-modules :

H1(LB/A ⊗B κB)∼→ H1(LB/κ ⊗B κB)

∼→ H1(LB0/A0 ⊗B0 κB)

([56, Ch.II, Prop.2.2.1]) which, according to proposition 4.8.20, imply that ψ is formally smooth(for the preadic topologies defined by the maximal ideals). By [30, Ch.0, Th.19.8.6(i)], f liftsto a ring homomorphism f : A0 → A, whence a commutative diagram

A0

ψ //

ϕf

B0

fB

B // B.

Then, by [30, Ch.0, Cor.19.3.11], the map fB lifts to a ring homomorphism fB : B0 → B.Notice now that both f and fB are local maps and induce isomorphisms on the residue fields; it


follows easily that, for suitable m,n ∈ N, they extend to surjective maps g and gB fitting into acommutative diagram

A0,(m)

ψ(m,n) //

g

B0,(m+n)

gB

A

ϕ // B

whence exact sequences ([56, Ch.II, Prop.2.1.2])

H1(LB0,(m+n)/A0,(m)⊗B0,(m+n)

M)→ H1(LB/A0,(m)⊗B M)→ H1(LB0,(m+n)/B ⊗B0,(m+n)

M)

H1(LB/A0,(m)⊗B M)→ H1(LB/A ⊗B M)→ Ω1

A/A0,(m)⊗AM = 0.

However, claim 4.8.33 applies to ψ, and together with corollary 4.8.25, it implies that the firstmodule of the first of these sequences vanishes; on the other hand, it is easily seen that the thirdB-module of the same sequence is finitely generated, so the same holds for the middle term. Byinspecting the second exact sequence, the claim follows. ♦

We may now conclude the proof of the theorem : let A∧ and B∧ be the completions of A andB; since ϕ is formally smooth, the same holds for its completion ϕ∧ : A∧ → B∧. First, we showthat Specϕ∧ is regular; to this aim, it suffices to check that H∧(M) := H1(LB∧/A∧ ⊗B∧ M)vanishes for every B∧-module M (corollary 4.8.25). However, we know already that H∧(M)is a B∧-module of finite type, if the same holds for M (claim 4.8.34); especially, for such M ,H∧(M) is separated for the adic topology of B∧ defined by the maximal ideal. Moreover,H∧(M) = 0, if M is the residue field of B∧ (proposition 4.8.20). Then the assertion followsfrom claim 4.8.32. Lastly, the natural morphism SpecA∧ → SpecA is regular by assumption,hence the same holds for the induced morphism SpecB∧ → SpecA (lemma 4.8.1(i)). Finally,since B∧ is a faithfully flat B-algebra, we conclude that Specϕ is regular, by virtue of lemma4.8.1(ii).

Proposition 4.8.35. Let A be a noetherian local ring, and ϕ : A → B an ind-etale local ringhomomorphism. Then :

(i) A is quasi-excellent if and only if the same holds for B.(ii) If A is excellent, the same holds for B.

Proof. Set f := Specϕ, and f∧ := Specϕ∧, where ϕ∧ : A∧ → B∧ is the map of completelocal rings obtained from ϕ. Notice first that, for every y ∈ SpecA, and every x ∈ f−1(y), thestalk Of−1y,x is an ind-etale κ(y)-algebra, i.e. is a separable algebraic extension of κ(y), hencef−1(y) is geometrically regular, and therefore f is regular and faithfully flat. Moreover, f is thelimit of a cofiltered system of formally etale morphisms, hence it is formally etale; a fortiori, Bis also formally smooth over A for the preadic topologies, so B∧ is formally smooth over A∧

for the preadic topologies, and consequently f∧ is a regular morphism (theorem 4.8.31).(i): If B is quasi-excellent, claim 4.8.5 and lemma 4.8.4(iv) imply that A is quasi-excellent.

Next, assume thatA is quasi-excellent; we have a commutative diagram (4.8.6), in which πA is aregular morphism, and then the same holds for πA f∧ = f πB (lemma 4.8.1(i)). Now, let y ∈SpecB be any point, and set z := f(y); we know that (f πB)−1(z) is a geometrically regularaffine scheme, so write it as the spectrum of a noetherian ring C. The stalk E := Of−1(z),y isa separable algebraic field extension of κ(z), hence π−1

B (y) ' SpecC ⊗B E, the spectrum ofa localization of C, so it is again geometrically regular, i.e. πB is a regular morphism, and weconclude by lemma 4.8.4(iii,iv).

(ii): By (i), it remains only to show that B is universally catenarian, provided the same holdsfor A. To this aim, it suffices to apply [33, Ch.IV, Lemma 18.7.5.1].


Proposition 4.8.36. In the situation of (4.7.29), suppose that ΦA is a finite ring homomorphism,and for every prime ideal p ⊂ A, set κ(p) := Ap/pAp (the residue field of the point p ∈ SpecA).Then

dimAp/qAp = dimκ(p) Ω1κ(p)/Fp − dimκ(q) Ω1

κ(q)/Fp

for every pair of prime ideals q ⊂ p ⊂ A.

Proof. To begin with, we notice :

Claim 4.8.37. Suppose that ΦA is a finite map. We have :(i) For every A-algebra B of essentially finite type, ΦB is finite as well.

(ii) Suppose moreover, that A is local, and let A∧ the completion of A. Then ΦA∧ =1A∧ ⊗A ΦA is finite, and Ω1

A∧/Fp = A∧ ⊗A Ω1A/Fp .

(iii) If A is local and reduced, then the same holds for A∧.(iv) Suppose moreover, that A is a local integral domain, and denote by K (resp. κ) the

field of fractions (resp. the residue field) of A. Then

(4.8.38) dimK Ω1K/Fp = dimκ Ω1

κ/Fp + dimA.

Proof of the claim. (i): Suppose first that B = A[X]; then it is easily seen that ΦB = ΦA ⊗FpΦFp[X], whence the contention, in this case. By an easy induction, we deduce that the claimholds as well for any free polynomial A-algebra of finite type. Next, let I ⊂ A be any ideal;since ΦA(I) ⊂ I , it is easily seen that ΦA/I = ΦA ⊗A A/I , so ΦA/I is finite, and therefore theassertion holds for any A-algebra of finite type. Lastly, let S ⊂ A be any multiplicative subset;since ΦA(S) ⊂ S, it is easily seen that

(4.8.39) ΦS−1A = S−1ΦA

and the assertion follows.(ii): Denote by mA the maximal ideal of A, and set B := A(Φ); by assumption, ΦA : A→ B

is a finite A-linear map, hence its mA-adic completion (ΦA)∧ : A∧ → B∧ equals 1A∧ ⊗A ΦA.Since ΦA(mA) = mp

A, we have a natural A∧-linear identification on mA-adic completions :

(4.8.40) B∧∼→ (A∧)(Φ)

and under this identification, (ΦA)∧ = ΦA∧ , whence the first assertion of (ii). Next, we no-tice that (4.8.40) yields a natural identification Ω1

A∧/Fp = Ω1B∧/Fp , and on the other hand, the

sequence of ring homomorphisms

Fp → A∧ΦA∧−−−→ B∧

yields natural identifications :

Ω1B∧/Fp = Ω1

B∧/A∧ = A∧ ⊗A Ω1B/A = A∧ ⊗A Ω1

B/Fp = B∧ ⊗B Ω1B/Fp = A∧ ⊗A Ω1

A/Fp

where the last equality is induced by (4.8.40) and the identity map A ∼→ B. This completes theproof of the second assertion.

(iii): Since A is reduced, ΦA is injective, and then the same holds for its completion (ΦA)∧.But we have seen that the latter is naturally identified with ΦA∧ , whence the claim.

(iv): Notice that Ω1K/Fp is a K-vector space of finite dimension, since its dimension equals

[K : Kp] ([30, Ch.0, Th.21.4.5]), and the latter is finite, by (i); the same argument applies tothe κ-vector space Ω1

κ/Fp . Denote by δ(A) the difference between the left and right hand-sideof (4.8.38); we have to show that δ(A) = 0. Let p ⊂ A∧ be any minimal prime ideal, so thatdimA∧/p = d := dimA. In view of (iii), L := (A∧)p is a field; taking into account (ii), we get

Ω1L/Fp = L⊗A∧ Ω1

A∧/Fp = L⊗A Ω1A/Fp = L⊗K Ω1

K/Fp .


Hence dimL Ω1L/Fp = dimK Ω1

K/Fp , so we see that

(4.8.41) δ(A) = δ(A∧/p) for any minimal prime ideal p ⊂ A∧.

We may then replaceA byA∧/p, after which we may assume thatA is a complete local domain.In this case, A contains a field mapping isomorphically onto κ, and there is a finite map of κ-algebras A′ := κ[[T1, . . . , Td]] → A ([61, Th.29.4(iii)]). Denote by K ′ the field of fractions ofA′, and notice that [K : K ′] = [K : Kp] · [Kp : K ′] = [K : Kp] · [Kp : K ′p]; on the otherhand, ΦK induces an isomorphism K

∼→ Kp, hence have as well [K : K ′] = [Kp : K ′p], so[K : Kp] = [K ′ : K ′p] and finally :

dimK′ Ω1K′/Fp = dimK Ω1

K/Fp

([30, Ch.0, Cor.21.2.5]). Since dimA′ = dimA, we conclude that δ(A) = δ(A′). Hence, itsuffices to check that δ(A′) = 0. Furthermore, set B := κ[T1, . . . , Td], and let m ⊂ B bethe maximal ideal generated by T1, . . . , Td; we have A′ = B∧m, so (4.8.41) further reduces toshowing that δ(Bm) = 0, which shall be left as an exercise for the reader. ♦

Now, in view of claim 4.8.37(i), we may replace A by Ap/qAp, and assume from start that Ais a local domain, that q = 0 and that p is the maximal ideal; in this case, the sought identity isgiven by claim 4.8.37(iv).

Theorem 4.8.42. With the notation of (4.7.29), suppose that A is noetherian. We have :(i) If ΦA is a finite ring homomorphism, then A is excellent.

(ii) Conversely, suppose thatA is a local Nagata ring, with residue field k, and that [k : kp]is finite. Then ΦA is a finite ring homomorphism.

Proof. (i): We reproduce the proof from [60, Appendix, Th.108].

Claim 4.8.43. If ΦA is finite, then A is quasi-excellent.

Proof of the claim. We check first the openness condition for the regular loci. To this aim, inlight of claim 4.8.37(i), we may assume that A is an integral domain, and it suffices to provethat the regular locus of SpecA is an open subset. Set B := A(Φ) (notation of (4.7.29)), and letp ⊂ A be any prime ideal; by theorem 4.7.30 and (4.8.39), the ring Ap is regular if and only if(ΦA)p is a flat ring homomorphism, if and only if Bp is a flat Ap-module. Since, by assumption,B is a finiteA-module, the latter holds if and only ifBp is a freeAp-module. Then, the assertionfollows from [61, Th.4.10].

Next, we show that A is a G-ring. By claim 4.8.37(i), we may assume that A is local. Fromclaim 4.8.37(ii) we see that the natural map

A(Φ)

L⊗A A∧ → (A∧)(Φ)

is an isomorphism in D(A-Mod). Then the assertion follows from [36, Lemma 6.5.13(i)] andcorollary 4.8.27. ♦

Now, it follows easily from proposition 4.8.36 and claim 4.8.37(i), that A is universally cate-narian; combining with claim 4.8.43, we obtain (i).

(ii): The assertion to prove is that A(Φ) is a finite A-module. To this aim, denote by I ⊂ Athe nilradical ideal; we remark :

Claim 4.8.44. It suffices to show that (A/I)(Φ) is a finite A-module.

Proof of the claim. Indeed, suppose that (A/I)(Φ) is a finite A-module; we shall deduce, byinduction on s, that (A/Is)(Φ) is a finite A-module, for every s > 0. Since A is noetherian,we have I t = 0 for t > 0 large enough, so the claim will follow. The assertion for s = 1is our assumption. Suppose therefore that s > 1, and that we already know the assertion fors − 1. Notice that Is/Is−1 is a finite A/I-module, hence a finite (A/I)(Φ)-module, by our


assumption. Since (A/Is−1)(Φ) is a finite A-module, we deduce easily that the assertion holdsfor s, as required. ♦

In view of claim 4.8.44, we may replace A by A/I (which is obviously still a Nagata ring),and assume from start that A is reduced. In this case, let p1, . . . , pt be the set of minimalprime ideals of A; we have a commutative diagram of ring homomorphisms :

AΦA //

A

∏ti=1 A/pi

∏ti=1 ΦA/pi // ∏t

i=1A/pi

whose vertical arrows are injective and finite maps. Suppose now that (A/pi)(Φ) is a finiteA/pi-module for every i = 1, . . . , t. Then

∏ti=1(A/pi)(Φ) is a finiteA-module, and therefore the same

holds for its A-submodule A(Φ). Thus, we are reduced to checking the sought assertion for eachof the quotients A/pi (which are still Nagata rings), and hence we may assume from start thatA is a domain. In this case, denote by K the field of fractions of A; by the definition of Nagataring, we are further reduced to checking that ΦK : K → K is a finite field extension, i.e. that[K : Kp] is finite. Denote A∧ the completion of A; we remark :

Claim 4.8.45. (i) The endomorphism ΦA∧ is a finite map.(ii) SpecK ⊗A A∧ is a geometrically reduced K-scheme.

Proof of the claim. (i): Indeed, A∧ is a quotient of a power series ring B := k[[T1, . . . , Tr]] ([30,Ch.0, Th.19.8.8(i)]), hence it suffices to show that ΦB is finite. However,Bp = kp[[T p1 , . . . , T

pr ]],

and [k : kp] is finite by assumption, whence the claim.(ii): This is [31, Ch.IV, Th.7.6.4]. ♦

Let p ⊂ A∧ be any minimal prime ideal; it follows from claim 4.8.45(ii) that L := Ap is aseparable field extension of K ([31, Ch.IV, Prop.4.6.1]), so the natural map

Ω1K/Fp ⊗K L→ Ω1

L/Z

is injective ([30, Ch.0, Cor.20.6.19(i)]); on the other hand, claim 4.8.45(i) implies that [L : Lp]is finite, i.e. Ω1

L/Fp is a finite-dimensional L-vector space ([30, Ch.0, Th.21.4.5]). We concludethat Ω1

K/Fp is a finite-dimensional K-vector space, whence the contention, again by loc.cit.

5. LOCAL COHOMOLOGY

5.1. Cohomology in a ringed space.

Definition 5.1.1. Let X be a topological space, F a sheaf on X . We say that F is qc-flabby ifthe restriction map F (V ) → F (U) is a surjection whenever U ⊂ V are quasi-compact opensubsets of X .

Lemma 5.1.2. Suppose that the topological space X fulfills the following two conditions :(a) X is quasi-separated (i.e. the intersection of any two quasi-compact open subsets of

X is quasi-compact).(b) X admits a basis of quasi-compact open subsets.

Let 0→ F ′ → F → F ′′ → 0 be a short exact sequence of abelian sheaves on X . Then :(i) If F ′ is qc-flabby, the induced sequence

0→ F ′(U)→ F (U)→ F ′′(U)→ 0

is short exact for every quasi-compact open subset U ⊂ X .(ii) If both F and F ′ are qc-flabby, the same holds for F ′′.


Proof. (i) is a variant of [39, Ch.II, Th.3.1.2]. Indeed, we may replace X by U , and therebyassume that X is quasi-compact. Then we have to check that every section s′′ ∈ F ′′(X) is theimage of an element of F (X). However, we can find a finite covering (Ui | i = 1, . . . , n) of X ,consisting of quasi-compact open subsets, such that s′′|Ui is the image of a section si ∈ F (Ui).For every k ≤ n, let Vk := U1 ∪ · · · ∪ Uk; we show by induction on k that s′′|Vk is in the imageof F (Vk); the lemma will follow for k = n. For k = 1 there is nothing to prove. Suppose thatk > 1 and that the assertion is known for all j < k; hence we can find t ∈ F (Vk−1) whoseimage is s′′|Vk−1

. The difference u := (t− sk)|Uk∩Vk−1lies in the image of F ′(Uk ∩ Vk−1); since

Uk ∩ Vk−1 is quasi-compact and F ′ is qc-flabby, u extends to a section of F ′(Uk). We canthen replace sk by sk + u, and assume that sk and t agree on Uk ∩ Vk−1, whence a section onVk = Uk ∪ Vk−1 with the sought property. Assertion (ii) is left to the reader.

Lemma 5.1.3. Let F : C → C ′ be a covariant additive left exact functor between abeliancategories. Suppose that C has enough injectives, and let M ⊂ C be a subcategory such that :

(a) For every A ∈ Ob(C ), there exists a monomorphism A→M , with M ∈ Ob(M ).(b) Every A ∈ Ob(C ) isomorphic to a direct factor of an object of M , is an object of M .(c) For every short exact sequence

0→M ′ →M →M ′′ → 0

in C with M ′ and M in M , the object M ′′ is also in M , and the sequence

0→ F (M ′)→ F (M)→ F (M ′′)→ 0

is short exact.Then, every injective object of C is in M , and we have RpF (M) = 0 for every M ∈ Ob(M )and every p > 0.

Proof. This is [43, Lemme 3.3.1]. For the convenience of the reader we sketch the easy proof.First, if I is an injective object of C , then by (a) we can find a monomorphism I → M withM in M ; hence I is a direct summand of M , so it is in M , by (b). Next, let M ∈ M , andchoose an injective resolution M → I• in C . For every i ∈ N let Zi := Ker(di : I i → I i+1);in order to prove that RpF (M) = 0 for every p > 0, we have to show that the sequences0→ F (Zi)→ F (I i)→ F (Zi+1)→ 0 are short exact for every i ∈ N. In turns, this will followfrom (c), once we know that Zi and I i are in M . This is already established for I i, and for Zi

it is derived easily from (c), via induction on i.

5.1.4. Let X be a topological space. To X we attach the site X whose objects are all thefamilies U := (Ui | i ∈ I) of open subsets of X (i.e. all the covering families – indexed by aset I taken within a fixed universe – of all the open subsets of X). The morphisms β : V :=(Vj | j ∈ J) → U in X are the refinements, i.e. the maps β : J → I such that Vj ⊂ Uβ(j) forevery j ∈ J . The covering U determines a space T (U) := qi∈IUi, the disjoint union of the opensubsets Ui, endowed with the inductive limit topology. A morphism β : V→ U is covering forthe topology of X if and only if the induced map T (V) → T (U) is surjective, and a family(Vi → U | i ∈ I) is covering if and only if Vi → U is covering for at least one i ∈ I . Noticethat all fibre products in X are representable.

5.1.5. For any object V of X , let CV be the full subcategory of X /V consisting of all coveringmorphisms U → V (notation of (1.1.2)). For any morphism V →W of X , we have a naturalfunctor CW → CV : U 7→ U ×W V, and the rule V 7→ CV is therefore a pseudo-functorX o → Cat from the opposite of X to the 2-category of categories. To any object U → V ofCV we attach a simplicial object U• of X , as follows ([4, Exp.V, §1.10]). For every n ∈ N, letUn := U×V · · · ×V U, the (n + 1)-th power of U; the face morphisms ∂i : Un → Un−1 (for alln ≥ 1 and i = 0, . . . , n) are the natural projections.


5.1.6. Let F be a presheaf on X ; the cosimplicial Cech complex associated to F and a cover-ing morphism U→ V, is defined as usual by setting :

Cn(U,F ) := F (Un)

with coface maps ∂i := F (∂i) : Cn−1(U,F )→ Cn(U,F ) (for all n ≥ 1 and i = 0, . . . , n). Itextends naturally to an augmented cosimplicial Cech complex :

C•aug(U,F ) := Cone(F (X)[0]→ C•(U,F )[−1]).

One defines a functor

FV : C oV → Set (U→ V) 7→ Equal(C0(U,F )

∂0//

∂1// C1(U,F )) .

Since CV is usually not a cofiltered category, the colimit over C oV is in general not well behaved

(e.g. not exact). However, suppose that U and U′ are two coverings of V, and ϕ, ψ : U→ U′ aretwo morphisms of coverings; it is well known that the induced maps of cosimplicial objects

C•(ϕ,F ), C•(ψ,F ) : C•(U′,F )→ C•(U,F )

are homotopic, hence they induce the same map FV(U′) → FV(U). Hence, the functor FV

factors through the filtered partially ordered set DV obtained as follows. We define a transitiverelation ≤ on the coverings of V, by the rule :

U ≤ U′ if and only if there exists a refinement U′ → U in CV.

Then DV is the quotient of Ob(CV) such that ≤ descends to an ordering on DV (in the senseof [16, Ch.III, §1, n.1]); one checks easily that the resulting partially ordered set (DV,≤) isfiltered. One can then introduce the functor

HomCat(Xo,Set)→ HomCat(X

o,Set) F 7→ F +

on the category of presheaves on X , by the rule :

(5.1.7) F +(V) := colimU∈DV

FV(U).

It is well known that F + is a separated presheaf, and if F is separated, then F + is a sheaf.Hence F ++ is a sheaf for every presheaf F , and it easy to verify that the functor F 7→ F ++

is left adjoint to the inclusion functor

(5.1.8) ι : X ∼ → HomCat(Xo,Set)

from the category of sheaves on X to the category of presheaves.

5.1.9. Notice that every presheaf F on X extends canonically to a presheaf on X ; namely forevery covering U := (Ui | i ∈ I) one lets F ′(U) :=

∏i∈I F (Ui); moreover, the functor F 7→

F ′ commutes with the sheafification functors F 7→ F ++ (on X and on X ). Furthermore, onehas a natural identification : Γ(X ,F ′) = Γ(X,F ).

The following lemma generalizes [39, Th.3.10.1].

Lemma 5.1.10. Let X be a topological space fulfilling conditions (a) and (b) of lemma 5.1.2.Then the following holds :

(i) For every filtered system F := (Fλ | λ ∈ Λ) of sheaves on X , and every quasi-compact subset U ⊂ X , the natural map :

colimλ∈Λ

Γ(U,Fλ)→ Γ(U, colimλ∈Λ

Fλ)

is bijective.


(ii) If moreover, F is a system of abelian sheaves, then the natural morphisms

colimλ∈Λ

RiΓ(U,Fλ)→ RiΓ(U, colimλ∈Λ

Fλ)

are isomorphisms for every i ∈ N and every quasi-compact open subset U ⊂ X .

Proof. (i): In view of (5.1.9), we regard F as a system of sheaves on X .We shall say that a covering U := (Ui | i ∈ I) is quasi-compact, if it consists of finitely many

quasi-compact open subsets Ui. Let P := colimλ∈Λ

ι(Fλ). Clearly colimλ∈Λ

Fλ = P++.

Claim 5.1.11. For every quasi-compact covering V, the natural map

P(V)→P+(V)

is a bijection.

Proof of the claim. Let U→ V be any covering morphism. If V is quasi-compact, condition (b)of lemma 5.1.2 implies that we can find a further refinement W → U with W quasi-compact,hence in (5.1.7) we may replace the partially ordered set DV by its full cofinal subset consistingof the equivalence classes of the quasi-compact coverings. Let U→ V be such a quasi-compactrefinement; say that U = (Ui | i = 1, . . . , n). Since filtered colimits are exact, we deduce:

Cn(U,P) = colimλ∈Λ

Cn(U,Fλ) for every n ∈ N.

Finally, since equalizers are finite limits, they commute as well with filtered colimits, so theclaim follows easily. ♦

In view of claim 5.1.11, assertion (i) is equivalent to :

Claim 5.1.12. The natural mapP+(V)→P++(V)

is a bijection for every quasi-compact covering V.

Proof of the claim. We compute P++(V) by means of (5.1.7), applied to the presheaf F :=P+; arguing as in the proof of claim 5.1.11 we may restrict the colimit to the cofinal subsetof DV consisting of the equivalence classes of all quasi-compact objects. Let U → V be sucha quasi-compact covering; since X is quasi-separated, also U ×V U is quasi-compact, henceP+(U) = P(U) and P+(U ×V U) = P(U ×V U), again by claim 5.1.11. And again, theclaim follows easily from the fact that equalizers commute with filtered colimits. ♦

(ii): Suppose that F is a system of abelian sheaves on X . For every λ ∈ Λ we choose aninjective resolution Fλ → I •

λ . This choice can be made functorially, so that the filtered systemF extends to a filtered system (Fλ → I •

λ | λ ∈ Λ) of complexes of abelian sheaves. Clearlycolimλ∈Λ

I •λ is a resolution of colim

λ∈ΛFλ. Moreover :

Claim 5.1.13. colimλ∈Λ

I •λ is a complex of qc-flabby sheaves.

Proof of the claim. Indeed, in view of (i) we have

Γ(U, colimλ∈Λ

I nλ ) = colim

λ∈ΛΓ(U,I n

λ )

for every n ∈ N and every quasi-compact open subset U ⊂ X . Then the claim follows from thewell known fact that injective sheaves are flabby (hence qc-flabby). ♦

One sees easily that if U is quasi-compact, the functor Γ(U,−) and the class M of qc-flabbysheaves fulfill conditions (a) and (b) of lemma 5.1.3, and (c) holds as well, in light of lemma5.1.2; therefore qc-flabby sheaves are acyclic for Γ(U,−) and assertion (ii) follows easily fromclaim 5.1.13.


5.1.14. Let Λ be a small cofiltered category with final object 0 ∈ Ob(Λ), and X0 a quasi-separated scheme; we consider the datum consisting of:

• Two cofiltered systems X := (Xλ | λ ∈ Λ) and Y := (Yλ | λ ∈ Λ) of X0-schemes,with affine transition morphisms :

ϕu : Xµ → Xλ ψu : Yµ → Yλ for every morphism u : µ→ λ in Λ.

• A cofiltered system of sheaves F := (Fλ | λ ∈ Λ) where Fλ is a sheaf on Yλ for everyλ ∈ Ob(Λ), with transition maps :

ψ−1u Fλ → Fµ for every morphism u : µ→ λ in Λ.

• A system g := (gλ : Yλ → Xλ | λ ∈ Λ) of quasi-compact and quasi-separated mor-phisms, such that gλ ψu = ϕu gµ for every morphism u : µ→ λ.

By [32, Ch.IV, Prop.8.2.3] the (inverse) limits of the systems X , Y and g are representable byX0-schemes and morphisms

X∞ := limλ∈Λ

Xλ Y∞ := limλ∈Λ

Yλ g∞ := limλ∈Λ

gλ : Y∞ → X∞

and the natural morphisms

ϕλ : X∞ → Xλ ψλ : Y∞ → Yλ.

are affine for every λ ∈ Λ. Moreover, F induces a sheaf on Y∞ :

F∞ := colimλ∈Λ

ψ−1λ Fλ.

Proposition 5.1.15. In the situation of (5.1.14), the following holds :(i) Suppose that X0 is quasi-compact. Then the natural map

colimλ∈Λ

Γ(Yλ,Fλ)→ Γ(Y∞,F∞)

is a bijection.(ii) If F is a system of abelian sheaves, then the natural morphisms :

colimλ∈Λ

ϕ−1λ Rigλ∗Fλ → Rig∞∗F∞

are isomorphisms for every i ∈ N.

Proof. (i): Arguing as in the proof of lemma 5.1.10(i), we regard each Fλ as a sheaf on thesite Tλ attached to Yλ by (5.1.4); similarly F∞ is a sheaf on the site T∞ obtained from Y∞.The assumption implies that Y∞, as well as Yλ for every λ ∈ Λ, are quasi-compact and quasi-separated. Then lemma 5.1.10(i) shows that the natural map :

colimλ∈Λ

Γ(Y∞, ψ−1λ Fλ)→ Γ(Y∞,F∞)

is a bijection. Let us set Gλ := ψ−1λ ι(Fλ) (notation of (5.1.8)). Quite generally, the presheaf

pull-back of a separated presheaf – under a map of topological spaces – is separated; hence Gλis a separated presheaf, so the global sections of its sheafification can be computed via the Cechcomplex : in the notation of (5.1.6), we have

Γ(Y∞,F∞) = colimλ∈Λ

colimU∞∈DY∞

Equal(C0(U∞,Gλ)// // C1(U∞,Gλ))

= colimU∞∈DY∞

colimλ∈Λ

Equal(C0(U∞,Gλ)//// C1(U∞,Gλ)) .

However, arguing as in the proof of claim 5.1.11, we see that we may replace the filtered setDY∞ by its cofinal system of quasi-compact coverings. If U∞ → Y∞ is such a quasi-compactcovering, we can find λ ∈ Ob(Λ), and a quasi-compact covering Uλ → Uλ such that Uλ ⊂Yλ is a quasi-compact open subset, and U∞ = Uλ ×Yλ Y∞ ([32, Ch.IV, Cor.8.2.11]). Up to


replacing λ by some λ′ ≥ λ, we can achieve that Uλ = Yλ, and moreover if Uµ → Yµ is anyother covering with the same property, then we can find ρ ∈ Λ with ρ ≥ µ, λ and such thatUλ ×Yλ Yρ = Uµ ×Yµ Yρ ([32, Ch.IV, Cor.8.3.5]). Likewise, if V → Yλ is a covering such thatV×Yλ Y∞ = U∞ ×Y∞ U∞, then, up to replacing λ by a larger index, we have V = Uλ ×Yλ Uλ.Hence, let us fix any such covering Uλ0 → Yλ0 and set Uµ := Uλ0 ×Yλ0

Yµ for every µ ≥ λ0; itfollows easily that

colimλ∈Λ

Equal(C0(U∞,Gλ)// // C1(U∞,Gλ)) = colim

µ≥λ0

Equal(C0(Uµ,Fµ) //// C1(Uµ,Fµ))

= colimµ≥λ0

Γ(Yµ,Fµ).

whence (i). To show (ii), we choose for each λ ∈ Λ an injective resolution Fλ → I •λ , having

care to construct I •λ functorially, so that F extends to a compatible system of complexes

(Fλ → I •λ | λ ∈ Λ).

Claim 5.1.16. I •∞ := colim

λ∈Λψ−1λ I •

λ is a complex of qc-flabby sheaves.

Proof of the claim. Indeed, let U∞ ⊂ V∞ be any two quasi-compact open subset of Y∞; weneed to show that the restriction map

Γ(V∞,I•∞)→ Γ(U∞,I

•∞)

is onto. By [32, Ch.IV, Cor.8.2.11] we can find λ ∈ Λ and quasi-compact open subsets Uλ ⊂Vλ ⊂ Yλ such that U∞ = ψ−1

λ Uλ, and likewise for V∞. Let us set Uu := ψ−1u Uλ for every

u ∈ Ob(Λ/λ), and likewise define Vu. Then, up to replacing Λ by Λ/λ, we may assume thatUµ ⊂ Vµ are defined for every µ ∈ Λ. By (i) the natural map

colimλ∈Λ

Γ(Uλ,Inλ )→ Γ(U∞,I

n∞)

is bijective for every n ∈ N, and likewise for V∞. Since an injective sheaf is flabby, the claimfollows easily. ♦

Using the assumptions and lemma 5.1.2, one checks easily that the functor g∞∗ and the classM of qc-flabby sheaves on Y∞ fulfill conditions (a)–(c) of lemma 5.1.3. Therefore we get anatural isomorphism in D(ZX∞-Mod) :

g∞∗I•∞∼→ Rg∞∗F∞.

To conclude, let V∞ → X∞ be any quasi-compact open immersion, which as usual we see asthe limit of a system (Vλ → Xλ | λ ∈ Λ) of quasi-compact open immersions; it then suffices toshow that the natural map

colimλ∈Λ

Γ(Vλ,I•λ )→ Γ(V∞,I

•∞)

is an isomorphism of complexes. The latter assertion holds by (i).

Corollary 5.1.17. Let f : Y → X be a quasi-compact and quasi-separated morphism ofschemes, F a flat quasi-coherent OX-module, and G any OY -module. Then the natural map

(5.1.18) F ⊗OX Rf∗G → Rf∗(f∗F ⊗OY G )

is an isomorphism.

Proof. The assertion is local on X , hence we may assume that X = SpecA for some ring A,and F = M∼ for some flat A-module M . By [57, Ch.I, Th.1.2], M is the colimit of a filteredfamily of free A-modules of finite rank; in view of proposition 5.1.15(ii), we may then assumethat M = A⊕n for some n ≥ 0, in which case the assertion is obvious.


Corollary 5.1.19. Consider a cartesian diagram of schemes

Y ′g′ //

f ′

Y

f

X ′

g // X

such that f is quasi-compact and quasi-separated, and g is flat. Then the natural map

(5.1.20) g∗Rf∗G → Rf ′∗g′∗G

is an isomorphism in D(OX′-Mod), for every OY -module G .

Proof. We easily reduce to the case where both X and X ′ are affine, hence g is an affine mor-phism. In this case, it suffices to show that g∗(5.1.20) is an isomorphism. However, we have anessentially commutative diagram

g∗(g∗Rf∗G )

α //

Rf∗G ⊗OX g∗OX′

g∗(Rf

′∗g′∗G )

∼ // Rf∗g′∗(g′∗G )

β // Rf∗(G ⊗OY g′∗OY ′)

whose left vertical arrow is g∗(5.1.20) and whose right vertical arrow is the natural isomorphismprovided by corollary 5.1.17 (applied to the flat OY -module F := g∗OY ′). Also, α and β are thenatural maps obtained as in [26, Ch.0, (5.4.10)]; it is easily seen that these are isomorphisms,for any affine morphism g and g′. The assertion follows.

Remark 5.1.21. Notice that, for an affine morphism f : Y → X , the map (5.1.18) is anisomorphism for any quasi-coherent OX-module F and any quasi-coherent OY -module G . Thedetails shall be left to the reader.

5.1.22. Let (X,A ) be a ringed space; one defines as in example 4.1.8(v) a total Hom cochaincomplex, which is a functor :

Hom•A : C(A -Mod)o × C(A -Mod)→ C(A -Mod)

on the category of complexes of A -modules. Recall that, for any two complexes M•, N•, andany object U of X , the group of n-cocycles in Hom•A (M•, N

•)(U) is naturally isomorphicto HomC(A|U -Mod)(M•|U , N

•|U [n]), and HnHom•A (M•, N

•)(U) is naturally isomorphic to thegroup of homotopy classes of maps M•|U → N•|U [n] (see example 4.1.7(i)). We also set :

(5.1.23) Hom•A := Γ Hom•A : C(A -Mod)o × C(A -Mod)→ C(Γ(A )-Mod).

The bifunctor Hom•A admits a right derived functor :

RHom•A : D(A -Mod)o × D+(A -Mod)→ D(A -Mod)

for whose construction we refer to [75, §10.7]. Likewise, one has a derived functor RHom•Afor (5.1.23), and there are natural isomorphisms of A -modules :

(5.1.24) H iRHom•A (M•, N•)∼→ HomD+(A -Mod)(M

•, N•[i])

for every i ∈ Z, and every bounded below complexes M• and N• of A -modules.

Lemma 5.1.25. Let (X,A ) be a ringed space, B an A -algebra. Then :(i) If I is an injective A -module, HomA (F ,I ) is flabby for every A -module F , and

HomA (B,I ) is an injective B-module.(ii) There is a natural isomorphism of bifunctors :

RΓ RHom•A∼→ RHom•A .


(iii) The forgetful functor D+(B-Mod)→ D+(A -Mod) admits the right adjoint :

D+(A -Mod)→ D+(B-Mod) : K• 7→ RHom•A (B, K•).

Proof. (i) : Let j : U ⊂ X be any open immersion; a section s ∈ HomA (F ,I )(U) is a mapof A|U -modules s : F|U → I|U . We deduce a map of A -modules j!s : j!F|U → j!I|U → I ;since I is injective, j!s extends to a map F → I , as required. Next, let B be an A -algebra;we recall the following :

Claim 5.1.26. The functor

A -Mod→ B-Mod : F 7→HomA (B,F )

is right adjoint to the forgetful functor.

Proof of the claim. We have to exhibit a natural bijection

HomA (N,M)∼→ HomB(N,HomA (B,M))

for every A -module M and B-module N . This is given by the following rule. To an A -linear map t : N → M one assigns the B-linear map t′ : N → HomA (B,M) such thatt′(x)(b) = t(x · b) for every x ∈ N(U) and b ∈ B(V ), where V ⊂ U are any two open subsetsof X . To describe the inverse of this transformation, it suffices to remark that t(x) = t′(x)(1)for every local section x of N . ♦

Since the forgetful functor is exact, the second assertion of (i) follows immediately fromclaim 5.1.26. Assertion (ii) follows from (i) and [75, Th.10.8.2].

(iii): Let K• (resp. L•) be a bounded below complex of B-modules (resp. of injectiveA -modules); then :

HomD+(A -Mod)(K•, L•) =H0Hom•C(A -Mod)(K

•, L•)

=H0Hom•C(B-Mod)(K•,HomA (B, L•))

= HomD+(B-Mod)(K•,HomA (B, L•))

where the last identity follows from (i) and (5.1.24).

Theorem 5.1.27. (Trivial duality) Let f : (Y,B)→ (X,A ) be a morphism of ringed spaces.For any two complexes M• in D−(A -Mod) and N• in D+(B-Mod), there are natural iso-morphisms :

(i) RHom•A (M•, Rf∗N•)∼→ RHom•B(Lf ∗M•, N

•) in D+(A (X)-Mod).(ii) RHom•A (M•, Rf∗N

•)∼→ Rf∗RHom•B(Lf ∗M•, N

•) in D+(A -Mod).

Proof. One applies lemma 5.1.25(ii) to deduce (i) from (ii).(ii) : By standard adjunctions, for any two complexes M• and N• as in the proposition, we

have a natural isomorphism of total Hom cochain complexes :

(5.1.28) Hom•A (M•, f∗N•)∼→ f∗Hom•B(f ∗M•, N

•).

Now, let us choose a flat resolution P•∼→M• of A -modules, and an injective resolution N• ∼→

I• of B-modules. In view of (5.1.28) and lemma 5.1.25(i) we have natural isomorphisms :

Rf∗RHom•B(Lf ∗M•, N•)∼→ f∗Hom•B(f ∗P•, I

•)∼←Hom•A (P•, f∗I

•)

in D+(A -Mod). It remains to show that the natural map

(5.1.29) Hom•A (P•, f∗I•)→ RHom•A (P•, f∗I

•)

is an isomorphism. However, we have two spectral sequences :

E1pq := HpHom•A (Pq, f∗I

•)⇒Hp+qHom•A (P•, f∗I•)

F 1pq := HpRHom•A (Pq, f∗I

•)⇒Hp+qRHom•A (P•, f∗I•)


and (5.1.29) induces a natural map of spectral sequences :

(5.1.30) E1pq → F 1

pq

Consequently, it suffices to show that (5.1.30) is an isomorphism for every p, q ∈ N, and there-fore we may assume that P• consists of a single flat A -module placed in degree zero. A similarargument reduces to the case where I• consists of a single injective B-module sitting in degreezero. To conclude, it suffices to show :

Claim 5.1.31. Let P be a flat A -module, I an injective B-module. Then the natural map :

HomA (P, f∗I)[0]→ RHom•A (P, f∗I)

is an isomorphism.

Proof of the claim. RiHom•A (P, f∗I) is the sheaf associated to the presheaf on X :

U 7→ RiHom•A|U (P|U , f∗I|U).

Since I|U is an injective B|U -module, it suffices therefore to show that RiHom•A (P, f∗I) = 0for i > 0. However, recall that there is a natural isomorphism :

RiHom•A (P, f∗I) ' HomD(A -Mod)(P, f∗I[−i]).

Since the homotopy category Hot(A -Mod) admits a left calculus of fractions, we deduce anatural isomorphism :

(5.1.32) RiHom•A (P, f∗I) ' colimE•→P

HomHot(A -Mod)(E•, f∗I[−i])

where the colimit ranges over the family of quasi-isomorphisms E• → P . We may furthermorerestrict the colimit in (5.1.32) to the subfamily of all suchE• → P whereE• is a bounded abovecomplex of flat A -modules, since this subfamily is cofinal. Every such map E• → P induces acommutative diagram :

HomA (P, f∗I[−i]) //

HomHot(A -Mod)(E•, f∗I[−i])

HomB(f ∗P, I[−i]) // HomHot(B-Mod)(f

∗E•, I[−i])

whose vertical arrows are isomorphisms. Since E• and P are complexes of flat A -modules,the induced map f ∗E• → f ∗P is again a quasi-isomorphism; therefore, since I is injective, thebottom horizontal arrow is an isomorphism, hence the same holds for the top horizontal one,and the claim follows easily.

Corollary 5.1.33. Let A be a ring, M• (resp. N•) a bounded below (resp. above) complexof A-modules. Set X := SpecA and denote by M•∼ (resp. N∼• ) the associated complex ofquasi-coherent OX-modules. Then the natural map :

RHom•A(N•,M•)→ RHom•OX (N∼• ,M

•∼)

is an isomorphism in D+(A-Mod).

Proof. We apply the trivial duality theorem 5.1.27 to the unique morphism

f : (X,OX)→ (pt, A)

of ringed spaces, where pt denotes the one-point space. Since f is flat and all quasi-coherentOX-modules are f∗-acyclic, the assertion follows easily.


5.1.34. Suppose that F is an abelian group object in a given topos X , and let (FilnF | n ∈ N)be a descending filtration by abelian subobjects of F , such that Fil0F = F . Set F n :=F/Filn+1F for every n ∈ N, and denote by gr•F the graded abelian object associated to thefiltered object Fil•F ; then we have :

Lemma 5.1.35. In the situation of (5.1.34), suppose furthermore that the natural map :

F → R limk∈N

F k

is an isomorphism in D(ZX-Mod). Then, for every q ∈ N the following conditions are equiva-lent :

(a) Hq(X,FilnF ) = 0 for every n ∈ N.(b) The natural map Hq(X,FilnF )→ Hq(X, grnF ) vanishes for every n ∈ N.

Proof. Obviously we have only to show that (b) implies (a). Moreover, let us set :

(FilnF )k := FilnF/Filk+1F for every n, k ∈ N with k ≥ n.

There follows, for every n ∈ N, a short exact sequence of inverse systems of sheaves :

0→ ((FilnF )k | k ≥ n)→ (F k | k ≥ n)→ F n−1 → 0

(where the right-most term is the constant inverse system which equals F n−1 in all degrees,with transition maps given by the identity morphisms); whence a distinguished triangle :

R limk∈N

(FilnF )k → R limk∈N

F k → F n−1[0]→ R limk∈N

(FilnF )k[1]

which, together with our assumption on F , easily implies that the natural map FilnF →R limk∈N(FilnF )k is an isomorphism in D(ZX-Mod). Summing up, we may replace the datum(F ,Fil•F ) by (FilnF ,Fil•+nF ), and reduce to the case where n = 0.

The inverse system (F n | n ∈ N) defines an abelian group object of the topos XN (see [36,§7.3.4]); whence a spectral sequence:

(5.1.36) Epq2 := lim

n∈NpHq(X,F n)⇒ Hp+q(XN,F •) ' Hp+q(X,R lim

n∈NF n).

By [75, Cor.3.5.4] we have Epq2 = 0 whenever p > 1, and, in view of our assumptions, (5.1.36)

decomposes into short exact sequences :

0→ limn∈N

1Hq−1(X,F n)αq−→ Hq(X,F )

βq−→ limn∈N

Hq(X,F n)→ 0 for every q ∈ N

where βq is induced by the natural system of maps (βq,n : Hq(X,F )→ Hq(X,F n) | n ∈ N).

Claim 5.1.37. If (b) holds, then βq = 0.

Proof of the claim. Suppose that (b) holds; we consider the long exact cohomology sequenceassociated to the short exact sequence :

0→ Filn+1F → FilnF → grnF → 0

to deduce that the natural map Hq(X,Filn+1F ) → Hq(X,FilnF ) is onto for every n ∈ N;hence the same holds for the natural map Hq(X,FilnF ) → Hq(X,F ). By considering thelong exact cohomology sequence attached to the short exact sequence :

(5.1.38) 0→ FilnF → F → F n → 0

we then deduce that βq,n vanishes for every n ∈ N, which implies the claim. ♦

Claim 5.1.39. If (b) holds, then the inverse system (Hq−1(X,F n) | n ∈ N) has surjectivetransition maps.


Proof of the claim. By considering the long exact cohomology sequence associated to the shortexact sequence : 0 → grnF → F n+1 → F n → 0, we are reduced to showing that theboundary map ∂q−1 : Hq−1(X,F n) → Hq(X, grnF ) vanishes for every n ∈ N. However,comparing with (5.1.38), we find that ∂q−1 factors through the natural map Hq(X,FilnF ) →Hq(X, grnF ), which vanishes if (b) holds. ♦

The lemma follows from claims 5.1.37 and 5.1.39, and [75, Lemma 3.5.3].

5.2. Quasi-coherent modules. Let X be any scheme, and F any abelian sheaf on X . Thesupport of F is the subset :

Supp F := x ∈ X |Fx 6= 0 ⊂ X.

Recall that an OX-module F is said to be f -flat at a point x ∈ X if Fx is a flat OY,f(x)-module.F is said to be f -flat over a point y ∈ Y if F is f -flat at all points of f−1(y). Finally, one saysthat F is f -flat if F is f -flat at all the points of X ([26, Ch.0, §6.7.1]).

5.2.1. We denote by OX-Mod (resp. OX-Modqcoh, OX-Modcoh, resp. OX-Modlfft) thecategory of all (resp. of quasi-coherent, resp. of coherent, resp. of locally free of finitetype) OX-modules. Also, we denote by D(OX-Mod)qcoh (resp. D(OX-Mod)coh) the fulltriangulated subcategory of D(OX-Mod) consisting of the complexes K• such that H iK•

is a quasi-coherent (resp. coherent) OX-module for every i ∈ Z. As usual, we shall usealso the variants D+(OX-Mod)qcoh (resp. D−(OX-Mod)qcoh, resp. Db(OX-Mod)qcoh, resp.D[a,b](OX-Mod)qcoh) consisting of all objects of D(OX-Mod)qcoh whose cohomology vanishesin sufficiently large negative degree (resp. sufficiently large positive degree, resp. outside abounded interval, resp. outside the interval [a, b]), and likewise for the corresponding subcate-gories of D(OX-Mod)coh.

5.2.2. There are obvious forgetful functors:

ιX : OX-Modqcoh → OX-Mod RιX : D(OX-Modqcoh)→ D(OX-Mod)qcoh

and we wish to exhibit right adjoints to these functors. To this aim, suppose first thatX is affine;then we may consider the functor:

qcohX : OX-Mod→ OX-Modqcoh F 7→ F (X)∼

where, for an OX(X)-module M , we have denoted as usual by M∼ the quasi-coherent sheafarising from M . If G is a quasi-coherent OX-module, then clearly qcohXG ' G ; moreover, forany other OX-module F , there is a natural bijection:

HomOX (G ,F )∼→ HomOX(X)(G (X),F (X)).

It follows easily that qcohX is the sought right adjoint.

5.2.3. Slightly more generally, let U be quasi-affine, i.e. a quasi-compact open subset of anaffine scheme, and choose a quasi-compact open immersion j : U → X into an affine schemeX . In this case, we may define

qcohU : OU -Mod→ OX-Mod F 7→ (qcohXj∗F )|U .

Since j∗ : OU -Mod → OX-Mod is right adjoint to j∗ : OX-Mod → OU -Mod, we havea natural isomorphism: HomOU (G ,F )

∼→ HomOX (j∗G , j∗F ) for every OU -modules G andF . Moreover, if G is quasi-coherent, the same holds for j∗G ([26, Ch.I, Cor.9.2.2]), whence anatural isomorphism G ' qcohUG , by the foregoing discussion for the affine case. Summingup, this shows that qcohU is a right adjoint to ιU , and especially it is independent, up to uniqueisomorphism, of the choice of j.


5.2.4. Next, suppose that X is quasi-compact and quasi-separated. We choose a finite coveringU := (Ui | i ∈ I) of X , consisting of affine open subsets, and set U :=

∐i∈I Ui, the X-scheme

which is the disjoint union (i.e. categorical coproduct) of the schemes Ui. We denote by U• thesimplicial covering such that Un := U×X · · ·×XU , the (n+1)-th power of U , with the face anddegeneracy maps defined as in (5.1.5); let also πn : Un → X be the natural morphism, for everyn ∈ N. Clearly we have πn−1∂i = πn for every face morphism ∂i : Un → Un−1. The simplicialscheme U• (with the Zariski topology on each scheme Un) can also be regarded as a fibred toposover the category ∆o (notation of [36, §2.2]); then the datum OU• := (OUn | n ∈ N) consistingof the structure sheaves on each Un and the natural maps ∂∗i OUn−1 → OUn for every n > 0 andevery i = 0, . . . , n (and similarly for the degeneracy maps), defines a ring in the associatedtopos Top(U•) (see [36, §3.3.15]). We denote by OU•-Mod the category of OU•-modules in thetopos Top(U•). The family (πn | n ∈ N) induces a morphism of topoi

π• : Top(U•)→ s.X

where s.X is the topos Top(X•) associated to the constant simplicial scheme X• (with itsZariski topology) such that Xn := X for every n ∈ N and such that all the face and degeneracymaps are 1X . Clearly the objects of s.X are nothing else than the cosimplicial Zariski sheaveson X . Especially, if we view a OX-module F as a constant cosimplicial OX-module, we maydefine the augmented cosimplicial Cech OX-module

(5.2.5) F → C •(U,F ) := π•∗ π∗•F

associated to F and the covering U; in every degree n ∈ N this is defined by the rule :

C n(U,F ) := πn∗π∗nF

and the coface operators ∂i are induced by the face morphisms ∂i in the obvious way.

Lemma 5.2.6. (i) The augmented complex (5.2.5) is aspherical for every OX-module F .(ii) If F is an injective OX-module, then (5.2.5) is homotopically trivial.

Proof. (See also e.g. [39, Th.5.2.1].) The proof relies on the following alternative descriptionof the cosimplicial Cech OX-module. Consider the adjoint pair :

(π∗, π∗) : OU-Mod // OX-Modoo

arising from our covering π : U → X . Let (> := π∗ π∗, η, µ) be the associated triple (see(4.5.3)), F any OX-module; we leave to the reader the verification that the resulting augmentedcosimplicial complex F → >•F – as defined in (4.5.2) – is none else than the augmented Cechcomplex (5.2.5). Thus, for every OX-module F , the augmented complex π∗F → π∗C •(U,F )is homotopically trivial (proposition 4.5.4); since π is a covering morphism, (i) follows. Fur-thermore, by the same token, the augmented complex π∗G → C •(U, π∗G ) is homotopicallytrivial for every OU-module G ; especially we may take G := π∗I , where I is an injectiveOX-module. On the other hand, when I is injective, the unit of adjunction I → >I is splitinjective; hence the augmented complex I → C •(U,I ) is a direct summand of the homotopi-cally trivial complex >I → C •(U,>I ), and (ii) follows.

Notice now that the schemes Un are quasi-affine for every n ∈ N, hence the functors qcohUn

are well defined as in (5.2.3), and indeed, the rule : (Fn | n ∈ N) 7→ (qcohUnFn | n ∈ N)yields a functor :

qcohU• : OU•-Mod→ OU•-Mod.

This suggests to introduce a quasi-coherent cosimplicial Cech OX-module :

qC •(U,F ) := π•∗ qcohU• π∗•F


for every OX-module F (regarded as a constant cosimplicial module in the usual way). Moreplainly, this is the cosimplicial OX-module such that :

qC n(U,F ) := πn∗ qcohUn π∗nF for every n ∈ N.

According to [26, Ch.I, Cor.9.2.2], the OX-modules qC n(U,F ) are quasi-coherent for all n ∈N. Finally, we define the functor :

qcohX : OX-Mod→ OX-Modqcoh F 7→ Equal(qC 0(U,F )∂0//

∂1// qC 1(U,F ))

Proposition 5.2.7. (i) In the situation of (5.2.4), the functor qcohX is right adjoint to ιX .(ii) Let Y be any other quasi-compact and quasi-separated scheme, f : X → Y any

morphism. Then the induced diagram of functors:

OX-Modf∗ //

qcohX

OY -Mod

qcohY

OX-Modqcohf∗ // OY -Modqcoh

commutes up to a natural isomorphism of functors.

Proof. For every n ∈ N and every OUn-module H , the counit of the adjunction yields a naturalmap of OUn-modules: qcohUnH → H . Taking H to be π∗nF on Un (for a given OX-moduleF ), these maps assemble to a morphism of cosimplicial OX-modules :

(5.2.8) qC •(U,F )→ C •(U,F )

and it is clear that (5.2.8) is an isomorphism whenever F is quasi-coherent. Let now ϕ : G →F be a map of OX-modules, with G quasi-coherent; after applying the natural transformation(5.2.8) and forming equalizers, we obtain a commutative diagram :

qcohXG∼ //

qcohXϕ

G

ϕ

qcohXF // F

from which we see that the rule: ϕ 7→ qcohXϕ establishes a natural injection:

(5.2.9) HomOX (G ,F )→ HomOX (G , qcohXF )

and since HomOX (G , (5.2.8)) is an isomorphism of cosimplicial OX-modules, (5.2.9) is actuallya bijection, whence (i).

(ii) is obvious, since both qcohY f∗ and f∗ qcohX are right adjoint to f ∗ ιY = ιX f ∗.

5.2.10. In the situation of (5.2.4), the functor qcohX is left exact, since it is a right adjoint,hence it gives rise to a left derived functor :

RqcohX : D+(OX-Mod)→ D+(OX-Modqcoh).

Proposition 5.2.11. Let X be a quasi-compact and quasi-separated scheme. Then :(i) RqcohX is right adjoint to RιX .

(ii) Suppose moreover, that X is semi-separated, i.e. such that the intersection of any twoaffine open subsets of X , is still affine. Then the unit of the adjunction (RιX , RqcohX)is an isomorphism of functors.


Proof. (i): The exactness of the functor ιX implies that qcohX preserves injectives; the assertionis a formal consequence : the details shall be left to the reader.

(ii): Let F • be any complex of quasi-coherent OX-modules; we have to show that the naturalmap F • → RqcohXF • is an isomorphism. Using a Cartan-Eilenberg resolution F • ∼→ I ••

we deduce a spectral sequence ([75, §5.7])

Epq1 := RpqcohXH

qF • ⇒ Rp+qqcohXF •

which easily reduces to checking the assertion for the cohomology of F •, so we may assumefrom start that F • is a single OX-module placed in degree zero. Let us then choose an injec-tive resolution F

∼→ I • (that is, in the category of all OX-modules); we have to show thatHpqcohXI • = 0 for p > 0. We deal first with the following special case :

Claim 5.2.12. Assertion (i) holds if X is affine.

Proof of the claim. Indeed, in this case, the chosen injective resolution of F yields a longexact sequence 0 → F (X) → I •(X), and therefore a resolution qcohXF := F (X)∼ →qcohXI • := I •(X)∼. ♦

For the general case, we choose any affine covering U of X and we consider the cochaincomplex of cosimplicial complexes qC •(U,I •).

Claim 5.2.13. For any injective OX-module I , the augmented complex:

qcohXI → qC •(U,I )

is homotopically trivial.

Proof of the claim. It follows easily from proposition 5.2.7(ii) that

qC •(U,F ) ' qcohX(C •(U,F ))

for any OX-module F . Then the claim follows from lemma 5.2.6(ii). ♦

We have a spectral sequence :

Epq1 := HpqC •(U,I q)⇒ Totp+q(qC •(U,I •))

and it follows from claim 5.2.13 that Epq1 = 0 whenever p > 0, and E0q = qcohXI q for every

q ∈ N, so the spectral sequence E•• degenerates, and we deduce a quasi-isomorphism

(5.2.14) qcohXI • ∼→ Tot•(qC •(U,I •)).

On the other hand, for fixed q ∈ N, we have qC q(U,I •) = πn∗qcohUnπ∗nI

•; since X is semi-separated, Un is affine, so the complex qcohUnπ

∗nI

• is a resolution of qcohUnπ∗nF = π∗nF , by

claim 5.2.12. Furthermore, πn : Un → X is an affine morphism, so qC q(U,I •) is a resolutionof πn∗π∗nF . Summing up, we see that Tot•(qC •(U,I •)) is quasi-isomorphic to C •(U,F ),which is a resolution of F , by lemma 5.2.6(i). Combining with (5.2.14), we deduce (ii).

Theorem 5.2.15. Let X be a quasi-compact and semi-separated scheme. The forgetful functor:

RιX : D+(OX-Modqcoh)→ D+(OX-Mod)qcoh

is an equivalence of categories, whose quasi-inverse is the restriction of RqcohX .

Proof. By proposition 5.2.11(ii) we know already that the composition RqcohX RιX is aself-equivalence of D+(OX-Modqcoh). For every complex F • in D+(OX-Mod), the counitof adjunction εF• : RιX RqcohXF • → F • can be described as follows. Pick an injectiveresolution α : F • → I •; then εF• is defined by the diagram :

qcohXI • ε• // I • F •αoo


where, for each n ∈ N, the map εn : qcohXI n → I n is the counit of the adjoint pair(ιX , qcohX). It suffices then to show that ε• is a quasi-isomorphism, when I • is an object ofD+(OX-Mod)qcoh. To this aim, we may further choose I • of the form Tot•(I ••), where I ••

is a Cartan-Eilenberg resolution of F • (see [75, §5.7]). We then deduce a spectral sequence :

Epq2 := RpqcohXH

qF • ⇒ Rp+qqcohXF •.

The double complex I •• also gives rise to a similar spectral sequence F pq2 , and clearly F pq

2 = 0whenever p > 0, and F 0q = HqF •. Furthermore, the counit of adjunction ε•• : qcohXI •• →I •• induces a morphism of spectral sequences ωpq : Epq

2 → F pq2 . Consequently, in order

to prove that the εF• is a quasi-isomorphism, it suffices to show that ωpq is an isomorphismfor every p, q ∈ N. This comes down to the assertion that RpqcohXG = 0 for every quasi-coherent OX-module G , and every p > 0. However, we have G = RιXG , so that RqcohXG =RqcohX RιXG , and then the contention follows from proposition 5.2.11(ii).

Lemma 5.2.16. Let X be a quasi-compact and quasi-separated scheme, U a quasi-compactopen subset of X , H a quasi-coherent OX-module, G a finitely presented quasi-coherent OU -module, and ϕ : G →H|U a OU -linear map. Then:

(i) There exists a finitely presented quasi-coherent OX-module F on X , and a OX-linearmap ψ : F →H , such that F|U = G and ψ|U = ϕ.

(ii) Especially, every finitely presented quasi-coherent OU -module extends to a finitely pre-sented quasi-coherent OX-module.

Proof. (i): Let (Vi | i = 1, . . . , n) be a finite affine open covering of X . For every i = 0, . . . , n,let us set Ui := U ∪ V1 ∪ · · · ∪ Vi; we construct, by induction on i, a family of finitely presentedquasi-coherent OUi-modules Fi, and morphisms ψi : Fi → H|Ui such that Fi+1|Ui = Fi andψi+1|Ui+1

= ψi for every i < n. For i = 0 we have U0 = U , and we set F0 := G , ψ0 := ϕ.Suppose that Fi and ψi have already been given. Since X is quasi-separated, the same holdsfor Ui+1, and the immersion j : Ui → Ui+1 is quasi-compact; it follows that j∗Fi and j∗H|Uiare quasi-coherent OUi-modules ([26, Ch.I, Prop.9.4.2(i)]). We let

M := j∗Fi ×j∗H|Ui H|Ui+1.

Then M is a quasi-coherent OUi+1-module admitting a map M →H|Ui+1

, and such that M|Ui =Fi. We can then find a filtered family of quasi-coherent OVi+1

-modules of finite presentation(Mλ | λ ∈ Λ), whose colimit is M|Vi+1

. Since Fi is finitely presented and Ui ∩ Vi+1 is quasi-compact, there exists λ ∈ Λ such that the induced morphism β : Mλ|Ui∩Vi+1

→ Fi|Ui∩Vi+1is

an isomorphism. We can thus define Fi+1 by gluing Fi and Mλ along β; likewise, ψi andthe induced map Mλ → M|Vi+1

→ H|Vi+1glue to a map ψi+1 as required. Clearly the pair

(F := Fn, ψ := ψn) is the sought extension of (G , ϕ).(ii): Let 0X be the final object in the category of OX-modules; to extend a finitely presented

quasi-coherent OU -module G , it suffices to apply (i) to the unique map G → 0X|U .

For ease of reference, we point out the following simple consequence of lemma 5.2.16.

Corollary 5.2.17. Let U be a quasi-affine scheme, E a locally free OU -module of finite type.Then we may find integers n,m ∈ N and a left exact sequence of OU -modules :

0→ E → O⊕mU → O⊕nU .

Proof. Set E ∨ := HomOU (E ,OU), and notice that E ∨ is a locally free OU -module of finite type,especially it is quasi-coherent and of finite presentation. By assumption, U is a quasi-compactopen subset of an affine scheme X , hence E ∨ extends to a quasi-coherent OX-module F offinite presentation (lemma 5.2.16(ii)); we have F = F∼, for some finitely presented A-moduleF ; we choose a presentation of F as the cokernel of an A-linear map A⊕n → A⊕m, for some


m,n ∈ N, whence a presentation of E ∨ as the cokernel of a morphism O⊕nU → O⊕mU , and afterdualizing again, we get the sought left exact sequence.

5.2.18. Let X be a scheme; for any quasi-coherent OX-module F , we consider the full sub-category C/F of the category OX-Modqcoh/F whose objects are all the maps ϕ : G → Fsuch that G is a finitely presented OX-module (notation of (1.1.12)). One verifies easily thatthis category is filtered, and we have a source functor :

ιF : C/F → OX-Modqcoh (G → F ) 7→ G

Proposition 5.2.19. With the notation of (5.2.18), suppose that X is quasi-compact and quasi-separated. Then, for any quasi-coherent OX-module F , the induced map

colimC/F

ιF → F

is an isomorphism.

Proof. Let F ′ be such colimit; clearly there is a natural map F ′ → F , and we have to showthat it is an isomorphism. To this aim, we can check on the stalk over the points x ∈ X , hencewe come down to showing :

Claim 5.2.20. Let F be any quasi-coherent OX-module. Then :(i) For every s ∈ Fx there exists a finitely presented quasi-coherent OX-module G , a

morphism ϕ : G → F and t ∈ Gx such that ϕx(t) = s.(ii) For every map ϕ : G → F with G finitely presented and quasi-coherent, and every

s ∈ Kerϕx, there exists a commutative diagram of quasi-coherent OX-modules :

Gϕ

!!CCCCCCCC

ψ

H // F

with H finitely presented and ψx(s) = 0.

Proof of the claim. (i): Let U ⊂ X be an open subset such that s extends to a section sU ∈F (U); we deduce a map ϕU : OU → F|U by the rule a 7→ a · sU for every a ∈ OU(U). In viewof lemma 5.2.16(i), the pair (ϕU ,OU) extends to a pair (ϕ,G ) with the sought properties.

(ii): We apply (i) to the quasi-coherent OX-module Kerϕ, to find a finitely presented quasi-coherent OX-module G ′, a morphism β : G ′ → Kerϕ and t ∈ Hx such that βx(t) = s. Thenwe let ψ : G → H := Coker(G ′

β−→ Kerϕ → G ) be the natural map. By construction, H isfinitely presented, ψx(s) = 0 and clearly ϕ factors through ψ.

Corollary 5.2.21. Let X be a coherent, quasi-compact and quasi-separated scheme, U ⊂ X aquasi-compact open subset. Then the induced functor

D[a,b](OX-Mod)coh → D[a,b](OU -Mod)coh

is essentially surjective, for every a, b ∈ N.

Proof. To start out, for any quasi-coherent OX-module F , define the category C/F as in5.2.18, and consider the full subcategory C ′/F (resp. C ′′/F ) of C/F whose objects areall the maps ϕ : G → F such that ϕ|U is an isomorphism (resp. an epimorphism). It is easilyseen that both C ′/F and C ′′/F are filtered categories, and we have :

Claim 5.2.22. With the foregoing notation, the following holds :(i) If F|U is a coherent OU -module, C ′/F is a cofinal subcategory of C/F .


(ii) For every object K• of Db(OX-Mod)coh and every c ∈ Z, the natural map

colimC/F

HomD(OX -Mod)(K•, ιF [c])→ HomD(OX -Mod)(K

•,F [c])

is an isomorphism (notation of (5.2.18)).

Proof of the claim. (i): Since U is quasi-compact, it is easily seen that C ′′/F is cofinal in C/F ,so we are reduced to checking that C ′/F is cofinal in C ′′/F . Hence, let ψ : G ′ → F be anyobject of C ′′/F ; since OX is coherent, K := Kerψ|U is a coherent OU -module, thereforewe may extend K to a coherent OX-module K ′ and the identity map of K to a morphismψ′ : K ′ → Kerψ of OX-modules (lemma 5.2.16(i)). Set G := G ′/ψ′(K ′); the induced mapG → F is an object of C ′/F , whence the contention.

(ii): We have two spectral sequences :

Epq2 : colim

C/FRpΓ(X,RqHom•OX (K•, ιF [c]))⇒ colim

C/FHomD(OX -Mod)(K

•, ιF [c+ p+ q])

F pq2 : RpΓ(X,RqHom•OX (K•,F [c]))⇒ HomD(OX -Mod)(K

•,F [c+ p+ q])

and a morphism of spectral sequences Epq2 → F pq

2 . Since the functors RpΓ commute withfiltered colimits, we deduce that it suffices to show that the natural morphism

colimC/F

RqHom•OX (K•, ιF [c]))→ RqHom•OX (K•,F [c])

is an isomorphism for every q ∈ Z. Then, a standard devissage argument further reduces to thecase where K• is concentrated in a single degree, so we come down to checking that the functorF 7→ ExtqOX (G ,F ) commutes with filtered colimits of quasi-coherent OX-modules, for everycoherent OX-module G and every q ∈ Z. To this aim, we may assume that X is affine, in whichcase – since OX is coherent – G admits a resolution L • → G consisting of free OX-modules offinite rank; the latter are acyclic for the functor HomOX , so we are left with the assertion thatthe functor F 7→ HqHomOX (L •,F ) commutes with filtered colimits, which is clear. ♦

Now, let j : U → X be the open immersion, F • an object of D[a,b](OU -Mod)coh. By [28,Ch.III, Prop.1.4.10, Cor.1.4.12] and a standard spectral sequence argument, it is easily seen that

H • := Rj∗F•

lies in Db(OX-Mod)qcoh. We may then replace H a by Coker (da−1 : H a−1 → H a), H b byKer (db : H b →H b), H i by 0 for i /∈ [a, b], and assume that H • lies in D[a,b](OX-Mod)qcoh

and j∗H • = F •. It then suffices to show the following :

Claim 5.2.23. Let H • be any object of D[a,b](OX-Mod)qcoh such that H •|U lies in the category

D[a,b](OU -Mod)coh. Then there exists an object G • of D[a,b](OX-Mod)coh, with a morphismϕ• : G • →H • in D(OX-Mod)qcoh such that ϕ•|U is an isomorphism.

Proof of the claim. We proceed by descending induction on a, and notice that in case a = b,the assertion follows immediately from claim 5.2.22(i) and proposition 5.2.19.

Thus, suppose that a < b, and that the assertion is already known for every object ofD[a+1,b](OX-Mod)qcoh fulfilling the stated condition. Define the truncation τ≥a+1H • as theunique object (up to isomorphism) of D[a+1,b](OX-Mod)qcoh fitting in a distinguished triangleof D(OX-Mod)qcoh

HaH •[−a]→H • → τ≥a+1H• → HaH •[1− a].

By inductive assumption, we may find an object G ′• of D[a+1,b](OX-Mod)coh and a morphismϕ′• : G ′• → τ≥a+1H • in D(OX-Mod)qcoh which restricts to an isomorphism on U . There


follows a morphism of distinguished triangles :

HaH •[−a] // H ′• //

β•

G ′• //

ϕ′•

HaH •[1− a]

HaH •[−a] // H • // τ≥a+1H • α• // HaH •[1− a]

for some object H ′• of D[a,b](OX-Mod)qcoh, and clearly β• restricts to an isomorphism on U .We may then replace H • by H ′•, and assume that τ≥a+1H • lies in D[a+1,b](OX-Mod)coh.

By claim 5.2.22, we may find an object ψ : G → HaH • of C ′/F such that α• factorsthrough ψ[1 − a] and a morphism ω• : τ≥a+1H • → G [1 − a] in D(OX-Mod)coh. ThenConeω•[−1] lies in D[a,b](OX-Mod)coh, and the induced morphism Coneω•[−1] → H • re-stricts to an isomorphism on U .

5.3. Duality for quasi-coherent modules. Let f : X → Y be a morphism of schemes; the-orem 5.1.27 falls short of proving that Lf ∗ is a left adjoint to Rf∗, since the former functor isdefined on bounded above complexes, while the latter is defined on complexes that are boundedbelow. This deficency has been overcome by N.Spaltenstein’s paper [71], where he showshow to extend the usual constructions of derived functors to unbounded complexes. On theother hand, in many cases one can also construct a right adjoint to Rf∗. This is the subject ofGrothendieck’s duality theory. We shall collect the statements that we need from that theory,and refer to the original source [46] for the rather elaborate proofs.

5.3.1. For any morphism of schemes f : X → Y , let f : (X,OX) → (Y, f∗OX) be thecorresponding morphism of ringed spaces. The functor

f ∗ : OX-Mod→ f∗OX-Mod

admits a left adjoint f ∗ defined as usual ([26, Ch.0, §4.3.1]). In case f is affine, f is flat and theunit and counit of the adjunction restrict to isomorphisms of functors :

1→ f ∗ f ∗ : f∗OX-Modqcoh → f∗OX-Modqcoh

f ∗ f ∗ → 1 : OX-Modqcoh → OX-Modqcoh.

on the corresponding subcategories of quasi-coherent modules ([27, Ch.II, Prop.1.4.3]). It fol-lows easily that, on these subcategories, f ∗ is also a right adjoint to f ∗.

Lemma 5.3.2. Keep the notation of (5.3.1), and suppose that f is affine. Then the functor :

f ∗ : D+(f∗OX-Mod)qcoh → D+(OX-Mod)qcoh

is right adjoint to the functor Rf ∗ : D+(OX-Mod)qcoh → D+(f∗OX-Mod)qcoh. Moreover, theunit and counit of the resulting adjunction are isomorphisms of functors.

Proof. By trivial duality (theorem 5.1.27), f ∗ is left adjoint toRf ∗ on D+(OX-Mod); it sufficesto show that the unit (resp. and counit) of this latter adjunction are isomorphisms for everyobject K• of D+(OX-Mod)qcoh (resp. L• of D+(f∗OX-Mod)qcoh). Concerning the unit, wecan use a Cartan-Eilenberg injective resolution of f ∗K•, and a standard spectral sequence, toreduce to the case whereK• = F [0] for a quasi-coherent f∗OX-module F ; however, the naturalmap f ∗ f ∗F → Rf ∗ f ∗F is an isomorphism, so the assertion follows from (5.3.1).

Remark 5.3.3. (i) Even when both X and Y are affine, neither the unit nor the counit ofadjunction in (5.3.1) is an isomorphism on the categories of all modules. For a counterexampleconcerning the unit, consider the case of a finite injection of domains A → B, where A islocal and B is semi-local (but not local); let f : X := SpecB → SpecA be the correspondingmorphism, and denote by F the f∗OX-module supported on the closed point, with stalk equal


to B. Then one verifies easily that f ∗ f ∗F is a direct product of finitely many copies ofF , indexed by the closed points of X . Concerning the counit, keep the same morphism f , letU := X \x, where x is a closed point and set F := j!OU ; then it is clear that f ∗ f ∗F issupported on the complement of the closed fibre of f .

(ii) Similarly, one sees easily that f ∗ is not a right adjoint to f ∗ on the category of all f∗OX-modules.

(iii) Incidentally, the analogous functor f ∗et defined on the etale (rather than Zariski) site is aright adjoint to fet∗ : OX,et-Mod→ f∗OX,et-Mod, and on these sites the unit and counit of theadjunctions are isomorphisms for all modules.

5.3.4. For the construction of the right adjoint f ! to Rf∗, one considers first the case where ffactors as a composition of a finite morphism followed by a smooth one, in which case f ! admitsa corresponding decomposition. Namely :

• In case f is smooth, we shall consider the functor :

f ] : D(OY -Mod)→ D(OX-Mod) K• 7→ f ∗K• ⊗OX ΛnOX

Ω1X/Y [n]

where n is the locally constant relative dimension function of f .• In case f is finite, we shall consider the functor :

f [ : D+(OY -Mod)→ D+(OX-Mod) K• 7→ f ∗RHom•OY (f∗OX , K•).

If f is any quasi-projective morphism, such factorization can always be found locally onX , andone is left with the problem of patching a family of locally defined functors ((f|Ui)

! | i ∈ I),corresponding to an open covering X =

⋃i∈I Ui. Since such patching must be carried out in

the derived category, one has to take care of many cumbersome complications.

5.3.5. Recall that a finite morphism f : X → Y is said to be pseudo-coherent if f∗OX is apseudo-coherent OY -module. This condition is equivalent to the pseudo-coherence of f in thesense of [8, Exp.III, Def.1.2].

Lemma 5.3.6. Let f : X → Y be a finite morphism of schemes. Then :(i) If f is finitely presented, the functor

F 7→ f ∗HomOY (f∗OX ,F )

is right adjoint to f∗ : OX-Modqcoh → OY -Modqcoh.(ii) If f is pseudo-coherent, the functor

K• 7→ f ∗RHom•OY (f∗OX , K•) : D+(OY -Mod)qcoh → D+(OX-Mod)qcoh

is right adjoint to

Rf∗ : D+(OX-Mod)qcoh → D+(OY -Mod)qcoh

(notation of (5.2.1)).

Proof. (i): Under the stated assumptions, f∗OX is a finitely presented OY -module (by claim5.7.8), hence the functor

F 7→HomOY (f∗OX ,F ) : OY -Mod→ f∗OX-Mod

preserves the subcategories of quasi-coherent modules, hence it restricts to a right adjoint for theforgetful functor f∗OX-Modqcoh → OY -Modqcoh (claim 5.1.26). Then the assertion followsfrom (5.3.1).

(ii) is analogous : since f∗OX is pseudo-coherent, the functor

K• 7→ RHom•OY (f∗OX , K•) : D+(OY -Mod)→ D+(f∗OX-Mod)


preserves the subcategories of complexes with quasi-coherent homology, hence it restricts toa right adjoint for the forgetful functor D+(f∗OX-Mod)qcoh → D+(OY -Mod)qcoh, by lemma5.1.25(iii). To conclude, it then suffices to apply lemma 5.3.2.

Proposition 5.3.7. Suppose that Xf−→ Y

g−→ Zh−→ W are morphisms of schemes. Then :

(i) If f and g are finite and f is pseudo-coherent, there is a natural isomorphism of functorson D+(OZ-Mod) :

ξf,g : (g f)[∼→ f [ g[.

(ii) If f and g f are finite and pseudo-coherent, and g is smooth of bounded fibre dimen-sion, there is a natural isomorphism of functors on D+(OZ-Mod) :

ζf,g : (g f)[∼→ f [ g].

(iii) If f , g and h g are finite and pseudo-coherent, and h is smooth of bounded fibredimension, then the diagram of functors on D+(OW -Mod) :

(h g f)[ζgf,h //

ξf,hg

(g f)[ h]

ξf,gh]

f [ (h g)[f[(ζg,h)

// f [ g[ h]

commutes.(iv) If, moreover

Yg //

h′

Z

h

W ′ j // W

is a cartesian diagram of schemes, such that h is smooth of bounded fibre dimensionand j is finite and pseudo-coherent, then there is a natural isomorphism of functors :

ϑj,h : g[ h] ∼→ h′] j[.

Proof. This is [46, Ch.III, Prop.6.2, 8.2, 8.6]. We check only (i). There is a natural commutativediagram of ringed spaces :

(X,OX)f //

f &&MMMMMMMMMM(Y,OY )

g //

g ((PPPPPPPPPPPP(Z,OZ)

(Y, f∗OX)

ϕ ((PPPPPPPPPPPP

α

OO

(Z, g∗OY )

γ

OO

(Z, (g f)∗OX)

β

OO

where :

(5.3.8) ϕ f = g f.

By claim 5.1.26, the (forgetful) functors α∗, β∗ and γ∗ admit right adjoints :

α[ : OY -Mod→ f∗OX-Mod : F 7→HomOY (f∗OX ,F )

β[ : g∗OY -Mod→ (g f)∗OX-Mod : F 7→Homg∗OY ((g f)∗OX ,F )

γ[ : OZ-Mod→ g∗OY -Mod : F 7→HomOZ (g∗OY ,F ).


Likewise, (γ β)∗ admits a right adjoint (γ β)[ and the natural identification

γ∗ β∗∼→ (γ β)∗

induces a natural isomorphisms of functors :

(5.3.9) (γ β)[ := HomOZ ((g f)∗OX ,−)∼→ β[ γ[.

(Notice that, in general, there might be several choices of such natural transformations (5.3.9),but for the proof of (iii) – as well as for the construction of f ! for more general morphisms f –it is essential to make an explicit and canonical choice.)

Claim 5.3.10. (i) α∗ ϕ∗ = g∗ β∗.(ii) The natural commutative diagram of sheaves :

g−1g∗OY//

OY

g−1(g f)∗OX // f∗OX

is cocartesian.

Proof of the claim. (i) is an easy consequence of (ii). Assertion (ii) can be checked on the stalks,hence we may assume that Z = SpecA, Y = SpecB and X = SpecC are affine schemes.Let p ∈ Y be any prime ideal, and set q := g(p) ∈ Z; then (g−1g∗OY )p = Bq (the Aq-moduleobtained by localizing the A-module B at the prime q). Likewise, (g−1(g f)∗OX)p = Cq and(f∗OX)p = Cp. Hence the claim boils down to the standard isomorphism : Cq⊗BqBp ' Cp. ♦

Let I• be a bounded below complex of injective OZ-modules; by lemma 5.1.25(i), γ[I• is acomplex of injective g∗OY -modules; taking into account (5.3.9) we deduce a natural isomor-phism of functors on D+(OZ-Mod) :

(5.3.11) R(γ β)[∼→ R(β[ γ[) = Rβ[ Rγ[.

Furthermore, (5.3.8) implies that g f ∗ = f ∗ ϕ∗. Combining with (5.3.11), we see that thesought ξf,g is a natural transformation :

f ∗ ϕ∗ Rβ[ Rγ[ → f ∗ Rα[ g∗ Rγ[

of functors on D+(OZ-Mod). Hence (i) will follow from :

Claim 5.3.12. (i) There exists a natural isomorphism of functors :

g∗OY -Mod→ f∗OX-Mod : ϕ∗ β[ ∼→ α[ g∗.(ii) The natural transformation :

R(α[ g∗)→ Rα[ g∗

is an isomorphism of functors : D+(g∗OY -Mod)→ D+(f∗OX-Mod).

Proof of the claim. First we show how to construct a natural transformation as in (i); this is thesame as exhibiting a map of functors : α∗ ϕ∗ β[ → g∗. In view of claim 5.3.10(i), the lattercan be defined as the composition of g∗ and the counit of adjunction β∗ β[ → 1g∗OY -Mod.

Next, recall the natural transformation :

(5.3.13) g∗Homg∗OY (F ,G )→HomOY (g∗F , g∗G ) for every g∗OY -modules F and G

(defined so as to induce the pull-back map on global Hom functors). Notice that the natural mapg∗g∗F → F (counit of adjunction) is an isomorphism for every quasi-coherent OY -module F .Especially, if F = g∗A for some quasi-coherent OY -module A , then (5.3.13) takes the form :

(5.3.14) g∗Homg∗OY (g∗A ,G )→HomOY (A , g∗G ).


Furthermore, by inspecting the definitions (and the proof of claim 5.1.26), one verifies easilythat the map ϕ∗ β[(G )→ α[ g∗(G ) constructed above is the same as the map (5.3.14), takenwith A = f∗OX . Notice that, since f is pseudo-coherent, f∗OX is even a finitely presentedOY -module; thus, in order to conclude the proof of (i), it suffices to show that (5.3.14) is anisomorphism for every finitely presented OY -module A and every g∗OY -module G . To thisaim, we may assume that Z and Y are affine; then we may find a presentation E := (O⊕pY →O⊕qY → A → 0). We apply the natural transformation (5.3.14) to E , thereby obtaining acommutative ladder with left exact rows; then the five-lemma reduces the assertion to the casewhere A = OY , which is obvious. To prove assertion (ii) we may again assume that Y and Zare affine, in which case we can find a resolution L• → f∗OX consisting of free OY -modulesof finite rank. For a given bounded below complex I• of injective g∗OY -modules, choose aresolution g∗I• → J• consisting of injective OY -modules. We deduce a commutative ladder :

g∗Hom•g∗OY (g∗ f∗OX , I•)µ1 //

λ1

Hom•OY (f∗OX , g∗I•)µ3 //

Hom•OY (f∗OX , J•)

λ2

g∗Hom•g∗OY (g∗L•, I

•)µ2 // Hom•OY (L•, g

∗I•)µ4 // Hom•OY (L•, J

•).

Since g is flat, λ1 is a quasi-isomorphism, and the same holds for λ2. The maps µ1 and µ2 areof the form (5.3.14), hence they are quasi-isomorphisms, by the foregoing proof of (i). Finally,it is clear that µ4 is a quasi-isomorphism as well, hence the same holds for µ3 µ1, and (ii)follows.

5.3.15. Let now f : X → Y be an embeddable morphism, i.e. such that it can be factored as acomposition f = g h where h : X → Z is a finite pseudo-coherent morphism, g : Z → Y issmooth and separated, and the fibres of g have bounded dimension. One defines :

f ! := h[ g].

Lemma 5.3.16. Let Xf−→ Y

g−→ Zh−→ W be three embeddable morphisms of schemes, such

that the compositions g f , h g and h g f are also embeddable. Then we have :(i) There exists a natural isomorphism of functors

ψg,f : (g f)! ∼→ f ! g!.

Especially, f ! is independent – up to a natural isomorphism of functors – of the choiceof factorization as a finite morphism followed by a smooth morphism.

(ii) The diagram of functors :

(h g f)!ψh,gf //

ψhg,f

(g f)! h!

ψg,fh!

f ! (h g)!

f !(ψh,g)// f ! g! h!

commutes.

Proof. The proof is a complicated verification, starting from proposition 5.3.7. See [46, Ch.III,Th.8.7] for details.

5.3.17. Let Xf−→ Y

g−→ Z be two finite and pseudo-coherent morphisms of schemes. The mapf \ : OY → f∗OX induces a natural transformation :

(5.3.18) Hom•OZ (g∗(f\),−) : β∗ (γ β)[ → γ[


(notation of the proof of proposition 5.3.7(i)) and we wish to conclude this section by exhibitinganother compatibility involving (5.3.18) and the isomorphism ξf,g. To this aim, we consider thecomposition of isomorphism of functors on D+(OZ-Mod)qcoh :

(5.3.19) Rf∗ g f ∗'&%$ !"#1−→ α∗ Rf∗ f ∗ ϕ∗

'&%$ !"#2−→ α∗ ϕ∗

'&%$ !"#3−→ g∗ β∗

where :/.-,()*+1 is deduced from (5.3.8) and the decomposition f = α f ./.-,()*+2 is induced from the counit of adjunction ε : Rf∗ f ∗ → 1 provided by lemma 5.3.2./.-,()*+3 is the identification of claim 5.3.10.

We define a morphism of functors as a composition :

(5.3.20) Rf∗ (g f)[ = Rf∗ g f ∗ R(γ β)['&%$ !"#4−→ g∗ β∗ R(γ β)[

'&%$ !"#5−→ g[

where :/.-,()*+4 is the isomorphism (5.3.19) R(γ β)[./.-,()*+5 is the composition of g∗ and the morphism β∗R(γβ)[ → Rγ[ deduced from (5.3.18).

We have a natural diagram of functors on D+(OZ-Mod)qcoh :

(5.3.21)R(g f)∗ (g f)[

'&%$ !"#6//

R(gf)∗(ξf,g)

Rg∗ Rf∗ (g f)['&%$ !"#9

//

Rg∗Rf∗(ξf,g)

Rg∗ g[

R(g f)∗ (f [ g[)'&%$ !"#7// Rg∗ Rf∗ (f [ g[) Rg∗ (Rf∗ f [) g[

'&%$ !"#8

OO

where :/.-,()*+6 and /.-,()*+7 are induced by the natural isomorphism R(g f)∗

∼→ Rg∗ Rf∗./.-,()*+8 is induced by the counit of the adjunction ε : Rf∗ f [ → 1./.-,()*+9 is Rg∗ (5.3.20).

Lemma 5.3.22. In the situation of (5.3.17), diagram (5.3.21) commutes.

Proof. The diagram splits into left and right subdiagrams, and clearly the left subdiagram com-mutes, hence it remains to show that the right one does too; to this aim, it suffices to considerthe simpler diagram :

Rf∗ (g f)[ //

Rf∗(ξf,g)

g[

Rf∗ f [ g[

99sssssssssss

whose horizontal arrow is (5.3.20), and whose upward arrow is deduced from the counit ε.However, the counit ε, used in /.-,()*+8 , can be expressed in terms of the counit ε : Rf ∗f ∗ → 1, usedin /.-,()*+2 , therefore we are reduced to considering the diagram of functors on D+(OZ-Mod)qcoh :

α∗ Rf ∗ f ∗ ϕ∗ R(γ β)['&%$ !"#a//

Rf∗(ξf,g)

α∗ ϕ∗ R(γ β)[

'&%$ !"#c

'&%$ !"#e// g∗ β∗ R(γ β)[

'&%$ !"#5

α∗ ϕ∗ Rβ[ Rγ[

'&%$ !"#d

α∗ Rf ∗ f ∗ Rα[ g['&%$ !"#b

// α∗ Rα[ g∗ Rγ[/.-,()*+f

// g[ = g∗ Rγ[


where :76540123a and 76540123b are induced from ε.76540123c is induced by the isomorphism (5.3.11).76540123d is induced by the isomorphism of claim 5.3.12(i).76540123e is induced by the identity of claim 5.3.10.76540123f is induced by the counit α∗ Rα[ → 1.

Now, by inspecting the construction of ξf,g one checks that the left subdiagram of the latterdiagram commutes; moreover, from claim 5.3.12(ii) (and from lemma 5.1.25(i)), one sees thatthe right subdiagram is obtained by evaluating on complexes of injective modules the analogousdiagram of functors on OZ-Mod :

α∗ ϕ∗ (γ β)[e//

c

g∗ β∗ (γ β)[

5

α∗ ϕ∗ β[ γ[

d

α∗ α[ g∗ γ[f

// g∗ γ[.

Furthermore, by inspecting the proof of claim 5.3.12(i), one verifies that f d is the same asthe composition

α∗ ϕ∗ β[ γ['&%$ !"#i−→ g∗ β∗ β[ γ[

/.-,()*+j−→ g∗ γ[

where 76540123j is deduced from the counit β∗ β[ → 1, and 76540123i is deduced from the identity of claim5.3.10. Hence we come down to showing that the diagram :

β∗ (γ β)[

β∗(5.3.9)

(5.3.18) // γ[

β∗ β[ γ[/.-,()*+k

66mmmmmmmmmmmmmmm

commutes in g∗OY -Mod, where 76540123k is deduced from the counit β∗ β[ → 1. This is an easyverification, that shall be left to the reader.

5.4. Depth and cohomology with supports. The aim of this section is to clarify some generalissues concerning local cohomology and the closely related notion of depth, in the contextof arbitrary schemes (whereas the usual references restrict to the case of locally noetherianschemes).

5.4.1. To begin with, let X be any topological space, i : Z → X a closed immersion, andj : U := X \Z → X the open immersion of the complement of Z. We let ZX be the constantabelian sheaf on X with value Z; hence ZX-Mod denotes the category of abelian sheaves onX . One defines the functor

ΓZ : ZX-Mod→ ZX-Mod F 7→ Ker (F → j∗j−1F )

as well as its composition with the global section functor :

ΓZ : ZX-Mod→ Z-Mod F 7→ Γ(X,ΓZF ).

It is clear that ΓZ and ΓZ are left exact functors, hence they give rise to right derived functors

RΓZ : D+(ZX-Mod)→ D+(ZX-Mod) RΓZ : D+(ZX-Mod)→ D+(Z-Mod).


Moreover, suppose that F → I • is an injective resolution; since injective sheaves are flabby,we obtain a short exact sequence of complexes

0→ ΓZI • → I • → j∗j−1I • → 0

whence a natural exact sequence of ZX-modules :

(5.4.2) 0→ ΓZF → F → j∗j−1F → R1ΓZF → 0 for every ZX-module F

and natural isomorphisms :

(5.4.3) Rq−1j∗j−1F

∼→ RqΓZF for all q > 1.

Lemma 5.4.4. (i) A flabby sheaf is ΓZ-acyclic.

(ii) Suppose that X fulfills conditions (a) and (b) of lemma 5.1.2, and that Z is a con-structible closed subset of X . Then :(a) Every qc-flabby sheaf on X is ΓZ-acyclic.(b) For every i ∈ N, the functor RiΓZ commutes with filtered colimits of abelian

sheaves.

Proof. (i) follows by inspecting (5.4.2), (5.4.3), together with the fact that flabby sheaves areacyclic for direct image functors such as j∗. Assertion (ii) is checked in the same way, usinglemmata 5.1.2, 5.1.3 and 5.1.10(ii).

5.4.5. Next, if (X,A ) is a ringed space, one can consider the restriction of ΓZ to the cate-gory A -Mod of A -modules, and the foregoing discussion carries over verbatim to such case.Moreover, the derived functor

RΓZ : D+(A -Mod)→ D+(A -Mod)

commutes with the forgetful functor D+(A -Mod) → D+(ZX-Mod). (Indeed, in view oflemma 5.4.4(i) one may compute RΓZF by a flabby resolution of F .)

Furthermore, let f : (Y,B) → (X,A ) be a morphism of ringed spaces. For every B-module F , the direct image f∗F is an A -module in a natural way, and the derived functorRf∗ : D+(B-Mod) → D+(A -Mod) commutes with the forgetful functors D+(A -Mod) →D+(ZX-Mod) and D+(B-Mod) → D+(ZY -Mod). (Again, one sees this easily when oneremarks that flabby resolutions compute Rf∗.)

5.4.6. In the situation of (5.4.5), let U := (Ut | t ∈ I) be a family of open subsets of X with⋃t∈I

Ut = U.

We attach to U an alternating Cech resolution, which is a complex R•(U) in C≤0(ZX-Mod)constructed as follows. Fix an arbitrary total ordering on I , and for every integer n > 0, letIn be the set of all sequences (t1, . . . , tn) of elements of I , such that t1 < · · · < tn. For everyt ∈ In, set Ut := Ut1 ∩ · · · ∩ Utn , and let jt : Ut → X be the open immersion. We let

R0(U) := ZX and Rn(U) :=⊕t∈In

jt!ZUt for every n > 0

The differential d1 : R1(U) → R0(U) is just the sum of the natural morphisms jt!ZUt → ZX ,for every t ∈ I . For n > 1, the differential dn : Rn(U)→ Rn−1(U) is the sum of the maps

dt : jt!ZUt → Rn−1(U) for every t ∈ In


defined as follows. Say that t = (t1, . . . , tn), and for every k = 1, . . . , n denote by t(k) ∈ In−1

the sequence obtained by deleting the k-th term from t; the inclusion Ut ⊂ Ut(k) induces amorphism of ZX-modules

d(k)t : jt!ZUt → jt(k)!ZUt(k)

⊂ Rn−1(U)

and we set

dt :=n∑k=1

(−1)k · d(k)t .

A direct computation shows that dn dn+1 = 0 for every n ∈ N. The name of R•(U) is justifiedby the following :

Lemma 5.4.7. With the notation of (5.4.6), the natural projection ZX → i∗ZZ induces a quasi-isomorphism of complexes of ZX-modules :

(5.4.8) R•(U)∼→ i∗ZZ [0].

Proof. For every finite subset J ⊂ I , set UJ := (Ut | t ∈ J), ZJ := X \⋃t∈J Ut, and denote by

iJ : ZJ → X the closed immersion. Notice the natural isomorphisms

R•(U)∼→ colim

J⊂IR•(UJ) i∗ZZ

∼→ colimJ⊂I

iJ∗ZZJ

where J ranges over all the finite subsets of I . Since (5.4.8) is likewise the colimit of thecorresponding system of maps R•(UJ)→ iJ∗ZZJ [0], we may assume from start that I is a finiteset. Now, the assertion can be checked at the stalks over the points of X , hence let x ∈ Xbe any such point, and R•(U)x the corresponding complex of Z-modules (the stalk of R•(U)at x). If x ∈ Z, then obviously R•(U)x is concentrated in degree 0, and the resulting map(5.4.8)x is clearly an isomorphism. If x /∈ Z, set J := t ∈ I | x ∈ Ut; it is easily seen thatR•(U)x = R•(UJ)x, so we may replace U by UJ , and assume that x ∈ Ut for every t ∈ I . Inthis case, a simple inspection shows that R•(U)x is naturally isomorphic to the Koszul complexK•(1I), where 1I = (1t | t ∈ I) is a sequence of units 1 ∈ Z, indexed by I . The complex (ofZ-modules) K•(1I) is obviously acyclic, so the claim follows.

5.4.9. Keep the notation of (5.4.6), and notice the natural isomorphism of A -modules :

HomZ(i∗ZZ ,F )∼→ ΓZ(F ) for every A -module F

whence a natural isomorphism of A (X)-modules :

HomZ(i∗ZZ ,F )∼→ ΓZ(F ) for every A -module F .

By lemma 5.4.7, we deduce natural isomorphisms in D+(A -Mod) (resp. in D+(A (X)-Mod))

RHom•Z(R•(U), K•)∼→ RΓZK

• ( resp. RHom•Z(R•(U), K•)∼→ RΓZK

• )

for every bounded below complex K• of A -modules. These isomorphisms allow to computeRΓZK

• as a Cech cohomology functor, as follows. For every A -module F , we set

C•alt(U•,F ) := Hom•Z(R•(U),F [0])

and we call it the alternating Cech complex of F relative to the covering U of U . Unwindingthe definitions, we see that the latter is the cochain complex :

(5.4.10) 0→ F (X)→∏t∈I

F (Ut)→ · · · →∏t∈In

F (Ut)→ · · ·


The differential dn of (5.4.10) is given by the following rule. In degree zero, d0 assigns to eachglobal section s ∈ F (X) the system of its restrictions (s|Ut | t ∈ I). In case n > 0, we have

dn(s•)t :=n+1∑k=1

(−1)k · (st(k))|Ut for every t ∈ In+1 and every s• ∈ Cn(U,F ).

Clearly, the rule : F 7→ C•alt(U,F ) defines a functor A -Mod→ C(A (X)-Mod). Now, pickany resolution K• ∼→ I • by a bounded below complex of injective A -modules; the foregoingyields a natural isomorphism :

RΓZK• ∼→ TotC•alt(U,I

•) in D+(A (X)-Mod).

5.4.11. We define a bifunctor

Hom•Z := ΓZ Hom•A : C(A -Mod)o × C(A -Mod)→ C(A -Mod).

The bifunctor Hom•Z admits a right derived functor :

RHom•Z : D(A -Mod)o × D+(A -Mod)→ D(A -Mod).

The construction can be outlined as follows. First, for a fixed complex K• of A -modules,one can consider the right derived functor of the functor G 7→ Hom•Z(K•,G ), which isdenoted RHom•Z(K•,−) : D+(A -Mod) → D(A -Mod). Next, one verifies that everyquasi-isomorphism K• → K ′• induces an isomorphism of functors : RHom•Z(K ′•,−)

∼→RHom•Z(K•,−), hence the natural transformation K• 7→ RHom•Z(K•,−) factors throughD(A -Mod), and this is the sought bifunctor RHom•Z . In case Z = X , one recovers thefunctor RHom•A of (5.1.22).

Lemma 5.4.12. In the situation of (5.4.5), the following holds :(i) The functor ΓZ is right adjoint to i∗i−1. More precisely, for any two A -modules F

and G we have natural isomorphisms :(a) HomA (F ,ΓZG ) 'HomA (i∗i

−1F ,G ).(b) RHom•A (F , RΓZG ) ' RHom•A (i∗i

−1F ,G ).(ii) If F is an injective (resp. flabby) A -module on X , then the same holds for ΓZF .

(iii) There are natural isomorphisms of bifunctors :

RHom•Z∼→ RΓZ RHom•A

∼→ RHom•A (−, RΓZ−).

(iv) If W ⊂ Z is any closed subset, there is a natural isomorphism of functors :

RΓW∼→ RΓW RΓZ .

(v) There is a natural isomorphism of functors :

Rf∗ RΓf−1Z∼→ RΓZ Rf∗ : D+(B-Mod)→ D+(A -Mod).

Proof. (i): To establish the isomorphism (a), one uses the short exact sequence

0→ j!j−1F → F → i∗i

−1F → 0

to show that any map F → ΓZG factors through H := i∗i−1F . Conversely, since j∗j−1H =

0, it is clear that every map H → G must factor through ΓZG . The isomorphism (b) is derivedeasily from (a) and (ii).

(iii): According to [75, Th.10.8.2], the first isomorphism in (iii) is deduced from lemmata5.4.4(i) and 5.1.25(ii). Likewise, the second isomorphism in (iii) follows from [75, Th.10.8.2],and assertion (ii) concerning injective sheaves.

(ii): Suppose first that F is injective; let G1 → G2 a monomorphism of A -modules andϕ : G1 → ΓZF any A -linear map. By (i) we deduce a map i∗i∗G1 → F , which extends to amap i∗i∗G2 → F , by injectivity of F . Finally, (i) again yields a map G2 → ΓZF extending ϕ.


Next, if F is flabby, let V ⊂ X be any open subset, and s ∈ ΓZF (V ); since s|V \Z = 0, wecan extend s to a section s′ ∈ ΓZF (V ∪ (X \Z)). Since F is flabby, s′ extends to a sections′′ ∈ F (X); however, by construction s′′ vanishes on the complement of Z.

(iv): Clearly ΓW = ΓW ΓZ , so the claim follows easily from (ii) and [75, Th.10.8.2].(v): Indeed, since flabby B-modules are acyclic for both f∗ and Γf−1Z , one can apply (ii) and

[75, Th.10.8.2] to naturally identify both functors with the right derived functor of ΓZ f∗.

Definition 5.4.13. Let K• ∈ D+(ZX-Mod) be any bounded below complex. The depth of K•

along Z is by definition :

depthZK• := supn ∈ N | RiΓZK

• = 0 for all i < n ∈ N ∪ +∞.Notice that

(5.4.14) depthWK• ≥ depthZK

• whenever W ⊂ Z.

More generally, suppose that Φ := (Zλ | λ ∈ Λ) is a family of closed subsets of X , and supposethat Φ is cofiltered by inclusion. Then, due to (5.4.14) it is reasonable to define the depth of K•

along Φ by the rule :depthΦK

• := sup depthZλK• | λ ∈ Λ.

5.4.15. Suppose now that X is a scheme and i : Z → X a closed immersion of schemes, suchthat Z is constructible inX . It follows that the open immersion j : X\Z → X is quasi-compactand separated, hence the functors Rqj∗ preserve the subcategories of quasi-coherent modules,for every q ∈ N ([28, Ch.III, Prop.1.4.10]). In light of (5.4.3) we derive that RqΓZ restricts to afunctor :

RqΓZ : OX-Modqcoh → OX-Modqcoh for every q ∈ N.

Lemma 5.4.16. In the situation of (5.4.15), the following holds :(i) The functor RΓZ restricts to a triangulated functor :

RΓZ : D+(OX-Mod)qcoh → D+(OX-Mod)qcoh

(notation of (5.2.2)).(ii) Let f : Y → X be an affine morphism of schemes,K• any object of D+(OY -Mod)qcoh.

Then we have the identity :

depthf−1ZK• = depthZRf∗K

•.

(iii) Let f : Y → X be a flat morphism of schemes, K• any object of D+(OX-Mod)qcoh.Then the natural map

f ∗RΓZK• → RΓf−1Zf

∗K•

is an isomorphism in D+(OY -Mod)qcoh. Especially, we have

depthf−1Zf∗K• ≥ depthZK

•

and if f is faithfully flat, the inequality is actually an equality.

Proof. (i): Indeed, if K• is an object of D+(OX-Mod)qcoh, we have a spectral sequence :

Epq2 := RpΓZH

qK• ⇒ Rp+qΓZK•.

(This spectral sequence is obtained from a Cartan-Eilenberg resolution of K• : see e.g. [75,§5.7].) Hence R•ΓZK

• admits a finite filtration whose subquotients are quasi-coherent; thenthe lemma follows from [28, Ch.III, Prop.1.4.17].

(ii): To start with, a spectral sequence argument as in the foregoing shows that Rf∗ restrictsto a functor D+(OY -Mod)qcoh → D+(OX-Mod)qcoh, and moreover we have natural identifi-cations : Rif∗K

• ∼→ f∗HiK• for every i ∈ Z and every object K• of D+(OY -Mod)qcoh. In


view of lemma 5.4.12(v), we derive natural isomorphisms : RiΓZRf∗K• ∼→ Rf∗R

iΓf−1ZK•

for every object K• of D+(OY -Mod)qcoh. The assertion follows easily.(iii): A spectral sequence argument as in the proof of (i) allows to assume that K• = F [0]

for some quasi-coherent OX-module F . In this case, let jX : X \Z → X and jY : Y \f−1Z → Y be the open immersions; considering the exact sequences f ∗(5.4.2) and f ∗(5.4.3),we reduce to showing that the natural map f ∗RjX∗j−1

X F → RjY ∗j∗Y f∗F is an isomorphism in

D+(OY -Mod)qcoh. The latter assertion follows from corollary 5.1.19.

Remark 5.4.17. Let X be an affine scheme, Z ⊂ X any closed subset, F • a bounded belowcomplex of quasi-coherent OX-modules, and U := (Ut | t ∈ I) a family of affine open subsetsof X such that

⋃t∈I Ut = X\Z. Then there exists a natural isomorphism

RΓZF • ∼→ TotC•alt(U,F•) in D+(OX-Mod)

(where C•alt denotes the alternating Cech complex). Indeed, pick any resolution F • ∼→ I • bya bounded below complex of injective OX-modules; by virtue of (5.4.9) it suffices to show thatthe natural map of double complexes

C•alt(U,F•)→ C•alt(U,I

•)

induces a quasi-isomorphism on total complexes. However, notice that the open subset Ut ⊂ Xis affine, for every n ∈ N and every t ∈ In (notation of (5.4.9)); hence, it suffices to checkthat the induced map F •(V ) → I •(V ) is a quasi-isomorphism, for every affine open subsetV ⊂ X . To this aim, consider the spectral sequence

Ep,q1 := Hp(V,F q)⇒ Hp+qI •(V ).

Since V is affine, we have Ep,q1 = 0 for every p > 0; on the other hand, the differential

d0,q : E0,q1 → E0,q+1

1 is nothing else than the differential dq(V ) : F q(V ) → F q+1(V ), forevery q ∈ Z, whence the claim.

5.4.18. Let X be a scheme, and x ∈ X any point. We consider the cofiltered family Φ(x) of allnon-empty constructible closed subsets of X(x) := Spec OX,x; clearly

⋂Z∈Φ(x) Z = x. Let

K• be any object of D+(ZX-Mod); we are interested in the quantities :

δ(x,K•) := depthxK•|X(x) and δ′(x,K•) := depthΦ(x)K

•|X(x).

Especially, we would like to be able to compute the depth of a complex K• along a closedsubset Z ⊂ X , in terms of the local invariants δ(x,K•) (resp. δ′(x,K•)), evaluated at thepoints x ∈ Z. This shall be achieved by theorem 5.4.20. To begin with, we remark :

Lemma 5.4.19. With the notation of (5.4.18), we have :(i) δ(x,K•) ≥ δ′(x,K•) for every x ∈ X and every complex K• of ZX-modules.

(ii) If the topological space |X| underlying X is locally noetherian, the inequality of (i) isactually an equality.

(iii) If F is a flat quasi-coherent OX-module, then δ′(x,F ) ≥ δ′(x,OX).

Proof. (i) follows easily from lemma 5.4.12(iv). (ii) follows likewise from lemma 5.4.12(iv)and the fact that, if |X| is locally noetherian, x is the smallest element of the family Φ(x).

(iii): According to [57, Ch.I, Th.1.2], the OX(x)-module F|X(x) is the colimit of a filteredsystem (Lλ | λ ∈ Λ) of free OX(x)-modules of finite rank. In view of lemma 5.4.4(iii), it iseasily seen that

depthZF ≥ infλ∈Λ

depthZLλ = depthZOX

for every closed onstructible subset Z ⊂ X . The assertion follows.

Theorem 5.4.20. With the notation of (5.4.18), let Z ⊂ X be any closed constructible subset,K• any object of D+(ZX-Mod). Then the following holds :


(i) (RΓZK•)|X(x) = RΓZ∩X(x)(K

•|X(x)) for every x ∈ Z.

(ii) depthZK• = inf δ(x,K•) | x ∈ Z = inf δ′(x,K•) | x ∈ Z.

Proof. We may assume that X is affine, hence separated; then, according to lemma 5.4.4(ii.a),cohomology with supports in Z can be computed by a qc-flabby resolution, hence we mayassume that K• is a complex of qc-flabby sheaves on X . In this case, arguing as in the proof ofclaim 5.1.13, one checks easily that the restriction K•|X(x) is a complex of qc-flabby sheaves onX(x). Thus, for the proof of (i), it suffices to show that the natural map in ZX-Mod :

(ΓZF )|X(x) → ΓZ∩X(x)(F|X(x))

is an isomorphism, for every object F of ZX-Mod.Now, for any point x ∈ X , we may view X(x) as the limit of the cofiltered system (Vλ | λ ∈

Λ) of all affine open neighborhoods of x in X . We denote by (jλ : Vλ\Z → Vλ | λ ∈ Λ) theinduced cofiltered system of open immersions, whose limit is the morphism j∞ : X(x)\Z →X(x); also, let ϕλ : X(x)→ Vλ be the natural morphisms. Then

F|X(x) = colimλ∈Λ

ϕ−1λ F|Uλ .

By proposition 5.1.15(ii), the natural map

colimλ∈Λ

ϕ−1λ jλ∗F|Vλ\Z → j∞∗F|X(x)\Z

is an isomorphism. We deduce natural isomorphisms :

colimλ∈Λ

ϕ−1λ ΓZ∩VλF|Vλ

∼→ ΓZ∩X(x)F|X(x).

However, we have as well natural isomorphisms :

ϕ−1λ ΓZ∩VλF|Vλ

∼→ (RΓZF )|X(x) for every λ ∈ Λ.

whence (i). Next, let d := depthZK•; in light of (i), it is clear that δ(x,K•) ≥ d for all x ∈ Z,

hence, in order to prove the first identity of (ii) it suffices to show :

Claim 5.4.21. Suppose d < +∞. Then there exists x ∈ Z such that RdΓxK•|X(x) 6= 0.

Proof of the claim. By definition of d, we can find an open subset V ⊂ X and a non-zero sections ∈ Γ(V,RdΓZK

•). The support of s is a closed subset S ⊂ Z. Let x be a maximal point of S.From lemma 5.4.12(iv) we deduce a spectral sequence

Epq2 := RpΓxR

qΓZ∩X(x)K•|X(x) ⇒ Rp+qΓxK

•|X(x)

and (i) implies that Epq2 = 0 whenever q < d, therefore :

RdΓxK•|X(x) ' R0ΓxR

dΓZ∩X(x)K•|X(x).

However, the image of s in RdΓZ∩X(x)K•|X(x) is supported precisely at x, as required. ♦

Finally, since Z is constructible, Z ∩ X(x) ∈ Φ(x) for every x ∈ X , hence δ′(x,K•) ≥ d.Then the second identity of (ii) follows from the first and lemma 5.4.19(i).

Corollary 5.4.22. Let Z ⊂ X be a closed and constructible subset, F an abelian sheaf onX , j : X \Z → X the induced open immersion. Then the natural map F → j∗j

∗F is anisomorphism if and only if δ(x,F ) ≥ 2, if and only if δ′(x,F ) ≥ 2 for every x ∈ Z.


5.4.23. Let now X be an affine scheme, say X := SpecA for some ring A, and Z ⊂ X aconstructible closed subset. In this case, we wish to show that the depth of a complex of quasi-coherent OX-modules along Z can be computed in terms of Ext functors on the category ofA-modules. This is the content of the following :

Proposition 5.4.24. In the situation of (5.4.23), let N be any finitely presented A-module suchthat SuppN = Z, and M• any bounded below complex of A-modules. We denote by M•∼ thecomplex of quasi-coherent OX-modules determined by M•. Then :

depthZM•∼ = sup n ∈ N | ExtjA(N,M•) = 0 for all j < n.

Proof. Let N∼ be the quasi-coherent OX-module determined by N .

Claim 5.4.25. There is a natural isomorphism in D+(A-Mod) :

RHom•A(N,M•)∼→ RHom•OX (N∼, RΓZM

•∼).

Proof of the claim. Let i : Z → X be the closed immersion. The assumption on N implies thatN∼ = i∗i

−1N∼; hence lemma 5.4.12(i.b) yields a natural isomorphism

(5.4.26) RHom•OX (N∼,M•∼)∼→ RHom•OX (N∼, RΓZM

•∼).

The claim follows by combining corollary 5.1.33 and (5.4.26). ♦

From claim 5.4.25 we see already that ExtjA(N,M•) = 0 whenever j < depthZM•∼. Sup-

pose that depthZM•∼ = p < +∞; in this case, claim 5.4.25 gives an isomorphism :

ExtpA(N,M•) ' HomOX (N∼, RpΓZM•∼) ' HomA(N,RpΓZM

•∼)

where the last isomorphism holds by lemma 5.4.16. To conclude the proof, we have to exhibita non-zero map from N to Q := RpΓZM

•∼. Let F0(N) ⊂ A denote the Fitting ideal of N ; thisis a finitely generated ideal, whose zero locus coincides with the support of N . More precisely,AnnA(N)r ⊂ F0(N) ⊂ AnnA(N) for all sufficiently large integers r > 0. Let now x ∈ Q beany non-zero element, and f1, . . . , fk a finite system of generators for F0(N); since x vanisheson X \Z, the image of x in Qfi is zero for every i ≤ k, i.e. there exists ni ≥ 0 such thatfnii x = 0 in Q. It follows easily that F0(N)n ⊂ AnnA(x) for a sufficiently large integer n, andtherefore x defines a map ϕ : A/F0(N)n → Q by the rule : a 7→ ax. According to [36, Lemma3.2.21], we can find a finite filtration 0 = Jm ⊂ · · · ⊂ J1 ⊂ J0 := A/F0(N)n such that eachJi/Ji+1 is quotient of a direct sum of copies of N . Let s ≤ m be the smallest integer such thatJs ⊂ Kerϕ. By restriction, ϕ induces a non-zero map Js−1/Js → Q, whence a non-zero mapN (S) → Q, for some set S, and finally a non-zero map N → Q, as required.

Remark 5.4.27. Notice that the existence of a finitely presentedA-moduleN with SuppN = Zis equivalent to the constructibility of Z.

5.4.28. In the situation of (5.4.23), let I ⊂ A be a finitely generated ideal such that V (I) = Z,and M• a bounded below complex of A-modules. Denote by M•∼ the complex of quasi-coherent modules on SpecA associated to M•. Then it is customary to set :

depthIM• := depthZM

•∼

and the depth along Z of M• is also called the I-depth of M•. Moreover, if A is a local ringwith maximal ideal mA, we shall often use the standard notation :

depthAM• := depthmAM

•∼

and this invariant is often called briefly the depth of M•. With this notation, theorem 5.4.20(ii)can be rephrased as the identity :

(5.4.29) depthIM• = inf

p∈V (I)depthAp

M•p .


5.4.30. Suppose now that f is a system of generators of the ideal I ⊂ A, and set Z :=V (I). In this generality, the Koszul complex K•(f) of (4.1.16) is not a resolution of the A-module A/I , and consequently H•(f ,M) (for an A-module M ) does not necessarily agreewith Ext•A(A/I,M). Nevertheless, the following holds.

Proposition 5.4.31. In the situation of (5.4.30), let M• be any object of D+(A-Mod), and set

d := sup i ∈ Z | Hj(f ,M•) = 0 for all j < i.Then we have :

(i) depthIM• = d.

(ii) There are natural isomorphisms :(a) Hd(f ,M•)

∼→ HomA(A/I,RdΓZM•∼)

∼→ ExtdA(A/I,M•), when d < +∞.(b) colim

m∈NK•(fm,M•)

∼→ RΓZM•∼ in D(A-Mod) (where the transition maps in the

colimit are the maps ϕ•fn of (4.1.23)).

Proof. (i): Let B := Z[t1, . . . , tr]→ A be the ring homomorphism defined by the rule : ti 7→ fifor i = 1, . . . , r; we denote ψ : X → Y := SpecB the induced morphism, J ⊂ B the idealgenerated by the system t := (ti | i = 1, . . . , r), and set W := V (J) ⊂ Y . From lemma5.4.12(v) we deduce a natural isomorphism in D(OY -Mod) :

(5.4.32) ψ∗RΓZM•∼ ∼→ RΓWψ∗M

•∼.

Hence we are reduced to showing :

Claim 5.4.33. d = depthJψ∗M•.

Proof of the claim. Notice that t is a regular sequence in B, hence Ext•B(B/J, ψ∗M•) '

H•(t, ψ∗M•); by proposition 5.4.24, there follows the identity :

depthJψ∗M• = sup n ∈ N | H i(t, ψ∗M

•) = 0 for all i < n.Clearly there is a natural isomorphism : H i(t, ψ∗M

•)∼→ H i(f ,M•), whence the assertion. ♦

(ii.a): From lemma 5.4.12(i.b) and corollary 5.1.33 we derive a natural isomorphism :

RHom•B(B/J, ψ∗M•)∼→ RHom•OY ((B/J)∼, RΓWψ∗M

•∼).

However, due to (5.4.32) and claim 5.4.33 we may compute :

RdHom•OY ((B/J)∼, RΓWψ∗M•∼)

∼→ HomB(B/J,RdΓWψ∗M•∼)

∼→ HomA(A/I,RdΓZM•∼).

The first claimed isomorphism easily follows. Similarly, one applies lemma 5.4.12(i.b) andcorollary 5.1.33 to compute Ext•A(A/I,M•), and deduces the second isomorphism of (ii.a)using (i).

(ii.b): For every i ≤ r, let Ui := SpecA[f−1i ] ⊂ X := SpecA, and set U := (Ui | i =

1, . . . , r); then U is an affine covering of X \Z. By inspecting the definitions we deduce anatural isomorphism of complexes :

(5.4.34) colimm∈N

K•(fm,M•)∼→ TotC•alt(U,M

•∼)

(notation of (5.4.9)) so the assertion follows from remark 5.4.17.

Corollary 5.4.35. In the situation of (5.4.28), let B be a faithfully flat A-algebra. Then :

depthIBB ⊗AM• = depthIM•.

Proof. It is a special case of lemma 5.4.16(iii). Alternatively, one remarks that

K•(g,M• ⊗A B) ' K•(g,M•)⊗A B for every finite sequence of elements g in A

which allows to apply proposition 5.4.31(ii.b).


Theorem 5.4.36. Let A → B be a local homomorphism of local rings, and suppose that themaximal ideals mA and mB of A and B are finitely generated. Let also M be an A-module andN a B-module which is flat over A. Then we have :

depthB(M ⊗A N) = depthAM + depthB/mAB(N/mAN).

Proof. Let f := (f1, . . . , fr) and g := (g1, . . . , gs) be finite systems of generators for mA andrespectively mB, the maximal ideal of B/mAB. We choose an arbitrary lifting of g to a finitesystem g of elements of mB; then (f ,g) is a system of generators for mB. We have a naturalidentification of complexes of B-modules :

K•(f ,g)∼→ Tot•(K•(f)⊗A K•(g))

whence natural isomorphisms :

K•((f ,g),M ⊗A N)∼→ Tot•(K•(f ,M)⊗A K•(g, N)).

A standard spectral sequence, associated to the filtration by rows, converges to the cohomologyof this total complex, and since N is a flat A-module, K•(g, N) is a complex of flat A-modules,so that the E1 term of this spectral sequence is found to be :

Epq1 ' Hq(f ,M)⊗A Kp(g, N)

and the differentials dpq1 : Epq1 → Ep+1,q

1 are induced by the differentials of the complexK•(g, N). Set κA := A/mA; notice that Hq(f ,M) is a κA-vector space, hence

Hq(f ,M)⊗A H•(g, N) ' H•(Hq(f ,M)⊗κA K•(g, N/mAN))

and consequently :Epq

2 ' Hq(f ,M)⊗κA Hp(g, N/mAN).

Especially :

(5.4.37) Epq2 = 0 if either p < d := depthAM or q < d′ := depthB/mAB(N/mAN).

Hence H i((f ,g),M ⊗A N) = 0 for all i < d + d′ and moreover, if d and d′ are finite,Hd+d′((f ,g),M ⊗A N) ' Edd′

2 6= 0. Now the sought identity follows readily from propo-sition 5.4.31(i).

Remark 5.4.38. By inspection of the proof, we see that – under the assumptions of theorem5.4.36 – the following identity has been proved :

Extd+d′

A (B/mB,M ⊗A N) ' ExtdA(A/mA,M)⊗A Extd′

B/mAB(B/mB, N/mAN)

where d and d′ are defined as in (5.4.37).

Corollary 5.4.39. Let f : X → S be a locally finitely presented morphism of schemes, F a f -flat quasi-coherent OX-module of finite presentation, G any quasi-coherent OS-module, x ∈ Xany point, i : f−1f(x)→ X the natural morphism. Then :

δ′(x,F ⊗OX f∗G ) = δ′(x, i∗F ) + δ′(f(x),G ).

Proof. Set s := f(x), and denote by

js : S(s) := Spec OS,s → S and fs : X(s) := X ×S S(s)→ S(s)

the induced morphisms. By inspecting the definitions, one checks easily that

δ′(x,F ⊗OX f∗G ) = δ′(x,F|X(s) ⊗OX(s)

f ∗sG|S(s)) and δ′(s,G ) = δ′(s,G|S(s)).

Thus, we may replace S by S(s) andX byX(s), and assume from start that S is a local schemeand s its closed point. Clearly we may also assume thatX is affine and finitely presented over S.Then we can write S as the limit of a cofiltered family (Sλ | λ ∈ Λ) of local noetherian schemes,and we may assume that f : X → S is a limit of a cofiltered family (fλ : Xλ → Sλ | λ ∈ Λ)


of morphisms of finite type, such that Xµ = Xλ ×Sλ Sµ whenever µ ≥ λ. Likewise, we maydescend F to a family (Fλ | λ ∈ Λ) of finitely presented quasi-coherent OXλ-modules, such thatFµ = ϕ∗µλFλ for every µ ≥ λ (where ϕµλ : Xµ → Xλ is the natural morphism). Furthermore,up to replacing Λ by some cofinal subset, we may assume that Fλ is fλ-flat for every λ ∈ Λ ([32,Ch.IV, Cor.11.2.6.1(ii)]). For every λ, µ ∈ Λ with µ ≥ λ, denote by ψλ : S → Sλ and ψµλ :Sµ → Sλ the natural morphisms, and let sλ := ψλ(s); given a constructible closed subschemeZλ ⊂ Sλ, set Z := ψ−1

λ Zλ (resp. Zµ := ψ−1µλZλ), which is constructible and closed in S (resp.

in Sµ). According to lemma 5.4.16(ii) we have depthZG = depthZµψµ∗G = depthZλψλ∗G ,whence the inequality :

δ′(sλ, ψλ∗G ) ≤ δ′(sµ, ψµ∗G ) ≤ δ′(s,G ) whenever µ ≥ λ.

On the other hand, every closed constructible subscheme Z ⊂ S is of the form ψ−1λ Zλ for some

λ ∈ Λ and Zλ closed constructible in Sλ ([28, Ch.III, Cor.8.2.11]), so that :

(5.4.40) δ′(s,G ) = supδ′(sλ, ψλ∗G ) | λ ∈ Λ.

Similarly, for every λ ∈ Λ let xλ be the image of x in Xλ and denote by iλ : f−1λ (sλ)→ Xλ the

natural morphism. Notice that f−1λ (sλ) is a scheme of finite type over Specκ(sλ), especially

it is noetherian, hence δ′(xλ, i∗λFλ) = δ(xλ, i∗λFλ), by lemma 5.4.19(ii); by the same token

we have as well the identity : δ′(x, i∗F ) = δ(x, i∗F ). Then, since the natural morphismf−1(s)→ f−1

λ (sλ) is faithfully flat, corollary 5.4.35 yields the identity :

(5.4.41) δ′(xλ, i∗λFλ) = δ′(x, i∗F ) for every λ ∈ Λ.

Next, notice that the local scheme X(x) is the limit of the cofiltered system of local schemes(Xλ(xλ) | λ ∈ Λ). Let Zλ ⊂ Xλ(xλ) be a closed constructible subscheme and set Z := ϕ−1

λ Zλ,where ϕλ : X(x)→ Xλ(xλ) is the natural morphism. Let Yλ be the fibre product in the cartesiandiagram of schemes :

Yλqλ //

pλ

Xλ(xλ)

S

ψλ // Sλ.

The morphism ϕλ factors through a unique morphism of S-schemes ϕλ : X(x) → Yλ, and ifyλ ∈ Yλ is the image of the closed point x of X(x), then ϕλ induces a natural identificationX(x)

∼→ Yλ(yλ). To ease notation, let us set :

H := F ⊗OX f∗G and Hλ := Fλ ⊗OXλ

f ∗λψλ∗G .

In view of theorem 5.4.20(i), there follows a natural isomorphism :

(5.4.42) RΓZH|X(x)∼→ ϕ∗λRΓq−1

λ (Zλ)(q∗λFλ|Xλ(xλ) ⊗OYλ

p∗λG ).

On the other hand, applying the projection formula (see remark 5.1.21) we get :

(5.4.43) qλ∗(q∗λFλ|Xλ(xλ) ⊗OYλ

p∗λG ) 'Hλ|Xλ(xλ).

Combining (5.4.42), (5.4.43) and lemma 5.4.12(v) we deduce :

RΓZH|X(x) ' ϕ∗λRΓZλHλ|Xλ(xλ).

Notice that the foregoing argument applies also in case S is replaced by some Sµ for someµ ≥ λ (and consequently X is replaced by Xµ); we arrive therefore at the inequality :

(5.4.44) δ′(xλ,Hλ) ≤ δ′(xµ,Hµ) ≤ δ′(x,H ) whenever µ ≥ λ.

Claim 5.4.45. δ′(x,H ) = sup δ′(xλ,Hλ) | λ ∈ Λ.


Proof of the claim. Due to (5.4.44) we may assume that d := δ′(xλ,Hλ) < +∞ for everyλ ∈ Λ. The condition means that RdΓZλHλ|Xλ(xλ) 6= 0 for every λ ∈ Λ and every closedconstructible subscheme Zλ ⊂ Xλ(xλ). We have to show that RdΓZH|X(x) 6= 0 for arbitrarilysmall constructible closed subschemes Z ⊂ X(x). However, given such Z, there exists λ ∈ Λand Zλ closed constructible in Xλ(xλ) such that Z = ϕ−1(Zλ) ([28, Ch.III, Cor.8.2.11]). Saythat:

S = SpecA Sλ = SpecAλ Xλ(xλ) = SpecBλ.

Hence Yλ ' SpecA⊗Aλ Bλ. Let also Fλ (resp. G) be a Bλ-module (resp. A-module) such thatF∼λ ' Fλ (resp. G∼ ' G ). Then H|Yλ ' (Fλ ⊗Aλ G)∼. Let mλ ⊂ Bλ and nλ ⊂ Aλ be themaximal ideals. Up to replacing Z by a smaller subscheme, we may assume that Zλ = V (mλ).To ease notation, set :

E1 := ExtaAλ(κ(sλ), G) E2 := ExtbBλ/nλB(κ(xλ), Fλ/nλBλ)

with a = δ′(sλ, ψλ∗G ) and b = d− a. Proposition 5.4.31(i),(ii) and remark 5.4.38 say that

0 6= E3 := ExtdBλ(κ(xλ), Fλ ⊗Aλ G) ' E1 ⊗κ(sλ) E2.

To conclude, it suffices to show that yλ ∈ SuppE3. However :

SuppE3 = (p−1λ SuppE1) ∩ (q−1

λ SuppE2).

Now, pλ(yλ) = s, and clearly s ∈ SuppE1, since G is an A-module and s is the closed point ofS; likewise, qλ(yλ) = xλ is the closed point of Xλ(xλ), therefore it is in the support of E2. ♦

We can now conclude the proof : indeed, from theorem 5.4.36 we derive :

δ′(xλ,Hλ) = δ′(x, i∗λFλ) + δ′(f(x), ψλ∗G ) for every λ ∈ Λ

and then the corollary follows from (5.4.40), (5.4.41) and claim 5.4.45.

5.4.46. Consider now a finitely presented morphism of schemes f : X → Y , and a sheaf ofideals I ⊂ OX of finite type. Let x ∈ X be any point, and set y := f(x); notice that f−1(y)is an algebraic κ(y)-scheme, so that the local ring Of−1(y),x is noetherian. We let I (y) :=i−1y I · Of−1(y), which is a coherent sheaf of ideals of the fibre f−1(y). The fibrewise I -depth

of X over Y at the point x is the integer:

depthI ,f (x) := depthI (y)xOf−1(y),x.

In case I = OX , the fibrewise I -depth shall be called the fibrewise depth at the point x, andshall be denoted by depthf (x). In view of (5.4.29) we have the identity:

(5.4.47) depthI ,f (x) = infdepthf (z) | z ∈ V (I (y)) and x ∈ z.

5.4.48. The fibrewise I -depth can be computed locally as follows. For a given x ∈ X , sety := f(x) and let U ⊂ X be any affine open neighborhood of x such that I|U is generated bya finite family f := (fi)1≤i≤r, where fi ∈ I (U) for every i ≤ r. Then proposition 5.4.31(i)implies that :

(5.4.49) depthI ,f (x) = supn ∈ N | H i(K•(f ,OX(U))x ⊗OY,y κ(y)) = 0 for all i < n.

Proposition 5.4.50. In the situation of (5.4.46), for every integer d ∈ N we define the subset:

LI (d) := x ∈ X | depthI ,f (x) ≥ d.Then the following holds:

(i) LI (d) is a constructible subset of X .(ii) Let Y ′ → Y be any morphism of schemes, set X ′ := X ×Y Y ′, and let g : X ′ → X ,

f ′ : X ′ → Y ′ be the induced morphisms. Then:

Lg∗I (d) = g−1LI (d) for every d ∈ N.


Proof. (ii): The identity can be checked on the fibres, hence we may assume that Y = Spec kand Y ′ = Spec k′ for some fields k ⊂ k′. Let x′ ∈ X ′ be any point and x ∈ X its image; sincethe map OX,x → OX′,x′ is faithfully flat, corollary 5.4.35 implies:

depthg∗I ,f ′(x′) = depthI ,f (x)

whence (ii).(i): (We refer here to the notion of constructibility defined in [25, Ch.0, Def.2.3.10], which

is local on X; this corresponds to the local constructiblity as defined in [28, Ch.0, Def.9.1.11].)We may assume that Y = SpecA, X = SpecB, I = I∼, where A is a ring, B is a finitelypresented A-algebra, and I ⊂ B is a finitely generated ideal, for which we choose a finitesystem of generators f := (fi)1≤i≤r. We may also assume that A is reduced. Suppose first thatA is noetherian; then, for every i ∈ N and y ∈ Y let us set:

NI (i) =⋃y∈Y

SuppH i(K•(f , B)⊗A κ(y)).

Taking into account (5.4.29) and (5.4.49), we deduce:

LI (d) =d−1⋂i=0

(X\NI (i)).

It suffices therefore to show that each subset NI (i) is constructible. For every j ∈ N we set:

Zj := Ker(dj : Kj(f , B)→ Kj+1(f , B)) Bj := Im(dj−1 : Kj−1(f , B)→ Kj(f , B)).

Using [32, Ch.IV, Cor. 8.9.5], one deduces easily that there exists an affine open subschemeU ⊂ Y , say U = SpecA′ for some flat A-algebra A′, such that A′ ⊗A Z•, A′ ⊗A B• andA′ ⊗A H•(K•(f , B)) are flat A′-modules. By noetherian induction, we can then replace Yby U and X by X ×Y U , and assume from start that Z•, B• and H•(K•(f , B)) are flat A-modules. In such case, taking homology of the complex K•(f , B) commutes with any basechange; therefore:

NI (i) =⋃y∈Y

SuppH i(f , B)⊗A κ(y) = SuppH i(f , B)

whence the claim, since the support of a B-module of finite type is closed in X .Finally, for a general ring A, we can find a noetherian subalgebra A′ ⊂ A, an A′-algebra B′

of finite type and a finitely generated ideal I ′ ⊂ B′ such that B = A ⊗A′ B and I = I ′B. LetI ′ be the sheaf of ideals on X ′ := SpecB′ determined by I ′; by the foregoing, LI ′(d) is aconstructible subset of X ′. Thus, the assertion follows from (ii), and the fact that a morphismof schemes is continuous for the constructible topology ([30, Ch.IV, Prop.1.8.2]).

5.5. Depth and associated primes.

Definition 5.5.1. Let X be a scheme, F a quasi-coherent OX-module.(i) Let x ∈ X be any point; we say that x is associated to F if there exists f ∈ Fx such

that the radical of the annihilator of f in OX,x is the maximal ideal of OX,x. If x is apoint associated to F and x is not a maximal point of Supp F , we say that x is animbedded point for F . We shall denote :

Ass F := x ∈ X | x is associated to F.

(ii) We say that the OX-module F satisfies condition S1 if every associated point of F isa maximal point of X .


(iii) Likewise, if X is affine, say X = SpecA, and M is any A-module, we denote byAssAM ⊂ X (or just by AssM , if the notation is not ambiguous) the set of primeideals associated to the OX-module M∼ arising from M . We say that M satisfiescondition S1 if the same holds for the OX-module M∼.

(iv) Let x ∈ X be a point, G a quasi-coherent OX-submodule of F . We say that G is ax-primary submodule of F if Ass F/G = x. We say that G is a primary submoduleof F if there exists a point x ∈ X such that G is x-primary.

(v) We say that a submodule G of the OX-module F admits a primary decomposition ifthere exist primary submodules G1, . . . ,Gn ⊂ F such that G = G1 ∩ · · · ∩ Gn.

Remark 5.5.2. (i) Our definition of associated point is borrowed from [45, Partie I, Def.3.2.1].It agrees with that of [44, Exp.III, Def.1.1] in case X is a locally noetherian scheme (see lemma5.5.18(ii)), but in general the two notions are distinct.

(ii) For a noetherian ringA and a finitely generatedA-moduleM , our condition S1 is the sameas that of [48]. It also agrees with that of [31, Ch.IV, Def.5.7.2], in case SuppM∼ = SpecA.

Lemma 5.5.3. Let X be a scheme, F a quasi-coherent OX-module. Then :(i) Ass F ⊂ Supp F .

(ii) Ass F|U = U ∩ Ass F for every open subset U ⊂ X .(iii) F = 0 if and only if Ass F = ∅.

Proof. (i) and (ii) are obvious. To show (iii), we may assume that F 6= 0, and we have to provethat Ass F is not empty. Let x ∈ X be any point such that Fx 6= 0, and pick a non-zero sectionf ∈ Fx; set I := AnnOX,x(f), let q be a minimal prime ideal of the quotient ring OX,x/I , andq ⊂ OX,x the preimage of q. Then q corresponds to a generization y of the point x in X; letfy ∈ Fy be the image of f . Then AnnOX,y(fy) = I · OY,y, whose radical is the maximal idealq · OX,y, i.e. y ∈ Ass F .

Proposition 5.5.4. Let X be a scheme, F a quasi-coherent OX-module. We have :(i) Ass F = x ∈ X | δ(x,F ) = 0.

(ii) If 0→ F ′ → F → F ′′ is an exact sequence of quasi-coherent OX-modules, then :

Ass F ′ ⊂ Ass F ⊂ Ass F ′ ∪ Ass F ′′.

(iii) If x ∈ X is any point, and G1, . . . ,Gn (for some n > 0) are x-primary submodules ofF , then G1 ∩ · · · ∩ Gn is also x-primary.

(iv) Let f : X → Y be a finite morphism of schemes. Then(a) The natural map :⊕

x∈f−1(y)

ΓxF|X(x) → Γyf∗F|Y (y)

is a bijection for all y ∈ Y .(b) Ass f∗F = f(Ass F ).

(v) Suppose that F is the union of a filtered family (Fλ | λ ∈ Λ) of quasi-coherent OX-submodules. Then : ⋃

λ∈Λ

Ass Fλ = Ass F .

(vi) Let f : Y → X be a flat morphism of schemes, and suppose that the topological space|X| underlying X is locally noetherian. Then Ass f ∗F ⊂ f−1Ass F .

Proof. (i) and (v) follow directly from the definitions.(ii): Consider, for every point x ∈ X the induced exact sequence of OX(x)-modules :

0→ ΓxF′x → ΓxFx → ΓxF

′′x .


Then :Supp ΓxF

′x ⊂ Supp ΓxFx ⊂ Supp ΓxF

′x ∪ Supp ΓxF

′′x

which, in light of (i), is equivalent to the contention.(iii): One applies (ii) to the natural injection : F/(G1 ∩ · · · ∩ Gn)→ ⊕ni=1F/Gi.(iv): We may assume that Y is a local scheme with closed point y ∈ Y . Let x1, . . . , xn ∈ X

be the finitely many points lying over y; for every i, j ≤ n, we let

πj : X(xj)→ Y and πij : X(xi)×X X(xj)→ Y

be the natural morphisms. To ease notation, denote also by Fi (resp. Fij) the pull back of of Fto X(xj) (resp. to X(xi)×X X(xj)). The induced morphism U := X(x1)q· · ·qX(xn)→ Xis faithfully flat, so descent theory yields an exact sequence of OY -modules :

(5.5.5) 0 // f∗F //∏n

j=1 πj∗Fj

p∗1 //

p∗2

//∏n

i,j=1 πij∗Fij.

where p1, p2 : U ×X U → U are the natural morphisms.

Claim 5.5.6. The induced maps :

Γyp∗1,Γyp

∗2 :

n∏j=1

Γyπj∗Fj→

n∏i,j=1

Γyπij∗Fij

coincide.

Proof of the claim. Indeed, it suffices to verify that they coincide after projecting onto eachfactor Gij := Γyπij∗Fij . But this is clear from definitions if i = j. On the other hand, if i 6= j,the image of πij in Y does not contain y, so the corresponding factor Gij vanishes. ♦

From (5.5.5) and claim 5.5.6 we deduce that the natural map

Γyf∗F →n∏j=1

Γyπj∗Fj

is a bijection. Clearly Γyπj∗Fj = ΓxFj , so both assertions follow easily.(vi): Let x ∈ X be any point; since |X| is locally noetherian, the subset x is closed and

constructible in X(x), and then f−1(x) is closed and constructible in Y ×X X(x). Henceδ(y, f ∗F ) ≥ δ(x,F ) for every y ∈ f−1(x) (lemma 5.4.16(iii) and theorem 5.4.20(ii)). Thenthe assertion follows from (i).

Corollary 5.5.7. In the situation of corollary 5.4.39, suppose furthermore that |S| is a locallynoetherian topological space. Then we have :

Ass F ⊗OX f∗G =

⋃s∈Ass G

Ass F ⊗OS,s κ(s).

Proof. Under the stated assumptions, it is easily seen that, for every x ∈ X (resp. s ∈ S), thesubset x (resp. s) is constructible inX (resp. in S). Then the assertion follows immediatelyfrom proposition 5.5.4(i) and theorem 5.4.20(ii).

Remark 5.5.8. Actually, it can be shown that the extra assumption on |S| is superflous : see[45, Part I, Prop.3.4.3].

Corollary 5.5.9. Let X be a quasi-compact and quasi-separated scheme, F a quasi-coherentOX-module and G ⊂ F a quasi-coherent submodule. Then :

(i) For every quasi-compact open subset U ⊂ X and every quasi-coherent primary OU -submodule N ⊂ F|U , with G|U ⊂ N , there exists a quasi-coherent primary OX-submodule M ⊂ F such that M|U = N and G ⊂ N .


(ii) G admits a primary decomposition if and only if there exists a finite open coveringX = U1∪· · ·∪Un consisting of quasi-compact open subsets, such that the submodulesG|Ui ⊂ F|Ui admit primary decompositions for every i = 1, . . . , n.

Proof. Say that N is x-primary for some point x ∈ U , and set Z := X \U . According to[26, Ch.I, Prop.9.4.2], we can extend N to a quasi-coherent OX-submodule M1 ⊂ F ; up toreplacing M1 by M1 + G , we may assume that G ⊂ M1. Since (F/M1)|U = F|U/N , itfollows from lemma 5.5.3(ii) that Ass F/M1 ⊂ x ∪ Z. Let M := ΓZ(F/M1), and denoteby M the preimage of M in F . There follows a short exact sequence :

0→M → F/M1 → F/M → 0.

Clearly R1ΓZM = 0, whence a short exact sequence

0→ ΓZM → ΓZ(F/M1)→ ΓZ(F/M )→ 0.

We deduce that depthZ(F/M ) > 0; then theorem 5.4.20(ii) and proposition 5.5.4(i) show thatZ ∩ Ass (F/M ) = ∅, so that M is x-primary, as required.

(ii): We may assume that a covering X = U1 ∪ · · · ∪ Un is given with the stated property.For every i ≤ n, let G|Ui = Ni1 ∩ · · · ∩Niki be a primary decomposition; by (i) we may extendevery Nij to a primary submodule Mij ⊂ F containing G . Then G =

⋂ni=1

⋂kij=1 Mij is a

primary decomposition of G .

Lemma 5.5.10. Let X be a quasi-separated and quasi-compact scheme, F a quasi-coherentOX-module of finite type. Then the submodule 0 ⊂ F admits a primary decomposition if andonly if the following conditions hold:

(a) Ass F is a finite set.(b) For every x ∈ Ass F there is a x-primary ideal I ⊂ OX,x such that the natural map

ΓxFx → Fx/IFx

is injective.

Proof. In view of corollary 5.5.9(i) we are reduced to the case where X is an affine scheme, sayX = SpecA, and F = M∼ for an A-module M of finite type. Suppose first that 0 admits aprimary decomposition :

(5.5.11) 0 =k⋂i=1

Ni.

Then the natural map M → ⊕ki=1M/Ni is injective, hence

(5.5.12) AssM ⊂ Assk⊕i=1

M/Ni ⊂k⋃i=1

AssM/Ni

by proposition 5.5.4(ii), and this shows that (a) holds. Next, if Ni and Nj are p-primary forthe same prime ideal p ⊂ A, we may replace both of them by their intersection (proposition5.5.4(iii)). Proceeding in this way, we achieve that the Ni appearing in (5.5.11) are primarysubmodules for pairwise distinct prime ideals p1, . . . , pk ⊂ A. By (5.5.12) we have AssM ⊂p1, . . . , pk. Suppose that p1 /∈ AssM , and setQ := Ker (M → ⊕ki=2M/Ni). For every j > 1we have Γpj(M/N1)∼ = 0, therefore :

ΓpjQ∼pj

= Ker (ΓpjM∼pj→ ⊕ki=1(M/Ni)

∼pj

) = 0.


Hence AssQ = ∅, by proposition 5.5.4(i), and then Q = 0 by lemma 5.5.3(iii). In otherwords, we can omit N1 from (5.5.11), and still obtain a primary decomposition of 0; iteratingthis argument, we may achieve that AssM = p1, . . . , pk. After these reductions, we see that

Γpj(M/Ni)∼pj

= 0 whenever i 6= j

and consequently :

Ker (ϕj : ΓpjM∼pj→ (M/Nj)pj) = 0 for every j = 1, . . . , k.

Now, for given j ≤ k, let f 1, . . . , fn be a system of non-zero generators for M/Nj; by assump-tion Ii := AnnA(f i) is a pj-primary ideal for every i ≤ n. Hence qj := AnnA(M/Nj) =⋂ni=1 Ii is pj-primary as well; since ϕj factors through (M/qjM)pj , we see that (b) holds.Conversely, suppose that (a) and (b) hold. Say that AssM = p1, . . . , pk; for every j ≤ k

we choose a pj-primary ideal qj such that the map ΓpjM∼pj→Mpj/qjMpj is injective. Clearly

Nj := Ker (M → Mpj/qjMpj) is a pj-primary submodule of M ; moreover, the induced mapϕ : M → ⊕kj=1M/Nj is injective, since Ass Kerϕ = ∅ (proposition 5.5.4(ii) and lemma5.5.3(iii)). In other words, 0 =

⋂kj=1Nj is a primary decomposition of 0.

Proposition 5.5.13. Let X be a quasi-compact and quasi-separated scheme, 0→ F ′ → F →F ′′ → 0 a short exact sequence of quasi-coherent OX-modules of finite type, and suppose that

(a) Ass F ′ ∩ Supp F ′′ = ∅.(b) The submodules 0 ⊂ F ′ and 0 ⊂ F ′′ admit primary decompositions.

Then the submodule 0 ⊂ F admits a primary decomposition.

Proof. The assumptions imply that Ass F ′ and Ass F ′′ are finite sets, (lemma 5.5.10), hencethe same holds for Ass F (proposition 5.5.4(ii)). Given any point x ∈ X and any x-primaryideal I ⊂ OX,x, we may consider the commutative diagram with exact rows:

0 // ΓxF′x

//

α

ΓxFx//

β

ΓxF′′x

γ

TorOX,x1 (I,F ′′

x ) // F ′x/IF

′x

// Fx/IFx// F ′′

x /IF′′x

Now, if x ∈ Ass F ′, assumption (a) implies that ΓxF′′x = 0 = Tor

OX,x1 (Ix,F ′′

x ), and by (b)and lemma 5.5.10 we can choose I such that α is injective; a little diagram chase then showsthat β is injective as well. Similarly, if x ∈ Ass F ′′, we have ΓxF

′x = 0 and we may choose I

such that γ is injective, which implies again that β is injective. To conclude the proof it sufficesto apply again lemma 5.5.10.

Proposition 5.5.14. Let Y be a quasi-compact and quasi-separated scheme, f : X → Y a finitemorphism and F a quasi-coherent OX-module of finite type. Then the OX-submodule 0 ⊂ Fadmits a primary decomposition if and only if the same holds for the OY -submodule 0 ⊂ f∗F .

Proof. Under the stated assumptions, we can apply the criterion of lemma 5.5.10. To startout, it is clear from proposition 5.5.4(iv.b) that Ass F is finite if and only if Ass f∗F is finite.Next, suppose that 0 ⊂ F admits a primary decomposition, let y ∈ Y be any point and setf−1(y) := x1, . . . , xn; we may also assume that Y and X are affine, and then for every j ≤ nwe can find an xj-primary ideal Ij ⊂ OX such that the map ΓxjFxj → Fxj/IjFxj is injective.Let I be the kernel of the natural map OY → f∗(OX/(I1 ∩ · · · ∩ In). Then I is a quasi-coherent


y-primary ideal of OY and we deduce a commutative diagram :

(5.5.15)

⊕nj=1 Γxj(Fx) //

α

Γy(f∗Fy)

β

⊕nj=1 Fxj/IFxj

// f∗Fy/If∗Fy

whose horizontal arrows are isomorphisms, in view of proposition 5.5.4(iv.a), and where α isinjective by construction. It follows that β is injective, so condition (b) of lemma 5.5.10 holdsfor the stalk f∗Fy, and since y ∈ Y is arbitrary, we see that 0 ⊂ f∗F admits a primarydecomposition. Conversely, suppose that 0 ⊂ f∗F admits a primary decomposition; then forevery y ∈ Y we can find a quasi-coherent y-primary ideal I ⊂ OY such that the map β of(5.5.15) is injective; hence α is injective as well, and again we deduce easily that 0 ⊂ F admitsa primary decomposition.

Proposition 5.5.16. Let X and F be as in lemma 5.5.10, i : Z → X a closed constructibleimmersion, and U := X\Z. Suppose that :

(a) The OU -submodule 0 ⊂ F|U and the OZ-submodule 0 ⊂ i∗F admit primary decom-positions.

(b) The natural map ΓZF → i∗i∗F is injective.

Then the OX-submodule 0 ⊂ F admits a primary decomposition.

Proof. We shall verify that conditions (a) and (b) of lemma 5.5.10 hold for F . To checkcondition (a), it suffices to remark that U ∩ Ass F = Ass F|U (which is obvious) and thatZ ∩ Ass F ⊂ Ass i∗F , which follows easily from our current assumption (b).

Next we check that condition (b) of loc.cit. holds. This is no problem for the points x ∈U ∩ Ass F , so suppose that x ∈ Z ∩ Ass F . Moreover, we may also assume that X is affine,say X = SpecA, so that Z = V (J) for some ideal J ⊂ A. Due to proposition 5.5.14 we knowthat 0 ⊂ G := i∗i

∗F admits a primary decomposition, hence we can find an x-primary idealI ⊂ A such that the natural map ΓxG → Gx/IGx ' Fx/(I + J)Fx is injective. ClearlyI + J is again a x-primary ideal; since Z is closed and constructible, theorem 5.4.20(i) and ourassumption (b) imply that the natural map ΓxF → ΓxG is injective, hence the same holdsfor the map ΓxF → Fx/(I + J)Fx, as required.

5.5.17. Let now A be a ring, and set X := SpecA. An associated (resp. imbedded) point forthe quasi-coherent OX-module OX is also called an associated (resp. imbedded) prime ideal ofA. We notice :

Lemma 5.5.18. Let M be any A-module. The following holds :(i) AssM is the set of all p ∈ SpecA with the following property. There exists m ∈ M ,

such that p is a maximal point of the closed subset Supp(m) (i.e. p is the preimage ofa minimal prime ideal of the ring A/AnnA(m)).

(ii) If p ∈ AssM , and p is finitely generated, there exists m ∈M with p = AnnA(m).

Proof. (i): Indeed, suppose that m ∈ M \0 is any element, and p is a maximal point ofV (AnnA(m)); denote by mp ∈Mp := M ⊗A Ap the image of m. Then :

(5.5.19) AnnAp(mp) = AnnA(m)⊗A Ap.

Hence Ap/AnnAp(mp) = (A/AnnA(m))⊗A Ap, and the latter is by assumption a local ring ofKrull dimension zero; it follows easily that the radical of AnnAp(mp) is pAp, i.e. p ∈ AssAM ,as stated.

Conversely, suppose that p ∈ AssAM ; then there exists mp ∈ Mp such that the radical ofAnnAp(mp) equals pAp. We may assume that mp is the image of some m ∈ M ; then (5.5.19)


implies that V (AnnA(m)) ∩ SpecAp = V (AnnAp(mp)) = p, so p is a maximal point ofV (AnnA(m)).

(ii): Suppose p ∈ AssM is finitely generated; by definition, there exists y ∈ Mp such thatthe radical of I := AnnAp(y) equals pAp. Since p is finitely generated, some power of pAp

is contained in I . Let t ∈ N be the smallest integer such that ptAp ⊂ I , and pick any non-zero element x in pt−1Apy; then AnnAp(x) = pAp. After clearing some denominators, wemay assume that x ∈ M . Let a1, . . . , ar be a finite set of generators for p; we may then findt1, . . . , tr ∈ A\p such that tiaix = 0 in M , for every i ≤ r. Set x′ := t1 · . . . · trx; then p ⊂AnnA(x′); however AnnAp(x

′) = pAp, therefore p = AnnA(x′), whence the contention.

5.5.20. Let now p ⊂ A be any prime ideal, and n ≥ 1; if A/pn does not admit imbeddedprimes, then the ideal pn is p-primary. In the presence of imbedded primes, this fails. Forinstance, we have the following :

Example 5.5.21. Let k be a field, k[x, y] the free polynomial algebra in indeterminates x and y;consider the ideal I := (xy, y2), and set A := k[x, y]/I . Let x and y be the images of x and yin A; then p := (y) ⊂ A is a prime ideal, and p2 = 0. However, m := AnnA(y) is the maximalideal generated by x and y, so the ideal 0 ⊂ A is not p-primary.

There is however a natural sequence of p-primary ideals naturally attached to p. To explainthis, let us remark, more generally, the following :

Lemma 5.5.22. Let A be a ring, p ⊂ A a prime ideal. Denote by ϕp : A→ Ap the localizationmap. The rule :

I 7→ ϕ−1p I

induces a bijection from the set of pAp-primary ideals of Ap, to the set of p-primary ideals of A.

Proof. Suppose that I ⊂ Ap is pAp-primary; since the natural map A/ϕ−1p I → Ap/I is in-

jective, it is clear that ϕ−1p I is p-primary. Conversely, suppose that J ⊂ A is p-primary; we

claim that J = ϕ−1p (Jp). Indeed, by assumption (and by lemma 5.5.3(iii)) we have (A/J)q = 0

whenever q 6= p; it follows easily that the localization map A/J → Ap/Jp is an isomorphism,whence the contention.

Definition 5.5.23. Keep the notation of lemma 5.5.22; for every n ≥ 0 one defines the n-thsymbolic power of p, as the ideal :

p(n) := ϕ−1p (pnAp).

By lemma 5.5.22, the ideal p(n) is p-primary for every n ≥ 1. More generally, for every A-module M , let ϕM,p : M → Mp be the localization map; then one defines the p-symbolicfiltration on M , by the rule :

Fil(n)p M := ϕ−1

M,p(pnMp) for every n ≥ 0.

The filtration Fil(•)p M induces a (linear) topology on M , called the p-symbolic topology.More generally, if Σ ⊂ SpecA is any subset, we define the Σ-symbolic topology on M , as

the coarsest linear topology TΣ on M such that Fil(n)p M is an open subset of TΣ, for every

p ∈ Σ and every n ≥ 0. If Σ is finite, it is induced by the Σ-symbolic filtration, defined by thesubmodules :

Fil(n)Σ M :=

⋂p∈Σ

Fil(n)p M for every n ≥ 0.


5.5.24. LetA be a ring,M anA-module, I ⊂ A an ideal. We shall show how to characterize thefinite subsets Σ ⊂ SpecA such that the Σ-symbolic topology on M agrees with the I-preadictopology (see theorem 5.5.33). Hereafter, we begin with a few preliminary observations. First,suppose that A is noetherian; then it is easily seen that, for every prime ideal p ⊂ A, and everypAp-primary ideal I ⊂ Ap, there exists n ∈ N such that pnAp ⊂ I . From lemma 5.5.22 wededuce that every p-primary ideal of A contains a symbolic power of p; i.e., every p-primaryideal is open in the p-symbolic topology of A. More generally, let Σ ⊂ SpecA be any subset,M a finitely generated A-module, and N ⊂ M a submodule; from the existence of a primarydecomposition for N ([61, Th.6.8]), we see that N is open in the Σ-symbolic topology ofM , whenever AssM/N ⊂ Σ. Especially, if AssM/InM ⊂ Σ for every n ∈ N, then theΣ-symbolic topology is finer than the I-preadic topology on M . On the other hand, clearlyInM ⊂ Fil(n)

p M for every p ∈ SuppM/IM and every n ∈ N, so the I-preadic topology on Mis finer than the SuppM/IM -symbolic topology. Summing up, if we have :

Σ0(M) :=⋃n∈N

AssM/InM ⊂ Σ ⊂ SuppM/IM

then the Σ-symbolic topology on M agrees with the I-preadic topology. Notice that the aboveexpression for Σ0(M) is a union of finite subsets ([61, Th.6.5(i)]), hence Σ0(M) is – a priori –at most countable; in fact, we shall show that Σ0(M) is finite. Indeed, for every n ∈ N, set :

grnIA := In/In+1 grnIM := InM/In+1M.

Then gr•IA := ⊕n∈NgrnIA is naturally a graded A/I-algebra, and gr•IM := ⊕n∈NgrnIM is agraded gr•IA-module. Let ψ : Spec gr•IA→ SpecA/I be the natural morphism, and set :

Σ(M) := ψ(Assgr•IA(gr•IM)).

Lemma 5.5.25. With the notation of (5.5.24), we have :

AssA/I(grnIM) ⊂ Σ(M) for every n ∈ N.

Proof. To ease notation, set A0 := A/I and B := gr•IA. Suppose that p ∈ AssA0(grnIM); bylemma 5.5.18(i), there exists m ∈ grnIM such that p is the preimage of a minimal prime ideal ofA0/AnnA0(m). However, if we regard m as a homogeneous element of the B-module gr•IM ,we have the obvious identity :

AnnA0(m) = A0 ∩ AnnB(m) ⊂ B

from which we see that the induced map

A0/AnnA0(m)→ B/AnnB(m)

is injective, hence ψ restricts to a dominant morphism V (AnnB(m)) → V (AnnA0(m)). Es-pecially, there exists q ∈ V (AnnB(m)) with ψ(q) = p; up to replacing q by a generization,we may assume that q is a maximal point of V (AnnB(m)), hence q is an associated prime forgr•IM , again by lemma 5.5.18(i).

5.5.26. An easy induction, starting from lemma 5.5.25, shows that AssAM/InM ⊂ Σ(M), forevery n ∈ N, therefore Σ0(M) ⊂ Σ(M). However, if A is noetherian, the same holds for gr•IA(since the latter is a quotient of an A/I-algebra of finite type), hence Σ(M) is finite, providedM is finitely generated ([61, Th.6.5(i)]), and a fortiori, the same holds for Σ0(M).

Remark 5.5.27. Another proof of the finiteness of Σ0(M) can be found in [21].


5.5.28. Next, we wish to show that actually there exists a smallest subset Σmin(M) ⊂ SpecAsuch that the Σmin(M)-symbolic topology on M agrees with I-preadic topology; after somesimple reductions, this boils down to the following assertion. Let Σ ⊂ SpecA be a finitesubset, and p, p′ ∈ Σ two elements, such that the Σ-symbolic topology on M agrees with boththe Σ \p-symbolic topology and the Σ\p′-symbolic topology; then these topologies agreeas well with the Σ \p, p′-symbolic topology. Indeed, given any subset Σ′ ⊂ SpecA withp ∈ Σ′, for the Σ′-symbolic and the Σ′ \p-symbolic topologies to agree, it is necessary andsufficient that, for every n ∈ N there exists m ∈ N such that :

Fil(m)Σ′\pM ⊂ Fil(n)

p M or, what is the same : Fil(m)Σ′\pMp ⊂ pnMp.

In the latter inclusion, we may then replace Σ′ \p by the smaller subset Σ′ ∩ SpecAp \p,without changing the two terms. Suppose now that p′ /∈ SpecAp; then if we apply the above,first with Σ′ := Σ, and then with Σ′ := Σ \p′, we see that the Σ-symbolic topology agreeswith the Σ\p-symbolic topology, if and only if the Σ\p′-symbolic topology agrees with theΣ\p, p′-symbolic topology, whence the contention. In case p′ ∈ SpecAp, we may assume thatp 6= p′, otherwise there is nothing to prove; then we shall have p /∈ SpecAp′ , so the foregoingargument still goes through, after reversing the roles of p and p′.

5.5.29. Finally, theorem 5.5.33 will characterize the subset Σmin(M) as in (5.5.26). To thisaim, for every prime ideal p ⊂ A, let A∧p denote the p-adic completion of A; we set :

AssA(I,M) := p ∈ SpecA | there exists q ∈ AssA∧pM⊗AA∧p such that

√q + IA∧p = pA∧p

(where, for any ideal J ⊂ A∧p we denote by√J ⊂ A∧p the radical of J , so the above condition

selects the points q ∈ SpecA∧p , such that q ∩ V (IA∧p ) = pA∧p ).

Lemma 5.5.30. Let A be a noetherian ring, M an A-module. Then we have :(i) depthAp

M⊗AAp = depthA∧pM⊗AA∧p for every p ∈ SpecA.

(ii) AssA(0,M) = AssAM .(iii) AssA(I,M) ⊂ V (I) ∩ SuppM .(iv) Suppose that M is the union of a filtered family (Mλ | λ ∈ Λ) of A-submodules. Then :

AssA(I,M) =⋃λ∈Λ

AssA(I,Mλ).

(v) AssA(I,M) contains the maximal points of V (I) ∩ SuppM .

Proof. (iii) is immediate, and in view of [61, Th.8.8], (i) is a special case of corollary 5.4.35.(ii): By definition, AssA(0,M) consists of all the prime ideals p ⊂ A such that pA∧p is an

associated prime ideal of M⊗AA∧p ; in light of proposition 5.5.4(i), the latter condition holdsif and only if depthA∧pM⊗AA

∧p = 0, hence if and only if depthAp

M⊗AAp = 0, by (i); toconclude, one appeals again to proposition 5.5.4(i).

(iv): In view of proposition 5.5.4(v), we have AssA∧pM⊗AA∧p =⋃λ∈Λ AssA∧pMλ⊗AA∧p ; the

contention is an immediate consequence.(v): In view of (iv), we are easily reduced to the case where M is a finitely generated A-

module, in which case SuppM = V (J) for some ideal J ⊂ A. Hence, suppose that p ⊂ A isa maximal point of SpecA/(I + J), in other words, the preimage of a minimal prime ideal ofA/(I + J); notice that SuppM⊗AA∧p = V (JA∧p ). Hence we have (J + I)A∧p ⊂ q + IA∧p forevery q ∈ SuppM⊗AA∧p , so the radical of q + IA∧p equals pA∧p , as required.

Lemma 5.5.31. LetA be a local noetherian ring, m its maximal ideal, and suppose thatA is m-adically complete. Let M be a finitely generated A-module, and (FilnM | n ∈ N) a descendingfiltration consisting of A-submodules of M . Then the following conditions are equivalent :


(a)⋂n∈N FilnM = 0.

(b) For every i ∈ N there exists n ∈ N such that FilnM ⊂ miM .

Proof. Clearly (b)⇒(a), hence suppose that (a) holds. For every i, n ∈ N, set

Ji,n := Im(FilnM →M/miM).

For given i ∈ N, the A-module M/miM is artinian, hence there exists n ∈ N such that Ji :=Ji,n = Ji,n′ for every n′ ≥ n. Assertion (b) then follows from the following :

Claim 5.5.32. If (a) holds, Ji = 0 for every i ∈ N.

Proof of the claim. By inspecting the definition, it is easily seen that the natural A/mi+1-linearmap Ji+1 → Ji is surjective for every i ∈ N, hence we are reduced to showing that J := lim

i∈NJi

vanishes. However, J is naturally a submodule of limi∈N

M/miM ' M , and if x ∈ M lies in J ,

then we have x ∈ FilnM + miM for every i, n ∈ N. Since FilnM is a closed subset for them-adic topology of M ([61, Th.8.10(i)]), we have

⋂i∈N(FilnM + miM) = FilnM , for every

n ∈ N, hence x ∈⋂n∈N FilnM , which vanishes, if (a) holds.

The following theorem generalizes [70, Ch.VIII, §5, Cor.5] and [47, Prop.7.1].

Theorem 5.5.33. Let A be a noetherian ring, I ⊂ A an ideal, M a finitely generated A-module, and Σ ⊂ SpecA/I a finite subset. Then the Σ-symbolic topology on M agrees withthe I-preadic topology if and only if AssA(I,M) ⊂ Σ.

Proof. Let p ∈ AssA(I,M)\Σ, and suppose, by way of contradiction, that the Σ-symbolictopology agrees with the I-adic topology, i.e. for every n ∈ N there exists m ∈ N such that :

Fil(m)Σ M ⊂ InM.

Let X := SpecAp and U := X\p; after localizing at the prime p, we deduce the inclusion :

(5.5.34) Fil(m)Σ∩UMp ⊂ InMp

(cp. the discussion in (5.5.26)). Let also M := M∼, the quasi-coherent OX-module associatedto M ; clearly we have ImM ⊂ Fil(m)

q M for every q ∈ SpecA/I , hence (5.5.34) implies theinclusion :

x ∈Mp | x|U ∈ ImM (U) ⊂ InMp.

Let A∧p (resp. M∧p ) be the p-adic completion of A (resp. of M ), f : X∧ := SpecA∧p → X

the natural morphism, and set U∧ := f−1U = X∧ \ pA∧p . Since f is faithfully flat, and U isquasi-compact, we deduce that :

(5.5.35) x ∈M∧p | x|U∧ ∈ Imf ∗M (U∧) ⊂ InM∧

p .

On the other hand, by assumption there exists q ∈ AssA∧pM∧p such that q∩V (IA∧p ) = pA∧p ,

hence we may find x ∈ M∧p whose support is q (lemma 5.5.18(ii)). It follows easily that the

image of x vanishes in f ∗M /Imf ∗M (U) for all m ∈ N, i.e. x|U ∈ Imf ∗M (U) for everym ∈ N, hence x ∈ InM∧

p for every n ∈ N, in view of (5.5.35). However, M∧p is separated for

the p-adic topology, a fortiori also for the I-preadic topology, so x = 0, a contradiction.Next, let p ∈ Σ\AssA(I,M), set Σ′ := Σ\p, and suppose that the Σ-symbolic topology

agrees with the I-adic topology; we have to prove that the latter agrees as well with the Σ′-symbolic topology. This amounts to showing that, for every n ∈ N there exists m ∈ N suchthat :

Fil(m)Σ′ M ⊂ Fil(n)

p M or, what is the same : Fil(m)Σ′ Mp ⊂ pnMp.


We may write :

Fil(m)Σ′ Mp = x ∈Mp | x|U ∈ (Fil

(m)Σ′ Mp)

∼(U) = x ∈Mp | x|U ∈ (Fil(m)Σ Mp)

∼(U)from which it is clear that the contention holds if and only if, for every n ∈ N there existsm ∈ N such that :

x ∈Mp | x|U ∈ ImM (U) ⊂ pnMp.

Arguing as in the foregoing, we see that the latter condition holds if and only if :

FilmM∧p := x ∈M∧

p | x|U∧ ∈ Imf ∗M (U∧) ⊂ pnM∧p .

In view of lemma 5.5.31, we are then reduced to showing the following :

Claim 5.5.36.⋂m∈N FilmM∧

p = 0.

Proof of the claim. Let x be an element in this intersection; for any q ∈ U∧∩V (IA∧p ), we havexq ∈

⋂m∈N q

m(M∧p )q, hence xq = 0, by [61, Th.8.10(i)]. In other words, Supp(x)∩U∩V (I) =

∅. Suppose that x 6= 0, and let q be any maximal point of Supp(x); by lemma 5.5.18(i), q isan associated prime, and the foregoing implies that q∩V (IA∧p ) = pA∧p , which contradictsthe assumption that p /∈ AssA(I,M).

Corollary 5.5.37. In the situation of theorem 5.5.33, AssA(I,M) is a finite set.

Proof. We have already found a finite subset Σ ⊂ SpecA/I such that the Σ-symbolic topologyon M agrees with the I-preadic topology (see (5.5.26)). The contention then follows straight-forwardly from theorem 5.5.33.

Example 5.5.38. Let k be a field with char k 6= 2, and let C ⊂ A2k := Spec k[X, Y ] be the

nodal curve cut by the equation Y 2 = X2 + X3, so that the only singularity of C is the nodeat the origin p := (0, 0) ∈ C. Let R := k[X, Y, Z], A := R/(Y 2 − X2 − X3); denote byπ : A3

k := SpecR → A2k the linear projection which is dual to the inclusion k[X, Y ] → R,

so that D := π−1C = SpecA. We define a morphism ϕ : A1k := Spec k[T ] → D by the

rule : T 7→ (T 2 − 1, T (T 2 − 1), T ) (i.e. ϕ is dual to the homomorphism of k-algebras suchthat X 7→ T 2 − 1, Y 7→ T (T 2 − 1) and Z 7→ T ). Let C ′ ⊂ D be the image of ϕ, with itsreduced subscheme structure. It is easy to check that the restriction of π maps C ′ birationallyonto C, so there are precisely two points p′0, p

′1 ∈ C ′ lying over p. Let n := I(C ′) ⊂ A,

the prime ideal which is the generic point of the (irreducible) curve C ′. We claim that the n-preadic topology on A does not agree with the n-symbolic topology. To this aim – in viewof theorem 5.5.33 – it suffices to show that p′0, p′1 ⊂ AssA(n, A). However, for any closedpoint p ∈ π−1(p), the p-adic completion A∧p admits two distinct minimal primes, correspondingto the two branches of the nodal conic C at the node p, and the corresponding irreduciblecomponents of B := SpecA∧p meet along the affine line V (Z). To see this, we may supposethat p = (X, Y, Z), hence A∧p ' k[[X, Y, Z]]/(Y 2 − X2(1 + X)), and notice that the latter isisomorphic to k[[S, Y, Z]]/(Y 2 − S2), via the isomorphism that sends Y 7→ Y , Z 7→ Z andS 7→ X(1+X)1/2 (the assumption on the characteristic of k ensures that 1+X admits a squareroot in k[[X]]). Now, say that p = p′0; then C ′p := C ′ ∩ B is contained in only one of the twoirreducible components of B. Let q ∈ B be the minimal prime ideal whose closure does notcontain C ′p; then q ∈ AssA∧p and q ∩ C ′p = pA∧p , therefore p′0 ∈ AssA(n, A), as stated.

Example 5.5.39. Let A be an excellent normal ring, I ⊂ A any ideal, and set Z := V (I).Then AssA(I, A) is the set Max(Z) of all maximal points of Z. Indeed, Max(Z) ⊂ AssA(I, A)by lemma 5.5.30(v). Conversely, suppose p ∈ AssA(I, A); the completion A∧p is still normal([61, Th.32.2(i)]), and therefore its only associated prime is 0, so the assumption means thatthe radical of IA∧p is pA∧p . Equivalently, dimA∧p /IA

∧p = 0, so dimAp/IAp = 0, which is the

contention.


Definition 5.5.40. Let X be a noetherian scheme, Y ⊂ X a closed subset, X the formal com-pletion ofX along Y ([26, Ch.I, §10.8]), and f : X→ X the natural morphism of locally ringedspaces. We say that the pair (X, Y ) satisfies the Lefschetz condition, if for every open subsetU ⊂ X such that Y ⊂ U , and every locally free OU -module E of finite type, the natural map :

Γ(U,E )→ Γ(X, f ∗E )

is an isomorphism. In this case, we also say that Lef(X, Y ) holds. (Cp. [44, Exp.X, §2].)

Lemma 5.5.41. In the situation of definition 5.5.40, suppose that Lef(X, Y ) holds, and letU ⊂ X be any open subset such that Y ⊂ U . Then :

(i) The functor :

OU -Modlfft → OX-Modlfft : E 7→ f ∗E

is fully faithful (notation of (5.2.1)).(ii) Denote by OU -Alglfft the category of OU -algebras, whose underlying OU -module is free

of finite type, and define likewise OX-Alglfft. Then the functor :

OU -Alglfft → OX-Alglfft : A 7→ f ∗A

is fully faithful.

Proof. (i): Let E and F be any two locally free OU -modules of finite type. We have :

HomOU (E ,F ) = Γ(U,HomOU (E ,F ))

and likewise we may compute HomOX(f ∗E , f ∗F ). However, the natural map :

f ∗HomOU (E ,F )→HomOX(f ∗E , f ∗F )

is an isomorphism of OX-modules. The assertion follows.(ii): An object of OU -Alglfft is a locally free OU -module A of finite type, together with

morphisms A ⊗OU A → A and 1A : OU → A of OU -modules, fulfilling the usual unitarity,commutativity and associativity conditions. An analogous description holds for the objects ofOX-Alglfft, and for the morphisms of either category. Since A ⊗OU A is again locally free offinite type, the assertion follows easily from (i) : the details are left to the reader.

Lemma 5.5.42. Let A be a noetherian ring, I ⊂ A an ideal, U ⊂ SpecA an open subset, U theformal completion of U along U ∩ V (I). Consider the following conditions :

(a) Lef(U,U ∩ V (I)) holds.(b) The natural map ρU : Γ(U,OU)→ Γ(U,OU) is an isomorphism.(c) The natural map ρ : A→ Γ(U,OU) is an isomorphism.

Then (c)⇒(b)⇔(a), and (c) implies that A is I-adically complete.

Proof. Clearly (a)⇒(b), hence we assume that (b) holds, and we show (a). Let V ⊂ U be anopen subset with U ∩ V (I) ⊂ V and E a coherent locally free OV -module. As A is noetherian,V is quasi-compact, so we may find a left exact sequence P• := (0 → E → O⊕mV → O⊕nV ) ofOV -modules (corollary 5.2.17). Since the natural map of locally ringed spaces f : U → U isflat, the sequence f ∗P is still left exact. Since the global section functors are left exact, therefollows a ladder of left exact sequences :

Γ(V, P•)→ Γ(U, f ∗P•)

which reduces the assertion to the case where E = OV ; the latter is covered by the following :

Claim 5.5.43. The natural map ρV : Γ(V,OV )→ Γ(U,OU) is an isomorphism.


Proof of the claim. The isomorphism ρU factors through ρV , hence the latter is a surjection.Suppose that s ∈ Ker ρV , and s 6= 0; then we may find x ∈ V such that the image sx of s inOV,x does not vanish. Moreover, we may find a ∈ A whose image a(x) in κ(x) does not vanish,and such that as is the restriction of an element of A; especially, as ∈ Γ(U,OU), and clearlythe image of as in Γ(V,OV ) lies in Ker ρV . Therefore, as = 0 in Γ(U,OU); however the imageasx of as in OU,x is non-zero by construction, a contradiction. This shows that ρV is injective,whence the claim. ♦

Finally, suppose that (c) holds; arguing as in the proof of claim 5.5.43, one sees that ρU isan isomorphism. Moreover, since Γ(U,OU) ' lim

n∈NΓ(U,OU/I

nOU), the morphism ρ factors

through the natural A-linear map i : A → A∧ to the I-adic completion of A. The compositionwith ρ−1 yields an A-linear left inverse s : A∧ → A to i. Set N := Ker s; clearly s issurjective, hence A∧ ' A ⊕ N . It follows easily that N/InN = 0 for every n ∈ N, especiallyN ⊂

⋂n∈N I

nA∧. Therefore N = 0, since A∧ is separated for the I-adic topology.

Proposition 5.5.44. Let ϕ : A → B be a flat homomorphism of noetherian rings, I ⊂ A anideal, U ⊂ SpecB an open subset. Set f := Specϕ : SpecB → SpecA, and assume that :

(a) B is complete for the IB-adic topology.(b) For every x ∈ V (I), we have : y ∈ f−1(x) | δ(y,Of−1(x)) = 0 ⊂ U .(c) For every x ∈ AssA(I, A), we have : y ∈ f−1(x) | δ(y,Of−1(x)) ≤ 1 ⊂ U .

Then Lef(U,U ∩ V (IB)) holds.

Proof. Set Σ := AssA(I, A), and let U be the formal completion of U along V (IB); by theorem5.5.33, the I-preadic topology on A agrees with the Σ-symbolic topology. Let also J be thefamily consisting of all ideals J ⊂ A such that AssA/J ⊂ Σ; it follows that the natural maps :

B → limJ∈J

B/JB Γ(U,OU)→ limJ∈J

Γ(U,OU/JOU)

are isomorphisms (see the discussion in (5.5.24)). In view of lemma 5.5.42, we are then reducedto showing :

Claim 5.5.45. The natural map B/JB → Γ(U,OU/JOU) is an isomorphism for every J ∈J .

Proof of the claim. Let f : Y := SpecB/JB → X := SpecA/J be the induced morphism; inview of corollary 5.4.22, it suffices to prove that δ(y,OY ) ≥ 2 whenever y ∈ Y \U . Thus, setx := f(y); by corollary 5.4.39 we have :

δ(y,OY ) = δ(y,Of−1(x)) + δ(x,OX).

Now, if δ(x,OX) = 1, notice that f(Y ) ⊂ V (J) ⊂ V (I), by lemmata 5.5.18(i) and 5.5.30(iii);hence (b) implies the contention in this case. Lastly, if δ(x,OX) = 0, then x ∈ AssA/J byproposition 5.5.4, hence we use assumption (c) to conclude.

5.6. Duality over coherent schemes. We say that a scheme X is coherent if OX is coherent.If X is coherent and F is an OX-module, we define the dual OX-module

F∨ := HomOX (F ,OX).

Notice that F∨ is coherent whenever F is. Moreover, for every morphism of OX-modulesϕ : F → G , we denote by ϕ∨ : G ∨ → F∨ the induced (transpose) morphism. As usual, thereis a natural morphism of OX-modules:

βF : F → F∨∨.

One says that F is reflexive at a point x ∈ X if there exists an open neighborhood U ⊂ Xof x such that F|U is a coherent OU -module, and βF |U is an isomorphism. One says that F isreflexive if it is reflexive at all points of X . We denote by OX-Rflx the full subcategory of the


category OX-Mod, consisting of all the reflexive OX-modules. It contains OX-Modlfft as a fullsubcategory (see (5.2.1)).

Lemma 5.6.1. Let X be a coherent scheme, x ∈ X any point, and F a coherent OX-module.Then the following conditions are equivalent:

(a) F is reflexive at the point x.(b) Fx is a reflexive OX,x-module, by which we mean that the coherent sheaf induced by

Fx on Spec OX,x is reflexive.(c) The map βF ,x : Fx → (F∨∨)x is an isomorphism.

Proof. Left to the reader.

Lemma 5.6.2. Suppose that X is a reduced coherent scheme, and F a coherent OX-module.Then βF∨ is an isomorphism of OX-modules and its inverse is β∨F . Especially, F∨ is reflexive.

Proof. The assumption on X implies that the only associated points of OX are the maximalpoints of X (i.e. OX has no imbedded points). It follows easily that, for every coherent OX-module G , the dual G ∨ satisfies condition S1 (see definition 5.5.1(ii)). Now, rather generally,let M be any OX-module; directly from the definitions one derives the identity:

(5.6.3) β∨M βM∨ = 1M∨ .

It remains therefore only to show that β∨F is a right inverse for βF∗ when F is coherent. SinceF∨∨∨ satisfies condition S1, it suffices to check that, for every maximal point ξ, the inducedmap on stalks

β∨F ,ξ : F∨∨∨ξ → F∨

ξ

is a right inverse for βF∨,ξ. However, since OX,ξ is a field, β∨F ,ξ is a linear map of OX,ξ-vectorspaces of the same dimension, hence it is an isomorphism, in view of (5.6.3).

Remark 5.6.4. The following observation is often useful. Suppose thatX is a coherent scheme,F an OX-module, reflexive at a given point x ∈ X . We can then choose a presentation O⊕nX,x →O⊕mX,x → F∨

x → 0, and after dualizing we deduce a left exact sequence

(5.6.5) 0→ Fx → O⊕mX,x

u−→ O⊕nX,x.

Especially, if OX,x satisfies condition S1 (in the sense of definition 5.5.1(iii)), then the sameholds for Fx. For the converse, suppose additionally that X is reduced, and let x ∈ X be apoint for which there exists a left exact sequence such as (5.6.5); then lemma 5.6.2 says that Fis reflexive at x : indeed, Fx ' (Cokeru∨)∨.

Lemma 5.6.6. (i) Let f : X → Y be a flat morphism of coherent schemes. The induced functorOY -Mod→ OX-Mod restricts to a functor

f ∗ : OY -Rflx→ OX-Rflx.

(ii) Let X0 be a quasi-compact and quasi-separated scheme, (Xλ | λ ∈ Λ) a cofilteredfamily of coherent X0-schemes with flat transition morphisms ψλµ : Xλ → Xµ suchthat the structure morphisms Xλ → X0 are affine, and set X := lim

λ∈ΛXλ. Then:

(a) X is coherent.(b) the natural functor: 2-colim

λ∈ΛoOXλ-Rflx→ OX-Rflx is an equivalence.

(iii) Suppose that f is surjective, and let F be any coherent OY -module. Then F is reflexiveif and only if f ∗F is a reflexive OX-module.

Proof. (i) follows easily from [36, Lemma 2.4.29(i.a)].(ii): For every λ ∈ Λ, denote by ψλ : X → Xλ the natural morphism. Let U ⊂ X be a quasi-

compact open subset, u : O⊕nU → OU any morphism of OU -modules; by [32, Ch.IV, Cor.8.2.11]


there exists λ ∈ Λ and a quasi-compact open subset Uλ ⊂ Xλ such that U = ψ−1(Uλ). By [32,Ch.IV, Th.8.5.2](i) we may then suppose that u descends to a homomorphism uλ : O⊕nUλ

→ OUλ ,whose kernel is of finite type, since Xλ is coherent. Since the transition morphisms are flat, wehave Keru = ψ∗λ(Keruλ), whence (ii.a). Next, using [32, Ch.IV, Th.8.5.2] one sees easily thatthe functor of (ii.b) is fully faithful and moreover, every reflexive OX-module F descends to acoherent OXλ-module Fλ for some λ ∈ Λ. For every µ ≥ λ let Fµ := ψ∗µλFλ; since βF is anisomorphism, loc.cit. shows that βFµ is already an isomorphism for some µ ≥ λ, whence (ii.b).

(iii): By virtue of (i), we may assume that f ∗F is reflexive, and we need to show that thesame holds for F . However, the natural map f ∗(F∨∨) → (f ∗F )∨∨ is an isomorphism ([28,Ch.0, Prop.12.3.5]), hence f ∗βF = βf∗F is an isomorphism. Since f is faithfully flat, wededuce that βF is an isomorphism, as stated.

Proposition 5.6.7. Let X be a coherent, reduced, quasi-compact and quasi-separated scheme,U ⊂ X a quasi-compact open subset. We have :

(i) The restriction functor OX-Rflx→ OU -Rflx is essentially surjective.(ii) Suppose furthermore, that δ′(x,OX) ≥ 2 for every x ∈ X \ U . Then the natural map

F → j∗j∗F

is an isomorphism, for every reflexive OX-module F .

Proof. (i): Given a reflexive OU -module F , lemma 5.2.16(ii) says that we can find a finitelypresented quasi-coherent OX-module G extending F∨; since X is coherent, G is a coherentOX-module, hence the same holds for G ∨, which extends F and is reflexive in light of lemma5.6.2.

(ii): Since the assertion is local on X , we can suppose that there exists a left exact sequence0 → F → O⊕mX → O⊕nX (see remark 5.6.4). Since the functor j∗ is left exact, it then sufficesto prove the contention for the sheaves O⊕mX and O⊕nX , and thus we may assume from startthat F = OX . Then, since X \U is constructible, corollary 5.4.22 applies and yields theassertion.

Corollary 5.6.8. Let X be a coherent scheme, f : X → S be a flat, locally finitely presentedmorphism, j : U → X a quasi-compact open immersion, F a reflexive OX-module. Supposethat

(a) depthf (x) ≥ 1 for every point x ∈ X\U , and(b) depthf (x) ≥ 2 for every maximal point η of S and every x ∈ (X\U) ∩ f−1(η).(c) OS has no imbedded points.

Then the natural morphism F → j∗j∗F is an isomorphism.

Proof. Since f is flat and OS has no imbedded points, corollary 5.4.39 and our assumptions (a)and (b) imply that δ′(x,OX) ≥ 2 for every x ∈ X\U , so the assertion follows from proposition5.6.7(ii).

5.6.9. Let X be any scheme. Recall that the rank of an OX-module F of finite type, is theupper semicontinuous function:

rk F : X → N x 7→ dimκ(x) Fx ⊗OX,x κ(x).

Clearly, if F is a is locally free OX-module of finite type, rk F is a continuous function on X .The converse holds, provided X is a reduced scheme. Moreover, if F is of finite presentation,rk F is a constructible function and there exists a dense open subset U ⊂ X such that rk Frestricts to a continuous function on U . We denote by PicX the full subcategory of OX-Modlfft

consisting of all the objects whose rank is constant equal to one (i.e. the invertible OX-modules).


In case X is coherent, we shall also consider the category DivX of generically invertible OX-modules, defined as the full subcategory of OX-Rflx consisting of all objects which are locallyfree of rank one on a dense open subset of X . If X is coherent, PicX is a full subcategory ofDivX .

Remark 5.6.10. (i) Let A be any integral domain, and set X = SpecA. Classically, one hasa notion of reflexive fractional ideal of A (see [61, p.80] or example 3.4.21). Suppose nowthat X is also coherent, in which case we have the notion of reflexive OX-module of (5.6). Weclaim that these two notions overlap on the subclass of reflexive fractional ideals of finite type:more precisely, let Div(A) be the full subcategory of A-Mod whose objects are the finitelygenerated reflexive fractional ideals of A. Then the essential image of the natural functor

(5.6.11) Div(A)→ OX-Mod M 7→M∼

is the category DivX , and (5.6.11) yields an equivalence of Div(A) with the latter category.Indeed, let F be any generically invertible OX-module, and set I := F (X); if K denotesthe field of fractions of A, then dimK I ⊗A K = 1. Let us then fix a K-linear isomorphismI ⊗A K

∼→ K, and notice that the induced A-linear map I → K is injective, since F is S1

(remark 5.6.4). We may then view I as a finitely generated A-submodule of K, and then it isclear that I is a fractional ideal ofA. Moreover, on the one hand F∨ is the coherent OX-moduleHomA(I, A)∼; on the other hand, the natural map HomA(I, A)→ HomK(I ⊗A K,K) = K isinjective, and its image is the fractional ideal I−1 (see (3.4.18)). We easily deduce that I is areflexive fractional ideal, and conversely it is easily seen that every OX-module in the essentialimage of (5.6.11) is generically invertible; since this functor is also obviously fully faithful, theassertion follows.

(ii) Let A be a coherent integral domain, and denote by coh.Div(A) the set of all coherentreflexive fractional ideals of A. It is easily seen that coh.Div(A) is a submonoid of Div(A), forthe natural monoid structure introduced in example 3.4.21. More generally, if X is a coherentintegral scheme, we may define a sheaf of monoids DivX on X , as follows. First, to any affineopen subset U ⊂ X , we assign the monoid DivX(U) := coh.Div(OX(U)). For each inclusionj : U ′ ⊂ U of affine open subset, notice that the restriction map OX(U)→ OX(U ′) is a flat ringhomomorphism, hence it gives a flat morphism of monoids OX(U) \ 0 → OX(U ′) \ 0. Wethen have an induced map of monoids DivX(j) : DivX(U) → DivX(U ′), by virtue of lemma3.4.27(iv). It is easily seen that the resulting presheaf DivX is a sheaf on the site of all affineopen subsets of X . By [26, Ch.0, §3.2.2], the latter extends uniquely to a sheaf of monoids onX , which we denote again DivX . We set

Div(X) := DivX(X).

If X is normal and locally noetherian (or more generally, if X is a Krull scheme, i.e. OX(U) isa Krull ring, for every affine open subset U ⊂ X) then DivX is an abelian sheaf, and Div(X)is an abelian group (proposition 3.4.25(i,iii)).

5.6.12. For future reference, it is useful to recall some preliminaries concerning the determinantfunctors defined in [55]. Let X be a scheme. We denote by gr.PicX the category of gradedinvertible OX-modules. An object of gr.PicX is a pair (L, α), where L is an invertible OX-module and α : X → Z is a continuous function. A homomorphism h : (L, α) → (M,β) isa homomorphism of OX-modules h : L → M such that hx = 0 for every x ∈ X with α(x) 6=β(x). We denote by gr.Pic∗X the subcategory of gr.PicX with the same objects, and whosemorphisms are the isomorphisms in gr.PicX . Notice that gr.PicX is a tensor category : thetensor product of two objects (L, α) and (M,β) is the pair (L, α)⊗(M,β) := (L⊗OXM,α+β).We denote by OX-Mod∗lfft the category whose objects are the locally free OX-modules of finite


type, and whose morphisms are the OX-linear isomorphisms. The determinant is the functor:

det : OX-Mod∗lfft → gr.Pic∗X F 7→ (ΛrkFOX

F, rkF ).

Let D(OX-Mod)perf be the category of perfect complexes of OX-modules; recall that, by def-inition, every perfect complex is locally isomorphic to a bounded complex of locally free OX-modules of finite type. The category D(OX-Mod)∗perf is the subcategory of D(OX-Mod)perf

with the same objects, and whose morphisms are the isomorphisms (i.e. the quasi-isomorphismsof complexes). The main theorem of chapter 1 of [55] can be stated as follows.

Lemma 5.6.13. ([55, Th.1]) With the notation of (5.6.12) there exists, for every scheme X , anextension of the determinant functor to a functor:

det : D(OX-Mod)∗perf → gr.PicX.

These determinant functors commute with every base change.

Proposition 5.6.14. Let X be a regular scheme. Then every reflexive generically invertibleOX-module is invertible.

Proof. The question is local on X , hence we may assume that X is affine. Let F be a generi-cally invertible OX-module, and U ⊂ X a dense open subset such that F|U is invertible. Denoteby Z1, . . . , Zt the irreducible components of Z := X\U whose codimension in X equals one,and for every i = 1, . . . , t, let ηi be the maximal point of Zi, and set Ai := OX,ηi . Since Fis S1 (remark 5.6.4), the stalk Fηi is a torsion-free Ai-module of finite type for every i ≤ t;however, Ai is a discrete valuation ring, hence Fηi is a free Ai-module, necessarily of rank one,for i = 1, . . . , t. Since F is coherent, it follows that there exists an open neighborhood Ui of ηiin X , such that F|Ui is a free O|Ui-module. Hence, we may replace U by U ∪U1 ∪ · · · ∪Ut andassume that every irreducible component of Z has codimension > 1, therefore δ′(x,OX) ≥ 2for every x ∈ Z. Now, F [0] is a perfect complex by Serre’s theorem ([75, Th.4.4.16]), sothe invertible OX-module det F is well defined (lemma 5.6.13). Let j : U → X be the openimmersion; in view of proposition 5.6.7, we deduce natural isomorphisms

F∼→ j∗j

∗F∼→ j∗det(j∗F [0])

∼→ j∗j∗det F [0]

∼← det F [0]

and the assertion follows.

We wish now to introduce a notion of duality better suited to derived categories of OX-modules (over a scheme X). Hereafter we only carry out a preliminary investigation of suchderived duality – the full development of which, will be the task of section 5.8.

Definition 5.6.15. Let X be a coherent scheme. A complex ω• in Db(OX-Mod)coh (notationof (5.2.1)) is called dualizing if it fulfills the following two conditions :

(a) The functor :

D : D(OX-Mod)o → D(OX-Mod) C• 7→ RHom•OX (C•, ω•)

restricts to a duality functor : D : Db(OX-Mod)ocoh → Db(OX-Mod)coh.(b) The natural transformation : ηC• : C• → D D(C•) restricts to a biduality isomor-

phism of functors on the category Db(OX-Mod)coh.

Remark 5.6.16. Let X be a coherent scheme, ω• an object of Db(O-Mod)coh, and define thefunctor D as in definition 5.6.15. A standard devissage argument shows that ω• is dualizing onX if and only if D(F [0]) lies in Db(OX-Mod)coh, and the biduality map F [0]→ D D(F [0])is an isomorphism for every OX-module F .


Example 5.6.17. Suppose that X is a noetherian regular scheme (i.e. all the stalks OX,x areregular rings). In light of Serre’s theorem [75, Th.4.4.16], every object of Db(OX-Mod)coh

is a perfect complex. It follows easily that the complex OX [0] is dualizing. For more generalschemes, the structure sheaf does not necessarily work, and the existence of a dualizing complexis a delicate issue. On the other hand, one may ask to what extent a complex is determined bythe properties (a) and (b) of definition 5.6.15. Clearly, if ω• is dualizing on X , then so is anyother complex of the form ω• ⊗OX L , where L is an invertible OX-module. Also, any shift ofω• is again dualizing. Conversely, the following proposition 5.6.24 says that any two dualizingcomplexes are related in such manner, up to quasi-isomorphism.

Lemma 5.6.18. Let (X,OX) be any locally ringed space, and P • and Q• two objects ofD−(OX-Mod) with a quasi-isomorphism :

P •L⊗OX Q

• ∼→ OX [0].

Then there exists an invertible OX-module L , a continuous function σ : |X| → Z and quasi-isomorphisms:

P •∼→ L [σ] and Q•

∼→ L −1[−σ].

Proof. It suffices to verify that, locally onX , the complexes P • andQ• are of the required form;indeed in this case X will be a disjoint union of open sets Un on which H•P • is concentratedin degree n and H•Q• is concentrated in degree −n. Note that OX-Mod is equivalent tothe product of the categories OUn-Mod and that the derived category of a product of abeliancategories is the product of the derived categories of the factors.

Claim 5.6.19. Let A be a (commutative) local ring, K• and L• two objects of D−(A-Mod)with a quasi-isomorphism

(5.6.20) K•L⊗A L•

∼→ A[0].

Then there exists s ∈ Z and quasi-isomorphisms :

K•∼→ A[s] L•

∼→ A[−s].

Proof of the claim. Set :

i0 := max(i ∈ Z | (H iK•) 6= 0) and j0 := max(i ∈ Z | (H iL•) 6= 0).

We may assume that Ki = 0 for every i > i0, and that L• is a bounded above complex offree A-modules. Then we may find a filtered system (K•λ | λ ∈ Λ) of complexes of A-modulesbounded from above, such that

• H i0(K•λ) is a finitely generated A-module, for every λ ∈ Λ;• the colimit of the system (K•λ | λ ∈ Λ) (in the category of complexes of A-modules) is

isomorphic to K•.

From (5.6.20) we get an isomorphismH0(K•⊗AL•)∼→ A, and it follows easily that the natural

map H0(K•λ ⊗A L•)→ H0(K• ⊗A L•) is surjective for some λ ∈ Λ. For such λ, we may thenfind a morphism in D−(A-Mod) :

(5.6.21) A[0]→ K•λL⊗A L•

whose composition with the natural map K•λL⊗A L• → K•

L⊗A L• is the inverse of (5.6.20).

Hence

(5.6.21)L⊗A K• : K• → (K•λ

L⊗A L•)

L⊗A K•

∼→ K•λL⊗A (L•

L⊗A K•)

∼→ K•λ


is a right inverse of the natural morphism K•λ → K• in D−(A-Mod). Especially, the inducedmap H i0K•λ → H i0K• is surjective, i.e. H i0K• is a finitely generated A-module. Likewise, wesee that Hj0L• is a finitely generated A-module. Now, notice that

Hk(K•L⊗A L•) '

H i0K• ⊗A Hj0L• for k = i0 + j0

0 for k > i0 + j0.

From Nakayama’s lemma it follows easily that H i0+j0(K•L⊗A L•) 6= 0, and then our as-

sumptions imply that i0 + j0 = 0 and H i0K• ⊗A Hj0L• ' A. One deduces easily thatH i0K• ' A ' Hj0L• (see e.g. [36, Lemma 4.1.5]). Furthermore, we can find a complexK•1 in D<i0(A-Mod) (resp. L•1 in D<j0(A-Mod)) such that :

K• ' A[−i0]⊕K•1 (resp. L• ' A[−j0]⊕ L•1)

whence a quasi-isomorphism :

ϕ : A[0]∼→ K•

L⊗A L•

∼→ A[0]⊕K•1 [−j0]⊕ L•1[−i0]⊕ (K•1L⊗A L•1).

However, by construction ϕ−1 restricts to an isomorphism on the direct summand A[0], there-fore K•1 ' 0 ' L•1 in D(A-Mod), and the claim follows. ♦

Now, for any point x ∈ X , let ix : x → X be the inclusion map, and set K•x := i∗xK• for

every complex K• of OX-modules. Notice that, if K• is a complex of flat OX-modules, then K•xis a complex of flat OX,x-modules ([4, Exp.V, Prop.1.6(1)]), therefore the rule K• 7→ K•x yieldsa well-defined functor D−(OX-Mod) → D−(OX,x-Mod), and moreover we have a naturalisomorphism

(5.6.22) K•xL⊗OX,x L

•x∼→ (K•

L⊗OX L

•)x

for every objects K•, L• of D−(OX-Mod). Especially, under the current assumptions, and inview of claim 5.6.19, we may find s ∈ Z, and an isomorphism of OX,x-modules

(5.6.23) HsP •x ⊗OX,x H−sQ•x

∼→ H0(P •xL⊗OX,x Q

•x)∼→ OX,x.

Since OX,x is local, we may thus find ax ∈ HsP •x and bx ∈ H−sQ•x such that (5.6.23) mapsax ⊗ bx to 1. Then ax and bx extend to local sections

a ∈ Γ(U,Ker(d : P s → P s−1)) b ∈ Γ(U,Ker(d : Q−s → P−s−1))

on some neighborhood U ⊂ X of x, and after shrinking U , we may assume that a ⊗ b getsmapped to 1, under the induced morphism of OU -modules

HsP •|U ⊗OU H−sQ•|U → H0(P •

L⊗OX Q

•)|U∼→ OU .

Then we obtain a well defined morphism in D−(OU -Mod)

ϕ : OU [s]→ P •|U (resp. ψ : OU [−s]→ Q•|U )

by the rule : t 7→ t · a (resp. t 7→ t · b) for every local section t of OU . Again by claim 5.6.19and (5.6.22) we deduce that ϕy : OU,y[s] → P •y is a quasi-isomorphism for every y ∈ U (andlikewise for ψy); . i.e. ϕ and ψ are the sought isomorphisms in D−(OU -Mod).

Proposition 5.6.24. Suppose that ω•1 and ω•2 are two dualizing complexes for the coherentscheme X . Then there exists an invertible OX-module L and a continuous function σ : |X| →Z such that

ω•2 ' ω•1 ⊗OX L [σ] in Db(OX-Mod)coh.


Proof. Denote by D1 and D2 the duality functors associated to ω1 and respectively ω2. Byassumption, we can find complexes P •, Q• in Db(OX-Mod)coh such that ω2 ' D1(P •) andω1 ' D2(Q•), and therefore

D2(F •) ' RHomOX (F •,D1(P •)) ' D1(F • L⊗OX P

•)

for every object F • of Db(OX-Mod)coh ([75, Th.10.8.7]).

Claim 5.6.25. Let C• be an object of D−(OX-Mod)coh, such that D1(C•) is in Db(OX-Mod).Then C• is in Db(OX-Mod)coh.

Proof of the claim. For given m,n ∈ N, the natural maps: τ−n]C• α−→ C•

β−→ τ[−mC• induce

morphisms

D1(τ[−mC•)

D1(β)−−−→ D1(C•)D1(α)−−−→ D1(τ−n]C

•)

Say that ω• ' τ[aω• for some integer a ∈ N. Then D1(τ−n]C

•) lies in D≥n+a(OX-Mod).Since D1(C•) is bounded, it follows that D1(β) = 0 for n large enough. Consider now thecommutative diagram :

τ[−nC• βα //

τ[−mC•

η

D1 D1(τ[−nC

•)D1D1(βα)

// D1 D1(τ[−mC•)

Since τ[−mC• is a bounded complex, η is an isomorphism in D(OX-Mod), so β α = 0

whenever n is large enough. Clearly this means that C• is bounded, as claimed. ♦

Applying claim 5.6.25 to C• := F • L⊗OX P • (which is in D−(OX-Mod)coh, since X is

coherent) we see that the latter is a bounded complex, and by reversing the roles of ω1 and ω2 it

follows that the same holds for F • L⊗OX Q

•. We then deduce isomorphisms in Db(OX-Mod)coh

D1 D2(F •) ' F • L⊗OX P

• and D2 D1(F •) ' F • L⊗OX Q

•.

Letting F • := OX [0] we derive :

OX [0] ' D2 D1 D1 D2(OX [0]) ' D2 D1(P •) ' P •L⊗OX Q

•.

Then lemma 5.6.18 says that P • ' E [τ ] for an invertible OX-module E and a continuousfunction τ : |X| → Z. Consequently : ω2 ' E ∨[−τ ] ⊗OX ω1, so the proposition holds withL := E ∨ and σ := −τ .

Proposition 5.6.26. LetA be a noetherian local ring, κ its residue field,M• a bounded complexof finitely generated A-modules. Set X := SpecA and suppose that there exists c ∈ Z such that

RHom•A(κ[0],M•) ' κ[c].

Then the complex of OX-modules M•∼ arising from M• is dualizing on X .

Proof. In view of remark 5.6.16 and corollary 5.1.33, it suffices to check :

Claim 5.6.27. For every finitely generated A-module N , the following holds :(i) D•(N) := RHom•A(N [0],M•) is a bounded complex.

(ii) The natural map

N [0]→ DD•(N) := RHom•A(D•(N),M•)

is an isomorphism.


Proof of the claim. We first show the claim for N = κ, in which case (i) holds by assumption.To check (ii), let M• ∼→ I• be a resolution consisting of a bounded below complex of injectiveA-modules, so that

D•(κ[0],M•)∼→ Hom•A(κ[0], I•)

∼→ I•[m]

where m ⊂ A is the maximal ideal, and Ik[m] denotes the submodule of m-torsion elements inIk, for every k ∈ Z. Under these isomorphisms, it is easily seen that the biduality map of (ii) isidentified with the unique one

κ→ H := HomHot(A-Mod)(I•[m], I•)

that sends 1 ∈ κ to the inclusion map j• : I•[m] → I• (details left to the reader). Now, it isclear that any morphism I•[m]→ I• in Hot(A-Mod) factors through j•, and on the other hand,our assumption implies that H ' κ. Hence, pick a morphism f • : I•[m] → I• representing agenerator for the A-module H , and write f • = j• g• for some endomorphism g• of I•[m]; inother words, f • is the image of j• under the A-linear map

HomHot(A-Mod)(g•, I•) : H → H

so the class of j• cannot vanish in H , and (ii) follows in this case.Next, we shall argue by induction on d := dim SuppN . If d = 0, then N is an A-module

of finite length, in which case we argue by induction on the length l of N . If l = 1, we haveN ' κ, so the assertions are already known. Suppose l > 1, and that both (i) and (ii) are alreadyknown for all A-modules of length < d; we may find an A-submodule N ′ ⊂ N such that bothN ′ and N ′′ := N/N ′ have length < d. From the inductive assumption for N ′ and N ′′, and theinduced distiguished triangle

D•(N ′)→ D•(N)→ D(N ′′)→ D•(N ′)[1]

we deduce that (i) holds for N . Likewise, since (ii) is known for both N ′ and N ′′, using the5-lemma we deduce easily that the same holds also for N .

Lastly, suppose that d > 0, and both (i) and (ii) are already known for all A-modules of finitetype whose support has dimension < d. Let N ′ := ΓmN ; both (i) and (ii) are already knownfor N ′, so the same devissage argument as in the foregoing reduces to showing the claim forN/N ′, i.e. we may assume that m /∈ AssN . Thus, let t ∈ m be any element such that the scalarmultiplication map t · 1N is injective, so we have a short exact sequence

0→ Nt−→ N → Nk := N/tkN → 0 for every k > 0.

Notice that dim SuppNk < d : indeed, if p is a minimal element of SuppN , then p ∈ AssN([61, Th.6.5(iii)]), hence t /∈ p, and therefore p /∈ SuppNk. By inductive assumption, both (i)and (ii) hold for Nk (for every k > 0), and by considering the induced distinguished triangle

D•(N)tk−−→ D•(N)→ D•(Nk)→ D•(N)[1]

we deduce that scalar multiplication by t is an isomorphism on HkD•(N) whenever |k| is largeenough; but the latter is an A-module of finite type, so it must vanish, by Nakayama’s lemma,and we conclude that (i) holds for N . Furthermore, by the same token we get an exact sequence

H iDD•(N)tk−−→ H iDD•(N)→ 0 for every i 6= 0


whence H iDD(N) = 0 for i 6= 0, again by Nakayama’s lemma. Lastly, for every k > 0consider the ladder with exact rows :

0 // Ntk //

α

N //

α

Nk//

0

0 // H0DD•(N)tk // H0DD•(N) // H0DD•(Nk) // 0

whose right-most vertical arrow is an isomorphism, by inductive assumption. A simple diagramchase then yields

H0DD•(N) = tk ·H0DD•(N) + α(N)

so α is surjective, by Nakayama’s lemma. To show the injectivity of α, let x ∈ N be any non-zero element, and choose k > 0 such that x /∈ tkN , so the image of x does not vanish in Nk,whence necessarily α(x) 6= 0, and the claim follows.

Lemma 5.6.28. Let f : X → Y be a morphism of coherent schemes, and ω•Y a dualizingcomplex on Y . We have :

(i) If f is finite and finitely presented, then f !ω•Y is dualizing on X .(ii) If Y is quasi-compact and quasi-separated, and f is an open immersion, then f ∗ω•Y is

dualizing on X .

Proof. (i): Denote by f : (X,OX) → (Y, f∗OX) the morphism of ringed spaces deduced fromf . For any object C• of D−(OX-Mod)coh we have natural isomorphisms :

D(C•) := RHomOX (C•, f !ω•Y )∼→ RHomOX (C•, f ∗RHomOY (f∗OX , ω

•Y ))

∼→ f ∗RHomf∗OX (f∗C•, RHomOY (f∗OX , ω

•Y ))

∼→ f ∗RHomOY (f∗C•, ω•Y ).

Hence, if C• is in Db(OX-Mod)coh, the same holds for D(C•), and we can compute :

D D(C•)∼→ f ∗RHomOY (RHomOY (f∗C

•, ω•Y ), ω•Y )∼→ f ∗f∗C

• ∼→ C•

and by inspecting the definitions, one verifies that the resulting natural transformation C• →D D(C•) is the biduality isomorphism. The claim follows.

(ii): Since OY is coherent, the natural map

f ∗RHomOY (G [0], ω•Y )→ RHomOX (f ∗G [0], f ∗ω•Y )

is an isomorphism in Db(OX-Mod)coh, for every OY -module G . Then the assertion followseasily from lemma 5.2.16(ii) and remark 5.6.16.

5.6.29. For every coherent, quasi-compact and quasi-separated scheme X , consider the cate-gory DualX whose objects are the pairs (U, ω•U), where U ⊂ X is an open subset, and ω•U isa dualizing complex on X . The morphisms (U, ω•U) → (U ′, ω•U ′) are the pairs (j, β), wherej : U → U ′ is an inclusion map of open subsets of X , and β : j∗ω•U ′

∼→ ω•U is an isomor-phism in Db(OU -Mod). Let XZar denote the full subcategory of Sch/X whose objects are the(Zariski) open subsets of X; it follows easily from lemma 5.6.28(ii), that the forgetful functor

(5.6.30) DualX → XZar (U, ω•U) 7→ U

is a fibration (see definition 1.4.1(ii)) With this notation, we have :

Proposition 5.6.31. Every descent datum for the fibration (5.6.30) is effective.


Proof. Let ((Ui, ω•i ); βij | i, j = 1, . . . , n) be a descent datum for the fibration (5.6.30); this

means that each (Ui, ω•i ) is an object of DualX , and if let Uij := Ui ∩Uj for every i, j ≤ n, then

βij : ω•i|Uij∼→ ω•j|Uij

are isomorphisms in Db(OUij -Mod), fulfilling a suitable cocycle condition (see (1.5.27)).We shall show, by induction on k = 1, . . . , n, that there exists a dualizing complex ω•Xk on

Xk := U1 ∪ · · · ∪ Uk, such that the descent datum ((Ui, ω•i ); βij | i, j = 1, . . . , k) is isomorphic

to the descent datum relative to the family (Ui | i = 1, . . . , k) determined by (Xk, ω•Xk

).For k = 1, there is nothing to prove. Next, suppose that k > 1, and ω•Xk−1

with the soughtproperties has already been constructed. If k − 1 = n, we are done; otherwise, let Vk :=Xk ∩ Uk+1, set Uk := (Ui,k+1 | i = 1, . . . , k), and C• := RHom•OVk

(ω•Xk|Vk , ω•k+1|Vk). The

system of morphisms (βi,k+1 | i = 1, . . . , k) determines a class ck in the Cech cohomologygroup H0(Uk, C

•). However, we have a spectral sequence (see lemma 5.2.6) :

Epq2 := Hp(Uk, H

qC•)⇒ Hp+qRHom•OVk(ω•Xk|Vk , ω

•k+1|Vk).

Claim 5.6.32. Epq2 = 0 for every q < 0.

Proof of the claim. More precisely, we check that HqC• = 0 for every q < 0. Indeed, bylemma 5.6.28(ii), that both ω•Xk|Vk and ω•k+1|Vk are dualizing on Vk. Moreover, on Ui,k+1 theyrestrict to isomorphic objects of Db(OUi,k+1

-Mod), for every i = 1, . . . , k. It follows easilyfrom proposition 5.6.24 that there exists an invertible OVk-module L and an isomorphism

ω•Xk|Vk∼→ ω•k+1|Vk ⊗OVk

L in Db(OVk-Mod).

Therefore, the biduality map induces an isomorphism

RHom•OVk(ω•Xk|Vk , ω

•k+1|Vk)

∼→ L [0]

whence the claim. ♦

From claim 5.6.32 we see that ck corresponds to an element ofH0RHom•OVk(ω•Xk|Vk , ω

•k+1|Vk),

and by construction, it is clear that this global section is an isomorphism ck : ω•Xk|Vk∼→ ω•k+1|Vk

in Db(OVk-Mod). By definition, ck is represented by a diagram of quasi-isomorphisms :

(5.6.33) ω•Xk|Vk ← T • → ω•k+1|Vk in C(OVk-Mod)

for some complex T •. Let T! be the OXk+1-module obtained as extension by zero of T , and

define likewise (ω•Xk)! and (ω•k+1)!. We let ω•Xk+1be the cone of the map of complexes

T! → (ω•Xk)! ⊕ (ω•k+1)!

deduced from (5.6.33). An easy inspection shows that this complex is dualizing on Xk+1, andit fulfills the stated condition.

Remark 5.6.34. The proof of proposition 5.6.31 is a special case of a general technique for“glueing perverse sheaves” developed in [7].

In the study of duality for derived categories of modules, the role of reflexive modules istaken by the more general class of Cohen-Macaulay modules. There is a version of this theoryfor noetherian regular schemes, and a relative variant, for smooth morphisms. Let us begin byrecalling the following :

Definition 5.6.35. Let (A,mA) be a local ring, M an A-module.(i) The dimension ofM , denoted dimAM , is the (Krull) dimension of SuppM ⊂ SpecA.

(By convention, the empty set has dimension −∞.)


(ii) If dimAM = depthAM , we say thatM is a Cohen-MacaulayA-module. The category

A-CM

is the full subcategory of A-Mod consisting of all finitely presented Cohen-MacaulayA-modules. For every n ∈ N, we let A-CMn be the full subcategory of A-CM whoseobjects are the Cohen-Macaulay A-modules of dimension n.

(iii) If A is a Cohen-Macaulay A-module, we say that A is a Cohen-Macaulay local ring.(iv) Let X be a scheme, F a quasi-coherent OX-module. We say that F is a Cohen-

Macaulay OX-module, if Fx is a Cohen-Macaulay OX,x-module, for every x ∈ X . Wesay that X is a Cohen-Macaulay scheme, if OX is a Cohen-Macaulay OX-module.

(v) Let B be another local ring, ϕ : A→ B a local ring homomorphism. The category

ϕ-CM

of ϕ-Cohen-Macaulay modules is the full subcategory ofB-Mod whose objects are allthe finitely presented B-modules N that are ϕ-flat, and such that N/mAN is a Cohen-Macaulay B-module. For every n ∈ N, we let ϕ-CMn be the full subcategory of ϕ-CMwhose objects are the ϕ-Cohen-Macaulay modules N with dimB N/mAN = n.

(vi) Let f : X → Y be a locally finitely presented morphism of schemes, F a quasi-coherent OX-module, x ∈ X any point, and set y := f(x). We say that F is f -Cohen-Macaulay at the point x, if Fx is a f \x-Cohen-Macaulay module. We say that f isCohen-Macaulay at the point x, if OX is f -Cohen-Macaulay at the point x (cp. [31,Ch.IV, Def.6.8.1]).

(vii) Let f and F be as in (vi). We say that F is f -Cohen-Macaulay (resp. that f isCohen-Macaulay) if F (resp. OX) is f -Cohen-Macaulay at every point x ∈ X .

(viii) Let f and F be as in (vi). The f -Cohen-Macaulay locus of F is the subset

CM(f,F ) ⊂ X

consisting of all x ∈ X such that f is Cohen-Macaulay at x. The subset CM(f,OX) isalso called the Cohen-Macaulay locus of f and is denoted briefly CM(f).

Lemma 5.6.36. Let f : X → Y be a locally finitely presented morphism of schemes, and F afinitely presented OX-module such that Supp F = X . We have :

(i) CM(f,F ) is an open subset of X ,(ii) The restriction CM(f)→ Y of f is locally equidimensional.

(iii) Let g : X ′ → X be another morphism of schemes, x′ ∈ X ′ a point such that g is etaleat x′. Then g(x′) ∈ CM(f,F ) if and only if x′ ∈ CM(f g, g∗F ).

(iv) If x ∈ CM(f), then δ′(x,OX) = δ′(f(x),OY ) + dim Of−1(fx),x (notation of (5.4.18)).

Proof. (i) and (ii) follow easily from [32, Ch.IV, Prop.15.4.3], and (iii) follows from corollary5.4.35. Lastly, (iv) is an immediate consequence of corollary 5.4.39.

Proposition 5.6.37. Let A be a regular local ring of dimension d, and M a finitely generatedCohen-Macaulay A-module. Then :

(i) ExtiA(M,A) = 0 for every i 6= c := d− dimM .(ii) The A-module D(M) := ExtcA(M,A) is Cohen-Macaulay.

(iii) The natural map M → Extc(D(M), A) is an isomorphism, and Supp D(M) =SuppM .

(iv) For every n ∈ N, the functor

D : A-CMn → A-CMon N 7→ Extd−nA (N,A)

is an equivalence of categories.


Proof. (i): According to [30, Ch.0, Prop.17.3.4], the projective dimension of M equals c, sowe may find a minimal free resolution 0 → Lc → · · · → L1 → L0 → M for M of lengthc. Hence, we need only prove the sought vanishing for every i < c. Set X := SpecA,Z := SuppM ⊂ X . By virtue of proposition 5.4.24, it suffices to show that depthZOX ≥ c.In light of (5.4.29), this comes down to showing that OX,z is a local ring of depth ≥ c, forevery z ∈ Z. The latter holds, since OX,z is a regular local ring of dimension ≥ c ([30, Ch.0,Cor.16.5.12]).

(ii): From (i) we deduce that L∨• := (L∨0 → · · · → L∨n), together with its natural augmenta-tion L∨n → D(M), is a free resolution of D(M); especially, the projective dimension of D(M)is ≤ c, and therefore

depthAD(M) ≥ dimM

again by [30, Ch.0, Prop.17.3.4]. On the other hand, it follows easily from [75, Prop.3.3.10]that Supp D(M) ⊂ SuppM , so D(M) is Cohen-Macaulay.

(iii): From the proof of (ii) we see that RHomA(D(M), A) is computed by the complexL∨∨• = L•, whence the first assertion. Invoking again [75, Prop.3.3.10], we deduce thatSuppM ⊂ Supp D(M); since the converse inclusion is already known, these two supportscoincide.

(iv) follows straightforwardly from (ii) and (iii).

Corollary 5.6.38. Let X be a regular noetherian scheme, and j : Y → X a closed immersion.We have :

(i) Y admits a dualizing complex ω•Y .(ii) If Y is a Cohen-Macaulay scheme, then we may find a dualizing complex ω•Y that is

concentrated in degree 0. Moreover, H0(ω•Y ) is a Cohen-Macaulay OY -module.

Proof. Indeed, lemma 5.6.28(i) shows that the complex ω•Y := j∗RHomOX (j∗OY ,OX) will do.According to proposition 5.6.37(i,ii), this complex fulfills the condition of (ii), up to a suitableshift.

5.6.39. Let f : X → S be a smooth quasi-compact morphism of schemes, F a finitely pre-sented quasi-coherent OX-module, x ∈ X any point, and s := f(x). Set

A := OS,s B := OX,x F := Fx B0 := B ⊗A κ(s) F0 := F ⊗A κ(s).

Theorem 5.6.40. In the situation of (5.6.39), suppose that F is a f \x-Cohen-Macaulay module.Then we may find an open neighborhood U ⊂ X of x in X such that the following holds:

(i) There exists a finite resolution of the OU -module F|U , of length n := dimB0−dimB F0

Σ• : 0→ Ln → Ln−1 → · · · → L0 → F|U → 0

consisting of free OU -modules of finite rank. Moreover, Σ• is universally OS-exact, i.e.for every coherent OS-module G , the complex Σ• ⊗OX f

∗G is still exact.(ii) The complex K• := RHom•OU (F|U ,OU) is concentrated in degree n.

(iii) The natural map F|U → RHomOU (K•,OU) is an isomorphism in D(OU -Mod).(iv) The B-module G := HnK•x is f \x-Cohen-Macaulay, and SuppG = SuppF .

Proof. (i): According to proposition 4.1.37, the B-module F admits a minimal free resolution

Σx,• : · · · → L2d2−−→ L1

d1−−→ L0ε−→ F → 0

that is universally A-exact; especially, di(Li) is a flat A-module, for every i ∈ N. It also followsthat Σx,• ⊗A κ(s) is a minimal free resolution of the B0-module F0. Since the latter is Cohen-Macaulay, and B0 is a regular local ring ([33, Ch.IV, Th.17.5.1]), the projective dimension ofthe B0-module F0 equals n ([30, Ch.0, Prop.17.3.4]), therefore dn(Ln) ⊗A κ(s) is a free B0-module, so dn(Ln) is a flat B-module (lemma 4.3.35); then we deduce that it is actually a free


B-module, as it is finitely presented. Since Σx,• is minimal, we conclude that Li = 0 for everyi > n. We may now extend Σx,• to a finite resolution Σ• of F|U by free OU -modules on someopen neighborhood U of x, and after replacing U by a smaller neighborhood of U , we mayassume that F|U is f|U -flat ([32, Ch.IV, Th.11.3.1]), hence Σ• is universally OS-exact, as stated.

(ii): Let L• := (Ln → · · · → L0) be the complex obtained after omitting F from the resolu-tion Σx,• (where L0 is placed in degree 0). Then K• is isomorphic to L∨• := HomB(L•, B). Onthe other hand, the universal exactness property of Σx,• implies that

HomB0(L• ⊗A κ(s), B0) = L∨• ⊗A κ(s)

computes RHom•B0(F0, B0). In view of proposition 5.6.37(i), we have ExtiB0

(F0, B0) = 0 forevery i 6= n. From this, a repeated application of [32, Ch.IV, Prop.11.3.7] shows that L∨• isconcentrated in degree n as well, and after shrinking U , we may assume that (ii) holds (detailsleft to the reader).

(iii): Let L• := (Ln → · · · → L0) be the complex obtained by omitting F|U form theresolution Σ•; the proof of (iii) shows that RHomOU (K•,OU) is computed by L ∨∨

• = L•,whence the contention.

(iv): By the same token, [32, Ch.IV, Prop.11.3.7] implies that Coker d∨i is a flat A-module,for every i = 1, . . . , n. Especially, G is a flat A-module, and the complex L∨• [−n], with itsnatural augmentation L∨n → G, is a universally A-exact and free resolution of the B-module G.Especially, the projective dimension of G is ≤ n, therefore

(5.6.41) depthBG ≥ dimB F0

by [30, Ch.0, Prop.17.3.4]. From (iii) we also see that the induced map

Fp → ExtnBp(Gp, Bp)

is an isomorphism, for every prime ideal p ⊂ B; therefore SuppF ⊂ SuppG. Symmetrically,the same argument yields SuppG ⊂ SuppF . Hence the supports of F and G agree. Lastly,combining with (5.6.41) we see that G is Cohen-Macaulay.

5.6.42. Keep the notation of (5.6.39), and let ϕ := f \x : A → B. From theorem 5.6.40(iii,iv)we deduce that, for every n ∈ N, the functor

Dϕ : ϕ-CMn → ϕ-CMon M 7→ ExtdimB0−n

B (M,B)

is an equivalence, and the natural map M → Dϕ Dϕ(M) is an isomorphism, for every ϕ-Cohen-Macaulay module M . We shall see later also a relative variant of corollary 5.6.38, in amore special situation (see proposition 5.8.5).

5.7. Schemes over a valuation ring. Throughout this section, (K, | · |) denotes a valued field,whose valuation ring (resp. maximal ideal, resp. residue field, resp. value group) shall bedenoted K+ (resp. mK , resp. κ, resp. Γ). Also, we let

S := SpecK+ and S/b := SpecK+/bK+ for every b ∈ mK

(so S/0 = S) and we denote by s := Specκ (resp. by η := SpecK) the closed (resp. generic)point of S. More generally, for every S-scheme X we let as well

X/b := X ×S S/b for every b ∈ mK .

A basic fact, which can be deduced easily from [45, Part I, Th.3.4.6], is that every finitelypresented S-scheme is coherent; we present here a direct proof, based on the following

Proposition 5.7.1. Let A be an essentially finitely presented K+-algebra, M a finitely gener-ated K+-flat A-module. Then M is a finitely presented A-module.


Proof. Let us writeA = S−1B for some finitely presentedK+-algebraB, and some multiplica-tive subset S ⊂ B; then we may find a finitely generated B-module MB with an isomorphismϕ : S−1MB

∼→ M of A-modules. Let M ′B be the image of MB in MB ⊗K+ K; since M is

K+-flat, ϕ induces an isomorphism S−1M ′B →M . It then suffices to show that M ′

B is a finitelypresented B-module; hence we may replace A by B and M by M ′

B, and assume that A is afinitely presented K+-algebra.

We are further easily reduced to the case where A = K+[T1, . . . , Tn] is a free polynomialK+-algebra. Pick a finite system of generators Σ for M ; define increasing filtrations Fil•A andFil•M onA andM , by letting FilkA be theK+-submodule of all polynomials P (T1, . . . , Tn) ∈A of total degree ≤ k, and setting FilkM := FilkA · Σ ⊂ M for every k ∈ N. We consider theRees algebra R(A)• and the Rees module R(M)• associated to these filtrations as in definition4.4.25(iii,iv). Say that Σ is a subset of cardinality N ; we obtain an A-linear surjection ϕ :A⊕N → M , and we set M ′

k := (FilkA)⊕N ∩ Kerϕ for every k ∈ N. Notice that the resultinggraded K+-module M ′

• :=⊕

k∈NM′k is actually a R(A)•-module and we get a short exact

sequence of graded R(A)•-modules

C• : 0→M ′• → R(A)⊕N• → R(M)• → 0.

Notice also that the K+-modules R(A)⊕Nk and R(M)k are torsion-free and finitely generated,hence they are free of finite rank (see [36, Rem.6.1.12(ii)]); then the same holds for M ′

k, forevery k ∈ N. It follows that the complex C•⊗K+ κ is still short exact. On the other hand, R(A)•is a K+-algebra of finite type (see example 4.4.28), hence R := R(A)• ⊗K+ κ is a noetherianring; therefore, M ′

• ⊗K+ κ is a graded R-module of finite type, say generated by the system ofhomogenenous elements x1, . . . , xt. Lift these elements to a system Σ′ := x1, . . . , xt ofhomogeneous elements ofM ′

•, and denote byM ′′• ⊂M ′

• the R(A)•-submodule generated by Σ′.By construction, M ′′

k ⊗K+ κ = M ′k ⊗K+ κ, hence M ′′

k = M ′k for every k ∈ N, by Nakayama’s

lemma. In other words, Σ′ is a system of generators forM ′•; it follows easily that the image of Σ′

in Kerϕ is also a system of generator for the latter A-module, and the proposition follows.

Corollary 5.7.2. Every essentially finitely presented K+-algebra is a coherent ring.

Proof. One reduces easily to the case of a free polynomial K+-algebra K+[T1, . . . , Tn], inwhich case the assertion follows immediately from proposition 5.7.1 : the details shall be leftto the reader.

Lemma 5.7.3. If A if a finitely presented K+-algebra, the subset MinA ⊂ SpecA of minimalprime ideals is finite.


Claim 5.7.4. The assertion holds when A is a flat K+-algebra.

Proof of the claim. Indeed, in this case, by the going-down theorem ([61, Th.9.5]) the minimalprimes lie in SpecA⊗K+ K, which is a noetherian ring. ♦

In general, A is defined over a Z-subalgebra R ⊂ K+ of finite type. Let L ⊂ K be the fieldof fractions of R, and set L+ := L ∩K+; then L+ is a valuation ring of finite rank, and thereexists a finitely presented L+-algebra A′ such that A ' A′ ⊗L+ K+. For every prime idealp ⊂ L+, let κ(p) be the residue field of L+

p , and set :

A′(p) := A′ ⊗L+ κ(p) K+(p) := K+ ⊗L+ κ(p).

Since SpecL+ is a finite set, it suffices to show that the subset MinA⊗L+κ(p) is finite for everyp ∈ SpecL+. However, A⊗L+ κ(p) ' A′(p)⊗κ(p) K

+(p), so this ring is a flat K+(p)-algebraof finite presentation. Let I be the nilradical ofK+(p); sinceK+(p)/I is a valuation ring, claim5.7.4 applies with A replaced by A⊗K+ K+(p)/I , and concludes the proof.


Lemma 5.7.5. Let (A,mA) be a local ring, K+ → A a local and essentially finitely presentedring homomorphism. Then there exists a local morphism ϕ : V → A of essentially finitelypresented K+-algebras, such that the following holds :

(i) V is a valuation ring, its maximal ideal mV is generated by the image of mK , and themap K+ → V induces an isomorphism Γ

∼→ ΓV of value groups.(ii) ϕ induces a finite field extension V/mV → A/mA.

Proof. Set X := SpecA and let x ∈ X be the closed point. Let us pick a1, . . . , ad ∈ Awhose classes in κ(x) form a transcendence basis over κ(s). The system (ai | i ≤ d) definesa factorization of the morphism Specϕ : X → S as a composition X

g−→ Y := AdS

h−→ S,such that ξ := g(x) is the generic point of h−1(s) ⊂ Y . The morphism g is essentially finitelypresented ([30, Ch.IV, Prop.1.4.3(v)]) and moreover :

Claim 5.7.6. The stalk OY,ξ is a valuation ring with value group Γ.

Proof of the claim. Indeed, set B := K+[T1, . . . , Td]; one sees easily that V := OX,ξ is thevaluation ring of the Gauss valuation | · |B : Frac(B)→ Γ ∪ 0 such that∣∣∣∣∣∑

α∈NdaαT

α

∣∣∣∣∣B

= max|aα| | α ∈ Nd

(where aα ∈ K+ and Tα := Tα11 · · ·Tαrr for all α := (α1, . . . , αd) ∈ Nd, and aα = 0 except for

finitely many α ∈ Nd). ♦In view of claim 5.7.6, to conclude the proof it suffices to notice that κ(x) is a finite extension

of κ(ξ).

Proposition 5.7.7. Let ϕ : K+ → A and ψ : A → B be two essentially finitely presented ringhomomorphisms. Then :

(i) If ψ is integral, B is a finitely presented A-module.(ii) If A is local, ϕ is local and flat, and A/mKA is a field, then A is a valuation ring and

ϕ induces an isomorphism Γ∼→ ΓA of value groups.

Proof. (i): The assumption means that there exists a finitely presented A-algebra C and a mul-tiplicative system T ⊂ C such that B = T−1C. Let I :=

⋃t∈T AnnC(t). Then I is the kernel

of the localization map C → B; the latter is integral by hypothesis, hence for every t ∈ Tthere is a monic polynomial P (X) ∈ C[X], say of degree n, such that P (t−1) = 0 in B, hencet′tn · P (t−1) = 0 in C, for some t′ ∈ T , i.e. t′(1 − ct) = 0 holds in C for some c ∈ C, inother words, the image of t is already a unit in C/I , so that B = C/I . Then lemma 4.3.41 saysthat V (I) ⊂ SpecC is closed under generizations, hence V (I) is open, by corollary 4.3.44 andlemma 5.7.3, and finally I is finitely generated, by lemma 4.3.38(ii). SoB is a finitely presentedA-algebra. Then (i) follows from the well known :

Claim 5.7.8. Let R be any ring, S a finitely presented and integral R-algebra. Then S is afinitely presented R-module.

Proof of the claim. Let us pick a presentation : S = R[x1, . . . , xn]/(f1, . . . , fm). By assump-tion, for every i ≤ n we can find a monic polynomial Pi(T ) ∈ R[T ] such that Pi(xi) = 0 in S.Let us set S ′ := R[x1, . . . , xn]/(P1(x1), . . . , Pn(xn)); then S ′ is a free R-module of finite rank,and there is an obvious surjection S ′ → S of R-algebras, the kernel of which is generated bythe images of the polynomials f1, . . . , fm. The claim follows. ♦

(ii): By lemma 5.7.5, we may assume from start that the residue field ofA is a finite extensionof κ. We can write A = Cp for a K+-algebra C of finite presentation, and a prime ideal p ⊂ Ccontaining mK ; under the stated assumptions, A/mKA is a finite field extension of κ, hence p is


a maximal ideal of C, and moreover p is isolated in the fibre g−1(s) of the structure morphismg : Z := SpecC → S. It then follows from [32, Ch.IV, Cor.13.1.4] that, up to shrinking Z, wemay assume that g is a quasi-finite morphism ([27, Ch.II, §6.2]). Let Kh+ be the henselizationof K+, and set Ch := C ⊗K+ Kh+. There exists precisely one prime ideal q ⊂ Ch lying overp, and Ch

q is a flat, finite and finitely presented Kh+-algebra by [33, Ch.IV, Th.18.5.11], so Chq

is henselian, by [33, Ch.IV, Prop.18.5.6(i)], and then Chq is the henselization of A, since the

natural map A→ Chq is ind-etale and induces an isomorphism Ch

q ⊗K+ κ ' κ(x). Finally, Chq is

also a valuation ring with value group Γ, by virtue of [36, Lemma 6.1.13 and Rem.6.1.12(vi)].To conclude the proof of (ii), it now suffices to remark :

Claim 5.7.9. Let V be a valuation ring, ϕ : R → V a faithfully flat morphism. Then R is avaluation ring and ϕ induces an injection ϕΓ : ΓR → ΓV of the respective value groups.

Proof of the claim. Clearly R is a domain, since it is a subring of V . To show that R is avaluation ring, it then suffices to prove that, for any two ideals I, J ⊂ R, we have either I ⊂ Jor J ⊂ I . However, since V is a valuation ring, we know already that either I · V ⊂ J · V orJ · V ⊂ I · V ; since ϕ is faithfully flat, the assertion follows. By the same token, we deducethat an ideal I ⊂ R is equal to R if and only if I · V = V , which implies that ϕΓ is injective(see [36, §6.1.11]).

Proposition 5.7.10. Let f : X → S be a flat and finitely presented morphism, F a coherentOX-module, x ∈ X any point, y := f(x), and suppose that f−1(y) is a regular scheme (thisholds e.g. if f is a smooth morphism). Let also n := dim Of−1(y),x. Then:

(i) hom.dimOX,xFx ≤ n+ 1.(ii) If F is f -flat at the point x, then hom.dimOX,xFx ≤ n.

(iii) If F is reflexive at the point x, then hom.dimOX,xFx ≤ n− 1.(iv) If f has regular fibres, and F is reflexive and generically invertible (see (5.6.9)), then

F is invertible.

Proof. To start out, we may assume that X is affine, and then it follows easily from lemma5.6.6(ii.a) that OX is coherent.

(ii): Since OX is coherent, we can find a possibly infinite resolution

· · · → E2 → E1 → E0 → Fx → 0

by free OX,x-modules Ei (i ∈ N) of finite rank; set E−1 := Fx and L := Im(En → En−1).It suffices to show that the OX,x-module L is free. For every OX,x-module M we shall letM(y) := M ⊗OX,x Of−1(y),x. Since L and Fx are torsion-free, hence flat K+-modules, theinduced sequence of Of−1(y),x-modules:

0→ L(y)→ En−1(y)→ · · ·E1(y)→ E0(y)→ Fx(y)→ 0

is exact; since f−1(y) is a regular scheme, L(y) is a free Of−1(y),x-module of finite rank. SinceL is also flat as a K+-module, [30, Prop.11.3.7] and Nakayama’s lemma show that any set ofelements of L lifting a basis of L(y), is a free basis of L.

(i): Locally onX we can find an epimorphism O⊕nX → F , whose kernel G is again a coherentOX-module, since OX is coherent. Clearly it suffices to show that G admits, locally on X , afinite free resolution of length ≤ n, which holds by (ii), since G is f -flat.

(iii): Suppose F is reflexive at x; by remark 5.6.4 we can find a left exact sequence

0→ Fx → O⊕mX,x

α−→ O⊕nX,x.

It follows that hom.dimOX,xFx = max(0, hom.dimOX,x(Cokerα)− 2) ≤ n− 1, by (i).(iv): Suppose that F is generically invertible, let j : U → X be the maximal open immersion

such that j∗F is an invertible OU -module, and set Z := X\U . Under the current assumptions,


X is reduced; it follows easily that U is the set where rk F = 1, and that j is quasi-compact(since the rank function is constructible : see (5.6.9)). It follows from (iii) that Z ∩ f−1(y)has codimension ≥ 2 in f−1(y), for every y ∈ S. Hence the conditions of corollary 5.6.8 arefulfilled, and furthermore F [0] is a perfect complex by (ii), so the invertible OX-module det Fis well defined (lemma 5.6.13). We deduce natural isomorphisms:

F∼→ j∗j

∗F∼→ j∗det(j∗F [0])

∼→ j∗j∗det F [0]

∼← det F [0]

whence (iv).

Proposition 5.7.11. Let X be a quasi-compact and quasi-separated scheme, n ∈ N an integer,and denote by π : PnX → X the natural projection. Then the map

H0(X,Z)⊕ PicX → PicPnX (r,L ) 7→ OPnX (r)⊗ π∗Lis an isomorphism of abelian groups.

Proof. To begin with, let us recall the following well known :

Claim 5.7.12. Let f : Y → T be a proper morphism of noetherian schemes, F a coherentf -flat OY -module, t ∈ T a point, p ∈ N an integer, and denote it : f−1(t) → Y the naturalimmersion. Suppose that Hp+1(f−1(t), i∗tF ) = 0. Then the natural map

(Rpf∗F )t → Hp(f−1(t), i∗tF )

is surjective.

Proof of the claim. In light of [28, Ch.III, Prop.1.4.15], we easily reduce to the case where T isa local scheme, say T = SpecA for some noetherian local ring A, and t is the closed point ofT . Set k := κ(t), and for every q ∈ N, consider the functor

F−q : A-Mod→ A-Mod M 7→ H0(T,Rpf∗(f∗M∼ ⊗OY F ))

(where, as usual, M∼ denotes the quasi-coherent OT -module arising from an A-module M ).Since F is f -flat, the system (F−q | q ∈ N) defines a homological functor. The assertion isthat F−p−1(k) = 0, in which case [29, Ch.III, Cor.7.5.3] (together with [28, Ch.III, Th.3.2.1and Th.4.1.5]) says that F−p−1(M) = 0 for every A-module M . Hence, F−p−1 is trivially anexact functor, and it follows that the natural map F−p(A) → F−p(k) is surjective ([29, Ch.III,Prop.7.5.4]), which is the claim. ♦

Now, the injectivity of the stated map is clear (details left to the reader). For the surjectivity,let G be an invertible OPnX -module, write X as the limit of a filtered system (Xλ | λ ∈ Λ) ofnoetherian schemes, and for every λ ∈ Λ, let pλ : PnX → PnXλ be the induced morphism; forsome λ ∈ Λ, we may find an invertible OPnXλ

-module Fλ with an isomorphism G∼→ p∗λGλ of

OPnX -modules. Clearly, it then suffices to check the sought surjectivity for the scheme Xλ; wemay therefore replace X by Xλ, and assume from start that X is noetherian.

Next, let x ∈ X be any point, and ix : Pnκ(x) → PnX the natural immersion; we have i∗xG 'OPn

κ(x)(rx) for some rx ∈ Z. Set F := G (−rx), and notice that H1(Pnκ(x), i

∗xF ) = 0; moreover,

Hx := H0(Pnκ(x), i∗xF ) is a one-dimensional κ(x)-vector space. From claim 5.7.12, it follows

that there exists an open neighborhood U of x in X , and a section s ∈ H0(π−1U,F ) mappingto a generator of Hx. The section s defines a map of OPnU -modules ϕ : OPnU (rx) → G|π−1U .Furthermore, since i∗xF ' OPn

κ(x), it is easily seen that ϕ restricts to an isomorphism on some

open subset of the form π−1U ′, where U ′ ⊂ U is an open neighborhood of x in X .Summing up, since x ∈ X is arbitrary, and since X is quasi-compact, we may find a finite

covering X = U1 ∪ · · · ∪ Ud consisting of open subsets, and for every i = 1, . . . , d an integerri and an isomorphism ϕi : OPnUi

(ri)∼→ G|π−1Ui of OPnUi

-modules. Then, for every i, j ≤ d such


that Uij := Ui ∩ Uj 6= ∅, the composition of the restriction of ϕi followed by the restriction ofϕ−1j , is an isomorphism

ϕij : OPnUij(ri)

∼→ OPnUij(rj)

of OPnUij-modules. This already implies that ri = rj whenever Uij 6= ∅, hence the rule :

x 7→ ri if x ∈ Ui yields a well defined continuous map r : X → Z. Lastly, ϕij is the scalarmultiplication by a section

uij ∈ H0(PnUij ,O×PnUij

) = H0(Uij,O×Uij

) for every i, j = 1, . . . , d.

The system (uij | i, j = 1, . . . , d) is a 1-cocycle whose class in Cech cohomology H1(U•,O×X )

corresponds to an invertible OX-module L , with a system of trivializations OUi∼→ L|Ui , for

i = 1, . . . , d; especially, ϕi can be regarded as an isomorphism (π−1L )|π−1Ui(ri)∼→ G|π−1Ui for

every i = 1, . . . , d. By inspecting the construction, it is easily seen that these latter maps patchto a single isomorphism of OPnX -modules π∗L (r)

∼→ G , as required.

Corollary 5.7.13. Let f : X → S be a smooth morphism of finite presentation, set Gm,X :=X ×S SpecK+[T, T−1], and denote by π : Gm,X → X the natural morphism. Then the map

PicX → PicGm,X L 7→ π∗L

is an isomorphism.

Proof. To start out, the map is injective, since π admits a section. Next, let j : Gm,X → P1X be

the natural open immersion, and denote again by π : P1X → X the natural projection. According

to theorem 5.2.15 and proposition 5.7.10(i), every invertible OGm,X -module extends to an invert-ible OP1

X-module (namely the determinant of a coherent extension). Since j∗OP1

X(n) ' OGm,X ,

the claim follows from proposition 5.7.11.

Remark 5.7.14. The proof of proposition 5.7.11 is based on the argument given in [62, Lecture13, Prop.3]. Of course, statements related to corollary 5.7.13 abound in the literature, and moregeneral results are available: see e.g. [6] and [73].

Proposition 5.7.15. Let f : X → S be a smooth morphism of finite type, j : U → X an openimmersion, and set Z := X\U . Then:

(i) Every invertible OU -module extends to an invertible OX-module.(ii) Suppose furthermore, that:

(a) For every point y ∈ S, the codimension of Z ∩ f−1(y) in f−1(y) is ≥ 1, and(b) The codimension of Z ∩ f−1(η) in f−1(η) is ≥ 2.

Then the restriction functors:

(5.7.16) OX-Rflx→ OU -Rflx PicX → PicU

are equivalences.

Proof. (i): Notice first that the underling space |X| is noetherian, since it is the union |f−1(s)|∪|f−1(η)| of two noetherian spaces. Especially, every open immersion is quasi-compact, and Xis quasi-separated. We may then reduce easily to the case where U is dense in X , in which casethe assertion follows from propositions 5.6.7(i) and 5.7.10(iv).

(ii): We begin with the following:

Claim 5.7.17. Under the assumptions of (ii), the functors (5.7.16) are faithful.

Proof of the claim. Since f is smooth, all the stalks of OX are reduced ([33, Ch.IV, Prop.17.5.7]).Since every point of Z is specialization of a point of U , the claim follows easily from remark5.6.4. ♦


Under assumptions (a) and (b), the conditions of corollary 5.6.8 are fulfilled, and one de-duces that (5.7.16) are full functors, hence fully faithful, in view of claim 5.7.17. The essentialsurjectivity of the restriction functor for reflexive OX-modules is already known, by proposition5.6.7(i).

Lemma 5.7.18. Let f : X → S be a flat, finitely presented morphism, and denote by U ⊂ Xthe maximal open subset such that the restriction f|U : U → S is smooth. Suppose that thefollowing two conditions hold:

(a) For every point y ∈ S, the fibre f−1(y) is geometrically reduced.(b) The generic fibre f−1(η) is geometrically normal.

Then the restriction functor induces an equivalence of categories (notation of (5.6.9)) :

(5.7.19) DivX∼→ PicU.

Proof. Let F be any generically invertible reflexive OX-module; it follows from proposition5.7.10(iv) that F|U is an invertible OU -module, so the functor (5.7.19) is well defined. Lety ∈ S be any point; condition (a) says in particular that the fibre f−1(y) is generically smoothover κ(y) ([61, Th.28.7]). Then it follows from [33, Ch.IV, Th.17.5.1] that U∩f−1(y) is a denseopen subset of f−1(y), and if x ∈ f−1(y)\U , then the depth of Of−1(y),x is≥ 1, since the latter isa reduced local ring of dimension ≥ 1. Similarly, condition (b) and Serre’s criterion [61, Ch.8,Th.23.8] say that for every x ∈ f−1(η)\U , the local ring Of−1(η),x has depth ≥ 2. Furthermore,[32, Ch.IV, Prop.9.9.4] implies that the open immersion j : U → X is quasi-compact; summingup, we see that all the conditions of corollary 5.6.8 are fulfilled, so that F = j∗j

∗F for everyreflexive OX-module F . This already means that (5.7.19) is fully faithful. To conclude, itsuffices to invoke proposition 5.6.7(i).

The rest of this section shall be concerned with some results that hold in the special casewhere the valuation of K has rank one.

Theorem 5.7.20. Suppose that K is a valued field of rank one. Let f : X → S be a finitelypresented morphism, F a coherent OX-module. Then :

(i) Every coherent proper submodule G ⊂ F admits a primary decomposition.(ii) Ass F is a finite set.

Proof. By lemma 5.5.10, assertion (ii) follows from (i). Using corollary 5.5.9(ii) we reduceeasily to the case where X is affine, say X = SpecA for a finitely presented K+-algebra A,and F = M∼, G = N∼ for some finitely presented A-modules N ⊂ M 6= 0. By consideringthe quotientM/N , we further reduce the proof to the case whereN = 0. Let us choose a closedimbedding i : X → SpecB, where B := K+[T1, . . . , Tr] is a free polynomial K+-algebra; byproposition 5.5.14, the submodule 0 ⊂ F admits a primary decomposition if and only if thesubmodule 0 ⊂ i∗F does. Thus, we may replaceA byB, and assume thatA = K+[T1, . . . , Tr],in which case we shall argue by induction on r. Let ξ denote the maximal point of f−1(s); byclaim 5.7.6, V := OX,ξ is a valuation ring with value group Γ. Then [36, Lemma 6.1.14] saysthat

Mξ ' V ⊕m ⊕ (V/b1V )⊕ · · · ⊕ (V/bkV )

for some m ∈ N and elements b1, . . . , bk ∈ mK \0. After clearing some denominators, wemay then find a map ϕ : M ′ := A⊕m⊕ (A/b1A)⊕ · · · ⊕ (A/bkA)→M whose localization ϕξis an isomorphism.

Claim 5.7.21. AssA/bA = ξ whenever b ∈ mK\0.

Proof of the claim. Clearly AssA/bA ⊂ f−1(s). Let x ∈ f−1(s) be a non-maximal point.Thus, the prime ideal p ⊂ A corresponding to x is not contained in mKA, i.e. there exists a ∈ p


such that |a|A = 1. Suppose by way of contradiction, that x ∈ AssA/bA; then we may findc ∈ A such that c /∈ bA but anc ∈ bA for some n ∈ N. The conditions translate respectively asthe inequalities :

|c|A > |b|A and |an|A · |c|A = |anc|A ≤ |b|Awhich are incompatible, since |an|A = 1. ♦

Since (Kerϕ)ξ = 0, we deduce from claim 5.7.21 and proposition 5.5.4(ii) that Ass Kerϕ =∅; therefore Kerϕ = 0, by lemma 5.5.3(iii). Moreover, the submodule N1 := A⊕m (resp.N2 := (A/b1A)⊕ · · · ⊕ (A/bkA)) of M ′ is either ξ-primary (resp. 0-primary), or else equalto M ′; in the first case, 0 ⊂ M ′ admits the primary decomposition 0 = N1 ∩ N2, and in thesecond case, either 0 is primary, or else M ′ = 0 has the empty primary decomposition. Now,if r = 0 then M = M ′ and we are done. Suppose therefore that r > 0 and that the theoremis known for all integers < r. Set M ′′ := Cokerϕ; clearly ξ /∈ SuppM ′′, so by proposition5.5.13 we are reduced to showing that the submodule 0 ⊂M ′′ admits a primary decomposition.We may then replace M by M ′′ and assume from start that ξ /∈ Supp F . Let F t ⊂ F bethe K+-torsion submodule; clearly F/F t is a flat K+-module, hence it is a finitely presentedOX-module (proposition 5.7.1), so F t is a finitely generated OX-module, by [17, Ch.10, §.1,n.4, Prop.6]. Hence we may find a non-zero a ∈ mK such that aF t = 0, i.e. the natural mapF t → F/aF is injective. Denote by i : Z := V (Ann F/aF )→ X the closed immersion.

Claim 5.7.22. There exists a finite morphism g : Z → Ar−1S .

Proof of the claim. Since ξ /∈ Supp F , we have I * mKA, i.e. we can find b ∈ I such that|b|A = 1. Set W := V (a, b) ⊂ X; it suffices to exhibit a finite morphism W → Ar−1

S . By[13, §5.2.4, Prop.2], we can find an automorphism σ : A/aA → A/aA such that σ(b) = u · b′,where u is a unit, and b′ ∈ (K+/aK+)[T1, . . . , Tr] is of the form b′ = T kr +

∑k−1j=0 ajT

jr , for

some a0, . . . , ak−1 ∈ (K+/aK+)[T1, . . . , Tr−1]. Hence the ring A/(aA+ bA) = A/(aA+ b′A)is finite over K+[T1, . . . , Tr−1], and the claim follows. ♦

Claim 5.7.23. The OX-submodule 0 ⊂ F/aF admits a primary decomposition.

Proof of the claim. Let G := (F/aF )|Z . By our inductive assumption the submodule 0 ⊂ g∗Gon Ar−1

S admits a primary decomposition; a first application of proposition 5.5.14 then showsthat the submodule 0 ⊂ G on Z admits a primary decomposition, and a second applicationproves the same for the submodule 0 ⊂ i∗G = F/aF . ♦

Finally, let j : Y := V (a) ⊂ X be the closed immersion, and set U := X \Y . By con-struction, the natural map ΓY F → j∗j

∗F is injective. Furthermore, U is an affine noetherianscheme, so [61, Th.6.8] ensures that the OU -submodule 0 ⊂ F|U admits a primary decomposi-tion. The same holds for j∗F , in view of claim 5.7.23. To conclude the proof, it remains onlyto invoke proposition 5.5.16.

Corollary 5.7.24. Let K be a valued field of rank one, f : X → S a finitely presented mor-phism, F a coherent OX-module, and suppose there is given a cofiltered family I := (Iλ | λ ∈Λ) such that :

(a) Iλ ⊂ OX is a coherent ideal for every λ ∈ Λ.(b) Iλ ·Iµ ∈ I whenever Iλ,Iµ ∈ I .

Then the following holds :(i) Fλ := AnnF (Iλ) ⊂ F is a submodule of finite type for every λ ∈ Λ.

(ii) There exists λ ∈ Λ such that Fµ ⊂ Fλ for every µ ∈ Λ.

Proof. We easily reduce to the case whereX is affine, sayX = SpecAwithA finitely presentedover K+, and for each λ ∈ Λ the ideal Iλ := Γ(X,Iλ) ⊂ A is finitely generated.


(i): For given λ ∈ Λ, let f1, . . . , fk be a finite system of generators of Iλ; then Fλ is thekernel of the map ϕ : F → F⊕k defined by the rule : m 7→ (f1m, . . . , fkm), which is finitelygenerated because A is coherent.

(ii): By theorem 5.7.20 we can find primary submodules G1, . . . ,Gn ⊂ F such that G1 ∩· · · ∩ Gn = 0; for every i ≤ n and λ ∈ Λ set Hi := F/Gi and Hi,λ := AnnHi

(Iλ). Since thenatural map F → ⊕ni=1Hi is injective, we have :

Fλ = F ∩ (H1,λ ⊕ · · · ⊕Hn,λ) for every λ ∈ Λ.

It suffices therefore to prove that, for every i ≤ n, there exists λ ∈ Λ such that Hi,µ = Hi,λ forevery µ ∈ Λ. Say that Hi is p-primary, for some prime ideal p ⊂ A; suppose now that thereexists λ such that Iλ ⊂ p; since Iλ is finitely generated, we deduce that InλHi = 0 for n ∈ Nlarge enough; from (b) we see that I n

λ ∈ I , so (ii) holds in this case. In case Iλ * p for everyλ ∈ Λ, we have AnnHi

(Iλ) = 0 for every λ ∈ Λ, so (ii) holds in this case as well.

Corollary 5.7.25. Suppose A is a valuation ring with value group ΓA, and ϕ : K+ → A is anessentially finitely presented local homomorphism from a valuation ring K+ of rank one. Then:

(i) If the valuation of K is not discrete, ϕ induces an isomorphism Γ∼→ ΓA.

(ii) If Γ ' Z and ϕ is flat, ϕ induces an inclusion Γ ⊂ ΓA, and (ΓA : Γ) is finite.

Proof. Suppose first that Γ ' Z; then A is noetherian, hence ΓA is discrete of rank one as well,and the assertion follows easily. In case Γ is not discrete, we claim that A has rank≤ 1. Indeed,suppose by way of contradiction, that the rank of A is higher than one, and let mA ⊂ A bethe maximal ideal; then we can find a, b ∈ mA\0 such that a−ib ∈ A for every i ∈ N. Letus consider the A-module M := A/bA; we notice that AnnM(ai) = a−ibA form a strictlyincreasing sequence of ideals, contradicting corollary 5.7.24(ii). Next we claim that ϕ is flat.Indeed, suppose this is not the case; then A is a κ-algebra. Let p ⊂ A be the maximal ideal;by lemma 5.7.5, we may assume that A/p is a finite extension of κ. Now, choose any finitelypresented quotient A of A supported at p; it follows easily that A is integral over K+, henceit is a finitely presented K+-module by proposition 5.7.7(i), and its annihilator contains mK ,which is absurd in view of [36, Lemma 6.1.14]. Next, since Γ is not discrete, one sees easilythat mKA is a prime ideal, and in light of the foregoing, it must then be the maximal ideal. Nowthe assertion follows from proposition 5.7.7(ii).

5.7.26. Let B be a finitely presented K+-algebra, A := (A,Fil•A) be a pair consisting of aB-algebra and a B-algebra filtration on A; let also M be a finitely generated A-module, andFil•M a good A-filtration on M (see definition 4.4.27). By definition, this means that the Reesmodule R(M)• of the filteredA-moduleM := (M,Fil•M) is finitely generated over the gradedRees B-algebra R(A)•. We have :

Proposition 5.7.27. In the situation of (5.7.26), suppose that A is a finitely presented B-algebra, M is a finitely presented A-module, and Fil•A is a good filtration. Then :

(i) R(A)• is a finitely presented B-algebra, and R(M)• is a finitely presented R(A)•-module.

(ii) If furthermore, Fil•A is a positive filtration (see definiion 4.4.27(ii)), then FiliM is afinitely presented B-module, for every i ∈ Z.

Proof. By lemma 4.4.31, there exists a finite system of generators m := (m1, . . . ,mn) of M ,and a sequence of integers k := (k1, . . . , kn) such that Fil•M is of the form (4.4.30).

(i): Consider first the case where A is a free B-algebra of finite type, say A = B[t1, . . . , tp],such that Fil•A is the good filtration associated to the system of generators t := (t1, . . . , tp)and the sequence of integers r := (r1, . . . , rp), and moreover M is a free A-module with basism. Then R(A)• is also a free B-algebra of finite type (see example 4.4.28). Moreover, for


every j ≤ n, let Mj ⊂ M be the A-submodule generated by mj , and denote by Fil•Mj thegood A-filtration associated to the pair (mj, kj); clearly Fil•M = Fil•M1⊕ · · · ⊕Fil•Mn,therefore R(M)• = R(M1)• ⊕ · · · ⊕ R(Mn)•, where M j := (Mj,Fil•Mj) for every j ≤ n.Obviously each R(A)•-module R(M j)• is free of rank one, so the proposition follows in thiscase.

Next, suppose that A is a free B-algebra of finite type, and M is arbitrary. Let F be afree A-module of rank n, e := (e1, . . . , en) a basis of F , and define an A-linear surjectionϕ : F → M by the rule ej 7→ mj for every j ≤ n. Then ϕ is even a map of filtered A-modules, provided we endow F with the good A-filtration Fil•F associated to the pair (e,k).More precisely, let N := Kerϕ; then the filtration Fil•M is induced by Fil•F , meaning thatFiliM := (N + FiliF )/N for every i ∈ Z. Obviously the natural map R(F )• → R(M)•is surjective, and its kernel is the Rees module R(N)• corresponding to the filtered A-moduleN := (N,Fil•N) with FiliN := N ∩ FiliF for every i ∈ Z. Let x := (x1, . . . , xs) be afinite system of generators of N , and choose a sequence of integers j := (j1, . . . , js) such thatxi ∈ FiljiN for every i ≤ s; denote by L the filtered A-module associated as in (4.4.29), to theA-module N and the pair (x, j). Thus, R(L)• is a finitely generated graded R(A)•-submoduleof R(N)•. To ease notation, set N := R(N)•/R(L)• and F := R(F )•/R(L)•. Recall (definition4.4.25(iii)) that R(A)• is a graded B-subalgebra of A[U,U−1]; then we have :

Claim 5.7.28. N =⋃n∈N AnnF (Un).

Proof of the claim. It suffices to consider a homogeneous element y := U iz ∈ R(F )i, for somesome z ∈ FiliF . Suppose that Uny ∈ R(L)i+n; especially, U i+nz ∈ R(N)i+n, so z ∈ N ,hence y ∈ R(N)i. Conversely, suppose that y ∈ R(N)i; write z = x1a1 + · · · + xsas for somea1, . . . , as ∈ A. Say that ar ∈ FilbrA for every r ≤ s, and set l := max(i, b1 + j1, . . . , bs + js).Then U l−iy = U lz ∈ R(L)l. ♦

By the foregoing, R(F )• is finitely presented over R(A)•. It follows that N is finitely gener-ated (corollary 5.7.24), hence R(N)• is a finitely generated R(A)•-module, so finally R(M)• isfinitely presented over R(A)•.

Lastly, consider the case of an arbitrary finitely presented B-algebra A, endowed with thegood B-algebra filtration associated to a system of generators x := (x1, . . . , xp) and the se-quence of integers r. We map the free B-algebra P := B[t1, . . . , tp] onto A, by the rule:tj 7→ xj for every j ≤ p. This is even a map of filtered algebras, provided we endow P with thegood filtration Fil•P associated to the pair (t, r); more precisely, Fil•P induces the filtrationFil•A on A, hence Fil•A is a good P -filtration. By the foregoing case, we then deduce thatR(A)• is a finitely presented R(P )•-module (where P := (P,Fil•P )), hence also a finitely pre-sented K+-algebra. Moreover, M is a finitely presented P -module, and clearly Fil•M is goodwhen regarded as a P -filtration, hence R(M)• is a finitely presented R(P )•-module, so also afinitely presented R(A)•-module.

(ii): The positivity condition implies that R(A)0 = B; then, taking into account (i), theassertion is just a special case of proposition 4.4.16(iii).

Theorem 5.7.29 (Artin-Rees lemma). Let A be an essentially finitely presented K+-algebra,I ⊂ A an ideal of finite type, M a finitely presented A-module, N ⊂ M a finitely generatedsubmodule. Then there exists an integer c ∈ N such that :

InM ∩N = In−c(IcM ∩N) for every n ≥ c.

Proof. We reduce easily to the case where A is a finitely presented K+-algebra; then, with allthe work done so far, we only have to repeat the argument familiar from the classical noetheriancase. Indeed, let us define a filtered K+-algebra A := (A,Fil•A) by the rule : FilnA := I−n

if n ≤ 0, and FilnA := A otherwise; also let M := (M,Fil•M) be the filtered A-module


such that FiliM := M · FiliA for every i ∈ Z. We endow Q := M/N with the filtrationFil•Q induced from Fil•M , so that the natural projection M → M/N yields a map of filteredA-modules M → Q := (Q,Fil•Q). By proposition 5.7.27, there follows a surjective mapof finitely presented graded R(A)•-modules : π• : R(M)• → R(Q)•, whose kernel in degreek ≤ 0 is the A-module I−kM ∩N . Hence we can find a finite (non-empty) system f1, . . . , fr ofgenerators for the R(A)•-module Ker π•; clearly we may assume that each fi is homogeneous,say of degree di; we may also suppose that di ≤ 0 for every i ≤ r, since Ker π0 generatesKer πn, for every n ≥ 0. Let d := min (di | i = 1, . . . , r); by inspecting the definitions, oneverifies easily that

R(A)i ·Ker πj ⊂ R(A)i+j−d ·Ker πd whenever i+ j ≤ d and 0 ≥ j ≥ d.

Therefore Ker πd+k = R(A)k ·Ker πd for every k ≤ 0, so the assertion holds with c := −d.

5.7.30. In the situation of theorem 5.7.29, letM be any finitely presentedA-module; we denoteby M∧ the I-adic completion of M , which is an A∧-module.

Corollary 5.7.31. With the notation of (5.7.30), the following holds :

(i) The functor M 7→M∧ is exact on the category of finitely presented A-modules.(ii) For every finitely presented A-module M , the natural map M ⊗A A∧ → M∧ is an

isomorphism.(iii) A∧ is flat over A. If additionally I ⊂ radA (the Jacobson radical of A), then A∧ is

faithfully flat over A.

Proof. (i): Let N ⊂ M be any injection of finitely presented A-modules; by theorem 5.7.29,the I-adic topology on M induces the I-adic topology on N . Then the assertion follows from[61, Th.8.1].

(ii): Choose a presentation A⊕p → A⊕q → M → 0. By (i) we deduce a commutativediagram with exact rows :

A⊕p ⊗A A∧ //

A⊕q ⊗A A∧ //

M ⊗A A∧ //

0

(A⊕p)∧ // (A⊕q)∧ // M∧ // 0

and clearly the two left-most vertical arrows are isomorphisms. The claim follows.(iii): The first assertion means that the functor M 7→M ⊗AA∧ is exact; this follows from (i)

and (ii), via a standard reduction to the case where M is finitely presented. Suppose next thatI ⊂ radA; to conclude, it suffices to show that the image of the natural map SpecA∧ → SpecAcontains the maximal spectrum MaxA ([61, Th.7.3]). This is clear, since A∧/IA∧ ' A/IA,and the natural map MaxA/IA→ MaxA is a bijection.

Theorem 5.7.32. Let B → A be a map of finitely presented K+-algebras. Then :

(i) If M is an ω-coherent A-module, M is ω-coherent as a B-module.(ii) If J is a coh-injective B-module, and I ⊂ B is a finitely generated ideal, we have :

(a) HomB(A, J) is a coh-injective A-module.(b)⋃n∈N AnnJ(In) is a coh-injective B-module.

(iii) If (Jn, ϕn | n ∈ N) is a direct system consisting of coh-injective B/In-modules Jn andB-linear maps ϕn : Jn → Jn+1 (for every n ∈ N), then colim

n∈NJn is a coh-injective

B-module.

Proof. (See (4.3.25) for the generalities on coh-injective and ω-coherent modules.)


(i): We reduce easily to the case where M is finitely presented over A. Let x1, . . . , xp be asystem of generators for the B-algebra A, and m1, . . . ,mn a system of generators for the A-module M . We let A := (A,Fil•A), where Fil•A is the good B-algebra filtration associatedto the pair x := (x1, . . . , xp) and r := (1, . . . , 1); likewise, let M be the filtered A-moduledefined by the good A-filtration on M associated to m := (m1, . . . ,mn) and k := (0, . . . , 0)(see definition 4.4.27). Then claim follows easily after applying proposition 5.7.27(ii) to thefiltered B-algebra A and the filtered A-module M .

(ii.a): Let N ⊂M be two coherent A-modules, and ϕ : N → HomB(A, J) an A-linear map.According to claim 5.1.26, ϕ corresponds by adjunction to a unique B-linear map ϕ : N → J ;on the other hand by (i), M , and M/N are ω-coherent B-modules, hence ϕ extends to a B-linear map ψ : M → J (lemma 4.3.27). Under the adjunction, ψ corresponds to an A-linearextension ψ : M → HomB(A, J) of ϕ.

(iii): Let M ⊂ N be two finitely presented B-modules, f : M → J := colimn∈N

Jn a B-linear

map. Since M is finitely generated, f factors through a map fn : M → Jn and the naturalmap Jn → J , provided n is large enough ([36, Prop.2.3.16(ii)]). By theorem 5.7.29 there existsc ∈ N such that In+cN ∩M ⊂ InM . Hence

fn+c := ϕn+c−1 · · · ϕn fn : M → Jn+c

factors through a uniqueB/In+c-linear map fn+c : M := M/(In+cN∩M)→ Jn+c; since In+c

is finitely generated, M is a coherent submodule of the coherent B-module N := N/In+cN ,therefore fn+c extends to a map g : N → Jn+c. The composition of g : N → Jn+c with theprojection N → N and the natural map Jn+c → J , is the sought extension of f .

(ii.b): Letting A := B/In in (ii.a), we deduce that Jn := AnnJ(In) is a coh-injective B/In-module, for every n ∈ N. Then the assertion follows from (iii).

5.8. Local duality. Throughout this section we let (K, | · |) be a valued field of rank one. Weshall continue to use the general notation of (5.7).

5.8.1. Let A be finitely presented K+-algebra, I ⊂ A an ideal generated by a finite systemf := (fi | i = 1, . . . , r), and denote by i : Z := V (I) → X := SpecA the natural closedimmersion. Let also I ⊂ OX be the ideal arising from I . For every n ≥ 0 there is a naturalepimorphism

i∗i−1OX → OX/I

n

whence natural morphisms in D(A-Mod) :

(5.8.2) RHom•OX (OX/In,F •)→ RHom•OX (i∗i

−1OX ,F•)∼→ RΓZF •

for any bounded below complex F • of OX-modules.

Theorem 5.8.3. In the situation of (5.8.1), let M• be any object of D+(A-Mod), and M•∼ thecomplex of OX-modules determined by M . Then (5.8.2) induces natural isomorphisms :

colimn∈N

ExtiA(A/In,M•)∼→ RiΓZM

•∼ for every i ∈ N.

Proof. For F • := M•∼, trivial duality (theorem 5.1.27) identifies the source of (5.8.2) withRHom•A(A/In,M•); then one takes cohomology in degree i and forms the colimit over n todefine the sought map. Next, by usual spectral sequence arguments, we may reduce to thecase where M• is a single A-module M sitting in degree zero (see e.g. the proof of proposition5.2.11(i)). By inspecting the definitions, one verifies easily that the morphism thus defined is thecomposition of the isomorphism of proposition 5.4.31(ii.b) and the map (4.1.27) (see the proofof proposition 5.4.31(ii.b)); then it suffices to show that the inverse system (HiK•(f

n) | n ∈ N)is essentially zero when i > 0 (lemma 4.1.28). By lemma 4.1.30, this will in turn follow,provided the following holds. For every finitely presented quotient B of A, and every b ∈ B,


there exists p ∈ N such that AnnB(bq) = AnnB(bp) for every q ≥ p. This latter assertion is aspecial case of corollary 5.7.24.

Corollary 5.8.4. In the situation of theorem 5.8.3, we have natural isomorphisms :

colimn∈N

ExtiA(In,M•)∼→ H i(X\Z,M•∼) for every i ∈ N.

Proof. Set U := X \Z. We may assume that M• is a complex of injective A-modules, inwhich case the sought map is obtained by taking colimits over the direct system of composedmorphisms :

HomA(In,M•)βn−→ Γ(U,M•)→ RΓ(U,M•∼)

where βn is induced by the identification (In)∼|U = OU and the natural isomorphism :

HomOU (OU ,M•∼) ' Γ(U,M•∼).

The complexRΓ(U,M•∼) is computed by a Cartan-Eilenberg injective resolutionM•∼ ∼→M •

of OX-modules, and then the usual arguments allow to reduce to the case where M• consists ofa single injective A-module M placed in degree zero. Finally, from the short exact sequence0→ In → A→ A/In → 0 we deduce a commutative ladder with exact rows :

0 // HomA(A/In,M) //

αn

M // HomA(In,M) //

βn

0

0 // ΓZM

∼ // M // Γ(U,M∼) // R1ΓZM∼ // 0

where αn is induced from (5.8.2). From theorem 5.8.3 it follows that colimn∈N

αn is an isomor-

phism, and RiΓZM∼ vanishes for all i > 0; hence colim

n∈Nβn is an isomorphism, and the con-

tention follows.

Proposition 5.8.5. Let Xf−→ Y

g−→ S be two finitely presented morphisms of schemes.(i) If g is smooth, OY [0] is a dualizing complex on Y .

(ii) If g is smooth and f is a closed immersion, then X admits a dualizing complex ω•X .(iii) If X and Y are affine and ω•Y is a dualizing complex on Y , then f !ω•Y is dualizing on

X (notation of (5.3.15)).(iv) Every finitely presented quasi-separated S-scheme admits a dualizing complex.

Proof. (i) is an immediate consequence of proposition 5.7.10(i).(ii): As in the proof of corollary 5.6.38, this follows directly from (i) and lemma 5.6.28(i).(iii): Let us choose a factorization f = pY i where i : X → An

Y is a finitely presented closedimmersion, and pY : An

Y → Y the smooth projection onto Y . In view of lemma 5.6.28(i) and(5.3.15), it suffices to prove the assertion for the morphism pY . To this aim, we pick a closedfinitely presented immersion h : Y → Am

S and consider the fibre diagram :

AnY

h′ //

pY

An+mS

pS

Y

h // AmS .

By (i) we know that the scheme AmS admits a dualizing complex ω•, and then lemma 5.6.28(i)

says that h!ω• is dualizing as well. By proposition 5.6.24 we deduce that ω•Y ' L [σ]⊗OY h!ω•

for some invertible OY -module L and some continuous function σ : |Y | → Z. Since pY issmooth, we can compute: p!

Y ω•Y ' (p!

Y h!ω•)⊗OAnYp∗Y L [σ], hence p!

Y ω•Y is dualizing if and

only if the same holds for p!Y h!ω•. By proposition 5.3.7(iv), the latter complex is isomorphic


to h′[ p!Sω• and again using lemma 5.6.28(i) we reduce to checking that p!

Sω• is dualizing,

which is clear from (i).(iv): Let f : X → S be a finitely presented morphism, with X quasi-separated. If X is

affine, (iii) says that f !OS[0] is dualizing on X . In the general case, let (Ui | i = 1, . . . , n) bea finite covering of X consisting of affine open subsets; for each i, j, k = 1, . . . , n, denote byfi : Ui → S the restriction of f , set Uij := Ui ∩ Uj , Uijk := Uij ∩ Uk, and let gij : Uij → Ui bethe inclusion map. We know that f !

iOS[0] is dualizing on Ui, for every i = 1, . . . , n; moreover,for every i, j = 1, . . . , n there exists a natural isomorphism

ψij : g∗ijf!iOS[0]

∼→ g∗jif!jOS[0]

fulfilling the cocycle condition

ψjk|Uijk ψij|Uijk = ψik|Uijk for every i, j, k = 1, . . . , n

(lemma 5.3.16). In other words, ((Ui, f!iOS[0]);ψij | i, j = 1, . . . , n) is a descent datum for the

fibration (5.6.30). Then the assertion follows from proposition 5.6.31.

Example 5.8.6. (i) For given b ∈ mK , let ib : S/b → S be the closed immersion. If b 6= 0, asimple computation yields a natural isomorphism in D(OS/b-Mod) :

i!bOS∼→ OS/b [−1].

By proposition 5.8.5(i,iii), we deduce that OS/b [0] is a dualizing complex on S/b, for every b ∈mK .

(ii) Next, let f : X → S/b be an affine finitely presented Cohen-Macaulay morphism, ofconstant fibre dimension n. Then ω•X := f !OS/b [0] is a dualizing complex on X , by (i) andproposition 5.8.5(iii). Moreover, ω•X is concentrated in degree−n, and H−nω•X is a finitely pre-sented f -Cohen-Macaulay OX-module. Indeed, let i : X → Y := Am

S/bbe a closed immersion

of S/b-schemes, and denote by g : Y → S/b the projection. Fix any x ∈ X , let y := i(x) and set

dx := dim Of−1(fx),x dy := dim Og−1(gy),y.

We have a natural isomorphism in D(OX-Mod)

ω•X∼→ i!OY [m]

∼→ i∗RHom•OY (i∗OX ,OY [m])

(lemma 5.3.16(i)), and by assumption i∗OX is a g-Cohen-Macaulay OY -module; by theorem5.6.40(ii), it follows that ω•X,x is concentrated in degree dy−dx−m, and its cohomology in thatdegree is f -Cohen-Macaulay, as asserted. Lastly, since the fibres of f and g are equidimensional([31, Ch.IV, Prop.5.2.1] and lemma 5.6.36(ii)), it is easily seen that dy − dx = m − n (detailsleft to the reader), whence the claim.

5.8.7. For any finitely presented morphism f : X → S we consider the map :

d : |X| → Z x 7→ tr. deg(κ(x)/κ(f(x))) + dim f(x).

Lemma 5.8.8. With the notation of (5.8.7), let x, y ∈ X , and suppose that x is a specializationof y. We have :

(i) x is an immediate specialization of y if and only if d(y) = d(x) + 1.(ii) If X is irreducible, d(x)− d(y) = dimX(y)− dimX(x).

(iii) X is catenarian and of finite Krull dimension.(iv) If X is irreducible and f is flat, then Of−1(fx),x is equidimensional.(v) If f is flat, then dim OX,x = dim Of−1(fx),x + dim OS,f(x).

(vi) If f is Cohen-Macaulay at the point x, then OX,x is equidimensional.


Proof. (i): In case f(x) = f(y), the assertion follows from [31, Ch.IV, Prop.5.2.1]. Hence,suppose that f(x) = s, f(y) = η; let Z be the topological closure of y in X , and endow Zwith its reduced subscheme structure; notice that Z is an S-scheme of finite type. By assump-tion, X is quasi-compact and quasi-separated, hence y is a pro-constructible subset ofX , andtherefore Z is the set of all specializations of y in X ([30, Ch.IV, Th.1.10.1]). Especially, y isthe unique maximal point of Zη := Z ∩ f−1(η); also, if x is an immediate specialization of y,then x is a maximal point of Zs := Z ∩ f−1(s), and by [32, Ch.IV, Lemme 14.3.10], the latterimplies that d(y) = d(x) + 1.

Conversely, suppose that d(y) = d(x) + 1; from [32, Ch.IV, Lemme 14.3.10] and [31, Ch.IV,Prop.5.2.1] we deduce that x is a maximal point of Zs; then x is an immediate specialization ofy inX , since otherwise x would be a specialization inX of a proper specialization y′ of y in Zη,and in this case [32, Ch.IV, Lemme 14.3.10] would say that d(y′) = d(x) + 1, i.e. d(y) = d(y′),contradicting [31, Ch.IV, Prop.5.2.1].

(iii) It is easily seen thatX has finite Krull dimension. Now, consider any sequence y0, . . . , ynof points of X , with y0 := y, yn := x, and such that yi+1 is an immediate specialization of yi,for i = 0, . . . , n − 1; from (i) we deduce that n = d(y) − d(x), especially n is independent ofthe chosen chain of specializations, so X is catenarian.

(ii): We reduce easily to the case where y is the maximal point of X , in which case we mayargue as in the proof of (iii) (details left to the reader).

(iv): This is trivial, if f(x) = η; hence let us assume that f(x) = s, and let y1, y2 be twomaximal generizations of x in f−1(s). We may find an affine open subset U ⊂ X such thaty1 ∈ U and y2 /∈ U ; arguing as in the proof of (i), we see that the maximal point z of U (whichis also the maximal point ofX) is an immediate generization of y1 in U , hence d(z) = d(y1)+1.Likewise, d(z) = d(y2) + 1, hence d(y1) = d(y2). In view of (i), the assertion follows easily.

(v): If f(x) = η, the identity is [31, Ch.IV, Cor.6.1.2]. Hence, suppose that f(x) = s. In thiscase, the proof of (iv) shows that the identity holds, when X is irreducible. In the general case,notice that – by the flatness assumption – every irreducible component of X intersect f−1(η),and it is therefore itself a flat S-scheme (with its reduced subscheme structure). Since f−1(η) isa noetherian scheme, it also follows that the set of irreducible components of X is finite; thus,let X1, . . . , Xn be the reduced irreducible components of X containing x, and fi : Xi → S(i = 1, . . . , n) the corresponding restrictions of f . Then

dim OX,x = max1≤i≤n

dim OXi,x and dim Of−1(s),x = max1≤i≤n

dim Of−1i (s),x

whence the contention.(vi): By assumption, Of−1(fx),x is a Cohen-Macaulay noetherian local ring, especially it is

equidimensional ([61, Th.17.3(i)]). Thus, the assertion already follows, in case f(x) = η.If f(x) = s, let y be any maximal generization of x in f−1(s); arguing as in the proof of(i), we see that every immediate generization of y in X is a maximal point of X , whence thecontention.

Lemma 5.8.9. (i) The K+-module K/K+ is coh-injective.(ii) Let A be a finitely presented K+-algebra, I ⊂ A a finitely generated ideal, J a coh-

injective K+-module. Then the A-module

JA := colimn∈N

HomK+(A/In, J)

is coh-injective.

Proof. (i): It suffices to show that Ext1K+(M,K/K+) = 0 whenever M is a finitely presented

K+-module. This is clear when M = K+, and then [36, Ch.6, Lemma 6.1.14] reduces to thecase where M = K+/aK+ for some a ∈ mK\0; in that case we can compute using the freeresolution K+ a−→ K+ →M , and the claim follows easily.


(ii): According to theorem 5.7.32(ii.a), the A-module J ′A := HomK+(A, J) is coh-injective.However, JA =

⋃n∈N AnnJ ′A(In), so the assertion follows from theorem 5.7.32(ii.b).

Theorem 5.8.10. Let f : X → S be a finitely presented affine morphism. Then for every pointx ∈ X , the OX,x-module

J(x) := R1−d(x)Γx(f!OS[0])|X(x)

is coh-injective, and we have a natural isomorphism in D(OX,x-Mod) :

RΓx(f!OS[0])|X(x) ' J(x)[d(x)− 1].

Proof. Fix x ∈ X and set d := d(x); in case f(x) = η, the assertion follows from [46, Ch.V,Cor.8.4 and following remark]. Hence, suppose that f(x) = s, the closed point of S, and saythat X = SpecA.

Claim 5.8.11. We may assume that x is closed in X , hence that κ(x) is finite over κ(s).

Proof of the claim. Arguing as in the proof of lemma 5.7.5, we can find a factorization of themorphism f as a composition X

g−→ Y := AdS

h−→ S, such that ξ := g(x) is the generic pointof h−1(s) ⊂ Y , the morphism g is finitely presented, and the stalk OY,ξ is a valuation ring.Moreover, let gy := g×Y 1Y (y) : X(y) := X×Y Y (y)→ Y (y); we have a natural isomorphism

(f !OS[0])|X(y) ' g!yOY (y)[d].

Then we may replace f by gy, and K+ by OY,ξ, whence the claim. ♦

Hence, suppose now that x is closed in X , let p ⊂ A be the maximal ideal corresponding tox, and p the image of p in A := A ⊗K+ κ; we choose a finite system of elements b1, . . . , br ∈OX(X) whose images in A generate p. Pick any non-zero a ∈ mK , and let I ⊂ A be the idealgenerated by the system (a, b1, . . . , br), and I ⊂ OX the corresponding coherent ideal; clearlyV (I) = x, hence theorem 5.8.3 yields a natural isomorphism :

(5.8.12) colimn∈N

RiHom•OX (OX/In, f !OS[0])

∼→ RiΓxf!OS[0] for every i ∈ Z

where the transition maps in the colimit are induced by the natural maps OX/I n → OX/I m,for every n ≥ m. However, from lemmata 5.3.6(ii), 5.3.16(i) and corollary 5.1.33 we deduceas well natural isomorphisms :

RHom•OX (OX/In, f !OS[0])

∼→ RHom•OS(f∗OX/In,OS)

∼→ RHom•K+(A/In, K+).

We wish to compute these Ext groups by means of lemma 4.3.27(iii); to this aim, let us remarkfirst that A/In is an integral K+-algebra for every n ∈ N, hence it is a finitely presented torsionK+-module, according to proposition 5.7.7(i). We may then use the coh-injective resolutionK+[0]→ (0→ K → K/K+ → 0) (lemmata 5.8.9(i), 4.3.27(iii)) to compute :

ExtiK+(A/In, K+) =

Jn := HomK+(A/In, K/K+) if i = 10 otherwise.

Now the theorem follows from lemma 5.8.9(i) and theorem 5.7.32(ii.a),(iii).We could also appeal directly to lemma 5.8.9(ii), provided we already knew that the foregoing

natural identifications transform the direct system whose colimit appears in (5.8.12), into thedirect system (Jn | n ∈ N) whose transition maps are induced by the natural maps A/In →A/Im, for every n ≥ m. For the sake of completeness, we check this latter assertion.


For every n ≥ m, let jn : Xn := SpecA/In → X and jmn : Xm → Xn be the natural closedimmersions. We have a diagram of functors :

jm∗ j!mn (f jn)!

ζ1 //

ε1

++WWWWWWWWWWWWWWWWWWWWWjn∗ (jmn∗ j!

mn) j!n f !

ε2

++VVVVVVVVVVVVVVVVVVVjm∗ j!

m f !ξ2oo

αmn

jm∗ (f jm)!

ξ1

OO

βmn // jn∗ (f jn)!ζ2 // jn∗ j!

n f !

where :• ζ1 and ζ2 are induced by the natural isomorphism ψf,jn : (f jn)! ∼→ j!

n f ! of lemma5.3.16(i).• ξ1 (resp. ξ2) is induced by the isomorphism ψfjn,jmn (resp. ψjn,jmn).• ε1 and ε2 are induced by the counit of adjunction jmn∗ j!

mn → 1D+(OXn -Mod).• βmn (resp. αmn) is induced by the natural map (f jn)∗OXn → (f jm)∗OXm (resp.

by the map jn∗OXn → jm∗OXm).It follows from lemma 5.3.22 that the two triangular subdiagrams commute, and it is also clearthat the same holds for the inner quadrangular subdiagram. Moreover, lemma 5.3.16(ii) yieldsa commutative diagram

(5.8.13)

(f jm)!

ψfjn,jmn

ψf,jm // j!m f !

ψjn,jmnf !

j!mn (f jn)!

j!mn(ψf,jn )// j!mn j!

n f !

such that jm∗(5.8.13) is the diagram :

jm∗ (f jm)!

ξ1

jm∗(ψf,jm )// jm∗ j!

m f !

ξ2

jm∗ j!mn (f jn)!

ζ1 // jm∗ j!mn j!

n f !.

We then arrive at the following commutative diagram :

RHom•OS(OX/I m,OS)RΓ(ψf,jm )

//

RΓ(βmn)

RHom•OX (OX/I m, f !OS[0])

RΓ(αmn)

RHom•OS(OX/I n,OS)RΓ(ψf,jn )

// RHom•OX (OX/I n, f !OS[0]).

Now, the maps RΓ(αmn) are the transition morphisms of the inductive system whose colimitappears in (5.8.12), hence we may replace the latter by the inductive system formed by the mapsRΓ(βmn). Combining with corollary 5.1.33, we finally deduce natural A-linear isomorphisms

colimn∈N

ExtiK+(A/In, K+)∼→ RiΓxf

!OS[0]

where the transition maps in the colimit are induced by the natural maps A/In → A/Im, forevery n ≥ m. Our assertion is an immediate consequence.

5.8.14. Let A be a local ring, essentially of finite presentation over K+. Set X := SpecA andlet x ∈ X be the closed point. Notice that X admits a dualizing complex. Indeed, one canfind a finitely presented affine S-scheme Y and a point y ∈ Y such that X ' Spec OY,y; byproposition 5.8.5(iv), Y admits a dualizing complex ω•Y , and since every coherent OX-moduleextends to a coherent OY -module, one verifies easily that the restriction of ω•Y is dualizing for


X . Hence, let ω• be a dualizing complex for X . It follows easily from theorem 5.8.10 andpropositions 5.8.5(iii) and 5.6.24, that there exists a unique c ∈ Z such that

J(x) := RcΓxω• 6= 0.

Moreover, J(x) is a coh-injective A-module, hence by lemma 4.3.27(iii), we obtain a welldefined functor

(5.8.15) Db(A-Modcoh)→ Db(A-Mod)o : C• 7→ Hom•A(C•, J(x)).

Furthermore, let Dbx(A-Modcoh) be the full subcategory of Db(A-Modcoh) consisting of all

complexes C• such that SuppH•C• ⊂ x. We have the following :

Corollary 5.8.16. (Local duality) In the situation of (5.8.14), the following holds :(i) The functor (5.8.15) restricts to an equivalence of categories :

D : Dbx(A-Modcoh)→ Db

x(A-Modcoh)o

and the natural transformation C• → D D(C•) is an isomorphism of functors.(ii) For every i ∈ Z there exists a natural isomorphism of functors :

RiΓx D∼→ D Rc−iΓ : Db(OX-Mod)coh → A-Modo

where D is the duality functor corresponding to ω•.(iii) Let c and D be as in (ii), I ⊂ A a finitely generated ideal such that V (I) = x, and

denote by A∧ the I-adic completion of A. Then for every i ∈ Z there exists a naturalisomorphism of functors :

D RiΓx D∼→ A∧ ⊗A Rc−iΓ : Db(OX-Mod)coh → A∧-Modocoh.

Proof. Let C• be any object of Dbx(A-Modcoh); we denote by i : x → X the natural closed

immersion, and C∼• the complex of OX-modules determined by C•. Obviously C∼• = i∗i−1C∼• ,

therefore :Hom•A(C•, J(x))

∼→ RHom•A(C•, J(x)) by lemma 4.3.27(iii)∼→ RHom•OX (C∼• , RΓxω

•[c]) by theorem 5.1.27∼→ RHom•OX (i∗i

−1C∼• , ω•[c]) by 5.4.12(i.b)

∼→ RHom•OX (C∼• , ω•[c])

which easily implies (i). Next, we compute, for any object F • of Db(OX-Mod)coh :

RiΓxD(F •)∼→ RiHom•OX (F •, RΓxω

•) by lemma 5.4.12(iii)∼→ RiHom•OX (F •, J(x)[−c])∼→ RiHom•A(RΓF •, J(x)[−c]) by theorem 5.1.27∼→ HomA(Rc−iΓF •, J(x)) by lemma 4.3.27(iii)

whence (ii). Finally, let F • be any object of Db(OX-Mod)coh; we compute :

A∧ ⊗A Rc−iΓF • ∼→ limn∈N

(A/In)⊗A Rc−iΓF • by corollary 5.7.31(ii)∼→ lim

n∈ND D((A/In)⊗A Rc−iΓF •) by (i)

∼→ limn∈N

D(HomA(A/In, RiΓxD(F •))) by (ii)∼→ D(colim

n∈NHomA(A/In, RiΓxD(F •)))

∼→ D(RiΓxD(F •))


so (iii) holds as well.

Corollary 5.8.17. In the situation of (5.8.14), set U := X \x and let F be a coherent OU -module, such that ΓyF = 0 for every closed point y ∈ U . Then Γ(U,F ) is a finitely presentedA-module.

Proof. By lemma 5.2.16(ii) we may find a coherent OX-module G such that G|U ' F . Usingcorollary 5.7.24 we see that ΓxG is a finitely generated submodule of G (X), hence F :=

G /ΓxG is a coherent OX-module that extends F , and clearly ΓxF = 0. There follows ashort exact sequence :

0→ F (X)→ Γ(U,F )→ R1ΓxF → 0.

We are thus reduced to showing that R1ΓxF is a finitely presented A-module. However,corollary 5.8.16(ii) yields a natural isomorphism :

R1ΓxF∼→ HomA(Extc−1

OX(F , ω•), J(x))

where ω• is a dualizing complex for X and c and J(x) are defined as in (5.8.14). In view ofcorollary 5.8.16(i), it then suffices to show that the finitely presented A-module Extc−1

OX(F , ω•)

is supported on x; equivalently, it suffices to verify that the stalks Rc−1HomOX (F , ω•)yvanish whenever y is a closed point of U . However, by [28, Ch.0, Prop.12.3.5], the latter arenaturally isomorphic to Extc−1

OX(y)(F|X(y), ω

•|X(y)), and ω•|X(y) is a dualizing complex for X(y).

According to (5.8.14), there is a unique integer c(y) such that J(y) := Rc(y)ω•|X(y) 6= 0 and inview of theorem 5.8.10 and lemma 5.8.8 we obtain :

(5.8.18) c(y) = c− 1.

Finally, (5.8.18) and corollary 5.8.16(iii) yield the isomorphism :

A∧ ⊗A Extc−1OX(y)

(F|X(y), ω•|X(y))

∼→ HomOX,y(ΓyF , J(y)) = 0.

To conclude, it suffices to appeal to corollary 5.7.31(iii).

Proposition 5.8.19. In the situation of (5.8.14), suppose additionally that the structure mor-phism f : X → S is flat, that f(x) = s and that f−1(s) is a regular scheme. Let j : U :=X\x → X be the open immersion, F a flat quasi-coherent OU -module. Then the followingtwo conditions are equivalent :

(a) Γ(U,F ) is a flat A-module.(b) δ(x, j∗F ) ≥ dim f−1(s) + 1.

Proof. If d := dim f−1(s) = 0, then A is a valuation ring (proposition 5.7.7(ii)), and U is thespectrum of the field of fractions of A, in which case both (a) and (b) hold trivially. Hence wemay assume that d ≥ 1. Suppose now that (a) holds, and set F := Γ(U,F ). In view of (5.4.3),we need to show that H i(U,F ) = 0 whenever 1 ≤ i ≤ d − 1. If F∼ denotes the OX-moduledetermined by F , then F∼|U = F . Moreover, by [57, Ch.I, Th.1.2], F is the colimit of a filteredfamily (Li | i ∈ I) of free A-modules of finite rank, hence H i(U,F ) = colim

i∈IH i(U,L∼i ) by

lemma 5.1.10(ii), so we are reduced to the case where F = OU , and therefore j∗F = OX(corollary 5.6.8). Since f−1(s) is regular, we have δ(x,Of−1(s)) = d; then, since the topologicalspace underlying X is noetherian, lemma 5.4.19(ii) and corollary 5.4.39 imply that δ(x,OX) ≥d+ 1, which is (b).

Conversely, suppose that (b) holds; we shall derive (a) by induction on d; the case d = 0having already been dealt with. Let mA (resp. mA) be the maximal ideal of A (resp. of A :=A/mKA). Suppose then that d ≥ 1 and that the assertion is known whenever dim f−1(s) < d.Pick t ∈ mA \m2

A, and let t ∈ mA be any lifting of t. Since f−1(s) is regular, t is a regular


element of A, so that t is regular in A and the induced morphism g : X ′ := SpecA/tA→ S isflat ([32, Ch.IV, Th.11.3.8]). Let j′ : U ′ := U ∩X ′ → X ′ be the restriction of j; our choice of tensures that g−1(s) is a regular scheme, so the pair (X ′,F/tF|U ′) fulfills the conditions of theproposition, and dim g−1(s) = d− 1. Furthermore, since F is a flat OU -module, the sequence0→ F

t−→ F → F/tF → 0 is short exact. Assumption (b) means that

(5.8.20) H i(U,F ) = 0 whenever 1 ≤ i ≤ d− 1.

Therefore, from the long exact cohomology sequence we deduce that H i(U,F/tF ) = 0 for1 ≤ i ≤ d− 2, i.e.

(5.8.21) δ(x, j′∗F/tF ) ≥ d.

The same sequence also yields a left exact sequence :

(5.8.22) 0→ H0(U,F )⊗A A/tAα−→ H0(U,F/tF )→ H1(U,F ).

Claim 5.8.23. H0(U,F )⊗A A/tA is a flat A/tA-module.

Proof of the claim. By (5.8.21) and our inductive assumption, H0(U,F/tF ) is a flat A/tA-module. In case d = 1, proposition 5.7.7(ii) shows that A/tA is a valuation ring, and then theclaim follows from (5.8.22), and [14, Ch.VI, §3, n.6, Lemma 1]. If d > 1, (5.8.20) implies thatα is an isomorphism, so the claim holds also in such case. ♦

Set V := SpecA[t−1] = X \ V (t) ⊂ U ; since F|V is a flat OV -module, the A[t−1]-module H0(U,F ) ⊗A A[t−1] ' H0(V,F|V ) is flat. Moreover, since t is regular on both Aand H0(U,F ), an easy calculation shows that TorAi (A/tA,H0(U,F )) = 0 for i > 0. Then thecontention follows from claim 5.8.23 and [36, Lemma 5.2.1].

Remark 5.8.24. Proposition 5.8.19 still holds – with analogous proof – if we replace X by thespectrum of a regular local ring A (and then dim f−1(s) + 1 is replaced by dimA in condition(b)). The special case where F is a locally free OU -module of finite rank has been studied indetail in [50].

We conclude this section with a result which shall be used in our discussion of almost purity.

5.8.25. For given b ∈ mK , let f : X → S/b be a morphism of finite presentation, x ∈ X a pointlying in the Cohen-Macaulay locus of f (see definition (5.6.35)(v)), F a finitely presentedOX(x)-module. Let also a ∈ OX,x, and set µa := a ·1F : F → F ; suppose that, for every pointy ∈ U := X(x)\x there is a coherent OX,y-module E, such that :

• E is f \y-Cohen-Macaulay and SuppE = X(y).• µa,y factors through a OX,y-linear map Fy → E.

Lemma 5.8.26. In the situation of (5.8.25), the following holds :(i) The OX,x-module a ·RiΓxF is finitely presented for every i < dim OX,x.

(ii) If dim OX,x > 1, the OX,x-module a · Γ(U,F ) is finitely presented.

Proof. (i): To ease notation, set A := OX,x and d := dimA. Fix a finitely generated idealI ⊂ A whose radical is the maximal ideal. We may assume that X is an affine scheme and f isCohen-Macaulay of constant fibre dimension n (lemma 5.6.36(i,ii)). In this case, X(x) admitsthe dualizing complex

ωX(x) := (f !OS/b [−n])|X(x)

which is a coherent OX(x)-module placed in degree zero ((5.8.14) and example 5.8.6(ii)). Wemay then define c ∈ N and J(x) as in (5.8.14), and by inspection we find that c = n− d(x) + 1if b = 0, and otherwise c = n−d(x). To compute this quantity, pick any irreducible component


Z of the fibre f−1(fx) containing x, and endow it with its reduced subscheme structure; thendimZ = n (lemma 5.6.36(ii)), and since Z is a scheme of finite type over κ(fx), we have

d(x) = n+ 1− dim Of−1(fx),x − dim OS,f(x)

([31, Ch.IV, Prop.5.2.1]). Now, if b = 0, lemma 5.8.8(v) applies and we obtain c = d. If b 6= 0,then f(x) = s, and dim Of−1(s),x = d, hence again c = d. Let also D be the functor (5.8.15).

Claim 5.8.27. For every finitely presented A-module M supported at the point x, and everyt ∈ A, we have a natural isomorphism : D(tM) ' tD(M).

Proof of the claim. It is a simple application of the exactness of the functor D. Indeed, from theleft exact sequence 0→ AnnM(t)→M

t−→M we deduce a right exact sequence :

D(M)t−→ D(M)→ D(AnnM(t))→ 0

i.e. D(AnnM(t)) ' D(M)/tD(M). Then the short exact sequence :

0→ AnnM(t)→Mτ−→ tM → 0

yields the exact sequence 0→ D(tM)→ D(M)D(τ)−−→ D(M)/tD(M)→ 0, and by inspecting

the definition one sees easily that D(τ) is the natural projection, whence the claim. ♦

We apply claim 5.8.27 and corollary 5.8.16(ii) to derive a natural isomorphism :

a ·RiΓxF ' D(a ·Hd−i(X(x),D(F ))) ' D(a · ExtiOX(x)(F , ωX(x))) for every i ∈ Z.

However – in view of lemma 5.6.36(i) and our assumptions – for any y ∈ U we may find anaffine open neighborhood V ⊂ X of y, a coherent f|V -Cohen-Macaulay OV -module E withSupp E = V , and a factorization Fy → Ey → Fy of µa,y; since ωX(x) is a coherent OX(x)-module, there results a factorization ([28, Ch.0, Prop.12.3.3, 12.3.5])

ExtiOX(x)(F , ωX(x))y → ExtiOV,y(Ey, ωX(x),y)→ ExtiOX(x)

(F , ωX(x))y.

Claim 5.8.28. ExtiOV,y(Ey, ωX(x),y) = 0 for every i > 0.

Proof of the claim. Let j : V → X be the open immersion, and set g := f j : V → S/b; by[28, Ch.0, Prop.12.3.3, 12.3.5], we are then reduced to showing that ExtiOV ′ (E , g

!OS/b [−n]) = 0for every i > 0. To this aim, pick a finitely presented S-immersion h : V → Y := Am

S/b, and let

p : Y → S/b be the projection; we deduce a natural isomorphism g!OS/b [−n]∼→ h!OY [m − n],

and by adjunction

(5.8.29) ExtiOV (E , g!OS/b [−n]) = ExtiOY (h∗E ,OY [m− n]).

Now, fix v ∈ V and set u := h(v); by theorem 5.6.40(ii), the complex RHom•OY (h∗E ,OY )uis concentrated in degree dim Op−1(pu),u − dim Of−1(fv),v, and since the fibres of f and p areequidimensional ([31, Ch.IV, Prop.5.2.1] and lemma 5.6.36(ii)) it is easily seen that this quantityequals m − n. After composing with the functor RiΓ, we deduce that the right-hand side of(5.8.29) vanishes for i > 0, as required. ♦

From claim 5.8.28, it follows that a · ExtiOX(x)(F , ωX(x)) is a coherent A-module ([28, Ch.0,

Prop.12.3.3]) supported at x for i > 0. Assertion (i) now follows from corollary 5.8.16(i).(ii): To start out, notice that ΓxF is an OX-module of finite type, by corollary 5.7.24; we

may then replace F by F/ΓxF and assume, without loss of generality, that :

(5.8.30) ΓxF = 0.


Let y1, . . . , yr ∈ U be the finitely many associated points of F that are closed in U (theorem5.7.20(ii)), and set Z := y1, . . . , yr; notice that Z is a closed subset of U , so that the OU -module ΓZF|U is of finite type, by corollary 5.7.24. Hence F := F|U/ΓZF|U is a coherentOU -module, and we obtain a short exact sequence :

0→ ΓZF|U → Γ(U,F )→ Γ(U,F )→ 0.

The OU -module F fulfills the assumptions of corollary 5.8.17, so that Γ(U,F ) is a finitelypresented OX,x-module. On the other hand, set F := Γ(X(x),F ); by corollary 5.8.4 we havea natural isomorphism :

colimn∈N

HomA(In, F )∼→ Γ(U,F )

and using (5.8.30) one sees easily that the transition maps in this colimit are injective. Hence,Γ(U, F ) is an increasing union of finitely presented A-modules ([26, Ch.0, §5.3.5]). Let M ⊂Γ(U,F ) be any finitely generated submodule that maps surjectively onto Γ(U,F ); it thenfollows that M is finitely presented, and clearly Γ(U,F ) = M + ΓZF|U , hence a · Γ(U,F ) =aM + a · ΓZF|U . Finally, since aM is finitely presented, it remains only to show :

Claim 5.8.31. a · ΓZF|U = 0.

Proof of the claim. Fix any y ∈ U , and let V ⊂ X be any open neighborhood of y with acoherent OV -module E as in the foregoing. After shrinking V , we may assume that µa|V ∩X(x)

factors through E . It follows that a ·ΓZ∩V F|V is a subquotient of ΓZ∩V E . We are then reducedto showing that ΓZ∩V E = 0. However, under the current assumptions dimU > 0, hence

(5.8.32) dim OU,yi > 0 for i = 1, . . . , r

(lemma 5.8.8(iii)). On the other hand, since E is f|V -Cohen-Macaulay with support V , itsassociated points are the maximal points of V ∩ f−1(η). Taking into account (5.8.32), weconclude that Z does not contain any associated point of E , as required.

5.9. Hochster’s theorem and Stanley’s theorem. The two main results of this section aretheorems 5.9.34 and 5.9.41, which were proved originally respectively by Hochster in [49], andby Stanley in [72]. Our proofs follow in the main the posterior methods presented by Bruns andHerzog in [22] (which in turns, partially rely on some ideas of Danilov). These results shall beused in section 6.5, in order to show that regular log schemes are Cohen-Macaulay. We beginwith some preliminaries from algebraic topology, which we develop only to the extent that isrequired for our special purposes: a much more general theory exists, and is well known toexperts (see e.g. [59, Ch.IX]).

For any topological space X , and any subset T ⊂ X , we denote by T the topological closureof T in X , endowed with the topology induced from X . We let H•(X) (resp. H•(X,T )) be thesingular homology groups of X (resp. of the pair (X,T )). Also, for every n ∈ N fix a Banachnorm ‖ · ‖ on Rn, and for every real number ρ > 0, let Bn(ρ) := v ∈ Rn | ‖v‖ < ρ, and setSn−1 := Bn(1) \ Bn(1) (so S−1 = ∅).

Definition 5.9.1. (i) A finite regular cell complex (or briefly : a cell complex) is the datum of atopological space X , together with a finite filtration

X−1 := ∅ ⊂ X0 ⊂ X1 ⊂ X2 ⊂ · · · ⊂ Xk := X

consisting of closed subspaces, such that X0 is a discrete topological space, and for everyi = 0, . . . , k we have a decomposition

X i \X i−1 =⋃λ∈Λi

eiλ

where :


(a) Λi is a finite set, and for every λ ∈ Λi there exists a homeomorphism

fλ : Bi(1)∼→ eiλ such that f−1

λ (eiλ) = Bi(1).

(b) eiλ ∩ eiµ = ∅ for any two distinct indices λ, µ ∈ Λi.(c) eiλ \ eiλ ⊂ X i−1 for every λ ∈ Λi.

(ii) The smallest k ∈ N such that Xk = X is called the dimension of X•, and is denoteddimX•. For every i = 0, . . . , k, the subsets eiλ are called the i-dimensional cells of X•.

(iii) A subcomplex of the cell complex X• is a closed subset Y ⊂ X which is a union ofcells of X . Then the filtration such that Y i := Y ∩X i for every i ≥ −1 defines a cell complexstructure Y • on Y .

Lemma 5.9.2. With the notation of definition 5.9.1, we have :(i) For every i = 0, . . . , dimX and q ∈ Z, we have natural isomorphisms of abelian

groups :

Hq(Xi, X i−1)

∼→

ZΛi if q = i0 otherwise.

(ii) More precisely, let (bλ | λ ∈ Λi) be the canonical basis of the free Z-module ZΛi; forevery λ ∈ Λi, the isomorphism of (i) maps the image of the induced map

Hifλ : Hi(Bi(1), Si−1)→ Hi(Xi, X i−1)

isomorphically onto the direct summand generated by bλ.

Proof. (i): This is obvious for i = 0. For i = 1, . . . , k := dimX• and every λ ∈ Λi, set

Y iλ := fλ(Bi(1/2)) Y i :=

⋃λ∈Λi

Y iλ Ai :=

⋃λ∈Λi

fλ(0)

and for any q ∈ Z, consider the natural group homomorphisms

(5.9.3)

Hq(Bi(1/2), Bi \0) α′ //

γ

Hq(Bi(1),Bi(1) \0)

Hq(Bi(1),Si−1)β′oo

Hq(Y

i, Y i \ Ai) α // Hq(Xi, X i \ Ai) Hq(X

i, X i−1).βoo

The map α is an isomorphism, by excision; the same holds for β, since X i−1 is a deformationretract of X i \ Ai. However, for every λ ∈ Λi we have natural isomorphisms

Hq(Yiλ , Y

iλ \ A)

∼→

Z if q = i0 otherwise.

whence the contention.(ii): Take q := i in (5.9.3); then clearly γ is a monomorphism whose image is the direct

summand Hi(Yiλ , Y

iλ \ A); on the other hand, arguing as in the foregoing, we see that both α′

and β′ are isomorphisms. The assertion follows easily.

Remark 5.9.4. (i) In the situation of lemma 5.9.2, notice that there is a natural bijection fromΛi to the set of connected components π0(X i \X i−1) of X i \X i−1, for every i = 0, . . . , dimX .

(ii) The direct sum decomposition ofHi(Xi, X i−1) provided by lemma 5.9.2(i) depends only

on the filtration X• (and not on the choice of homeomorphisms fλ). Indeed, this is clear, sincethe map Hifλ factors as a composition :

Hi(Bi(1),Si−1)∼→ Hi(e

iλ, e

iλ \ eiλ)

lλ−−→ Hi(Xi, X i−1)

where lλ is the homomorphism induced by the obvious map of pairs (eiλ, eiλ\eiλ)→ (X i, X i−1).


(iii) In view of (ii), the maps lλ admit natural left inverse homomorphisms

pλ : Hi(Xi, X i−1)→ Hi(e

iλ, e

iλ \ eiλ)

such that

(5.9.5) pµ lλ = 0 for every λ, µ ∈ Λi with λ 6= µ.

These maps can be described as follows. For any λ ∈ Λi, the natural map of pairs (X i, X i−1)→(X i, X i \ eiλ) induces a homomorphism

mλ : Hi(Xi, X i−1)→ Hi(X

i, X i \ eiλ).

Arguing by excision as in the proof of lemma 5.9.2(ii), it is easily seen that qλ := mλ lλ is anisomorphism (details left to the reader), and we set pλ := q−1

λ mλ. Obviously pλ is left inverseto lλ, and in oder to check (5.9.5) it suffices to show that mµ lλ = 0 for every µ 6= λ. However,mµ lλ is the homomorphism arising from the map of pairs (eiλ, e

iλ \ eiλ) → (X i, X i \ eiµ), so

the assertion follows, after remarking that eiλ ⊂ X i \ eiµ.

5.9.6. We attach to any cell complex X• a complex of abelian groups C•(X•), as follows.• For i < 0 and for i > k := dimX•, we set Ci(X

•) := 0, and for i = 1, . . . , k we let

Ci(X•) := Hi(X

i, X i−1).

Sometimes we write just Ci(X), unless it is useful to stress which filtration X• on X we areconsidering. For every i > 0, the differential di : Ci(X

•)→ Ci−1(X•) is the composition

Hi(Xi, X i−1)

∂i−−→ Hi−1(X i−1)ji−1−−−→ Hi−1(X i−1, X i−2)

where ∂i is the boundary operator of the long exact homology sequence associated to the pair(X i, X i−1), and ji−1 is the homomorphism induced by the obvious map of pairs (X i−1,∅) →(X i−1, X i−2). In order to check that di di+1 = 0 for every i ∈ Z, recall that every element ofHi+1(X i+1, X i) is the class c of a singular (i + 1)-chain c of X i+1, whose boundary ∂i+1c is asingular i-chain of X i; then di+1(c) is the class of ∂i+1c in Hi(X

i, X i−1), and di di+1(c) is theclass of ∂i ∂i+1c in Hi−1(X i−1, X i−2), so it vanishes.• It is also useful to consider an augmented version of the above complex; namely, let us set

C −1(X•) := Z and C i(X•) := Ci(X

•) for every i 6= −1

with differential d0 given by the rule : d0(b0λ) = 1 for every λ ∈ Λ0.

Proposition 5.9.7. With the notation of (5.9.6), there exist natural isomorphisms of abeliangroups

HqC•(X•)∼→ Hq(X) for every q ∈ N.

Proof. For every topological space T , let C•(T ) denote the complex of singular chains of T .The filtration X• induces a finite filtration of complexes

C•(X0) ⊂ C•(X

1) ⊂ C•(X2) ⊂ · · · ⊂ C•(X)

whence a convergent spectral sequence

E1pq := Hp+q(X

p, Xp−1)⇒ Hp+q(X)

(see [75, Th.5.5.1]). By direct inspection, it is easily seen that the differential d1p,0 : E1

p,0 →E1p−1,0 agrees with the differential dp of C•(X•), for every p ∈ N. On the other hand, lemma

5.9.2 shows that this spectral sequence degenerates, whence the contention.


5.9.8. Keep the notation of definition 5.9.1, and set Λ0 := X0. For every i = 0, . . . , dimX andevery λ ∈ Λi, the abelian group Hi,λ := Hi(e

iλ, e

iλ \ eiλ) is free of rank one; if i > 0, we fix

one of the two generators of this group, and we denote by biλ its image in Ci(X•) (see remark

5.9.4(ii)). For i = 0, each e0λ is a point, hence H0,λ admits a canonical identification with Z,

and we let b0λ be the image of 1, under the resulting map Z ∼→ H0,λ → C0(X•). The system of

classes (biλ | 0 ≤ i ≤ dimX,λ ∈ Λi) is called an orientation for X•. We may write

di(biλ) =

∑µ∈Λi−1

[biλ : bi−1µ ]bi−1

µ for every i ≤ dimX• and every λ ∈ Λi

for a system of uniquely determined integers [biλ : bi−1µ ], called the incidence numbers of the

cells eiλ and ei−1µ (relative to the chosen orientation of X•). With this notation, we may state :

Lemma 5.9.9. The incidence numbers of a cell complex X• fulfill the following conditions :(i)∑

µ∈Λi−1[biλ : bi−1

µ ] · [bi−1µ : bi−2

ν ] = 0 for every i ≥ 2, every λ ∈ Λi and every ν ∈ Λi−2.(ii) Let λ ∈ Λ1 be any index, and say that e1

λ \ e1λ = e0

µ ∪ e0ρ. Then [b1

λ : b0µ] + [b1

λ : b0ρ] = 0.

Proof. Condition (i) translates the identity di−1 di = 0 for the differential d• of the complexC•(X•). The identity of (ii) follows by a simple inspection of the definition of d1 : details leftto the reader.

5.9.10. The generalities of the previous paragraphs shall be applied to the following situation.Let (V, σ) be a strictly convex polyhedral cone such that 〈σ〉 = V , and set d := dimR V . Weattach to σ a cell complex C•σ as follows. Fix u0 ∈ σ∨ such that σ ∩ Keru0 = 0 (corollary3.3.14), let σ be the topological interior of σ (in V ) and set

Cσ := σ ∩ u−10 (1) Cσ := σ ∩ u−1

0 (1).

Let v1, . . . , vk be a minimal system of generators for σ; we may assume that u0(vi) = 1 fori = 1, . . . , k, in which case

(5.9.11) Cσ =

k∑i=1

λivi | λ1, . . . , λk ≥ 0 andk∑i=1

λi = 1

which shows that Cσ is a compact topological space (with the topology inherited from V ).

Lemma 5.9.12. There exists a homeomorphism Bd−1(1)∼→ Cσ that maps Bd−1(1) onto Cσ.

Proof. Set v0 := k−1 · (v1 + · · · + vk) and W := Keru0. It suffices to show that there existsa homeomorphism Bd−1(1)

∼→ Dσ := Cσ − v0 ⊂ W that maps Bd−1(1) onto Dσ := Cσ − v0.However, pick a minimal system u1, . . . , ut of generators of σ∨ (corollary 3.3.12(i)); since σspans V , for every i = 1, . . . , t there exists j ≤ k such that ui(vj) > 0. Hence – after replacingthe ui by suitable scalar multiples – we may assume that ui(v0) = 1 for every i = 1, . . . , t, andthen lemma 3.3.2 implies that

Dσ = v ∈ W | ui(v) ≥ −1 for every i = 1, . . . , t.

Also, proposition 3.3.11(i) implies that

Dσ = v ∈ W | ui(v) > −1 for every i = 1, . . . , t.

For every v ∈ W , set µ(v) := min(ui(v) |i = 1, . . . , t). It is easily seen that µ(v) < 0 for everyv ∈ W \ 0; indeed, if µ(w) ≥ 0, then the subset λw | λ ∈ R+ lies in Dσ, and since thelatter is compact, it follows that w = 0. Moreover we have

(5.9.13) (R+w) ∩Dσ = λw | λ ∈ [0,−1/µ(w)] for every w ∈ W \ 0.


Fix a Banach norm ‖ · ‖V on V , and consider the mapping

ϕ : Dσ → W such that ϕ(0) := 0 and ϕ(v) := −µ(v)

‖v‖· v for v 6= 0.

It is easily seen that ϕ is injective; since µ is a continuous mapping, the same follows for ϕ, andsince Dσ is compact, we conclude that ϕ induces a homeomorphism Dσ

∼→ ϕ(Dσ). However,from (5.9.13) it follows that

ϕ(Dσ) = v ∈ W | ‖v‖ ≤ 1 and ϕ(Dσ) = v ∈ W | ‖v‖ < 1whence the claim.

5.9.14. Keep the notation of (5.9.10); we consider the finite filtration C•σ of Cσ, defined asfollows. For every i = −1, . . . , d − 1, we let Ci

σ ⊂ Cσ be the union of the subsets τ ∩ u−10 (1),

where τ ranges over the (finite) set of all faces of σ of dimension i + 1. Clearly C−1σ = ∅,

Cd−1σ = Cσ, and Ci

σ is a closed subset of Cσ, for every i = 0, . . . , d − 1. Moreover, it followseasily from lemma 5.9.12 and proposition 3.3.11(i) that the datum of Cσ and its filtration C•σis a cell complex. With the notation of definition 5.9.1, the indexing set Λi can be taken tobe the set of all (i + 1)-dimensional faces of σ, for every i = 0, . . . , d − 1 : indeed, if τ issuch a face, denote by τ the relative interior of τ (see example 3.3.16(iii)); then it is clear thateiτ := τ ∩ Cσ 6= ∅ and eiτ = τ ∩ Cσ.

Remark 5.9.15. (i) Notice that the cell complex C•σ is independent – up to homeomorphism –of the choice of u0. Indeed, say that u′0 ∈ σ∨ is any other linear form such that σ∩Keru′0 = 0,and let C ′σ := σ ∩ u′−1

0 (1). We define a homeomorphism ψ : C ′σ∼→ Cσ, by the rule :

v 7→ u0(v)−1 · v for every v ∈ C ′σ.It is easily seen that ψ restricts to homeomorphisms C ′iσ

∼→ Ciσ for every i = 0, . . . , d− 1.

(ii) For any integer i = 0, . . . , d − 3, and any cells eiτ , ei+2λ of C•σ such that eiτ ⊂ ei+2

λ ,there exist exactly two (i + 1)-dimensional cells ei+1

µ , ei+1ρ such that eiτ ⊂ ei+2

µ ∩ ei+1ρ and

ei+1µ ∪ ei+1

ρ ⊂ ei+2λ : indeed, this assertion is a direct translation of claim 3.3.9(ii).

Proposition 5.9.16. With the notation of (5.9.8) and (5.9.14), the following holds for everyi = 0, . . . , d− 2 :

(i) If τ ∈ Λi+1 and µ ∈ Λi is not a facet of τ , we have [bi+1τ : biµ] = 0.

(ii) If τ ∈ Λi+1 and µ ∈ Λi is a facet of τ , then [bi+1τ : biµ] ∈ 1,−1.

(iii) In the situation of remark 5.9.15(ii), we have :

[bi+2λ : bi+1

µ ] · [bi+1µ : biτ ] + [bi+2

λ : bi+1ρ ] · [bi+1

ρ : biτ ] = 0.

Proof. (i): Consider the commutative diagram

(5.9.17)

Hi+1(ei+1τ , ei+1

τ \ ei+1τ )

∂ //

lτ

Hi(ei+1τ \ ei+1

τ )jτ //

g

Hi(Ciσ, C

iσ \ eiµ)

Hi+1(Ci+1σ , Ci

σ)∂′ // Hi(C

iσ)

ji // Hi(Ciσ, C

i−1σ )

mµ

OO

where lτ and mµ are defined as in remark 5.9.4(ii,iii) and g is induced by the inclusion mapei+1τ \ ei+1

τ → Ciσ, the maps ∂ and ∂′ are the boundary operators of the long exact sequences

attached to the pairs (ei+1τ , ei+1

τ \ ei+1τ ) and (Ci+1

σ , Ciσ), and ji (resp. jτ ) is deduced from the

obvious map of pairs (ei+1τ \ ei+1

τ ,∅) → (Ciσ, C

iσ \ eiµ) (resp. (Ci

σ,∅) → (Ciσ, C

i−1σ )). In light

of remark 5.9.4(iii), we need to check that mµ ji−1 ∂′ lτ = 0, and to this aim, it suffices toshow that jτ = 0. However, the assumption implies that µ ∩ τ is a (proper) face of both µ andτ ; this translates as the identity (ei+1

τ \ ei+1τ ) ∩ eiµ = ∅, whence the claim.


(ii): Clearly C•τ is a cell subcomplex of C•σ, and ei+1τ and eiµ are cells of this subcomplex.

Moreover, any orientation for C•σ restricts to an orientation for C•τ , and the resulting incidencenumber of ei+1

τ and eiµ is the same for either cell complex. Thus, we may replace σ by τ , inwhich case the map g of (5.9.17) is the identity.

Claim 5.9.18. If σ = τ and i > 0, both ∂ and kµ := mµ ji in (5.9.17) are isomorphisms.

Proof of the claim. For ∂, we use the long exact homology sequence of the pair (ei+1τ , ei+1

τ \ei+1τ ):

since i ≥ 0, lemma 5.9.12 implies that Hi+1(ei+1τ ) = 0, so ∂ is injective; since i > 0, the same

argument shows that ∂ is surjective. Next, kµ is the homomorphism deduced from the longexact homology sequence of the pair (Ci

τ , Ciτ \ eiµ), and we remark that Ci

τ \ eiµ is contractible :indeed, eiµ is homeomorphic to Bi(1) (lemma 5.9.12), hence Ci

τ \ eiµ is a retraction of Ciτ \ x,

for any x ∈ eiµ, and the latter is homeomorphic to Ri (again by lemma 5.9.12). If i > 1, wededuce already that kµ is an isomorphism. For i = 1, the same argument shows that kµ isinjective, so its cokernel is a cyclic torsion group that injects into H0(C1

τ \ e1µ) ' Z, hence it

must vanish as well. ♦Since (5.9.17) commutes, claim 5.9.18 yields the assertion, in case i > 0. If i = 0, C1

τ isisomorphic to B1(1), and e0

µ is one of the two points of B1(1) \ B1(1), so the assertion can bechecked easily, by inspecting the definitions.

(iii) follows from (i), lemma 5.9.9(i) and remark 5.9.15(ii).

The following result says that the properties of proposition 5.9.16 completely characterizethe incidence numbers of C•σ.

Proposition 5.9.19. Keep the notation of (5.9.14), and consider a system of integers

(βiλµ | i = 1, . . . , d− 1, λ ∈ Λi, µ ∈ Λi−1)

such that :(a) If λ ∈ Λi and µ ∈ Λi−1 is not a facet of λ, then βiλµ = 0.(b) If λ ∈ Λi and µ ∈ Λi−1 is a facet of λ, then βiλµ ∈ 1,−1.(c) If µ and τ are the two 1-dimensional facets of the 2-dimensional face λ of σ, then

β1λµ + β1

λτ = 0.

(d) Let λ ∈ Λi+1 and µ ∈ Λi−1 be two faces, such that µ is a face of λ, and denote by τand ρ the two facets of λ that contain µ. Then

βi+1λτ β

iτµ + βi+1

λρ βiρµ = 0.

Then there exists a unique orientation of C•σ(biλ | i = 0, . . . , d− 1, λ ∈ Λi)

such that for every i = 1, . . . , d− 1 we have :

(5.9.20) [biλ : bi−1µ ] = βiλµ for every λ ∈ Λi, µ ∈ Λi−1.

Proof. We construct the orientation classes biλ fulfilling condition (5.9.20), by induction on i.For i = 0, the condition is empty, and the classes b0

λ are prescribed by (5.9.8). For i = 1, and agiven λ ∈ Λ1, denote by τ and ρ the two faces of λ; clearly the condition [b1

λ : b0τ ] = β1

λτ (andassumption (b)) determines b1

λ univocally, and in view of (c) and lemma 5.9.9(ii), for this choiceof orientation of e1

λ we have as well [b1λ : b0

ρ] = β1λρ. Moreover, if µ ∈ Λ0 \ τ, ρ, then (5.9.20)

is verified as well, by virtue of (a) and proposition 5.9.16(i).Now, suppose that i > 1 and that we have already constructed orientation classes bjµ as

sought, for every j < i and every µ ∈ Λj . Let λ ∈ Λi be any face, and fix a facet τ of λ; again,the identity

(5.9.21) [biλ : biτ ] = βiλτ


determines biλ, and it remains to check that – with this choice of biλ – condition (5.9.20) holds forevery µ ∈ Λi−1 \ τ, and by virtue of (a) and proposition 5.9.16(i), it suffices to consider thefacets µ of λ. However, notice that the system of orientation classes (bjµ | j = 0, . . . , i− 1, µ ⊂λ) amounts to an orientation of Ci−1

λ (regarded as a cell subcomplex of C•σ), and the incidencenumbers for the complex (Ci−1

λ )• relative to these classes agree with the incidence numbers ofC•σ, relative to the same classes. Especially, the sums

c :=∑

µ∈Λi−1

βiλµbi−1µ c′ :=

∑µ∈Λi−1

[biλ : bi−1µ ]bi−1

µ

are well defined elements of Ci−1(Ci−1λ ), and in fact :

di−1(c) =∑

µ∈Λi−1

βiλµ · di−1(bi−1µ )

=∑

µ∈Λi−1

βiλµ ·∑

ρ∈Λi−2

[bi−1µ : bi−2

ρ ]bi−2ρ

=∑

µ∈Λi−1

∑ρ∈Λi−2

βiλµβi−1µρ b

i−2ρ (by inductive assumption)

= 0 (by (d))

and a similar calculation yields di−1(c′) = 0 as well. However, Ci−1λ is homeomorphic to Si−1

(lemma 5.9.12), hence

Ker(di−1 : Ci−1(Ci−1λ )→ Ci−2(Ci−1

λ )) ' Z

which, in view of (b), implies that c = ±c′. Taking into account (5.9.21), we see that actuallyc = c′, whence the claim. The uniqueness of the orientation fulfilling condition (5.9.20) can bechecked easily by induction on i : the details shall be left to the reader.

5.9.22. Let now L be a free abelian group of finite rank d, and (LR, σ) a strictly convex L-rational polyhedral cone (see (3.3.20)), such that 〈σ〉 = LR; set

P := L ∩ σ Fλ := L ∩ λ Pλ := F−1λ P for every face λ of σ

so P and its localization Pλ are fine and saturated monoids (proposition 3.3.22(i)), and Fλ is aface of P , for every such λ. Let R be any ring, and if λ, µ ⊂ σ are any two faces with λ ⊂ µ,denote by

jλµ : R[Pλ]→ R[Pµ]

the natural localization map; notice that, if 0 ⊂ σ is the unique 0-dimensional face, then P0 = P ,and j0µ is the localization map R[P ] → R[Pµ]. We attach to P the complex C •P of R[P ]-modules such that :

C 0P := R[P ] and C i

P :=⊕λ∈Λi−1

R[Pλ] for every i = 1, . . . , dimP

with differentials given by the rule :

di(xλ) :=∑µ∈Λiλ⊂µ

[biµ : bi−1λ ] · jλµ(xλ) for every i ≥ 1, λ ∈ Λi−1 and xλ ∈ R[Pλ]

andd0(x) :=

∑µ∈Λ0

j0µ(x) for every x ∈ R[P ]


where (bjτ | j = 0, . . . dimσ − 1, τ ∈ Λj) is a chosen orientation for C•σ. Taking into accountlemma 5.9.9(ii) and proposition 5.9.16(iii), it is easily seen that di+1 di = 0 for every i ≥ 0(essentially, the differential di is the transpose of the differential di−1 of C •(C•σ)).

Set as well X := SpecR[P ], and notice that Z := SpecR〈P/mP 〉 ' SpecR is a closedsubset of X . For any R[P ]-module M , we denote as usual by M∼ the quasi-coherent OX-module arising from M .

Theorem 5.9.23. With the notation of (5.9.22), for every R[P ]-module M we have a naturalisomorphism

RΓZM∼ ∼→M ⊗R[P ] C

•P in D+(R[P ]-Mod).

Proof. For every face λ of σ, set Uλ := SpecR[Pλ], and denote by gλ : Uλ → X the openimmersion. We consider the chain complex R• of ZX-modules such that :

R0 := ZX Ri :=⊕λ∈Λi−1

gλ!ZUλ for every i = 1, . . . , dim P .

The differential d1 : R1 → R0 is just the sum of the natural morphisms gλ!ZUλ → ZX , for λranging over the one-dimensional faces of σ. For i > 1, the differential di is the sum of themaps

dλ :=∑µ∈Λiµ⊂λ

[biλ : bi−1µ ] · dλµ! : gλ!ZUλ → Rn−1 for every λ ∈ Λi−1

where dλµ : gλ!ZUλ → gµ!ZUµ ⊂ Rn−1 is induced by the inclusion Uλ ⊂ Uµ. With this notation,a simple inspection of the definitions yields a natural identification

(5.9.24) M ⊗R[P ] C•P∼→ Hom•Z(R•,M

∼[0]) in C(R[P ]-Mod).

On the other hand, let λ1, . . . , λn be the one-dimensional faces of σ; then λi is L-rational (see(3.3.20)), and Fλi is a fine, sharp and saturated monoid of dimension one (propositions 3.3.22(i)and 3.4.7(ii)), so Fλi ' N for i = 1, . . . , n (theorem 3.4.16(ii)). For each i = 1, . . . , n, let yi bethe unique generator of Fλi , and denote by I ⊂ P the ideal generated by y1, . . . , yn; we remark

Claim 5.9.25. The radical of I is mP .

Proof of the claim. For any x ∈ mP , pick a subset S ⊂ 1, . . . , n such that x =∑

i∈S aiyi,with ai > 0 for every i ∈ S. Let N ∈ N be large enough such that Nai ≥ 1 for every i ∈ S,and pick bi ∈ N such that Nai ≥ bi ≥ 1 for every i ∈ S. It follows that Nx−

∑i∈S biyi ∈ P ,

and the claim follows. ♦Now, let i : Z → X be the closed immersion; we have :

Claim 5.9.26. The natural map ZX → i∗ZZ induces an isomorphism

(5.9.27) R•∼→ i∗ZZ [0] in D+(ZX-Mod).

Proof of the claim. The assertion can be checked on the stalks at the points of X , hence letx ∈ X be any such point; if x ∈ Z, claim 5.9.25 easily implies that (R•)x is concentratedin degree zero, and then (5.9.27)x is clearly an isomorphism of complexes of Z-modules. Ifx /∈ Z, let p ⊂ R[P ] be the prime ideal corresponding to x, and λ ⊂ σ the unique face suchthat P \ Fλ = p ∩ P ; a simple inspection of the construction shows that

(R•)x∼→ C •(C

•λ)

(notation of (5.9.6)). But proposition 5.9.7 and lemma 5.9.12 imply that C •(C•λ) is acyclic,whence the claim. ♦

In view of (5.9.24) and claim 5.9.26, we are reduced to showing that the natural map

Hom•Z(R•,M∼[0])→ RHom•Z(R•,M

∼[0])


is an isomorphism in D+(R[P ]-Mod). This can be done by a spectral sequence argument,along the lines of remark 5.4.17 (which indeed includes a special case of the situation we areconsidering here, namely the case where P is a free monoid : the details shall be left to thereader).

5.9.28. Notice that C •P is a complex of L-gradedR-modules, hence its cohomology is L-gradedas well, and we wish next to compute the graded terms gr•H

•(C •P ). To this aim, we make thefollowing :

Definition 5.9.29. Let V be a finite dimensional R-vector space, X ⊂ V any subset, and z ∈ Vany point. Then :

(a) We say that a point x ∈ X is visibile from z, if tz+ (1− t)x | 0 ≤ t ≤ 1∩X = x.(b) We say that a subset S ⊂ X is visibile from z, if every point of S is visible from z.

Lemma 5.9.30. In the situation of (5.9.14), let z ∈ V \ Cσ be any point, and denote by S theset of points of Cσ that are visible from z. Then :

(i) S is a subcomplex of C•σ.(ii) S (with the topology induced from V ) is homeomorphic to Be(1), for e ∈ d−2, d−1.

Proof. Pick a system of generators ρ1, . . . , ρn for σ∨, and choose ρ0 ∈ σ∨ so that Cσ = σ ∩ρ−1

0 (1); we may assume that, for some integer k ≤ n we have

ρi(z) < 0 if and only if 1 ≤ i ≤ k.

Say that y ∈ Cσ, so that ρi(y) ≥ 0 for every i = 1, . . . , n, and set yt := tz + (1 − t)y forevery t ∈ [0, 1]; then clearly ρi(yt) ≥ 0 for every i = k + 1, . . . , n. Now, if ρ0(z) 6= 1, we getρ0(yt) 6= 1 for every t 6= 0, therefore the whole ofCσ is visible from z, in which case (i) is trivial,and (ii) follows from 5.9.12. Hence we may assume that ρ0(z) = 1, so ρ0(yt) = 1 for everyt ∈ [0, 1]. Suppose now that ρi(y) > 0 for some i ≤ k; then there exists a unique ti ∈]0, 1[ suchthat ρi(yti) = 0. Hence, if s := min(ρi(y) | i = 1, . . . , k) > 0, let t := min(ti | i = 1, . . . , k); itfollows that ρi(yt) ≥ 0 for every i = 1, . . . , n, so yt ∈ Cσ, which says that y is not visible fromz. Conversely, if s = 0, then it follows easily that y is visible from z. We conclude that

S = Cσ ∩k⋃i=1

Ker ρi

which shows (i). Next, set W := Ker ρ0, and denote by τz : ρ−10 (1)

∼→ W the translation mapgiven by the rule : x 7→ x − z for every x ∈ ρ−1

0 (1). To conclude, it suffices to check thatS ′ := τz(S) is homeomorphic to Bd−2(1). To this aim, denote by λ the convex cone in Wgenerated by τz(Cσ). Explicitly, if v1, . . . , vk ∈ σ have been chosen so that (5.9.11) holds, thenλ is the cone generated by v1− z, . . . , vk − z; especially, λ is a polyhedral cone, and it is easilyseen that 〈λ〉 = W . Moreover, λ is strictly convex; indeed, otherwise there exist real numbersa1, . . . , ak ≥ 0, with ai > 0 for at least an index i ≤ k, such that

∑ki=1 ai · (vi − z) = 0, i.e.∑k

i=1 aivi = (∑k

i=1 ai) · z, which is absurd, since z /∈ σ. Pick u ∈ λ∨ such that λ ∩Keru = 0,and set Cλ := λ∩u−1(1); by lemma 5.9.12, the subset Cλ is homeomorphic to Bd−2(1). Lastly,let π : W \ Keru → u−1(1) be the radial projection (so π(w) is the intersection point of Rwwith u−1(1), for every w ∈ W \Keru). It is easily seen that π maps S ′ bijectively onto Cλ, sothe restriction of π is a homeomorphism S ′

∼→ Cλ, as required.

Lemma 5.9.31. In the situation of (5.9.22), let λ ⊂ σ be any face. For every l ∈ LR we have:

(i) The set of points of σ that are visible from l is a union of faces of σ.


(ii) Suppose l ∈ L, and endow R[Pλ] with its natural L-grading. Then

grlR[Pλ] =

Rl if λ ⊂ σ is not visible from l0 otherwise.

Proof. Let ρ1, . . . , ρn be a system of generators of σ∨; we may assume that, for some integerk ≤ n we have

ρi(l) < 0 if and only if 1 ≤ i ≤ k.

Arguing as in the proof of lemma 5.9.30, we check easily that the set of points of σ visible froml equals S := σ ∩

⋃ki=1 Ker ρi, which already shows (i).

(ii): Suppose first that λ ⊂ σ is not visible from l; then there exists x ∈ λ \ S, and since Fλgenerates λ (see (3.3.20)), we may assume that x ∈ Fλ (details left to the reader). A simpleinspection then shows that there exists a sufficiently large N ∈ N such that l+Nx ∈ σ, whencel ∈ R[Pλ], and so grlR[Pλ] = Rl. Conversely, if the latter identity holds, then there existsx ∈ Fλ such that l + x ∈ P , whence ρi(x) > 0 for i = 1, . . . , k, so x is not visible from l.

5.9.32. In the situation of (5.9.22), let us fix a linear form u0 ∈ σ∨ such that σ ∩ Keru0 = 0,and define σ and Cσ as in (5.9.10). For any l ∈ LR, denote by Sl the set of points of σ that arevisible from l, so Sl is a union of faces of σ, by lemma 5.9.31(i), and therefore Cl := Sl ∩Cσ isa subcomplex of C•σ.

Proposition 5.9.33. With the notation of (5.9.32), suppose l ∈ L. Then the following holds :(i) If −l ∈ σ, the complex of R-modules grlC

•P is isomorphic to R[−d].

(ii) If −l /∈ σ, there is a natural isomorphism of complexes of R-modules :

grlC•P∼→ Hom•Z(C •(C

•σ)/C •(C

•l ), R[−1]).

Moreover, in this case, both C •(C•σ) and C •(C•l ) are acyclic complexes.

Proof. (i): If −l ∈ σ, then l ∈ Pλ if and only if λ = σ, so the assertion is clear.(ii): The sought identification of complexes follows from lemma 5.9.31(ii), by a direct in-

spection of the constructions (and indeed, this holds even if −l ∈ σ : details left to the reader).Moreover, it is already known from proposition 5.9.7 and lemma 5.9.12 that C •(C•σ) is acyclic.• Next, if l ∈ P , then clearly Sl = ∅, so the assertion for C •(C•l ) is trivial in this case.• Thus, suppose that l /∈ P ; if furthermore −l /∈ P , then the convex cone τ generated by P

and l is still strictly convex, so we may find u1 ∈ τ∨ such that u1(l) = 1 and σ∩Keru1 = 0. SetC ′σ := σ ∩ u−1

1 (1) and C ′l := Sl ∩C ′σ. In remark 5.9.15(i) we have exhibited a homeomorphismC ′•σ

∼→ C•σ that preserves the respective cell complex structures, and a simple inspection showsthat this homeomorphism maps C ′l onto Cl. On the other hand, it is easily seen that C ′l is alsothe set of points of C ′σ that are visible from l (indeed, the segment that joins any point of C ′σto l lies in u−1

1 (1), so its intersection with σ equals its intersection with C ′σ : details left to thereader). By virtue of lemma 5.9.30(ii) (and proposition 5.9.7), it follows that C •(C ′•l ) is acyclic,so the same holds for C •(C•l ).• Lastly, suppose −l ∈ P \ σ; we let ρ1, . . . , ρn be a system of generators of σ∨, and k ≤ n

an integer such that ρi(l) < 0 if and only if 1 ≤ i ≤ k. Denote by τ∨ (resp. µ∨) the convex conein L∨R generated by (ρk+1, . . . , ρn) (resp. by (−ρ1, . . . ,−ρk)), and let τ (resp. µ) be the dual ofτ∨ (resp. of µ∨) in LR; with this notation, we have l ∈ τ ∩µ. Especially, τ ∩µ 6= ∅, and sinceτ is dense in τ (by proposition 3.3.11(i)), we deduce that τ ∩µ 6= ∅ as well. Pick z ∈ τ ∩µ;by inspecting the proof of lemma 5.9.31(i), it is easily seen that Sl = Sz. But by construction,−z /∈ σ, therefore – arguing as in the previous case – we conclude that C •(C•z ) = C •(C•l ) isacyclic.

Theorem 5.9.34 (Hochster). Let R be a Cohen-Macaulay noetherian ring, P a fine, sharp andsaturated monoid. Then R[P ] is a Cohen-Macaulay ring.


Proof. In view of [61, p.181, Cor.] we may assume that R is a field. We argue by inductionon d := dimP . If d = 0, we have P = 0, and there is nothing to show. Suppose that d > 0and that the assertion is already known for every Cohen-Macaulay ring R and every monoid asabove, of dimension < d. Set L := P gp, and let σ ⊂ LR be the unique convex polyhedral conesuch that P = L ∩ σ. The ideal n := R[mP ] is maximal in R[P ], and proposition 5.9.33 andtheorem 5.9.23 show that

(5.9.35) depthR[P ]n = dim P.

On the other hand, we have the more general :

Claim 5.9.36. Let F be a field, P a fine monoid, A an integral domain which is an F -algebraof finite type, and F ′ the field of fractions of A. Then we have :

(i) For every maximal ideal m ⊂ A, the Krull dimension of Am equals the transcendencedegree of F ′ over F .

(ii) For every maximal ideal m ⊂ F [P ], the Krull dimension of F [P ]m equals rkZ Pgp.

Proof of the claim. (i): This is a straightforward consequence of [61, Th.5.6].(ii): Choose an isomorphism : P gp ∼→ L⊕ T , where T is the torsion subgroup of P gp, and L

is a free abelian group of finite rank; there follows an induced isomorphism (2.3.52) :

F [P gp]∼→ F [L]⊗F F [T ].

Let B be the maximal reduced quotient of F [P gp] (so the kernel of the projection F [P gp]→ Bis the nilpotent radical); we deduce that B is a direct product of the type

∏ni=1 F

′i [L], where

each F ′i is a finite field extension of F . By (i), the Krull dimension of F ′i [L]m equals r :=rkZP

gp for every maximal ideal m ⊂ F ′i [L], hence every irreducible component of SpecB hasdimension r. Let also C be the maximal reduced quotient of F [P ]; the natural map C → Bis an injective localization, obtained by inverting a finite system of generators of P , hence theinduced morphism SpecB → SpecC is an open immersion with dense image. Let Z be any(reduced) irreducible component of SpecC; again by (i) it follows that every non-empty opensubset of Z has dimension equal to dimZ, so necessarily the latter equals r. ♦

From (5.9.35), corollary 3.4.10(i) and claim 5.9.36(ii) we see already that R[P ]n is a Cohen-Macaulay ring. Next, let λ1, . . . , λk be the one-dimensional faces of σ, and define Pλi as in(5.9.22), for every i = 1, . . . , k. In light of claim 5.9.25, we have

SpecR[P ] \ n =k⋃i=1

SpecR[Pλi ]

so it remains to check that R[Pλi ] is Cohen-Macaulay for every i = 1, . . . , k. However, we mayfind a decomposition Pλi

∼→ Qi × Gi, where Gi is a free abelian group of finite type, and Qi

is a fine, sharp and saturated monoid of dimension d − 1 (lemma 3.2.10). Set Si := R[Gi];then R[Pλi ] = Si[Qi], and Si is a Cohen-Macaulay ring ([61, Th.17.7]), so the claim follows byinductive assumption.

5.9.37. In the situation of (5.9.22), we endow R[P ] with its natural L-grading, and denote byA := (R[P ], gr•R[P ]) the resulting L-graded R-algebra. Let (M, gr•M) be an L-graded A-module (see definition 4.4.9(ii)); we set

M † :=⊕l∈L

HomR(grlM,R)

which is naturally an R-submodule of M∗ := HomR(M,R) : indeed, any R-linear mapgrlM → R yields a linear form M → R, after composition with the projection M → grlM .Notice that M∗ is naturally an R[P ]-module; namely for any linear form f : M → R and any


x ∈ P , one defines x · f : M → R by the rule : x · f(m) := f(xm) for every m ∈M . Then, itis easily seen that M † is an R[P ]-submodule of M∗; more precisely, we have

(5.9.38) x · HomR(grlM,R) ⊂ HomR(grl−xM,R) for every x ∈ P .In light of (5.9.38), it is convenient to define an L-grading on M † by the rule :

grlM† := HomR(gr−lM,R) for every l ∈ L

and then (M †, gr•M†) is naturally an L-graded A-module. Clearly, the rule M 7→ M † yields a

functor from the category A-Mod of L-graded A-modules to A-Modo.

Example 5.9.39. If S ⊂ L is any P -submodule, notice that L \ (−S) is also a P -submodule ofL, and set

S† := L/(L \ (−S))

where the quotient is a pointed P -module, as in remark 2.3.17(iii). Explicitly, S† is the set(−S) ∪ 0S† (where the zero element 0S† should not be confused with the neutral element 0of the abelian group L); the P -module structure on S† is determined by the rule :

x · s :=

x+ s if x+ s ∈ −S0S† otherwise for every s ∈ −S.

Then it is easily seen that there exists a natural isomorphism of L-graded A-modules :

R[S]†∼→ R〈S†〉

(notation of (3.1.31)). On the other hand, the natural map R[S]→ (R[S]∗)∗ induces an isomor-phism R[S]

∼→ (R[S]†)† whence an isomorphism of L-graded A-modules

(5.9.40) R[S]∼→ R〈S†〉†.

Theorem 5.9.41 (Stanley, Danilov). In the situation of (5.9.22), take R to be any field, and setP := L ∩ σ. We have :

(i) There exists a natural isomorphism :

RHom•R[P ](R,R[P ])∼→ R[−d] in D(R[P ]-Mod)

(here R is regarded as an R[P ]-module, via the augmentation map R[P ]→ R).(ii) The complex of coherent OX-modules R[P ]∼[0] is dualizing on X .

Proof. (i): From proposition 5.9.33 we deduce a map of complexes of L-graded A-modules

ϕ• : C •P → R〈(P )†〉[−d]

(notation of example 5.9.39) with grlϕ• a quasi-isomorphism of complexes of R-modules, for

every l ∈ L. Since grlC•P is a finite dimensional R-vector space for every l ∈ L, there follows

– in view of (5.9.40) – a quasi-isomorphism of complexes of L-graded A-modules

(5.9.42) R[P ][0]∼→ (C •P )†[−d].

Claim 5.9.43. With the foregoing notation, we have :(i) ExtiR[P ](R, (C

jP )†) = 0 for every i ∈ N and every j = 1, . . . , d.

(ii) (C 0P )† is the injective hull of the R[P ]-module R.

Proof of the claim. (i): Let λ ⊂ σ be any face with dimλ > 0; it suffices to check that Ei :=ExtiR(P ](R,R[Pλ]

†) = 0 for every i ∈ N. To this aim, pick any x ∈ Fλ \ 0; since R[Pλ]† =

R〈P †λ〉 (example 5.9.39), we see that scalar multiplication by x is an automorphism on R[Pλ]†,

hence also on Ei. On the other hand, scalar multiplication by x is the zero endomorphism onR, hence also on Ei, and the claim follows.

(ii) is a special case of example 4.3.33 : details left to the reader. ♦


From (5.9.42) we get a convergent spectral sequence

Epq1 := ExtpR[P ](R, (C

d−qP )†)⇒ Extp+qR[P ](R,R[P ])

and claim 5.9.43 implies that Epq1 = 0 unless p = 0 and q = d; moreover, claim 4.3.24 says that

E0,d1 ' R, whence the contention.(ii): To ease notation, set S := SpecR and ωP := R[P ]∼; in view of [46, Ch.V, Cor.2.3], it

suffices to check that the complex of coherent OX(x)-modules ωP (x)[0] is dualizing onX(x), forevery x ∈ X (notation of definition 2.4.17(iii)). Hence, fix any such point x, let p ⊂ R[P ] be theprime ideal corresponding to x, and denote by λ ⊂ σ the unique face such that P \Fλ = p∩P ,so that x ∈ Uλ := SpecR[Pλ]. We may find a decomposition P ∼→ F gp

λ ×Q, where Q is also afine, sharp and saturated monoid (lemma 3.2.10), whence an isomorphism of S-schemes

Uλ∼→ SpecR[F gp

λ ]×S Y where Y := SpecR[Q]

and by construction, the induced projection p : Uλ → Y maps x to the maximal ideal of R[Q]generated by the maximal ideal mQ of Q. Let τ ⊂ Qgp

R be the unique polyhedral cone such thatQ = τ ∩Qgp, set Q := Q∩ τ , and define the coherent OY -module ωQ := R[Q]∼. It is easilyseen that there is a natural identification

ωP |Uλ∼→ p∗ωQ.

In view of [46, Ch.V, Th.8.3], it then suffices to check that ωQ(p(x)) is dualizing on Y (p(x)).Thus, we may replace P by Q, and assume from start that p is the augmentation ideal of R[P ].In this case, the assertion follows from (i) and proposition 5.6.26 (details left to the reader).

6. LOGARITHMIC GEOMETRY

6.1. Log topoi. Henceforth, all topoi under consideration will be locally ringed and withenough points, and all morphisms of topoi will be morphisms of locally ringed topoi (see(2.4.13)). The purpose of this restriction is to insure that we obtain the right notions, whenwe specialize to the case of schemes.

6.1.1. Let T := (T,OT ) be a locally ringed topos. Recall ([52, §1.1]) that a pre-log structureon T is the datum of a pair (M,α), where M is a T -monoid, and α : M → OT is a morphismof T -monoids, called the structure map of M , and where the monoid structure on OT is inducedby the multiplication law (hence, by the multiplication in the ring OT (U), for every object U ofT ). The generalities of [36, (6.4.1)-(6.4.8)] actually carry over verbatim to any locally ringedtopos T , hence we shall recall briefly the main definitions and constructions that we need, andrefer to loc.cit. for further details.

6.1.2. A morphism (M,α) → (N, β) of pre-log structures on T , is a map γ : M → N ofT -monoids, such that β γ = α. We denote by pre-logT the category of pre-log structures onT . A morphism of locally ringed topoi f : T → S induces a pair of adjoint functors :

(6.1.3) f ∗ : pre-logS → pre-logT f∗ : pre-logT → pre-logS.

A pre-log structure (M,α) on T is said to be a log structure if α restricts to an isomorphism:

α−1O×T∼→M× ∼→ O×T .

The datum of a locally ringed topos (T,OT ) and a log structure on T is also called, for short, alog topos. We denote by logT the full subcategory of pre-logT consisting of all log structureson T . When there is no danger of ambiguity, we shall often omit mentioning explicitly the mapα, and therefore only write M to denote a pre-log or a log structure. The category logT admitsan initial object, namely the log structure (O×T , j), where j is the natural inclusion; this is calledthe trivial log structure. logT admits a final object as well : this is (OT ,1OT ).


Lemma 6.1.4. Let T be a topos with enough points, ϕ : M → N a morphism of integralT -monoids inducing isomorphisms M× ∼→ N × and M ] ∼→ N ]. Then ϕ is an isomorphism.

Proof. This can be checked on the stalks, hence we are reduced to the corresponding assertionsfor a morphism M → N of monoids. Moreover, M ] is just the set-theoretic quotient of M bythe translation action of M× (lemma 2.3.31(iii)), so the assertion is straightforward, and shallbe left as an exercise for the reader.

Definition 6.1.5. Let γ : (M,α) → (N, β) be a morphism of pre-log structures on the locallyringed topos T , and ξ a T -point.

(i) We say that (M,α) is integral (resp. saturated) if M is an integral (resp. integral andsaturated) T -monoid.

(ii) We say that γ is flat (resp. saturated) at the point ξ, if γξ : M ξ → N ξ is a flat morphismof monoids (resp. a saturated morphism of integral monoids) (see remark 2.3.23(vi)).

(iii) We say that γ is flat (resp. saturated), if γ is a flat morphism of T -monoids (resp. a sat-urated morphism of integral T -monoids) (see definition 3.2.28). In view of proposition2.3.26 (resp. corollary 3.2.29), this is the same as saying that γ is flat (resp. saturated)at every T -point.

6.1.6. The forgetful functor :

logT → pre-logT : M 7→Mpre-log

admits a left adjoint :

pre-logT → logT : (M,α) 7→ (M,α)log

such that the resulting diagram :

(6.1.7)

α−1(O×T ) //

M

O×T

// M log

is cocartesian in the category of pre-log structures. One calls M log the log structure associatedto M . Composing with the adjunction (6.1.3), we deduce a pair of adjoint functors :

f ∗ : logS → logT f∗ : logT → logS

for any map of locally ringed topoi f : T → S. Explicitly, if N is any log structure on S, thenf ∗N is the push-out in the cocartesian diagram of T -monoids :

f ∗O×S//

f ∗(Npre-log)

O×T

// f ∗N.

It follows easily that the induced map of T -monoids :

(6.1.8) f ∗(N ])→ (f ∗N)]

is an isomorphism. As a consequence, for every point ξ of T , the natural map N f(ξ) → (f ∗N)ξis a local morphism of monoids.


6.1.9. Let (M,α) be a pre-log structure on T . The morphism α extends to a unique morphismof pointed T -monoids α : M → OT , whence a new pre-log structure

(M,α) := (M, α).

Clearly, (M,α) is a log structure if and only if the same holds for (M,α). More precisely, forany pre-log structure M there is a natural isomorphism of log structures :

(M)log ∼→ (M log).

Furthermore, for any morphism f : T → S of topoi, we have natural isomorphisms of pre-log(resp. log) structures

f ∗(N)∼→ (f ∗N) f∗(M)

∼→ (f∗M)

for every pre-log (resp. log) structure N on S and M on T (details left to the reader).

Example 6.1.10. (i) Let T → S be a morphism of topoi. Since the initial object of a categoryis the empty coproduct, it follows formally that the inverse image f ∗(O×S , j) of the trivial logstructure on S, is the trivial log structure on T .

(ii) Let T be a topos, and jU : T/U → T an open subtopos (see example 2.2.6(ii)). Considerthe subsheaf of monoids M ⊂ OT such that :

M(V ) := s ∈ OT (V ) | s|U×V ∈ O×T (U × V ) for every object V of T .

Then it is easily seen that the natural map M → OT is a log structure on T . This log structureis (naturally isomorphic to) the extension jU∗O×U of the trivial log structure on T/U .

(iii) Let U be any object of the topos T , and M a log structure on T . Since OT/U = OT |U , itis easily seen that the natural morphism of pre-log structures

(Mpre-log)|U → (M |U)pre-log

is an isomorphism.(iv) Let β : M → OT be a pre-log structure on a topos T . Then β−1(0) ⊂M is an ideal, and

β factors uniquely through the natural map M →M/β−1(0), and a pre-log structure

(M,β)red := (M/β−1(0), β)

called the reduced pre-log structure associated to M . As usual, we shall often write just M red

instead of (M,β)red. We say that β is reduced if the induced morphism of pre-log structuresM →M red is an isomorphism.

Suppose now that M is a log structure; then it is easily seen that the same holds for M red.More precisely, since the tensor product is right exact (see (2.3.19)), for any pre-log structureM the natural morphism of log structures

(M red)log → (M log)red

is an isomorphism.

Lemma 6.1.11. Let γ : (M,α)→ (N, β) be a morphism of pre-log structures on T . We have :(i) If M is integral (resp. saturated), then the same holds for M log.

(ii) The unit of adjunction M →M log is a flat morphism.(iii) If γ is flat (resp. saturated) at a T -point ξ, the same holds for the induced morphism

γlog : M log → N log of log structures.(iv) Especially, if γ is flat (resp. saturated), the same holds for γlog.

Proof. In view of lemma 2.3.46(ii) and proposition 2.3.26, both (i) and (ii) can be checked onstalks. Taking into account lemma 2.3.46(i), we are reduced to showing the following. LetP be a monoid, A a ring, β : P → (A, ·) a morphism of monoids; then the natural mapP → P ′ := P ⊗β−1A× A

× is flat, and if P is integral (resp. saturated), the same holds for P ′.


The first assertion follows easily from example 3.1.23(vi), and the second follows from remark3.2.5(i) (resp. from corollary 3.2.25(ix) and proposition 3.2.26).

(iii): The map γlog can be factored as the composition of

γ′ := γ ⊗β−1O×TO×T : N log → P := M ⊗β−1O×T

O×T

and the natural unit of adjunction γ′′ : P → P log = M log. If γξ is flat, the same clearly holds forγ′ξ, and (ii) says that γ′′ is flat, hence γlog

ξ is flat in this case. Lastly, suppose that γξ is saturated,and we wish to show that γlog

ξ is saturated. Set P := α−1ξ O×T,ξ and Q := β−1

ξ O×T,ξ. Then theinduced map (P−1M ξ)

] → (Q−1N ξ)] is saturated (lemma 3.2.12(ii,iii)). But the latter is the

same as the morphism (γlogξ )], and then also γξ is saturated, again by lemma 3.2.12(iii).

Lemma 6.1.12. Let f : T ′ → T be a morphism of topoi, ξ a T ′-point, and γ : (M,α)→ (N, β)a morphism of integral log structures on T . The following conditions are equivalent :

(a) γ is flat (resp. saturated) at the T -point f(ξ).(b) f ∗γ is flat (resp. saturated) at the T ′-point ξ.(c) γ]ξ is a flat (resp. saturated) morphism of monoids.

Proof. The equivalence of (a) and (c) follows from corollary 3.1.49 (resp. lemma 3.2.12(ii)).By the same token, (b) holds if and only if (f ∗γ)]ξ is flat (resp. saturated); in light of theisomorphism (6.1.8), the latter condition is equivalent to (c).

6.1.13. For any locally ringed topos T , let us write the objects of Mnd/Γ(T,OT ) in the form(M,ϕ), where M is any monoid, and ϕ : M → Γ(T,OT ) a morphism of monoids. There is anobvious global sections functor :

Γ(T,−) : pre-logT →Mnd/Γ(T,OT ) : (N,α) 7→ (Γ(T,N),Γ(T, α))

which admits a left adjoint :

Mnd/Γ(T,OT )→ pre-logT : (M,ϕ) 7→ (M,ϕ)T := (MT , ϕT ).

Indeed,MT is the constant sheaf on (T,CT ) with valueM , and ϕT is the composition of the mapof constant sheaves MT → Γ(T,OT )T induced by ϕ, with the natural map Γ(T,OT )T → OT .Again, we shall often just write MT to denote this pre-log structure.

After taking associated log structures, we deduce a left adjoint :

(6.1.14) Mnd/Γ(T,OT )→ logT : (M,ϕ) 7→M logT := (M,ϕ)log

T

to the global sections functor. M logT is called the constant log structure associated to (M,ϕ).

Definition 6.1.15. Let T be a locally ringed topos, (M,α) a log structure on T .

(i) A chart for M is an object (P, β) of Mnd/Γ(T,OT ), together with a map of pre-log structures ωP : (P, β)T → M , inducing an isomorphism on the associated logstructures. (Notation of (6.1.13).)

(ii) We say that a chart (P, β) is finite (resp. integral, resp. fine, resp. saturated) if P is afinitely generated (resp. integral, resp. fine, resp. integral and saturated) monoid.

(iii) Let ϕ : M → N be a morphism of log structures on T . A chart for ϕ is the datum ofcharts :

ωP : (P, β)T →M and ωQ : (Q, γ)T → N


for M , respectively N , and a morphism of monoids ϑ : Q→ P , fitting into a commu-tative diagram :

QT

ϑlogT //

ωQ

P logT

ωP

N

ϕ // M.

We say that such a chart is finite (resp. integral, resp. fine, resp. saturated) if themonoids P and Q are finitely generated (resp. integral, resp. fine, resp. integral andsaturated). We say that the chart is flat (resp. saturated), if ϑ is a flat morphism ofmonoids (resp. a saturated morphism of integral monoids).

(iv) We say that M is quasi-coherent (resp. coherent) if there exists a covering family(Uλ → 1T | λ ∈ Λ) of the final object 1T in (T,CT ), and for every λ ∈ Λ, a chart (resp.a finite chart) (Pλ, βλ) for M |Uλ .

(v) We say that (M,α) is quasi-fine (resp. fine) if it is integral and quasi-coherent (resp.and coherent).

(vi) Let ξ be any T -point. We say that a chart (P, β) is local (resp. sharp) at the point ξ, ifthe morphism βξ : P → OT,ξ is local (resp. if P is sharp and βξ is local).

Lemma 6.1.16. Let f : T → S be a morphism of locally ringed topoi, Q a log structure on S,and ξ any point of S. The following holds :

(i) If Q is quasi-coherent (resp. coherent, resp. integral, resp. saturated, resp. quasi-fine,resp. fine), then the same holds for f ∗Q.

(ii) Suppose that Q is an integral log structure. Then Q] is an integral S-monoid, and Q issaturated if and only if Q] is a saturated S-monoid.

(iii) Suppose that Q is quasi-coherent. Then Q is integral (resp. integral and saturated,resp. fine, resp. fine and saturated) if and only if there exists a covering family (Uλ →1S | λ ∈ Λ) of the final object of S, and for every λ ∈ Λ, an integral (resp. integral andsaturated, resp. fine, resp. fine and saturated) chart (Pλ)Uλ → Q|Uλ .

(iv) If P is any coherent log structure on S, and ω : P ξ → Qξ is a map of monoids, then :(a) There exists a neighborhood U of ξ and a morphism ϑ : P |U

∼→ Q|U of logstructures, such that ϑξ = ω.

(b) Moreover, for any two morphisms ϑ, ϑ′ : P |U∼→ Q|U with the property of (a), we

may find a smaller neighborhood V → U of ξ such that ϑ|V = ϑ′|V .(c) Especially, if Q is also coherent, and ω is an isomorphism, we may find ϑ and a

small enough U as in (a), such that ϑ is an isomorphism.(v) If M is a finitely generated monoid, and ω : M → OS,ξ a morphism of monoids, then

we may find a neighborhood U of ξ and a morphism ϑ : MU → OU of S/U -monoids,such that ϑξ = ω.

Proof. (i): If Q is the constant log structure associated to a map of monoids α : Q→ Γ(S,OS),then f ∗Q is the constant log structure associated to Γ(S, f \) α (where f \ : OS → f∗OT isthe natural map). The assertions concerning quasi-coherent or coherent log structures are astraightforward consequence. Next, suppose that Q is integral (resp. saturated); we wish toshow that f ∗Q is integral (resp. saturated). To this aim, let M := f ∗(Qpre-log); by lemma2.3.45(i), M is an integral (resp. saturated) T -monoid; then the assertion follows from lemma6.1.11(i).

(ii): By lemma 2.3.46(ii) the assertion can be checked on stalks. Hence, suppose that Q isintegral; thenQξ is integral by loc.cit., consequently, the same holds forQξ/Q

×ξ (lemma 2.3.38).

The second assertion follows from lemma 3.2.9(ii).


(iii): Suppose first that Q is quasi-coherent and integral. Hence, there is a covering family(Uλ → 1S | λ ∈ Λ), and for every λ ∈ Λ a monoid Mλ, a pre-log structure αλ : (Mλ)Uλ → OUλ ,and an isomorphism ((Mλ)Uλ , αλ)

log ∼→ Q|Uλ; whence a cocartesian diagram of S-monoids, as

in (6.1.7) :

(6.1.17)

N := α−1λ (O×Uλ) //

(Mλ)Uλ

O×Uλ// Q|Uλ

.

The induced diagram (6.1.17)int of integral S-monoids is still cocartesian. Moreover, sinceQ|Uλ

is integral, αλ factors through a unique map βλ : ((Mλ)Uλ)int → Q|Uλ→ OUλ , and the

morphism in S underlying the induced morphism of S-monoids N int → N ′ := β−1λ (O×Uλ) is an

epimorphism (this can be checked easily on the stalks). Furthermore, ((Mλ)Uλ)int ' (M intλ )Uλ

(see (2.3.49)). It follows that the natural map

Q|Uλ→ O×Uλ qN ′ (M

intλ )Uλ ' (M int

λ )logUλ

is an isomorphism, so the claim holds with Pλ := M intλ . If Q is fine, we can find Mλ as above

which is also finitely generated, in which case the resulting Pλ shall be fine.Suppose additionally, that Q is saturated. By the previous case, we may then find a covering

family (Uλ → eS | λ ∈ Λ), and for every λ ∈ Λ an integral monoid Mλ, a pre-log structure αλ :

(Mλ)Uλ → OUλ , and an isomorphism ((Mλ)Uλ , αλ)log ∼→ Q|Uλ

; whence a cocartesian diagram

(6.1.17) consisting of integral S-monoids. The induced diagram (6.1.17)sat is still cocartesian;one may then argue as in the foregoing, to obtain a natural isomorphism : Q|Uλ

∼→ (M satλ )log

Uλ.

Furthermore, if Mλ is finitely generated, the same holds forM satλ (corollary 3.4.1(ii)), hence the

chart thus obtained shall be fine and saturated, in this case.Conversely, if a family (Pλ | λ ∈ Λ) of integral (resp. saturated) monoids can be found

fulfilling the condition of (iii), then (Pλ)Uλ is an integral (resp. saturated) pre-log structure onT/Uλ (example 2.3.47(ii)), hence the same holds for its associated log structure Q|Uλ (lemma6.1.11(i)), and therefore also for Q (lemmata 2.3.46(ii), 2.2.16 and example 6.1.10(iii)). More-over, if each Pλ is fine, then Q is fine as well.

(iv.a): We may assume that P admits a finite chart α : MS → P , for some finitely generatedmonoid M , denote by β : Q → OS the structure map of Q, and set ω′ := ω αξ : M → Qξ.According to lemma 3.1.7(ii), the morphism ω′ factors through a map ω′′ : M → Γ(U ′, Q),for some neighborhood U ′ of ξ. By adjunction, ω′′ determines a morphism of S/U ′-monoidsψ : MU ′ → Q|U ′ whence a morphism of pre-log structures

(6.1.18) (MU ′ , β|U ′ ψ)→ (Q|U ′ , β|U ′).

Let us make the following general observation :

Claim 6.1.19. Let N be a finite monoid, F any S-monoid, f, g : NS → F two morphisms ofS-monoids, such that fξ = gξ. Then there exists a neighborhood U of ξ in S such that f|U = g|U .

Proof of the claim. By adjunction, the morphisms f and g correspond to unique maps ofmonoids Γ(f),Γ(g) : N → Γ(S, F ); since N is finite and fξ = gξ, we may find a neigh-borhood U of ξ such that the maps N → F (U) induced by Γ(f) and Γ(g) coincide. Again byadjunction, we deduce a unique morphism of S/U -monoids NU → F|U , which by constructionis just the restriction of both f and g. ♦

Let γ : P → OS be the structure map of P ; we apply claim 6.1.19 with S replaced by S/U ′,to deduce that there exists a small enough neighborhood U → U ′ of ξ such that the restriction


(γα)|U : M|U → OS|U agrees with β|U ψ|U . Then it is clear that the morphism of log structureassociated to (6.1.18)|U yields the sought extension ϑ of ω.

(iv.b): By assumption we have the identity : ϑξ = ϑ′ξ; however, any morphism of log struc-tures P |U → Q|U is already determined by its restriction to the image of any finite local chartMU → P |U . Hence the assertion follows from claim 6.1.19.

(iv.c): We apply (iv.a) to ω−1 to deduce the existence of a morphism σ : Q|U → P |U suchthat σξ = ω−1 on some neighborhood U of ξ. Hence, (ϑ σ)ξ = 1Qξ and (σ ϑ)ξ = 1P ξ . By(iv.b), these identities persist on some smaller neighborhood.

(v): The proof is similar to that of (iv.a), though simpler : we leave it as an exercise for thereader.

Definition 6.1.20. (i) A morphism (T,M) → (S,N) of topoi with pre-log (resp. log) struc-tures, is a pair f := (f, log f) consisting of a morphism of locally ringed topoi f : T → S, anda morphism

log f : f ∗N →M

of pre-log structures (resp. log structures) on T . We say that f is log flat (resp. saturated) iflog f is a flat (resp. saturated) morphism of pre-log structures.

(ii) Let (f, log f) as in (i) be a morphism of log topoi, ξ a T -point; we say that f is strict atthe point ξ, if log fξ is an isomorphism. We say that f is strict, if it is strict at every T -point.

(iii) A chart for ϕ is the datum of charts

ωP : (P, β)T →M and ωQ : (Q, γ)S → N

for M , respectively N , and a morphism of monoids ϑ : Q → P , such that (f ∗ωQ, ωP , ϑ) is achart for the morphism log f . We say that such a chart (ωQ, ωP , ϑ) is finite (resp. fine, resp. flat,resp. saturated) if the same holds for the corresponding chart (f ∗ωQ, ωP , ϑ) of log f .

Remark 6.1.21. (i) Let f : (T,M) → (S,N) be a morphism of log topoi, g : T ′ → Ta morphism of topoi, and f ′ : (T ′, g∗M) → (S,N) the composition of f and the naturalmorphism of log topoi (T ′, g∗M) → (T,M); let also ξ be a T ′-point. Then f ′ is strict at thepoint ξ if and only if f is strict at the point g(ξ). Indeed, f ′ is strict at ξ if and only if (log f ′)]ξ isan isomorphism (lemma 6.1.4), if and only if (log f)]g(ξ) is an isomorphism (by (6.1.8)), if andonly if log fg(ξ) is an isomorphism (again by lemma 6.1.4).

(ii) For any log topos (T,M), let us set (T,M) := (T,M). Then (T,M) is a log topos(see (6.1.9)), and clearly, every morphism f : (T,M) → (S,N) of log topoi extends naturallyto a morphism f : (T,M) → (S,N) of log topoi.

6.1.22. The category of log topoi admits arbitrary 2-limits. Indeed, if T := ((Tλ,Mλ) | λ ∈ Λ)is any pseudo-functor (from a small category Λ, to the category of log topoi), the 2-limit of Tis the pair (T,M), where T is the 2-limit of the system of ringed topoi (Tλ | λ ∈ Λ), and Mis the log structure on T obtained as follows. For every λ ∈ Λ, let pλ : T → Tλ be the naturalprojection; take the colimit M ′ of the induced system of pre-log structures (p∗λMλ | λ ∈ Λ), andthen let M be the log structure associated to M ′.

6.1.23. Consider a 2-cartesian diagram of log topoi :

(T ′,M ′)g //

f ′

(T,M)

f

(S ′, N ′)

h // (S,N).

The following result yields a relative variant of the isomorphism (6.1.8) :


Lemma 6.1.24. In the situation of (6.1.23), the morphism

g∗Coker(log f)→ Coker(log f ′)

induced by log g is an isomorphism of T ′-monoids.

Proof. Indeed, denote by β : M → OT and γ : N ′ → OS′ the log structures of T and S ′. Fixany T ′-point ξ, let ξ := g(ξ′), and set

P := M ξ ⊗Nf(ξ)N ′f ′(ξ′) and ρ := βξ ⊗ γf ′(ξ′) : P → OT ′ .

Then M ′ξ′ = P ⊗ρ−1OT ′,ξ′

O×T ′,ξ′ , and it is easily seen that ρ−1OT ′,ξ′ = O×T,ξ ⊗O×S,f(ξ)

O×S′,f ′(ξ′).

Therefore (M ′ξ′)

] = P/ρ−1OT ′,ξ′ = M ]ξ ⊗N]

f(ξ)N ′]f ′(ξ′), and

Coker(log f ′ξ′) = Coker(N ′]f ′(ξ′) →M ]ξ ⊗N]

f(ξ)N ′]f ′(ξ′)) = Coker(N ]

f(ξ) →M ]ξ)


6.1.25. We consider now a special situation, which will be encountered in proposition 6.1.28.Namely, let Q be a monoid, and H ⊂ Q× a subgroup. Let also G be an abelian group, ρ : G→Qgp a group homomorphism, and set :

Hρ := G×Qgp H Qρ := G×Qgp Q.

The natural inclusion H → Q and the projection Qρ → Q determine a unique morphism :

(6.1.26) Qρ ⊗Hρ H → Q.

Lemma 6.1.27. In the situation of (6.1.25), suppose furthermore that the composition :

Gρ−→ Qgp → (Q/H)gp

is surjective. Then (6.1.26) is an isomorphism.

Proof. Set G′ := G ⊕ H , and let ρ′ : G′ → Qgp be the unique group homomorphism thatextends ρ and the natural map H → Q → Qgp. Under the standing assumptions, ρ′ is clearlysurjective. Define Qρ′ and Hρ′ as in (6.1.25); there is a natural isomorphism of monoids :Qρ′

∼→ Qρ ⊕ H , inducing an isomorphism Hρ′∼→ Hρ ⊕ H , and defined as follows. To every

g ∈ G, h ∈ H , q ∈ Q such that [(g, h), q] ∈ Qρ′ , we assign the element [(g, h−1q), a] ∈ Qρ⊕H(details left to the reader). Under this isomorphism, the projection Hρ′ → H is identified withthe map Hρ ⊕H → H given by the rule : (h1, h2) 7→ πH(h1) · h2, where πH : Hρ → H is theprojection. It then follows that the natural map :

Qρ ⊗Hρ H → Qρ′ ⊗Hρ′ H

is an isomorphism. Thus, we may replace G and ρ by G′ and ρ′, which allows to assume fromstart that ρ is surjective. However, we have natural isomorphisms :

Ker(Qρ

πQ−−→ Q) ' Ker ρ ' Ker(HρπH−−→ H).

Moreover, the set underlyingQ (resp. H) is the set-theoretic quotient of the setQρ (resp. Hρ) bythe translation action of Ker ρ; hence the natural maps Qρ/Ker πQ → Q and Hρ/Ker πH → Hare isomorphisms (lemma 2.3.31(ii)). We can then compute :

Qρ ⊗Hρ H ' (Qρ/Ker πQ)⊗Hρ/KerπH H ' Q⊗H H ' Q

as stated.


Proposition 6.1.28. Let T be a locally ringed topos, ξ any T -point, and M a coherent (resp.fine) log structure on T . Suppose that G is a finitely generated abelian group with a grouphomomorphism G→Mgp

ξ such that the induced map G→ (M ])gpξ is surjective. Set

P := G×MgpξM ξ.

Then the induced morphism P → M ξ extends to a finite (resp. fine) chart PU → M |U on someneighborhood U of ξ.


Claim 6.1.29. Let Y be any locally ringed topos, ξ a Y -point, and α : QY → OY the constantpre-log structure associated to a map of monoids ϑ : Q → Γ(Y,OY ), where Q is finitelygenerated. Set S := α−1

ξ O×Y,ξ ⊂ QY,ξ = Q. Then :

(i) S and S−1Q are finitely generated monoids.(ii) There exists a neighborhood U of ξ such that α|U factors as the composition of the

natural map of sheaves of monoids jU : QU → (S−1Q)U , and a (necessarily unique)pre-log structure αS : (S−1Q)U → OU .

(iii) The induced map of log structures jlogU : Qlog

U → (S−1Q)logU is an isomorphism.

(iv) α−1S,ξ(O

×U,ξ) = (S−1Q)× is a finitely generated group.

Proof of the claim. (i) follows from lemma 3.1.20(iv).(ii): Since O×Y,ξ is the filtered colimit of the groups Γ(U,O×U ), where U ranges over the neigh-

borhoods of ξ, lemma 3.1.7(ii) and (i) imply that the induced map S → O×Y,ξ factors throughΓ(U,O×Y ) for some neighborhood U of ξ. Then the composition of ϑ and the natural mapΓ(Y,OY )→ Γ(U,OU) extends to a unique map S−1Q→ Γ(U,OU), whence the sought pre-logstructure αS on U .

(iii): Let N be any log structure on U ; it is clear that every morphism of pre-log structuresQU → N factors uniquely through (S−1Q)U , whence the contention.

(iv): Indeed, by construction we have : α−1S,ξ(O

×U,ξ) = Sgp. ♦

Let Y be a neighborhood of ξ such that M |Y admits a finite local chart α : QY → OY ; welift ξ to some Y -point, ξY , and choose a neighborhood U ∈ Ob(T/Y ) of ξY , as provided byclaim 6.1.29. We may then replace T by U , ξ by ξY and α by the chart αS of claim 6.1.29(ii),which allows to assume from start that S := α−1

ξ (O×T,ξ) is a finitely generated group. Moreover,let H := Ker(S → O×T,ξ); clearly αξ : Q → OT,ξ factors through the quotient Q′ := Q/H ,hence we may find a neighborhood U of ξ such that α|U factors through a (necessarily unique)map of pre-log structures αH : Q′U → OU . Furthermore, if N is any log structure on U , everymap of pre-log structures QU → N factors through Q′U , so that αH is a chart for M |U . We maytherefore replace T by U and α by αH , which allows to assume additionally, that αξ is injectiveon the subgroup S. Now, for any finitely generated subgroup H ⊂ O×T,ξ with S ⊂ H , we setM ξ,H := H qS Q; clearly, the monoids M ξ,H are finitely generated, and moreover :

M ξ = colimS⊂H⊂O×T,ξ

M ξ,H .

Furthermore, we deduce a natural sequence of maps of monoids :

(6.1.30) 1 →M ξ,HϕH−−→M ξ

ψH−−→ O×T,ξ/H → 1.

Claim 6.1.31. For every subgroup H as above, the sequence (6.1.30) is exact, i.e. M ξ,H is thekernel of ψH , and O×T,ξ/H is the cokernel of ϕH .

Proof of the claim. By lemma 2.3.29(iii), the assertion concerning KerψH can be verified onthe underlying map of sets; however, lemma 2.3.31(ii) says that the set M ξ is the set-theoretic


quotient (Q × O×T,ξ)/S, for the natural translation action of S, and a similar description holdsfor M ξ,H , therefore KerϕH is the set-theoretic quotient (H ×Q)/S, as required.

The assertion concerning CokerϕH holds by general categorical nonsense. ♦Let ε : M ξ → Mgp

ξ be the natural map; claim 6.1.31 implies that the sequence of abeliangroups (6.1.30)gp is right exact, and then a little diagram chase shows that :

(6.1.32) ε−1(ImϕgpH ) = M ξ,H whenever S ⊂ H ⊂ O×T,ξ.

Since G is finitely generated, we may find H as above, large enough, so that Mgpξ,H contains the

image of G. In view of (6.1.32), we deduce that the natural map

G×Mgpξ,H

M ξ,H → P

is an isomorphism, so P is finitely generated, by corollary 3.4.2; moreover P is integral when-ever M ξ is. Then, lemma 6.1.16(iv.a) implies that the natural map P → M ξ extends to amorphism of log structures ϑ : P log

U → M |U on some neighborhood U of ξ. It remains to showthat ϑ restricts to a chart for M |V , on some smaller neighborhood V of ξ. To this aim, it sufficesto show that the map of stalks ϑξ is an isomorphism (lemma 6.1.16(iv.c)). The latter assertionfollows from lemma 6.1.27.

Proposition 6.1.28 is the basis of several frequently used tricks that allow to construct “good”charts for a given coherent log structure (and for a morphism of such structures), or to “improve”given charts. We conclude this section with a selection of these tricks.

Corollary 6.1.33. Let T be a topos, ξ a T -point, M a fine log structure on T . Then there existsa neighborhood U of ξ in T , and a chart PU →M |U such that :

(a) P gp is a free abelian group of finite rank.(b) The induced morphism of monoids P → OT,ξ is local.

Proof. Choose a group homomorphism G := Z⊕n → Mgpξ (for some integer n ≥ 0), such

that the induced map G → (M ])gpξ is surjective, and set P := G ×Mgp

ξM ξ. By proposition

6.1.28, the induced map P → M ξ extends to a chart PU → M |V , for some neighborhood Uof ξ. According to example 2.3.36(v), P gp is a subgroup of G, whence (a). Next, claim 6.1.29implies that, after replacing P by some localization (which does not alter P gp), and U by asmaller neighborhood of ξ, we may achieve (b) as well.

Corollary 6.1.34. Let T be a topos, ξ a T -point, M a coherent log structure on T , and supposethat M ξ is integral and saturated. Then we have :

(i) There exists a neighborhood U of ξ in T , and a fine and saturated chart PU → M |Uwhich is sharp at the point ξ.

(ii) Especially, there exists a neighborhood U of ξ in T , such that M |U is a fine and satu-rated log structure.

Proof. (i): By lemma 3.2.10, we may find a decomposition M ξ = P ×M×ξ , for a sharp sub-

monoid P ⊂ M ξ. Set G := P gp; we deduce an isomorphism G∼→ Mgp

ξ /M×ξ , and clearly

P = G ×MgpξM ξ. By proposition 6.1.28, it follows that the induced map P → M ξ extends to

a chart β : PU → M |U on a neighborhood U of ξ. By construction, β is sharp at the T -point ξ;moreover, since M ξ is saturated, it is easily seen that the same holds for P . Finally, P is finitelygenerated, by corollary 3.4.2.

(ii): This follows immediately from (i) and lemma 6.1.16(iii).

Theorem 6.1.35. Let T be a locally ringed topos, ξ a T -point, f : M → N a morphism ofcoherent (resp. fine) log structures on T . Then :

(i) There exists a neighborhood U of ξ, such that f|U admits a finite (resp. fine) chart.


(ii) More precisely, given a finite (resp. fine) chart ωP : PT → M , we may find a neigh-borhood U of ξ, a finite (resp. fine) monoid Q, and a chart of f|U of the form

(ωP |U , ωQ : QU → N |U , ϑ : P → Q).

(iii) Moreover, if f is a flat (resp. saturated) morphism of fine log structures and (ωP , ωQ, ϑ)is a fine chart for f , then we may find a neighborhood U of ξ, a localization map j :Q→ Q′, and a flat (resp. saturated) and fine chart for f|U of the form (ωP |U , ωQ′ , jϑ),such that(a) ωQ|U = ωQ′ jU .(b) The chart ωQ′ is local at the point ξ.

Proof. Up to replacing T by T/U ′i for a covering (U ′i → 1T | i ∈ I) of the final object, we mayassume that we have finite (resp. fine) charts ωP : PT → M and Q′T → N (lemma 6.1.16(iii)),whence a morphism of pre-log structures :

ω : PTωP−−→M

f−→ N.

Let ξ be any T -point; there follow maps of monoids ϕ : P → M ξ → N ξ and ψ : Q′ → N ξ.Set G := (P ⊕ Q′)gp, and apply proposition 6.1.28 to the group homomorphism G → Ngp

ξ

induced by ϕ and ψ; for Q := G×NgpξN ξ, we obtain a finite (resp. fine) local chart QU → N |U

on some neighborhood U of ξ. Then, ϕ and the natural map P → G determine a unique mapP → Q, whence a morphism ω′ : PU → QU → N |U of pre-log structures; by construction,ω′ξ : P = PU,ξ → N ξ is none else than ϕ. By lemma 6.1.16(iv.b), we may then find a smallerneighborhood V → U of ξ such that ω′|V = ω|V . This proves (i) and (ii).

Next, we suppose that f is flat (resp. saturated) and both M , N are fine, and we wish to show(iii). In view of claim 6.1.29, we may find – after replacing T by a neighborhood of ξ – a finechart for f of the form (ωP ′ , ωQ′ , ϑ

′), such that :• P ′ and Q′ are localizations of P and Q, and ϑ′ is induced by ϑ;• ωP = ωP ′ (jP )T and ωQ = ωQ′ (jQ)T , where jP : P → P ′ and jQ : Q→ Q′ are the

localization maps;• the induced maps Q′] →M ]

ξ and P ′] → N ]ξ are isomorphisms.

Now, by proposition 2.3.26 (resp. corollary 3.2.29), the map fξ is flat (resp. saturated), hence thesame holds for the induced map M ]

ξ → N ]ξ, by corollary 3.1.49(i) (resp. by lemma 3.2.12(iii)).

Then the map P ′] → Q′] induced by ϑ′ is flat (resp. saturated) as well, so the same holds forϑ′, by corollary 3.1.49(ii) (resp. again by lemma 3.2.12(iii)). Finally, jP ϑ′ : P → Q′ is flat(resp. saturated), by example 3.1.23(iii) (resp. by lemma 3.2.12(i)), and (ωP , ωQ′ , jP ϑ′) is achart for f with the sought properties.

Corollary 6.1.36. Let T be a topos, ξ a T -point, and ϕ : M → N a saturated morphism of fineand saturated log structures on T . We have :

(i) There exists a neighborhood U of ξ, and a fine and saturate chart (ωP , ωQ, ϑ : P → Q)of ϕ|U , such that ωP and ωQ are sharp at the point ξ.

(ii) More precisely, suppose that (ωP , ωQ, ϑ : P → Q) is a fine and saturated chart for ϕ,such that M is sharp at the point ξ, and ωQ is local at ξ. Then there exists a sectionσ : Q] → Q of the projection Q→ Q], such that (ωP , ωQ σT , ϑ]) is a chart for ϕ.

Proof. After replacing T by some neighborhood of ξ, we may assume that M admits a chartωP : PT → M which is fine, saturated, and sharp at the point ξ (corollary 6.1.34(i)). Then,by theorem 6.1.35(iii), we may find a neighborhood U of ξ, and a fine and saturated chart(ωP |U , ωQ, ϑ : P → Q) for ϕ|U , such that ωQ is local at ξ, so that ϑ is also a local morphism.Hence, it suffices to show assertion (ii).

(ii): We notice the following :


Claim 6.1.37. Let ϑ : P → Q a local and saturated morphism of fine and saturated monoids,and suppose that P is sharp. Then there exists a section σ : Q] → Q of the projection Q→ Q],such that ϑ(P ) lies in the image of σ.

Proof of the claim. Pick an isomorphism β : Q∼→ Q] ×Q× as in lemma 3.2.10, and denote by

ψ : P → Q× the composition of ϑ with the induced projection Q → Q×. The morphism ϑ] isstill local and saturated (lemma 3.2.12(iii)), hence corollary 3.2.32(ii) implies that ϑ]gp extendsto an isomorphism P gp ⊕ L ∼→ Q]gp, where L is a free abelian group of finite rank. Thus, wemay extend ψgp to a group homomorphism ψ′ : Q]gp → Q×. Define an automorphism α ofQ] ×Q×, by the rule : (x, g) 7→ (x, g · ψ′(x)−1). The restriction σ : Q] → Q of (α β)−1 willdo. ♦

With the notation of claim 6.1.37 it is easily seen that ωQ σT is still a chart for N , hence(ωP , ωQ σT , ϑ]) is a chart for ϕ as required.

Corollary 6.1.38. Let f : (T,M)→ (S,N) a morphism of log topoi with coherent (resp. fine)log structures, and suppose that N admits a finite (resp. fine) chart ωQ : QS → N . Let also ξbe any T -point; we have :

(i) There exists a neighborhood U of ξ, and a finite (resp. fine) chart (ωQ|U , ωP , ϑ : Q →P ) for the morphism f|U .

(ii) Moreover, if M and N are fine and f is log flat (resp. saturated) then, on some neigh-borhood U of ξ, we may also find a chart (ωQ|U , ωP , ϑ) which is flat (resp. saturated)and fine.

Proof. This is an immediate consequence of theorem 6.1.35.

6.2. Log schemes. We specialize now to the case of a scheme X . Whereas in [36, §6.4] weconsidered only pre-log structures on the Zariski site of a scheme, hereafter we shall treat uni-formly the categories of log structures on the topoi X∼et and X∼Zar (notation of (2.4.13)).

6.2.1. Henceforth, we choose τ ∈ et,Zar (see (2.4.18)), and whenever we mention a topol-ogy on a scheme X , it will be implicitly meant that this is the topology Xτ (unless explicitlystated otherwise). Let X be a scheme; a pre-log structure (resp. a log structure) on X is apre-log structure (resp. a log structure) on the topos X∼τ . The datum of a scheme X and a logstructure onX is called briefly a log scheme. It is known that a morphism of schemesX → Y isthe same as a morphism of locally ringed topoiX∼τ → Y ∼τ , hence we may define a morphism oflog schemes (X,M)→ (Y,N) as a morphism of log topoi (X∼τ ,M)→ (Y ∼τ , N) (and likewisefor morphisms of schemes with pre-log structures). We denote by pre-logτ (resp. logτ ) thecategory of schemes with pre-log structures (resp. of log schemes) on the chosen topology τ .We denote by :

int.logτ sat.logτ qcoh.logτ coh.logτthe full subcategory of the category logτ , consisting of all log schemes with integral (resp.integral and saturated, resp. quasi-coherent, resp. coherent) log structures.

A scheme with a quasi-fine (resp. fine, resp. quasi-fine and saturated, resp. fine and saturated)log structure is called, briefly, a quasi-fine log scheme (resp. a fine log scheme, resp. a qfs logscheme, resp. a fs log scheme), and we denote by

qf .logτ f .logτ qfs.logτ fs.logτ

the full subcategory of logτ consisting of all quasi-fine (resp. fine, resp. qfs, resp. fs) logschemes on the topology τ . In case it is clear (or indifferent) which topology we are dealingwith, we will usually omit the subscript τ . There is an obvious (forgetful) functor :

F : log→ Sch


to the category of schemes, and it is easily seen that this functor is a fibration. For every schemeX , we denote by logX the fibre category F−1(X) i.e. the category of all log structures on X(or logXτ , if we need to specify the topology τ ). The same notation shall be used also for thevarious subcategories : so for instance we shall write int.logX for the full subcategory of allintegral log structures on X . Moreover, we shall say that the log scheme (X,M) is locallynoetherian if the underlying scheme X is locally noetherian.

6.2.2. Most of the forthcoming assertions hold in both the etale and Zariski topoi, with the sameproof. However, it may occasionally happen that the proof of some assertion concerning X∼τ(for τ ∈ et,Zar), is easier for one choice or the other of these two topologies; in such cases,it is convenient to be able to change the underlying topology, to suit the problem at hand. Thisis sometimes possible, thanks to the following general considerations.

The morphism of locally ringed topoi u of (2.4.15) induces a pair of adjoint functors :

u∗ : logZar → loget u∗ : loget → logZar

as well as analogous adjoint pairs for the corresponding categories of sheaves of monoids (resp.of pre-log structures) on the two sites. It follows formally that u∗ sends constant log struc-tures to constant log structures, i.e. for every scheme X , and every object M := (M,ϕ) ofMnd/Γ(X,OX) we have a natural isomorphism :

u∗(XZar,MlogXZar

) ' (Xet,MlogXet

).

More generally, lemma 6.1.16(i) shows that u∗ preserves the subcategories of quasi-coherent(resp. coherent, resp. integral, resp. fine, resp. fine and saturated) log structures.

Proposition 6.2.3. (i) The functor u∗ on log structures is faithful and conservative.(ii) The functor u∗ restricts to a fully faithful functor :

u∗ : int.logZar → int.loget.

(iii) Let (Xet,M) be any log scheme. Then the counit of adjunction u∗u∗(Xet,M) →(Xet,M) is an isomorphism if and only if the same holds for the counit of adjunctionu∗u∗(M

])→M ].

Proof. (i): Let (XZar,M) be any log structure; set (Xet,M et) := u∗(X,M), and denote byu\X : u∗OXZar

→ OXetthe natural map of structure rings. Since u is a morphism of locally

ringed topoi, we have :(u\X)−1O×Xet

= u∗(O×XZar).

It follows easily that M et is the push-out in the cocartesian diagram :

u∗M× α //

O×Xet

u∗M

β // M et.

However, for every geometric point ξ of X , the natural map OX,|ξ| → OX,ξ is faithfully flat,hence αξ is injective, and therefore also βξ, in light of lemma 2.3.31(ii). The faithfulness of u∗

is an easy consequence. Next, let f : M → N be a morphism of log structures on XZar, set(Xet, N et) := u∗(X,N), and suppose that u∗f : M et → N et is an isomorphism; we wish toshow that f is an isomorphism. However, β induces an isomorphism of monoids :

(6.2.4) u∗(M ])∼→M ]

et

and likewise for N ; it follows already that f induces an isomorphism M ] ∼→ N ]. To conclude,it suffices to invoke lemma 6.1.4.


(ii): Let us suppose that M is integral. According to [10, Prop.3.4.1], it suffices to show thatthe unit of adjunction (X,M)→ u∗(Xet,M et) is an isomorphism. Now, from the isomorphism(6.2.4) we deduce a commutative diagram :

M ] //

(u∗M et)]

u∗u∗(M ]) // u∗(M

]et)

whose bottom arrow is an isomorphism, and whose left vertical arrow is an isomorphism aswell, by lemma 2.4.26(iii) (and again [10, Prop.3.4.1]). We claim that also the right verticalarrow is an isomorphism. Indeed, since u∗ is left exact, the latter arrow is a monomorphism,hence it suffices to show it is an epimorphism; however, since M et is an integral log structure(lemma 6.1.16(i)), it is easily seen that the projection M et →M ]

et is a O×Xet-torsor (in the topos

X∼et/M]et). Then the contention follows from the exact sequence of pointed sheaves (2.4.12),

and the vanishing result of lemma 2.4.26(iv).Summing up, we conclude that the top horizontal arrow in the above diagram is an isomor-

phism, so the assertion follows from lemma 6.1.4.(iii): Set (Xet, (u∗M)et) := u∗u∗(Xet,M). To begin with, lemma 2.4.26(iv) easily implies

that the natural morphism (u∗M)] → u∗(M]) is an isomorphism; together with the general

isomorphism (6.1.8), this yields a short exact sequence of Xet-monoids :

0→ O×Xet→ (u∗M)et → u∗u∗(M

])→ 0

which easily implies the assertion : the details shall be left to the reader.

We shall prove later on some more results in the same vein (see corollary 6.2.22).

6.2.5. Arguing as in (6.1.22), we see easily that all finite limits are representable in the categoryof log schemes. The rule X 7→ (X,O×X ) defines a fully faithful inclusion of the category ofschemes into the category of log schemes. Hence, we shall regard a scheme as a log schemewith trivial log structure. Especially, if (X,M) is any log scheme, and Y → X is any morphismof schemes, we shall often use the notation :

(6.2.6) Y ×X (X,M) := (Y,O×Y )×(X,O×X ) (X,M).

Especially, if ξ is any τ -point of X , the localization of (X,M) at ξ is the log scheme

(X(ξ),M(ξ)) := X(ξ)×X (X,M)

(see definition 2.4.17(ii,iii)). If τ = et, this operation is also called the strict henselization of Xat ξ.

Definition 6.2.7. (i) For every integer n ∈ N, we have the subset :

(X,M)n := x ∈ X | dimM ξ ≤ n for every τ -point ξ → X localized at x.Especially (X,M)0 consists of all points x ∈ X such that M ξ = O×X,ξ for every τ -point ξ of Xlocalized at x; this subset is called the trivial locus of (X,M), and is also denoted (X,M)tr.

(ii) If f : (X,M) → (Y,N) is a morphism of log schemes, we denote by Str(f) ⊂ X thestrict locus of f , which is the subset consisting of all points x ∈ X such that f is strict at everyτ -point localized at x (see definition 6.1.20(ii)).

Remark 6.2.8. Let f : (X,M)→ (Y,N) be any morphism of log schemes.(i) Since log f induces local morphisms on stalks, it is easily seen that f restricts to a map

ftr : (X,M)tr → (Y,N)tr.

(ii) Especially, we have (X,M)tr ⊂ Str(f).


6.2.9. Let X := (Xi | i ∈ I) be a cofiltered family of quasi-separated schemes, with affinetransition morphisms fϕ : Xj → Xi, for every morphism ϕ : j → i in I . Denote by X the limitof X , and for each i ∈ I let πi : X → Xi be the natural projection.

Lemma 6.2.10. In the situation of (6.2.9), let X := ((Xi,M i) | i ∈ I) be a cofiltered sys-tem of log schemes, with transition morphisms (fϕ, log fϕ) : (Xi,M i) → (Xj,M j) for everymorphism ϕ : i→ j in I . We have :

(i) The limit of the system X exists in the category log.(ii) Let (X,M) denote the limit of the system X . If Xi is quasi-compact for every i ∈ I ,

then the natural map

colimi∈I

Γ(Xi, N i)→ Γ(X,N)

is an isomorphism.

Proof. (i): Let X be the limit of the system of schemes X , and endow X with the sheaf ofmonoids M := colim

i∈Iπ∗iM i, where π∗ is the pull-back functor for sheaves of monoids (see

(2.3.43)), and the transition maps π∗jM j → π∗iM i are induced by the morphisms log fϕ :f ∗ϕM j → M i, for every ϕ : i → j in I . Then the structure maps of the log structures M i

induce a well defined morphism of X-monoids M → OX , and we claim that the resultingscheme with pre-log structure (X,M) is actually a log scheme. Indeed, the assertion can bechecked on the stalks, and notice that, for every τ -point ξ of X we have a natural identification

OX,ξ∼→ colim

i∈IOXi,πi(ξi).

(This is clear for τ = Zar, and for τ = et one uses [33, Ch.IV, Prop.18.8.18(ii)]); it then sufficesto invoke lemma 2.3.46(i). Lastly, it is easily seen that (X,M) is a limit of the system X : thedetails shall be left to the reader.

(ii): In view of the explicit construction in (i), the assertion follows immediately from propo-sition 5.1.15.

Example 6.2.11. Let X be a scheme. For the following example we choose to work with theetale topology Xet on X . A divisor on X is a closed subscheme D ⊂ X which is regularlyembedded in X and of codimension 1 ([33, Ch.IV, Def.19.1.3, §21.2.12]). Suppose moreoverthat X is noetherian, let D ⊂ X be a divisor, and denote by (Di | i ∈ I) the irreducible reducedcomponents of D. We say that D is a strict normal crossing divisor, if :

• OX,x is a regular ring, for every x ∈ D.• D is a reduced subscheme.• For every subset J ⊂ I , the (scheme theoretic) intersection

⋂j∈J Dj is regular of pure

codimension ]J in X .A closed subscheme D of a noetherian scheme X is a normal crossing divisor if, for everyx ∈ X there exists an etale neighborhood f : U → X of x such that f−1D is a strict normalcrossing divisor in U .

Suppose that D is a normal crossing divisor of a noetherian scheme X , and let j : U :=X\D → X be the natural open immersion. We claim that the log structure j∗O×U is fine (this isthe direct image of the trivial log structure on Uet : see example 6.1.10(ii)). To see this, let ξ beany geometric point of X localized at a point of D; up to replacing X by an etale neighborhoodof ξ, we may assume that D is a strict normal crossing divisor; we can assume as well that Xis affine and small enough, so that the irreducible components (Dλ | λ ∈ Λ) are of the formV (Iλ), where Iλ ⊂ A := Γ(X,OX) is a principal divisor, say generated by an element xλ ∈ A,for every λ ∈ Λ. We claim that j∗O×U is the constant log structure associated to the pre-logstructure :

α : N(Λ)X → OX : eλ 7→ xλ


where (eλ | λ ∈ Λ) is the standard basis of N(Λ). Indeed, let S ⊂ Λ be the largest subset suchthat the image of ξ lies in DS :=

⋂λ∈S Dλ, we have xλ ∈ O×X,x for all λ /∈ S, so that the push-

out of the induced diagram of stalks O×X,ξ ← α−1O×X,ξ → N(Λ) is the same as the push-out PSof the analogous diagram O×X,ξ ← α−1

S O×X,ξ → N(S), where αS : N(S)X → OX is the restriction

of α. Suppose that a ∈ OX,ξ and a is invertible on X(ξ)\DS; the minimal associated primesof A/(a) are all of height one, and they must therefore be found among the prime ideals Axλ,with λ ∈ S. It follows easily that a is of the form u ·

∏λ∈S x

kλλ for certain kλ ∈ N and u ∈ O×X,ξ.

Therefore, the natural map βξ : PS → (j∗O×U )ξ is surjective. Moreover, the family (xλ | λ ∈ S)

is a regular system of parameters of OX,ξ ([30, Ch.0, Prop.17.1.7]), therefore the natural mapSymn

κ(ξ)(mξ/m2ξ) → mn

ξ /mn+1ξ is an isomorphism for every n ∈ N (here mξ ⊂ OX,ξ is the

maximal ideal); it follows easily that βξ is also injective.

Example 6.2.12. Suppose that X is a regular noetherian scheme, and D ⊂ X a divisor on X;let U := X \D. If D is not a normal crossing divisor, the log structure M := j∗O

×U on Xet

is not necessarily fine. For a counterexample, let K be an algebraically closed field, C ⊂ A2K

a nodal cubic; take D ⊂ X := A3K to be the (reduced, affine) cone over the cubic C, with

vertex x0 ∈ A3, and pick a geometric point ξ localized at x0. It is easily seen that, away fromthe vertex, D is a normal crossing divisor, hence M |X\x0 is a fine log structure on X \x0,by example 6.2.11. More precisely, let y0 ∈ C be the unique singular point, L ⊂ D the linespanned by x0 and y0, and η a geometric point localized at the generic point of L. By inspectingthe argument in example 6.2.11, we find that :

Mη ' N⊕2 ⊕ O×X,η

(indeed, an isomorphism is obtained by choosing a, b ∈ OC,y0 such that V (a) and V (b) are thetwo branches of the cubic C in an etale neighborhood of y0). On the other hand, let p ⊂ A :=K[T1, T2, T3] be the prime ideal corresponding to x0, and I ⊂ Ap the ideal defining the closedsubschemeX(x0)∩D inX(x0); we claim that I ·OX,ξ is still a prime ideal, necessarily of heightone. Indeed, let B := Ap/I , and denote by A∧ (resp. B∧) the p-adic completion of Ap (resp.of B); then B∧ is also the completion of the reduced ring OX,ξ/I , hence it suffices to show thatSpecB∧ is irreducible. However, we may assume that C ⊂ SpecK[T1, T2] is the affine cubicdefined by the ideal J ⊂ K[T1, T2] generated by T 3

1 − T 22 + T1T2. Then I is generated by the

element f := T 31 − T 2

2 T3 + T1T2T3; also, p = (T1, T2, T3), so that A∧ ' K[[T1, T2, T3]] andB∧ ' A∧/(f). Suppose SpecB∧ is not irreducible. This means that V (f) ⊂ SpecA∧ is aunion V (f) = Z1 ∪ · · · ∪ Zn of n ≥ 2 irreducible components Zi. Since A∧ is a local regularring, each such irreducible component Zi is a divisor, defined by some principal prime ideal qiin A∧. Let ai be a generator for qi; then f admits factorizations of the form f = aibi for somenon-invertible bi ∈ A∧. Fix some i, and set a := ai, b := bi; since f is homogeneous of degree3, we must have a ∈ pk\pk+1 for either k = 1 or k = 2 (k 6= 0 since a is not a unit, and k 6= 3,since b is not a unit); then b ∈ p3−k\p4−k. Write a = a′ + a′′ and b = b′ + b′′, where a′′ ∈ pk+1,b′′ ∈ p4−k and a′ (resp. b′) is homogeneous of degree k (resp. 3− k). Then f = ab = a′b′ + c,where c ∈ p4 and a′b′ is homogeneous of degree 3. This means that f = a′b′ is a factorizationof f inA. However, f is irreducible inA, a contradiction. (Instead of this elementary argument,one can appeal to [63, Th.43.20], which runs as follows. If R is a local domain, then there isa natural bijection between the set of minimal prime ideals of the henselization Rh of R andthe set of maximal ideals of the normalization of R in its ring of fractions. In our case, thenormalization Dν of D is the cone over the normalization of C, hence the only point of Dν

lying over x0 is the vertex of Dν .)It follows that any choice of a generator of I yields an isomorphism :

M ξ ' N⊕ O×X,ξ.


Suppose now that – in an etale neighborhood V of ξ – the log structure M is associated to apre-log structure α : PV → OV , for some monoid P ; hence M |V is the push-out of the diagramO×V ← α−1(O×V )→ PV , whence isomorphisms :

P/α−1ξ (O×X,ξ) 'M ξ/O

×X,ξ ' N P/α−1

η (O×X,η) 'Mη/O×X,η ' N⊕2.

But clearly α−1ξ (O×X,ξ) ⊂ α−1

η (O×X,η), so we would have a surjection of monoids N → N⊕2,which is absurd.

On the other hand, we remark that the log structure j∗O×U on the Zariski site XZar is fine :indeed, one has a global chart NXZar

→ j∗O×U , provided by the equation defining the divisor D.

6.2.13. Let R be a ring, M a monoid, and set S := SpecR[M ]. The unit of adjunction εM :M → R[M ] can be regarded as an object (M, εM) of Mnd/Γ(S,OS), whence a constant logstructure M log

S on S (see (6.1.13)). The rule

M 7→ Spec(R,M) := (S,M logS )

is clearly functorial in M . Namely, to any morphism λ : M → N of monoids, we attach themorphism of log schemes

Spec(R, λ) := (SpecR[λ], λlogSpecR[N ]) : Spec(R,N)→ Spec(R,M).

Likewise, if P is a pointed monoid, SpecR〈P 〉 is a closed subscheme of SpecR[P ], and wemay define

Spec〈R,P 〉 := Spec(R,P )×SpecR[P ] SpecR〈P 〉.Lastly, if M is a non-pointed monoid, notice the natural isomorphism of log schemes

Spec〈R,M〉∼→ Spec(R,M).

(Notation of remark 6.1.21(ii) : the details shall be left to the reader.)

Lemma 6.2.14. With the notation of (6.2.13), let a ∈M be any element, and setMa := S−1a M ,

where Sa := an | n ∈ N. ThenUa := SpecR[Ma] is an open subscheme of S, and the inducedmorphism of log schemes :

Spec(R,Ma)→ Spec(R,M)×S Uais an isomorphism.

Proof. Let βS : MS → OS and βUa : (Ma)Ua → OUa be the natural charts, and denote byϕ : M →Ma the localization map. For every τ -point ξ of Ua, we have the identity :

βS,ξ = βUa,ξ ϕ : M → OS,ξ.

Let Q := β−1Ua,ξ

O×Ua,ξ. The assertion is a straightforward consequence of the following :

Claim 6.2.15. The induced commutative diagram of monoids :

ϕ−1Q //

M

Q // Ma

is cocartesian.

Proof of the claim. Let b ∈ M , and suppose that βUa,ξ(a−1b) ∈ O×Ua,ξ. Since βUa,ξ(a) ∈O×Ua,ξ, we deduce that the same holds for βUa,ξ(b), i.e. b ∈ ϕ−1Q. Let Q′ := ϕ(ϕ−1Q); weconclude that Q = S−1

a Q′, the submonoid of Ma generated by Q′ and a−1. The claim followseasily.


6.2.16. In the same vein, let X be a R-scheme, and (M,ϕ) any object of Mnd/Γ(X,OX).The map ϕ induces, via the adjunction of (2.3.50), a homomorphism of R-algebras R[M ] →Γ(X,OX), whence a map of schemes f : X → S := SpecR[M ], inducing a morphism

(6.2.17) X ×S Spec(R,M)→ (X, (M,ϕ)logX )

of log schemes.

Lemma 6.2.18. In the situation of (6.2.16), we have :(i) The map (6.2.17) is an isomorphism.

(ii) The log scheme Spec(R,M) represents the functor

log→ Set : (Y,N) 7→ HomZ-Alg(R,Γ(Y,N))× HomMnd(P,Γ(Y,N)).

Proof. (i): The log structure f ∗(M logS ) of Spec(R,M)×S X represents the functor :

F : logX → Set : N 7→ HomMnd/Γ(S,OS)((M, εM),Γ(S, f∗N)).

However, if N is any log structure on X , the pre-log structure (f∗N)pre-log is the same asf∗(N

pre-log) (see [36, (6.4.8)]). From the explicit construction of direct images for pre-logstructures, and since the global sections functor is left exact (because it is a right adjoint), wededuce a cartesian diagram of monoids :

Γ(S, f∗N) //

Γ(S,OS)

Γ(X,N) // Γ(X,OX).

It follows easily that F is naturally isomorphic to the functor given by the rule :

N 7→ HomMnd/Γ(X,OX)((M,ϕ),Γ(X,N)).

The latter is of course the functor represented by (M,ϕ)logX .

(ii) can now be deduced formally from (i), or proved directly by inspecting the definitions.

6.2.19. From (6.2.13) it is also clear that the rule (R,M) 7→ Spec(R,M) defines a functor

Z-Algo × -Mono → log

which commutes with fibre products; namely, say that

(R′,M ′)← (R,M)→ (R′′,M ′′)

are two morphisms of Z-Alg × -Mon; then there is a natural isomorphism of log schemes :

Spec(R′ ⊗R R′′,M ′ ⊗M M ′′)∼→ Spec(R′,M ′)×Spec(R,M) Spec(R′′,M ′′).

For the proof, one compares the universal properties characterizing these log schemes, usinglemma 6.2.18(ii) : details left to the reader.

6.2.20. Let X be a scheme, M a sheaf of monoids on Xτ . We say that M is locally constantif there exists a covering family (Uλ → X | λ ∈ Λ) and for every λ ∈ Λ, a monoid Pλ andan isomorphism of sheaves of monoids M |Uλ ' (Pλ)Uλ . We say that M is constructible if, forevery affine open subsetU ⊂ X we can find finitely many constructible subsetsZ1, . . . , Zn ⊂ Usuch that :

• U = Z1 ∪ · · · ∪ Zn• M |Zi is a locally constant sheaf of monoids, for every i = 1, . . . , n.


If moreover, M ξ is a finitely generated monoid for every τ -point ξ of X , we say that M is offinite type. If ϕ : M → N is a morphism of constructible sheaves of monoids on Xτ , then it iseasily seen that Kerϕ, Cokerϕ and Imϕ are also constructible. Moreover, if M and N are offinite type, then the same holds for Cokerϕ and Imϕ.

Suppose that M is a sheaf of monoids on Xτ , and for every x ∈ X choose a τ -point xlocalized at x; the rank of M is the function

rkM : X → N ∪ ∞ x 7→ dimQMgpx ⊗Z Q.

It is clear from the definitions that the rank function of a constructible sheaf of monoid of finitetype is constructible on X .

Lemma 6.2.21. Let X be a scheme, ϕ : Q→ Q′ a morphism of coherent log structures on Xτ .Then :

(i) The sheaves Q] and Cokerϕ are constructible of finite type.(ii) (X,Q)n is an open subset of X , for every integer n ≥ 0. (See definition 6.2.7(i).)

(iii) The rank functions of Q] and Cokerϕ are (constructible and) upper semicontinuous.

Proof. Suppose that Q admits a finite chart α : MX → Q. Pick a finite system of generatorsΣ ⊂M , and for every S ⊂ Σ, set :

ZS :=⋂s∈S

D(s) ∩⋂

t∈Σ\S

V (t)

where, as usual D(s) (resp. V (s)) is the open (resp. closed) subset of the points x ∈ X suchthat the image s(x) ∈ κ(x) of s is invertible (resp. vanishes). Clearly each ZS is a constructiblesubset of X , and their union equals X . Moreover, for every S ⊂ Σ, and every τ -point ξsupported on ZS , the submonoid Nξ := α−1

ξ (O×X,ξ) ⊂ M is a face of M (lemma 3.1.20(i)),hence it is the submonoid 〈S〉 generated by Σ ∩ Nξ = S (lemma 3.1.20(ii)). It follows easilythat Q]

|ZS ' (M/〈S〉)|ZS .More generally, suppose that Q is coherent. We may assume that X is quasi-compact. Then,

by the foregoing, we may find a finite set Λ and a covering family (fλ : Uλ → X | λ ∈ Λ)

of X , such that Q]|Uλ is a constructible sheaf of monoids on Uλ. Since fλ is finitely presented,

it maps constructible subsets to constructible subsets ([33, Ch.IV, Th.1.8.4]); it follows easilythat the restriction of Q] to fλ(Uλ) is constructible, therefore Q] is constructible. Next, noticethat Cokerϕ = Cokerϕ]; by the foregoing, Q′] is constructible as well, so the same holds forCokerϕ.

Next, Let x ∈ X be any point, and ξ a τ -point of X localized at x; by theorem 6.1.35(i) wemay find a neighborhood f : U → X of ξ in Xτ and a finite chart (ωP , ωP ′ , ϑ) for ϕ|U . Byclaim 6.1.29, we may assume, after replacing U by a smaller neighborhood of ξ, that ωP andωP ′ are local at the point ξ, in which case Q]

ξ = P ] and Cokerϕξ = Cokerϑ. Let r : X → N(resp. r′ : U → N) denote the rank function of Q] (resp. of Q]

|U ); then it is clear that r′ = r f .On the other hand, for every y ∈ U and every τ -point η of U localized at y, the stalk Q]

η is aquotient of P ], hence r′(y) ≤ r(x) and dimQη ≤ dimQξ. Since f is an open mapping, thisshows (ii), and also that the rank function of Q] is upper semicontinuous. Likewise, let s (resp.s′) denote the rank function of Cokerϕ (resp. Cokerϕ|U ); then s′ = s f , and Cokerϕη isa quotient of Cokerϑ for every τ -point η of U ; the latter implies that s′(y) ≤ s(x) for everyy ∈ U , which shows that s is upper semicontinuous.

Corollary 6.2.22. Let X be a scheme, M a log structure on XZar, and set

(Xet,M et) := u∗(XZar,M).


Then M is integral (resp. coherent, resp. fine, resp. fine and saturated) if and only if the sameholds for M et.

Proof. It has already been remarked that (Xet,M et) is coherent (resp. fine, resp. fine andsaturated) whenever the same holds for (XZar,M); furthermore, the proof of proposition 6.2.3(i)shows that the natural map on stalks M |ξ| → M et,ξ is injective for every geometric point ξ ofX , therefore M is integral whenever the same holds for M et.

Next, we suppose that M et is coherent, and we wish to show that M is coherent.Let x ∈ X be any point, ξ a geometric point localized at x. By assumption there exists a

finitely generated monoid P ′, with a morphism α : P ′ →M et,ξ inducing an isomorphism :

P ′ ⊗β−1M×et,ξM×

et,ξ →M et,ξ 'Mx ⊗M×x O×X,ξ.

It follows easily that we may find a finitely generated submonoid Q ⊂Mx, such that the imageof α lies in (Q ·M×

x )⊗M×x O×X,ξ, and therefore the natural map :

(Q ·M×x )⊗M×x O×X,ξ →M et,ξ

is surjective. Then lemma 2.3.31(ii) implies that Q ·M×x = Mx, in other words, the induced

map Q→M ]x = M ]

et,ξ is surjective. Set

P := Qgp ×Mgpet,ξ

M et,ξ.

In this situation, proposition 6.1.28 tells us that the induced map βξ : P → M et,ξ extends to anisomorphism of log structures βlog : P log

Uet→M et|Uet

on some etale neighborhood U → X of ξ.On the other hand, it is easily seen that the diagram of monoids :

Mx//

Mgpx

M et,ξ

// Mgpet,ξ

is cartesian, therefore βξ factors uniquely through a morphism β′x : P →Mx. The latter extendsto a morphism of log structures β′ log : P log

U ′Zar→ M |U ′Zar

on some (Zariski) open neighborhoodU ′ of x in X (lemma 6.1.16(iv.a),(v)). By inspecting the construction, we find a commutativediagram of monoids :

(u∗P logU ′Zar

)ξ //

(u∗β′ log)ξ

P logUet,ξ

βlogξ

(u∗M)ξ M et,ξ

(where u∗ is the functor on log structures of (6.2.2)) whose horizontal arrows and right verticalarrow are isomorphisms; it follows that (u∗β′ log)ξ is an isomorphism as well, therefore u∗β′ log

restricts to an isomorphism on some smaller etale neighborhood f : U ′′ → U ′ of ξ (lemma6.1.16(iv.c)). Since f is an open morphism, we deduce that the restriction of (u∗β′ log)ξ isalready an isomorphism on f(U ′′)et, and f(U ′′) is a (Zariski) open neighborhood of x in X . Fi-nally, in light of proposition 6.2.3(i), we conclude that the restriction of β′ log is an isomorphismon f(U ′′)Zar, so M is coherent, as stated.

Lastly, we suppose that M et is fine and saturated, and we wish to show that M is saturated.However, the assertion can be checked on the stalks, hence let ξ be any geometric point of X;the proof of proposition 6.2.3 shows that the natural map M ]

|ξ| → M ]et,ξ is bijective, so the

assertion follows from lemma 3.2.9(ii).


Corollary 6.2.23. The category coh.log admits all finite limits. More precisely, the limit (inthe category of log structures) of a finite system of coherent log structures, is coherent.

Proof. Let Λ be a finite category, X := ((Xλ,Mλ) | λ ∈ Λ) an inverse system of schemeswith coherent log structures indexed by Λ. Denote by Y the limit of the system (Xλ | λ ∈ Λ)of underlying schemes, and by πλ : Y → Xλ the natural morphism, for every λ ∈ Λ. Itis easily seen that the limit of X is naturally isomorphic to the limit of the induced systemY := ((Y, π∗λMλ) | λ ∈ Λ), and in view of lemma 6.1.16(i), we may therefore replace X byY , and assume that X is a finite inverse system in the category logX , for some scheme X(especially, the underlying maps of schemes are 1X , for every morphism λ→ µ in Λ).

It suffices then to show that the push-out of two morphisms g : N → M , h : N → M ′ ofcoherent log structures on X , is coherent. However, notice that the assertion is local on the siteXτ , hence we may assume that N admits a finite chart QX → N . Then, thanks to theorem6.1.35(ii), we may further assume that both f and g admit finite charts of the form QX → PXand respectively QX → P ′X , for some finitely generated monoids P and P ′. We deduce anatural map (P qQ P ′)X

∼→ PX qQX P ′X →M qN M ′, which is the sought finite chart.

Lemma 6.2.24. In the situation of (6.2.9), suppose that Xi is quasi-compact for every i ∈ I ,and that there exists i ∈ I , and a coherent log structure N i on Xi, such that the log structureN := π∗iN i on Xτ admits a finite chart β : QX → N . Then there exists a morphism ϕ : j → iin I and a chart βj : QXj → N j := f ∗ϕN i for N j , such that π∗jβj = β.

Proof. After replacing I by I/i, we may assume that N j is well defined for every j ∈ I , andi is the final object of I . In this case, notice that (X,N) is the limit of the cofiltered system oflog schemes ((Xj, N j) | j ∈ I). We begin with the following :

Claim 6.2.25. In order to prove the lemma, it suffices to show that, for every τ -point ξ of Xthere exists j(ξ) ∈ I , a neighborhood Uξ → Xj(ξ) of πj(ξ)(ξ) in Xj(ξ),τ , and a chart βj(ξ) :QUξ → N j(ξ)|Uξ such that (1Uξ ×Xj(ξ) πj(ξ))∗βj(ξ) = β.

Proof of the claim. Clearly we may assume that Uξ is an affine scheme, for every τ -point ξ ofX .Under the stated assumptions, X is quasi-compact, hence we may find a finite set ξ1, . . . , ξnof τ -points of X , such that (Uξk ×Xj(ξk)

X → X | k = 1, . . . , n) is covering in Xτ . Since I iscofiltered, we may then find j ∈ I and morphisms ϕk : j → j(ξk) for every k = 1, . . . , n. Afterreplacing each βj(ξk) by f ∗ϕkβj(ξk), we may assume that j(ξk) = j for every k ≤ n.

In this case, set βk := βj(ξk), and Uk := Uξk for every k ≤ n; let also Ukl := Uk ×Xj Ul,U∼kl := Ukl ×Xj X for every k, l ≤ n, and denote by πkl : U∼kl → Ukl the natural projection.Hence, for every k, l ≤ n we have a morphism of Ukl-monoids :

βkl := βk|Ukl : QUkl → N j|Ukl

and by construction we have π∗klβkl = π∗lkβlk under the natural identification : U∼kl∼→ U∼lk .

Notice now that Ukl is quasi-compact and quasi-separated for every k, l ≤ n, hence the naturalmap

colimi∈I

Γ(Ukl ×Xj Xi, N i)→ Γ(U∼kl , N)

is an isomorphism (lemma 6.2.10(ii)). On the other hand, βkl is given by a morphism of monoidsbkl : Q → Γ(Ukl, N j), and likewise π∗klβkl is given by the morphism Q → Γ(U∼kl , N) obtainedby composiiton of bkl and the natural map Γ(Ukl, N j) → Γ(U∼kl , N). By lemma 3.1.7(ii), wemay then find a morphism j′ → j in I such that the following holds. Set

Vk := Uk ×Xj Xj′ Vkl := Vk ×Xj′ Vl for every k, l ≤ n


and let pk : Vk → Uk be the projection for every k ≤ n; let also β′k := p∗kβk, which is a chartQVk → N j′|Vk for the restriction of N j′ . Then

(6.2.26) β′k|Vkl = β′l|Vkl for every k, l ≤ n

under the natural identification Vkl∼→ Vlk. By construction, the system of morphisms (Vk ×Xj′

X → X | k = 1, . . . , n) is covering in Xτ ; after replacing j′ by a larger index, we may thenassume that the system of morphisms (Vk → Xj′ | k = 1, . . . , n) is covering in Xj′,τ ([32,Ch.IV, Th.8.10.5]). In this case, (6.2.26) implies that the local charts β′k glue to a well definedchart βj′ : QXj′

→ N j′ , and a direct inspection shows that we have indeed π∗j′βj′ = β. ♦

Now, denote by α : QX → OX the composition of β and the structure map of N , and let ξ bea τ -point of X; we have a natural isomorphism

N ξ = N i,πi(ξ) ⊗N×i,πi(ξ)

O×X,ξ∼→ colim

j∈IN i,πi(ξ) ⊗N×

i,πi(ξ)O×Xj ,πj(ξj)

∼→ coliminI

N j,πj(ξ)

([33, Ch.IV, Prop.18.8.18(ii)]). It follows that βξ and αξ factor through morphisms of monoidsβξ,j : Q → N j,πj(ξ) and αξ,j : Q → OXj ,πj(ξ) for some j ∈ I (lemma 3.1.7(ii)), and again wemay replace I by I/j, after which we may assume that βξ,j and αξ,j are defined for every j ∈ I .Then αξ,j extends to a pre-log structure αj : QUj → OUj on some neighborhood Uj → Xj ofπj(ξ) in Xj,τ (lemma 6.1.16(v)), and we may also assume that βξ,j extends to a morphism ofpre-log structures βj : (QUj , αj)→ N j|Uj (lemma 6.1.16(iv.a)). Notice as well that

N ]j,ξ ' N ]

ξ and β−1ξ,jN

×j,πj(ξ)

= β−1ξ N×ξ for every j ∈ I

and since βξ induces an isomorphism Q/β−1ξ N×ξ

∼→ N ]ξ, we deduce that βj,ξ (which is the same

as βξ,j) induces an isomorphismQ/α−1j,ξO

×Xj ,πj(ξ)

∼→ N ]j,ξ for every j ∈ I . In turn, it then follows

from lemma 6.1.4 that βj,ξ induces an isomorphism (QUj , αj)logπj(ξ)

∼→ N j,πj(ξ).Next, by lemma 6.1.16(iv.b,c) we may find, for every j ∈ I , a neighborhood U ′j → Uj of ξ in

Xτ , such that the restriction (QU ′j, αj|U ′j)→ N j|U ′j of βj is a chart for N j|U ′j .

By construction, the morphism ((1Uj ×Xj πj)∗βj)ξ : Q→ N ξ is the same as βξ, so by lemma6.1.16(iv.b) we may find a neighborhood Vj → U ′j ×Xj X of ξ in Xτ , such that

((1Uj ×Xj πj)∗βj)|Vj = β|Vj and ((1Uj ×Xj πj)∗αj)|Vj = α|Vj .

Claim 6.2.27. In the situation of (6.2.9), let Y → X be an object of the site Xτ , with Y quasi-compact and quasi-separated. We have :

(i) There exists i ∈ I , an object Yi → Xi of Xi,τ , and an isomorphism of X-schemesY∼→ Yi ×Xi X .

(ii) Moreover, if Y → X is covering in Xτ , then we may find i ∈ I and Yi → Xi as in (i),which is covering in Xi,τ .

Proof of the claim. (i) is obtained by combining [32, Ch.IV, Th.8.8.2(ii)] and [33, Ch.IV,Prop.17.7.8(ii)] (and [32, Ch.IV, Cor.8.6.4] if τ = Zar). Assertion (ii) follows from (i) and[32, Ch.IV, Th.8.10.5]. ♦

By claim 6.2.27(i), we may assume that Vj = U ′′j ×Xj X for some neighborhood U ′′j → U ′jof πj(ξ) in Xj,τ . To conclude, it suffices to invoke claim 6.2.25.

Proposition 6.2.28. In the situation of (6.2.9), suppose that Xi is quasi-compact for everyi ∈ I . Then the natural functor :

(6.2.29) 2-colimI

coh.logXi → coh.logX

is an equivalence.

Proof. To begin with, we show :


Claim 6.2.30. The functor (6.2.29) is faithful. Namely, for a given i ∈ I , let M i and N i be twocoherent log structures on Xi, and fi, gi : M i → N i two morphisms, such that π∗i fi agrees withπ∗i gi. Then there exists a morphism ψ : j → i in I such that f ∗ψfi = f ∗ψgi.

Proof of the claim. Set M := π∗iM i, and define likewise the coherent log structure N onX . For any τ -point ξ of X , pick a neighborhood Uξ → Xi of πi(ξ) in Xi, and a finite chartβ : PUξ → M i|Uξ for the restriction of M i. The morphisms fi|Uξ and gi|Uξ are determined bythe induced maps ϕ := Γ(Uξ, fi) β and γ := Γ(Uξ, gi) β, and the assumption means that thecomposition of ϕ and the natural map Γ(Uξ, N i)→ Γ(Uξ ×Xi X,N) equals the composition ofγ with the same map.

It then follows from lemmata 6.2.10(ii) and 3.1.7(ii) that there exists a morphism ψ : i(ξ)→ iin I such that the composition of ϕ with the natural map Γ(Uξ, N i) → Γ(Uξ ×Xi Xi(ξ), f

∗ψN)

equals the composition of γ with the same map; in other words, if we set U ′ξ := Uξ ×Xi Xi(ξ),we have f ∗ψfi|U ′ξ = f ∗ψgi|U ′ξ . Next, since X is quasi-compact, we may find finitely many τ -pointsξ1, . . . , ξn such that the family (U ′ξk ×Xi(ξk)

X → X) is covering in Xτ . Then, by [32, Ch.IV,Th.8.10.5] we may find j ∈ I and morphisms ψk : j → i(ξk) in I , for k = 1, . . . , n, such thatthe induced family (U ′′k := U ′ξk ×Xi(ξk)

Xj → Xj) is covering in Xj,τ . By construction we have

f ∗ψkψfi|U ′′k = f ∗ψkψgi|U ′′k for k = 1, . . . , n

therefore f ∗ψkψfi = f ∗ψkψgi, as required. ♦

Claim 6.2.31. The functor (6.2.29) is full. Namely, let i ∈ I and M i, N i as in claim 6.2.30,and suppose that f : π∗iM i → π∗iN i is a given morphism of log structures; then there existsa morphism ψ : j → i in I , and a morphism of log structures fj : f ∗ψM i → f ∗ψN i such thatπ∗j fj = f .

Proof of the claim. Indeed, by theorem 6.1.35(i) we may find a covering family (Uλ → X | λ ∈Λ) in Xτ , and for each λ ∈ Λ a finite chart

βλ : Pλ,Uλ →M |Uλ γλ : Qλ,Uλ → N |Uλ ϑλ : Pλ → Qλ

for the restriction f|Uλ . Clearly we may assume that each Uλ is affine, and since X is quasi-compact, we may assume as well that Λ is a finite set; in this case, claim 6.2.27 implies thatthere exists a morphism j → i in I , a covering family (Uj,λ → Xj | λ ∈ Λ), and isomorphismof X-schemes Uλ

∼→ Uj,λ×Xj X for every λ ∈ Λ. Next, lemma 6.2.24 says that, after replacingj by some larger index, we may assume that for every λ ∈ Λ there exist charts

βj,λ : Pλ,Uj,λ →M j := f ∗ψM i and γj,λ : Qλ,Uj,λ → N j := f ∗ψN i

with π∗jβj,λ = βλ and π∗jγj,λ = γλ. Set fj,λ := ϑlogλ,Uj,λ

: M j → N j; by construction we have :

(1Uj,λ ×Xj πj)∗fj,λ = f|Uλ for every λ ∈ Λ.

Next, for every λ, µ ∈ Λ, let Uj,λµ := Uj,λ ×Xj Uj,µ; we deduce, for every λ, µ ∈ Λ, twomorphisms of log structures fj,λ|Uj,λµ , fj,µ|Uλµ : M j|Uλµ → N j|Uλµ , which agree after pull back toUj,λµ×XjX . By applying claim 6.2.30 to the cofiltered system of schemes (Uj,λµ×XjXj′ | j′ ∈I/j), we may then achieve – after replacing j by some larger index – that fj,λ|Uj,λµ = fj,µ|Uλµ ,in which case the system (fj,λ | λ ∈ Λ) glues to a well defined morphism fj as sought. ♦

Finally, let us show that (6.2.29) is essentially surjective. Indeed, let M be a coherent logstructure on X; let us pick a covering family U := (Uλ → X | λ ∈ Λ) and finite chartsβλ : Pλ,Uλ → M |Uλ for every λ ∈ Λ. As in the foregoing, we may assume that each Uλ isaffine, and Λ is a finite set, in which case, according to claim 6.2.27(ii) we may find i ∈ I and acovering family (Ui,λ → Xi | λ ∈ Λ) with isomorphisms of X-schemes Ui,λ ×Xi X

∼→ Uλ for


every λ ∈ Λ; for every j ∈ I/i and every λ ∈ Λ, let us set Uj,λ := Ui,λ×XiXj . The compositionαλ : Pλ,Uλ → OUλ of βλ and the structure map ofM |Uλ is determined by a morphism of monoids

Pλ → Γ(Uλ,OUλ) = colimi∈I/i

Γ(Uj,λ,OUj,λ).

Then, as usual, lemma 3.1.7(ii) implies that there exists j ∈ I/i such that αλ descends to apre-log structure Pλ,Uj,λ → OUj,λ on Uj,λ, whose associated log structure we denote by M j,λ.For every λ, µ ∈ Λ, let Uj,λµ := Uj,λ ×Xj Uj,µ; by construction we have isomorphisms

(6.2.32) (1Uj,λµ ×Xj πj)∗M j,λ|Uj,λµ∼→ (1Uj,µλ ×Xj πj)∗M j,µ|Uj,µλ for every λ, µ ∈ Λ.

By applying claim 6.2.31 to the cofiltered system (Uj,λµ×XjXj′ | j′ ∈ I/j), we can then obtain– after replacing j by a larger index – isomorphisms ωλµ : M j,λ|Uj,λµ

∼→ Mj,µ|Uj,µλ for everyλ, µ ∈ Λ, whose pull-back to Uj,λµ ×Xj X are the isomorphisms (6.2.32). Lastly, in view ofclaim 6.2.30, we may achieve – after further replacement of j by a larger index – that the systemD := (M j,λ, ωλµ | λ, µ ∈ Λ) is a descent datum for the fibration F of (6.2.1), whose pull-backto X is isomorphic to the natural descent datum for M , associated to the covering family U .Then D glues to a log structure M j , such that π∗jM j 'M .

Corollary 6.2.33. In the situation of (5.1.14), suppose that X0 is quasi-compact, and let

(g∞, log g∞) : (Y∞, N∞)→ (X∞,M∞)

be a morphism of log schemes with coherent log structures. Then there exists λ ∈ Λ, and amorphism

(gλ, log gλ) : (Yλ, Nλ)→ (Xλ,Mλ)

of log schemes with coherent log structures, such that log g∞ = ψ∗λ log gλ.

Proof. IfX0 is quasi-compact, the same holds for allXλ and Yλ, so the assertion is an immediateconsequence of proposition 6.2.28.

Corollary 6.2.34. In the situation of (5.1.14), let N0 and M0 be coherent log structures onY0, and respectively X0, and (g0, log g0) : (Y0, N0) → (X0,M0) a morphism of log schemes.Suppose also that X0 is quasi-compact (as well as quasi-separated); then we have :

(i) If the morphism of log schemes

(g∞, ψ∗0 log g0) : Y∞ ×Y0 (Y0, N0)→ X∞ ×X0 (X0,M0)

admits a chart (ω∞, ω′∞, ϑ : P → Q), there exists a morphism u : λ → 0 in Λ such

that the morphism of log schemes

(gλ, ψ∗u log g0) : Yλ ×Y0 (Y0, N0)→ Xλ ×X0 (X0,M0)

admits a chart (ωλ, ω′λ, ϑ).

(ii) If (g∞, ψ∗0 log g0) is a log flat (resp. saturated) morphism of fine log schemes, there

exists a morphism u : λ→ 0 in Λ such that (gλ, ψ∗u log g0) is a log flat (resp. saturated)

morphism of fine log schemes.

Proof. (i): Since X0 is quasi-compact, the same holds for every Xλ and Yλ, as well as for X∞and Y∞. In view of lemma 6.2.24, we may then find a morphism v : µ → 0, such that ω∞ andω′∞ descend to charts ωv : PXµ → ϕ∗vM0 and ω′v : QYµ → ψ∗vN0. We deduce two morphismsof pre-log structures PYµ → ψ∗vN0, namely

β1 := ψ∗v(log g0) g∗µωv and β2 := ω′v ϑYµand by construction we have ψ∗µβ

log1 = ψ∗µβ

log2 . By proposition 6.2.28 it follows that there exists

w : λ → µ such that ψ∗wβlog1 = ψ∗wβ

log2 . The latter means that (ϕ∗wωv, ψ

∗wω′v, ϑ) is a chart for


the morphism of log schemes (gλ, ψ∗vw log g0) : (Yλ, ψ

∗vwN0)→ (Xλ, ϕ

∗vwM0), i.e. the claim

holds with u := v w.(ii): Suppose therefore that (g∞, ψ

∗0 log g0) is a log flat (resp.saturated) morphism, and let

U := (Ui → X∞ | i ∈ I) be a covering family for X∞,τ , such that Ui × (X0,M0) admits afinite (resp. fine) chart, and Ui is affine for every i ∈ I . Since X∞ is quasi-compact, we mayassume that I is a finite set, and then there exists λ ∈ Λ such that U descends to a coveringfamily Uλ := (Uλ,i → Xλ | i ∈ I) for Xλ,τ (claim 6.2.27(ii)). After replacing Λ by Λ/λ,we may assume that λ = 0, in which case we set Uλ,i := U0,i ×X0 Xλ for every object λof Λ, and every i ∈ I . Clearly, it suffices to show that there exists u : λ → 0 such thatUλ,i×Xλ (gλ, ψ

∗u log g0) is flat (resp. saturated) for every i ∈ I . Set Y ′λ,i := Yλ×XλUλ,i for every

λ ∈ Λ; we may then replace the cofiltered system X and Y , by respectively (Uλ,i | λ ∈ Λ) and(Y ′λ,i | λ ∈ Λ), which allows to assume from start, that X∞ ×X0 (X0,M0) admits a finite (resp.fine) chart. In this case, lemma 6.2.24 allows to further reduce to the case where M0 admits afinite (resp. fine) chart.

In this case, by theorem 6.1.35(iii), we may find a covering family V := (Vj → Y∞ | j ∈ J)for Y∞,τ , consisting of finitely many affine schemes Vj , and for every j ∈ J , a flat (resp.saturated) and fine chart for the induced morphism Vj ×Y0 (Y0, N0)→ X∞ ×X0 (X0,M0). Asin the foregoing, after replacing Λ by some category Λ/λ, we may assume that V descends to acovering family V0 := (V0,j → Y0 | j ∈ J) for Y0,τ , in which case we set Vλ,j := V0,j×Y0 Yλ forevery λ ∈ Λ. Clearly, it suffices to show that there exists λ ∈ Λ such that the induced morphismVλ,j ×Y0 (Y0, N0)→ Xλ×X0 (X0,M0) is log flat (resp. saturated). Thus, we may replace Y bythe cofiltered system (Vλ,j | λ ∈ Λ), which allows to assume that (g∞, ψ

∗0 log g0) admits a flat

(resp. saturated) and fine chart. In this case, the assertion follows from (i).

Proposition 6.2.35. The inclusion functors :

qf .log→ qcoh.log qfs.log→ qf .log

admit right adjoints :

qcoh.log→ qf .log : (X,M) 7→ (X,M)qf qf .log→ qfs.log : (X,M) 7→ (X,M)qfs.

Proof. Let (X,M) be a scheme with quasi-coherent (resp. quasi-fine) log structure. We needto construct a morphism of log schemes

ϕ : (X,M)qf → (X,M) (resp. ϕ : (X,M)qfs → (X,M))

such that (X,M)qf (resp. (X,M)qfs) is a quasi-fine (resp. qfs) log scheme, and the followingholds. Every morphism of log schemes ψ : (Y,N) → (X,M) with (Y,N) quasi-fine (resp.qfs), factors uniquely through ϕ. To this aim, suppose first that M admits a chart (resp. aquasi-fine chart) α : PX →M . By lemma 6.2.18(i), α determines an isomorphism

(X,M)∼→ Spec(Z, P )×SpecZ[P ] X.

Since N is integral (resp. and saturated), the morphism (Y,N) → Spec(Z, P ) induced by ψfactors uniquely through Spec(Z, P int) (resp. Spec(Z, P sat)) (lemma 6.2.18(ii)). Taking intoaccount lemma 6.1.16(iii), it follows easily that we may take

(X,M)qf := Spec(Z, P int)×Z[P ] X (X,M)qfs := Spec(Z, P sat)×Z[P ] X

Next, notice that the universal property of (X ′,M ′) := (X,M)qf (resp. of (X ′,M ′) :=(X,M)qfs) is local on Xτ : namely, suppose that (X ′,M ′) has already been found, and letU → X be an object of Xτ , with a morphism (Y,N)→ (U,M |U) from a quasi-fine (resp. qfs)log scheme; there follows a unique morphism

(Y,N)→ (U,M |U)×(X,M) (X ′,M ′)∼→ U ×X (X ′,M ′)


(notation of (6.2.6)). Thus (U,M |U)qf ' U×X (X,M)qf (resp. (U,M |U)qfs ' U×X (X,M)qfs,)since the latter is a quasi-fine (resp. qfs) log scheme. Therefore, for a general quasi-coherent(resp. quasi-fine) log structure M , choose a covering family (Uλ → X | λ ∈ Λ) such that(Uλ,M |Uλ) admits a chart for every λ ∈ Λ; it follows that the family U := ((Uλ,M |Uλ)qf | λ ∈Λ), together with the natural isomorphisms :

Uµ ×X (Uλ,M |Uλ)qf ' Uλ ×X (Uµ,M |Uµ)qf for every λ, µ ∈ Λ

is a descent datum for the fibred category over Xτ , whose fibre over any object U → X isthe category of affine U -schemes endowed with a log structure (resp. likewise for the familyU := ((Uλ,M |Uλ)qfs | λ ∈ Λ)). Using faithfully flat descent ([42, Exp.VIII, Th.2.1]), onesees that U comes from a quasi-fine (resp. qfs) log scheme which enjoys the sought universalproperty.

Remark 6.2.36. (i) By inspecting the proof of proposition 6.2.35, we see that the quasi-fine logscheme associated to a coherent log scheme, is actually fine, and the qfs log scheme associatedto a fine log scheme, is a fs log scheme (one applies corollary 3.4.1(i)). Hence we obtainfunctors

coh.log→ f .log : (X,M) 7→ (X,M)f f .log→ fs.log : (X,M) 7→ (X,M)fs

which are right adjoint to the inclusion functors f .log→ coh.log and fs.log→ f .log.(ii) Notice as well that, for every log scheme (X,M) with quasi-coherent (resp. fine) log

structure, the morphism of schemes underlying the counit of adjunction (X,M)qf → (X,M)(resp. (X,M)fs → (X,M)) is a closed immersion (resp. is finite).

(iii) Furthermore, let f : Y → X be any morphism of schemes; the proof of proposition6.2.35 also yields a natural isomorphism of (Y, f ∗M)-schemes :

(Y, f ∗M)qf ∼→ Y ×X (X,M)qf (resp. (Y, f ∗M)qfs ∼→ Y ×X (X,M)qfs).

(iv) Let (X,M) be a quasi-fine log scheme, and suppose that X is a normal, irreduciblescheme, and (X,M)tr is a dense subset ofX . Denote byXqfs the scheme underlying (X,M)qfs;then we claim that the projection Xqfs → X (underlying the counit of adjunction) admits anatural section :

σX : X → Xqfs.

The naturality means that, if f : (X,M)→ (Y,N) is any morphism of quasi-fine log schemes,where Y is also normal and irreducible, and (Y,N)tr is dense in Y , then the induced diagramof schemes :

(6.2.37)X

σX //

f

Xqfs

fqfs

Y

σY // Y qfs

commutes, and therefore it is cartesian, by virtue of (iii). Indeed, suppose first that X is affine,say X = SpecA for some normal domain A, and M admits an integral chart, given by amorphism β : P → A, for some integral monoid P ; we have to exhibit a ring homomorphismP sat ⊗P A→ A, whose composition with the natural map A→ P sat ⊗P A is the identity of A.The latter is the same as the datum of a morphism of monoids P sat → A whose restriction to Pagrees with β. However, since the trivial locus of M is dense in X , the image of P in A doesnot contain 0, hence β extends to a group homomorphism βgp : P gp → Frac(A)×; since A isintegrally closed in Frac(A), we have βgp(P sat) ⊂ A, as required. Next, suppose that U → Xis an object of Xτ , with U also affine and irreducible; then U is normal and the trivial locus of


(U,M |U) is dense in U . Thus, the foregoing applies to (U,M |U) as well, and by inspecting theconstructions we deduce a natural identification :

σU = 1U ×X σX .

Lastly, for a general (X,M), we can find a covering family (Uλ → X | λ → Λ) in Xτ , suchthat Uλ is affine, and (Uλ,M |Uλ) admits an integral chart for every λ ∈ Λ; proceeding as above,we obtain a system of morphisms (σλ : Uλ → Uqfs | λ ∈ Λ), as well as natural identifications :

1Uµ ×X σλ = 1Uλ ×X σµ for every λ, µ ∈ Λ.

In other words, we have defined a descent datum for the category fibred over Xτ , whose fibreover any object U → X is the category of all morphisms of schemes U → Uqfs. By invokingfaithfully flat descent ([42, Exp.VIII, Th.2.1]), we see that this descent datum yields a morphismσX : X → Xqfs such that 1Uλ ×X σX = σλ for every λ ∈ Λ. The verification that σX is asection of the projection Xqfs → X , and that (6.2.37) commutes, can be carried out locally onXτ , in which case we can assume that M admits a chart as above, and one can check explicitlythese assertions, by inspecting the constructions.

6.3. Logarithmic differentials and smooth morphisms. In this section we introduce the log-arithmic version of the usual sheaves of relative differentials, and we study some special classesof morphisms of log schemes.

Definition 6.3.1. Let (X,Mα−→ OX) and (Y,N

β−→ OY ) be two schemes with pre-log structures,and f : (X,M) → (Y,N) a morphism of schemes with pre-log structures. Let also F be anOX-module. An f -linear derivation of M with values in F is a pair (∂, log ∂) consisting ofmaps of sheaves :

∂ : OX → F log ∂ : logM → F

such that :• ∂ is a derivation (in the usual sense).• log ∂ is a morphism of sheaves of (additive) monoids on Xτ .• ∂ f \ = 0 and log ∂ log f = 0.• ∂ α(m) = α(m) · log ∂(m) for every object U of Xτ , and every m ∈M(U).

The set of all f -linear derivations with values in f shall be denoted by :

Der(Y,N)((X,M),F ).

The f -linear derivations shall also be called (Y,N)-linear derivations, when there is no dangerof ambiguity.

6.3.2. The set Der(Y,N)((X,M),F ) is clearly functorial in F , and moreover, for any object Uof Xτ , any s ∈ OX(U), and any f -linear derivation (∂, log ∂), the restriction (s · ∂|U , s · log ∂|U)is an element of Der(Y,N)((U,M |U),F ), hence the rule U 7→ Der(Y,N)((U,M |U),F ) definesan OX-module :

Der(Y,N)((X,M),F ).

In case M = O×X and N = O×Y are the trivial log structures, the f -linear derivations of M arethe same as the usual f -linear derivations, i.e. the natural map

(6.3.3) Der(Y,O×Y )((X,O×X ),F )→ DerY (X,F )

is an isomorphism. In the category of usual schemes, the functor of derivations is representedby the module of relative differentials. This construction extends to the category of schemeswith pre-log structures. Namely, let us make the following :


Definition 6.3.4. Let (X,Mα−→ OX) and (Y,N

β−→ OY ) be two schemes with pre-log structures,and f : (X,M) → (Y,N) a morphism of schemes with pre-log structures. The sheaf oflogarithmic differentials of f is the OX-module :

Ω1X/Y (logM/N) := (Ω1

X/Y ⊕ (OX ⊗Z logMgp))/R

where R is the OX-submodule generated locally on Xτ by local sections of the form :• (dα(a),−α(a)⊗ log a) with a ∈M(U)• (0, 1⊗ log a) with a ∈ Im((f−1N)(U)→M(U))

where U ranges over all the objects of Xτ (here we use the notation of (3.1)). For arbitrarya ∈M(U), the class of (0, 1⊗ log a) in Ω1

X/Y (logM/N) shall be denoted by d log a.

6.3.5. It is easy to verify that Ω1X/Y (logM/N) represents the functor

F 7→ Der(Y,N)((X,M),F )

on OX-modules. Consequently, (6.3.3) translates as a natural isomorphism of OX-modules :

(6.3.6) Ω1X/Y

∼→ Ω1X/Y (log O×X /O

×Y ).

Furthermore, let us fix a scheme with pre-log structure (S,N), and define the category :

pre-log/(S,N)

as in (1.1.2). Also, let Mod.pre-log/(S,N) be the category whose objects are all the pairs((X,M),F ), where (X,M) is a (S,N)-scheme, and F is an OX-module. The morphisms

((X,M),F )→ ((Y,N),G )

in Mod.pre-log/(S,N) are the pairs (f, ϕ) consisting of a morphism f : (X,M) → (Y,N)of (S,N)-schemes, and a morphism ϕ : f ∗G → F of OX-modules. With this notation, weclaim that the rule :

(6.3.7) (X,M) 7→ ((X,M),Ω1X/S(logM/N))

defines a functor pre-log/(S,N) → Mod.pre-log/(S,N). Indeed, slightly more generally,consider a commutative diagram of schemes with pre-log structures :

(6.3.8)

(X,M)g //

f

(X ′,M ′)

f ′

(S,N)

h // (S ′, N ′).

An OX-linear map :

(6.3.9) g∗Ω1X′/S′(logM ′/N ′)

dg−→ ΩX/S(logM/N)

is the same as a natural transformation of functors :

(6.3.10) Der(S,N)((X,M),F )→ Der(S′,N ′)((X′,M ′), g∗F )

on all OX-modules F . The latter can be defined as follows. Let (∂, log ∂) be an (S,N)-linearderivation of M with values in F ; then we deduce morphisms :

∂′ : OX′g\−→ g∗OX

g∗∂−−→ g∗F log ∂′ : M ′ log g−−→ g∗Mg∗ log ∂−−−−→ g∗F

and it is easily seen that (∂′, log ∂′) is a (S ′, N ′)-linear derivation of M ′ with values in g∗F .


6.3.11. Consider two morphisms (X,M)f−→ (Y,N)

g−→ (Z, P ) of schemes with pre-log struc-tures. A direct inspection of the definitions shows that :

Der(Y,N)((X,M),F ) = Ker(Der(Z,P )((X,M),F )→ Der(Z,P )((Y,N), f∗F ))

for every OX-module F , whence a right exact sequence of OX-modules :

(6.3.12) f ∗Ω1Y/Z(logN/P )

df−→ Ω1X/Z(logM/P )→ Ω1

X/Y (logM/N)→ 0

extending the standard right exact sequence for the usual sheaves of relative differentials.

Proposition 6.3.13. Suppose that the diagram (6.3.8) is cartesian. Then the map (6.3.9) is anisomorphism.

Proof. If (6.3.8) is cartesian, X is the scheme X ′×S′ S, and M is the push-out of the diagram :

f−1N ← (f ′ g)−1N ′ϕ−→ g∗M ′.

Suppose now that log ∂ : M ′ → g∗F and ∂ : OX′ → g∗F define a f ′-linear derivation ofM ′. By adjunction, we deduce morphisms α : g−1OX′ → F and β : g−1M ′ → F . Byconstruction, we have β ϕ = 0, hence β extends uniquely to a morphism log ∂′ : M → Fsuch that log ∂′ log f = 0. Likewise, α extends by linearity to a unique f -linear derivation∂ : OX → F . One checks easily that (∂′, log ∂′) is a f -linear derivation of M , and that everyf -linear derivation of M with values in F is obtained in this fashion.

6.3.14. The functor (6.3.7) admits a left adjoint. Indeed, let ((X,Mα−→ OX),F ) be any object

of Mod.pre-log/(S,N); we define an (S,N)-scheme (X ⊕F ,M ⊕F ) as follows. X ⊕Fis the spectrum of the OX-algebra OX ⊕F , whose multiplication law is given by the rule :

(s, t) · (s′, t′) := (ss′, st′ + s′t) for every local section s of OX and t of F .

Likewise, we define a composition law on the sheaf M ⊕F , by the rule :

(m, t) · (m′, t′) := (mm′, α(m) · t′+α(m′) · t) for every local section m of M and t of F .

Then M ⊕F is a sheaf of monoids, and α extends to a pre-log structure α⊕ 1F : M ⊕F →OX ⊕F . The natural map OX → OX ⊕F is a morphism of algebras, whence a natural mapof schemes π : X ⊕F → X , which extends to a morphism of schemes with pre-log structures(X ⊕F ,M ⊕F ) → (X,M), by letting log π : π∗M → M ⊕F be the map induced by thenatural monomorphism (notice that π∗ induces an equivalence of sites Xτ

∼→ (X ⊕F )τ ).Now, let (Y, P ) be any (S,N)-scheme, and :

ϕ : F := ((X,M),F )→ Ω := ((Y, P ),Ω1Y/S(logP/N))

a morphism in Mod.pre-log/(S,N). By definition, ϕ consists of a morphism f : (X,M) →(Y, P ) and an OX-linear map f ∗Ω1

Y/S(logP/N) → F , which is the same as a (S,N)-linearderivation :

∂ : OY → f∗F log ∂ : P → f∗F .

In turns, the latter yields a morphism of (S,N)-schemes :

(6.3.15) (Y ⊕ f∗F , P ⊕ f∗F )→ (Y, P )

determined by the map of algebras :

OY → OY ⊕ f∗F s 7→ (s, ∂s) for every local section s of OY

and the map of monoids :

P 7→ P ⊕ f∗F p 7→ (p, log ∂p) for every local section p of P .


Finally, we compose (6.3.15) with the natural morphism

(X ⊕F ,M ⊕F )→ (Y ⊕ f∗F , P ⊕ f∗F )

that extends f , to obtain a morphism gϕ : (X ⊕F ,M ⊕F )→ (Y, P ). We leave to the readerthe verification that the rule ϕ 7→ gϕ establishes a natural bijection :

HomMod.pre-log/(S,N)(F ,Ω)∼→ Hompre-log/(S,N)((X ⊕F ,M ⊕F ), (Y, P )).

6.3.16. In the situation of definition 6.3.4, let (∂, log ∂) be an f -linear derivation of M withvalues in an OX-module F . Consider the map :

∂′ : O×X → F : u 7→ u−1 · ∂u for all local sections u of O×X .

By definition, ∂′α : α−1M → F agrees with the restriction of log ∂; in view of the cocartesiandiagram (6.1.7), we deduce that log ∂ extends uniquely to an f -linear derivation log ∂log ofM log.Let f log : (X,M log)→ (Y,N log) be the map deduced from f ; a similar direct verification showsthat log ∂log is a f log-linear derivation. There follow natural identifications :

(6.3.17) Ω1X/Y (log(M/N)) = Ω1

X/Y (log(M log/N)) = Ω1X/Y (log(M log/N log)).

Moreover, if M and N are log structures, the natural map :

(6.3.18) OX ⊗Z logMgp → Ω1X/Y (log(M/N)) a⊗ b 7→ a · d log(b)

is an epimorphism. Indeed, we have da = d(a+ 1) for every local section a ∈ OX(U) (for anyetale X-scheme U ), and locally on Xτ , either a or 1 + a is invertible in OX (this holds certainlyon the stalks, hence on appropriate small neighborhoods U ′ → U ); hence da lies in the imageof (6.3.18).

Example 6.3.19. Let R be a ring, ϕ : N →M be any map of monoids, and set :

S := SpecR S[M ] := SpecR[M ] S[N ] := SpecR[N ].

Also, let f : Spec(R,M) → Spec(R,N) be the morphism of log schemes induced by ϕ (see(6.2.13)). With this notation, we claim that (6.3.18) induces an isomorphism :

OS[M ] ⊗Z Cokerϕgp ∼→ Ω1S[M ]/S[N ](logM log

S[M ]/NlogS[N ])

To see this, we may use (6.3.12) to reduce to the case where N = 1. Next, notice that thefunctor

Mndo → pre-log/(S,O×S ) : M 7→ Spec(R,M)

commutes with limits, and the same holds for the functor (6.3.7), since the latter is a rightadjoint. Hence, we may assume thatM is finitely generated; then lemma 3.1.7(i) further reducesto the case where M = N⊕n for some integer n ≥ 0, and even to the case where n = 1. SetX := S[N]; it is easy to see that a S-linear derivation of Nlog

X with values in an OX-module F ,is completely determined by a map of additive monoids N → Γ(X,F ), and the latter is thesame as an OX-linear map OX → F , whence the contention.

Lemma 6.3.20. Let f : (X,M) → (Y,N) be a morphism of schemes with quasi-coherent logstructures. Then :

(i) The OX-module Ω1X/Y (logM/N) is quasi-coherent.

(ii) If M is coherent, X is noetherian, and f : X → Y is locally of finite type, thenΩ1X/Y (logM/N) is a coherent OX-module.

(iii) If both M and N are coherent, and f : X → Y is locally of finite presentation, thenΩ1X/Y (logM/N) is a quasi-coherent OX-module of finite presentation.


Proof. Applying the right exact sequence (6.3.12) to the sequence (X,M)f−→ (Y,N) →

(Y,O×Y ), we may easily reduce to the case where N = O×Y is the trivial log structure on Y .In this case, N admits the chart given by the unique map of monoids : 1 → Γ(Y,OY ), and fadmits the chart 1 → P , whenever M admits a chart P → Γ(X,OX).

Hence, everything follows from the following assertion, whose proof shall be left to thereader. Suppose that ϕ : A → B is a ring homomorphism, M (resp. N ) is the constant logstructure on X := SpecB (resp. on Y := SpecA) associated to a map of monoids α : P → B(resp. β : Q → A), and f : (X,M) → (Y,N) is defined by ϕ admits a chart ϑ : Q →P . Then Ω1

X/Y (logM/N) is the quasi-coherent OX-module L∼, associated to the B-moduleL := (Ω1

B/A ⊕ (B ⊗Z Pgp))/R, where R is the submodule generated by the elements of the

form (0, 1 ⊗ log ϑ(q)) for all q ∈ Q, and those of the form (dα(m),−α(m) ⊗ logm), for allm ∈M .

6.3.21. Let us fix a log scheme (Y,N). To any pair of (Y,N)-schemes X := (X,M), X ′ :=(X ′,M ′), we attach a contravariant functor

HY (X ′, X) : (X ′τ )o → Set

by assigning, to every object U of X ′τ , the set of all morphisms (U,M ′|U)→ (X,M) of (Y,N)-

schemes. It is easily seen that HY (X ′, X) is a sheaf on X ′τ . Any morphism ϕ : (X ′′,M ′′) →(X ′,M ′) (resp. ψ : (X,M)→ (X ′′,M)) of (Y,N)-schemes induces a map of sheaves :

ϕ∗ : ϕ∗HY (X ′, X)→HY (X ′′, X) (resp. ψ∗ : HY (X ′, X)→HY (X ′, X ′′))

in the obvious way.

Definition 6.3.22. With the notation of (6.3.21) :(i) We say that a morphism i : (T ′, L′) → (T, L) of log schemes is a closed immersion

(resp. an exact closed immersion) if the underlying morphism of schemes T ′ → T is aclosed immersion, and log i : i∗L → L′ is an epimorphism (resp. an isomorphism) ofT ′τ -monoids.

(ii) We say that a morphism i : (T ′, L′) → (T, L) of log schemes is an exact nilpotentimmersion if i is an exact closed immersion, and the ideal I := Ker(OT → i∗OT ′) isnilpotent.

(iii) We say that a morphism f : (X,M) → (Y,N) of log schemes is formally smooth(resp. formally unramified, resp. formally etale) if, for every exact nilpotent immersioni : T ′ → T of fine (Y,N)-schemes, the induced map of sheaves i∗ : i∗HY (T,X) →HY (T ′, X) is an epimorphism (resp. a monomorphism, resp. an isomorphism).

(iv) We say that a morphism f : (X,M)→ (Y,N) of log schemes is smooth (resp. unram-ified, resp. etale) if the underlying morphism X → Y is locally of finite presentation,and f is formally smooth (resp. formally unramified, resp. formally etale).

Example 6.3.23. Let (S, P ) be a log scheme, (f, log f) : (X,M) → (Y,N) be a morphismof S-schemes, such that Y is a separated S-scheme. The pair (1(X,M), (f, log f)) induces amorphism

Γf : (X,M)→ (X ′,M ′) := (X,M)×S (Y,N)

the graph of f . Then it is easily seen that Γf is a closed immersion of log schemes. Indeed,the morphism of schemes underlying Γf is a closed immersion ([26, Ch.I, 5.4.3]) and it is easilyseen that the morphism log Γf : Γ∗fM

′ → M is an epimorphism on the underlying sheaves ofsets, so it is a fortiori an epimorphism of Xτ -monoids.

Proposition 6.3.24. Let f : (X,M) → (Y,N), g : (Y,N) → (Z, P ), and h : (Y ′, N ′) →(Y,N) be morphisms of log schemes. Denote by P either one of the properties : ”formally


smooth”, ”formally unramified”, ”formally etale”, ”smooth”, ”unramified”, ”etale”. Thefollowing holds :

(i) If f and g enjoy the property P, then the same holds for g f .(ii) If (f, log f) enjoys the property P, then the same holds for

(f, log f)×(Y,N) (Y ′, N ′) : (X,M)×(Y,N) (Y ′, N ′)→ (Y ′, N ′).

(iii) Let (jλ : Uλ → X | λ ∈ Λ) be a covering family inXτ ; endow Uλ with the log structure(M)|Uλ and suppose that fλ := (f jλ, (log f)|Uλ) : (Uλ, (M)|Uλ) → (Y,N) enjoysthe property P, for every λ ∈ Λ. Then f enjoys the property P as well.

(iv) An open immersion of log schemes is etale.(v) A closed immersion of log schemes is formally unramified.

Proof. (i), (ii) and (iv) are left to the reader.(iii): To begin with, if each fλ is locally of finite presentation, the same holds for f , by [33,

Ch.IV, lemme 17.7.5], hence we may assume that P is either ”formally smooth” or ”formallyunramified”. (Clearly, the case where P is ”formally etale” will follow.)

Now, let i : T ′ → T be an exact nilpotent immersion of (Y,N)-schemes, and ξ a τ -pointof T ′. By inspecting the definitions, it is easily seen that the stalk HY (T,X)ξ is the union ofthe images of the stalks HY (T, Uλ)ξ, for every λ ∈ Λ, and likewise for HY (T ′, X)ξ. It readilyfollows that f is formally smooth whenever all the fλ are formally smooth.

Lastly, suppose that all the fλ are formally unramified, and let sξ, tξ ∈ HY (T,X)i(ξ) be twosections whose images in HY (T ′, X)ξ agree; after replacing T by a neighborhood of i(ξ) in Tτ ,we may assume that sξ, tξ are represented by two (Y,N)-morphisms s, t : T → (X,M) suchthat s i = t i. Choose λ ∈ Λ such that Uλ is a neighborhood of s i(ξ) = t i(ξ); this meansthat there exists a neighborhood p′ : U ′ → T ′ of ξ, and a morphism sU ′ : U ′ → Uλ lifiting s i(and thus, also t i). Then we may find a neighborhood p : U → T of i(ξ) which identifies p′

with p×T 1T ′ ([33, Ch.IV, Th.18.1.2]), and since jλ is etale, we may furthermore find morphismssU , tU : U → Uλ such that jλ sU = s i = t i = jλ tU . Set iU := i ×T 1U : U ′ → U ;by construction, sU and tU yield two sections of i∗UHY (U,Uλ)ξ, whose images in HY (U ′, Uλ)ξcoincide. Since fλ is formally unramified, it follows that – up to replacing U by a neighborhoodof i(ξ) inUτ – we must have sU = tU , so sξ = tξ, and we conclude that f is formally unramified.

(v): Consider a commutative diagram of log schemes :

(T ′, L′)h′ //

i

(X,M)

f

(T, L)

h // (Y,N)

where f is a closed immersion, and i is an exact closed immersion. We are easily reduced toshowing that there exists at most a morphism (g, log) : (T, L) → (X,M) such that f g = hand h′ = g i. Since f : X → Y is a closed immersion of schemes, there exists at mostone morphism of schemes g : T → X lifting h and extending h′ ([33, Ch.IV, Prop.17.1.3(i)]).Hence we may assume that such a g is given, and we need to check that there exists at mostone morphism log g : g∗M → L whose composition with g∗(log f) : h∗N → g∗M equalslog h. However, by assumption log f is an epimorphism, hence the same holds for g∗(log f)(see (1.1.31)), whence the contention.

Corollary 6.3.25. Let f and g be as proposition 6.3.24. We have :(i) If g f is formally unramified, the same holds for f .

(ii) If g f is formally smooth (resp. formally etale) and g is formally unramified, then fis formally smooth (resp. formally etale).


(iii) Suppose that g is formally etale. Then f is formally smooth (resp. formally unramified,resp. formally etale) if and only if the same holds for g f .

Proof. (i): Let Y =⋃λ∈Λ be an affine open covering of Y , and for each λ ∈ Λ, let fλ :

Uλ×Y (X,M)→ (Uλ, N |Uλ) be the restriction of f ; in light of proposition 6.3.24(iii) it sufficesto show that each fλ is formally unramified, and on the other hand, the restriction g fλ :Uλ ×Y (X,M) → (Z, P ) of g f is formally unramified, by proposition 6.3.24(i),(iv). Itfollows that we may replace f and g respectively by fλ and gλ, which allows to assume that Yis affine, especially separated, so that g is a separated morphism of schemes. In such situation,one may – in view of example 6.3.23 – argue as in the proof of [33, Ch.IV, Prop.17.1.3] : thedetails shall be left to the reader.

(ii) is a formal consequence of the definitions (cp. the proof of [33, Ch.IV, Prop.17.1.4]), and(iii) follows from (ii) and proposition 6.3.24(i).

Proposition 6.3.26. Let f : (X,M) → (Y,N) be a morphism of log schemes, and i an exactclosed immersion of (Y,N)-schemes, defined by an ideal I := Ker(OT → i∗OT ′) with I 2 =0. For any global section s : T ′ →HY (T ′, X), denote by Ts the morphism :

i∗HY (T,X)×HY (T ′,X) T′ → T ′

deduced from i∗ : i∗HY (T,X) → HY (T ′, X). Let also U ⊂ T ′ be the image of Ts (i.e. thesubset of all t′ ∈ T ′ such that Ts,ξ 6= ∅ for every τ -point ξ localized at t′), and suppose thatU 6= ∅. We have :

(i) U is an open subset of T ′, and Ts|U is a torsor for the abelian sheaf

G := HomOT ′ (s∗Ω1

X/Y (log(M/N)),I )|U .

(ii) If f is a smooth morphism of log schemes with coherent log structures, we have :(a) The OX-module Ω1

X/Y (log(M/N)) is locally free of finite type.(b) If T ′ is affine, Ts is a trivial G -torsor.

Proof. (i): To any τ -point ξ of U , and any two given local sections h and g of Ts,ξ, we assignthe f -linear derivation of M s(ξ) with values in s∗Iξ given by the rule :

a 7→ h∗(a)− g∗(a) for every a ∈ OX,s(ξ)

d logm 7→ log u(m) := u(m)− 1 for every m ∈M s(ξ)

where u(m) is the unique local section of Ker(OT,ξ → i∗OT ′,ξ) such that

log(g(m) · u(m)) = log h(m).

We leave to the reader the laborious, but straightforward verification that the above map iswell-defined, and yields the sought bijection between Ts,ξ and Gξ.

(ii.b) is standard : first, since f is smooth, we have U = T ′; by lemma 6.3.20(iii), the OX-module Ω1

X/Y (log(M/N)) is quasi-coherent and finitely presented, hence G is quasi-coherent;however, the obstruction to gluing local sections of a G -torsor lies in H1(Tτ ,G ) (see (2.4.11));the latter vanishes whenever T is affine.

(ii.a) We may assume that X is affine, say X = SpecA; then Ω1X/Y (log(M/N)) is the quasi-

coherent OX-module arising from a finitely presented A-module Ω (lemma 6.3.20(iii)). Let Ibe any A-module, and set :

TI := X ⊕ I∼ LI := M ⊕ I∼

(notation of (6.3.14)). Let i : X → TI (resp. π : TI → X) be the natural closed immersion(resp. the natural projection); then (TI , LI) is a fine log scheme, and π (resp. i) extends to amorphism of log schemes (π, log π) : (TI , LI)→ (X,M) (resp. (i, log i) : (X,M)→ (TI , LI))with log π : π∗M → LI (resp. log i : i∗LI → M ) induced by the obvious inclusion (resp.


projection) map. Notice that (i, log i) is an exact immersion and (I∼)2 = 0, hence (i) says thatthe set T (TI) of all morphisms g : (TI , LI)→ (X,M) such that

f g = f π g i = 1(X,M)

is in bijection with HomA(Ω, I). Moreover, any map ϕ : I → J of A-modules induces amorphism iϕ : (TJ , LJ) → (TI , LI) of log schemes, and if ϕ is surjective, iϕ is an exactnilpotent immersion. Furthermore, we have a commutative diagram of sets :

T (TI)i∗ϕ //

T (TJ)

HomA(Ω, I)

ϕ∗ // HomA(Ω, J)

whose vertical arrows are bijections, and where i∗ϕ (resp. ϕ∗) is given by the rule g 7→ g iϕ,(resp. ψ 7→ ϕ ψ). Assertion (ii.b) implies that ι∗ϕ is surjective when f is smooth and ϕ issurjective, hence the same holds for ϕ∗, i.e. Ω is a projective A-module, as stated.

Corollary 6.3.27. Let (f, log f) : (X,M)→ (Y,N) be a morphism of log schemes. We have :(i) If M and N are fine log structures, and f is strict (see definition 6.1.20(ii)), then

(f, log f) is smooth (resp. etale) if and only if the underlying morphism of schemesf : X → Y is smooth (resp. etale).

(ii) Suppose that (f, log f) is a smooth (resp. etale) morphism of log schemes on the Zariskisites of X and Y , and either :(a) M and N are both integral log structures,(b) or else N is coherent (on YZar) and M is fine (on XZar).

Then the induced morphism of log schemes on etale sites :

u∗(f, log f) := (f, u∗X log f) : u∗X(X,M)→ u∗Y (Y,N)

is smooth (resp. etale). (Notation of (6.2.2).)(iii) Suppose that (f, log f) is a morphism of log schemes on the Zariski sites of X and Y ,

and M is an integral log structure on XZar. Suppose also that u∗(f, log f) is smooth(resp. etale). Then the same holds for (f, log f).

Proof. (i): Suppose first that (f, log f) is smooth (resp. etale). Let i : T ′ → T be a nilpotentimmersion of affine schemes, defined by an ideal I ⊂ OT such that I 2 = 0; let s : T ′ → Xand t : T → Y be morphisms of schemes such that f s = t i. By lemma 6.1.16(i), the logstructures L := t∗N and L′ := s∗M are fine, and by choosing the obvious maps log s and log t,we deduce a commutative diagram :

(6.3.28)

(T ′, L′)s //

i

(X,M)

f

(T, L)

t // (Y,N).

Then proposition 6.3.26(ii.b) says that there exists a morphism (resp. a unique morphism) ofschemes g : T → X such that g i = s and f g = t, i.e. f is smooth (resp. etale).

The converse is easy, and shall be left as an exercise for the reader.(ii): Suppose first that (f, log f) is smooth, N is coherent and M is fine. By proposition

6.3.24(iii), the assertion to prove is local on Xet, hence we may assume that f admits a chart,given by a morphism of finite monoids P → P ′ and commutative diagrams :

P ′XZar→M ω : PYZar

→ N


(theorem 6.1.35(i)). Now, consider a commutative diagram of log schemes on etale sites :

(6.3.29)

(T ′, L′)s //

i

u∗X(X,M)

u∗f

(T, L)t // u∗Y (Y,N)

where i is an exact nilpotent immersion of fine log schemes, defined by an ideal I ⊂ OT suchthat I 2 = 0. Since the assertion to prove is local on Tet, we may assume that T is affine and– in view of theorem 6.1.35(ii) – that t admits a chart, given by a morphism of finite monoidsϕ : P → Q, and commutative diagrams :

(6.3.30)

PTet= t∗PYet

t∗u∗Y ω //

ϕT

t∗u∗YN

log t

P //

ϕ

Γ(Y,OY )

t\

QTet

β // L Qψ // Γ(T,OT )

Since i is an exact closed immersion, it follows that the morphism :

QT ′et

i∗β−−→ i∗Llog i−−→ L′

is a chart for L′. Especially, (T ′, L′) is isomorphic to (T ′et, QlogT ′ ), the constant log structure

deduced from the morphism i\ ψ : Q → Γ(T ′,OT ′). The latter is also the log schemeu∗T ′(TZar, Q

logT ′ ). From proposition 6.2.3(ii) we deduce that there exists a unique morphism sZar :

(TZar, QlogT ′ )→ (X,M), such that u∗sZar = s. On the other hand, by inspecting (6.3.30) we find

that there exists a unique morphism tZar : (TZar, QlogT ) → (Y,N) such that u∗tZar = t. These

morphisms can be assembled into a diagram :

(6.3.31)

(T ′Zar, QlogT ′ )

iZar

sZar // (X,M)

f

(TZar, Q

logT )

tZar // (Y,N)

where iZar is an exact closed immersion. By construction, the diagram u∗(6.3.31) is naturallyisomorphic to (6.3.29); especially, (6.3.31) commutes (proposition 6.2.3(i)). Now, since f issmooth, proposition 6.3.26(ii.b) implies that there exists a morphism v : (TZar, Q

logT )→ (X,M)

such that f v = tZar and v iZar = sZar; then u∗v provides an appropriate lifting of t, whichallows to conclude that u∗(f, log f) is smooth.

Next, suppose that (f, log f) is etale (and we are still in case (b) of the corollary); thenthere exists a unique morphism v with the properties stated above. However, from proposition6.2.3(i),(ii) we deduce easily that the natural map :

HY ((TZar, QlogT ), (X,M))(T )→Hu∗(Y,N)((T, L), u∗(X,M))(T )

is a bijection, and the same holds for the analogous map for T ′. In view of the foregoing,this shows that the map Hu∗(Y,N)((T, L), u∗(X,M))(T ) → Hu∗(Y,N)((T

′, L′), u∗(X,M))(T ′)is bijective, whenever t admits a chart and T is affine. We easily conclude that u∗(f, log f) isetale.

The case where both N and M are both integral, is similar, though easier : we may assumethat L admits a chart, in which case (T, L) is of the form u∗(TZar, Q

logT ) for some finite integral

monoid Q; then (T ′, L′) admits a similar description, and again, by appealing to proposition6.2.3(ii) we deduce that (6.3.29) is of the form u∗(6.3.31), in which case we conclude as in theforegoing.


Conversely, suppose that u∗(f, log f) is smooth, and consider a commutative diagram (6.3.28)of log schemes on Zariski sites, with i an exact closed immersion, defined by a sheaf of idealsI ⊂ OT such that I 2 = 0. By applying everywhere the pull-back functors from Zariski toetale sites, we deduce a commutative diagram u∗(6.3.28) of log schemes on etale sites, andit is easy to see that u∗(i, log i) is again an exact nilpotent immersion. According to proposi-tion 6.3.26(ii.b), after replacing T by any affine open subset, we may find a morphism of logschemes g : u∗T (T, L) → u∗X(X,M), such that g u∗i = u∗s and u∗f g = t. By proposition6.2.3(i),(ii) there exists a unique morphism g′ : (T, L) → (X,M) such that u∗g′ = g, andnecessarily g′ i = s and f g′ = t. We conclude that (f, log f) is smooth.

Finally, if u∗(f, log f) is etale, the morphism g exhibited above is unique, and therefore thesame holds for g′, so (f, log f) is etale as well.

Proposition 6.3.32. In the situation of (6.3.11), suppose that the log structures M,N,P arecoherent, and consider the following conditions :

(a) f is smooth (resp. etale).(b) df is a locally split monomorphism (resp. an isomorphism).

Then (a)⇒(b), and if g f is smooth, then (b)⇒(a).

Proof. We may assume that the schemes under consideration are affine, say X = SpecA,Y = SpecB, Z = SpecC; then (6.3.12) amounts to an exact sequence of A-modules :

A⊗B Ω(g)df−→ Ω(g f)

ω−→ Ω(f)→ 0.

On the other hand, let i : (T ′L′) → (T, L) be an exact closed immersion of (Y,N)-schemes,defined by an ideal I ⊂ OT with I 2 = 0. Suppose that s : T ′ → HY (T ′, X) is a globalsection, i.e. a given morphism of (Y,N)-schemes (T ′L′)→ (X,M). In this situation, we havea natural sequence of morphisms :

H1 := HY (T,X)α−→H2 := HZ(T,X)

f∗−→H3 := HZ(T, Y )

where α is the obvious monomorphism. Let us consider the pull-back of these sheaves alongthe global section s :

T := T ′ ×HY (T ′,X) i∗H1 T ′ := T ′ ×HZ(T ′,X) i

∗H2 T ′′ := T ′ ×HZ(T ′,Y ) i∗H3.

By proposition 6.3.26(i), any choice of a global section of T (T ) determines a commutativediagram of sets :

(6.3.33)

HomA(Ω(f),I (T ))ω∗ //

HomA(Ω(g f),I (T ))df∗ //

HomB(Ω(g),I (T ))

T (T ) // T ′(T )

f∗ // T ′′(T )

whose vertical arrows are bijections.In order to show that df is a locally split monomorphism (resp. an isomorphism), it suffices

to show that the map df ∗ := HomA(df, I) is surjective for every A-module I . To this aim, wetake (T, L) := (TI , LI), (T ′, L′) := (X,M), and let i be the nilpotent immersion defined by theideal I ⊂ A ⊕ I , as in the proof of proposition 6.3.26(ii.a). If I ⊂ OT is the correspondingsheaf of ideals, then I (T ) = I; moreover, the inclusions A → A ⊕ I and M ⊂ M ⊕ Idetermine a morphism h : (TI , LI) → (X,M), which is a global section of T (T ). Havingmade these choices, consider the resulting diagram (6.3.33) : if f is etale, f∗ is an isomorphism,and when f is smooth, f∗ is surjective (proposition 6.3.26(ii.b)), whence the contention.

For the converse, we may suppose that df is a split monomorphism, hence df ∗ in (6.3.33) is asplit surjection. We have to show that T (T ) is not empty, and by assumption (and proposition


6.3.26(ii.b)) we know that T ′(T ) 6= ∅. Choose any h ∈ T ′(T ); then t and f∗h are twoelements of T ′′(T ), so we may find ϕ ∈ HomB(Ω(g),I (T )) such that ϕ+f∗h = t (where thesum denotes the action of HomB(Ω(g),I (T )) on its torsor T ′′). Then we may write ϕ = ψdffor some ψ : Ω(g f)→ I (Y ), and it follows easily that ψ + h lies in T (T ). Finally, if df isan isomorphism, we have Ω(f) = 0, hence T (T ) contains exactly one element.

Proposition 6.3.34. Let R be a ring, ϕ : P → Q a morphism of finitely generated monoids,such that Kerϕgp and the torsion subgroup of Cokerϕgp (resp. Kerϕgp and Cokerϕgp) arefinite groups whose orders are invertible in R. Then, the induced morphism

Spec(R,ϕ) : Spec(R,Q)→ Spec(R,P )

is smooth (resp. etale).

Proof. To ease notation, set

f := Spec(R,ϕ) (X,M) := Spec(R,Q) (Y,N) := Spec(R,P ).

Clearly f is finitely presented. We have to show that, for every commutative diagram like(6.3.28) with i an exact nilpotent immersion of fine log schemes, there is, locally on Tτ at leastone morphism (resp. a unique morphism) h : (T, L)→ (X,M) such that f h = h i.

Let I := Ker(OT → i∗OT ′); by considering the I -adic filtration on OT , we reduce easily tothe case where I 2 = 0, and then we may embed I in L via the morphism :

I → O×T ⊂ L x 7→ 1 + x.

(Here I is regarded as a sheaf of abelian groups, via its addition law.) Since i is exact, thenatural morphism L/I → i∗L

′ is an isomorphism, whence a commutative diagram :

(6.3.35)

Llog i //

i∗L′

Lgp (log i)gp

// i∗(L′)gp ' Lgp/I

and since L is integral, one sees easily that (6.3.35) is cartesian (it suffices to consider the stalksover the τ -points).• On the other hand, suppose first that both Kerϕgp and Cokerϕgp are finite groups whose

order is invertible in R, hence in I ; then a standard diagram chase shows that we may find aunique map g : P gp

T → Lgp of abelian sheaves that fits into a commutative diagram :

(6.3.36)

QgpT

//

ϕgpT

t∗Ngp (log t)gp

// Lgp

(log i)gp

P gpT

//

g

33gggggggggggggggggggggggggggggi∗s∗Mgp i∗(log s)gp

// i∗(L′)gp.

• More generally, we may write : Cokerϕgp ' G ⊕ H , where H is a finite group withorder invertible in R, and G is a free abelian group of finite rank. The direct summand Glifts to a direct summand G′ ⊂ P gp. Extend ϕgp to a map ψ : Qgp ⊕ G′ → P gp, by therule : (x, g) 7→ ϕgp(x) · g. Given any τ -point ξ of T , we may extend the morphism Qgp =QgpT,ξ → Lgp

ξ in (6.3.36)ξ, to a map ω : Qgp ⊕ G′ → Lgpξ whose composition with (log i)gp

ξ

agrees with the composition of ψ and the bottom map P gp = P gpT,ξ → i∗(L

′)gpξ of (6.3.36)ξ.

By the usual arguments, ω extends to a map of abelian sheaves ϑ : (Qgp ⊕ G′)U → (Lgp)|Uon some neighborhood U → T of ξ, and if U is small enough, the composition (log i)gp

|U ϑagrees with the composition of ψU and the bottom map β : P gp

U → i∗(L′gp)|U of (6.3.36)|U . We

may then replace T by U , and since Kerψ = Kerϕgp, and Cokerψ = H , the same diagram


chase as in the foregoing shows that we may again find a morphism g : P gpT → Lgp fitting into

a commutative diagram :

(Qgp ⊕G′)Tϑ //

ψT

Lgp

(log i)gp

P gpT

β //

g88qqqqqqqqqqqqi∗L′gp

In either case, in view of (6.3.35), the morphisms

PT → P gpT

g−→ Lgp and PT → i∗s∗M

i∗ log s−−−→ i∗L′

determine a unique morphism PT → L, which induces a morphism of log schemes (T, L) →(X,M) with the sought property.

Theorem 6.3.37. Let f : (X,M)→ (Y,N) be a morphism of fine log schemes. Assume we aregiven a fine chart β : QY → N of N . Then the following conditions are equivalent :

(a) f is smooth (resp. etale).(b) There exists a covering family (gλ : Uλ → X | λ ∈ Λ) in Xet, and for every λ ∈ Λ, a

fine chart (β, (Pλ)Uλ → g∗λM,ϕλ : Q→ Pλ) of the induced morphism of log schemes

f|Uλ := (f gλ, g∗λ log f) : (Uλ, g∗λM)→ (Y,N)

such that :(i) Kerϕgp

λ and the torsion subgroup of Cokerϕgpλ (resp. Kerϕgp

λ and Cokerϕgpλ ) are

finite groups of orders invertible in OUλ .(ii) The natural morphism of Y -schemes pλ : Uλ → Y ×SpecZ[Q] SpecZ[Pλ] is etale.

Proof. Suppose first that τ = et, so M and N are log structures on etale sites. Then the logstructure g∗λM on (Uλ)et is just the restriction of M , and g∗λ log f is the restriction of log f to(Uλ)et. In case τ = Zar, the log structure g∗λM can be described as follows. Form the logscheme (X,M ′) := u∗X(X,M) (notation of (6.2.2)), take the restriction M ′

|Uλ of M ′ to (Uλ)et,and push forward to the Zariski site to obtain uX∗(M ′

|Uλ) = g∗λM (by proposition 6.2.3(ii)).Since f is smooth (resp. etale) if and only if u∗f is (corollary 6.3.27(ii),(iii)), we conclude thatthe assertion concerning f holds if and only if the corresponding assertion for u∗f does. Hence,it suffices to consider the case of log structures on etale sites. Set S := SpecZ[Q].

(b)⇒(a): Taking into account lemma 6.2.18(i), we deduce a commutative diagram of logschemes :

(Uλ,M |Uλ) ∼ //

gλ

Spec(Z, Pλ)×SpecZ[Pλ] Uλpλ // Spec(Z, Pλ)×SpecZ[Q] Y

πλ

(X,M)f // (Y,N)

∼ // Spec(Z, Q)×SpecZ[Q] Y.

It follows by corollary 6.3.27(i) (resp. by propositions 6.3.24(ii) and 6.3.34) that pλ (resp. πλ) issmooth. Hence f gλ is smooth, by proposition 6.3.24(i). Finally, f is smooth, by proposition6.3.24(iii).

(a)⇒(b): Suppose that f is smooth, and fix a geometric point ξ of X . Since (6.3.18)ξ is asurjection, we may find elements t1, . . . , tr ∈ M ξ such that (d log ti | i = 1, . . . , r) is a basisof the free OX,ξ-module Ω1

X/Y (logM/N)ξ (proposition 6.3.26(ii.a)). Moreover, the kernel of(6.3.18)ξ is generated by sections of the form :

• 1⊗ log a where a ∈ N ′ := Im(log fξ : N f(ξ) →M ξ)•∑s

j=1 α(mi)⊗ logmi where m1, . . . ,ms ∈M ξ and∑s

j=1 dα(mj) = 0 in Ω1X/Y


whence a well-defined OX,ξ-linear map :

(6.3.38) Ω1X/Y (logM/N)ξ → κ(ξ)⊗Z (Mgp

ξ /(M×ξ ·N

′)) : d log a 7→ 1⊗ a.Consider the map of monoids :

ϕ : P1 := N⊕r ⊕Q→M ξ

which is given by the rule : ei 7→ ti on the canonical basis e1, . . . , er of N⊕r, and on the

summand Q it is given by the map Qβξ−→ N f(ξ)

log fξ−−−→ M ξ. Since (6.3.38) is a surjection, wesee that the same holds for the induced map

κ(ξ)⊗Z Pgp1

1κ(ξ)⊗Zϕgp

−−−−−−→ κ(ξ)⊗Z Mgpξ → κ(ξ)⊗Z (Mgp

ξ /M×ξ ).

It follows that the cokernel of the map ϕ : P gp1 → Mgp

ξ /M×ξ induced by ϕgp, is a finite group

(lemma 6.2.21(i)) annihilated by an integer nwhich is invertible in OX,ξ. Letm1, . . . ,ms ∈Mgpξ

be finitely many elements, whose images in Mgpξ /M

×ξ generate Cokerϕ; therefore we may find

u1, . . . , us ∈ M×ξ , and x1, . . . , xs ∈ P gp

1 , such that mni · ui = ϕ(xi) for every i ≤ s. However,

since OX,ξ is strictly henselian, M×ξ ' O×X,ξ is n-divisible, hence we may find v1, . . . , vs in M×

ξ

such that ui = vni for i = 1, . . . , n. Define group homomorphisms :

Z⊕s γ−→ Z⊕s ⊕ P gp1

δ−→Mgpξ

by the rules : γ(ei) = (−nei, xi) and δ(ei, y) = mivi · ϕ(y) for every i = 1, . . . , s and everyy ∈ P gp

1 . It is easily seen that δ factors through a group homomorphism h : G := Coker γ →M ξ; moreover the natural map P gp

1 → G is injective, and its cokernel is annihilated by n;furthermore, the induced map G→ M ]

ξ is surjective. Let P := h−1M ξ. Then, the natural mapQgp → P gp is injective, and the torsion subgroup of P gp/Qgp is annihilated by n. We deducethat the rule : x 7→ d log h(x) for every x ∈ P gp, induces an isomorphism :

(6.3.39) OX,ξ ⊗Z (P gp/Qgp)∼→ Ω1

X/Y (logM/N)ξ.

It follows that we may find an etale neighborhood U → X of ξ, such that (6.3.39) extends to anisomorphism of OU -modules :

(6.3.40) OU ⊗Z (P gp/Qgp)∼→ Ω1

U/Y (logM |U/N).

Next, proposition 6.1.28 says that, after replacing U by a smaller etale neighborhood of ξ, therestriction h|P : P →M ξ extends to a chart PU →M |U , whence a strict morphism

p : (U,M |U)→ (Y ′, P logY ′ ) := (Y,N)×Spec(Z,Q) Spec(Z, P ).

as sought. Taking into account corollary 6.3.27(i), it remains only to show :

Claim 6.3.41. p is etale.

Proof of the claim. By proposition 6.3.13 and example 6.3.19, we have natural isomorphisms

Ω1Y ′/Y (logP log

Y ′ /N)∼→ OY ′ ⊗Z (P gp/Qgp).

In view of (6.3.40), it follows that the map

dp : p∗Ω1Y ′/Y (logP log

Y ′ /N)→ Ω1U/Y (logM |U/N)

is an isomorphism, and then the claim follows from proposition 6.3.32.

Corollary 6.3.42. Keep the notation of theorem 6.3.37. Suppose that f is a smooth morphismof fs log schemes, and that Q is fine, sharp and saturated. Then there exists a covering family(gλ : Uλ → X) in Xet, and fine and saturated charts (β, (Pλ)Uλ → g∗λM,ϕλ : Q→ Pλ) of f|Uλfulfilling conditions (b.i) and (b.ii) of the theorem, and such that moreover ϕλ is injective, andP×λ is a torsion-free abelian group, for every λ ∈ Λ.


Proof. Notice that, under the stated assumptions, Qgp is a torsion-free abelian group; hencetheorem 6.3.37 already implies the existence of a covering family (gλ : Uλ → X) in Xet, andof fine charts (β, ω′λ : (P ′λ)Uλ → g∗λM,ϕ′λ : Q→ P ′λ) such that ϕ′λ is injective and Coker(ϕ′λ)

gp

is a finite group of order invertible in OUλ , for every λ ∈ Λ. Since g∗λM is saturated (lemma6.1.16(i)), it is clear that ω′λ factors through a morphism (P ′λ)

satUλ→ g∗λM of pre-log structures,

which is again a chart, so we may assume that P ′λ is fine and saturated, for every λ ∈ Λ(corollary 3.4.1(ii)), in which case we may find an isomorphism of monoids P ′λ = Pλ×G, whereP×λ is torsion-free, and G is a finite group (lemma 3.2.10). Let d be the order of Coker(ϕ′λ)

gp,and denote by ϕλ the composition of ϕ′λ and the projection P ′λ → Pλ; sinceQgp is a torsion-freeabelian group, we deduce a short exact sequence :

(6.3.43) 0→ G→ Coker(ϕ′λ)gp → Cokerϕgp

λ → 0

and notice that ϕλ is also injective. We may assume that Uλ and Y are affine, say Uλ = SpecBλ

and Y = SpecA, and since d is invertible in OUλ , we reduce easily – via base change by a finitemorphism Y ′ → Y – to the case where A contains the subgroup µd ⊂ Q× of d-th power rootsof 1. The chart ω′λ determines a morphism of monoids P ′λ → Bλ, and the map f ] : A → Bλ

factors through the natural ring homomorphism

A→ A⊗R[Q] R[P ′λ]∼→ A⊗R[Q] (R[Pλ]⊗R R[G]) where R := Z[d−1,µd].

On the other hand, let Γ := HomZ(G,µd); then we have a natural decomposition

R[G] '∏χ∈Γ

eχR[G]

where eχ is the idempotent of R[G] defined as in (8.6.11) (cp. the proof of theorem 8.6.23(i));each factor is a ring isomorphic toR, whence a corresponding decomposition of Uλ as a disjointunion of Y -schemes Uλ =

∐χ∈Γ Uλ,χ. We are then further reduced to the case where Uλ = Uλ,χ

for some character χ of G. In view of (6.3.43), it is easily seen that the composition

Qgp (ϕ′λ)gp

−−−→ (P ′λ)gp → G

χ−→ µd

extends to a well defined group homomorphism χ : P gpλ → µd, whence a map χUλ : Pλ,Uλ →

µd,Uλ ⊂ g∗λM of sheaves on Uλ,τ . Define ωλ : Pλ,Uλ → g∗λM by the rule : s 7→ ω′λ(s) · χUλ(s)for every local section s of Pλ,Uλ . It is easily seen that ωλ is again a chart for g∗λM (e.g. one mayapply lemma 6.1.4). Lastly, a direct inspection shows that (β, ωλ, ϕλ) is a chart of f|Uλ with thesought properties.

6.3.44. Let (Yi | i ∈ I) be a cofiltered system of quasi-compact and quasi-separated schemes,with affine transition morphisms, and suppose that 0 is an initial object of the indexing categoryI . Suppose also that g0 : X0 → Y0 is a finitely presented morphism of schemes. Let Xi :=X0×Y0 Yi for every i ∈ I , and denote gi : Xi → Yi the induced morphism. Let also g : X → Ybe the limit of the family of morphisms (gi | i ∈ I).

Corollary 6.3.45. In the situation of (6.3.44), suppose that (g, log g) : (X,M) → (Y,N) isa smooth morphism of fine log schemes. Then there exists i ∈ I , and a smooth morphism(gi, log gi) : (Xi,M i)→ (Yi, N i) of fine log schemes, such that log g = π∗i log gi.

Proof. First, using corollary 6.2.33, we may find i ∈ I such that (g, log g) descends to a mor-phism (gi, log gi) of log schemes with coherent log structures. After replacing I by I/i, we maythen suppose that i = 0, in which case we set (Xi,M i) := Xi × (X0,M0), and define likewise(Yi, N i) for every i ∈ I .

Next, arguing as in the proof of corollary 6.2.34(ii), we may assume that N0 admits a finechart. In this case, also N admits a fine chart, and then we my find a covering family U :=(Uλ → X | λ ∈ Λ) forXτ , such that the induced morphism (Uλ,M |Uλ)→ (Y,N) admits a chart


fulfilling conditions (i) and (ii) of theorem 6.3.37. Moreover, under the current assumptions, Xis quasi-compact, hence we may assume that Λ is a finite set, in which case there exists i ∈ Isuch that U descends to a covering family (Ui,λ → Xi | λ ∈ Λ) for Xi,τ (claim 6.2.27(ii)), andas usual, we may then reduce to the case where i = 0. It then suffices to show that there existsi ∈ I such that the induced morphism U0,λ ×X0 (Xi,M i) → (Yi, N i) is smooth (proposition6.3.24(iii)). Thus, we may replace the system (Xi | i ∈ I) by the system of schemes (U0,λ ×X0

Xi | i ∈ I), and assume from start that (g, log g) admits a fine chart fulfilling conditions (i) and(ii) of theorem 6.3.37. In this case, corollary 6.2.34(i) and [32, Ch.IV, Prop.17.7.8(ii)] implymore precisely that there exists i ∈ I such that the morphism (Xi,M i) → (Yi, N i) fulfillsconditions (i) and (ii) of theorem 6.3.37, whence the contention.

6.4. Logarithmic blow up of a coherent ideal. This section introduces the logarithmic ver-sion of the scheme-theoretic blow up of a coherent ideal. We begin by recalling a few generali-ties on graded algebras and their homogeneous prime spectra; next we introduce the logarithmicvariants of these constructions. Finally, the logarithmic blow up shall be exhibited as the loga-rithmic homogeneous spectrum of a certain graded algebra, naturally attached to any sheaf ofideals in a log structure.

6.4.1. Let A := ⊕n∈NAn be a N-graded ring, and set A+ := ⊕n>0An, which is an ideal ofA. Following [27, Ch.II, (2.3.1)], one denotes by ProjA the set consisting of all graded primeideals of A that do not contain A+, and one endows ProjA with the topology induced from theZariski topology of SpecA. For every homogeneous element f ∈ A+, set :

D+(f) := D(f) ∩ ProjA

where as usual, D(f) := SpecAf ⊂ SpecA. The system of open subsets D+(f), for franging over the homogeneous elements of A+, is a basis of the topology of ProjA ([27, Ch.II,Prop.2.3.4]). Clearly :

(6.4.2) D+(fg) = D+(f) ∩D+(g) for every homogeneous f, g ∈ A+.

For any homogeneous element f ∈ A+, let also A(f) ⊂ Af be the subring consisting of allelements of degree zero (for the natural Z-grading of Af ); in other words :

A(f) :=∑k∈N

Ak · f−k ⊂ Af .

The topological space ProjA carries a sheaf of rings O , with isomorphisms of ringed spaces :

ωf : (D+(f),O|D+(f))∼→ SpecA(f) for every homogeneous f ∈ A+.

and the system of isomorphisms ωf is compatible, in an obvious way, with the inclusions :

jf,g : D+(fg) ⊂ D+(f)

as in (6.4.2), and with the natural homomorphisms A(f) → A(fg). Especially, the locally ringedspace (ProjA,O) is a separated scheme.

6.4.3. Let A′ := ⊕n∈NA′n be another N-graded ring, and ϕ : A → A′ a homomorphism ofgraded rings (i.e. ϕ(An) ⊂ A′n for every n ∈ N). Following [27, Ch.II, (2.8.1)], we let :

G(ϕ) := ProjA′ \ V (ϕ(A+)).

This open subset of ProjA′ is also the same as the union of all the open subsets of the formD+(ϕ(f)), where f ranges over the homogeneous elements of A+. The restriction to G(ϕ) ofSpecϕ : SpecA′ → SpecA, is a continuous map aϕ : G(ϕ)→ ProjA. Moreover, we have theidentity :

aϕ−1(D+(f)) = D+(ϕ(f)) for every homogeneous f ∈ A+.


Furthermore, the homomorphism ϕ induces a homomorphism ϕ(f) : A(f) → A′(ϕ(f)), whence amorphism of schemes :

Φf : D+(ϕ(f))→ D+(f).

Let g ∈ A+ be another homogeneous element; it is easily seen that :

jf,g Φfg = (Φf )|D+(ϕ(fg)).

It follows that the locally defined morphisms Φf glue to a unique morphism of schemes :

Projϕ : G(ϕ)→ ProjA

such that the diagram of schemes :

(6.4.4)

SpecA′(ϕ(f))

ωϕ(f)

Specϕ(f) // SpecA(f)

ωf

D+(ϕ(f))

(Projϕ)|D+(ϕ(f))// D+(f)

commutes for every homogeneous f ∈ A+ ([27, Ch.II, Prop.2.8.2]). Notice that G(ϕ) =ProjA′, whenever ϕ(A+) generates the ideal A′+.

6.4.5. To ease notation, set Y := ProjA. Let M := ⊕n∈ZMn be a Z-graded A-module (i.e.Ak · Mn ⊂ Mk+n for every k ∈ N and n ∈ Z); for every homogeneous f ∈ A+, denoteby M(f) ⊂ Mf the submodule consisting of all elements of degree zero (for the natural Z-grading of Mf ). Clearly M(f) is a A(f)-module in a natural way, whence a quasi-coherentOD+(f)-module M∼

(f); these modules glue to a unique quasi-coherent OY -module M∼ ([27,Ch.II, Prop.2.5.2]). Especially, for every n ∈ Z, let A(n) be the Z-graded A-module such thatA(n)k := An+k for every k ∈ Z (with the convention that Ak = 0 if k < 0). We set :

OY (n) := A(n)∼.

Any element f ∈ An induces a natural isomorphism of D+(f)-modules :

OY (n)|D+(f)∼→ OD+(f)

([27, Ch.II, Prop.2.5.7]). Hence, on the open subset

Un(A) :=⋃f∈An

D+(f)

the sheaf OY (n) restricts to an invertible OUn(A)-module. Especially, if A1 generates the idealA+, the OY -modules OY (n) are invertible, for every n ∈ Z.

6.4.6. In the situation of (6.4.3), let M := ⊕n∈ZMn be a Z-graded A-module. Then M ′ :=M ⊗A A′ is a Z-graded A′-module, with the grading defined by the rule :

M ′n :=

∑j+k=n

Im(Mj ⊗Z A′k →M ′) for every n ∈ Z

([27, Ch.II, (2.1.2)]). Then [27, Ch.II, Prop.2.8.8] yields a natural morphism of OG(ϕ)-modules:

νM : (Projϕ)∗M∼ → (M ′)∼|G(ϕ).

Moreover, set :G1(ϕ) :=

⋃f∈A1

D+(ϕ(f))

and notice that G1(ϕ) ⊂ U1(A′) ∩ G(ϕ); by inspecting the proof of loc.cit. we see that the re-striction νM |G1(ϕ) is an isomorphism. Especially, νM is an isomorphism whenever A1 generatesthe ideal A+. It is also easily seen that G1(ϕ) = U1(A′) if ϕ(A+) generates A′+.


For any f ∈ A1, the restriction (νM)|D+(ϕ(f)) can be described explicitly : namely, we havenatural identifications

ω∗f (M∼|D+(f))

∼→M∼(f) ω∗ϕ(f)(M

′)∼|D+(ϕ(f))∼→ (M ′)∼(ϕ(f))

and in view of (6.4.4), the morphism (νM)|D+(ϕ(f)) is induced by the A′(ϕ(f))-linear map :

M(f) ⊗A(f)A′(ϕ(f)) →M ′

(ϕ(f))

given by the rule :

(mk · f−k)⊗ (a′j · ϕ(f)−j) 7→ (mk ⊗ a′j) · ϕ(f)−j−k for all k, j ∈ Z, mk ∈Mk, a′j ∈ A′j .

6.4.7. The foregoing results can be improved somewhat, in the following special situation. LetR → R′ be a ring homomorphism, A a N-graded R-algebra (hence the structure morphismR → A is a ring homomorphism R → A0); the ring A′ := R′ ⊗R A is naturally a N-gradedR′-algebra, and the induced map ϕ : A→ A′ is a homomorphism of graded rings. In this case,obviously ϕ(A+) generates the ideal A′+, hence G(ϕ) = ProjA′, and indeed, Projϕ inducesan isomorphism of SpecR′-schemes :

Y ′∼→ SpecR′ ×SpecR Y

where again Y := ProjA and Y ′ := ProjA′. Moreover, for every Z-graded A-module M , thecorresponding morphism νM is an isomorphism, regardless of whether or not A1 generates A+

([27, Ch.II, Prop.2.8.10]). Especially, νA(n) is a natural identification ([27, Ch.II, Cor.2.8.11]) :

(Projϕ)∗OY (n)∼→ OY ′(n) for every n ∈ Z.

6.4.8. LetX be a scheme, A := ⊕n∈NAn a N-graded quasi-coherent OX-algebra on the Zariskisite of X; we let A+ := ⊕n>0An. According to [27, Ch.II, Prop.3.1.2], there exists an X-scheme π : Proj A → X , with natural isomorphisms of U -schemes :

ψU : U ×X Proj A∼→ Proj A (U)

for every affine open subset U ⊂ X , and the system of isomorphisms ψU is compatible, in anobvious way, with inclusions U ′ ⊂ U of affine open subsets. For any integer d > 0, everyf ∈ Γ(X,Ad) defines an open subset D+(f) ⊂ Proj A , such that :

D+(f) ∩ π−1U = D+(f|U) ⊂ Proj A (U) for every affine open subset U ⊂ X.

6.4.9. To ease notation, set Y := Proj A , and let again π : Y → X be the natural morphism.Let M := ⊕n∈ZMn be a Z-graded A -module, quasi-coherent as a OX-module; for every affineopen subset U ⊂ X , the graded A (U)-module M (U) yields a quasi-coherent Oπ−1U -moduleM∼

U , and every inclusion of affine open subsets U ′ ⊂ U induces a natural isomorphism ofOπ−1U ′-modules : M∼

U |U ′∼→ M∼

U ′ . Therefore the locally defined modules M∼U glue to a well

defined quasi-coherent OY -module M∼.Especially, for every n ∈ Z, denote by A (n) the Z-graded A -module such that A (n)k :=

An+k for every k ∈ Z, with the convention that An = 0 whenever n < 0. We set:

OY (n) := A (n)∼ and M∼(n) := M∼ ⊗OY OY (n).

Denote by Un(A ) ⊂ Y the union of the open subsets Un(A (U)), for U ranging over the affineopen subsets of Y ; from the discussion in (6.4.5), it clear that the restriction OY (n)|Un(A ) is aninvertible OUn(A )-module. This open subset can be described as follows. For every x ∈ X , let :

(6.4.10) An(x) := An,x ⊗OX,x κ(x) and set A (x) := ⊕n∈NAn(x)

which is a N-graded κ(x)-algebra; then :

(6.4.11) Un(A ) = y ∈ Y | An(π(y)) * p(y)where p(y) ⊂ A (π(y)) denotes the prime ideal corresponding to the point y.


6.4.12. Moreover, for every Z-graded A -module M , and every n ∈ Z, there exists a naturalmorphism of OY -modules :

(6.4.13) M∼(n)→M (n)∼

where M (n) is the Z-graded A -module given by the rule : M (n)k := Mn+k for every k ∈ N([27, Prop.3.2.16]). The restriction of (6.4.13) to the open subset U1(A ) is an isomorphism([27, Ch.II, Cor.3.2.8]). Especially, we have natural morphisms of OY -modules :

(6.4.14) OY (n)⊗OY OY (m)→ OY (n+m) for every n,m ∈ Z

whose restrictions to U1(A ) are isomorphisms. Furthermore, we have a natural morphismM0 → π∗M∼ of OX-modules ([27, Ch.II, (3.3.2.1)]), whence, by adjunction, a morphism ofOY -modules :

(6.4.15) π∗M0 →M∼.

Applying (6.4.15) to the modules Mn = M (n)0, and taking into account the isomorphism(6.4.13), we deduce a natural morphism of OU1(A )-modules :

(6.4.16) (π∗Mn)|U1(A ) →M∼(n)|U1(A )

which can be described as follows. Let U ⊂ X be any affine open subset; for every f ∈ A1(U),the restriction of (6.4.16) to D+(f) ⊂ π−1U is given by the morphisms

Mn(U)⊗OX(U) A (U)(f) →M (n)(U)(f) :=∑k∈Z

Mk+n(U) · f−k ⊂M (U)f .

induced by the scalar multiplication Mn ⊗OX Ak → Mn+k. Especially, we have natural mor-phisms of OY -modules :

(6.4.17) π∗An → OY (n) for every n ∈ N

whose restrictions to U1(A ) are epimorphisms. An inspection of the definition, also shows thatthe diagram of OY -modules :

(6.4.18)

π∗An ⊗OY π∗Am

//

π∗An+m

OY (n)⊗OY OY (m) // OY (n+m)

commutes for every n,m ∈ N, where the top horizontal arrow is induced by the graded mul-tiplication An ⊗OX Am → An+m, the vertical arrows are the maps (6.4.17), and the bottomhorizontal arrow is the map (6.4.14).

6.4.19. Next, let A ′ := ⊕n∈NA ′n be another N-graded quasi-coherent OX-algebra on the Zariski

site of X , and ϕ : A → A ′ a morphism of graded OX-algebras; for every affine open subsetU ⊂ X , we deduce a morphism ϕU : A (U) → A ′(U) of graded OX(U)-algebras, whence anopen subset G(ϕU) ⊂ Proj A ′(U) as in (6.4.3). If V ⊂ U is a smaller affine open subset, thenatural isomorphism

V ×U Proj A ′(U)∼→ Proj A ′(V )

induces an identification V ×U G(ϕU)∼→ G(ϕV ), hence there exists a well defined open subset

G(ϕ) ⊂ Proj A ′ such that the morphisms ProjϕU glue to a unique morphism of X-schemes :

Projϕ : G(ϕ)→ Proj A .

If A ′+ is generated – locally on X – by ϕ(A+), we have G(ϕ) = Proj A ′.


Moreover, if M is a Z-graded quasi-coherent A -module, the morphisms νM (U) assembleinto a well defined morphism of OG(ϕ)-modules :

νM : (Projϕ)∗M∼ → (M ′)∼|G(ϕ)

where the grading of M ′ := M ⊗A A ′ is defined as in (6.4.6). Likewise, the union of thesubsets G1(ϕU), for U ranging over the affine open subsets of X , is an open subset :

(6.4.20) G1(ϕ) ⊂ U1(A ′) ∩G(ϕ)

such that the restriction νM |G1(ϕ) is an isomorphism. Especially, set Y ′ := Proj A ′; we have anatural morphism :

(6.4.21) νA (n) : (Projϕ)∗OY (n)→ OY ′(n)|G(ϕ)

which is an isomorphism, if A1 generates A+ locally on X . Again, we have G1(ϕ) = U1(A ′)whenever ϕ(A+) generates A ′

+, locally on X .

6.4.22. The discussion in (6.4.7) implies that any morphism of schemes f : X ′ → X induces anatural isomorphism of X ′-schemes ([27, Ch.II, Prop.3.5.3]) :

(6.4.23) Proj f ∗A∼→ X ′ ×X Proj A

and the description (6.4.11) implies that (6.4.23) restricts to an isomorphism :

Un(f ∗A )∼→ X ′ ×X Un(A ) for every n ∈ N.

Furthermore, set Y ′ := Proj f ∗A , and let πY : Y ′ → Y be the morphism deduced from(6.4.23); the discussion in (6.4.7) implies as well that, for any Z-graded quasi-coherent A -module M , there is a natural isomorphism :

(f ∗M )∼∼→ π∗Y M∼

([27, Ch.II, Prop.3.5.3]). Especially, f induces a natural identification ([27, Ch.II, Cor.3.5.4]) :

(6.4.24) OY ′(n)∼→ π∗Y OY (n) for every n ∈ Z.

6.4.25. Keep the notation of (6.4.9), and let CX be the category whose objects are all the pairs(ψ : Z → X,L ), where ψ is a morphism of schemes and L is an invertible OZ-module onthe Zariski site of Z; the morphisms (ψ : Z → X,L ) → (ψ′ : Z ′ → X,L ′) are the pairs(β, h), where β : Z → Z ′ is a morphism of X-schemes, and h : β∗L ′ ∼→ L is an isomorphismof OZ-modules (with composition of morphisms defined in the obvious way). Consider thefunctor:

FA : C oX → Set

which assigns to any object (ψ,L ) of CX , the set consisting of all homomorphisms of gradedOZ-algebras :

g : ψ∗A → Sym•OZL

which are epimorphisms on the underlying OZ-modules (here Sym•OZL denotes the symmetricOZ-algebra on the OZ-module L ); on a morphism (β, h) as in the foregoing, and an elementg′ ∈ FA (ψ′,L ′), the functor acts by the rule :

FA (β, h)(g′) := (Sym•OZh) β∗g′.

Lemma 6.4.26. The object (π : U1(A )→ X,OY (1)|U1(A )) of CX represents the functor FA .


Proof. Given an object (ψ : Z → X,L ) of CX , and g ∈ FA (ψ,L ), set :

P(L ) := Proj Sym•OZL .

According to [27, Ch.II, Cor.3.1.7, Prop.3.1.8(iii)], the natural morphism πZ : P(L ) → Z isan isomorphism. On the other hand, since g is an epimorphism, we have G(g) = P(L ); taking(6.4.23) into account, we deduce a morphism of Z-schemes :

Proj g : P(L )→ Y ′ := Z ×X Proj A

which is the same as a morphism of X-schemes :

P(g) : Z → Proj A .

We need to show that the image of P(g) lies in the open subset U1(A ); to this aim, we mayassume that both X and Z are affine, say X = SpecR, Z = SpecS, in which case A isthe quasi-coherent algebra associated to a N-graded R-algebra A, L is the invertible moduleassociated to a projective rank one S-module L, and g : S ⊗R A → Sym•SL is a surjectivehomomorphism of R-algebras. Then locally on Z, L is generated by elements of the formg(1 ⊗ t), for some local sections t of A1, and up to replacing Z by an affine open subset, wemay assume that t ∈ A1 is an element such that t′ := g(1 ⊗ t) generates the free S-module L.In this situation, we have P(L ) = D+(t′), and the isomorphism πZ : P(L )

∼→ Z is induced bythe natural identification:

(6.4.27) S = S[t′](t′).

Likewise, OP(L )(n) is the OP(L )-module associated to the S[t′]-moduleL⊗n⊗SS[t′]∼→ S[t′](n),

hence (6.4.27) induces a natural identification :

(6.4.28) π∗ZL ⊗n ∼→ OP(L )(n) for every n ∈ N.Moreover, P(g) is the same as the morphism Φt : D+(t′) → D+(t) (notation of (6.4.3));especially the image of P(g) lies in U1(A ), as required. From this description, we also canextract an explicit expression for Φt; namely, it is induced by the map of R-algebras :

A(t) → S such that ak · t−k 7→ g(1⊗ ak) · t′−k

for every k ∈ N, and every ak ∈ Ak. Next, letting n := 1 in (6.4.21) and (6.4.24), we obtain anatural isomorphism of OP(L )-modules :

OP(L )(1)∼→ (Proj g)∗OY ′(1)

∼→ (Proj g)∗ π∗Y OY (1)∼→ π∗Z P(g)∗OY (1)

(notice that, since by assumption g is an epimorphism, we have G1(g) = U1(Sym•OZL ) =P(L ), hence νϕ∗A (1) is an isomorphism). Composition with (6.4.28) yields an isomorphism :

h(g) : P(g)∗OY (1)∼→ L

of OZ-modules, whence a morphism in CX

(P(g), h(g)) : (ψ,L )→ (π|U1(A ),OY (1)|U1(A )).

In case X and Z are affine, and L is associated to a free module L, generated by an elementof the form t′ := g(1⊗ t) as in the foregoing, we can describe explicitly h(g); namely, a directinspection of the construction shows that in this case h(g) is induced by the map of S-modules

S ⊗A(t)A(1)(t) → L : s⊗ ak · t1−k 7→ s · g(1⊗ ak) · (t′)1−k for every s ∈ S, ak ∈ Ak.

Conversely, let β : Z → U1(A ) be a morphism of X-schemes, and h : β∗OY (1)|U1(A )∼→ L

an isomorphism of OZ-modules. In view of the natural isomorphisms (6.4.14), we deduce, forevery n ∈ N, an isomorphism :

h⊗n : β∗OY (n)|U1(A )∼→ L ⊗n.


Combining with the epimorphisms (6.4.17) :

ωn : (π∗An)|U1(A ) → OY (n)|U1(A )

we may define the epimorphism of OZ-modules :

(6.4.29) g(β, h) :=⊕n∈N

h⊗n β∗(ωn) : ψ∗A → Sym•OZL

which, in view of (6.4.18), is a homomorphism of graded OZ-algebras, i.e. g(β, h) ∈ F (ψ,L ).This homomorphism can be described explicitly, locally on Z : namely, say again that X =SpecR, Z = SpecS, L = L∼ for a free S-module of rank one, and A = A∼ for someN-graded R-algebra A; suppose moreover that the image of β lies in an open subset D+(t) ⊂U1(A ), for some t ∈ A1. Then β comes from a ring homomorphism β\ : A(t) → S, h is anS-linear isomorphism S ⊗A(t)

A(1)(t)∼→ L, and t′ := h(1 ⊗ t) is a generator of L; moreover,

ωn is the epimorphism deduced from the map :

An ⊗R A(t) → A(n)(t) : an ⊗ bk · t−k 7→ anbk · t−k for every an ∈ An, bk ∈ AkBy inspecting the construction, we see therefore that g is the direct sum of the morphisms :

gn : S ⊗R An → L⊗n : s⊗ an 7→ s · β\(an · t−n) · t′⊗n for every s ∈ S, an ∈ An.Finally, it is easily seen that the natural transformations :

(6.4.30) g 7→ (P(g), h(g)) and (β, h) 7→ g(β, h)

are inverse to each other : indeed, the verification can be made locally on Z, hence we mayassume that X and Z are affine, and L is free, in which case one may use the explicit formulaeprovided above.

6.4.31. We wish now to consider the logarithmic counterparts of the notions introduced in theforegoing. To begin with, let X be any scheme, N a monoid, and P a (commutative) N -gradedmonoid of the topos Xτ (see definition 2.3.8). Notice that P× =

∐n∈N(P× ∩ P n), hence the

sheaf of invertible sections of a N -graded monoid on X is a N -graded abelian sheaf.

Definition 6.4.32. Let X be a scheme, and β : M → OX a log structure on Xτ .(i) A N-graded O(X,M)-algebra is a datum (A , P , α, βA ) consisting of a N-graded OX-

algebra A := ⊕n∈NAn (on the site Xτ ), a graded monoid P on Xτ , and a commutativediagram :

Mα //

β

P

βA

OX // A

where βA restricts to a morphism of graded monoids P →∐

n∈N An (and the compo-sition law on the target is induced by the multiplication law of A ), α is a morphismof monoids M → (P )0, and the bottom map is the natural morphism OX → A0.We say that the N-graded O(X,M)-algebra (A , P , α, βA ) is quasi-coherent, if A is aquasi-coherent OX-algebra.

(ii) A morphism (g, log g) : (A , P , α, βA ) → (A ′, P ′, α′, β′A ) of N-graded O(X,M)-alge-bras is a commutative diagram :

P

βA

log g // P ′

β′A

Ag // A ′


where log g is a morphism of N-graded monoid such that log g α = α′, and g is amorphism of OZ-algebras.

6.4.33. Let M be a monoid, S an M -module; we say that an element s ∈ S is invertible, if thetranslation map M → S : m 7→ m · s (for all m ∈ M ) is an isomorphism. It is easily seen thatthe subset S× consisting of all invertible elements of S, is naturally an M×-module.

Let M be a sheaf of monoids on Xτ , and N an M -module; by restriction of scalars, N isnaturally an M×-module. We define the M×-submodule N × ⊂ N , by the rule :

N ×(U) := N (U)× for every object U of Xτ .

Conversely, for any M×-module P , the extension of scalars N ⊗M× M defines a functorM×-Mod → M -Mod. It is easily seen that the latter restricts to an equivalence from the fullsubcategory M×-Inv of M×-torsors to the subcategory (M -Inv)× of M -Mod whose objectsare all invertible M -modules, and whose morphisms are the isomorphisms of M -modules (seedefinition 2.3.6(iv)); the functor N 7→ N × provides a quasi-inverse (M -Inv)× →M×-Inv.

Especially, if (X,M) is a log scheme, we see that the category of invertible M -modules isequivalent to that of invertible O×X -modules, hence also to that of invertible OX-modules.

If ϕ : (Z,N)→ (X,M) is a morphism of log schemes, and N a M -module, we let :

(6.4.34) ϕ∗N := ϕ−1N ⊗ϕ−1M N.

Clearly, ϕ∗N is an invertible N -module, whenever N is an invertible M -module.

6.4.35. Keep the notation of definition 6.4.32(i); by faithfully flat descent, the restriction of Ato the Zariski site of X is again a quasi-coherent OX-algebra, which we denote again by A . Wemay then set Y := Proj A , and let π : Y → X be the natural morphism. The composition ofπ−1βA and the morphism (6.4.17), yields a map on the site Yτ :

(6.4.36) π−1P n → OY (n) for every n ∈ N.

Set OY (•) :=∐

n∈Z OY (n); it is easily seen that the morphisms (6.4.14) induce a naturalZ-graded monoid structure on OY (•), and the coproduct of the maps (6.4.36) amounts to amorphism of graded monoids :

ω• : π−1P → OY (•).We let Q be the push-out in the cocartesian diagram :

(6.4.37)

ω−1• (OY (•)×) //

π−1P

OY (•)× // Q.

Clearly Q is naturally a Z-graded monoid, in such a way that all the arrows in (6.4.37) aremorphisms of Z-graded monoids. For every n ∈ Z, let Qn be the degree n subsheaf of Q; themap ω• and the natural inclusion OY (•)× → OY (•) determine a unique morphismQ→ OY (•),whose restriction in degree zero is a pre-log structure :

β∼A : Q0 → OY .

Clearly α induces a unique morphism α∼ : π−1M → Q0, such that the diagram of monoids :

π−1Mα∼ //

π−1β

Q0

β∼A

π−1OXπ\ // OY


commutes. Denote by P∼ the log structure associated to β∼A ; the homogeneous spectrum of thequasi-coherent N-graded algebra (A , P , α, βA ) is defined as the (X,M)-scheme :

Proj(A , P ) := (Y, P∼).

We also let :U1(A , P ) := U1(A )×Y Proj(A , P ).

Furthermore, for every n ∈ Z we have a natural morphism of Q0-monoids :

(6.4.38) Q0 ⊗O×YOY (n)× → Qn

and it is easily seen that (6.4.38)|U1(A ) is an isomorphism. We set :

P∼(n) := (P∼ ⊗Q0 Qn)|U1(A ) for every n ∈ Z.

Hence (6.4.38) induces a natural isomorphism :

(6.4.39) (P∼ ⊗O×YOY (n)×)|U1(A )

∼→ P∼(n).

Especially, P∼(n) is an invertible P∼|U1(A )-module, for every n ∈ Z. From (6.4.39), we alsodeduce natural isomorphisms of P∼|U1(A )-modules :

(6.4.40) P∼(n)⊗P∼ P∼(m)∼→ P∼(n+m) for every n,m ∈ Z

and of OU1(A )-modules :

(6.4.41) P∼(n)⊗P∼ OU1(A )∼→ OY (n)|U1(A ) for every n ∈ Z.

Additionally, the morphism π−1P n → Qn deduced from (6.4.37), yields a natural map ofP∼|U1(A )-modules :

(6.4.42) λn : (π∗P n)|U1(A ) → P∼(n) for every n ∈ N.

Example 6.4.43. Let (Z, γ : N → OZ) be a log scheme, L an invertible N -module, and set :

βA (L ) := Sym•NL ⊗N γ : Sym•NL → A (L ) := Sym•OZ (L ⊗N OZ)

which is a morphisms of N-graded monoids (notation of example 2.3.10). Clearly A (L ) isalso a N-graded quasi-coherent OZ-algebra. Denote also :

αL : N → Sym•NL

the natural morphism that identifies N to Sym0NL ; then the datum

(A (L ), Sym•NL , αL , βA (L ))

is a quasi-coherent O(Z,N)-algebra, and a direct inspection of the definitions shows that theinduced morphism of log schemes :

(6.4.44) π(Z,N) : P(L ) := Proj(A (L ), Sym•NL )→ (Z,N)

is an isomorphism. Furthermore, we have natural isomorphisms as in (6.4.28) :

π∗(Z,N)(L⊗n ⊗N OZ)

∼→ OP(L )(n) for every n ∈ Z.

Let P∼L → OP(L ) be the log structure of P(L ); there follows a natural identification :

(6.4.45) π∗(Z,N)L⊗n ∼→ P∼L ⊗O×P(L )

OP(L )(n)×∼→ P∼L (n) for every n ∈ Z

where the last isomorphism is (6.4.39), in view of the fact that U1(A (L )) = Proj A (L ).


Example 6.4.46. (i) Let (Z,N) be a log scheme, and n ∈ N any integer. We define an N-grading on N⊕n, by setting

grkN⊕n := a• := (a1, . . . , an) | a1 + · · ·+ an = k for every k ∈ N.

We then define the N-graded monoid

Sym•NN⊕n := N⊕nZ ×N

whose N-grading is deduced in the obvious way from the foregoing grading of N⊕n. The logstructure γ : N → OZ extends naturally to a map of N-graded Z-monoids :

Sym•Nγ⊕n : Sym•NN

⊕n → Sym•OZO⊕nZ .

Namely, if e1, . . . , en is the canonical basis of the free OZ-module O⊕nZ , then SymkOZ

O⊕nZ is afree OZ-module with basis

ea• := ea11 · · · eann | a• ∈ grkN⊕n

and Sym•Nγ⊕n is given by the rule : (a•, x) 7→ γ(x)·ea• for every a• ∈ N⊕n, every τ -open subset

of Z, and every section x ∈ N(U). Clearly Sym•Nγ⊕n defines an N-graded O(Z,N)-algebra

Sym•(OZ ,N)(OZ , N)⊕n := (Sym•OZO⊕nZ , Sym•NN⊕n) for every n ∈ N.

We set :Pn(Z,N) := Proj Sym•(OZ ,N)(OZ , N)⊕n+1

and we call it the projective n-dimensional space over (Z,N).(ii) Denote by N∼ the log structure of Pn(Z,N), and by N∼(k) the N∼-modules defined as in

(6.4.39), for every k ∈ Z. By simple inspection we get a commutative diagram of monoids :

Γ(Z, SymkNN

⊕n+1)Γ(Z,Symk

Nγ⊕n+1)

//

Γ(Z, SymkOZ

O⊕n+1Z )

Γ(Pn(Z,N), N

∼(k)) // Γ(Pn(Z,N),OPn(Z,N)

(k))

for every k ∈ N

whose right vertical arrow is an isomorphism. Especially, the natural basis of the free N(Z)-module Γ(Z, Sym1

NN⊕n+1) = N(Z)⊕n+1 yields a distinguished system of n+ 1 elements

ε0, . . . , εn ∈ Γ(Pn(Z,N), N∼(1)).

On the other hand, we have as well the distinguished system of global sections

T0, . . . , Tn ∈ Γ(Pn(Z,N),OPn(Z,N)

(1))

corresponding to the natural basis of Γ(Z, Sym1OZ

O⊕n+1Z ) = OZ(Z)⊕n+1. For each i = 0, . . . , n,

the largest open subset Ui ⊂ Pn(Z,N) such that Ti ∈ Γ(Ui,OPn(Z,N)

(1)×) is the complement of thehyperplane where Ti vanishes. Moreover, notice that

T−1i Tj ∈ N∼(Ui) for every i, j = 0, . . . , n.

With this notation, the isomorphism (6.4.39) yields the identification :

εj = T−1i Tj ⊗ Ti on Ui for every i, j = 0, . . . , n

from which we also see that, for every i = 0, . . . , n, the open subset Ui is the largest such thatεi ∈ Γ(Ui, N

∼(1)×). By the same token, we obtain :

(6.4.47) (Pn(Z,N), N∼)tr = U0 ∩ · · · ∩ Un ' Gn

m,Z .


6.4.48. Let (Z,N) be a log scheme, (ϕ, logϕ) : (A , P ) → (A ′, P ′) a morphism of quasi-coherent N-graded O(Z,N)-algebras. We let (notation of (6.4.19)) :

G(ϕ, logϕ) := G(ϕ)×Proj A ′ Proj(A ′, P ′).

Denote also by π : Y := Proj A → Z and π′ : Y ′ := Proj A ′ → Z ′ the natural projections;there follows, on the one hand, a morphism of N-graded monoids :

(6.4.49) π′−1(logϕ) : (Projϕ)−1(π−1P )→ (π′−1P ′)|G(ϕ)

and on the other hand, a morphism of Z-graded monoids :

(6.4.50) (Projϕ)−1OY (•)× → OY ′(•)×|G(ϕ)

deduced from (6.4.21). Define the Z-graded monoid Q on Yτ as in (6.4.37), and the analogousZ-graded monoid Q′ on Y ′τ . Then (6.4.49) and (6.4.50) determine a unique morphism of Z-graded monoids :

ϑ : (Projϕ)−1Q→ Q′|G(ϕ)

and by construction, the restriction of ϑ in degree zero is a morphism of pre-log structures :

ϑ0 : (Projϕ)∗((Q)0, β∼A )→ ((Q′)0, β

∼A ′)

whence a morphism of (Z,N)-schemes :

Proj(ϕ, logϕ) : G(ϕ, logϕ)→ Proj(A , P ).

Moreover, on the one hand, (6.4.21) induces an isomorphism of O×Y ′-modules :

ν×A (n) : (O×Y ′ ⊗(Projϕ)−1O×Y(Projϕ)−1OY (n)×)|G1(ϕ)

∼→ OY ′(n)×|G1(ϕ) for every n ∈ Z

(notation of (6.4.20)). On the other hand, for every n ∈ Z, the morphism of (Projϕ)−1(Q)0-modules ϑn determines a morphism of P ′∼-modules :

(6.4.51) ϑ∼n : Proj(ϕ, log ϕ)∗P∼(n)|G1(ϕ) → P ′∼(n)|G1(ϕ)

and by inspecting the construction, it is easily seen that the isomorphism (6.4.39) (and the corre-sponding one for P ′∼(n)) identifies ϑ∼n with ν×A (n) ⊗O×

Y ′P ′∼; especially, ϑ∼n is an isomorphism.

6.4.52. Let ψ : (Z ′, N ′) → (Z,N) be a morphism of log schemes, and (A , P , α, βA ) a N-graded quasi-coherent O(Z,N)-algebra. We may view P as a N -module, via the morphism α,hence we may form the N ′-module ψ∗P , as in (6.4.34). Moreover, by remark (3.1.25)(i), ψ∗Pis a N-graded sheaf of monoids on Z ′τ , such that

(6.4.53) ψ∗(A , P ) := (ψ∗A , ψ∗P , ψ∗α, ψ∗βA )

is a N-graded quasi-coherent O(Z′,N ′)-algebra, and in view of the isomorphism (3.1.26), weobtain a natural isomorphism of (Z ′, N ′)-schemes:

(6.4.54) Projψ∗(A , P )∼→ (Z ′, N ′)×(Z,N) Proj(A , P ).

Furthermore, denote by π(A ,P ) : U1(ψ∗(A , P )) → U1(A , P ) the morphism deduced from(6.4.54), and by πY : Y ′ := Projψ∗A → Y := Proj A the underlying morphism of schemes.From (6.4.24) we obtain natural isomorphisms :

(O×Y ′ ⊗π−1Y O×Y

π−1Y OY (n)×)|U1(ψ∗A )

∼→ OY ′(n)×|U1(ψ∗A ) for every n ∈ Z

and the latter induce natural identifications :

(6.4.55) π∗(A ,P )P∼(n)

∼→ (ψ∗P )∼(n) for every n ∈ Z.


6.4.56. Keep the notation of (6.4.35), and let log C(X,M) be the category whose objects are thepairs ((Z,N),L ), where (Z,N) is a (X,M)-scheme, and L is an invertible N -module. Themorphisms ((Z,N),L )→ ((Z ′, N ′),L ′) are the pairs (ϕ, h), where ϕ : (Z,N)→ (Z ′, N ′) isa morphism of (X,M)-schemes, and h : ϕ∗L ′ ∼→ L is an isomorphism of N -modules (withcomposition of morphisms defined in the obvious way). There is an obvious forgetful functor :

p : log C(X,M) → CX : ((Z,N),L ) 7→ (Z,L ⊗N OZ)

and the functor FA can be lifted to a functor :

F(A ,P ) : log C(X,M) → Set

which assigns to any object ((Z, γ : N → OZ),L ) the set consisting of all morphisms ofN-graded quasi-coherent O(Z,N)-algebras :

ψ∗Plog g //

ψ∗βA

Sym•NL

βA (L )

ψ∗A

g // A (L )

where ψ : (Z,N) → (X,M) is the structural morphism, and g is an epimorphism on theunderlying OZ-modules.

Proposition 6.4.57. The object (U1(A , P ), P∼(1)) of log C(X,M) represents the functor F(A ,P ).

Proof. Given an object (ψ : (Z,N)→ (X,M),L ) of log C(X,M), and any element (g, log g) ∈F(A ,P )((Z,N),L ), define P(L ) as in (6.4.44); there follows a morphism of (Z,N)-schemes :

Proj(g, log g) : P(L )→ Projψ∗(A , P ).

In view of (6.4.44) and (6.4.54), this is the same as a morphism of (X,M)-schemes :

P(g, log g) : (Z,N)→ Proj(A , P )

and arguing as in the proof of lemma 6.4.26, we see that the image of P(g, log g) lands inU1(A , P ). Next, combining (6.4.45), (6.4.51) and (6.4.55), we deduce a natural isomorphism :

π∗(Z,N) P(g, log g)∗P∼(1)∼→ Proj(g, log g)∗ π∗(A ,P )P

∼(1)∼→ Proj(g, log g)∗(ψ∗P )∼(1)∼→ P∼L (1)∼→ π∗(Z,N)L

whence an isomorphism h(g, log g) : P(g, log g)∗P∼(1)∼→ L , and the datum :

(P(g, log g), h(g, log g)) : ((Z,N),L )→ (U1(A , P ), P∼(1))

is a well defined morphism of log C(X,M).Conversely, let ϕ := (β, log β) : (Z,N) → U1(A , P ) be a morphism of (X,M)-schemes,

and h : ϕ∗P∼(1)∼→ L an isomorphism of N -modules. In view of (6.4.40), we deduce an

isomorphism :h⊗n : ϕ∗P∼(n)

∼→ L ⊗n for every n ∈ Z.Combining with (6.4.42), we may define the map of N -modules :

log g(ϕ, h) :=⊕n∈N

h⊗n β∗(λn) : ψ∗P → Sym•NL .

On the other hand, in view of (6.4.41), we have an isomorphism of OZ-modules :

h⊗N OZ : β∗OY (1)|U1(A )∼→ ϕ∗P∼(1)⊗N OZ

∼→ L ⊗N OZ


(where, as usual, Y := Proj A ). We let (notation of (6.4.29)) :

g(ϕ, h) := g(β, h⊗N OZ)

and notice that the pair (g(ϕ, h), log g(ϕ, h)) is an element of F(A ,P )((Z,N),L ). Summingup, we have exhibited two natural transformations :

ϑ : F(A ,P ) ⇒ Homlog C(X,M)(−, (U1(A , P ), P∼(1))) (g, log g) 7→P(g, log g)

σ : Homlog C(X,M)(−, (U1(A , P ), P∼(1)))⇒ F(A ,P ) (ϕ, h) 7→ (g(ϕ, h), log g(ϕ, h))

and it remains to show that these transformations are isomorphisms of functors. However, thelatter fit into an essentially commutative diagram of natural transformations :

F(A ,P )ϑ +3

Homlog C(X,M)(−, (U1(A , P ), P∼(1))) σ +3

F(A ,P )

FA p +3 HomCX (p(−), (U1(A ),OY (1)|U1(A ))) +3 FA p

whose bottom line is given by the natural transformations (6.4.30). Moreover, given an isomor-phism h of invertible N -modules, the discussion in (6.4.33) leads to the identity :

(6.4.58) h = (h⊗N OZ)× ⊗OZ×N.

Likewise, we have natural identifications :

SymnNL = (Symn

OZL ⊗N OZ)× ⊗O×Z

N for every n ∈ Z

which show that βA and g determine uniquely log g. This – and an inspection of the proof oflemma 6.4.26 – already implies that σϑ is the identity automorphism of the functor F(A ,P ). Fi-nally, let ϕ = (β, log β) and h as in the foregoing, so that (ϕ, h) is a morphism in log C(X,M); toconclude we have to show that (ϕ′, h′) := ϑ σ(ϕ, h) equals (ϕ, h). Say that ϕ′ := (β′, log β′);by the above (and by the proof of lemma 6.4.26) we already know that β = β′, and (6.4.58)implies that h = h′. Hence, it remains only to show that log β = log β′, which can be checkeddirectly on the stalks over the τ -points of Z : we leave the details to the reader.

Example 6.4.59. (i) Let ψ : (Z ′, N ′) → (Z,N) be a morphism of log schemes, n ∈ Nany integer, and Pn(Z,N) the n-dimensional projective space over (Z,N), defined as in example6.4.46. A simple inspection of the definitions yields a natural isomorphism of O(Z′,N ′)-algebras

Sym•(OZ′ ,N ′)(OZ′ , N′)⊕n

∼→ ψ∗Sym•(OZ ,N)(OZ , N)⊕n for every n ∈ N

whence a natural isomorphism of (Z ′, N ′)-schemes :

Pn(Z′,N ′)∼→ (Z ′, N ′)×(Z,N) Pn(Z,N) for every n ∈ N.

(ii) Let L be any invertible N -module; notice that a morphism of N-graded monoids

Sym•NN⊕n → Sym•NL

which is the identity map in degree zero, is the same as the datum of a sequence

(βi : N → L | i = 1, , . . . , n)

of morphisms of N -modules, and the latter is the same as a sequence (b1, . . . , bn) of globalsections of L . Since Sym1

OZO⊕n+1Z generates Sym•OZOn+1

Z , proposition 6.4.57 and (i) implythat Pn(Z,N) represents the functor log C(Z,N) → Set that assigns to any pair ((X,M),L ) theset of all sequences (b0, . . . , bn) of global sections of L . The bijection

(6.4.60) Homlog C(Z,N)(((X,M),L ), (Pn(Z,N), N

∼(1)))∼→ Γ(X,L )n+1


can be explicited as follows. Let (ϕ, h) : ((X,M),L )→ (Pn(Z,N), N∼(1)) be a given morphism

in log C(Z,N); then h : ϕ∗N∼(1)→ L is an isomorphism of M -modules, which induces a mapon global sections

Γ(h) : Γ(Pn(Z,N), N∼(1)))→ Γ(X,L ).

However,N∼(1) admits a distiguished system of global sections ε0, . . . , εn (example 6.4.46(ii)),and the bijection (6.4.60) assigns to (ϕ, h) the sequence (Γ(h)(ε0), . . . ,Γ(h)(εn)).

(iii) Given a sequence b• := (b0, . . . , bn) as in (ii), denote by

fb• : (X,M)→ Pn(Z,N)

the corresponding morphism. A direct inspection of the definitions shows that the formation offb• is compatible with arbitrary base changes h : (Z ′, N ′)→ (Z,N). Namely, set (X ′,M ′) :=(Z ′, N ′) ×(Z,N) (X,M), let g : (X ′,M ′) → (X,M) be the induced morphism, L ′ := g∗L ,and suppose that L ′ is also invertible (which always holds, if M ′ is an integral log structure);the sequence b• pulls back to a corresponding sequence b′• := (b′0, . . . , bn) of global sections ofL ′, and there follows a cartesian diagram of log schemes :

(X ′,M ′)fb′• //

g

Pn(Z′,N ′)Pnh

(X,M)fb• // Pn(Z,N).

The details shall be left to the reader.

Definition 6.4.61. Let (X,M) be a log scheme, and I ⊂M an ideal of M (see (1.2.22)).(i) Let f : (Y,N)→ (X,M) be a morphism of log schemes; then f−1I is an ideal in the

sheaf of monoids f−1M ; we let :

IN := log f(f−1I ) ·N

which is the smallest ideal of N containing the image of f−1I .(ii) A logarithmic blow up of the ideal I is a morphism of log schemes

ϕ : (X ′,M ′)→ (X,M)

which enjoys the following universal property. The ideal IM ′ is invertible, and everymorphism of log schemes (Y,N) → (X,M) such that IN ⊂ N is invertible, factorsuniquely through ϕ.

Remark 6.4.62. (i) Keep the notation of definition 6.4.61. By the usual general nonsense, it isclear that the blow up (X ′,M ′) is determined up to unique isomorphism of (X,M)-schemes.

(ii) Moreover, let f : Y → X be a morphism of schemes. Then we claim that the naturalprojection :

(Y ′,M ′Y ) := Y ×X (X ′,M ′)→ (Y,MY ) := Y ×X (X,M)

is a logarithmic blow up of IMY , provided IM ′Y is an invertible ideal; especially, this holds

whenever M ′ is an integral log structure (lemma 6.1.16(i)). The proof is left as an exercise forthe reader.

6.4.63. Let (X,M) be a log scheme, I ⊂ M an ideal; we shall show the existence of thelogarithmic blow up of I , under fairly general conditions. To this aim, we introduce the gradedblow up OX-algebra :

B(X,M,J ) :=⊕n∈N

I n ⊗M OX


where I n ⊗M OX is the sheaf associated to the presheaf U 7→ I n(U) ⊗M(U) OX(U) onXτ . Here I 0 := M , and for every n > 0 we let I n be the sheaf associated to the presheafU 7→ I (U)n on Xτ . The graded multiplication law of the blow up OX-algebra is induced bythe multiplication I n ×I m → I n+m, for every n,m ∈ N.

The natural map I n → I n ⊗M OX induces a morphism of sheaves of monoids :∐n∈N

I n → B(X,P ,J ).

The latter defines a N-graded O(X,M)-algebra, which we denote B(X,M,I ).

6.4.64. Suppose first that I is invertible; then it is easily seen that, locally on Xτ , I is gener-ated by a regular local section (see example 2.3.36(i)), hence the same holds for the power I n,for every n ∈ N. Therefore I n is a locally free M -module of rank one, and we have a naturalisomorphism :

B(X,M,I )∼→ (A (I ), Sym•MI )

(notation of example (6.4.43)). It follows that in this case, the natural projection :

π(X,M,I ) : Proj B(X,M,I )→ (X,M)

is an isomorphism of log schemes.

6.4.65. The formation of B(X,M,I ) is obviously functorial with respect to morphisms oflog schemes; more precisely, such a morphism f : (Y,N) → (X,M) induces a morphism ofN-graded O(Y,N)-algebras:

B(f,I ) : f ∗B(X,M,I )→ B(Y,N,IN)

(notation of (6.4.53)) which is an epimorphism on the underlying OY -modules. Moreover, ifg : (Z,Q)→ (Y,N) is another morphism, we have the identity :

(6.4.66) B(f g,I ) = B(g,IN) g∗B(f,I ).

Furthermore, the construction of the blow up algebra is local for the topology ofXτ : if U → Xis any object of Xτ , we have a natural identification

B(U,M |U ,I|U)∼→ B(X,M,I )|U .

In the presence of charts for the log structure M , we can give a handier description for the blowup algebra; namely, we have the following :

Lemma 6.4.67. Let X be a scheme, α : P → OX a pre-log structure, β : P → P log the naturalmorphism of pre-log structures. Let also I ⊂ P be an ideal, and set I P log := β(I ) · P log

(which is the ideal of P log generated by the image of I ). Then :(i) There is a natural isomorphism of graded OX-algebras :⊕

n∈N

I n ⊗P OX∼→ B(X,P log,I P log).

(ii) Especially, suppose (X,M) is a log scheme that admits a chart β : PX → M , letI ⊂ P be an ideal, and set IM := β(IX) ·M . Then B(X,M, IM) is a N-gradedquasi-coherent O(X,M)-algebra.

(iii) In the situation of (ii), set (S, P logS ) := Spec(Z, P ) (see (6.2.13)), and denote by f :

(X,M)→ (S, P logS ) the natural morphism. Then the map :

B(f, IP logS ) : f ∗B(S, P log

S , IP logS )→ B(X,M, IM)

is an isomorphism of N-graded O(X,M)-algebras.


Proof. (i): Since the functor (2.3.51) commutes with all colimits, we have a natural isomor-phism of sheaves of rings on Xτ :

Z[P log]∼→ Z[P ]⊗Z[α−1O×X ] Z[O×X ].

We are therefore reduced to showing that the natural morphism

Z[I n]⊗Z[α−1O×X ] Z[O×X ]→ I nZ[P log]

is an isomorphism for every n ∈ N. The latter assertion can be checked on the stalks over theτ -points of X; to this aim, we invoke the more general :

Claim 6.4.68. Let G be an abelian group, ϕ : H → P ′ and ψ : H → G two morphisms ofmonoids, I ⊂ P ′ an ideal. Then the natural map

(6.4.69) Z[I]⊗Z[H] Z[G]→ IZ[P ′ ⊗H G]

is an isomorphism.

Proof of the claim. Recall that Z[I] = IZ[P ′], and the map (6.4.69) is induced by the naturalidentification: Z[P ′] ⊗Z[H] Z[G]

∼→ Z[P ′ ⊗H G] (see (2.3.52)). Especially, (6.4.69) is clearlysurjective, and it remains to show that it is also injective. To this aim, notice first that ψ factorsthrough the unit of adjunction η : H → Hgp; the morphism Z[η] : Z[H]→ Z[Hgp] = H−1Z[H]is a localization map (see (2.3.53)), especially it is flat, hence (6.4.69) is injective when G =Hgp and ψ = η. It follows easily that we may replace H by Hgp, P ′ by P ′ ⊗H Hgp, I byI · (P ′ ⊗H Hgp), and therefore assume that H is a group. Let L := ψ(H); arguing as in theforegoing, we may consider separately the case where ψ is the surjection H → L and the casewhere ψ is the injection i : L → G. However, one sees easily that Z[i] : Z[L] → Z[G] is a flatmorphism, hence it suffices to consider the case where ψ is a surjective group homomorphism.Set K := Kerψ; we have a natural identification : P ′ ⊗H G ' P ′ ⊗K 1, hence we mayfurther reduce to the case where G = 1. Then the contention is that the augmentation mapZ[K]→ Z induces an isomorphism ω : Z[I]⊗Z[K] Z

∼→ IZ[P ′/K]. From lemma 2.3.31(ii), wederive easily that IZ[P ′/K] = Z[I/K], where I/K is the set-theoretic quotient of I for the K-action deduced from ϕ. However, any set-theoretic section I/K → I of the natural projectionI → I/K yields a well-defined surjection Z[I/K]→ Z[I]⊗Z[K] Z whose composition with ωis the identity map. The claim follows. ♦

(ii): Let U be an affine object of Xτ , say U = SpecA; from (i), we see that B(X,M, IM)|Uis the quasi-coherent OU -algebra associated to the A-algebra ⊕n∈NZ[In]⊗Z[P ] A.

(iii): In view of (i) we know already that B(f, IP logS ) induces an isomorphism on the under-

lying OX-algebras; hence, by lemma 6.2.18(i), it remains to show that B(f, IP logS ) induces an

isomorphism :f ∗(InP log

S )∼→ Inf ∗(P log

S ) for every n ∈ N.Let γ : f−1P log

S → f−1OS → OX be the natural map; after replacing I by In, we come downto showing that the natural map :

f−1(IP logS )⊗γ−1O×X

O×X → I · (f−1P logS ⊗γ−1O×X

O×X )

is an isomorphism. This assertion can be checked on the stalks over the τ -points of X; if x issuch a point, let G := O×X,x, H := γ−1

x G and P ′ := P logS,f(x). The map under consideration is the

natural morphism of P ′-modules :

ω : (IP ′)⊗H G→ I(P ′ ⊗H G)

and it suffices to show that Z[ω] is an isomorphism; however, the latter is none else than (6.4.69),so we may appeal to claim 6.4.68 to conclude.


6.4.70. We wish to generalize lemma 6.4.67(ii) to log structures that do not necessarily admitglobal charts. Namely, suppose now thatM is a quasi-coherent log structure onX , and I ⊂Mis a coherent ideal (see definition 2.3.6(v)). For every τ -point ξ of X , we may then find aneighborhood U of ξ in Xτ , a chart β : PU → M |U , and local sections s1, . . . , sn ∈ I (U)which form a system of generators for I|U . We may then write si,ξ = ui · β(xi) for certainx1, . . . , xn ∈ P and u1, . . . , un ∈ O×X,ξ. Up to shrinking U , we may assume that u1, . . . , un ∈O×X (U), and it follows that I|U = I ·M |U , where I ⊂ P is the ideal generated by x1, . . . , xn. Inother words, locally on Xτ , the datum (X,M,I ) is of the type considered in lemma 6.4.67(ii);therefore the blow up OX-algebra B(X,M,I ) is quasi-coherent. We may then consider thenatural projection :

(6.4.71) π(X,M,I ) : BlI (X,M) := Proj B(X,M,I )→ (X,M).

Next, let f : (Y,N) → (X,M) be a morphism of log schemes; it is easily seen that INis a coherent ideal of N , hence B(Y,N,IN) is quasi-coherent as well, and since the mapB(f,I ) of (6.4.65) is an epimorphism on the underlying OY -modules, we have :

G(B(f,I )) = BlIN(Y,N)

(notation of (6.4.48)) whence a closed immersion of (Y,N)-schemes :

Proj B(f,I ) : BlIN(Y,N)→ (Y,N)×(X,M) BlI (X,M)

([27, Ch.II, Prop.3.6.2(i)] and (6.4.54)), which is the same as a morphism of (X,M)-schemes :

ϕ(f,I ) : BlIN(Y,N)→ BlI (X,M).

Moreover, if g : (Z,Q)→ (Y,N) is another morphism, (6.4.66) induces the identity :

(6.4.72) ϕ(f,I ) ϕ(g,IN) = ϕ(f g,I ).

Example 6.4.73. In the situation of lemma 6.4.67(iii), notice that OS is a flat Z[PS]-algebra,and therefore :

B(S, P logS , IP log

S ) =⊕n∈N

InOS.

Moreover, (6.4.54) specializes to a natural isomorphism of (X,M)-schemes :

(6.4.74) BlIM(X,M)∼→ X ×S BlIP log

S(S, P log

S ).

We wish to give a more explicit description of the log structure of BlIP logS

(S, P logS ). To begin

with, recall that S ′ := Proj B(S, P logS , IP log

S ) admits a distinguished covering by (Zariski) affineopen subsets : namely, for every a ∈ I , consider the localization

Pa := T−1a P where Ta := an | n ∈ N

and let Qa ⊂ Pa be the submonoid generated by the image of P and the subset a−1b | b ∈ I;then

S ′ =⋃a∈I

SpecZ[Qa].

Hence, let us setUa := Spec(Z, Qa) for every a ∈ I.

We claim that these locally defined log structures glue to a well defined log structure Q on thewhole of S ′τ . Indeed, let a, b ∈ I; we have

Ua,b := SpecZ[Qa] ∩ SpecZ[Qb] = SpecZ[Qa ⊗P Qb]

and it is easily seen that Qa⊗P Qb = Qa[b−1a], i.e. the localization of Qa obtained by inverting

its element a−1b, and this is of course the same as Qb[a−1b]. Then lemma 6.2.14 implies that


the log structures of Ua and Ub agree on Ua,b, whence the contention. It is easy to check that theresulting log scheme is precisely BlIP log

S(S, P log

S ) : the details shall be left to the reader.

Proposition 6.4.75. Let (X,M) be a log scheme with quasi-coherent log structure, and I ⊂M a coherent ideal. Then, the morphism (6.4.71) is a logarithmic blow up of I .

Proof. Let f : (Y,N) → (X,M) be a morphism of log schemes, and suppose that IN is aninvertible ideal of N ; in this case, we have already remarked (see (6.4.64)) that the projectionπ(Y,N,IN) : BlIN(Y,N)→ (Y,N) is an isomorphism. We deduce a morphism :

(6.4.76) ϕ(f,I ) π−1(Y,N,IN) : (Y,N)→ BlI (X,M).

To conclude, it remains to show that (6.4.76) is the only morphism of log schemes whose com-position with π(X,M,I ) equals f . The latter assertion can be checked locally on Xτ , hencewe may assume that M admits a chart PX → M , such that I = IM for some finitelygenerated ideal I ⊂ P . In this case, in view of (6.4.74), the set of morphisms of (X,M)-schemes (Y,N) → BlI (X,M) is in natural bijection with the set of (S, P log

S )-morphisms(Y,N) → B := BlIP log

S(S, P log

S ) (notation of example 6.4.73). In other words, we may as-

sume that (X,M) = (S, P logS ), and I = IP log

S . Then f is determined by log f , i.e. by a mapβ : P → N(Y ). Let a1, . . . , ak be a system of generators for I; for each i = 1, . . . , k, we letUi ⊂ Y be the subset of all y ∈ Y for which there exists a τ -point ξ of Y with |ξ| = y and

(6.4.77) aiN ξ = IN ξ.

Notice that, if y ∈ Ui, then (6.4.77) holds for every τ -point ξ of Y localized at y (details left tothe reader).

Claim 6.4.78. The subset Ui is open in Yi for every i = 1, . . . , k, and Y =⋃ki=1 Ui.

Proof of the claim. Say that y ∈ Ui, and let ξ be a τ -point of Y localized at y, such that (6.4.77)holds. This means that, for every j = 1, . . . , k, there exists uj ∈ N ξ such that aj = ujai. Then,we may find a τ -neighborhood h : U ′ → of ξ such that this identity persists in N(U ′); thus,h(U ′) ⊂ Ui. Since h is an open map, this shows that Ui is an open subset.

Next, let ξ be any τ -point of Y ; by assumption, we have IN ξ = bN ξ for some b ∈ N ξ; thismeans that for every i = 1, . . . , k there exists ui ∈ N ξ such that ai = uib. Since a1, . . . , akgenerate I , we must have ui ∈ N×ξ for at least an index i ≤ k, in which case |ξ| ∈ Ui, and thisshows that the Ui cover the whole of Y , as claimed. ♦

It is easily seen that, for every i = 1, . . . , k, any morphism (Ui, N |Ui) → B of (S, P logS )-

schemes factors through the open immersion Spec (Z,Qai) → B (where Qa, for an elementa ∈ P , is defined as in example 6.4.73). Conversely, by construction β extends to a uniquemorphism of monoids Qai → N(Ui). Summing up, there exists at most one morphism of(S, P log

S )-schemes (Ui, N |Ui)→ B. In light of claim 6.4.78, the proposition follows.

6.4.79. Keep the notation of proposition 6.4.75; by inspecting the construction, it is easily seenthat the log structure M ′ of BlI (X,M) is quasi-coherent, and if M is coherent (resp. quasi-fine, resp. fine), then the same holds for M ′. However, simple examples show that M ′ mayfail to be saturated, even in cases where (X,M) is a fs log scheme. Due to the prominent roleplayed by fs log schemes, it is convenient to introduce the special notation :

sat.BlI (X,M) := (BlI (X,M))qfs

for the saturated logarithmic blow up of a coherent ideal I in a quasi-fine log structure M(notation of proposition 6.2.35). Clearly the projection sat.BlI (X,M) → (X,M) is a finalobject of the category of saturated (X,M)-schemes in which the preimage of I is invertible. If(X,M) is a fine log scheme, sat.BlI (X,M) is a fs log scheme. Moreover, for any morphism


of schemes f : Y → X , let MY := f ∗M ; from remarks (6.4.62)(ii) and 6.2.36(iii), we deducea natural isomorphism of (Y,MY )-schemes :

(6.4.80) sat.BlIMY(Y,MY )

∼→ Y ×X sat.BlI (X,M).

Theorem 6.4.81. Let (X,M) be a quasi-fine log scheme with saturated log structure, I ⊂Man ideal, ξ a τ -point of X . Suppose that, in a neighborhood of ξ, the ideal I is generated byat most two sections, and denote :

ϕ : BlI (X,M)→ (X,M) (resp. ϕsat : sat.BlI (X,M)→ (X,M))

the logarithmic (resp. saturated logarithmic) blow up of I . Then :(i) If Iξ is an invertible ideal of M ξ, the natural morphisms

ϕ−1(ξ)→ Specκ(ξ) ϕ−1sat(ξ)→ Specκ(ξ)

are isomorphisms.(ii) Otherwise, ϕ−1(ξ) is a κ(ξ)-scheme isomorphic to P1

κ(ξ); furthermore, the same holdsfor the reduced fibre ϕ−1

sat(ξ)red, provided M is fine.

Proof. After replacing X by a τ -neighborhood of ξ, we reduce to the case where M admits anintegral and saturated chart α : PX →M (lemma 6.1.16(iii)), and if M is a fs log structure, wemay also assume that the chart α is fine and sharp at ξ (corollary 6.1.34(i)). Furthermore, wemay assume that I is generated by at most two elements of M(X), and if Iξ is principal, wemay assume that the same holds for I . In the latter case, since M is integral, I is invertible,hence ϕ and ϕsat are isomorphisms, so (i) follows already.

Now, suppose that Iξ is not invertible, and let a′, b′ ∈ M(X) be a system of generators forI ; we can write a′ = α(a) · u, b′ = α(b) · v for some a, b ∈ P and u, v ∈ κ×. Set t := a−1b,and let J ⊂ P be the ideal generated by a and b; clearly JM = I , and Pa, Pb 6= J .

Consider first the case whereX = Specκ, where κ is a field (resp. a separably closed field, incase τ = et). In this situation, a pre-log structure on Xτ is the same as a morphism of monoidsβ : P → κ, the associated log structure is the induced map of monoids

βlog : P ⊗P0 κ× → κ where P0 := β−1κ×

and α is the natural map P → P ⊗P0 κ×. After replacing P by its localization P−1

0 P , we mayalso assume that P0 = P×. Let

(S, P logS ) := Spec(κ, P ) J∼ := JP log

S (Y,N) := BlJ∼(S, P logS ).

Denote by ε : κ[P ] → κ the homomorphism of κ-algebras induced by β via the adjunction(2.3.50), and set I := Ker ε. In view of lemma 6.4.67(i) and (6.4.80), we have natural cartesiandiagrams of κ-schemes :

(6.4.82)

ϕ−1(ξ) //

(Y,N)

ϕ−1sat(ξ)

//

(Y,N)sat

|ξ| Spec ε // S |ξ| Spec ε // S.

On the other hand, (a, b) can be regarded as a pair of global sections of J∼N , so the universalproperty of example 6.4.59(ii) yields a morphism of (S, P log

S )-schemes :

f(a,b) : (Y,N)→ P1

(S,P logS ).

In light of example 6.4.59(i), the assertion concerning ϕ−1(ξ) will then follow from the :

Claim 6.4.83. The morphism |ξ| ×S f(a,b) is an isomorphism of κ-schemes.


Proof of the claim. Let Qa ⊂ P gp (resp. Qb ⊂ P gp) be the submonoid generated by P and t(resp. by P and t−1); by inspecting example 6.4.73, we see that Y is covered by two affine opensubsets :

Ua := Specκ[Qa] Ub := Specκ[Qb]

and Ua ∩ Ub = Specκ[Qa ⊗P Qb]. On the other hand, P1

(S,P logS )

is covered as well by two

affine open subsets U ′0 and U ′∞, both isomorphic to Specκ[P ⊕ N], and such that U ′0 ∩ U ′∞ =Specκ[P ⊕ Z], as usual. Moreover, a direct inspection shows that f(a,b) restricts to morphisms

Ua → U ′0 Ub → U ′∞

induced respectively by the maps of κ-algebras

ω0 : κ[P ⊕ N]→ κ[Qa] ω∞ : κ[P ⊕ N]→ κ[Qb]

such that ω0(x, n) = x · tn and ω∞(x, n) = x · t−n for every (x, n) ∈ P ⊕ N. We show thatω0 := ω0 ⊗κ[P ] κ[P ]/I is an isomorphism; the same argument will apply also to ω∞, so theclaim shall follow.

Indeed, clearly the κ[P ]-algebra κ[Qa] is generated by t, hence ω0 is surjective, and then thesame holds for ω0. Next, set Ha := I ·κ[Qa]; it is easily seen that Ha consists of all sums of theform

∑nj=0 cjt

j , for arbitrary n ∈ N, with cj ∈ I for every j = 0, . . . , n. Clearly, an elementp(T ) ∈ κ[N] = κ[T ] lies in Kerω0 if and only if p(t) ∈ Ha, so we come down to the followingassertion. Let c0, . . . , cn ∈ κ[P ] such that

(6.4.84)n∑j=0

cjtj = 0

in κ[Qa]; then cj lies in the ideal κ[β−1(0)] of κ[P ], for every j = 0, . . . , n. Since P is integral,(6.4.84) is equivalent to the identity :

∑nj=0 cja

n−jbj = 0 in κ[P ]. For every x ∈ P , denote byπx : κ[P ]→ κ the κ-linear projection such that πx(x) = 1, and πx(y) = 0 for every y ∈ P \x.

Suppose, by way of contradiction, that ci /∈ κ[β−1(0)] for some i ≤ n, hence πx(ci) 6= 0 forsome x ∈ P0; since P0 = P×, we may replace cj by x−1cj , for every j ≤ n, and assume thatπ1(ci) 6= 0, hence πan−ibi(cian−ibi) 6= 0 (again, using the assumption that P is integral). Thus,there exists j 6= i with j ≤ n, such that πan−ibi(cjan−jbj) 6= 0, and we may then find an elementc ∈ P such that πan−ibi(can−jbj) = 1, i.e. can−jbj = an−ibi; up to swapping the roles of a andb, we may assume that i > j, in which case we may write ti−j = c; since P is saturated, itfollows that t ∈ P , hence J is generated by a, which is excluded. ♦

Next, we assume that M is a fs log structure (and X is still Specκ), and we consider themorphism ϕsat. As already remarked, we may assume that α is sharp at ξ, and P fine andsaturated; the sharpness condition amounts to saying that β(x) = 0 for every x ∈ P \ 1,therefore I is the augmentation ideal of the graded κ-algebra κ[P ]. By inspecting the proof ofproposition 6.2.35, we see that (Y,N)sat is covered by two affine open subsets

U fsa := Specκ[Qsat

a ] U fsb := Specκ[Qsat

b ]

and U fsa ∩ U fs

b = Specκ[Qsata ⊗P Qsat

b ]. Since J is not principal, we have t /∈ P , and since P issaturated, we deduce that t is not a torsion element of P gp; as the latter is a finitely generatedabelian group, it follows that we may find a unimodular element u ∈ P gp such that t lies in thesubmonoid Nu ⊂ P gp generated by u; this condition means that t = uk for some k ∈ N, and Nuis not properly contained in another rank one free submonoid of P gp. Write u = a′−1b′ for somea′, b′ ∈ P , let J ′ ⊂ P be the ideal generated by a′ and b′, and Ra′ (resp. Rb′) the submonoid ofP gp generated by P and u (resp. by P and u−1); clearly Rsat

a′ = Qsata , and Rsat

b′ = Qsatb . Denote

by N ′ the log structure of (Y,N)sat, and J ′ := J ′N ′; it is easily seen that

J ′|U fsa

= a′N ′|U fsa

J ′|U fsb

= b′N ′|U fsb


hence J ′ is invertible, and example 6.4.59(ii) yields a morphism of (S, P logS )-schemes

f(a′,b′) : (Y,N)sat → P1

(S,P logS ).

In light of example 6.4.59(i), the assertion concerning ϕ−1sat(ξ) will then follow from the :

Claim 6.4.85. (|ξ| ×S f(a′,b′))red : ϕ−1sat(ξ)red → P1

κ is an isomorphism of κ-schemes.

Proof of the claim. As in the proof of claim 6.4.83, the morphism f(a′,b′) restricts to morphismsU fsa → U ′0 and U fs

b → U ′∞ induced by maps of κ-algebras :

ω0 : κ[P ⊕ N]→ κ[Qsata ] ω∞ : κ[P ⊕ N]→ κ[Qsat

b ]

such that ω0(x, n) = x · un and ω∞(x, n) = x · u−n for every (x, n) ∈ P ⊕ N, and again, itsuffices to show that

(ω0 ⊗κ[P ] κ[P ]/I)red : κ[T ]→ (κ[Rsata ]/Iκ[Rsat

a ])red

is an isomorphism (where for any ring A, we denote Ared be the maximal reduced quotientof A, i.e. Ared := A/nil(A), where nil(A) is the nilpotent radical of A). We break the latterverification in two steps : first, let us check that the map

ω′0 : κ[T ]→ κ[Ra]/Iκ[Ra] p(T ) 7→ p(u) (mod Iκ[Ra])

is an isomorphism. Indeed, ω′0 is induced by the map of monoids ϕ : N→ Ra such that n 7→ un

for every n ∈ N; if t = uk, the map ϕ fits into the cocartesian diagram :

Nψ //

kN

Qa

j

N

ϕ // Ra

where kN is the k-Frobenius map, ψ is given by the rule : n 7→ tn for every n ∈ N, and j is thenatural inclusion. Hence :

ω′0 = (κ[ψ]⊗κ[P ] κ[P ]/I)⊗κ[Tk] κ[T ].

However, the proof of claim 6.4.83 shows that κ[ψ] ⊗κ[P ] κ[P ]/I is an isomorphism, whencethe contention. Lastly, let show that the natural map

ω′′0 : κ[Ra]/Iκ[Ra]→ (κ[Rsata ]/Iκ[Rsat

a ])red

is an isomorphism. Indeed, it is clear that the natural map ω′′0 : κ[Ra]→ κ[Rsata ] is integral and

injective, hence Specω′′0 is surjective; therefore Specω′′0 is still surjective and integral. However,the foregoing shows that κ[Ra]/Iκ[Ra] is reduced, so we deduce that ω′′0 is injective. To showthat ω′′0 is surjective, it suffices to show that the classes of the generating system Rsat

a ⊂ κ[Rsata ]

lie in the image of ω′′0. Hence, let x ∈ Rgpa , with xm ∈ Ra for some m > 0, so that xm = y · un

for some n ∈ N and y ∈ P . If y 6= 1, we have y ∈ I , hence the image of xm vanishes inκ[Rsat

a ]/Iκ[Rsata ], and the image of x vanishes in the reduced quotient; finally, if y = 1, the

identity xm = un implies that m divides n, since u is unimodular; hence x = un/m and theimage of x agrees with ω′′0(un/m). ♦

Finally, let us return to a general quasi-fine log scheme (X,M); the theorem will follow fromthe more precise :

Claim 6.4.86. In the situation of the theorem, suppose moreover that(a) M admits a saturated chart α : PX →M(b) I = JM , where J ⊂ P is an ideal generated by two elements a, b ∈ P(c) Iξ is not invertible.

Then we have :


(i) There exists a morphism of (X,M)-schemes : BlI (X,M) → P1(X,M) inducing an

isomorphism of κ(ξ)-schemes ϕ−1(ξ)∼→ P1

κ(ξ).(ii) If furthermore, P is fine (and saturated) and α is sharp at ξ, then there exists a morphism

of (X,M)-schemes : sat.BlI (X,M) → P1(X,M) inducing an isomorphism of κ(ξ)-

schemes ϕ−1sat(ξ)red

∼→ P1κ(ξ).

Proof of the claim. (i): Denote by N the log structure of BlI (X,M); the elements a, b defineglobal sections of the invertibleN -module IN , and we claim that the corresponding morphismof (X,M)-schemes f(a,b) : BlI (X,M) → P1

(X,M) will do. Indeed, set (|ξ|,M ξ) := |ξ| ×X(X,M), and recall that there exists natural isomorphisms

|ξ| ×X P1(X,M)

∼→ P1(|ξ|,Mξ)

|ξ| ×X BlI (X,M)∼→ BlIMξ

(|ξ|,M ξ)

(example 6.4.59(i) and remark 6.4.62(ii)). Denote by N ξ the log structure of BlIMξ(|ξ|,M ξ);

by example 6.4.59(iii), the base change

|ξ| ×X f : BlIMξ(|ξ|,M ξ)→ P1

(|ξ|,Mξ)

is the unique morphism f(a,b) of (|ξ|,M ξ)-schemes corresponding to the pair (a, b) of globalsections of IN ξ obtained by pulling back the pair (a, b). Therefore, in order to check that|ξ|×Xf is an isomorphism, we may replace from start (X,M) by (|ξ|,M ξ) (whose log structureis still quasi-fine, by lemma 6.1.16(i), and assume that X = Specκ, where κ is a field (resp. aseparably closed field, in case τ = et), in which case the assertion is just claim 6.4.83.

(ii): Denote byN ′ the log structure of sat.BlI (X,M), define a′, b′ and J ′ as in the foregoing,and set again J ′ := J ′N ′. Again, it is easily seen that J ′ is an invertible N ′-module, andthe pair (a′, b′) yields a morphism f(a′,b′) : sat.BlI (X,M) → P1

(X,M) which fulfills the soughtcondition. Indeed, denote by N ′ξ the log structure of sat.BlIMξ

(|ξ|,M ξ); in light of (6.4.80)and example 6.4.59(iii), the base change

|ξ| ×X f(a′,b′) : sat.BlIMξ(|ξ|,M ξ)→ P1

(|ξ|,Mξ)

is the unique morphism f(a′,b′) of (|ξ|,M ξ)-schemes corresponding to the pair (a′, b′) of globalsections of J ′N ′ξ obtained by pulling back the pair (a′, b′). Thus, the assertion is just claim6.4.85.

6.5. Regular log schemes. In this section we introduce the logarithmic version of the classicalregularity condition for locally noetherian schemes. This theory is essentially due to K.Kato([53]), and we mainly follow his exposition, except in a few places where his original argumentsare slightly flawed, in which cases we supply the necessary corrections.

6.5.1. Let A be a ring, P a monoid. Recall that mP is the maximal (prime) ideal of P (see(3.1.11)). The mP -adic filtration of P is the descending sequence of ideals :

· · · ⊂ m3P ⊂ m2

P ⊂ mP

where mnP is the n-th power of m in the monoid P(P ) (see (3.1.1)). It induces a mP -adic

filtration Fil•M on any P -module M and any A[P ]-algebra B, defined by letting FilnM :=mnPM and FilnB := A[mn

P ] ·B, for every n ∈ N.

Lemma 6.5.2. Suppose that P is fine. We have :

(i) The mP -adic filtration is separated on P .(ii) If P is sharp, P \mn

P is a finite set, for every n ∈ N.


Proof. (i): Indeed, choose A to be a noetherian ring, set J :=⋂n≥0A[mn

P ] and notice that J isgenerated by m∞P :=

⋂n∈Nm

nP . On the other hand, J is annihilated by an element of 1 +A[mP ]

([61, Th.8.9]). Thus, suppose x ∈ m∞P , and pick y ∈ A[mP ] such that (1 − y)x = 0; wemay write y = a1t1 + · · · + artr for certain a1, . . . , ar ∈ A and t1, . . . , tr ∈ mP . Thereforex = a1xt1 + · · ·+ arxtr in A[P ], which is absurd, since P is integral.

Assertion (ii) is immediate from the definition.

6.5.3. Keep the assumptions of lemma 6.5.2. It turns out that P can actually be made into agraded monoid, albeit in a non-canonical manner. We proceed as follows. Let ε : P → P sat

the inclusion map, T ⊂ P sat the torsion subgroup, set Q := P sat/T , and let π : P sat → Q bethe natural surjection. We may regard logQ as a submonoid of the polyhedral cone QR, lyingin the vector space Qgp

R , as in (3.4.6). Since QR is a rational polyhedral cone, the same holdsfor Q∨R, hence we may find a Q-linear form γ : logQgp ⊗Z Q → Q, which is non-negativeon logQ, and such that QR ∩ Keru ⊗Q R is the minimal face of QR, i.e. the R-vector spacespanned by the image of Q×. If we multiply γ by some large positive integer, we may achievethat γ(logP ) ⊂ N. We set :

grγnP := (γ π ε)−1(n) for every n ∈ N.

It is clear that grγnP · grγmP ⊂ grγn+mP , hence

P =∐n∈N

grγnP

is a N-graded monoid, and consequently :

A[P ] =⊕n∈N

A[grγnP ]

is a graded A-algebra. Moreover, it is easily seen that grγ0P = P×. More generally, for x ∈ P ,let µ(x) be the maximal n ∈ N such that x ∈ mn

P ; then there exists a constant C ≥ 1 such that :

γ(x) ≥ µ(x) ≥ C−1γ(x) for every x ∈ Pso that the mP -adic filtration and the filtration defined by grγ•P , induce the same topology onP and on A[P ]. As a corollary of these considerations, we may state the following “regularitycriterion” for fine monoids :

Proposition 6.5.4. Let P be an integral monoid such that P ] is fine. Then we have

rkP×mP/m2P ≥ dimP

(notation of example 2.3.18) and the equality holds if and only if P ] is a free monoid.

Proof. (Notice that mP/m2P is a free pointed P×-module, since P× obviously acts freely on

mP \m2P .) Since mP ]/m

2P ]

= (mP/m2P )⊗P P ], we may replace P by P ], and assume from start

that P is sharp and fine. Then, the rank of mP/m2P equals the cardinality of the set Σ := mP \

m2P , which is finite, by lemma 6.5.2. We have a surjective morphism of monoids ϕ : N(Σ) → P ,

that sends the basis of N(Σ) bijectively onto Σ ⊂ P (corollary 3.1.10). The sought inequalityfollows immediately, and it is also clear that we have equality, in case P is free. Conversely, ifequality holds, ϕgp must be a surjective group homomorphism between free abelian group ofthe same finite rank (corollary 3.4.10(i)), so it is an isomorphism, and then the same holds forϕ.

Proposition 6.5.5. Let P be a fine and sharp monoid, A a noetherian local ring. Set SP :=1 + A[mP ]; then we have :

dimS−1P A[P ] = dimA+ dimP.


Proof. To begin with, the assumption that P is sharp implies that S−1P A[P ] is local. Next,

notice that A[P ] is a free A-module, hence the natural map A → S−1P A[P ] is a flat and local

ring homomorphism. Let k be the residue field of A; in view of [61, Th.15.1(ii)], we deduce :

dimS−1P A[P ] = dimA+ dimS−1

P k[P ].

Hence we are reduced to showing the stated identity for A = k. However, clearly S−1P k[P ] =

k[P ]m, where m is the maximal ideal generated by the image of mP , hence it suffices to applyclaim 5.9.36(ii) and corollary 3.4.10(i), to conclude.

6.5.6. Let A be a ring, and P a fine and sharp monoid. We define :

A[[P ]] := limn∈N

A〈P/mnP 〉.

Alternatively, this is the completion of A[P ] for its A[mP ]-adic topology. In view of the finite-ness properties of the mP -adic filtration (lemma 6.5.2(ii)), one may present A[[P ]] as the ring offormal infinite sums

∑σ∈P aσ · σ, with arbitrary coefficients aσ ∈ A, where the multiplication

and addition are defined in the obvious way. Moreover, we may use a morphism of monoidsγ : logP → N as in (6.5.3), to see that :

(6.5.7) A[[P ]] =∏n∈N

A[grγnP ]

where A[grγnP ] · A[grγmP ] ⊂ A[grγn+mP ] for every m,n ∈ N. So any element x ∈ A[[P ]] canbe decomposed as an infinite sum

x =∑n∈N

grγnx.

The term grγ0x ∈ grγ0A = A does not depend on the chosen γ : it is the constant term of x, i.e.the image of x under the natural projection A[[P ]]→ A.

Corollary 6.5.8. Let P be a fine and sharp monoid, A a noetherian local ring. Then :(i) For any local morphism P → A (see (3.1.11)), we have the inequality:

dimA ≤ dimA/mPA+ dimP.

(ii) dimA[P ] = dimA[[P ]] = dimA+ dimP .

Proof. (i): Set A0 := A/mPA, and B := S−1P A0[P ], where SP ⊂ A0[P ] is the multiplicative

subset 1 + A0[mP ]; if we denote by gr•A (resp. gr•B) the graded A0-algebra associated to themP -adic filtration on A (resp. on B), we have a natural surjective homomorphism of gradedA0-algebras :

gr•B → gr•A.

Hence dimA = dim gr•A ≤ dim gr•B = dimB, by [61, Th.15.7]. Then the assertion followsfrom proposition 6.5.5.

(ii): Again we consider the ringB := S−1P A[P ], where SP := 1+A[mP ]. One sees easily that

the graded ring associated to the mP -adic filtration on B is just the ring algebra A[P ], whencethe first stated identity, taking into account [61, Th.5.7]. The same argument applies as well tothe mP -adic filtration on A[[P ]], and yields the second identity.

As a first application, we have the following combinatorial version of Kunz’s theorem 4.7.30that characterizes regular rings via their Frobenius endomorphism.

Theorem 6.5.9. Let P be a monoid such that P ] is fine, k > 1 an integer, and suppose that theFrobenius endomorphism kP : P → P is flat (see example 3.5.10(i)). Then P ] is a free monoid.


Proof. First, we remark that k]P : P ] → P ] is still flat (corollary 3.1.49(i)). Moreover, k]P isinjective. Indeed, suppose that xk = yk ·u for some x, y ∈ P and u ∈ P×; from theorem 3.1.42we deduce that there exist b1, b2, t ∈ P such that

b1x = b2y 1 = bk1t u = bk2t.

Especially, b1, b2 ∈ P×, so the images of x and y agree in P ]. Hence, we may replace P by P ],and assume that kP is flat and injective, in which case Z[kP ] : Z[P ] → Z[P ] is flat (theorem3.2.3), integral and injective, hence it is faithfully flat. Now, letR be the colimit of the system ofrings (Rn | n ∈ N), where Rn := Z[P ], and the transition map Rn → Rn+1 is Z[kP ] for everyn ∈ N. The induced map j : R0 → R is still faithfully flat; moreover, let p be any prime divisorof k, and notice that j pP = j (where pP is the p-Frobenius map). It follows that pP is flat andinjective as well, so Fp[pP ] : Fp[P ] → Fp[P ] is a flat ring homomorphism (again, by theorem3.2.3), and then the same holds for the induced map Fp[[pP ]] : Fp[[P ]] → Fp[[P ]]. By Kunz’stheorem, we deduce that Fp[[P ]] is a regular local ring, with maximal ideal m := Fp[[mP ]];notice that the images of the elements of mP \m2

P yield a basis for the Fp-vector space m/m2.Say that mP \m2

P = x1, . . . , xs; it follows that the continuous ring homomorphism

Fp[[T1, . . . , Ts]]→ Fp[[P ]] Ti 7→ xi for i = 1, . . . , s

is an isomorphism. From the discussion of (6.5.6), we immediately deduce that P ' N⊕s, asrequired.

6.5.10. Now we wish to state and prove the combinatorial versions of the Artin-Rees lemma,and of the so-called local flatness criterion (see e.g. [61, Th.22.3]). Namely, let P be a pointedmonoid, such that P ] is finitely generated; let also (A,mA) be a local noetherian ring, N afinitely generated A-module, and :

α : P → (A, ·)a morphism of pointed monoids. The following is our version of the Artin-Rees lemma :

Lemma 6.5.11. In the situation of (6.5.10), let J ⊂ P be an ideal, M a finitely generatedP -module, M0 ⊂M a submodule. Then there exists c ∈ N such that :

(6.5.12) JnM ∩M0 = Jn−c(J cM ∩M0) for every n > c.

Proof. Set M := M/P×, M0 := M0/P× and J := J/P×, the set-theoretic quotients for the

respective natural P×-actions. Notice that J is an ideal of P ] and M0 ⊂ M is an inclusion ofP -modules. Moreover, any set of generators of the ideal J (resp. of the P ]-module M0) liftsto a set of generators for J (resp. for the P -module M0). Furthermore, it is easily seen that(6.5.12) is equivalent to the identity JnM ∩M0 = Jn−c(J cM ∩M0). Hence, we are reducedto the case where P = P ] is a finitely generated monoid. Then Z[P ] is noetherian, Z[M ] is aZ[P ]-module of finite type, and we notice that :

Z[JnM ∩M0] = JnZ[M ] ∩ Z[M0] Z[Jn−c(J cM ∩M0)] = Jn−c(J cZ[M ] ∩ Z[M0]).

Thus, the assertion follows from the standard Artin-Rees lemma [61, Th.8.5].

Proposition 6.5.13. In the situation of (6.5.10), suppose moreover that P ] is fine (see remark2.3.14(vi)), and let mα := α−1mA. Then the following conditions are equivalent :

(a) N is α-flat (see definition 3.1.34).(b) Tor

Z〈P 〉i (N,Z〈M〉) = 0 for every i > 0 and every integral pointed P -module M .

(c) TorZ〈P 〉1 (N,Z〈P/mα〉) = 0.

(d) The natural map :

(mnα/m

n+1α )⊗P N → mn

αN/mn+1α N

is an isomorphism of A-modules, for every n ∈ N (notation of (3.1.33)).


Proof. The assertion (a)⇔(b) is just a restatement of proposition 3.1.40(i), and holds in greatergenerality, without any assumption on either A or the pointed integral monoid P .

As for the remaining assertions, let S := α−1(A×); since the localization P → S−1P is flat,the natural maps :

TorZ〈P 〉i (N,Z〈M〉)→ Tor

Z〈S−1P 〉i (N,Z〈S−1M〉)

are isomorphisms, for every i ∈ N and every P -moduleM . Also, notice that the two P -modulesappearing in (d) are actually S−1P -modules (and the natural map is Z〈S−1P 〉-linear). Hence,we can replace everywhere P by S−1P , which allows to assume that α is local, i.e. mα = mP .

Next, obviously (b)⇒(c).(c)⇒(d): For every n ∈ N, we have a short exact sequence of pointed P -modules :

0→ mnP/m

n+1P → P/mn+1

P → P/mnP → 0.

It is easily seen that mnP/m

n+1P is a free P/mP -module (in the category of pointed modules),

so the assumption implies that TorZ〈P 〉1 (N,mn

P/mn+1P ) = 0 for every n ∈ N. By looking at the

induced long Tor-sequences, we deduce that the natural map

TorZ〈P 〉1 (N,P/mn+1

P )→ TorZ〈P 〉1 (N,P/mn

P )

is injective for every n ∈ N. Then, a simple induction shows that, under assumption (c), allthese modules vanish. The latter means that the natural map :

mnP ⊗P N → mn

PN

is an isomorphism, for every n ∈ N. We consider the commutative ladder with exact rows :

mn+1P ⊗P N //

mnP ⊗P N //

(mnP/m

n+1P )⊗P N //

0

mn+1P N // mn

PN // mnPN/m

n+1P N // 0.

By the foregoing, the two left-most vertical arrows are isomorphisms, hence the same holds forthe right-most, whence (c).

(d)⇒(c): We have to show that the natural map u : mP ⊗P N → mPN is an isomorphism.To this aim, we consider the mP -adic filtrations on these two modules; for the associated gradedmodules one gets :

grn(mPN) = mnPN/m

n+1P N grn(mP ⊗P N) = (mn

P/mn+1P )⊗P N

for every n ∈ N; whence maps of A-modules :

grn(mP ⊗P N)grnu−−→ grn(mPN).

which are isomorphism by assumption. To conclude, it suffices to show :

Claim 6.5.14. For every ideal I ⊂ P , the mPA-adic filtration is separated on the A-moduleI ⊗P N .

Proof of the claim. Indeed, notice that the ideal I/P× ⊂ P ] is finitely generated (proposition3.1.9(ii)), hence the same holds for I , so I ⊗P N is a finitely generated A-module. Then thecontention follows from [61, Th.8.10]. ♦

(c)⇒(b): We argue by induction on i. For i = 1, suppose first that M = P/I for some idealI ⊂ P (notice that any such quotient is an integral pointed P -module); in this case, the assertionto prove is that the natural map v : I ⊗P N → IN is an isomorphism. However, consider themP -adic filtration on P/I; for the associated graded module we have :

grn(P/I) = (mnP ∪ I)/(mn+1 ∪ I) for every n ∈ N


and it is easily seen that this is a free pointed P×-module, for every n ∈ N. Hence, by inspectingthe long exact Tor-sequences associated to the exact sequences

0→ grn(P/I)→ P/(mn+1P ∪ I)→ P/(mn

P ∪ I)→ 0

our assumption (c), together with a simple induction yields :

(6.5.15) TorZ〈P 〉1 (N,Z〈P/(mn

P ∪ I)〉) for every n ∈ N.

Now, fix n ∈ N; in light of lemma 6.5.11, there exists k ≥ n such that mkP ∩ I ⊂ mn

P I . Wededuce surjective maps of A-modules :

I ⊗P N →I

mkP ∩ I

⊗P N →I

mnP I⊗P N

∼→ I ⊗P NmnP (I ⊗P N)

.

On the other hand, (6.5.15) says that the natural map (mnP ∪ I) ⊗P N → (mn

P ∪ I)N is anisomorphism, so the same holds for the induced composed map :

I

mkP ∩ I

⊗P N∼→ mk

P ∪ ImkP

⊗P N →(mk

P ∪ I)N

mkPN

.

Consequently, the kernel of v is contained in mnP (I ⊗P N); since n is arbitrary, we are reduced

to showing that the mP -adic filtration is separated on I ⊗P N , which is claim 6.5.14.Next, again for i = 1, let M be an arbitrary integral P -module. In view of remark 3.1.27(i),

we may assume that M is finitely generated; moreover, remark 3.1.27(ii), together with an easyinduction further reduces to the case where M is cyclic, in which case, according to remark3.1.27(iii), M is of the form P/I for some ideal I , so the proof is complete in this case.

Finally, suppose i > 1 and assume that the assertion is already known for i − 1. We maysimilarly reduce to the case where M = P/I for some ideal I as in the foregoing; to conclude,we observe that :

TorZ〈P 〉i (N,Z〈P/I〉) ' Tor

Z〈P 〉i−1 (N,Z〈I〉).

Since obviously I is an integral P -module, the contention follows.

As a corollary, we have the following combinatorial going-down theorem, which is proved inthe same way as its commutative algebra counterpart.

Corollary 6.5.16. In the situation of (6.5.10), assume that P ] is fine, and that A is α-flat. Letp ⊂ q be two prime ideals of P , and q′ ⊂ A a prime ideal such that q = α−1q′. Then thereexists a prime ideal p′ ⊂ q′ such that p = α−1p′.

Proof. Let β : Pq → Aq′ be the morphism induced by α; it is easily seen that Aq′ is β-flat,and moreover (Pp)

] = (P ])]p is still fine (lemma 3.1.20(iv)). Hence we may replace α by β,which (in view of (3.1.11)) allows to assume that q (resp. q′) is the maximal ideal of P (resp.of A). Next, let P0 := P/p, A0 := A/pA and denote by α0 : P0 → A0 the morphism inducedfrom α; it is easily seen that A0 is α0-flat : for instance, the natural map (mn

P/mn+1P )⊗P A0 →

mnPA0/m

n+1P A0 is of the type f ⊗A A0, where f is the map in proposition 6.5.13(c), thus if

the latter is bijective, so is the former. Moreover P ]0 is a quotient of P ], hence it is again fine.

Therefore we may replace P by P0 and A by A0, which allows to further assume that p = 0,and it suffices to show that there exists a prime ideal q′ ⊂ A, such that α−1q′ = 0. SetΣ := P \ 0; it is easily seen that the natural morphism P → Σ−1P is injective; moreover, itscokernel C (in the category of pointed P -modules) is integral, so that Tor

Z〈P 〉1 (A,Z〈C〉) = 0

by assumption. It follows that the localization map A→ Σ−1A is injective, especially Σ−1A 6=0, and therefore it contains a prime ideal q′′. The prime ideal q′ := q′′ ∩ A will do.


Lemma 6.5.17. Let A be a noetherian local ring, N an A-module of finite type, α : P → Aand β : Q → A two morphisms of pointed monoids, with P ] and Q] both fine. Suppose that αand β induce the same constant log structure on SpecA (see (6.1.13)). Then N is α-flat if andonly if it is β-flat.

Proof. Let ξ be a τ -point localized at the closed point of X := SpecA, set B := OX,ξ andlet ϕ : A → B be the natural map. Let also M be the push-out of the diagram of monoidsP ← (ϕ α)−1B× → B× deduced from α; then M ' P log

X,ξ , the stalk at the point ξ of theconstant log structure on Xτ associated to α. Since ϕ is faithfully flat, it is easily seen that N isα-flat if and only if N ⊗A B is ϕ α-flat.

Hence we may replace A by B, α by ϕ α, N by N ⊗AB, and Q by M , after which we mayassume that Q = P qα−1(A×) A

×; especially, there exists a morphism of monoids γ : P → Qsuch that α = β γ, and moreover β is a local morphism.

Next, set S := α−1(A×); clearly γ extends to a morphism of monoids γ′ : S−1P → Q,and α and β γ′ induce the same constant log structure on Xτ . Arguing as in the proof ofproposition 6.5.13, we see that N is P -flat if and only if it is S−1P -flat. Hence, we may replaceP by S−1P , which allows to assume that γ induces an isomorphism P qP× A×

∼→ Q, thereforealso an isomorphism P ] ∼→ Q]. The latter implies that mQ = mPQ; moreover, notice that themorphism of monoids P× → A× is faithfully flat, so the natural map :

TorZ〈P 〉1 (N,Z〈P/mP 〉)→ Tor

Z〈Q〉1 (N,Z〈(P/mP )⊗P Q〉)→ Tor

Z〈Q〉1 (N,Z〈Q/mQ〉)

is an isomorphism. The assertion follows.

Lemma 6.5.18. Let M be an integral monoid, A a ring, ϕ : M → A a morphism of monoids,and set S := SpecA. Suppose that A is ϕ-flat. Then the log structure (M,ϕ)log

S on Sτ is thesubsheaf of monoids of OS generated by O×S and the image of M .

Proof. To ease notation, set M := (M,ϕ)logS ; let ξ be any τ -point of S. Then the stalk M ξ is

the push-out of the diagram : O×S,ξ ← ϕ−1ξ (O×S,ξ) → M where ϕξ : M → OS,ξ is deduced from

ϕ. Hence M ξ is generated by O×S,ξ and the image of M , and it remains only to show that thestructure map M ξ → OS,ξ is injective. Therefore, let a, b ∈M and u, v ∈ O×S,ξ such that :

(6.5.19) ϕξ(a) · u = ϕξ(b) · v.We come down to showing :

Claim 6.5.20. There exist c, d ∈M such that :

ϕξ(c), ϕξ(d) ∈ O×S,ξ ac = bd ϕξ(c) · v = ϕξ(d) · u.

Proof of the claim. Let mξ ⊂ OS,ξ be the maximal ideal, and set p := ϕ−1ξ (mξ), so that ϕξ

extends to a local morphism ϕp : Mp → OS,ξ. Since OS,ξ is a flat A-algebra, OS,ξ is ϕ-flat,and consequently it is faithfully ϕp-flat (lemma 3.1.36). Then, assumption (6.5.19) leads to theidentity:

aMp ⊗Mp OS,ξ = ϕξ(a) · OS,ξ = ϕξ(b) · OS,ξ = bMp ⊗Mp OS,ξwhence aMp = bMp, by faithful ϕp-flatness. It follows that there exist x, y ∈ Mp such thatax = b and by = a, hence axy = a, which implies that xy = 1, since Mp is an integral module.The latter means that there exist c, d ∈M \p such that ac = bd in M . We deduce easily that

ϕξ(a) · ϕξ(c) · v = ϕξ(a) · ϕξ(d) · u.Thus, to complete the proof, it suffices to show that ϕξ(a) is regular in OS,ξ. However, themorphism of M -modules µa : M → M : m 7→ am (for all m ∈ M ) is injective, hence thesame holds for the map µa ⊗M OS,ξ : OS,ξ → OS,ξ, which is just multiplication by ϕ(a).


6.5.21. Let (X,M) be a locally noetherian log scheme, with coherent log structure (on thesite Xτ , see (6.2.1)), and let ξ be any τ -point of X . We denote by I(ξ,M) ⊂ OX,ξ the idealgenerated by the image of the maximal ideal of M ξ, and we set :

d(ξ,M) := dim OX,ξ/I(ξ,M) + dimM ξ.

Lemma 6.5.22. In the situation of (6.5.21), suppose furthermore that (X,M) is a fs log scheme.Then we have the inequality :

dim OX,ξ ≤ d(ξ,M).

Proof. According to corollary 6.1.34(i), there exists a neighborhood U → X of ξ in Xτ , anda fine and saturated chart α : PU → M |U , which is sharp at the point ξ. Especially, P ' M ]

ξ,therefore dimP = dimM ξ (corollary 3.4.10(ii)). Notice that OX,ξ is a noetherian local ring(this is obvious for τ = Zar, and follows from [33, Ch.IV, Prop.18.8.8(iv)] for τ = et), hence toconclude it suffices to apply corollary 6.5.8(i) to the induced map of monoids P → OX,ξ.

Definition 6.5.23. Let (X,M) be a locally noetherian fs log scheme, ξ a τ -point of X .(i) We say that (X,M) is regular at the point ξ, if the following holds :

(a) the inequality of lemma (6.5.22) is actually an equality, and(b) the local ring OX,ξ/I(ξ,M) is regular.

(ii) We denote by (X,M)reg ⊂ X the set of points x such that (X,M) is regular at any(hence all) τ -points of X localized at x.

(iii) We say that (X,M) is regular, if (X,M)reg = X .(iv) Suppose that K is a field, and X a K-scheme. We say that the K-log scheme (X,M)

is geometrically regular, if E ×K (X,M) is regular, for every field extension E of K.

Remark 6.5.24. (i) Certain constructions produce log structures M → OX that are morphismsof pointed monoids. It is then useful to extend the notion of regularity to such log structures.We shall say that (X,M) is a pointed regular log scheme, if there exists a log structure N onX , such that M = N (notation of (6.1.9)), and (X,N) is a regular log scheme.

(ii) Likewise, if K is a field, X a K-scheme, and M a log structure on X , we shall say thatthe K-log scheme (X,M) is geometrically pointed regular, if M = N for some log structureN on X , such that (X,N) is geometrically regular.

6.5.25. Let (X,M) be a locally noetherian fs log scheme, ξ a τ -point of X , and O∧X,ξ thecompletion of OX,ξ. The next result is the logarithmic version of the classical characterizationof complete regular local rings ([61, Th.29.7 and Th.29.8]).

Theorem 6.5.26. With the notation of (6.5.25), the log scheme (X,M) is regular at the point ξif and only if there exist :

(a) a complete regular local ring (R,mR), and a local ring homomorphism R→ O∧X,ξ;(b) a fine and saturated chart PX(ξ) →M(ξ) which is sharp at the closed point ξ of X(ξ),

such that the induced continuous ring homomorphism

R[[P ]]→ O∧X,ξ

is an isomorphism if OX,ξ contains a field, and otherwise it is a surjection, with kernelgenerated by an element ϑ ∈ R[[P ]] whose constant term lies in mR\m2

R.

Proof. Suppose first that (a) and (b) hold. If R contains a field, then it follows that

dim OX,ξ = dim O∧X,ξ = dimR + dimP

by corollary 6.5.8(ii); moreover, in this case I(ξ,M) = mPOX,ξ, hence OX,ξ/I(ξ,M) =OX,ξ/mPOX,ξ, whose completion is O∧X,ξ/mPO∧X,ξ ' R so that OX,ξ/I(ξ,M) is regular ([30,Ch.0, Prop.17.3.3(i)]). Furthermore,

dimR = dim O∧X,ξ/mPO∧X,ξ = dim OX,ξ/I(ξ,M)


([61, Th.15.1]). Hence (X,M) is regular at the point ξ. If R does not contain a field, we obtaindim OX,ξ = dimR + dimP − 1. On the other hand, let ϑ0 be the image of ϑ in mR; then wehave O∧X,ξ/mPO∧X,ξ ' R/ϑ0R, which is regular of dimension dim R − 1, and again we invoke[30, Ch.0, Prop.17.3.3(i)] to see that (X,M) is regular at ξ.

Conversely, suppose that (X,M) is regular at ξ. Suppose first that OX,ξ contains a field;then we may find a field k ⊂ O∧X,ξ mapping isomorphically to the residue field of O∧X,ξ ([61,Th.28.3]). Pick a sequence (t1, . . . , tr) of elements of OX,ξ whose image in the regular localring OX,ξ forms a regular system of parameters. Let also P a fine saturated monoid for whichthere exists a neighborhood U → X of ξ and a chart PU → M |U , sharp at the point ξ. Therefollows a map of monoids α : P → OX,ξ, and necessarily the image of mP lies in the maximalideal of OX,ξ, and generates I(ξ,M). We deduce a continuous ring homomorphism

k[[P × N⊕r]]→ O∧X,ξ

which extends α, and which maps the generators T1, . . . , Tr of N⊕r onto respectively t1, . . . , tr.This map is clearly surjective, and by comparing dimensions (using corollary 6.5.8(ii)) one seesthat it is an isomorphism. Then the theorem holds in this case, with R := k[[N⊕r]].

Next, if OX,ξ does not contain a field, then its residue characteristic is a positive integer p,and we may find a complete discrete valuation ring V ⊂ O∧X,ξ whose maximal ideal is pV , andsuch that V/pV maps isomorphically onto the residue field of O∧X,ξ ([61, Th.29.3]). Again, wechoose a morphism of monoids α : P → OX,ξ as in the foregoing, and a sequence (t1, . . . , tr)of elements of OX,ξ lifting a regular system of parameters for OX,ξ/I(ξ,M), by means of whichwe define a continuous ring homomorphism

ϕ : V [[P × N⊕r]]→ O∧X,ξ

as in the previous case. Again, it is clear that ϕ is surjective. The image of the ideal J generatedby the maximal ideal of P × N⊕r is the maximal ideal of O∧X,ξ; in particular, there exists x ∈ Jsuch that ϑ := p− x lies in Kerϕ. If we let R := V [[N⊕r]], it is clear that ϑ ∈ mR \m2

R.

Claim 6.5.27. Let A be a ring, π a regular element of A such that A/πA is an integral domain,P a fine and sharp monoid. Let also ϑ be an element of A[[P ]] whose constant term is π (see(6.5.6)). Then A[[P ]]/(ϑ) is an integral domain.

Proof of the claim. To ease notation, set A0 := A/πA, B := A[[P ]] and C := B/ϑB. Choose adecomposition (6.5.7), and set FilγnB :=

∏i≥nA[[grγi P ]] for every n ∈ N. Filγ•B is a separated

filtration by ideals of B, and we may consider the induced filtration Filγ•C on C. First, weremark that Filγ•C is also separated. This comes down to checking that⋂

n≥0

ϑB + FilγnB = ϑB.

To this aim, suppose that, for a given x ∈ B we have identities of the type x = ϑyn + zn, withyn ∈ B and zn ∈ FilγnB for every n ∈ N. Then, since π is regular, an easy induction shows thatgrγi (yn) = grγi (ym) whenever n,m > i, and moreover x = ϑ ·

∑i∈N grγi (yi+1), which shows the

contention. It follows that, in order to show that C is a domain, it suffices to show that the sameholds for the graded ring grγ•C associated to Filγ•C. However, notice that :

FilγnB ∩ ϑB = ϑ · FilγnB for every n ∈ N.(Indeed, this follows easily from the fact that π is a regular element : the verification shall beleft to the reader). Hence, we may compute :

grγnC =FilγnB + ϑB

Filγn+1 + ϑB' FilγnB

Filγn+1 + ϑFilγn' A[grγnP ]/ϑA[grγnP ] ' A0[grγnP ].

Thus, grγ•A ' A0[P ], which is a domain, since by assumption A0 is a domain. ♦


From claim 6.5.27(ii) we deduce that R[[P ]]/(ϑ) is an integral domain. Then, again bycomparing dimensions, we see that ϕ factors through an isomorphism R[[P ]]/(ϑ)

∼→ O∧X,ξ.

Remark 6.5.28. Resume the notation of (6.5.25), and suppose that OX,ξ/I(ξ,M) is a regularlocal ring. Let PX(ξ) → M(ξ) be a chart as in theorem 6.5.26(b), and α : P → OX,ξ thecorresponding morphism of monoids. Moreover, if OX,ξ contains a field, let V denote a coeffi-cient field of O∧X,ξ, and otherwise, let V be a complete discrete valuation ring whose maximalideal is generated by p, the residue characteristic of OX,ξ. In either case, pick a ring homomor-phism V → O∧X,ξ inducing an isomorphism of V/pV onto the residue field of OX,ξ. Let as wellt1, . . . , tr ∈ OX,ξ be any sequence of elements whose image in OX,ξ/I(ξ,M) forms a regularsystem of parameters, and extend the map α to a morphism of monoids P × N⊕r → OX,ξ, bythe rule : ei 7→ ti, where e1, . . . , er is the natural basis of N⊕r. Then by inspecting the proof oftheorem 6.5.26, we see that the induced continuous ring homomorphism :

V [[P × N⊕r]]→ O∧X,ξ

is always surjective, and if V is not a field, its kernel contains an element ϑ whose constant termin V is ϑ0 = p. If V is a field (resp. if V is a discrete valuation ring) then (X,M) is regular atthe point ξ, if and only if this map is an isomorphism (resp. if and only if the kernel of this mapis generated by ϑ).

Corollary 6.5.29. Let (X,M) be a regular log scheme. Then the scheme X is normal andCohen-Macaulay.

Proof. Let x ∈ X be any point, and ξ a τ -point localized at x; we have to show that OX,x isCohen-Macaulay; in light of [33, Ch.IV, Cor.18.8.13(a)] (when τ = et), it suffices to show thatthe same holds for OX,ξ. Then [61, Th.17.5] further reduces to showing that the completion O∧X,ξis Cohen-Macaulay; the latter follows easily from theorems 6.5.26 and 5.9.34(i). Next, in orderto prove that X is normal, it suffices to show that OX,x is regular, whenever x has codimensionone in X ([61, Th.23.8]). Again, by [33, Ch.IV, Cor.18.8.13(c)] (when τ = et) and [30, Ch.0,Prop.17.1.5], we reduce to showing that O∧X,ξ is regular for such a point x. However, for a pointof codimension one we have r := dimM ξ ≤ 1. If r = 0, then I(ξ,M) = 0, hence OX,ξ isregular. Lastly, if r = 1, we see that M ]

ξ ' N (theorem 3.4.16(iii)); consequently, there existsa regular local ring R and an isomorphism O∧X,ξ ' R[[N]]/(ϑ), where ϑ = 0 if OX,ξ contains afield, and otherwise the constant term of ϑ lies in mR \ m2

R. Since R[[N]] is again regular, theassertion follows in either case.

Corollary 6.5.30. Let i : (X ′,M ′) → (X,M) be an exact closed immersion of regular logschemes (see definition 6.3.22(i)). Then the underlying morphism of schemes X ′ → X is aregular closed immersion.

Proof. Let ξ be a τ -point of X , and denote by J the kernel of i\ξ : OX,ξ → OX′,ξ. To easenotation, let as well A := OX,ξ/I(ξ,M) and A′ := OX′,ξ/I(ξ,M ′) Since log i : i∗M → M ′ isan isomorphism, i\ξ induces an isomorphism :

OX,ξ/(J + I(ξ,M))∼→ A′.

Since A and A′ are regular, there exists a sequence of elements t1, . . . , tk ∈ J , whose imagein A forms the beginning of a regular system of parameters and generate the kernel of theinduced map A → A′ ([30, Ch.0, Cor.17.1.9]). Extend this sequence by suitable elements ofOX,ξ, to obtain a sequence (t1, . . . , tr) whose image in A is a regular system of parameters. Wededuce a surjection ϕ : V [[P × N⊕r]] → O∧X,ξ as in remark 6.5.28, where V is either a fieldor a complete discrete valuation ring. Denote by (e1, . . . , er) the natural basis of N⊕r. Now,suppose first that V is a complete discrete valuation ring; then Kerϕ is generated by an element


ϑ ∈ V [[P × N⊕r]], whose constant term is a uniformizer in V ; we deduce easily from claim6.5.27 that (e1, . . . , er, ϑ) is a regular sequence in V [[P × N⊕r]]; hence the same holds for thesequence (ϑ, e1, . . . , er), in view of [61, p.127, Cor.]. This implies that (t1, . . . , tr) is a regularsequence in O∧X,ξ, hence (t1, . . . , tk) is a regular sequence in OX,ξ, which is the contention. Thecase where V is a field is analogous, though simpler : the details shall be left to the reader.

Remark 6.5.31. In the situation of remark 6.5.28, suppose that P is a free monoid of finite rankd, let f1, . . . , fd ⊂ OX,ξ be the image of the (unique) system of generators of P , and for everyi ≤ d let Zi ⊂ X be the zero locus of fi; then

⋃di=1 Zi is a strict normal crossing divisor in

the sense of example 6.2.11. The proof is analogous to that of corollary 6.5.30 : the sequence(f1, . . . , fd, ϑ) is regular in V [[P ⊕ N⊕r]], hence also its permutation (ϑ, f1, . . . , fd) is regular,whence the claim (the details shall be left to the reader).

Proposition 6.5.32. Resume the situation of (6.5.25). Then the log scheme (X,M) is regularat the τ -point ξ if and only if the following two conditions hold :

(a) The ring OX,ξ/I(ξ,M) is regular.(b) There exists a morphism of monoids P → OX,ξ from a fine monoid P , whose associated

constant log structure on X(ξ) is the same as M(ξ), and such that OX,ξ is P -flat.

Proof. Suppose first that (X,M) is regular at the point ξ; then by definition (a) holds. Next,we may find a fine monoid P and a local morphism α : P → A := OX,ξ whose associatedconstant log structure on X(ξ) is the same as M(ξ), and a regular local ring R, with a ringhomomorphism R → A such that the induced continuous map R[[P ]] → A∧ fulfills condition(b) of theorem 6.5.26 (where A∧ is the completion of A). Since the completion map ϕ : A →A∧ is faithfully flat, A is α-flat if and only if A∧ is ϕ α-flat; in light of proposition 3.1.39(i),it then suffices to show that, for every ideal I ⊂ P , the natural map

A∧L⊗P P/I → A∧ ⊗P P/I

is an isomorphism in D−(A∧-Mod) (notation of (3.1.33)). To this aim we remark :

Claim 6.5.33. For any ideal I ⊂ P , the natural morphism

R[P ]L⊗P P/I → R〈P/I〉

is an isomorphism in D−(R[P ]-Mod).

Proof of the claim. We consider the base change spectral sequence for Tor :

E2ij := Tor

Z[P ]i (TorZj (R,Z[P ]),Z〈P/I〉)⇒ TorZi+j(R,Z〈P/I〉)

([75, Th.5.6.6]). Clearly E2ij = 0 whenever j > 0, whence isomorphisms :

TorZ〈P 〉i (R〈P 〉,Z〈P/I〉) ∼→ TorZi (R,Z〈P/I〉)

for every i ∈ N. The claim follows easily. ♦

In light of claim 6.5.33 we deduce natural isomorphisms :

A∧L⊗P P/I

∼→ A∧L⊗R[P ] (R[P ]

L⊗P P/I)

∼→ A∧L⊗R[P ] R〈P/I〉

in D−(A∧-Mod). Now, ifA contains a field, we haveA∧ ' R[[P ]], which is a flatR[P ]-algebra([61, Th.8.8]), and the contention follows. If A does not contain a field, the complex

0→ R[[P ]]ϑ−→ R[[P ]]→ A∧ → 0

is a R[P ]-flat resolution of A∧. Since R[[P ]]/IR[[P ]] ' limn∈N

R〈P/(I ∪ mnP )〉, we come down

to the following :


Claim 6.5.34. Let ϑ ∈ R[P ] be any element whose constant term ϑ0 is a regular element in R.Then, for every ideal I ⊂ P , the image of ϑ in R〈P/I〉 is a regular element.

Proof of the claim. In view of (6.5.3), P can be regarded as a graded monoid P =∐

n∈N Pn,so R[P ] is a graded algebra, and I =

∐n∈N(I ∩ Pn) is a graded ideal. Thus, 〈P/I〉 is a graded

R-algebra as well, and the claim follows easily (details left to the reader). ♦Conversely, suppose that conditions (a) and (b) hold. By virtue of lemma 6.5.17, we may

assume that P is fine, sharp and saturated, and that the map P → OX,ξ is local. Accordingto remark 6.5.28, we may find a regular local ring R of dimension ≤ 1 + dimA/mPA, and asurjective ring homomorphism :

ϕ : R[[P ]]→ A∧

Suppose first that dimR = 1 + dimA/mPA; then Kerϕ contains an element ϑ whose constantterm ϑ0 lies in mR \m2

R, and ϕ induces an isomorphism R0 := R/ϑ0R∼→ A∧/mPA

∧. We haveto show that ϕ induces an isomorphism ϕ : R[[P ]]/(ϑ)

∼→ A∧. To this aim, we consider themP -adic filtrations on these rings; for the associated graded rings we have :

grnR[[P ]]/(ϑ) ' R0〈mnP/m

n+1P 〉 grnA

∧ = mnPA∧/mn+1

P A∧.

Hence grnϕ is the natural map A∧/mPA∧ ⊗P (mn

P/mn+1P ) → mn

PA∧/mn+1

P A∧, and the latteris an isomorphism, since A∧ is P -flat. The assertion follows in this case. The remainingcase where dimR = dimA/mPA is similar, but easier, so shall be left to the reader, as anexercise.

Corollary 6.5.35. Let (X,M) be a log scheme, ξ a τ -point of X , and suppose that (X,M) isregular at ξ. Then the following conditions are equivalent :

(a) OX,ξ is a regular local ring.(b) M ]

ξ is a free monoid of finite rank.

Proof. (b)⇒(a) : Indeed, it suffices to show that O∧X,ξ is regular ([30, Ch.0, Prop.17.3.3(i)]); bytheorem 6.5.26, the latter is isomorphic to eitherR[[P ]] orR[[P ]]/ϑR[[P ]], whereR is a regularlocal ring, P := M ξ, and the constant term of ϑ lies in mR \ m2

R. Since, by assumption, P isa free monoid of finite rank, it is easily seen that rings of the latter kind are regular ([30, Ch.0,Cor.17.1.8]).

(a)⇒(b) : By proposition 6.5.32, R := OX,ξ is P -flat, for a morphism P := M ]ξ → R that

induces the log structure M(ξ) on X(ξ). It follows that

mPR/m2PR = (mP/m

2P )⊗P R = R⊕r

where r := rkP×mP/m2P is the cardinality of mP \ m2

P . In view of [33, Ch.IV, Prop.16.9.3,Cor.16.9.4, Cor.19.1.2], we deduce that the image of mP \ m2

P is a regular sequence of R oflength r, hence dimP = dimR − dimR/mPR = r, since (X,M) is regular at ξ. Then theassertion follows from proposition 6.5.4.

Corollary 6.5.36. Let (X,M) be a regular log scheme, ξ any τ -point of X , and p ⊂ M ξ anideal. We have :

(i) If p is a prime ideal, pOX,ξ is a prime ideal of OX,ξ, and ht p = ht pOX,ξ.(ii) If p is a radical ideal, pOX,ξ is a radical ideal of OX,ξ.

Proof. To ease notation, set A := OX,ξ. Pick a morphism β : P → M ξ as in proposition6.5.32; by corollary 6.1.34(i) and lemma 6.5.17, we may further assume that P is fine, sharpand saturated. Let q := β−1p ⊂ P ; then pA = qA.

(i): Since p is a prime ideal, q is a prime ideal of P , and in order to prove that pA is a primeideal, it suffices therefore to show that the completion (A/qA)∧ of A/qA is an integral domain.However, remark 6.5.28 implies that (A/qA)∧ ' B/ϑB, where B := R[[P\q]], with (R,mR) a


regular local ring, and ϑ is either zero, or else it is an element whose constant term ϑ0 ∈ R liesin mR\m2

R. The assertion is obvious when ϑ = 0, and otherwise it follows from claim 6.5.27.Next, by going down (corollary 6.5.16), we see that :

ht pA ≥ ht p.

On the other hand, notice that we have a natural identification SpecM ξ∼→ SpecP , especially

ht p = ht q, hence dimA/pA = dim(A/qA)∧ = dimR+ dim(P\q)− ε, where ε is either 0 or1 depending on whether R does or does not contain a field (corollary 6.5.8(ii)). Thus :

ht pA ≤ dimA− dimA/pA = dimP − dim(P \q) = ht p

(corollary 3.4.10(iii)), which completes the proof.(ii): In this case, q is a radical ideal of P , so it can be written as a finite intersection of prime

ideals of P (lemmata 3.1.15 and 3.1.20(iii)); then the assertion follows from (i) and lemma3.1.37 (details left to the reader).

Lemma 6.5.37. Let X be a scheme, M a fs log structure on XZar, and ξ a geometric point ofX . Then (X,M) is regular at the point |ξ| if and only if u∗X(X,M) is regular at ξ.

Proof. Set (Y,N) := u∗X(X,M) (notation of (6.2.2) : of course Y = X , but the sheaf OY isdefined on the site Xet, hence B := OY,ξ is the strict henselization of A := OX,|ξ|), and letα : M |ξ| → B be the induced morphism of monoids; since α is local, it is easily seen that N ξ isisomorphic to the push-out of the diagramM |ξ| ←M×

|ξ| → B×, especially dimM |ξ| = dimN ξ.Moreover, I(ξ,N) = I(|ξ|,M)B, so that B0 := B/I(ξ,N) is the strict henselization of A0 :=A/I(|ξ|,M) ([33, Ch.IV, Prop.18.6.8]); hence A0 is regular if and only if the same holds for B0

([33, Ch.IV, Cor.18.8.13]). Finally, dimA = dimB and dimA0 = dimB0 ([61, Th.15.1]). Thelemma follows.

6.5.38. Let (Y,N) be a log scheme, y a τ -point of Y , and α : QlogY → N a fine and saturated

chart which is sharp at y. Let also ϕ : Q → P be an injective morphism of monoids, with Pfine and saturated, and such that P× is a torsion-free abelian group. Define (X,M) as the fibreproduct in the cartesian diagram of log schemes :

(6.5.39)

(X,M)f //

(Y,N)

h

Spec(Z, P )Spec(Z,ϕ)

// Spec(Z, Q)

where h is induced by α; especially, h is strict.

Lemma 6.5.40. In the situation of (6.5.38), Let x be any τ -point of X such that f(x) = y, andsuppose that (Y,N) is regular at y. Then (X,M) is regular at x.

Proof. To begin with, we show :

Claim 6.5.41. The natural map :

OX,xL⊗P P/I → OX,x ⊗P P/I

is an isomorphism in D−(OX,x -Mod), for every ideal I ⊂ P (notation of (3.1.33)).


Proof of the claim. It suffices to show that this map is an isomorphism in D−(OY,y -Mod), andin the latter category we have a commutative diagram :

(6.5.42)

(OY,y

L⊗Q P )

L⊗P P/I

//OY,y

L⊗Q P/I

(OY,y ⊗Q P )L⊗P P/I // (OY,y ⊗Q P )⊗P P/I ∼ // OY,y ⊗Q P/I.

Since OX,x is a localization of OY,y ⊗Q P , we are reduced to showing that the bottom arrowof (6.5.42) is an isomorphism. However, the top arrow of (6.5.42) is always an isomorphism.Moreover, on the one hand, since (Y,N) is regular at y, the ring OY,y is Q-flat (proposition6.5.32), and on the other hand, since ϕ is injective, P/I is an integral Q-module, so the twovertical arrows are isomorphisms as well, and the claim follows. ♦

Claim 6.5.43. OX,x/I(x,M) is a regular ring.

Proof of the claim. Let β : P → A := OX,x be the morphism deduced from h, and set :

S := β−1(O×X,x) I(x, P ) := P \S.

Then the Z[P ]-algebra A is a localization of the Z[P ]-algebra

B := S−1(OY,y ⊗Q P )

and it suffices to show that B/I(x,M) is regular.It is easily seen that I(x,M) = I(x, P )A and ϕ(mQ) ⊂ I(x, P ); on the other hand, since α

is sharp at the point y, we have I(y,N) = mQOY,y, and Q \ mQ = 1. Let p be the residuecharacteristic of OY,y; there follow isomorphisms of Z(p)-algebras :

B/I(x,M)B ' S−1OY,y ⊗Q P/I(x, P ) ' OY,y/I(y,N)⊗Z(p)Z(p)[S

gp].

By assumption, OY,y/I(y,N) is regular, hence we are reduced to showing that Z(p)[Sgp] is a

smooth Z(p)-algebra ([33, Ch.IV, Prop.17.5.8(iii)]). However, under the current assumptionsP gp is a free abelian group of finite rank, hence the same holds for Sgp, and the contentionfollows easily. ♦

In light of proposition 3.1.39(i), claims 6.5.41 and 6.5.43 assert that conditions (a) and (b)of proposition 6.5.32 are satisfied for the τ -point x of (X,M), so the latter is regular at x, asstated.

Theorem 6.5.44. Let f : (X,M)→ (Y,N) be a smooth morphism of locally noetherian fs logschemes, ξ a τ -point of X , and suppose that (Y,N) is regular at the point f(ξ). Then (X,M)is regular at the point ξ.

Proof. In case τ = Zar, lemma 6.5.37 and corollary 6.3.27(ii) reduce the assertion to the cor-responding one for u∗f . Hence, we may assume that τ = et. Next, since the assertion is localon Xτ , we may assume that N admits a fine and saturated chart α : Qlog

Y → N which is sharpat f(ξ) (corollary 6.1.34(i)); then, by corollary 6.3.42, we may further assume that there existsan injective morphism of fine and saturated monoids ϕ : Q → P , such that P× is torsion-free,and a cartesian diagram as in (6.5.39). Then the assertion follows from lemma 6.5.40.

Corollary 6.5.45. Let f : (X,M) → (Y,N) be a smooth morphism of locally noetherian finelog schemes. Then, for every y ∈ (Y,N)tr, the log scheme Specκ(y) ×Y (X,M) is geometri-cally regular.


Proof. The trivial log structure on Specκ(y) is obviously saturated, so the same holds for thelog structure of Specκ(y) ×Y (X,M). Hence, the assertion is an immediate consequence oftheorem 6.5.44.

Theorem 6.5.46. Let (X,M) be a locally noetherian fs log scheme. Then the subset (X,M)reg

is closed under generization.

Proof. Let ξ be a τ -point of X , with support x ∈ (X,M)reg, and let η be a generization of ξ,whose support is a generization y of x. We have to show that (X,M) is regular at the point η.Since the assertion is local on X , we may assume that (X,M) admits a fine and saturated chartβ : PX → M , sharp at the point ξ (corollary 6.1.34(i)). Let α : P → OX,η be the morphismdeduced from βη, and set p := α−1mη, where mη ⊂ OX,η is the maximal ideal. We consider thecartesian diagram of log schemes :

(X ′,M ′) //

(X,M)

g

Spec〈Z, P/p〉 // Spec(Z, P )

where g is induced by β (notation of (6.2.13)). Clearly ξ and η induce τ -points on X ′, whichwe denote by the same names. We have natural identifications:

OX,ξ/I(ξ,M)∼→ OX′,ξ/I(ξ,M ′) M ′

η = O×X′,η OX,η/I(η,M)∼→ OX′,η

([33, Ch.IV, Prop.18.6.8]). Moreover, from proposition 6.5.32 and lemma 6.5.17 we know thatOX,ξ is P -flat; then corollary 6.5.16 yields the inequality :

dim OX′,ξ = dim OX,ξ ⊗P P/p ≥ dim OX,ξ − ht(p) = dim OX,ξ/I(ξ,M) + dimP/p

in other words : dim OX′,ξ = d(ξ,M ′) (notation of definition 6.5.23), so (X ′,M ′) is regular atξ. By the same token, OX,η is Pp-flat; from proposition 6.5.32, it follows that (X,M) is regularat η if and only if the same holds for (X ′,M ′).

Hence we may replace (X,M) by (X ′,M ′), and P by P \p, after which we may assume thaty ∈ (X,M)tr. In this case :

(6.5.47) α(P ) ⊂ O×X,η

and we have to show that OX,η is a regular local ring, or equivalently, that OX,y is regular ([33,Ch.IV, Cor.18.8.13(c)]).

Denote by Y the topological closure of y in X , endowed with its reduced subscheme struc-ture. According to [27, Ch.II, Prop.7.1.7] (and [33, Ch.IV, Prop.18.8.8(iv)] in case τ = et), wemay find a local injective ring homomorphism j : OY,ξ → V , where V is a discrete valuationring. Let also β : P → OY,ξ be the morphism deduced from β. Then (6.5.47) implies thatj β(P ) ⊂ V \0, hence j β extends to a homomorphism of groups P gp → K×, whereK× := (V \0)gp is the multiplicative group of the field of fractions of V ; after compositionwith the valuation K× → Z of V , there follows a group homomorphism

ϕ : P → Z.Notice also that j β is a local morphism, since the same holds for j and β; consequently,ϕ(P ) 6= 0. Set Q := ϕ−1N; then ϕ(Q) is a non-trivial submonoid of N, and dimQ =dimQ/Kerϕgp = dimϕ(Q) = 1. Set

T := SpecV (S,Q) := Spec(Z, Q).

Notice that Q is saturated and fine (corollary 3.4.2), so there exists an isomorphism :

Z⊕r × N ∼→ Q for some r ∈ N


(theorem 3.4.16(iii)); the latter determines a chart :

(6.5.48) NS → Q

which is sharp at every τ -point of S localized outside the trivial locus Spec(Z, Q)tr. We considerthe cartesian diagram of log schemes :

(X ′, g′∗Q)g′ //

f ′

Spec(Z, Q)

f

(X,M)

g // Spec(Z, P )

where f is the morphism of log schemes induced by the inclusion map ψ : P → Q. Notice thatf is a smooth morphism (proposition 6.3.34); moreover, the restriction

ftr : Spec(Z, Q)tr → Spec(Z, P )tr

of f , is just the morphism SpecZ[ψgp] : SpecZ[Qgp] → SpecZ[P gp], hence it is an isomor-phism of schemes. It follows that f ′ is a smooth morphism (proposition 6.3.24(ii)), and itsrestriction f ′tr to the trivial loci, is an isomorphism.

The homomorphism j induces a morphism h : T → X , such that the closed point of T mapsto x and the generic point maps to y. By construction g h lifts to a morphism of schemesh′ : T → S, and the pair (h, h′) determines a morphism T → X ′. Let x′, y′ ∈ X ′ be theimages of respectively the closed point and the generic point of T , and choose τ -points ξ′ and η′

localized at x′ and respectively y′; then the image of x′ in X is the point x, therefore (X ′, g′∗Q)is regular at the point ξ′, by theorem 6.5.44. Furthermore, g′(ξ′) lies outside Spec(Z, Q)tr,hence (6.5.48) induces a chart NX′ → g′∗Q, which is sharp at ξ′. In light of corollary 6.5.35,we deduce that OX′,x′ is a regular ring, and then the same holds also for OX′,y′ ([30, Ch.0,Cor.17.3.2]). However, y′ lies in the trivial locus of (X ′, g′∗Q), and its image in X is y, so bythe foregoing the natural map OX,y → OX′,y′ is an isomorphism, and the contention follows.

6.5.49. Let (X,M) be any log scheme, ξ a τ -point of X . There follows a continuous map

ψξ : X(ξ)→ Tξ := SpecM ξ

that sends the closed point of X(ξ) to the closed point of Tξ. For every point p ∈ Tξ, let p bethe topological closure of p in Tξ; then

X(p) := ψ−1ξ p

is a closed subset of X(ξ), which we endow with its reduced subscheme structure, and we set

(X(p),M(p)) := ((X(ξ),M(ξ))×X X(p))red

(notation of example 6.1.10(iv)). Notice that Up := ψ−1ξ (p) is an open subset of X(p), for every

p ∈ Tξ. We call the family(Up | p ∈ Tξ)

of locally closed subschemes of X(ξ), the logarithmic stratification of (X(ξ),M(ξ)). Forinstance, U∅ = (X(x),M(x))tr. More generally, it is clear from the definition that

(6.5.50) Up = (X(p),M(p))tr for every p ∈ Tξ.

Corollary 6.5.51. In the situation of (6.5.49), suppose that (X,M) is regular at ξ. We have :(i) The log scheme (X(p),M(p)) is pointed regular, for every p ∈ Tξ.

(ii) The scheme X(p) is irreducible, and its codimension in X equals the height of p in Tξ,for every p ∈ Tξ.

(iii) The scheme Up is regular and irreducible, for every p ∈ Tξ.


Proof. (i): By theorem 6.5.46, it suffices to show that (X(p),M(p)) is pointed regular at ξ, forevery p ∈ Tξ. However, say that X(ξ) = SpecA, and let P → M(ξ) be a fine and saturatedchart, sharp at the τ -point ξ. Then X(p) = SpecA0, where A0 := A/pA, and M(p) = N,where N is the fs log structure deduced from the induced map β : P \p→ A0. By proposition6.5.32 and lemma 6.5.17, the ring A is P -flat; then A0 is β-flat, and the assertion follows fromproposition 6.5.32.

Next, (ii) is a rephrasing of corollary 6.5.36(i), and (iii) follows from (i),(ii) and (6.5.50), byvirtue of corollary 6.5.35.

Proposition 6.5.52. Let (X, β : M → OX) be a regular log scheme, set U := (X,M)tr, anddenote by j : U → X the open immersion. Then the morphism β induces identifications:

M∼→ j∗O

×U ∩ OX Mgp ∼→ j∗O

×U .

Proof. Notice that the scheme X is normal (corollary 6.5.29), hence both j∗O×U and OX aresubsheaves of the sheaf i∗OX0 , where X0 is the subscheme of maximal points of X , and i :X0 → X is the natural morphism; so we may intersect these two sheaves inside the latter.

In view of lemma 6.5.18 and proposition 6.5.32, we know already that β is injective, andclearly the image of β lands in j∗O×U , so it remains only to show that β (resp. βgp) induces anepimorphism onto j∗O×U ∩ OX (resp. onto j∗O×U ). The assertions can be checked on the stalks,hence let ξ be any τ -point of X; to begin with, we show :

Claim 6.5.53. The induced map βgpξ : Mgp

ξ → (j∗O×U )ξ is a surjection.

Proof of the claim. Set A := OX,ξ; notice that (X(ξ),M(ξ))tr is the complement of the unionof the finitely many closed subsets of the form Zp := SpecA/pA, where p ⊂ M ξ runs overthe prime ideals of height one. Due to corollary 6.5.36(i), each Zp is an irreducible divisor inSpecA. Now, let s ∈ (j∗O

×U )ξ; then the divisor of s is of the form

∑htp=1 np[p], for some

np ∈ Z. By lemmata 6.1.16(ii) and 6.2.21(i), M ]ξ is a fine and saturated monoid; then lemma

3.4.22 and proposition 3.4.32 say that the fractional ideal I :=⋂

ht p=1 mnpp of M ξ is reflexive,

and therefore the fractional ideal IA of A is reflexive as well (lemma 3.4.27(ii)). Since Ais normal, by considering the localizations of IOX,ξ at the prime ideals of height one of A,we deduce easily that IA = sA ([14, Ch.VII, §4, n.2, Cor. du Th.2]). Thus, we may writes =

∑ni=1 β

gpξ (xi)ai for certain ai ∈ A and xi ∈ I (for i = 1, . . . , n). Then we must have

βgpξ (xi)ai /∈ s · mξ for at least one index i ≤ n (where mξ ⊂ A is the maximal ideal); for suchi, it follows that s−1 · βgp

ξ (xi) ∈ A×, whence s ∈Mgpξ , as required. ♦

Now, let s ∈ (j∗O×U )ξ ∩ A; by claim 6.5.53, we may find x ∈ Mgp

ξ such that βgpξ (x) = s. To

conclude, it suffices to show that x ∈ M ξ. By theorem 3.4.16(i), we are reduced to showingthat x ∈ (M ξ)p for every prime ideal p ⊂ M ξ of height one. Set q := pA; then q is a primeideal of height one (corollary 6.5.36(i)), and βξ extends to a well defined map of monoidsβp : (M ξ)p → Aq. Furthermore, Aq is a discrete valuation ring (corollary 6.5.29), whosevaluation we denote v : Aq \ 0 → N. In view of theorem 3.4.16(ii), we see that x ∈ (M ξ)p ifand only if (v βp)gp(x) ≥ 0. However, clearly v(s) ≥ 0, whence the contention.

Remark 6.5.54. Let reg.logτ denote the full subcategory of logτ whose objects are the regularlog schemes. As an immediate consequence of proposition 6.5.52 (and of remark 6.2.8(i)) wesee that the forgetful functor F of (6.2.1) restricts to a fully faithful functor

reg.logτ → Open (X,M) 7→ ((X,M)tr → X)

where Open is full subcategory of Morph(Sch) whose objects are the open immersions.


6.6. Resolution of singularities of regular log schemes. Most of this section concerns resultsthat are special to the class of log schemes over the Zariski topology; these are then applied toetale fs log structures, after we have shown that every such log structure admits a logarithmicblow up which descends to the Zariski topology (proposition 6.6.52). The same statement –with an unnecessary restriction to regular log schemes – can be found in the article [64] byW.Niziol : see theorem 5.6 of loc.cit.. We mainly follow her treatment, except for fleshing outsome details, and correcting some inaccuracies.

Therefore, we let here τ = Zar, and all log structures considered in this section until (6.6.48)are defined on the Zariski sites of their underlying schemes.

6.6.1. Let f : (Y,N) → (X,M) be any morphism of log schemes; we remark that f is amorphism of monoidal spaces, i.e. for every y ∈ Y , the map M f(y) → Ny induced by log f , islocal. Indeed, it has been remarked in (6.1.6) that the natural map M f(y) → (f ∗M)y is local,and on the other hand, a section s ∈ (f ∗M)y is invertible if and only if its image in OY,y isinvertible, if and only if log fy(s) is invertible in Ny, whence the contention. We shall denote

f ] : (Y,N)] → (X,M)]

the morphism of sharp monoidal spaces induced by f in the obvious way (i.e. the underlyingcontinuous map is the same as the continuous map underlying f , and log f ] : f ∗M ] → N ] is(log f)] : see definition 3.5.1).

6.6.2. We let K be the category whose objects are all data of the form X := ((X,M), F, ψ),where (X,M) is a log scheme, F is a fan, and ψ : (X,M)] → F is a morphism of sharpmonoidal spaces, such that logψ : ψ∗OF →M ] is an isomorphism. The morphisms :

((Y,N), F ′, ψ′)→ X

in K are all the pairs (f, ϕ), where f : (Y,N) → (X,M) is a morphism of log schemes, andϕ : F ′ → F is a morphism of fans, such that the diagram

(Y,N)]f] //

ψ′

(X,M)]

ψ

F ′

ϕ // F

commutes. Especially, notice that lemma 6.1.4 implies the identity :

(6.6.3) Str(f) = ψ′−1Str(ϕ).

(notation of definitions 3.5.1(ii) and 6.2.7(ii)). We shall say that an object X is locally noether-ian, if the same holds for the scheme X . Likewise, a morphism (f, ϕ) in K is quasi-compact,(resp. quasi-separated, resp. separated, resp. locally of finite type, resp. of finite type) if thesame holds for the morphism of schemes underlying f . We say that f is etale, if the same holdsfor the morphism of log schemes underlying f . Also, the trivial locus of X is defined as thesubset (X,M)tr of X .

6.6.4. There is an obvious (forgetful) functor :

F : K → Sch ((X,M), F, ψ) 7→ X

which is a fibration. More precisely, every morphism of schemes S ′ → S induces a base changefunctor (notation of (1.1.16)) :

FK /S → FK /S ′ X 7→ S ′ ×S Xunique up to natural isomorphism of functors. Namely, for X := ((X,M), F, ψ) one lets

S ′ ×S X := (S ′ ×S (X,M), F, ψ′)


where ψ′ := ψ π], and π : S ′ ×S (X,M)→ (X,M) is the natural projection.

Example 6.6.5. (i) Let P be a monoid,R a ring, and set (S, P logS ) := Spec(R,P ) (see (6.2.13)).

The unit of adjunction εP : P → R[P ] determines a unique morphism of sharp monoidal spaces

ψP : Spec(R,P )] → TP := (SpecP )]

(proposition 3.5.6), and we claim that S := (Spec(R,P ), TP , ψP ) is an object of K .The assertion can be checked on the stalks, hence let ξ be a point of S; then ξ corresponds

to a prime ideal pξ ⊂ R[P ], and by inspecting the definitions, we see that ψP (ξ) = ε−1P (pξ) =

qξ := P ∩ pξ ∈ SpecP . Again, a direct inspection shows that the morphism

(logψP )ξ : OTP ,qξ → (P logS )]ξ

is none else than the natural isomorphism P ]qξ

∼→ P logS,ξ /O

×S,ξ deduced from εP and the natural

identificationsP logS,ξ = O×S,ξ ⊗P\qξ P = O×S,ξ ⊗P×qξ Pqξ .

(ii) The construction of S is clearly functorial in P . Namely, say that λ : P → Q is amorphism of monoids, and set S ′ := SpecR[Q], TQ := (SpecQ)]. There follows a morphism

(6.6.6) S ′ := (Spec(R,Q), TQ, ψQ)→ S

in K , whose underlying morphism of log schemes is Spec(R, λ), and whose underlying mor-phism of fans is just (Specλ)].

6.6.7. Example 6.6.5 can be globalized to more general log schemes, at least under some addi-tional assumptions. Namely, let (X,M) be a regular log scheme. For every point x of X , letmx ⊂ OX,x be the maximal ideal; we set :

F (X) := x ∈ X | I(x,M) = mx(notation of (6.5.21)), and we endow F (X) with the topology induced from X . The fan of(X,M) is the sharp monoidal space

F (X,M) := (F (X),M |F (X))].

We wish to show that F (X,M) is indeed a fan. Let U ⊂ X be any open subset; to begin with,it is clear that F (U,M |U) is naturally an open subset of F (X,M); hence the contention is localon X , so we may assume that X is affine, say X = SpecA for a noetherian ring A, and that Madmits a finite chart PX → M ; denote by β : P → A the induced morphism of monoids. Letnow ξ be a point of X , and pξ ⊂ A the corresponding prime ideal; set qξ := β−1pξ ∈ SpecP .By inspecting the definitions, it is easily seen that I(ξ,M) = qξApξ . Therefore, for any primeideal q ⊂ P , let V (q)max be the finite set consisting of all the maximal points of the closedsubset V (q) := SpecA/qA (i.e. the minimal prime ideals of A/qA); in light of corollary6.5.36(i), it follows easily that

(6.6.8) F (X) =⋃

q∈SpecP

V (q)max

especially, F (X) is a finite set (lemma 3.1.20(iii)). Let t ∈ F (X) be any element, and denoteby U(t) ⊂ F (X) the subset of all generizations of t in F (X); as a corollary, we see that U(t)is an open subset of F (X). Moreover, (6.6.8) also implies that :

U(t) =⋃

q∈SpecP

(Spec OX,t/qOX,t)max.

However, corollary 6.5.36(i) says that Spec OX,t/qOX,t is irreducible for every q ∈ SpecP ,therefore the set U(t) is naturally identified with a subset of SpecP . Furthermore, arguing as


in example 6.6.5(i) we find a natural isomorphism P ]qt

∼→ OF (X,M),t. Moreover, if t′ ∈ U(t) isany other point, the specialization map OF (X,M),t → OF (X,M),t′ corresponds – under the aboveisomorphism – to the natural morphism P ]

qt → P ]q′t

induced by the localization map Pqt →Pq′t . This shows that the open monoidal subspace (U(t),M |U(t)) is naturally isomorphic to(SpecPqt)

], hence F (X,M) is a fan, as stated.

Remark 6.6.9. (i) The discussion in (6.6.7) shows more precisely that, if :(a) (U,M) is a log scheme with U = SpecA affine,(b) there exists a morphism β : P → A from a finitely generated monoid P , inducing the

log structure M on U , and(c) β(q)A is a prime ideal for every q ∈ SpecP

then F (U,M) is naturally identified with (SpecP )]. In this situation, denote by

fβ : (U,M)] → TP := (SpecP )]

the morphism of sharp monoidal spaces deduced from β (proposition 3.5.6); by inspecting thedefinitions, we see that the associated map log fβ : f ∗βOTP → M ] is an isomorphism. Via theforegoing natural identification, there results a morphism πU : (U,M)] → F (U,M) which canbe described without reference to P . Indeed, let x ∈ U be any point, and p ∈ SpecA thecorresponding prime ideal; by inspecting the definitions we find that

πU(x) = I(x,M) which is the largest prime ideal in SpecAp ∩ F (U,M).

Especially, πU(x) is a generization of x, and the inverse of (log πU)x : OF (U,M),πU (x)∼→ M ]

x

is induced by the specialization map Mx → MπU (x) (which induces an isomorphism on theassociated sharp quotient monoids).

(ii) Finally, it follows easily from corollary 6.5.36(i) and [32, Ch.IV, Cor.8.4.3], that ev-ery regular log scheme (X,M) admits an affine open covering X =

⋃i∈I Ui, such that each

(Ui,M |Ui) fulfills conditions (a)–(c) above, and the intrinsic description in (i) shows that themorphisms πUi glue to a well defined morphism πX : (X,M)] → F (X,M) of sharp monoidalspaces, such that the datum

K (X,M) := ((X,M), F (X,M), πX)

is an object of K .(iii) Notice that π−1

X (t) is an irreducible locally closed subset of X of codimension equalto the height of t, for every t ∈ F (X,M); also π−1

X (t) is a regular scheme, for its reducedsubscheme structure. Indeed, we have already observed that t is the unique maximal point ofπ−1X (t), and then the assertion follows immediately from corollary 6.5.51(ii,iii). Furthermore,

the inclusion map j : F (X,M)→ X is a continuous section of πX , and notice that jπX(U) =j−1U for every open subset U ⊂ X , since j maps every t ∈ F (X,M) to the unique maximalpoint of π−1

X (t). It follows easily that πX is an open map.

6.6.10. Denote by Kint the full subcategory of K whose objects are the data ((X,M), F, ψ)such that M is an integral log structure, and F is an integral fan. There is an obvious functor

Kint → int.Fan : ((X,M), F, ψ) 7→ F

to the category of integral fans, which shall be used to construct useful morphisms of logschemes, starting from given morphisms of fans. This technique rests on the following threeresults :

Lemma 6.6.11. Let ((X,M), F, ψ) be an object of Kint, with F locally fine. Then, for eachpoint x ∈ X there exists an open neighborhood U ⊂ X of x, a fine chart QU →M |U (for somefine monoid Q depending on x), and an isomorphism of monoids Q] ∼→ OF,ψ(x).


Proof. The assertion is local onX , hence we may assume that F = (SpecP )] for a fine monoidP . In this case, ψ is determined by the corresponding map

β : PX →M ].

Indeed, for any x ∈ X , the point ψ(x) is the prime ideal β−1x (mx) ⊂ P , where mx ⊂M ]

x is themaximal ideal; moreover :

OF,ψ(x) = P/Sx where Sx := β−1x (1)

and – under this identification – the isomorphism logψx : OF,ψ(x)∼→M ]

x is deduced from βx inthe obvious way.

Now, let x ∈ X be any point; after replacing X by the open subset ψ−1U(ψ(x)), we mayassume that P = OF,ψ(x) (notation of (3.5.16)), and by assumption logψx : P → M ]

x is anisomorphism. Pick a surjection α : Z⊕r → P gp, and let Q be the pull-back in the cartesiandiagram :

Q //

Z⊕r

(logψx)gpα

M ]x

// Mgpx /M

×x .

By construction, α restricts to a morphism of monoids ϑ : Q → P , inducing an isomorphismQ] ∼→ P , whence an isomorphisms of fans (SpecP )]

∼→ (SpecQ)], and Q is fine, by corollary3.4.2. Moreover, the choice of a lifting Z⊕r →Mgp

x of (logψx)gp α determines a morphism

βx : Q→M ]x ×Mgp

x /M×xMgp

x = Mx

which lifts logψx ϑ (here it is needed that M is integral). Next, after replacing X by an openneighborhood of x, we may assume that βx extends to a map of pre-log structures β : QX →Mlifting β (lemma 6.1.16(iv.b),(v)). It remains only to show that β is a chart for M , whichcan be checked on the stalks. Thus, let y ∈ X be any point, and set Sy := β−1

y M×y ; with

the foregoing notation, the stalk QlogX,y of the induced log structure is naturally isomorphic to

(S−1y Q× O×X,y)/S

gpy , therefore

(QlogX,y)

] ' Q/Sy ' P/Sy

and the induced map P/Sy → M ]y is again deduced from βy, so it is an isomorphism; then the

same holds for βlogy : Qlog

X,y →My (lemma 6.1.4).

Lemma 6.6.12. Let µ : P → P ′ be a morphism of integral monoids such that µgp is surjective,ϕ : (SpecP ′)] → (SpecP )] the induced morphism of affine fans, and denote by

λ : P → Q := P gp ×P ′gp P ′

the map of monoids determined by µ and the unit of adjunction P → P gp. Then :(i) The natural projection Q → P ′ induces an isomorphism ω : (SpecP ′)] → (SpecQ)]

of affine fans, such that (Specλ)] ω = ϕ.(ii) The induced morphism (6.6.6) in Kint (with R := Z) is int.Fan-cartesian.

(iii) If P and P ′ are fine, the morphism (6.6.6) is etale.

Proof. (See (1.4) for generalities concerning inverse images and cartesian morphisms relativeto a functor.) Notice first that, since µgp is surjective, the projection Q → P ′ induces anisomorphism Q] ∼→ P ′], whence (i).

(ii): Notice that the log structure of Spec(Z, Q) is integral, by lemma 6.1.16(iii). Next, setF ′ := (SpecP ′)], define S, S ′ as in example 6.6.5(ii) (withR := Z), let g : ((Y,N), F ′′, ψY )→S be any morphism of Kint, and ϕ′ : F ′′ → F ′ a morphism of integral fans, such that the


image of g in int.Fan equals ϕ ϕ′; we must show that g factors through a unique morphismh : ((Y,N), F ′′, ψY ) → S ′, whose image in int.Fan equals ϕ′. As usual, we may reduceto the case where Y = SpecA is affine, F ′′ = SpecP ′′ is an affine fan for a sharp integralmonoid P ′′, and ψY is given by a map of sheaves P ′′Y → N ]. In such situation, the morphism(Y,N) → Spec(Z, P ) underlying g is determined by a morphism of monoids P → N(Y ), orwhich is the same, a map of sheaves γY : PY → N ; likewise, ϕ′ is given by a morphism ofmonoids P ′ → P ′′, and composing with ψY , we get a map of sheaves αY : P ′Y → N ]. Finally,the condition that g lies over ϕ ϕ′ translates as the commutativity of the following diagram ofsheaves :

P gpY

//

γgpY

P ′gpY

P ′Yoo

αY

Ngp // Ngp/N× N ].oo

There follows a unique morphism of sheaves :

(6.6.13) QY → Ngp ×Ngp/N× N] = N

(here it is needed that N is integral) such that the diagram :

PY //

γY

QY//

P ′Y

αY

N N // N ]

commutes. Then (6.6.13) determines a morphism of log schemes hY : (Y,N) → Spec(Z, Q),such that the pair (hY , ϕ

′) is the unique morphism h in Kint with the sought properties.(iii): If both P and P ′ are fine, so is Q (corollary 3.4.2), and by construction, λgp is an

isomorphism. Then the assertion follows from theorem 6.3.37.

Proposition 6.6.14. Let X := ((X,M), F, ψ) be an object of Kint. Let also ϕ : F ′ → F be anintegral partial subdivision, with F locally fine and F ′ integral. We have :

(i) (X,M) is a fine log scheme.(ii) If F is saturated, (X,M) is a fs log scheme.

(iii) X admits an inverse image over ϕ (relative to the functor of (6.6.10)).(iv) If ϕ is finite, then the cartesian morphism (f, ϕ) : ϕ∗X → X , is quasi-compact.(v) If F ′ is locally fine, the morphism (f, ϕ) is quasi-separated and etale.

Proof. (i): This is just a restatement of lemma 6.6.11.(ii): In light of (i), we only have to show that Mx is a saturated monoid, for every x ∈ X .

Since by assumption M ]x is saturated, the assertion follows from lemma 3.2.9(ii).

Claim 6.6.15. In order to show (iii)–(v), we may assume that :(a) F = (SpecP )] for a fine monoid P , and X is an affine scheme.(b) The map of global sections P → Γ(X,M ]) determined by ψ, comes from a morphism

of pre-log structures β : PX →M which is a fine chart for M .

Proof of the claim. To begin with, suppose that ((X ′,M ′), F ′, ψ′) is the sought preimage of X;let U ⊂ X be any open subset, and V ⊂ F an open subset such that ψ(U) ⊂ V . Then it iseasily seen that the object

ϕ∗X ×X (U, V ) := ((f−1U,M ′|f−1U), ϕ−1V, ψ′|f−1U)

is a preimage of ((U,M |U), V, ψ|U) over the restriction ϕ−1V → V of ϕ. Now, suppose thatwe have found an affine open covering X =

⋃i∈I Ui, and for every i ∈ I an affine open subset


Vi ⊂ F with ψ(Ui) ⊂ Vi, such that the object U i := ((Ui,M |Ui), Vi, ψ|Ui) admits a preimageover the restriction ϕi : ϕ−1Vi → Vi of ϕ; then the foregoing implies that there are naturalisomorphisms:

ϕ∗iU i ×U i (Uij, Vij)∼→ ϕ∗jU j ×Uj (Uij, Vij)

for every i, j ∈ I , where Uij := Ui ∩Uj and Vij := Vi ∩Vj . Thus, we may glue all these inverseimages along these isomorphisms, to obtain the sought inverse image of X .

Moreover, if ϕ is finite, the same will hold for the restrictions ϕi, and if each cartesian mor-phism ϕ∗iU i → U i is quasi-compact, the same will hold also for (f, ϕ) ([25, Ch.I, §6.1]).Likewise, if each morphism ϕ∗iU i → U i is etale and quasi-separated, then the same will holdfor (f, ϕ) ([25, Ch.I, Prop.6.1.11] and proposition 6.3.24(iii)).

Therefore, we may replace X by any Ui, and F by the corresponding Vi, which reduces theproof of (iii)–(v) to the case where condition (a) is fulfilled. Lastly, in light of lemma 6.6.11 wemay suppose that the open subsets Ui are small enough, so that also condition (b) is fulfilled. ♦

In view of claim 6.6.15, we shall assume henceforth that conditions (a) and (b) are fulfilled.Let S := (Spec(Z, P ), TP , ψP ) be the object of Kint considered in example 6.6.5(i) (withR := Z); in this situation, β determines a morphism of schemes fβ : X → S, and in view oflemma 6.2.14 we haveX ' X×SS (notation of (6.6.4)). Therefore, if we find an inverse imageS ′ for S over ϕ, the object X ×S S ′ will provide the sought inverse image of X . Thus, in orderto show (iii)–(v), we may further reduce to the case where X = S ([25, Ch.I, Prop.6.1.5(iii),Prop.6.1.9(iii)] and proposition 6.3.24(ii)).

Claim 6.6.16. In order to prove (iii)–(v), we may assume that F ′ is affine.

Proof of the claim. Indeed, say that F ′ =⋃i∈I Vi is an open covering, and let ϕi : Vi → TP be

the restriction of ϕ; suppose that we have found, for each i ∈ I , an inverse image ϕ∗iS of S overϕi; again, it follows easily that an inverse image for S over ϕ can be constructed by gluing theobjects ϕ∗iS. This already implies that, in order to show (iii), we may assume that F ′ is affine.

Now, suppose that ϕ is finite, so we may find an open covering as above, such that further-more each Vi is affine, and I is a finite set. Suppose that each of the corresponding morphismsϕ∗iS → S is quasi-compact; by the foregoing, the schemes underlying the objects ϕ∗iS give afinite open covering of the scheme underlying ϕ∗S, and then it is clear that (iv) holds.

Next, suppose furthermore that each morphism ϕ∗iS → S is etale. It follows that themorphism ϕ∗S → S is also etale (proposition 6.3.24(ii)); then, since S is noetherian, thescheme underlying ϕ∗S is locally noetherian ([25, Ch.I, Prop.6.2.2]), and therefore the mor-phism ϕ∗S → S is quasi-separated ([25, Ch.I, Cor.6.1.13]). ♦

In view of claim 6.6.16, we shall assume that F ′ = (SpecP ′)] is affine as well, for a sharpand integral P ′, and that ϕ is given by a morphism λ : P → P ′ inducing a surjection on the as-sociated groups (details left to the reader). Then assertions (iii) and (iv) are now straightforwardconsequences of lemma 6.6.12(i,ii), and (v) follows from lemma 6.6.12(iii), after one remarksthat, when F ′ is fine, one may choose for P ′ a fine monoid.

Remark 6.6.17. Keep the assumptions of proposition 6.6.14, and let t ∈ F0 a point of F ofheight zero (see (3.5.16)). Notice that the inclusion map jt : t → F is an open immersion,hence the fibre Xt := ψ−1(t) ⊂ X is open; indeed, it is clear from the definitions, that ψ−1(F0)is precisely the trivial locus of X . Moreover, let t′ ∈ ϕ−1(t) since the group homomorphismOgpF,t → Ogp

F ′,t′ is surjective, we see that t′ is of height zero in F ′, and ϕ restricts to an isomor-phism of fans (t′,OF,t′)

∼→ (t,OF,t). Set X t := j∗tX (whose underlying scheme is Xt), anddefine likewise ϕ∗X t′ := (ϕ jt′)∗X (whose underlying scheme is an open subset of the triviallocus of ϕ∗X). We deduce that :

• The trivial locus of ϕ∗X is the preimage of (X,M)tr.


• For every t′ ∈ F ′0, the restriction of (f, ϕ) : ϕ∗X t′ → X t is an isomorphism.

Corollary 6.6.18. In the situation of proposition 6.6.14, let ((X ′,M ′), F ′, ψ′) := ϕ∗X . Thefollowing holds :

(i) If F ′ = F sat, and ϕ : F sat → F is the counit of adjunction, then (X ′,M ′) = (X,M)fs,and f is the counit of adjunction.

(ii) If ϕ is the blow up of a coherent ideal I of OF , then f is the blow up of IM , theunique ideal of M whose image in M ] equals ψ−1I .

(iii) Suppose moreover, that (X,M) is regular, F ′ is saturated , and X = K (X,M)(notation of remark 6.6.9(iii)). Then (X ′,M ′) is regular, and ϕ∗X is isomorphic toK (X ′,M ′).

Proof. To start with, we remark that the assertions are local on X . Indeed, this is clear for (i),and for (iii) it follows easily from remark 6.6.9(i,,ii); concerning (ii), suppose that X =

⋃i∈I Ui

is an open covering, such that ϕ∗(Ui,M |Ui) is the blow up of the ideal IM |Ui , for every i ∈ I .For every i, j ∈ I set Uij := Ui ∩ Uj; by the universal property of the blow up, there are uniqueisomorphisms of (Uij,M |Uij)-schemes :

Uij ×Ui ϕ∗(Ui,M |Ui)∼→ Uij ×Uj ϕ∗(Uj,M |Uj).

Then both ϕ∗(X,M) and the blow up of IM are necessarily obtained by gluing along theseisomorphisms, so they are isomorphic.

Thus, we may assume that F = (SpecP )] for some fine monoid P , and (X,M) = X ×S(S, P log

S ) for some morphism of schemes X → S, where as usual (S, P logS , F, ψP ) is defined

as in example 6.6.5(i). In this case, (i) follows from remarks 6.2.36(iii), 3.5.8(ii) and lemma6.6.12(ii).

Likewise, remark 6.4.62(ii) allows to reduce (ii) to the case where X = (S, P logS , F, ψP ), and

I = I∼ for some ideal I ⊂ P ; in which case we conclude by inspecting the explicit descriptionin example 6.4.73.

(iii): From proposition 6.6.14(ii,v) and theorem 6.5.44 we already see that (X ′,M ′) is regular.Next, we are easily reduced to the case whereX is affine, sayX = SpecA, and F ′ = (SpecQ)],for some saturated monoid Q, and by lemma 6.6.12, we may assume that ϕ is induced by amorphism of monoids λ : P → Q such that λgp is an isomorphism, and (X ′,M ′) = ϕ∗X =X ×S S ′, where S ′ := (Spec(Z, Q), F ′, ψQ) is defined as in example 6.6.5(ii). Let q ⊂ Q beany prime ideal, and set p := q ∩ P ; in light of remark 6.6.9(i), it then suffices to show thatQ/q⊗P A = Q/q⊗P/p A/pA is an integral domain. To this aim, let us remark :

Claim 6.6.19. Let λ : P → Q be a morphism of fine monoids, such that λgp is an isomorphism,F ⊂ Q any face, and denote λF : F ∩ P → P the inclusion map. Then :

(i) The natural map CokerλgpF → P gp/(F ∩ P )gp is injective.

(ii) If moreover, P is saturated, then CokerλgpF is a free abelian group of finite rank.

Proof of the claim. The map of (i) is obtained via the snake lemma, applied to the ladder ofabelian groups :

0 // (F ∩ P )gp //

λgpF

P gp //

λgp

P gp/(F ∩ P )gp //

0

0 // F gp // Qgp // Qgp/F gp // 0.

taking into account that both Kerλgp and Cokerλgp vanish. Then (i) is obvious, and (ii) comesdown to checking that P gp/(F ∩ P )gp is torsion-free, in case P is fine and saturated. But sinceF ∩ P is a face of P , the latter assertion is an easy consequence of proposition 3.4.7. ♦


Now, set F := Q \ q; we come down to checking that B := F ⊗F∩P A/pA is an integraldomain, and remark 6.6.9(i) tells us that A/pA is a domain. However, A/pA is (F ∩ P )-flat(proposition 6.5.32 and lemma 6.5.17), hence the natural map

B → C := F gp ⊗(F∩P )gp Frac(A/pA)

is injective; on the other hand, claim 6.6.19(ii) implies that C = A[CokerλgpF ] is a free (poly-

nomial) A-algebra, whence the contention.

Example 6.6.20. In the situation of example 3.5.27, take T := SpecP for a fine monoid P ,and define S as in example 6.6.5(i). The k-Frobenius map kP (example 3.5.10(i)) induces anendomorphism kS := (Spec(R,kP ),kT ) of S in K . By proposition 6.6.14(iii) and example3.5.27, there exists a unique morphism g fitting into a commutative diagram of Kint :

(6.6.21)

ϕ∗S //

g

S

kS

ϕ∗S // S

(where the horizontal arrows are the cartesian morphisms). Say that ϕ∗S = ((Y,N), F, ψ);then we have g = (g,kF ) for a unique endomorphism g := (g, log g) of the log scheme (Y,N).Let U := SpecP ′ ⊂ F be any open affine subset ; since kF is the identity on the underlyingtopological spaces, g restricts to an endomorphism g|ψ−1U of ψ−1U×Y (Y,N). In view of lemma6.6.12, the latter log scheme is of the form S ′ as in example 6.6.5(ii), with Q := P gp ×P ′gp P ′.Then g|ψ−1U is induced by an endomorphism ν of Q, fitting into a commutative diagram :

P //

kP

Q

ν

P // Q

whose horizontal arrows are the natural injections. Since Q ⊂ P gp, it is clear that ν = kQ.Especially, g : Y → Y is a finite morphism of schemes. Furthermore, for every point y ∈ Y ,we have a commutative diagram of monoids :

(6.6.22)

N g(y)

log gy

// N ]g(y)

log g]y

OF,ψ(y)∼oo

kψ(y):=(logkF )ψ(y)

Ny

// N ]y OF,ψ(y).

∼oo

6.6.23. Let (K, | · |) be a valued field, Γ and K+ respectively the value group and the valuationring of | · |; denote by 1 ∈ Γ the neutral element, and by Γ+ ⊂ Γ the submonoid consisting ofall elements ≤ 1. Set S := SpecK+; we consider the log scheme (S,O∗S ), where O∗S ⊂ OS isthe subsheaf such that O∗S (U) := OS(U)\0 for every open subset U ⊂ S. To this log schemewe associate the object of Kint :

f(K, | · |) := ((S,O∗S ), Spec Γ+, ψΓ)

where ψΓ is the morphism of monoidal spaces arising from the isomorphism

Γ+∼→ (K+\0)/(K+)×

deduced from the valuation | · |. It is well known that ψΓ is a homeomorphism (see e.g. [36,§6.1.26]). We shall use objects of this kind to state some separation and properness criteriafor log schemes whose existence is established via proposition 6.6.14. To this aim, we need to


digress a little, to prove the following auxiliary results, which are refinements of the standardvaluative criteria for morphisms of schemes.

6.6.24. Let f : X → Y be a morphism of schemes, R an integral ring, and K the field offractions of R. Let us denote by X(R)max ⊂ X(R) the set of morphisms SpecR → X whichmap SpecK to a maximal point of X . There follows a commutative diagram of sets :

(6.6.25)

X(R)max//

X(K)max

Y (R) // Y (K).

Proposition 6.6.26. Let f : X → Y be a morphism of schemes. The following conditions areequivalent :

(a) f is separated.(b) f is quasi-separated, and for every valued field (K, | · |), the map of sets :

(6.6.27) X(K+)max → X(K)max ×Y (K) X(K+)max

deduced from diagram (6.6.25) (with R := K+, the valuation ring of | · |), is injective.

Proof. (a)⇒ (b) by the valuative criterion of separation ([25, Ch.I, Prop.5.5.4]). Conversely, weshall show that if (b) holds, then the assumptions of the criterion of loc.cit. are fulfilled. Indeed,let (L, | · |L) be any valued field, with valuation ring L+, and suppose we have two morphismsσ1, σ2 : SpecL+ → X whose restrictions to SpecL agree, and such that f σ1 = f σ2.

Let s, η ∈ SpecL+ be respectively the closed point and the generic point, set x := σ1(η) =σ2(η) ∈ X , and denote by ϕ : OX,x → L the ring homomorphism corresponding to the restric-tion of σ1 (and σ2); then x admits two specializations xi := σi(s) ∈ X (for i = 1, 2) such that ϕsends the image Ai ⊂ OX,x of the specialization map OX,xi → OX,x, into L+, and the maximalideal of Ai into the maximal ideal mL of L+, for both i = 1, 2. Moreover, y := f(x1) = f(x2).

Denote by B ⊂ OX,x the smallest subring containing A1 and A2, and set p := B ∩ ϕ−1mL.Then ϕ(Bp) ⊂ L+ as well, and Bp dominates both A1 and A2. Now, let t ∈ X be a maximalpoint which specializes to x; by [25, Ch.I, Prop.5.5.2] we may find a valued field (E, | · |E)with a local ring homomorphism OX,t → E, such that the valuation ring E+ of | · |E dominatesthe image of the specialization map OX,x → OX,t. Let κ(E) be the residue field of E+, andBp ⊂ κ(E) the image of Bp. By the same token, we may find a valuation ring V ⊂ κ(E) withfraction field κ(E), and dominating Bp; then it is easily seen that the preimage K+ ⊂ E+ ofV is a valuation ring with field of fractions K = E, which dominates the image of Bp in OX,t([61, Th.10.1(iv)]). Hence, K+ dominates the images of OX,xi , for both i = 1, 2, and therefore,also the image of OY,y; in other words, in this way we obtain two elements in X(K+)max whoseimages agree in Y (K+), and whose restrictions SpecK → X coincide. By assumption, thesetwo K+-points must then coincide, especially x1 = x2, and therefore σ1 = σ2, as required.

Proposition 6.6.28. Let f : X → Y be a quasi-compact morphism of schemes. The followingconditions are equivalent :

(a) f is universally closed.(b) For every valued field (K, | · |), the corresponding map (6.6.27) is surjective.

Proof. (a)⇒ (b) by the valuative criterion of [25, Ch.I, Prop.5.5.8]. Conversely, we will showthat (b) implies that the conditions of loc.cit. are fulfilled. Hence, let (L, | · |L) be any valuedfield, with valuation ring L+, and suppose that we have a morphism σ : SpecL → X , whosecomposition with f extends to a morphism SpecL+ → Y ; we have to show that σ extendsto a morphism SpecL+ → X . Denote by η, s ∈ SpecL+ respectively the generic and the


closed point, let x ∈ X be the image of η, and y ∈ Y the image of s. Then σ corresponds toa ring homomorphism σ\ : OX,x → L, and L+ dominates the image of the map OY,y → OX,xdetermined by f . Moreover, σ\ factors through the residue field κ(x) of OX,x. Then κ(x)+ :=κ(x)∩L+ is a valuation ring with fraction field κ(x), and we are reduced to showing that thereexists a specialization x′ ∈ X of x, such that f(x′) = y, and such that κ(x)+ dominates theimage of the specialization map OX,x′ → OX,x.

Let now t ∈ X be a maximal point which specializes to x; by [25, Ch.I, Prop.5.5.8] we mayfind a valued field (E, | · |E) with a local ring homomorphism OX,t → E, whose valuation ringE+ dominates the image of the specialization map OX,x → OX,t. Let κ(E) be the residue fieldof E+; the induced map OX,x → κ(E) factors through κ(x), and by [61, Th.10.2] we mayfind a valuation ring V ⊂ κ(E) with field of fractions κ(E), which dominates κ(x)+. Thepreimage K+ ⊂ E+ of V is a valuation ring with field of fractions K = E ([61, Th.10.1]).By construction, K+ dominates the image of OX,y, in which case assumption (b) says that thereexists a specialization x′ of x such that f(x′) = y, and such that K+ dominates the image ofthe specialization map OX,x′ → OX,t. A simple inspection then shows that κ(x)+ dominates theimage of OX,x′ in OX,x, as required.

Corollary 6.6.29. Let f : X → Y be a morphism of schemes which is quasi-separated and offinite type. Then the following conditions are equivalent :

(a) f is proper.(b) For every valued field (K, | · |), the corresponding diagram (6.6.25) (with R := K+ the

valuation ring of | · |) is cartesian.

Proof. It is immediate from propositions 6.6.26 and 6.6.28.

6.6.30. We are now ready to return to log schemes. Resume the situation of (6.6.23), let X :=((X,M), F, ψ) be any object of K , and denote by α : M → OX the structure map of M .

Suppose we are given a morphism σ : S → X of schemes, and we ask whether there existsa morphism of log structures β : σ∗M → O∗S , such that the pair (σ, β) is a morphism of logschemes (S,O∗S )→ (X,M). By definition, this holds if and only if the composition

β : σ∗Mσ∗α−−→ σ∗OX

σ\−→ OS

factors through O∗S . Moreover, in this case the factorization is unique, so that σ determines βuniquely. Let η ∈ S be the generic point; we claim that the stated condition is fulfilled, if andonly if t := σ(η) lies in (X,M)tr (see definition 6.2.7(i)). Indeed, if t lies in the trivial locus,M t = O×X,t, so certainly the image of βt : M t → OS,η lies in O∗S,η = K×. Now, if s ∈ S is anyother point, the composition

Mσ(s)

βσ(s)−−−→ OS,s → OS,η = K

factors through M t, hence its image lies in K× ∩ OS,s = O∗S,s, whence the contention. Con-versely, if β factors through O∗S , it follows especially that the image of the stalk of the structuremap M t → OX,t lies in the preimage of O∗S,η = O×S,η, and the latter is just O×X,t, so t lies in thetrivial locus.

Next, let f : (S,O∗S ) → (X,M) be any morphism of log schemes. We claim that thereexists a unique morphism of fans ϕ := (ϕ, logϕ) : Spec Γ+ → F , such that the pair (f, ϕ) is amorphism f(K, | · |) → X in K . Indeed, since ψΓ is a homeomorphism, there exists a uniquecontinuous map ϕ on the underlying topological spaces, such that ϕ ψΓ = ψ f , and forthe same reason, the map log f ] : f ∗M/O×S → O∗S /O

×S is of the form ψ∗Γ(logϕ) for a unique

morphism of sheaves logϕ as sought. Then logϕ will be a local morphism, since the sameholds for log f (see (6.6.1)).


Summing up, we have shown that the natural map

HomK (f(K, | · |), X)→ X(K+) : (σ, ϕ) 7→ σ

is injective, and its image is the set of all the morphisms S → X which map η into (X,M)tr.

Proposition 6.6.31. LetX := ((X,M), F, ψ) be an object of Kint, and ϕ : F ′ → F an integralpartial subdivision, with F and F ′ locally fine. We have :

(i) If the induced map F ′(N) → F (N) is injective, the cartesian morphism ϕ∗X → X isseparated.

(ii) If ϕ is a proper subdivision, the cartesian morphism ϕ∗X → X is proper, and inducesan isomorphism of schemes (ϕ∗X)tr

∼→ (X,M)tr.

Proof. The assertions are local on X (cp. the proof of claim 6.6.15), so we may assume –by lemma 6.6.11 – that M admits a chart PX → M , and F = (SpecP )]. In this case, letS := SpecZ[P ], and denote by S the object of Kint attached to P , as in example 6.6.5(i)(with R := Z); then (X,M) is isomorphic to X ×S (S, P log

S ), and if fS : ϕ∗S → S is thecartesian morphism over ϕ, then the cartesian morphism f : ϕ∗X → X is given by the pair(1X×SfS, ϕ) (see the proof of proposition 6.6.14(iii)). Thus, we may replaceX by S ([25, Ch.I,Prop.5.3.1(iv)]), in which case lemma 6.6.12 shows that the log scheme (X ′,M ′) underlyingϕ∗S admits an open covering consisting of affine log schemes of the form Spec(Z, Q), for afine monoid Q. Notice that, for such Q we have :

Spec(Z, Q)tr = Z[Qgp]

which is a dense open subset of SpecZ[Q].(i): According to proposition 6.6.14(v), the morphism f is quasi-separated, so we may apply

the criterion of proposition 6.6.26. Indeed, let (K, | · |) be any valued field, and suppose thatσi ∈ X ′(K+)max (for i = 1, 2) are two sections, whose images in X ′(K)max ×S(K) S(K+)coincide; we have to show that σ1 = σ2. However, we have just seen that the maximal pointsof X ′ lie in (X ′,M ′)tr, so the discussion of (6.6.30) shows that both σi extend uniquely tomorphisms σ′i : f(K, | · |) → ϕ∗S in K , and it suffices to show that σ′1 = σ′2. By definition,the datum of σ′i is equivalent to the datum of a morphism σ′′i : f(K, | · |)→ S, and a morphismof fans ϕ′i : Spec Γ+ → F ′. Again by (6.6.30), the morphisms σ′′1 and σ′′2 agree if and only ifthey induce the same morphisms of schemes; the latter holds by assumption, since σ1 and σ2

yield the same element of S(K+). On the other hand, in view of (b) and proposition 3.5.24, theelements ϕ1, ϕ2 ∈ F ′(Γ+) coincide if and only if their images in F (Γ+) coincide; but again, thislast condition holds since the images of both σi agree in S(K+).

(ii): In view of (i) and proposition 6.6.14(iv),(v) we know already that f is separated andof finite type, so we may apply the criterion of corollary 6.6.29. Hence, let σ ∈ S(K+) be asection, and x ∈ X ′(K)max aK-rational point such that f(x) is the image of σ in S(K); in viewof (i), it suffices to show that σ lifts to a section σ ∈ X ′(K+)max, whose image in X ′(K) is x.Since x ∈ (X ′,M ′)tr, remark 6.6.17 implies that f(x) ∈ (S, P log

S )tr, and then the discussionin (6.6.30) says that σ underlies a unique morphism (σ′, β) : f(K, | · |) → S. By proposition3.5.26, the element β ∈ F (Γ+) lifts to an element β′ ∈ F ′(Γ+), and finally the pair ((σ, β), β′)determines a unique morphism f(K, | · |) → ϕ∗S, whose underlying morphism of schemes isthe sought σ. Lastly, notice that the map F ′(1) → F (1) induced by ϕ, is bijective bypropositions 3.5.24 and 3.5.26 (where 1 is the monoid with one element), which means thatϕ restricts to a bijection on the points of height zero; then remark 6.6.17 implies the secondassertion of (ii).

Theorem 6.6.32. Let (X,M) be a regular log scheme. Then there exists a smooth morphism oflog schemes f : (X ′,M ′) → (X ′,M), whose underlying morphism of schemes is proper and


birational, and such that X ′ is a regular scheme. More precisely, f restricts to an isomorphismof schemes f−1Xreg → Xreg on the preimage of the open locus of regular points of X .

Proof. We use the object X := ((X,M), F (X,M), πX) attached to (X,M) as in remark6.6.9(ii). Indeed, it is clear that F (X,M) is locally fine and saturated, hence theorem 3.6.31yields an integral, proper, simplicial subdivision ϕ : F ′ → F (X,M) which restricts to anisomorphism ϕ−1F (X,M)sim

∼→ F (X,M)sim. Take (f, ϕ) : ϕ∗X → X to be the cartesianmorphism over ϕ, and denote by (X ′,M ′) the log scheme underlying ϕ∗X; it follows alreadyfrom proposition 6.6.31(ii) that f is proper on the underlying schemes. Next, corollary 6.5.35shows that Xreg is π−1F (X,M)sim, so f restricts to an isomorphism f−1Xreg

∼→ Xreg. Further-more, f is etale, by proposition 6.6.14(v), hence the log scheme (X ′,M ′) is regular (theorem6.5.44). Finally, again by corollary 6.5.28 we see that X ′ is regular.

6.6.33. Let now (Y,N) be a regular log scheme, such that F (Y,N) is affine (notation of remark6.6.9(ii)), say isomorphic to (SpecP )], for some fine, sharp and saturated monoid P . Let I ⊂ Pbe an ideal generated by two elements a, b ∈ P , and denote by f : (Y ′, N ′) → (Y,N) thesaturated blow up of the ideal IN of N (see (6.4.79)). Set U ′ := (Y ′, N ′)tr, U := (Y,N)tr, anddenote j : U → Y , j′ : U ′ → Y ′ the open immersions. In this situation we have :

Lemma 6.6.34. (i) H1(Y ′, N ′]gp) = 0.(ii) Suppose morever, that R1j′∗O

×U ′ = 0. Then R1j∗O

×U = 0.

Proof. (i): Since N ′]gp = π∗XOgpF (Y ′,N ′)

, claim 6.6.39(ii) and remark 6.6.9(iii) reduce to showing

(6.6.35) H1(F (Y ′, N ′),OgpF (Y ′,N ′)

) = 0.

However, by corollary 6.6.18(iii), the fan F (Y ′, N ′) is the saturated blow up of the idealIOF (Y,N) of OF (Y,N), hence it admits the affine covering

F (Y ′, N ′) = (SpecP [a−1b]sat)] ∪ (SpecP [b−1a]sat)].

Notice now that every affine fan is a local topological space, hence the left hand-side of (6.6.35)can be computed as the Cech cohomology of Ogp

F (Y ′,N ′)relative to this covering ([39, II, 5.9.2]).

However, the intersection of the two open subsets is (SpecP [a−1b, b−1a])], and clearly therestriction map

H0(SpecP [a−1b]sat,OgpF (Y ′,N ′)

)→ H0(SpecP [a−1b, b−1a],OgpF (Y ′,N ′)

)

is surjective. The assertion is an immediate consequence.(ii): Let y ∈ Y be any point; according to (6.4.80), the morphism

f ×Y Y (y) : (Y ′, N ′)×Y Y (y)→ (Y (y), N(y))

is the saturated blow up of the ideal IN(y); on the other hand, let U(y) := U ×Y Y (y),U ′(y) := U ′ ×Y Y (y) and denote by jy : U(y) → Y (y) and j′y : U ′(y) → Y ′ ×Y Y (y)

the open immersions; in light of proposition 5.1.15(ii), it suffices to show that R1jy∗OU(y) = 0,and the assumption implies that R1j′y∗OU ′(y) = 0. Summing up, we may replace Y be Y (y),and assume from start that Y is local, and y is its closed point. From the assumption we get :

H1(Y ′, j′∗O×U ′) = H1(U ′,O×Y ′) = PicU ′.

On the other hand, recall that N ′]gp = j′∗O×U ′/O

×Y ′ (proposition 6.5.52); combining with (i), we

deduce that the natural mapPicY ′ → PicU ′

is surjective. Set Y ′0 := f−1(y) ⊂ Y ′, endow Y ′0 with its reduced subscheme structure, and leti : Y ′0 → Y ′ be the closed immersion. If I is an invertible ideal of P , then f is an isomorphism,in which case the assertion is obvious. We may then assume that I is not invertible, in which


case claim 6.4.86(ii) says that there exists a morphism of (Y,N)-schemes h : (Y ′, N ′)→ P1(Y,N)

inducing an isomorphism of κ(y)-schemes

(6.6.36) (h×Y Specκ(y))red : Y ′0∼→ P1

κ(y).

Let us remark :

Claim 6.6.37. Let S be a noetherian local scheme, s the closed point of S, and f : X → S aproper morphism of schemes. Suppose that dimX(s) ≤ 1, and H1(X(s),OX(s)) = 0. Thenthe natural map

PicX → PicX(s)

is injective.

Proof of the claim. Say that S = SpecA for a local ring A, and denote by mA ⊂ A the maximalideal. For every k ∈ N, set Sn := SpecA/mk+1

A , and let in : Xn := X ×S Sn → X be theclosed immersion. Let L be any invertible OX-module, and suppose that i∗0L ' OX0; we haveto show that L ' OX . We notice that, for every k ∈ N, the natural map

H0(Xk+1, i∗k+1L )→ H0(Xk, i

∗kL )

is surjective : indeed, its cokernel is an A-submodule of H1(X0,mkAL /mk+1

A L ), and sincedimX0 ≤ 1, the natural map

(mkA/m

k+1A )⊗κ(s) H

1(X0, i∗0L )

∼→ H1(X0, (mkA/m

k+1A )⊗κ(s) L )→ H1(X0,m

kAL /mk+1

A L )

is surjective; on the other hand, our assumptions imply that H1(X0, i∗0L ) = 0, whence the

contention. Let A∧ be the mA-adic completion of A; taking into account [28, Ch.III, Th.4.1.5],we deduce that the natural map

H0(X,L )⊗A A∧ → H := H0(X0, i∗L )

is a continuous surjection, for the mA-adic topologies. Since H is a discrete space for thistopology, and since the image of H0(X,L ) is dense in the mA-adic topology of H0(X,L )⊗AA∧, we conclude that the restriction map H0(X,L ) → H is surjective as well. Let s ∈ H bea global section of i∗0L whose image in i∗0Lx is a generator of the latter A0-module, for everyx ∈ X0, and pick s ∈ H0(X,L ) whose image in H equals s. It remains only to check that, forevery x ∈ X , the A-module Lx is generated by the image sx of s. However, since X is proper,every x ∈ X specializes to a point of X0, hence we may assume that x ∈ X0, in which case oneconcludes easily, by appealing to Nakayama’s lemma (details left to the reader). ♦

Combining claim 6.6.37 and [58, Prop.11.1(i)], we see that the induced map

i∗ : PicY ′ → PicY ′0

is injective. Let now L be any invertible OU -module; we have to show that L extends to aninvertible OY -module (which is then isomorphic to OY ). However, notice that f restricts toan isomorphism g : U ′

∼→ U , hence g∗L is an invertible OU ′-module, and by the foregoingthere exists an invertible OY ′-module L ′ such that L ′

|U ′ ' g∗L . In light of the isomorphism(6.6.36), there exists an invertible OP1

Y-module L ′′ such that i∗L ′ ' i∗h∗L ′′. Therefore

L ′ ' h∗L ′′.

Now, on the one hand, claim 6.6.37 implies that L ′′ = OP1Y

(n) for some n ∈ N; on theother hand, (6.4.47) implies that h restricts to a morphism of schemes htr : U ′ → Gm,Y ,and L ′′

|Gm,Y = OP1Y

(n)|Gm,Y = OGm,Y , so finally g∗L = OU ′ , hence L = OU , whence thecontention.

The following result complements proposition 6.5.52.


Theorem 6.6.38. Let (X,M) be any regular log scheme, set U :=: (X,M)tr and denote byj : U → X the open immersion. Then we have :

R1j∗O×U = 0.

Proof. We begin with the following general :

Claim 6.6.39. Let π : T1 → T2 be a continuous open and surjective map of topological spaces,such that π−1(t) is an irreducible topological space (with the subspace topology) for everyt ∈ T2. Then, we have :

(i) For every sheaf S on T2, the natural map S → π∗π∗S is an isomorphism.

(ii) For every abelian sheaf S on T2, the natural map S[0] → Rπ∗π∗S is an isomorphism

in D(ZT2-Mod).

Proof of the claim. (i): Since π is open, π∗S is the sheaf associated to the presheaf : U 7→S(πU), for every open subset U of T1. We show, more precisely, that this presheaf is alreadya sheaf; since π is surjective, the claim shall follow immediately. Now, let U ⊂ T1 be an opensubset, and (Ui | i ∈ I) a family of open subsets of X covering U ; for every i, j ∈ I , setUij := Ui ∩ Uj . It suffices to show that S(πU) is the equalizer of the two maps :∏

i∈I S(πUi)// //∏

i,j∈I S(πUij).

Since S is a sheaf, the latter will hold, provided we know that πUi ∩ πUj = πUij for everyi, j ∈ I . Hence, let t ∈ πUi ∩ πUj; this means that π−1(t) ∩ Ui 6= ∅ and π−1(t) ∩ Uj 6= ∅.Since π−1(t) is irreducible, we deduce that π−1(t) ∩ Uij 6= ∅, as required.

(ii): The proof of (i) also shows that, for every flabby abelian sheaf J on T2, the abelian sheafπ∗J is flabby on T1. Hence, if S is any abelian sheaf on T2, we obtain a flabby resolution ofπ∗S of the form π∗J•, by taking a flabby resolution S → J• of S on T2. According to lemma5.1.3, there is a natural isomorphism

π∗π∗J

∼→ Rπ∗π∗S

in D(ZT2-Mod). Then the assertion follows from (i). ♦

After these preliminaries, let us return to the log scheme (X,M), and its associated objectX := ((X,M), F (X,M), πX). The assertion to prove is local on X , hence we may assumethat F (X,M) = (SpecP )], for some sharp, fine and saturated monoid P . Next, by theorem6.6.32 (and its proof) there exists an integral proper simplicial subdivision ϕ : F ′ → F (X,M),such that the log scheme (X ′,M ′) underlying ϕ∗X is regular, X ′ is regular, and the morphismX ′ → X restricts to an isomorphism U ′ := (X ′,M ′)tr → U . In this situation, we mayfind a further subdivision ϕ′ : F ′′ → F ′ such that both ϕ′ and ϕ ϕ′ are compositions ofsaturated blow up of ideals generated by at most two elements of P (example 3.6.15(iii)). Saythat ϕ ϕ′ = ϕr ϕr−1 · · · ϕ1, where each ϕi is a saturated blow up of the above type. Byproposition 6.6.14(iii), we deduce a sequence of morphisms of log schemes

(X1,M1)g1−−→ · · · → (Xr−1,M r−1)

gr−1−−−→ (Xr,M r)gr−−→ (Xr+1,M r+1) := (X,M)

each of which is the blow up of a corresponding ideal, and by the same token, ϕ′ induces amorphism g : (X1,M1) → (X ′,M ′) of (X,M)-schemes. For every i = 1, · · · , r + 1, setUi := (Xi,M i)tr, and let ji : Ui → Xi be the open immersion; especially, Ur+1 = U . We shallshow, by induction on i, that

(6.6.40) R1ji∗O×Ui

= 0 for i = 1, . . . , r + 1.

Notice first that the stated vanishing translates the following assertion. For every x ∈ Xi andevery invertible OUi-module L , there exists an open neighborhood Ux of x in Xi such that


L|Ux∩Ui extends to an invertible OUx-module. However, it follows immediately from proposi-tions 5.6.7 and 5.6.14, that every invertible OU ′-module extends to an invertible OX′-module.Since g restricts to an isomorphism U1 = g−1U ′

∼→ U ′, we easily deduce that every invert-ible OU1-module extends to an invertible OX1-module (namely : if L is such a module, extendg|U1∗L to an invertible OX′-module, and pull the extension back to X1, via g∗). Summing up,we see that (6.6.40) holds for i = 1.

Next, suppose that (6.6.40) has already been shown to hold for a given i ≤ r; by lemma6.6.34(ii), it follows that (6.6.40) holds for i+ 1, so we are done.

6.6.41. Let X be a scheme, M a fine log structure on the Zariski site of X , x a point of X ,and notice that every fractional ideal of Mx is finitely generated (lemma 3.4.22(iv)). Say thatX(x) = SpecA; in view of lemma 3.4.27(ii) there is a natural map of abelian groups

Div(Mx)→ Div(A).

Composing with the map I 7→ I∼ as in (5.6.11), we get, by virtue of loc.cit., a map

(6.6.42) Div(Mx)→ DivX(x).

Corollary 6.6.43. With the notation of (6.6.41), suppose furthermore that (X,M) is regular atthe point x. Then (6.6.42) induces an isomorphism

Div(Mx)∼→ Div(X(x)).

Proof. By virtue of proposition 3.4.37(ii) (and remark 5.6.10(i)), the map under investigationis already known to be injective. To show surjectivity, let K be the field of fractions of A, andL ⊂ K any reflexive fractional ideal of A; we may then regard L := L∼ as a coherent OX-submodule of i∗OX0 , where i : X0 → X is the inclusion map of the subscheme X0 := SpecK.By proposition 5.6.14, the OU -module L|U is invertible, hence it extends to an invertible OX(x)-module L ′, by virtue of theorem 6.6.38. Since X(x) is local, L ′ is a free OX(x)-module, andtherefore L|U ' OU . Thus, pick a ∈ L (U) ⊂ K which generates L|U ; after replacing L bya−1L, we may assume that L|U = OU as subsheaves of i∗OX0 . Let Σ be the set of points ofheight one of X(x) contained in X(x) \ U ; for each y ∈ Σ, the maximal ideal my of OX(x),y isgenerated by a single element ay, and there exists ky ∈ Z such that akyy generates the OX(x),y-submodule Ly of K. To ease notation, let P := Mx, and denote ψ : X(x) → SpecP thenatural continuous map; also, let mψ(y) be the maximal ideal of the localization Pψ(y) for everyy ∈ X(x), and set

I :=⋂y∈Σ

mkyψ(y).

In light of lemma 3.4.22(i,ii) and proposition 3.4.32, it is easily seen that I/P× is a reflexivefractional ideal of the fine and saturated monoid P ], so I is a reflexive fractional ideal of P .Then IA ⊂ K is a reflexive fractional ideal of A. Set L ′′ := (IA)∼ ⊂ i∗OX0; then L ′′

|U =

L|U and L ′′y = Ly for every y ∈ Σ. It follows that, for every y ∈ Σ, there exists an open

neighborhood Uy of y in X(x) such that L ′′|Uy = L|Uy . Let U ′ := U ∪

⋃y∈Σ Uy, and denote by

j′ : U ′ → X(x) the open immersion. Notice that δ′(y,OX) ≥ 2 for every y ∈ X \U ′ (corollary6.5.29). In light of proposition 5.6.7(ii) (and remark 5.6.10(i)), we deduce

L = j′∗L|U ′ = j′∗L′′|U ′ = L ′′



6.6.44. Let M be a log structure on the Zariski site of a local scheme X , such that (X,M) is aregular log scheme. Let x ∈ X be the closed point, say that X = SpecB for some local ringB, and let β : P → B a chart for M which is sharp at x. As usual, if M is any B-module, wedenote M∼ the quasi-coherent OX-module arising from M .

Theorem 6.6.45. In the situation of (6.6.44), suppose as well that dimP = 2 and dimX = 2.Then every indecomposable reflexive OX-module is isomorphic to (IB)∼, for some reflexivefractional ideal I of P . (Notation of (3.4.26).)

Proof. Set Q := Div+(P ) and define ϕ : P → Q as in example 3.4.46. The chart β defines amorphism ψ : (X,M)→ Spec(Z, P ), and we let (X ′,M ′) be the fibre product in the cartesiandiagram

(X ′,M ′) //

f

Spec(Z, Q)

Spec(Z,ϕ)

(X,M)

ψ // Spec(Z, P ).

Arguing as in the proof of claim 7.3.36, it is easily seen that X ′ is a local scheme and (X ′,M ′)is regular. Moreover, X ′ is a regular scheme, since Q is a free monoid (corollary 6.5.35), and fis a finite morphism of Kummer type (lemma 7.3.7).

Claim 6.6.46. The OX-module f∗OX′ is isomorphic to a finite direct sum (I1B ⊕ · · · ⊕ IkB)∼,where I1, . . . , Ik are reflexive fractional ideals of P , and OX is a direct summand of f∗OX′ .

Proof of the claim. In view of example 3.4.46, we see that f∗OX′ = (Q⊗P B)∼ is the direct sumof the OX-modules (grγQ ⊗P B)∼, where γ ranges over the elements of Qgp/P gp, and gr•Qdenotes the ϕ-grading of Q. But for each such γ, the natural map grγQ ⊗P B → grγQ · B isan isomorphism, since B is P -flat (lemma 6.5.17 and proposition 6.5.32). This shows the firstassertion, and the second is clear as well, since gr0Q = P . ♦

Denote by x′ the closed point of X ′; from corollary 6.5.36(i) we see that U := (X,M)1 isthe complement of x and U ′ := (X ′,M ′)1 is the complement of x′ (notation of definition6.2.7(i)). Let also j : U → X and j′ : U ′ → X ′ be the open immersions.

Claim 6.6.47. (i) The restriction g : U ′ → U of f is a flat morphism of schemes.(ii) The restriction functor j∗ : OX-Rflx→ OU -Rflx is an equivalence (notation of (5.6)).

Proof of the claim. (i): It suffices to check that the restriction Spec(Z, Q)1 → Spec(Z, P )1 ofSpec(Z, ϕ) is flat. However, we have the affine open covering

Spec(Z, P )1 = SpecZ[Pp1 ] ∪ SpecZ[Pp2 ]

where p1, p2 ⊂ P are the two prime ideals of height one (see example 3.4.17(i)). Hence, we arereduced to showing that the morphism of log schemes underlying

Spec(Z, ϕpi) : Spec(Z, Qpi)→ Spec(Z, Ppi)

is flat for i = 1, 2. However, it is clear that Qpi is an integral Ppi-module, hence it suffices tocheck that Qpi is a flat Ppi-module, for i = 1, 2 (proposition 3.1.52), or equivalently, that Q]

pi

is a flat P ]pi-module (corollary 3.1.49(ii)). The latter assertion follows immediately from the

discussion of (3.4.43).(ii): From proposition 5.6.7 we see that j∗ is full and essentially surjective. Moreover, it

follows from remark 5.6.4 that every reflexive OX-module is S1, so j∗ is also faithful (detailsleft to the reader). ♦

In light of claim 6.6.47(ii), it suffices to show that every indecomposable reflexive OU -moduleF is isomorphic to (IB)∼|U , for some reflexive fractional ideal I of P . However, for such F ,


claim 6.6.47(i) and lemma 5.6.6(i) imply that g∗F is a reflexive OU ′-module. From proposition5.6.7 and corollary 5.4.22 we deduce that δ′(x′, j′∗g

∗F ) ≥ 2, so j′∗g∗F is a free OX′-module

of finite rank ([30, Ch.0, Prop.17.3.4]), and finally, g∗F is a free OU ′-module of finite rank.Taking into account claim 6.6.46, it follows that g∗g∗F = F ⊗OU g∗OU ′ is a direct sum ofOU -modules of the type (IB)∼|U , for various I ∈ Div(P ); moreover, F is a direct summandof g∗g∗F . Then, we may find a decomposition g∗g∗F = F1 ⊕ · · · ⊕Ft such that Fi is anindecomposable OU -module for i = 1, . . . , t, and F1 = F (details left to the reader : noticethat – since reflexive OU -modules are S1 – the length t of such a decomposition is bounded bythe dimension of the κ(η)-vector space (g∗g

∗F )η, where η is the maximal point of X). On theother hand, notice that

EndOU ((IB)∼|U) = EndB(IB) (claim 6.6.47(ii))

= (IB : IB) (by (3.4.35))

= (I : I)B (lemma 3.4.27(i))=B (proposition 3.4.37(i))

for every I ∈ Div(P ). Now the contention follows from theorem 4.3.3.

6.6.48. Let now (Xet,M) be a quasi-coherent log scheme on the etale site ofX . Pick a coveringfamily (Uλ | λ ∈ Λ) of X in Xet, and for every λ ∈ Λ, a chart Pλ,Uλ →M |Uλ . The latter induceisomorphisms of log schemes (Uλ,M |Uλ)

∼→ u∗(Uλ,Zar, PlogUλ,Zar

) (see (6.2.2)); in other words,every quasi-coherent log structure on Xet descends, locally on Xet, to a log structure on theZariski site. However, (Xet,M) may fail to descend to a unique log structure on the whole ofXZar. Our present aim is to show that, at least under a few more assumptions on M , we mayfind a blow up (X ′et,M

′) → (Xet,M) such that (X ′et,M′) descends to a log structure on X ′Zar.

To begin with, for every λ ∈ Λ let

Tλ := (SpecPλ)] and Sλ := SpecZ[Pλ].

Also, let Sλ := (Spec(Z, Pλ), Tλ, ψPλ) be the object of K attached to Pλ, as in example6.6.5(i). From the isomorphism

(Uλ,Zar, u∗M |Uλ)∼→ Uλ ×Sλ Spec(Z, Pλ)

we deduce an object

Uλ := Uλ ×Sλ Sλ = ((Uλ,Zar, u∗M |Uλ), Tλ, ψλ).

Suppose now that M is a fs log structure; then we may choose for each Pλ a fine and saturatedmonoid (lemma 6.1.16(iii)). Next, suppose thatX is quasi-compact; in this case we may assumethat Λ is a finite set, hence S := Pλ | λ ∈ Λ is a finite set of monoids, and consequentlywe may choose a finite sequence of integers c(S ) fulfilling the conditions of (3.6.25) relativeto the category S -Fan. Then, for every λ ∈ Λ we have a well defined integral roof ρλ :Tλ(Q+) → Q+, and we denote by fλ : T (ρλ) → Tλ the corresponding subdivision. Therefollows a cartesian morphism

f ∗λUλ → Uλ

whose underlying morphism of log schemes is a saturated blow up of the ideal Iρλu∗M |Uλ(proposition 3.6.13 and corollary 6.6.18). Next, for λ, µ ∈ Λ, set Uλµ := Uλ ×X Uµ and let

Uλµ := ((Uλµ, u∗M |Uλµ), Tλ, ψλµ)

where ψλµ is the composition of ψλ and the projection Uλµ → Uλ. Denote by (U∼λ ,M∼λ ) (resp.

(U∼λµ,M∼λµ)) the log scheme underlying f ∗λUλ (resp. f ∗λUλµ). Also, for any λ, µ, γ ∈ Λ, let

πλµγ : Uλµ ×X Uγ → Uλµ be the natural projection.

Lemma 6.6.49. In the situation of (6.6.48), we have :


(i) There exists a unique isomorphism of log schemes gλµ : (U∼λµ,M∼λµ)

∼→ (U∼µλ,M∼µλ)

fitting into a commutative diagram :

(U∼λµ,M∼λµ)

gλµ //

(U∼µλ,M∼µλ)

Uλµ Uµλ

whose vertical arrows are the saturated blow up morphisms.(ii) There exist natural isomorphisms of OUλµ-modules

ωλµ : g∗λµOUµλ(1)∼→ OUλµ(1)

such that (π∗µγλωµλ) (π∗λµγωλµ) = π∗λγµωλγ for every λ, µ, γ ∈ Λ.

Proof. (i): By the universal property of the saturated blow up, it suffices to show that :

(6.6.50) Iρλu∗M |Uλµ = Iρµu∗M |Uµλ on Uλµ = Uµλ.

The assertion is local on Uλµ, hence let x ∈ Uλµ be any point; we get an isomorphism ofu∗Mx-monoids :

OTλ,ψλµ(x)∼→ OTµ,ψµλ(x)

whence an isomorphism of fans U(ψλµ(x))∼→ U(ψµλ(x)) (notation of (3.5.16)). Therefore, the

subset U(x) := ψ−1λµU(ψλµ(x))∩ψ−1

µλU(ψµλ(x)) is an open neighborhood of x in Uλµ, and bothψλµ and ψµλ factor through the same morphism of monoidal spaces :

ψ(x) : (U(x), (u∗M])|U(x))→ F (x) := (Spec u∗Mx)

]

and open immersions F (x) → Tλ and F (x) → Tµ. It then follows from (3.6.25) that thepreimage of Iρλ on F (x) agrees with the preimage of Iρµ , whence the contention.

(ii): By inspecting the definitions, it is easily seen that the epimorphism (6.4.17) identifiesnaturally OU∼λµ

(1) to the ideal IρλµOU∼λµof OU∼λµ

generated by the image of Iρλu∗M |Uλµ . Hencethe assertion follows again from (6.6.50).

6.6.51. Lemma 6.6.49 implies that

((U∼λ ,OU∼λ (1)), (gλµ, ωλµ) | λ, µ ∈ Λ)

is a descent datum – relative to the faithfully flat and quasi-compact morphism∐

λ∈Λ Uλ →X – for the fibred category of schemes endowed with an ample invertible sheaf. Accordingto [42, Exp.VIII, Prop.7.8], such a datum is effective, hence it yields a projective morphismπ : X∼ → X together with an ample invertible sheaf OX∼(1) on X∼, with isomorphismsgλ : X∼ ×X Uλ

∼→ U∼λ of Uλ-schemes and π∗λOX∼(1)∼→ g∗λOU∼λ (1) of invertible modules.

Then the datum (u∗M∼λ , u

∗ log gλµ | λ, µ ∈ Λ) likewise determines a unique sheaf of monoidsM∼ on X∼et , and the structure maps of the log structures Mλ glue to a well defined morphismof sheaves of monoids M∼ → OX∼et

, so that (X∼et,M∼) is a log scheme, and the projection π

extends to a morphism of log schemes (π, log π) : (X∼et,M∼)→ (Xet,M).

Proposition 6.6.52. In the situation of (6.6.51), the counit of adjunction

u∗u∗(X∼et,M

∼)→ (X∼et,M∼)

is an isomorphism.


Proof. (This is the counit of the adjoint pair (u∗, u∗) of (6.1.6), relating the categories oflog structures on X∼Zar and X∼et .) Recall that there exist natural epimorphisms (Q⊕d+ )T (ρλ) →OT (ρλ),Q (see (3.6.27)), which induce epimorphisms of U∼λ,Zar-monoids

ϑλ : (Q⊕d+ )U∼λ,Zar→ (M∼

λ )]Q for every λ ∈ Λ.

The compatibility with open immersions expressed by (3.6.28) implies that the system of maps(u∗ϑλ | λ ∈ Λ) glues to a well defined epimorphism of X∼et-monoids :

ϑ : (Q⊕d+ )X∼et→ (M∼)]Q.

In view of lemma 2.4.26(ii), it follows that the counit of adjunction :

u∗u∗(M∼)]Q → (M∼)]Q

is an isomorphism. By applying again lemma 2.4.26(ii) to the monomorphism (M∼)] →(M∼)]Q, we deduce that also the counit

u∗u∗(M∼)] → (M∼)]

is an isomorphism. Then the assertion follows from proposition 6.2.3(iii).

Corollary 6.6.53. Let (Xet,M) be a quasi-compact regular log scheme. Then there existsa smooth morphism of log schemes (X ′et,M

′) → (Xet,M) whose underlying morphism ofschemes is proper and birational, and such that X ′ is a regular scheme. More precisely, frestricts to an isomorphism (X ′et,M

′)tr → (Xet,M)tr on the trivial loci.

Proof. Given such (Xet,M), we construct first the morphism π : (X∼et,M∼) → (Xet,M)

as in (6.6.51). Since π ×X 1Uλ is proper for every λ ∈ Λ, if follows that π is proper ([31,Ch.IV, Prop.2.7.1]). Likewise, notice that each morphism U∼λ → Uλ induces an isomorphism(U∼λ ,M

∼λ )tr

∼→ (Uλ, u∗M |Uλ)tr (remark 6.6.17). It follows easily that π restricts an isomor-phism on the trivial loci. Hence, we may replace (Xet,M) by (X∼et,M

∼et). Then, by corollary

6.3.27(ii) and proposition 6.6.52 we are further reduced to showing that there exists a propermorphism π′ : (X ′Zar,M

′) → u∗(X∼et,M

∼) with X ′ regular, restricting to an isomorphism onthe trivial loci. However, in light of lemma 6.5.37 (and again, proposition 6.6.52), the sought π′

is provided by the more precise theorem 6.6.32.

6.7. Local properties of the fibres of a smooth morphism. Resume the situation of example6.6.5(ii), and to ease notation set ϕ := (Specλ)], and (f, log f) := Spec(R, λ). Suppose now,that λ : P → Q is an integral, local and injective morphism of fine monoids. Then f : S ′ → Sis flat and finitely presented (theorem 3.2.3). Moreover :

Lemma 6.7.1. In the situation of (6.7), for every point s ∈ S, the fibre f−1(s) is either empty,or else it is pure-dimensional, of dimension

dim f−1(s) = dimQ− dimP = rkZ Cokerλgp.

Proof. To begin with, notice that λ−1Q× = P×, whence the second stated identity, in view ofcorollary 3.4.10(i). To prove the first stated identity, we easily reduce to the case where R is afield. Notice that the image of f is an open subset U ⊂ S ([31, Ch.IV, Th.2.4.6]), especially U(resp. S ′) is pure-dimensional of dimension rkZP

gp (resp. rkZQgp) by claim 5.9.36(ii). Hence,

fix any closed point s ∈ S, and set X := f−1(s). From [31, Ch.IV, Cor.6.1.2] we deduce that,for every closed point s′ ∈ X , the Krull dimension of OX,s′ equals r := rkZCokerλgp. Moreprecisely, say that Z ⊂ X is any irreducible component; we may find a closed point s′ ∈ Zwhich does not lie on any other irreducible component of X , and then the foregoing impliesthat the dimension of Z equals r, as stated.


Next, let s ∈ U be any point, and denote K the residue field of OU,s, and π : R[P ] → Kthe natural map. Let y ∈ SpecK[P ] be the K-rational closed point such that a(y) = π(a)for every a ∈ P ; then the image of y in S equals s, and if we let fK := SpecK[λ], we havean isomorphism of K-schemes f−1

K (y)∼→ f−1(s). The foregoing shows that f−1

K (y) is pure-dimensional of dimension r, hence the same holds for f−1(s).

Now, let us fix s ∈ S, such that f−1(s) 6= ∅, and suppose that ψP (s) = mP is the closedpoint of TP . For every q ∈ ϕ−1(mP ), the closure q of q in TQ is the image of the naturalmap SpecQ/q→ TQ. We deduce natural isomorphisms of schemes :

S0 := ψ−1P mp

∼→ SpecR〈P/mP 〉 S ′q := ψ−1Q q

∼→ SpecR〈Q/q〉

under which, the restriction fq : S ′q → S0 of f is identified with SpecR〈λq〉, where λq :P/mP → Q/q is induced by λ. The latter is an integral and injective morphism as well (corol-lary 3.1.51). Explicitly, set F := Q\q, and let λF : P× → F be the restriction of λ; thenλq = (λF ), and lemma 6.7.1 yields the identity :

(6.7.2) dim f−1q (s) = dimQ/q = dimQ− ht q.

Also, notice that TQ is a finite set under the current assumptions (lemma 3.1.20(iii)), and clearly

f−1(s) =⋃

q∈Max(ϕ−1mP )

f−1q (s)

where, for a topological space T , we denote by Max(T ) the set of maximal points of T . There-fore, for every irreducible component Z of f−1(s) there must exist q ∈ Max(ϕ−1mP ) such thatZ ⊂ f−1

q (s), and especially,

(6.7.3) dim f−1(s) = dim f−1q (s).

Conversely, we claim that (6.7.3) holds for every q ∈ Max(ϕ−1mP ). Indeed, suppose thatdim f−1

q (s) is strictly less than dim f−1(s) for one such q, and letZ be an irreducible componentof f−1

q (s); let also Z ′ be an irreducible component of f−1(s) containing Z. By the foregoing,there exists q′ ∈ Max(ϕ−1mP ) with Z ′ ⊂ f−1

q′ (s). Set q′′ := q∪ q′; then clearly q′′ ∈ ϕ−1(mP ),and

q′′ = q ∩ q′.Especially, Z ⊂ f−1

q′′ (s); however, it follows from (6.7.2) that dim f−1q′′ (s) < dim f−1

q (s) (sinceht q < ht q′′); but this is absurd, since dimZ = dim f−1

q (s) (lemma 6.7.1). The same countingargument also shows that every maximal point of f−1(s) gets mapped necessarily to a maximalpoint of ϕ−1(mP ); in other words, we have shown that ψQ restricts to a surjective map :

Max(f−1s)→ Max(ϕ−1mP ).

More precisely, let s′ ∈ f−1(s) be a point such that ψQ(s′) = mQ, the closed point of TQ. Thenclearly s′ ∈ f−1

q (s) for every q ∈ Max(ϕ−1mP ), and it follows that the foregoing surjectionrestricts further to a surjective map :

(6.7.4) f−1s′ (s)→ Max(ϕ−1mP ).

On the other hand, since logψQ is an isomorphism, we have

(QlogS′ )

]s′ = OTQ,q = Q]

q for every s′ ∈ ψ−1Q (q)

(where s′ denotes any τ -point of S ′ localized at s′); explicitly, if F = Q\q, then Q]q = Q/F ;

likewise, we get (P logS )]s = P/ϕ−1F . Taking into account (6.7.2), we deduce :

dim f−1(f(s′))− dim f−1q (f(s′)) = rkZ Coker (log f)gp

s′ for every s′ ∈ ψ−1Q (q).


Thus, for every n ∈ N, let

Un := s′ ∈ S ′ | rkZ Coker (log f)gps′ = n.

The foregoing implies that for every s ∈ S, the subset U0∩ f−1(s) is open and dense in f−1(s),and Un ∩ f−1(s) is either empty, or else it is a subset of pure codimension n in f−1(s).

6.7.5. In the situation of (6.7), suppose moreover that log f is saturated; notice that this condi-tion holds if and only if logϕ is saturated, if and only if the same holds for λ (lemma 3.2.12(iii)).Then, corollary 3.2.32(ii) says that Coker(log f ])gp

s′ is torsion-free for every s′ ∈ S ′; especially,Coker(log f ])gp

s′ vanishes for every s′ ∈ U0. Then corollary 3.2.32(i) implies that log f ]s′ is anisomorphism for every s′ ∈ U0, in which case the same holds for log fs′ (lemma 6.1.4); in otherwords, U0 is the strict locus of f (see definition 6.2.7(ii)).

Lemma 6.7.6. Let K be an algebraically closed field of characteristic p, and χ : P → (K, ·) alocal morphism of monoids. Let also λ : P → Q be as in (6.7.5). We have :

(i) The K-algebra Q⊗P K is Cohen-Macaulay.(ii) Suppose moreover, that either p = 0, or else the order of the torsion subgroup of

Γ := Cokerλgp is not divisible by p. Then Q ⊗P K is reduced (i.e. its nilradical istrivial).

Proof. (i): Since χ is a local morphism, we have χ−1(0) = mP , and χ is determined by itsrestriction P× → K×, which is a homomorphism of abelian groups. Notice thatK× is divisible,hence it is injective in the category of abelian groups; since the unit of adjunction P → P sat islocal (lemma 3.2.9(iii)), it follows that χ extends to a local morphism χ′ : P sat → K. Noticethat Q ⊗P K = Qsat ⊗P sat K (lemma 3.2.12(iv)), hence we may replace P by P sat and Q byQsat, which allows to assume from start that Q is saturated. In this case, by theorem 5.9.34(i),both K[P ] and K[Q] are Cohen-Macaulay; since K[λ] is flat, theorem 5.4.36 and [31, Ch.IV,Cor.6.1.2] imply that Q⊗P K is Cohen-Macaulay as well.

(ii): Let Q =⊕

γ∈ΓQγ be the λ-grading of Q (see remark 3.2.5(iii)). Under the currentassumptions, λ is exact (lemma 3.2.30(ii)), consequently Q0 = P (remark 3.2.5(v)). Moreover,Qγ is a finitely generated P -module, for every γ ∈ Γ (corollary 3.4.3), hence either Qγ = ∅,or else Qγ is a free P -module of rank one, say generated by an element uγ (remark 3.2.5(iv)).Furthermore, Qk

γ = Qkγ for every integer k > 0 and every γ ∈ Γ (proposition 3.2.31). Thus,wheneverQγ 6= ∅, the element uγ⊗1 is a basis of theK-vector spaceQγ⊗PK, and (uγ⊗1)k isa basis of Qkγ ⊗P K, for every k > 0; especially, uγ ⊗ 1 is not nilpotent. In view of proposition4.4.12(ii), the contention follows.

6.7.7. Let f : (X,M) → (Y,N) be a smooth and log flat morphism of fine log schemes. Forevery n ∈ N, let U(f, n) ⊂ X be the subset of all x ∈ X such that rkZCoker (log f ]x)

gp = n,for any τ -point x localized at x. By lemma 6.2.21(iii), U(f, n) is a locally closed subset (resp.an open subset) of X , for every n > 0 (resp. for n = 0).

Theorem 6.7.8. In the situation of (6.7.7), we have :(i) f is a flat morphism of schemes.

(ii) For all y ∈ Y , every connected component of f−1(y), is pure dimensional, and f−1(y)∩Un is either empty, or else it has pure codimension n in f−1(y), for every n ∈ N.

(iii) Moreover, if f is saturated, we have :(a) The strict locus Str(f) of f is open in X , and Str(f)∩ f−1(y) is a dense subset of

f−1(y), for every y ∈ Y .(b) f−1(y) is geometrically reduced and Cohen-Macaulay, for every y ∈ Y .(c) Coker(log f ]x)

gp is a free abelian group of finite rank, for every τ -point x of X .


Proof. Let ξ be any τ -point of X; according to corollary 6.1.33, there exists a neighborhood Vof ξ, and a fine chart PV → N |V , such that P gp is a free abelian group of finite rank, and theinduced morphism of monoids P → OY,ξ is local.

By lemma 6.1.12, [31, Ch.IV, Th.2.4.6, Prop.2.5.1] and [33, Ch.IV, Prop.17.5.7], it sufficesto prove the theorem for the morphism f ×Y V : (X,M) ×Y V → (V,N |V ). Hence we mayassume from start that N admits a fine chart β : PY → N , such that P gp is torsion-free and theinduced map P → OY,ξ is local.

Next, by theorem 6.3.37, remark 6.1.21(i), [31, Ch.IV, Th.2.4.6] and [33, Ch.IV, Prop.17.5.7](and again lemma 6.1.12) we may further assume that f admits a fine chart (β, ωQ : QX →M,λ), such that λ is injective, the torsion subgroup of Cokerλgp is a finite group whose orderis invertible in OX , and the induced morphism of schemes X → Y ×SpecZ[Q] SpecZ[P ] is etale.Moreover, by theorem 6.1.35(iii), we may assume – after replacing Q by a localization, and Xby a neighborhood of ξ in Xτ – that the morphism λ : P → Q is integral (resp. saturated, if fis saturated), and the morphism Q→ OX,ξ induced by ωP,ξ is local, so λ is local as well.

In this case, the same sort of reduction as in the foregoing shows that, in order to prove (i)and (ii), it suffices to consider a morphism f as in (6.7), for which these assertions have alreadybeen established. Likewise, in order to show (iii), it suffices to consider a morphism f as in(6.7.5). For such a morphism, assertions (iii.a) and (iii.c) are already known, and (iii.b) is animmediate consequence of lemma 6.7.6.

Theorem 6.7.8(iii) admits the following partial converse :

Proposition 6.7.9. Let f : (X,M) → (Y,N) be a smooth and log flat morphism of fs logschemes, such that f−1(y) is geometrically reduced, for every y ∈ Y . Then f is saturated.

Proof. Fix a τ -point ξ of X; the assertion can be checked on stalks, hence we may assume thatN admits a fine and saturated chart β : PY → N (lemma 6.1.16(iii)), such that the inducedmorphism αP : P → OY,f(ξ) is local (claim 6.1.29). Then, by corollary 6.3.42, we may findan etale morphism g : U → X and a τ -point ξ′ of U with g(ξ′) = ξ, such that the inducedmorphism of log schemes fU : (U, g∗M) → (Y,N) admits a fine and saturated chart (β, ωQ :QU → g∗M,λ), where λ is injective, and the induced ring homomorphism

(6.7.10) Q⊗P OY,f(ξ) → OU,ξ′

is etale. By [33, Ch.IV, Prop.17.5.7], the fibres of fU are still geometrically reduced, hencewe are reduced to the case where U = X and ξ = ξ′. Furthermore, after replacing Q by alocalization, and X by a neighborhood of ξ, we may assume that the map αQ : Q → OX,ξinduced by ωQ,ξ is local and λ is integral (theorem 6.1.35(iii) and lemma 3.2.9(i)). Lastly, let Kbe the residue field of OY,f(ξ); our assumption on f−1(y) means that the ringA := OX,ξ⊗OY,f(ξ)

Kis reduced.

We shall apply the criterion of proposition 3.2.31. Thus, let Q =⊕

γ∈ΓQγ be the λ-gradingof Q and notice as well that λ is a local morphism (since the same holds for αP and αQ),therefore it is exact (lemma 3.2.30(ii)); consequently Q0 = P (remark 3.2.5(v)). Moreover, Qγ

is a finitely generated P -module, for every γ ∈ Γ (corollary 3.4.3), hence either Qγ = ∅, orelse Qγ is a free P -module of rank one.

We have to prove that Qkγ = Qkγ for every integer k > 0 and every γ ∈ Γ. In case Qγ = ∅,

this is the assertion that Qkγ = ∅ as well. However, since Q is saturated, the same holds forΓ′ := (λP )−1Q/(λP )gp (lemma 3.2.9(i,ii)), so it suffices to remark that Γ′ ⊂ Γ is precisely thesubmonoid consisting of all those γ ∈ Γ such that Qγ 6= ∅.

Therefore, fix a generator uγ for every P -module Qγ 6= ∅, and by way of contradiction,suppose that there exist γ ∈ Γ′ and k > 0 such that ukγ does not generate the P -module Qkγ;this means that there exists a ∈ mP such that ukγ = a · ukγ . Now, notice that the induced


morphism of monoids β : P → K is local, especially β(a) = 0, therefore (uγ ⊗ 1)k = 0 in theK-algebra Q ⊗P K. Denote by I ⊂ Q ⊗P K the annihilator of uγ ⊗ 1, and notice that, since(6.7.10) is flat, IA is the annihilator of the image u′ of uγ ⊗ 1 in A.

On the other hand, it is easily seen that I is the graded ideal generated by (uµ ⊗ 1 | µ ∈ Γ′′)where Γ′′ ⊂ Γ′ is the subset of all µ such that uγ · uµ is not a generator of the P -module Qγ+µ.Clearly uµ /∈ Q× for every µ ∈ Γ′′, therefore the image of uµ⊗1 in A lies in the maximal ideal.Therefore IA 6= A, i.e. u′ is a non-zero nilpotent element, a contradiction.

6.7.11. Let (X,M) be any log scheme, and x any geometric point, localized at a point x ∈ X .Suppose that y ∈ X is a generization of x, and y a geometric point localized at y; then, arguingas in (2.4.22), we may extend uniquely any strict specialization morphism X(y) → X(x) to amorphism of log schemes (X(y),M(y))→ (X(x),M(x)) fitting into a commutative diagram

(X(y),M(y)) //

(X(y),M(y))

(X(x),M(x)) // (X(x),M(x))

whose right vertical arrow is induced by the natural isomorphism

(X(y),M(y))∼→ (X(x),M(x))×X(x) X(y).

A simple inspection shows that the induced morphism

Γ(X(x),M(x))] → Γ(X(y),M(y))]

is naturally identified with the morphism M ]x → M ]

y obtained from the specialization mapMx →My.

6.7.12. Let g : (X,M)→ (Y,N) be a morphism of log schemes, x any geometric point of X ,and set y := g(x). The log structures M → OX and N → OY , and the map log gx induce acommutative diagram of continuous maps

(6.7.13)

X(x)gx //

ψx

Y (y)

ψy

SpecMxϕx // SpecNy

(notation of 2.4.19), and notice that ψx (resp. ψy) maps the closed point ofX(x) (resp. of Y (y))to the closed point tx ∈ SpecMx (resp. ty ∈ SpecNy).

Proposition 6.7.14. In the situation of (6.7.12), suppose that g is a smooth morphism of finelog schemes, and moreover :

(a) either g is a saturated morphism(b) or (X,M) is a fs log scheme.

Then the following holds :(i) The map ψx restricts to a bijection :

Max(g−1x (y))

∼→ Max(ϕ−1x (ty)).

(ii) For every irreducible component Z of g−1x (y), set

(Z,M(Z)) := (X(x),M(x))×X(x) Z.

Then the κ(y)-log scheme (Z,M(Z)red) is geometrically pointed regular. (Notation ofexample 6.1.10(iv) and remark 6.5.24(ii).)


Proof. By corollary 6.1.33, there exists a neighborhood U → Y of y, and a fine chart β : PU →N |U such that βy is a local morphism, and P gp is torsion-free. Now, let

gx := (gx, log gx) : (X(x),M(x))→ (Y (y), N(y))

be the morphism of log schemes induced by g (notation of 6.7.11); by theorem 6.3.37, we mayfind a fine chart for gx of the type (i∗yβ, ω, λ : P → Q), where λ is injective, and the order ofthe torsion subgroup of Cokerλgp is invertible in OX,x. Moreover, set R := OY (y),y; then theinduced map X(x) → SpecQ ⊗P R is ind-etale, and – after replacing Q by some localization– we may assume that ωx : Q → OX(x),x is local (claim 6.1.29), hence the same holds for λ.Furthermore, under assumption (a) (resp. (b)), we may also suppose that Q is saturated, bylemmata 3.2.9(ii) and 6.1.16(ii) (resp. that λ is saturated, by theorem 6.1.35(iii)).

Let us now define f : S ′ → S as in (6.7); notice that ωx induces a closed immersion Y (y)→S, and we have a natural identification of Y (y)-schemes :

SpecQ⊗P R = Y (y)×S S ′.Denote by s the image of y in S, and s′ the image of x in Y (y) ×S S ′ ⊂ S ′; there follows anisomorphism of Y (y)-schemes :

X(x)∼→ Y (y)×S(s) S

′(s′)

([33, Ch.IV, Prop.18.8.10]). Moreover, our chart induces isomorphisms :

SpecMx∼→ TQ SpecNy

∼→ TP

which identify ϕx to the map ϕ : TQ → TP of (6.7). In view of these identifications, we see that(6.7.13) is the restriction to the closed subset X(x), of the analogous diagram :

S ′(s′)fs′ //

ψs′

S(s)

ψs

TQϕ // TP .

We are thus reduced to the case where X = S ′, Y = S and g = f . Moreover, let s (resp. s′)be the support of s (resp. of s′); the morphism π : S ′(s′) → S ′(s′) is flat, hence it restricts to asurjective map

Max(f−1s′ s)→ Max(f−1

s′ s).

In order to prove (i), it suffices then to show that the map Max(f−1s′ s)→ Max(ϕ−1mP ), defined

as the composition of the foregoing map and the surjection (6.7.4), is injective. This boilsdown to the assertion that, for every q ∈ Max(ϕ−1mP ) the point s′ lies on a unique irreduciblecomponent of the fibre π−1(f−1

q,s′s). However, let β : P → κ(s) be the composition of the chartP → R and the projection R → κ(s); then f−1

q (s) = SpecQ/q ⊗P κ(s). Since π is ind-etale,the assertion will follow from [33, Ch.IV, Prop.17.5.7] and corollary 6.5.29, together with :

Claim 6.7.15. The log scheme Wq := Spec〈Z, Q/q〉 ×SpecZ[P ] Specκ(s) is geometricallypointed regular.

Proof of the claim. To ease notation, set F := Q \ q; by assumption λ−1F = P×, so that

Wq = (W ′q) where W ′

q := Spec(Z, F )×Spec(Z,P×) Specκ(s)

(notation of (6.2.13); notice that the log structures of Spec(Z, P×) and Specκ(s) are trivial).Moreover, let λF : P× → F be the restriction of λ; then λgp

F is injective, and

CokerλgpF ⊂ Cokerλgp

(corollary 3.2.33(i)), hence the order of the torsion subgroup of CokerλgpF is invertible in OS,s.

Moreover, if λ is saturated, then the same holds for λF (corollary 3.2.33(ii)), and it is easily


seen that, if Q is saturated, the same holds for F . Consequently, the morphism Spec(Z, λF ) issmooth (proposition 6.3.34). The same then holds for the morphism W ′

q → Specκ(s) obtainedafter base change of Spec(Z, λF ) along the morphism h : Spec(Z, P×) → Specκ(s) inducedby β (proposition 6.3.24(ii)).

Now we notice that under either of the assumptions (a) or (b), W ′q is a fs log scheme. Indeed,

under assumption (b), this follows by remarking that Specκ(s) is trivially a fs log scheme, andSpec(Z, λF ) is saturated. Under assumption (a), Spec(Z, F ) is a fs log scheme, and it sufficesto observe that h is a strict morphism. Lastly, since the morphismW ′

q → Specκ(s) is obviouslysaturated, we apply corollary 6.5.45 to conclude. ♦

(ii): In light of the foregoing, we see that, for any irreducible component Z of g−1x (y), there

exists a unique q(Z) ∈ Max(ϕ−1mP ) such that Z is isomorphic to the strict henselization off−1q(Z)(s), at the geometric point s′. Notice now that the log structure of Wq(Z) is reduced, by

virtue of claim 6.7.15 and proposition 6.5.52; then, a simple inspection of the definitions showsthat (Z,M(Z)red) is isomorphic to the strict henselization Wq(Z)(s

′). Invoking again claim6.7.15, we deduce the contention.

6.7.16. In the situation of (6.7.12), suppose moreover that Y is a normal scheme, and (Y,N)tr

is a dense subset of Y . Let η be a geometric point of Y (y), localized at the generic point η. Let

(Uq | q ∈ SpecMx)

be the logarithmic stratification of (X(x),M(ξ)) (see (6.5.49)). Notice that ψx(g−1(η)) lies inthe preimage Σ ⊂ SpecMx of the maximal point ∅ of SpecNy; therefore, g−1

x (η) is the unionof the subsets Uq ×Y |η|, for all q ∈ Σ.

Proposition 6.7.17. In the situation of (6.7.16), suppose that g is a smooth morphism of finelog schemes. Then the following holds :

(i) X is a normal scheme.(ii) The scheme g−1

x (η) is normal and irreducible.(iii) For every q ∈ Σ, the κ(η)-scheme Uq ×Y |η| is non-empty, geometrically normal and

geometrically irreducible, of pure dimension dim g−1x (η)− ht q.

(iv) Especially, set W := (X(x)\U∅)×Y |η|; then ψx induces a bijection :

Max(W )∼→ q ∈ Σ | htq = 1.

(v) For every w ∈ Max(W ), the stalk Og−1x (η),w is a discrete valuation ring.

Proof. Set R := OY (y),y; arguing as in the proof of proposition 6.7.14, we may find :• a local, flat and saturated morphism λ : P → Q of fine monoids, such that the order of

the torsion subgroup of Cokerλgp is invertible in R;• local morphisms of monoids P → R, Q → OX(x),x which are charts for the log struc-

tures deduced from N and respectively M , and such that the induced morphism ofY (y)-schemes X(x)→ SpecQ⊗P R is ind-etale.

By [33, Ch.IV, Prop.17.5.7], we may then assume that (X,M) (resp. (Y,N)) is the schemeSpecQ ⊗P R (resp. SpecR), endowed with the log structure deduced from the natural mapQ → Q ⊗P R (resp. the chart P → R), and g is the natural projection. Suppose first that Ris excellent, and let R′ be the normalization of R in a finite extension K ′ of Frac(R); then R′

is also strictly local and noetherian, and if y′ denotes the closed point of Y ′ := SpecR′, thenthe residue field extension κ(y) ⊂ κ(y′) is purely inseparable. Set (X ′,M ′) := (X,M)×Y Y ′,(Y ′, N ′) := (Y,N) ×Y Y ′, and let g′ : (X ′,M ′) → (Y ′, N ′) be the induced morphism oflog schemes; it follows especially that the restriction g′−1(y′) → g−1(y) is a homeomorphismon the underlying topological spaces. Hence, there is a geometric point x′ of X ′, unique up


to isomorphism, whose image in X agrees with x, and we easily deduce an isomorphism ofY -schemes ([33, Ch.IV, Prop.18.8.10])

(6.7.18) X ′(x′)∼→ X(x)×Y Y ′.

Let η′ be the generic point of Y ′; by assumption, N ′ is trivial in a Zariski neighborhood of η′,hence (X ′, N ′)×Y ′ |η′| is a fs log schemes (since g is saturated), and then the same log schemeis also regular (theorem 6.5.44), therefore g′−1(η′) is a normal scheme (corollary 6.5.29). On theother hand, R′ is a Krull domain (see remark 7.1.27), and g′ is flat with reduced fibres (theorem6.7.8(iii.b)), so X ′ is a noetherian normal scheme (claim 7.1.32), and consequently the sameholds for X ′(x′) ([33, Ch.IV, Prop.18.8.12(i)]); especially, the latter is irreducible, so the sameholds for X ′(x′) ×Y ′ |η′|. In view of (6.7.18), it follows that X(x) ×Y |η′| is also normal andirreducible. Since K ′ is arbitrary, this completes the proof of (i) and (ii), in this case.

Next, if R is any normal ring, we may write R as the union of a filtered family (Ri | i ∈ I)of excellent normal local subrings ([31, Ch.IV, (7.8.3)(ii,vi)]). For each i ∈ I , denote by yithe image of y in SpecRi; then the strict henselization Rsh

i of Ri at yi is also a subring of R,so we may replace Ri by Rsh

i , which allows to assume that each Ri is strictly local, normaland noetherian ( [33, Ch.IV, Prop.18.8.8(iv), Prop.18.8.12(i)]). Up to replacing I by a cofinalsubset, we may assume that the image of P lies in Ri, for every i ∈ I . For each i ∈ I , setXi := SpecQ⊗P Ri, Yi := SpecRi, and endow Xi (resp. Yi) with the log structure M i (resp.N i) deduced from the natural map Q → Q ⊗P Ri (resp. P → Ri). There follows a systemof morphisms of log schemes gi : (Xi,M i) → (Yi, N i) for every i ∈ I , whose limit is themorphism g. Moreover, since (Y,N)tr is dense in Y , the image of P lies in R\0, hence it liesin Ri \0 for every i ∈ I , and the latter means that (Yi, N i)tr is dense in Yi, for every i ∈ I .Let ηi (resp. xi) be the image of η (resp. of x) in Yi (resp. in Xi); moreover, for each i ∈ I ,let xi ∈ Xi be the support of xi. By the previous case, we know that g−1

i,xi(ηi) is normal and

irreducible. However, X (resp. g−1x (η)) is the limit of the system of schemes (Xi | i ∈ I) (resp.

(g−1i,xi

(ηi) | i ∈ I)), so (i) and (ii) follow. (Notice that the colimit of a filtered system of integral(resp. normal) domains, is an integral (resp. normal) domain : exercise for the reader.)

(iii): For every q ∈ SpecQ = SpecMx, set Xq := SpecQ/q ⊗P R; since the chart Q →OX(x),x is local, it is clear that x ∈ Xq for every such q, and :

Xq(x) =⋃

p∈SpecQ/q

Up.

If ϕ(q) = ∅, the induced map P → Q/q is still flat (corollary 3.1.51), hence the projectionXq(x) → Y is a flat morphism of schemes, especially g−1

x (η) ∩Xq(x) 6= ∅. Furthermore, thesubset Xq(x) is pure-dimensional, of codimension ht q in g−1

x (η), by (6.7.2) and [31, Ch.IV,Cor.6.1.4]. It follows that Uq is a dense open subset of Xq(x). To conclude, it remains onlyto show that each Xq(x) is geometrically normal and geometrically irreducible; however, letjq : Xq → X be the closed immersion; the induced morphism of log schemes gq : (Xq, j

∗qM)→

(Y,N) is also smooth, hence the assertion follows from (ii).(iv) is a straightforward consequence of (iii).(v): Notice that A := Og−1

x (η),w is ind-etale over the noetherian ring Q ⊗P κ(η), hence itsstrict henselization is noetherian, and then A itself is noetherian ([33, Ch.IV, Prop.18.8.8(iv)]).Since X is normal, and w is a point of height one in g−1

x (η), we conclude that A is a discretevaluation ring.

7. ETALE COVERINGS OF SCHEMES AND LOG SCHEMES

7.1. Acyclic morphisms of schemes. For any scheme X , we shall denote by :

Cov(X)


the category whose objects are the finite etale morphismsE → X; the morphisms (E → X)→(E ′ → X) are the X-morphisms of schemes E → E ′. By faithfully flat descent, Cov(X) isnaturally equivalent to the subcategory of X∼et consisting of all locally constant constructiblesheaves. If f : X → Y is any morphism of schemes, and ϕ : E → Y is an object Cov(Y ),then f ∗ϕ := ϕ ×Y X : E ×Y X → X is an object of Cov(X); more precisely, we have afibration :

(7.1.1) Cov→ Sch

over the category of schemes, whose fibre, over any scheme X , is the category Cov(X).

Lemma 7.1.2. Let f be a morphism of schemes, and suppose that either one of the followingconditions holds :

(a) f is integral and surjective.(b) f is faithfully flat.(c) f is proper and surjective.

Then f is of universal 2-descent for the fibred category (7.1.1).

Proof. This is [4, Exp.VIII, Th.9.4].

Lemma 7.1.3. In the situation of definition 5.5.40, suppose that Lef(X, Y ) holds. Let U ⊂ Xbe any open subset such that Y ⊂ U , and denote by j : Y → U the closed immersion. Then theinduced functor

j∗ : Cov(U)→ Cov(Y )

is fully faithful.

Proof. Let Cov(X) be the full subcategory of OX-Alglfft consisting of all finite etale OX-algebras (notation of lemma 5.5.41(ii)); the category Cov(U) is a full subcategory of the cat-egory OU -Alglfft, so lemma 5.5.41(ii) already implies that the functor Cov(U) → Cov(X) isfully faithful, hence we are reduced to showing that the functor :

Cov(X)→ Cov(Y ) A 7→ Spec A ⊗OXOY

is fully faithful. To this aim, let I ⊂ OX be the ideal defining the closed immersion Y ⊂ X;consider the direct system of schemes

(Yn := Spec OX/In+1 | n ∈ N)

and let Cov(Y•) be the category consisting of all direct systems (En | n ∈ N) of schemes,such that En is finite etale over Yn, and such that the transition maps En → En+1 induceisomorphisms En

∼→ En+1 ×Yn+1 Yn for every n ∈ N. The morphisms in Cov(Y•) are themorphisms of direct systems of schemes. We have a natural functor :

Cov(X)→ Cov(Y•) A 7→ (Spec A /I n+1A | n ∈ N)

which is fully faithful, by [26, Ch.I, Cor.10.6.10(ii)]. Finally, the functor Cov(Y•)→ Cov(Y )given by the rule (En | n ∈ N) → E0 is an equivalence, by [33, Ch.IV, Th.18.1.2]. The claimfollows.

7.1.4. Consider now a cofiltered family S := (Sλ | λ ∈ Λ) of affine schemes. Denote by S thelimit of S , and suppose moreover that Λ admits a final element 0 ∈ Λ. Let f0 : X0 → S0 be afinitely presented morphism of schemes, and set :

Xλ := X0 ×S0 Sλ fλ := f0 ×S0 Sλ : Xλ → Sλ for every λ ∈ Λ.

Set as well X := X0 ×S0 S and f := f0 ×S0 S : X → S. For every λ ∈ Λ, let pλ : S → Sλ(resp. p′λ : X → Xλ) be the natural morphism.


The functors p′∗λ : Cov(Xλ) → Cov(X) define a pseudo-cocone in the 2-category Cat,whence a functor :

(7.1.5) 2-colimλ∈Λ

Cov(Xλ)→ Cov(X).

Lemma 7.1.6. In the situation of (7.1.4), the functor (7.1.5) is an equivalence.

Proof. It is a rephrasing of [32, Th.8.8.2, Th.8.10.5] and [33, Ch.IV, Prop.17.7.8(ii)].

Lemma 7.1.7. Let X be a scheme, j : U → X an open immersion with dense image, andf : X ′ → X an integral surjective and radicial morphism. The following holds :

(i) The morphism f induces an equivalence of sites

f ∗ : X ′et → Xet

and the functor f ∗ : Cov(X)→ Cov(X ′) is an equivalence.(ii) Suppose that X is separated. Then :

(a) The functor j∗ : Cov(X)→ Cov(U) is faithful.(b) If moreover, X is reduced and normal, and has finitely many maximal points, j∗ is

fully faithful, and its essential image consists of all the objects ϕ of Cov(U) suchthat ϕ×X X(x) lies in the essential image of the pull-back functor :

Cov(X(x))→ Cov(X(x)×X U)

for every geometric point x of X . (Notation of definition 2.4.17(ii).)(iii) Furthermore, if X is locally noetherian and regular, and X \U has codimension ≥ 2

in X , then j∗ is an equivalence.

Proof. (i) follows from [4, Exp.VIII, Th.1.1].(ii.a): Indeed, let ϕ : E → X be any finite etale morphism; it is easily seen that E is a

separated scheme, and since ϕ is flat and U is dense in X , we see that E ×X U is a densesubscheme of E, so the claim is clear.

(ii.b): Choose a covering X =⋃i∈I Vi consisting of affine open subsets, and let

g : X ′ :=∐i∈I

Vi → X

be the induced morphism; set also j′i := j ×X Vi for every i ∈ I . By lemma 7.1.2, g isof universal 2-descent for the fibred category Cov. On the other hand, X ′′ := X ′ ×X X ′ isseparated, and the induced open immersion j′′ : U ×X X ′′ → X ′′ has dense image, hencethe corresponding functor j′′∗ is faithful, by (i). By corollary 1.5.35(ii), the full faithfulnessof j∗ follows from the full faithfulness of the pull-back functor j′∗ corresponding to the openimmersion j′ := j ×X X ′. The latter holds if and only if the same holds for all the pull-backfunctors j′∗i . Hence, we may replace X by Vi, and assume from start that X is affine, sayX = SpecA. Let η1, . . . , ηs ∈ X be the maximal points. Under the current assumptions,A is the product of s domains, and its total ring of fractions FracA is the product of fieldsκ(η1)× · · · × κ(ηs). Let E → X and E ′ → X be two objects of Cov(X), and h : E ×X U →E ′ ×X U a morphism; we may write E = SpecB, E ′ = SpecB′ for finite etale A-algebrasB and B′, and the restrictions hηi := h ×U X(ηi) : E(ηi) → E ′(ηi) induce a map of FracA-algebras

h\η :=s∏i=1

h\ηi : B′ ⊗A FracA→ B ⊗A FracA.

On the other hand, by [33, Ch.IV, Prop.17.5.7], B (resp. B′) is the normalization of A inB ⊗A FracA (resp. in B′ ⊗A FracA). It follows that h\η restricts to a map B′ → B, and thecorresponding morphism E → E ′ is necessarily an extension of h. This shows that j∗ is fullyfaithful. To proceed, we make the following general remark.


Claim 7.1.8. Let Z be a scheme, V0 ⊂ Z an open subset, and ϕ : E → V0 an object ofCov(V0). Suppose that, for every open subset V ⊂ Z containing V0, the pull-back functorCov(V ) → Cov(V0) is fully faithful. Let F be the family consisting of all the data (V, ψ, α)where V ⊂ X is any open subset with V0 ⊂ V , ψ : EV → V is a finite etale morphism, and α :ψ−1V0

∼→ E is an isomorphism of V0-schemes. F is partially ordered by the relation such that(V, ψ, α) ≥ (V ′, ψ′, α′) if and only if V ′ ⊂ V and there is an isomorphism β : ψ−1V ′

∼→ EV ′of V ′-schemes, such that α′ β|ψ−1V ′ = α|ψ−1V ′ . Then F admits a supremum.

Proof of the claim. Using Zorn lemma, it is easily seen that F admits maximal elements, and itremains to show that any two maximal elements (V, ψ, α) and (V ′, α′, ψ) are isomorphic; to seethis, set V ′′ := V ∩ V ′ : by assumption, the isomorphism α′−1 α : ψ−1V0

∼→ ψ′−1V0 extendsto an isomorphism of V ′′-schemes ψ−1V ′′

∼→ ψ′−1V ′′, using which one can glue EV and EV ′to obtain a datum (V ∪ V ′, ψ′′, α′′) which is larger than both our maximal elements, hence it isisomorphic to both. ♦

Claim 7.1.9. In the situation of claim 7.1.8, suppose that Z is reduced and normal, and hasfinitely many maximal points. Let (Umax, ψ, α) be a supremum for F , and z a geometric pointof Z, such that the morphism ϕ ×Z Z(z) extends to a finite etale covering of Z(z); then thesupport of z lies in Umax.

Proof of the claim. By lemma 7.1.6, there exists an etale neighborhood g : Y → Z of z, withY affine, a finite etale morphism ϕY : EY → Y , and an isomorphism h : ϕ×U Y ' ϕY ×Z U .We have a natural essentially commutative diagram :

Cov(gY )α //

Desc(Cov, g) //

Cov(Y )

Cov(U ∩ gY )

β // Desc(Cov, g ×Z U) // Cov(Y ×Z U)

where α and β are equivalences, by lemma 7.1.2. Moreover, let Y ′ := Y ×Z Y and Y ′′ :=Y ′ ×Z Y ; by the foregoing, the functors

Cov(Y ′)→ Cov(Y ′ ×Z U) and Cov(Y ′′)→ Cov(Y ′′ ×Z U)

are fully faithful, hence the right square subdiagram is 2-cartesian (corollary 1.5.35(iii)). Thus,the datum (ϕ×Z gY, ϕY , h) determines an object ϕ′ of Cov(gY ), together with an isomorphismϕ′×Z U ' ϕ×Z gY , which we may use to glue ϕ and ϕ′ to a single object ϕ′′ of Cov(U ∪gY ).The claim follows. ♦

The foregoing shows that the assumptions of claim 7.1.8 are fulfilled, with Z := X , V0 := Uand any object ϕ of Cov(U), hence there exists a largest open subset Umax ⊂ X over which ϕextends. However, claim 7.1.9 shows that Umax = X , so the proof of (ii.b) is complete.

(iii): In view of (ii.b) and [33, Ch.IV, Cor.18.8.13], we are reduced to the case where X isa regular local scheme, and it suffices to show that j∗ is essentially surjective. We argue byinduction on the dimension n of X\U . If n = 0, then X\U is the closed point, in which caseit suffices to invoke the Zariski-Nagata purity theorem ([44, Exp.X, Th.3.4(i)]). Suppose n > 0and that the assertion is already known for smaller dimensions. Let ϕ be a given finite etalecovering of U , and x a maximal point of X \U ; then X(x)\U = x, so ϕ|U∩X(x) extendsto a finite etale morphism ϕx over X(x). In turns, ϕx extends to an affine open neighborhoodV ⊂ X of X , and up to shrinking V , this extension ϕ′ agrees with ϕ on U ∩ V , by lemma7.1.6. Hence we can glue ϕ and ϕ′, and replace U by U ∪V . Repeating the procedure for everymaximal point of X\U , we reduce the dimension of X\U ; then we conclude by the inductiveassumption.


Definition 7.1.10. Let f : X → S be a morphism of schemes, x a geometric point of Xlocalized at x ∈ X; and set s := f(x), s := f(x). Let also n ∈ N be any integer, and L ⊂ Nany non-empty set of prime numbers.

(i) We say that x is strict if κ(x) is a separable closure of κ(x). (Notation of (2.4.16).)(ii) We associate to x a strict geometric point xst, constructed as follows. Let κ(xst) :=

κ(x)s, the separable closure of κ(x) inside κ(x); then the inclusion κ(xst) ⊂ κ(x)defines a morphism of schemes :

(7.1.11) |x| → |xst| := Specκ(xst)

and x is the composition of (7.1.11) and a unique strict geometric point

xst : |xst| → X

localized at x.(iii) Let f : X → Y be a morphism of schemes; we call f(x)st the strict image of x in Y .

Notice that the natural identification |x| = |f(x)| induces a morphism of schemes :

|xst| → |f(x)st|.(iv) Denote by fx : X(x) → S(s) the morphism of strictly local schemes induced by f .

We say that f is locally (−1)-acyclic at the point x, if the scheme f−1x (ξ) is non-empty

for every strict geometric point ξ of S(s).(v) We say that f is locally 0-acyclic at the point x, if the scheme f−1

x (ξ) is non-empty andconnected for every strict geometric point ξ of S(s).

(vi) We say that a group G is a finite L-group if G is finite and all the primes dividingthe order of G lie in L. We say that G is an L-group if it is a filtered union of finiteL-groups.

(vii) We say that f is locally 1-aspherical for L at the point x, if we have :

H1(f−1x (ξ)et, G) = 1

for every strict geometric point ξ of S(s), and every L-group G (where 1 denotes thetrivial G-torsor).

(viii) We say that f is locally (−1)-acyclic (resp. locally 0-acyclic, resp. locally 1-asphericalfor L), if f is locally (−1)-acyclic (resp. locally 0-acyclic, resp. locally 1-asphericalfor L) at every point of X .

(ix) We say that f is (−1)-acyclic (resp. 0-acyclic) if the unit of adjunction : F → f∗f∗F

is a monomorphism (resp. an isomorphism) for every sheaf F on Set.

See section 2.1 for generalities about G-torsors for a group object G on a topos T . Here weshall be mainly concerned with the case where T is the etale topos X∼et of a scheme X , and Gis representable by a group scheme, finite and etale over X . In this case, using faithfully flatdescent one can show that any G-torsor is representable by a principal G-homogeneous space,i.e. a finite, surjective, etale morphism E → X with a G-action G → AutX(E) such that theinduced morphism of X-schemes

G× E → E ×X Eis an isomorphism.

7.1.12. If GX is the constant X∼et-group arising from a finite group G and X is non-empty andconnected, rightGX-torsors are also understood asG-valued characters of the etale fundamentalgroup ofX . Indeed, let ξ be a geometric point ofX; recall ([42, Exp.V, §7]) that π := π1(Xet, ξ)is defined as the automorphism group of the fibre functor

Fξ : Cov(X)→ Set (Ef−→ X) 7→ f−1ξ.


We endow π with its natural profinite topology, as in (1.6.7), so that Fξ can be viewed as anequivalence of categories

Fξ : Cov(X)→ π-Set

(see (1.6.7)).

Lemma 7.1.13. In the situation of (7.1.12), there exists a natural bijection of pointed sets :

H1(Xet, GX)∼→ H1

cont(π,G)

from the pointed set of right GX-torsors, to the first non-abelian continuous cohomology groupof P with coefficients in G (see (1.6.1)).

Proof. Let f : E → X be a right GX-torsor, and fix a geometric point s ∈ Fξ(E); given anyσ ∈ π, there exists a unique gs,σ ∈ G such that

s · gs,σ = σE(s).

Any g ∈ G determines a X-automorphism gE : E → E, and by definition, the automorphismFξ(gE) on Fξ(E) commutes with the left action of any element τ ∈ π; however Fξ(gE) is justthe right action of g on Fξ(E), hence we may compute :

s · gs,τ · gs,σ = (τE(s)) · gs,σ = τE(s · gs,τ ) = τE · σE(s)

so the rule σ 7→ gs,σ defines a group homomorphism ρs,f : π → G which is clearly continuous.We claim that the conjugacy class of ρs,f does not depend on the choice of s. Indeed, if s′ ∈Fξ(E) is another choice, there exists a (unique) element h ∈ G such that h(s) = s′; arguing asin the foregoing we see that σE commutes with the right action of h on Fξ(E). In other words,σE(s) = h−1 σE(s′), so that gs′,σ = h gs,σ h−1.

Therefore, denote by ρf the conjugacy class of ρs,f ; we claim that ρf depends only on theisomorphism class of the GX-torsor E. Indeed, any isomorphism t : E

∼→ E ′ of right GX-torsors induces a bijection Fξ(t) : Fξ(E)

∼→ Fξ(E′), equivariant for the action of G, and for any

σ ∈ π we have Fξ(t) σE = σE′ Fξ(t), whence the assertion.Conversely, given a continuous group homomorphism ρ : π → G, let us endow G with

the induced left π-action, and right G-action. Then G is an object of π-Set, to which therecorresponds a finite etale morphism Eρ → X , with an isomorphism Eρ×X |ξ|

∼→ G of sets withleft π-action. Since the right action of G is π-equivariant, we have a corresponding G-actionby X-automorphisms on Eρ, so Eρ is G-torsor, and its image under the map of the lemma isclearly the conjugacy class of ρ.

Finally, in order to show that the map of the lemma is injective, it suffices to prove that, forany right GX-torsor (f : E → X,ϕ : E × G → E) and any s ∈ Fξ(E), there exists anisomorphism of right GX-torsors Eρs,f

∼→ E. However, s and ρs,f determine an identificationof sets with left π-action :

(7.1.14) G∼→ Fξ(E)

whence an isomorphism t : Eρs,f∼→ E in Cov(X). Moreover, (7.1.14) also identifies the

right G-action on Fξ(E) to the natural right G-action on G; the latter is π-equivariant, hence itinduces a right G-action ϕ′ : E ×G→ E, such that t is G-equivariant. To conclude, it sufficesto show that ϕ = ϕ′. In view of [33, Ch.IV, Cor.17.4.8], the latter assertion can be checked onthe stalks over the geometric point ξ, where it holds by construction.

Remark 7.1.15. (i) In the situation of (7.1.12), let E → X be any right GX-torsor, andρE : π → G the corresponding representation. Then the proof of lemma 7.1.13 shows thatthe left π-action on Fξ(E) is isomorphic to the left π-action on G induced by ρE; especially, ρEis surjective if and only if π acts transitively on Fξ(E), if and only if the scheme E is connected


(since a decomposition of E into connected components corresponds to a decomposition ofFξ(E) into orbits for the π-action).

(ii) Let ϕ : G′ → G be a homomorphism of finite groups, and

H1cont(π, ϕ) : H1

cont(π,G′)→ H1

cont(π,G)

the induced map. Denote by r (resp. l) the right (resp. left) translation action of G on itself. Letalso E ′ → X be a principal G′-homogeneous space, given by a map ρ : G′ → AutX(E ′), anddenote by c′ ∈ H1

cont(π,G′) the class of E ′. Then the class c := H1

cont(π, ϕ)(c) can be describedgeometrically as follows. The scheme E ′ ×G admits an obvious right G-action, induced by r.Moreover, it admits as well a right G′-action : namely, to any element g ∈ G′, we assign theX-automorphism ρg × lϕ(g−1) : E ′ × G ∼→ E ′ × G. Set E := (E × G)/G′; it is easily seenthat E is a principal G-homogeneous space, and its class is precisely c (the detailed verificationshall be left as an exercise for the reader).

(iii) Consider a commutative diagram of schemes

E ′g //

f ′ AAAAAAAA E

f~~~~~~~~~~

X

where f (resp. f ′) is a G-torsor (resp. a G′-torsor) for a given finite group G (resp. G′). Denoteby ρ : G→ AutX(E) and ρ′ : G′ → AutX(E ′) the respective actions. We have :

(a) Suppose that there exists a group homomorphism ϕ : G′ → G, such that ρ ϕ = g ρ′.Then g induces a G′-equivariant map

Fξ(E′)→ Res(ϕ)Fξ(E)

whence an isomorphism (E ′ × G)/G′∼→ G of G-torsors. Hence, let c (resp. c′)

denote any representative of the equivalence class of E (resp. E ′) in H1cont(π,G) (resp.

H1cont(π,G

′)); in view of (ii), it follows that

H1(π, ϕ)(c′) = c.

In other words, the induced diagram of continuous group homomorphisms

π1(Xet, ξ)c′

zzuuuuuuuuuuc

$$IIIIIIIIII

G′ϕ // G

commutes, up to composition with an inner automorphism of G. (Details left to therreader.)

(b) If E is connected, a group homomorphism ϕ : G → G′ fulfilling the condition of (a)exists and is unique up to composition with an inner automorphism of G. Indeed, fixany e′ ∈ Fξ(E

′) and let e := f(e′); if g′ ∈ G′, define ϕ(g′) as the unique g ∈ Gsuch that f(e′ · g′) = e · g; also, in view of (i) we may pick σg′ ∈ π1(X, ξ) such thatσg′ · e′ = e′ · g′, and notice that σg′ · e = e · ϕ(g′) for every g′ ∈ G′. Now, if h′ ∈ G′ isany other element, we may compute :

f(e′ · g′h′) = f(σg′ · e′ · h′) = σg′ · f(e′ · h′) = σg′ · e · ϕ(h′) = e · ϕ(g′) · ϕ(h′)

whence ϕ(g′h′) = ϕ(g′) · ϕ(h′), as required.

Lemma 7.1.16. Let f : X → Y be a morphism of schemes, F , G two sheaves on Yet. Then :


(i) If F is locally constant and constructible, the natural map :

ϑf : f ∗HomY ∼et(F ,G )→HomX∼et

(f ∗F , f ∗G )

is an isomorphism.(ii) If f is 0-acyclic, the functor

f ∗ : Cov(Y )→ Cov(X) : (E → Y ) 7→ (E ×Y X → X)

is fully faithful.

Proof. (i): Suppose we have a cartesian diagram of schemes :

X ′g′ //

f ′

X

f

Y ′

g // Y.

Then, according to (2.3.2), we have a natural isomorphism :

ϑg′ g′∗ϑf ⇒ ϑf ′ f ′∗ϑg(an invertible 2-cell, in the terminology of (1.3.2)). Now, if g – and therefore g′ – is a coveringmorphism, ϑg and ϑg′ are isomorphisms, and g′∗ϑf is an isomorphism if and only if the sameholds for ϑf . Summing up, in this case ϑf is an isomorphism if and only if the same holds forϑf ′ . Thus, we may choose g such that g∗ is a constant sheaf, and after replacing f by f ′, wemay assume that F = SY is the constant sheaf associated to a finite set S. Since the functors

HomY ∼et(−,G ) : (Y ∼et )o → Y ∼et and f ∗ : Y ∼et → X∼et

are left exact, we may further reduce to the case where S = 1 is the set with one element,in which case F = 1Y is the final object of Y ∼et , and f ∗F = 1X is the final object of X∼et .Moreover, we have a natural identification :

HomY ∼et(1Y ,G )

∼→ G : σ 7→ σ(1)

and likewise for HomX∼et(1Y , f

∗G ). Using the foregoing characterization, it is easily checkedthat, under these identifications, ϑf is the identity map of f ∗G , whence the claim.

(ii): It has already been remarked that Cov(Y ) is equivalent to the category of locally con-stant constructible sheaves on Yet, and likewise for Cov(X). Let E and F be two objects ofCov(Y ); we have natural bijections :

HomCov(X)(f∗E, f ∗F )

∼→Γ(X,HomX∼et(f ∗E, f ∗F ))

∼→Γ(Y, f∗f∗HomY ∼et

(E,F )) by (i)∼→Γ(Y,HomY ∼et

(E,F )) since f is 0-acyclic∼→HomCov(Y )(E,F )

as stated.

Lemma 7.1.17. Let f : X → S be a quasi-compact morphism of schemes, and suppose that :

(a) f is locally (−1)-acyclic.(b) For every strict geometric point ξ of S, the induced morphism fξ : f−1(ξ) → |ξ| is

0-acyclic (i.e. f has non-empty geometrically connected fibres).

Then f is 0-acyclic.


Proof. Let F be a sheaf on Set. For every strict geometric point ξ of S, we have a commutativediagram :

(7.1.18)

Fξ

εξ //

α

(f∗f∗F )ξ

Γ(|ξ|, ξ∗F )f∗ξ // Γ(f−1(ξ), f ∗ξ ξ∗F )

where ε : F → f∗f∗F is the unit of adjunction. The map α is an isomorphism, and the

same holds for f ∗ξ , since fξ is 0-acyclic. Hence εξ is injective, which shows already that fis (−1)-acyclic. It remains to show that εξ is surjective. Hence, let t ∈ (f∗f

∗F )ξ be anysection. From (7.1.18) we see that there exists a section t′ ∈ Fξ such that the images of t andεξ(t

′) agree on Γ(f−1(ξ), f ∗ξ ξ∗F ). We may find an etale neighborhood g : U → S of ξ,such that t′ (resp. t) extends to a section t′U ∈ F (U) (resp. tU ∈ Γ(X ×S U, f ∗F )). LetXU := X ×S U , fU := f ×S U : XU → U , and for every geometric point x of XU , denote byf ∗x : FfU (x) → f ∗UFx the natural isomorphism (2.4.21). We set

V := x ∈ XU | tU,x = f ∗x(t′U,f(x))

where x is any geometric point of X localized at x, and tU,x ∈ f ∗Fx (resp. t′U,f(x) ∈ FfU (x))denotes the image of tU (resp. of t′U ). Clearly V is an open subset of XU , and we have :

Claim 7.1.19. (i) V = f−1U fU(V ).

(ii) fU(V ) ⊂ U is an open subset.

Proof of the claim. (i): Given a point u ∈ U , choose a strict geometric point u localized at u,and set s := g(u)st; by assumption, the morphism fs : f−1(s) → |s| is 0-acyclic, hence theimage of tU in Γ(f−1(s), f ∗s s∗F ) is of the form f ∗s t

′′, for some t′′ ∈ Fs. It follows thatV ∩ f−1

U (u) is either the whole of f−1U (u) or the empty set, according to whether t′′ agrees or

not with the image of t′U in Fs = g∗Fu.(ii): The subset X \V is closed, especially pro-constructible; since f is quasi-compact, we

deduce that fU(X \V ) is a pro-constructible subset of U ([30, Ch.IV, Prop.1.9.5(vii)]). It thenfollows from (i) that fU(V ) is ind-constructible, hence we are reduced to showing that fU(V )is closed under generizations ([30, Ch.IV, Th.1.10.1]). To this aim, since V is open, it sufficesto show that fU is generizing, i.e. that the induced maps XU(x) → U(u) are surjective, forevery u ∈ U and every x ∈ f−1

U (u). However, choose a geometric point x localized at x, andlet u := fU(x); since the natural maps XU(x) → XU(x) and U(u) → U(u) are surjective, itsuffices to show that the same holds for the map fU,x : XU(x) → U(u). The image of x (resp.U ) in X (resp. in S) is a geometric point which we denote by the same name; since the naturalmaps XU(x) → X(x) and U(u) → S(u) are isomorphisms, we are reduced to showing thatfx : X(x)→ S(u) is surjective, which holds, since f is locally (−1)-acyclic. ♦

Set W := fU(V ); in view of claim 7.1.19, W is an etale neighborhood of ξ, and the naturalmap F (W ) → f ∗F (U) sends the restriction t′U |W of t′U to the restriction tU |V of tU , whencethe claim.

7.1.20. Consider now a cartesian diagram of schemes :

(7.1.21)X ′

g′ //

f ′

X

f

S ′

g // S


where g is a local morphism of strictly local schemes, and denote by s (resp. by s′) the closedpoint of S (resp. of S ′). Let x′ ∈ f ′−1(s′) be any point, x′ a geometric point of X ′ localized atx′, and set x := g′(x′), x := g′(x′). Then g′ induces a morphism of S ′-schemes :

(7.1.22) X ′(x′)→ X(x)×S S ′.

Lemma 7.1.23. In the situation of (7.1.20), suppose that g is an integral morphism. Then :(i) The induced morphism f ′−1(s′) → f−1(s) induces a homeomorphism on the underly-

ing topological spaces.(ii) (7.1.22) is an isomorphism.

Proof. If g is integral, κ(s′) is a purely inseparable algebraic extension of κ(s), hence the mor-phism T ′ := Specκ(s′) → T := Specκ(s) is radicial, and the same holds for the inducedmorphisms :

f ′−1(s′)∼→ f−1(s)×T T ′ → f−1(s) f−1

x (s)×T T ′ → f−1x (s)

([26, Ch.I, Prop.3.5.7(ii)]). Especially (i) holds, and therefore the natural map X ′(x′) →X(x) ×S S ′ is an isomorphism; we see as well that f−1

x (s) ×T T ′ is a local scheme. Thenthe assertion follows from [33, Ch.IV, Rem.18.8.11].

Proposition 7.1.24. Let f : X → S and g : S ′ → S be morphisms of schemes, with g quasi-finite, and set X ′ := X ×S S ′. Suppose that f is locally (−1)-acyclic (resp. locally 0-acyclic);then the same holds for f ′ := f ×S S ′ : X ′ → S ′.

Proof. Let s′ be any geometric point of S ′, and set s := g(s′). Denote by s ∈ S (resp. s′ ∈ S ′)the support of s (resp. s′); then f is locally (−1)-acyclic at the points of f−1(s), if and onlyif fs := f ×S S(s) : X ×S S(s) → S(s) enjoys the same property at the points of f−1

s (s).Likewise, f ′ is locally (−1)-acyclic (resp. locally 0-acyclic) at the points of f ′−1(s′), if andonly if f ′s′ : X ′ ×S′ S ′(s′) → S ′(s′) enjoys the same property at the points of f−1

s (s). Hence,we may replace g by gs′ : S ′(s′)→ S(s), and f by the induced morphism X ×S S(s)→ S(s),which allows to assume that g is finite ([33, Ch.IV, Th.18.5.11]), hence integral. Let ξ′ (resp.x′) be any strict geometric point of S ′ (resp. of f ′−1(s′)), and let ξ := g(ξ′)st, (resp. let x be theimage of x′ in X); we have natural morphisms :

f ′−1x′ (ξ′)

α−→ X(x)×S ξ′β−→ f−1

x (ξ).

However, α is an isomorphism, by lemma 7.1.23(ii), and β is a radicial morphism, since the fieldextension κ(ξ) ⊂ κ(ξ′) is purely inseparable ([26, Ch.I, Prop.3.5.7(ii)]). The claim follows.

Lemma 7.1.25. Let S be a strictly local scheme, s ∈ S the closed point, f : X → S a morphismof schemes, x (resp. ξ) a strict geometric point of f−1(s) (resp. of S). Then we may find :

(a) A cartesian diagram (7.1.21), with S ′ strictly local, irreducible and normal.(b) A strict geometric point x′ of f ′−1(s′) with g′(x′)st = x.(c) A strict geometric point ξ′ of S ′ localized at the generic point of S ′, with g(ξ′)st = ξ,

and such that (7.1.22) induces an isomorphism :

(7.1.26) f ′−1x′ (ξ′)

∼→ f−1x (ξ).

Proof. Denote by Z ⊂ S the closure of the image of ξ, endow Z with its reduced subschemestructure, set Y := X ×S Z ⊂ X , and let hx : Y (x) → Z be the natural morphism. ThenZ is a strictly local scheme ([33, Ch.IV, Prop.18.5.6(i)]). Moreover, the closed immersionY → X induces an isomorphism of Z-schemes : Y (x)

∼→ X(x) ×S Z (lemma 7.1.23(ii)). Byconstruction, ξ factors through a strict geometric point ξ′ of Z, and we deduce an isomorphism :h−1x (ξ′)

∼→ f−1x (ξ) of Z-schemes. Thus, we may replace (S,X, ξ) by (Z, Y, ξ′), and assume that

S is the spectrum of a strictly local domain, and ξ is localized at the generic point of S. Say that


S = SpecA, and denote by Aν the normalization of A in its field of fractions F . Then Aν is theunion of a filtered family (Aλ | λ ∈ Λ) of finite A-subalgebras of F ; since A is henselian, eachAλ is a product of henselian local rings, hence it is a local henselian ring, so the same holds forAν . Moreover, the residue field κ(s′) of Aν is an algebraic extension of the residue field κ(s) ofA, which is separably closed, hence κ(s′) is separably closed, i.e. Aν is strictly henselian, so wemay fulfill condition (a) by taking S ′ := SpecAν . Condition (b) holds as well, due to lemma7.1.23(i). Finally, it is clear that ξ lifts to a unique strict geometric point ξ′ of S ′, and it followsfrom lemma 7.1.23(ii) that (7.1.26) is an isomorphism, as required.

Remark 7.1.27. In the situation of lemma 7.1.25, suppose furthermore that S is noetherian. Adirect inspection reveals that the scheme S ′ exhibited in the proof of the lemma, is the spectrumof the normalization of a noetherian domain. Quite generally, the normalization of a noetheriandomain is a Krull domain ([63, Th.33.10]).

7.1.28. In the situation of (7.1.4), let x ∈ X be any point, and s := f(x). Let also ξ be a strictgeometric point of S. We deduce a compatible system of points xλ := p′λ(x) ∈ Xλ, whence acofiltered system of local schemes

X := (Xλ(xλ) | λ ∈ Λ).

For every λ ∈ Λ, set ξλ := pλ(ξ)st. This yields a compatible system of strict geometric points

(ξλ | λ ∈ Λ), such that :ξ∼→ lim

λ∈Λξλ.

Choose a geometric point x of X localized at x, and set likewise xλ := p′λ(x); then the sys-tem X lifts to a system X sh := (Xλ(xλ) | λ ∈ Λ), whose limit is naturally isomorphicto X(x) ([33, Ch.IV, Prop.18.8.18(ii)]). Furthermore, X sh induces a natural isomorphism ofκ(ξ)-schemes :

(7.1.29) f−1x (ξ)

∼→ limλ∈Λ

f−1λ,xλ

(ξλ)

where, as usual, fx : X(x) → S (resp. fλ,xλ : Xλ(xλ) → Sλ) is deduced from f (resp. fromfλ). These remarks, together with the following lemma 7.1.30, and the previous lemma 7.1.25,will allow in many cases, to reduce the study of the fibres of fx, to the case where the base S isstrictly local, excellent and normal.

Lemma 7.1.30. Let S be a strictly local normal scheme. Then there exists a cofiltered familyS := (Sλ | λ ∈ Λ) consisting of strictly local normal excellent schemes, such that :

(a) S is isomorphic to the limit of S .(b) The natural morphism S → Sλ is dominant for every λ ∈ Λ.

Proof. Say that S = SpecA, and write A as the union of a filtered family A := (Aλ | λ ∈ Λ)of excellent noetherian local subrings, which we may assume to be normal, by [31, Ch.IV,(7.8.3)(ii),(vi)]. Proceeding as in (7.1.28), we choose a compatible family of geometric pointssλ localized at the closed points of SpecAλ, for every λ ∈ Λ; using these geometric points,we lift A to a filtered family (Ash

λ | λ ∈ Λ) of strict henselizations, whose colimit is natu-rally isomorphic to A. Moreover, each Ash

λ is noetherian, normal and excellent ([33, Ch.IV,Prop.18.8.8(iv), Prop.18.8.12(i)] and proposition 4.8.35(ii)). Let η be the generic point of S,hλ : S → Sλ := SpecAsh

λ the natural morphism, and ηshλ := hλ(η) for every λ ∈ Λ. The

cofiltered system (Sλ | λ ∈ Λ) fulfills condition (a). Moreover, by construction, the image ofηshλ in SpecAλ is the generic point ηλ; then ηsh

λ is the generic point of Sλ, since the latter is theonly point of Sλ lying over ηλ. Hence (b) holds as well.

Proposition 7.1.31. Let f : X → S be a flat morphism of schemes. We have :(i) f is locally (−1)-acyclic.


(ii) Suppose moreover, that f has geometrically reduced fibres, and :(a) either f is locally finitely presented,(b) or else, S is locally noetherian.

Then f is locally 0-acyclic.

Proof. Let x ∈ X be any point, set s := f(x), choose a geometric point x of X localized at x,set s := f(x), and let ξ be any strict geometric point of S(s).

(i): If f is flat, the induced morphism fx : X(x) → S(s) is faithfully flat; especially, fx issurjective ([61, Th.7.3(i)]).

(ii): We shall use the following :

Claim 7.1.32. Let A be a Krull domain, F the field of fractions of A, B a flat A-algebra, andsuppose that B ⊗A κ(p) is reduced, for every prime ideal p ∈ SpecA of height one. Then B isintegrally closed in B ⊗A F .

Proof of the claim. See [61, §12] for the basic generalities on Krull domains; especially, [61,Th.12.6] asserts that, if A is a Krull domain, the natural sequence of A-modules :

E : 0→ A→ F →⊕

ht p=1

F/Ap → 0

is short exact, where p ∈ SpecA ranges over the prime ideals of height one. By flatness, E ⊗ABis still exact, hence B =

⋂ht p=1 B ⊗A Ap (where the intersection takes place in B ⊗A F ). We

are therefore reduced to the case where A is a discrete valuation ring. Let t denote a chosengenerator of the maximal ideal of A, and suppose that x ∈ B ⊗A F is integral over B, so thatxn + b1x

n−1 + · · ·+ bn = 0 in B⊗A F , for some b1, . . . , bn ∈ B; let also r ∈ N be the minimalinteger such that we have x = t−rb for some b ∈ B. We have to show that r = 0; to this aim,notice that bn + trb1b

n−1 + · · · + trnbn = 0 in B; if r > 0, it follows that the image b of b inB/tB satisfies the identity : bn = 0, therefore b = 0, since by assumption, B/tB is reduced.Thus b = tb′ for some b′ ∈ B, and x = t1−rb′, contradicting the minimality of r. ♦

Now, in the general situation of (ii), setX ′ := X×S S(s). The natural morphismX(x)→ Xfactors uniquely through a morphism of S(s)-schemes j : X(x) → X ′, and if we denote byx′ the image in X ′ of x, then j induces an isomorphism of S(s)-schemes : X(x)

∼→ X ′(x′).Hence, f is locally 0-acyclic at the point x, if and only if the induced morphism X ′ → S(s)is locally 0-acyclic at the support x′ of x′, so we may replace S by S(s), and assume that S isstrictly local, when (a) holds, and even strictly local and noetherian, when (b) holds.

We have to show that f−1x (ξ) is connected, and by lemma 7.1.25 and remark 7.1.27, we are

further reduced to the case where S = S(s) = SpecA is strictly local and normal, ξ is localizedat the generic point of S, and moreover :

(a’) either f is finitely presented,(b’) or else, A is a (not necessarily noetherian) Krull domain.

Claim 7.1.33. In case (b’) holds, f−1x (ξ) is connected.

Proof of the claim. Indeed, let F be the field of fractions of A; the field κ(ξ) is algebraic overF , hence it suffices to show that X(x) ×S SpecK is connected for every finite field extensionF ⊂ K ([32, Ch.IV, Prop.8.4.1(ii)]). Let AK be the normalization of A in K; then AK isagain a Krull domain ([14, Ch.VII, §1, n.8, Prop.12]), and B := Osh

X,x ⊗A AK is a flat AK-algebra. Notice that the geometric fibres of the induced morphism fx,K : SpecB → SpecAKare cofiltered limits of schemes that are etale over the fibres of f ; since the fibres of f aregeometrically reduced, it follows that the same holds for the fibres of fx,K . HenceB is integrallyclosed in B⊗AF (claim 7.1.32); especially these two rings have the same idempotents, whencethe contention. ♦


Finally, suppose that (a’) holds. By lemma 7.1.30, the scheme S is the limit of a cofilteredfamily (Sλ | λ ∈ Λ) of strictly local excellent and normal schemes, such that the natural mapsS → Sλ are dominant. By [32, Ch.IV, Th.8.8.2(ii)], we may find λ ∈ Λ, a finitely presentedmorphism fλ : Xλ → Sλ, and an isomorphism of S-schemes : X ∼→ Xλ ×Sλ S. We setXµ := Xλ ×Sλ Sµ, for every µ ∈ Λ with µ ≥ λ; then, up to replacing Λ by a cofinal subset,we may assume that fµ : Xµ → Sµ is defined for every µ ∈ Λ. By [32, Ch.IV, Cor.11.2.6.1],after replacing Λ by a cofinal system, we may assume that fλ is flat for every λ ∈ Λ. For everyµ ∈ Λ, let Zµ ⊂ Sµ be the subset consisting of all points s ∈ Sµ such that the fibre f−1

µ (s)is not geometrically reduced; by [32, Ch.IV, Th.9.7.7], Zµ is a constructible subset of Sµ. Byassumption, we have ⋂

µ∈Λ

p−1µ Zµ = ∅

and it is clear that p−1µ Zµ ⊂ p−1

λ Zλ whenever µ ≥ λ. Then, Zµ = ∅ for some µ ∈ Λ ([30, Ch.IV,Prop.1.8.2, Cor.1.9.8]), hence we may replace Λ by a still smaller cofinal subset, and achievethat all the fλ have geometrically reduced fibres. For every λ ∈ Λ, let xλ ∈ Xλ be the imageof the point x. Arguing as in (7.1.28), we obtain a compatible system of strict geometric pointsξλ of Sλ (resp. xλ of Xλ), such that pλ(ξ) factors through ξλ; whence an isomorphism (7.1.29).Thus, f−1

x (ξ) is reduced if and only if f−1λ,xλ

(ξλ) is reduced for every sufficiently large λ ∈ Λ([32, Ch.IV, Prop.8.7.2]). Furthermore, since pλ is dominant, ξλ is localized at the generic pointof Sλ, for every λ ∈ Λ. Thus, we are reduced to the case where S = S(s) is the spectrum ofa strictly local noetherian normal domain A, and ξ is localized at the generic point of S; sincesuch A is a Krull domain ([61, Th.12.4(i)]), this is covered by claim 7.1.33.

Example 7.1.34. (i) Let A be an excellent local ring, and A∧ the completion of A. Then thenatural morphism :

f : SpecA∧ → SpecA

is locally 0-acyclic. Indeed, this follows from proposition 7.1.31(ii) (and from the excellenceassumption, which includes the geometric regularity of the formal fibres of A).

(ii) Suppose additionally, that A is strictly local. Then f is 0-acyclic. To see this, we applythe criterion of lemma 7.1.17 : indeed, since f is flat, it is (−1)-acyclic (proposition 7.1.31(i));it remains to show that f has geometrically connected fibres, and since A∧ is strictly local ([33,Ch.IV, Prop.18.5.14]), this is the same as showing that f is locally 0-acyclic at the closed pointof SpecA∧, which has already been remarked in (i).

(iii) More generally, f is 0-acyclic whenever A is excellent and henselian. Indeed, in thiscase the argument of (ii) again reduces to showing that f has geometrically connected fibres.However, consider the natural commutative diagram :

(7.1.35)

Spec (A∧)sh //

f sh

SpecA∧

f

SpecAsh // SpecA.

Since A is henselian, Ash is the colimit of a filtered family of finite etale and local A-algebras.Since A and A∧ have the same residue field, it follows easily that A∧ ⊗A Ash is the colimit ofa filtered family of finite etale and local A∧-algebras, hence it is strictly henselian, and there-fore (7.1.35) is cartesian, especially the geometric fibres of f are connected if and only if thesame holds for the geometric fibres of f sh, and the latter are reduced (even regular), since A isexcellent. Hence, we come down to showing that f sh is locally 0-acyclic at the closed point ofSpec (A∧)sh, which holds again by proposition 7.1.31(ii).

For future use, we point out the following


Proposition 7.1.36. Let g : X → Y be a flat morphism of excellent noetherian schemes, withX strictly local, and Y normal. Let U ⊂ X be an open subset, and Z ⊂ Y a closed subscheme.Suppose that :

(i) g−1(z) ⊂ U for every maximal point z of Z.(ii) U ∩ g−1(z) is a dense open subset of g−1(z), for every z ∈ Z.

(iii) The fibres g−1(z) are reduced, for every z ∈ Z.Then the induced functor Cov(U)→ Cov(U ×Y Z) is fully faithful.

Proof. Indeed, say that X = SpecB, Y = SpecA, Z = V (I) for some ideal I ⊂ A, anddenote by B∧ the mB-adic completion of the local ring B (where mB ⊂ B denotes the maximalideal). Let also f : SpecB∧ → Y be the induced morphism, and U∧ ⊂ SpecB∧ the preimageof U . In light of example 7.1.34(ii) and lemma 7.1.16(ii), it suffices to show that the inducedfunctor Cov(U∧) → Cov(U∧ ×Y Z) is fully faithful. In view of lemma 7.1.3, we are furtherreduced to checking that conditions (a)–(c) of proposition 5.5.44 hold for the induced ringhomomorphism ϕ : A → B∧, the open subset U∧, and the ideal I . However, by example5.5.39, we have AssA(I, A) = Max(Z), hence (c) follows trivially from our assumption (i).Next, since B∧ is a faithfully flat B-algebra, assumption (ii) implies that U∧∩f−1(z) is a denseopen subset of f−1(z), for every z ∈ Z. Moreover, since B is excellent, the natural morphismSpecB∧ → X is regular, so the same holds for the induced morphism f−1(z) → g−1(z), andthen our assumption (iii) implies – together with [61, Th.32.3(i)] – that f−1(z) is reduced, forevery z ∈ Z, whence condition (b). Lastly, we check condition (a), i.e. we show that B∧ isI-adically complete. Indeed, let C be the I-adic completion of B∧; the natural map B∧ → C isinjective, and it admits a left inverse, constructed as follows. Let a := (an | n ∈ N) be a givensequence of elements of B∧, which is Cauchy for the I-adic topology; then a is also Cauchy forthe mB-adic topology, and it is easily seen that the limit l of a in the mB-adic topology dependsonly on the class [a] of a in C, so we get a well defined ring homomorphism λ : C → B∧ bythe rule : [a] 7→ l, and clearly λ is the sought left inverse. It remains to check that λ is injective;thus, suppose that l = 0, and that [a] 6= 0; this means that there existsN ∈ N such that an /∈ IN ,for every n ∈ N. Now, the induced sequence (an | n ∈ N) of elements of B∧/IN is stationary,and on the other hand, it converges mB-adically to 0; therefore an = 0 for every sufficientlylarge n ∈ N, a contradiction.

7.1.37. Let A be a noetherian normal ring, and endow the A-algebra A[[t]] with its t-adic topol-ogy. Let

ϕ : X := Spf A[[t]]→ X := SpecA[[t]] π : X → S := SpecA i : S → X

be respectively the natural morphism of locally ringed spaces, the natural projection, and theclosed immersion determined by the ring homomorphismA[[t]]→ A given by the rule : f(t) 7→f(0), for every f(t) ∈ A[[t]]. Let also U0 ⊂ SpecA be an open subset, U := π−1U0 andU := ϕ−1U . Finally, denote by E a locally free OU -module of finite rank, and set E ∧ := ϕ∗|UE ,which is a locally free OU-module of finite rank.

Lemma 7.1.38. In the situation of (7.1.37), suppose that S \U0 has codimension ≥ 2 in S.Then:

(i) The natural mapΓ(U,E )→ Γ(U,E ∧)

is an isomorphism of A[[t]]-modules.(ii) The pull-back functor :

i∗|U0: Cov(U)→ Cov(U0) : (E → U) 7→ (E ×U U0 → U0)

is an equivalence.


Proof. (i): To begin with, set Z := S\U0; since the morphism π is flat, hence generizing ([61,Th.9.5]), the closed subset X \U = π−1Z has codimension ≥ 2 in X . Since A and A[[t]] areboth normal, we deduce :

(7.1.39) depthX\V OX ≥ 2 depthZOS ≥ 2

(theorem 5.4.20(ii) and [61, Th.23.8]); therefore (corollary 5.4.22) :

(7.1.40) Γ(U,OX) = Γ(X,OX) = A[[t]].

Next, the short exact sequences of OX-modules :

0→ i∗OS → OX/tn+1OX → OX/t

nOX → 0 for every n ∈ N

induce exact sequences

(7.1.41) RjΓZi∗OS → RjΓZOX/tn+1OX → RjΓZOX/t

nOX for every n, j ∈ N.

Then (7.1.39) and (7.1.41) yield inductively :

depthZOX/tnOX ≥ 2 for every n ∈ N

and again corollary 5.4.22 implies :

(7.1.42) Γ(U,OX/tnOX) = A[t]/tnA[t] for every n ∈ N.

Since U is quasi-compact, we may find a left exact sequence P := (0 → E → O⊕mU → O⊕nU )of OU -modules (corollary 5.2.17). Since ϕ is a flat morphism of locally ringed spaces, thesequence ϕ∗P is still left exact. Since the global section functors are left exact, we are thenreduced to the case where E = OU . Then we may write :

E ∧ = OU = limn∈N

OU/tnOU

where, for each n ∈ N, we regard OU/tnOU as a sheaf of (pseudo-discrete) rings on U =V (t) ⊂ U . The functor Γ(U,−) is a right adjoint, hence commutes with limits, and we deducean isomorphism :

Γ(U,E ∧)∼→ lim

n∈NΓ(U,OU/t

nOU).

(This is even a homeomorphism, provided we view the target as a limit of rings with the discretetopology.) Taking (7.1.42) into account, we obtain Γ(U,E ∧) = A[[t]] which, together with(7.1.40), implies the contention.

(ii): Notice first that (i) and lemma 5.5.42 imply that Lef(U, i(U0)) holds (see definition5.5.40). Since the pull-back functor π∗|U : Cov(U0)→ Cov(U) is a right quasi-inverse to i|U0 ,it is clear that the latter is essentially surjective. The full faithfulness is a special case of lemma7.1.3.

7.2. Local asphericity of smooth morphisms of schemes. Let S be a strictly local scheme,s ∈ S the closed point, f : X → S a smooth morphism, x any geometric point of f−1(s), anddenote by fx : X(x)→ S the induced morphism of strictly local schemes. For any open subsetU ⊂ S we have a base change functor :

(7.2.1) f ∗x : Cov(U)→ Cov(f−1x U) (E → U) 7→ (E ×U f−1

x U).

Theorem 7.2.2. In the situation of (7.2), we have :(i) The functor (7.2.1) is fully faithful.

(ii) Suppose moreover that S is excellent and normal, and that S\U has codimension ≥ 2in S. Then (7.2.1) is an equivalence of categories.


Proof. (i): In view of lemma 7.1.16(ii) it suffices to show that fx is 0-acyclic (since in that case,the same will obviously hold also for its restriction f−1

x U → U ). To begin with, fx is locally(−1)-acyclic, by proposition 7.1.31(i), hence it remains only to show that f is locally 0-acyclicat the point x (lemma 7.1.17). The latter assertion follows from proposition 7.1.31(ii) and [33,Ch.IV, Th.17.5.1].

(ii): In light of (i), it suffices to show that (7.2.1) is essentially surjective, under the assump-tions of (ii). We argue by induction on the relative dimension n of f . Let x ∈ X be the supportof x. We may find an open neighborhood U ⊂ X of x, and an etale morphism of S-schemesϕ : U → An

S ([33, Ch.IV, Cor.17.11.4]). Let x′ := ϕ(x); there follows an isomorphism ofS-schemes : X(x)

∼→ AnS(x′), hence we may assume from start that X = An

S , and f is thenatural projection. Especially, the theorem holds for n = 0. Suppose then, that n > 0, and thatthe theorem is already known when the relative dimension is < n. Write f as the compositionf = h g, where

g : X ' An−1S ×S A1

S → A1S and h : A1

S → S

are the natural projections; set x1 := g(x), and U1 := h−1x1U , (where hx1 : S1 := A1

S(x1)→ S isthe morphism induced by h). We have S1\U1 = h−1

x1(S\U), and since flat maps are generizing

([61, Th.9.5]) we easily see that the codimension of S1 \U1 in S1 equals the codimension ofS\U in S. From our inductive assumption, we deduce that the base change functor Cov(U1)→Cov(f−1

x U) is essentially surjective, and hence it suffices to show that the same holds for thefunctor Cov(U)→ Cov(U1). Thus, we are reduced to the case where X = A1

S . Suppose now,that E → f−1

x U is a finite etale morphism; we can write fx as the limit of a cofiltered familyof smooth morphisms (fλ : Yλ → S | λ ∈ Λ), where each Yλ is an affine etale A1

S-scheme.Then f−1

x U is the limit of the family (Yλ×S U | λ ∈ Λ). By [32, Ch.IV, Th.8.8.2(ii), Th.8.10.5]and [33, Ch.IV, Prop.17.7.8], we may find a λ ∈ Λ, a finite etale morphism Eλ → Yλ ×S Uand an isomorphism of f−1

x U -schemes : Eλ ×Yλ A1S(x)

∼→ E. Denote by y ∈ Yλ the imageof the closed point of A1

S(x), and by y the geometric point of Yλ obtained as the image of x(the latter is viewed naturally as a geometric point of A1

S(x)); by construction, y lies in theclosed fibre Y0 := Yλ ×S Specκ(s), which is an etale A1

κ(s)-scheme, and we may therefore finda specialization z ∈ Y0 of y, with z a closed point. Pick a geometric point z of Y0 localized atz, and a strict specialization map Yλ(z) → Yλ(y) as in (2.4.22); there follows a commutativediagram :

A1S(x) ' Yλ(y) //

Yλ(z)

Yλ(y) // Yλ(z).

The finite etale covering Eλ ×Yλ Yλ(y)→ Yλ(y)×S U lies in the essential image of the functor

Cov(Yλ(z)×S U)→ Cov(Yλ(y)×S U) C 7→ C ×Yλ(z) Yλ(y).

It follows that E → f−1x U lies in the essential image of the functor

Cov(f−1λ,zU)→ Cov(f−1

x U) C 7→ C ×Y (z) Y (y) ' C ×Y (z) A1S(x)

and therefore it suffices to show that the pull-back functor Cov(U) → Cov(f−1z U) is essen-

tially surjective. In other words, we may replace x by z, and assume throughout that x is aclosed point of A1

S .

Claim 7.2.3. Under the current assumptions, we may find a strictly local normal scheme T , withclosed point t, a finite surjective morphism g : T → S, and a finite morphism of Specκ(s)-schemes :

Specκ(t)→ Specκ(x).


Proof of the claim. Since κ(x) is a finite extension of κ(s), it is generated by finitely manyalgebraic elements u1, . . . un, and an easy induction allows to assume that n = 1. In this case,one constructs first a scheme T ′ by taking any lifting of the minimal polynomial of u1 : forthe details, see e.g. [28, Ch.0, (10.3.1.2.)], which shows that the resulting T is local, finite andflat over S, so T ′ maps surjectively onto S. Next, we may replace T ′ by its maximal reducedsubscheme, which is still strictly local and finite over S. Next, since S is excellent, the nor-malization (T ′)ν of T ′ is finite over S ([31, Ch.IV, Scholie 7.8.3(vi)]); let T be any irreduciblecomponent of (T ′)ν ; by [33, Ch.IV, Prop.18.8.10], T fulfills all the sought conditions. ♦

Choose g : T → S as in claim 7.2.3; since the residue field extension κ(s) → κ(t) isalgebraic and purely inseparable, there exists a unique point x′ ∈ A1

S(x)×S T lying over t, andwe may find a unique strict geometric point x′ of A1

S(x)×S T localized at x′, and lying over x.In view of [33, Ch.IV, Prop.18.8.10], there follows a natural isomorphism of T -schemes :

A1S(x)×S T

∼→ A1T (x′).

Denote by fx′ := fx ×S T : A1T (x′) → T the natural projection, and set UT := g−1U ; since

the morphism g : T → S is generizing ([61, Th.9.4(ii)]), it is easily seen that T \UT hascodimension ≥ 2 in T .

Let F : Cov → Sch be the fibred category (7.1.1). We have a natural essentially commuta-tive diagram of categories :

(7.2.4)

Cov(U) //

Desc(F, g ×S U)

δ

Cov(f−1x U) // Desc(F, g ×S f−1

x U)

where, for any morphism of schemes h, we have denoted by Desc(F, h) the category of descentdata for the fibred category F , relative to the morphism h.

According to lemma 7.1.2, the morphism g is of universal 2-descent for the fibred categoryF , so the horizontal arrows in (7.2.4) are equivalences. Hence, the theorem will follow, oncewe know that δ is essentially surjective. However, we have :

Claim 7.2.5. (i) Set U ′T := UT ×S T and U ′′T := U ′T ×S T . The pull-back functors :

Cov(U ′T )→ Cov(A1T (x′)×T U ′T ) Cov(U ′′T )→ Cov(A1

T (x′)×T U ′′T )

are fully faithful.(ii) Suppose that the pull-back functor

Cov(UT )→ Cov(f−1x′ UT )

is essentially surjective. Then the same holds for the functor δ.

Proof of the claim. (i): Let z′′ be any geometric point of X ′′ := A1T ×S T ×S T whose strict

image in A1T is x′, and let z′ be the image of z′′ in X ′ := A1

T ×S T ; by lemma 7.1.23(ii), thenatural morphisms :

X ′(z′)→ A1T (x′)×S T X ′′(z′′)→ A1(x′)×S T ×S T

are isomorphisms (notice that T ×S T is also strictly local). Then the claim follows fromassertion (i) of the theorem, applied to the projections X ′ → T ×S T and X ′′ → T ×S T ×S T .

(ii): Recall that an object of Desc(F, g ×S f−1x′ U) consists of a finite etale morphism E ′T →

f−1x′ UT and a X ′-isomorphism β′ : E ′T ×S T

∼→ T ×S E ′T fulfilling a cocycle condition onE ×S T ×S T . By assumption, ET descends to a finite etale morphism ET → UT ; then (i)implies that β′ descends to a U ′T -isomorphism β : ET ×S T

∼→ T ×S ET , and the cocycleidentity for β′ descends to a cocycle identity for β. ♦


In view of claim 7.2.5, we may replace (S, U, x) by (T, UT , x′), and therefore assume that x

is a κ(s)-rational point of A1κ(s). In this case, any choice of coordinate t on A1

S yields a sectionσx : S → A1

S(x) of the natural projection, such that σx(s) = x. To conclude the proof of thetheorem, it suffices to show that the pull-back functor :

Cov(f−1x U)

σ∗x−→ Cov(U)

is fully faithful.Say that S = SpecA; then the scheme A1

S(x) is the spectrum of At, the henselizationof A[t] along the ideal mt generated by t and the maximal ideal m of A. Let A∧ (resp.At∧) be the m-adic (resp. mt-adic) completion of A (resp. of At), and notice the naturalisomorphism:

At∧/tAt∧ ∼→ A∧

(indeed, it is easy to check that At∧ ' A∧[[t]]), whence a natural diagram of schemes :

X∧ := SpecAt∧ g′ //

π

SpecAtfx

S∧ := SpecA∧g //

σ

OO

SpecA

σx

OO

(where π is the natural projection) whose horizontal arrows commute with both the downwardarrows and the upward ones. Set U∧ := g−1U ; by example 7.1.34(ii) and lemma 7.1.16(ii), thepull-back functors

g∗ : Cov(U)→ Cov(U∧) g′∗ : Cov(f−1x U)→ Cov(π−1U∧)

are fully faithful. Consequently, we are easily reduced to showing that the pull-back functor :Cov(π−1U∧)

σ∗−→ Cov(U∧) is an equivalence. The latter holds by lemma 7.1.38(ii).

Example 7.2.6. As an application of theorem 7.2.2, suppose that K ⊂ E is an extension ofseparably closed fields, VK a geometrically normal and strictly local K-scheme, U ⊂ VK anopen subset, and ξ a geometric point of VE := VK ×K E, whose image in VK is supported onthe closed point. Then the induced functor

Cov(U)→ Cov(U ×VK VE(ξ))

is fully faithful, and it is an equivalence in case VK\U has codimension ≥ 2 in VK .Indeed, let Ka (resp. Ea) be an algebraic closure of K (resp. E), and choose a homomor-

phism Ka → Ea extending the inclusion of K into E. Then both VKa := VK ×K Ka andVEa(ξ) := VE(ξ) ×E Ea are still normal and strictly local (lemma 7.1.23(ii)), and the inducedfunctors

Cov(U)→ Cov(U ×K Ka) Cov(U ×VK VE(ξ))→ Cov(U ×VK VEa(ξ))

are equivalences (lemma 7.1.7(i)). It then suffices to show that the induced functor

Cov(U ×K Ka)→ Cov(U ×VK VEa(ξ))

has the asserted properties. Hence, we may replace K by Ka and E by Ea, and assume fromstart that K ⊂ E is an extension of algebraically closed fields. In this case, E can be writtenas the colimit of a filtered family (Rλ | λ ∈ Λ) of smooth K-algebras; correspondingly, VE isthe limit of a cofiltered system (Vλ | λ ∈ Λ) of smooth VK-schemes, and – by lemma 7.1.6 –Cov(U ×VK VE(ξ)) is the 2-colimit of the system of categories

Cov(U ×VK Vλ(ξλ)) (λ ∈ Λ)

(where, for each λ ∈ Λ, we denote by ξλ the image of ξ in Vλ). Now the contention followsdirectly from theorem 7.2.2.


Theorem 7.2.7. Let f : X → S is a smooth morphism of schemes, L ⊂ N be a set of primes,and suppose that all the elements of L are invertible in OS . Then f is 1-aspherical for L.

Proof. Let x be any geometric point of X , s := f(x), and η a strict geometric point of S(s). Toease notation, set T := X(x), let fx : T → S be the natural map, and Tη := f−1

x (η); we haveto show that H1(Tη,et, G) = 1 for every L-group G. Arguing as in the proof of proposition7.1.31, we reduce to the case where S = S(s). Then, by lemma 7.1.25, we can further assumethat S is normal and η is localized at the generic point η of S. By lemma 7.1.30, S is the limitof a cofiltered system (Sλ | λ ∈ Λ) of strictly local, normal and excellent schemes, and as usual,after replacing Λ by a cofinal subset, we may assume that f (resp. η) descends to a compatiblesystem of morphisms (fλ : Xλ → Sλ | λ ∈ Λ), (resp. of strict geometric points ηλ localizedat the generic point of Sλ). By [33, Ch.IV, Prop.17.7.8(ii)], there exists λ ∈ Λ such that fµis smooth for every µ ≥ λ. Then, in view of [4, Exp.VII, Rem.5.14] and the isomorphism(7.1.29), we may replace f by fλ, and η by ηλ, and assume from start that S is strictly local,normal and excellent, and G is a finite L-group.

Let ϕ : Eη → Tη be a principal G-homogeneous space; we come down to showing thatEη admits a section Tη → Eη. By [33, Ch.IV, Prop.17.7.8(ii)] and [32, Ch.IV, Th.8.8.2(ii),Th.8.10.5], we may find a finite separable extension κ(η) ⊂ L, and a principal G-homogeneousspace

ϕL : EL → TL := T ×S SpecL ρL : G→ AutTL(EL)

such thatϕ = ϕL ×SpecL Specκ(η) ρ = ρL ×SpecL Specκ(η).

Say that S = SpecA, denote by AL the normalization of A in L, and set SL := SpecAL. ThenSL is again normal and excellent ([31, Ch.IV, (7.8.3)(ii),(vi)]), and the residue field of AL isan algebraic extension of the residue field of A, hence it is separably closed, so SL is strictlylocal as well. Thus, we may replace S by SL, and assume that Eη descends to a principalG-homogeneous space Eη → Tη := f−1

x (η) on Tη. Next, we may write η as the limit ofthe filtered system of affine open subsets of S, so that – by the same arguments – we find anaffine open subset U ⊂ S and a principal G-homogeneous space EU → TU := f−1

x U , witha G-equivariant isomorphism of Tη-schemes : EU ×TU Tη

∼→ Eη. Denote by D1, . . . , Dn theirreducible components of S \U which have codimension one in S, and for every i ≤ n, setD′i := f−1

x Di. Let also ηT be the generic point of T .

Claim 7.2.8. For given i ≤ n, let y be the generic point of Di, and z a maximal point of D′i. Wehave:

(i) T and EU are normal schemes, and T (y) is regular.(ii) D′i is a closed subset of pure codimension one in T .

(iii) Let my (resp. mz) be the maximal ideal of OS,y (resp. of OT,z); then my · OT,z = mz.(iv) Let t ∈ A be any element such that t · OS,y = my. Then there exist an integer m > 0

such that (m, charκ(s)) = 1, a finite etale covering

Ey → T (y)[t1/m] := T (y)×S SpecA[T ]/(Tm − t)

and an isomorphism of T (y)[t1/m]×T TU -schemes :

Ey ×T TU∼→ EU ×T T (y)[t1/m].

Proof of the claim. (i): Since S is normal by assumption, the assertion for T and EU followsfrom [33, Ch.IV, Prop.17.5.7, Prop.18.8.12(i)]. Next, set W := X(y); since OS,y is a discretevaluation ring, W is a regular scheme ([33, Ch.IV, Prop.17.5.8(iii)]). For any w ∈ T (y) ⊂W , the natural map OT (y),w → W (w)sh is faithfully flat, and W (w)sh is regular ([33, Ch.IV,Cor.18.8.13]), therefore OT (y),w is regular, by [30, Ch.0, Prop.17.3.3(i)].


(ii): Say that p ⊂ A is the prime ideal of height one such that V (p) = Di; to ease notation, letalso B := Osh

X,x. Let q1, . . . , qk ⊂ SpecB be the set of maximal points of D′i. Using the factthat flat morphisms are generizing ([61, Th.9.5]), one verifies easily that A ∩ qj = p for everyj ≤ k. Fix j ≤ k, and set q := qj . Since A is normal, Ap is a discrete valuation ring, hencepAp is a principal ideal, say generated by t ∈ Ap; therefore qBq is the minimal prime idealof Bq containing t, so qBq has height at most one, by Krull’s Hauptidealsatz ([61, Th.13.5]).However, a second application of [61, Th.9.5] shows that the height of q in B cannot be lowerthan one, hence q has height one, which is the contention.

(iii): From (i) and (ii) we see that OS,y and OT,z are discrete valuation rings; then the assertionfollows easily from [33, Ch.IV, Th.17.5.1].

(iv): To begin with, since T (y) is regular, it decomposes as a disjoint union of connectedcomponents, in natural bijection with the set of maximal points of D′i. Let Z ⊂ T (y) bethe connected open subscheme containing z; it suffices to show that there exists a finite etalecovering :

EZ → Z[t1/m] := Z ×T (y) T (y)[t1/m]

with an isomorphism of Z[t1/m] ×T TU -schemes : EZ ×T TU∼→ EU ×T Z[t1/m]. By (iii) we

have t·OT,z = mz. Notice that T (z)×T TU = T (ηT ), andEηT := EU×T T (z) is a disjoint unionof spectra of finite separable extensions L1, . . . , Lk of κ(ηT ). Moreover, EηT is a principal G-homogeneous space over T (ηT ), i.e. every Lj is a Galois extension of κ(ηT ), with Galois groupGj := Gal(Lj/κ(ηT )) ⊂ G. Since G is an L-group, the same holds for Gj , hence EU ×T Z istamely ramified along the divisor z (the topological closure of z ⊂ Z), and the assertionfollows from Abhyankar’s lemma [42, Exp.XIII, Prop.5.2]. ♦

Claim 7.2.9. There exist :

(a) a finite dominant morphism S ′ → S, such that both S ′ and T ′ := T ×S S ′ are strictlylocal and normal;

(b) an open subset U ′ ⊂ S ′, such that S ′\U ′ has codimension ≥ 2 in S ′;(c) a finite etale morphism E ′ → T ′U ′ := T ×S U ′, with an isomorphism of T ′U ′-schemes :

E ′ ×T Tη ' Eη ×T T ′U ′ .

Proof of the claim. For every i ≤ n, let yi be the maximal point of Di, and choose ti ∈ A whoseimage in OS,yi generates the maximal ideal. Choose also mi ∈ N with (mi, charκ(s)) = 1

and such that there exists a finite etale covering Ei → T (yi)[t1/mii ] extending the etale covering

EU ×T T (yi)[t1/mii ] (claim 7.2.8(ii.d)). Let S ′ be the normalization of SpecA[t

1/m1

1 , . . . , t1/mns ].

Then S ′ is finite over S, hence it is excellent ([31, Ch.IV, (7.8.3)(ii,vi)]), and strictly local (cp.the proof of lemma 7.1.25). Set E ′η := Eη ×S S ′, T ′ := T ×S S ′; since the geometric fibres offx are connected (proposition 7.1.31(ii)), the same holds for the geometric fibres of the inducedmorphism T ′ → S ′, therefore T ′ is connected, and then it is also strictly local, by the usualarguments. Notice also that T ′ is the limit of a cofiltered family of smooth S ′-schemes, henceit is reduced and normal ([33, Ch.IV, Prop.17.5.7]). Say that E ′η = SpecC, T = SpecB,T ′ = SpecB′, and let C ′ be the integral closure of B′ in C. Notice that C ⊗B κ(ηT ) is a finiteproduct of finite separable extensions of the field B′ ⊗B κ(ηT ), and consequently the naturalmorphism ϕ′ : ET ′ := SpecC ′ → T ′ is finite ([61, §33, Lemma 1]). Define :

U ′ := y ∈ S ′ | ϕ′ ×S′ S ′(y) : ET ′(y)→ T ′(y) is etale.

Let now y ∈ U ′ any point; then S ′(y) is the limit of the cofiltered family (Uλ | λ ∈ Λ) ofaffine open neighborhoods of y in S ′, and ϕ′ ×S′ S ′(y) the limit of the system of morphisms(ϕ′λ := ϕ′ ×S′ Uλ | λ ∈ Λ); we may then find λ ∈ Λ such that ϕ′λ is etale ([33, Ch.IV,


Prop.17.7.8(ii)]), hence Uλ ⊂ U ′, which shows that U ′ is open. Furthermore, from [33, Ch.IV,Prop.17.5.7] it follows that EU ×T T ′ is normal, whence an isomorphism of T ′-schemes :

ET ′ ×S U ' EU ×T T ′

(cp. the proof of lemma 7.1.7(ii.b)) especially, U ×S S ′ ⊂ U ′. Likewise, by construction wehave natural morphisms : T ′(yi)→ T (yi)[t

1/mii ], and using the fact that all the schemes in view

are normal we deduce isomorphisms of T ′(yi)-schemes :

ET ′(yi)∼→ Ei ×T (yi)[t

1/mii ]

T ′(yi).

Thus, U ′ contains all the points of S ′ of codimension ≤ 1, since the image in S of any suchpoint lies in U ∪ y1, . . . , yn. The morphism E ′ := ET ′ ×S′ U ′ → T ′U ′ fulfills conditions(a)-(c). ♦

Now, choose S ′ → S, U ′ ⊂ S ′, and E ′ → T ′U ′ as in claim 7.2.9; since the correspondingT ′ is local, there exists a unique point x′ ∈ X ′ := X ×S S ′ lying over x; pick a geometricpoint x′ of X ′ localized at x′, and lying over x; it then follows from [33, Ch.IV, Prop.18.8.10]that the natural morphism X ′(x′) → T ′ is an isomorphism. In such situation, theorem 7.2.2says that there exists a finite etale covering E → U ′ with an isomorphism of T ′U ′-schemes :E ×U ′ T ′U ′

∼→ E ′, whence an isomorphism of Tη-schemes :

Eη ' E(η)×Specκ(η) Tη.

Since κ(η) is separably closed, the etale morphism E(η) → Specκ(η) admits a section, hencethe same holds for ϕ, as claimed.

7.2.10. Let f : X → S be a morphism of schemes, and j : U ⊂ X an open immersion suchthat Uη := U ∩ f−1(η) 6= ∅ for every η ∈ Smax, where Smax ⊂ S denotes the subset of allmaximal points of S. We deduce a natural essentially commutative diagram of functors :

D(S, f, U) :

Cov(X)j∗ //

∏η ι∗η

Cov(U)∏η ι∗η|U

∏η∈Smax

Cov(f−1η)

∏η j∗η //∏

η∈SmaxCov(Uη)

where jη : Uη → f−1(η) is the restriction of j and ιη : f−1(η) → X is the natural immersion.Let us say that U ⊂ X is fibrewise dense, if f−1(s) ∩ U is dense in f−1(s), for every s ∈ S.Then we have :

Theorem 7.2.11. In the situation of (7.2.10), suppose that f is smooth, and U is fibrewisedense. The following holds :

(i) The restriction functor j∗ is fully faithful.(ii) The diagram D(S, f, U) is 2-cartesian.

(iii) If furthermore, f−1Smax ⊂ U , then j∗ is an equivalence.

Proof. Assertion (ii) means that the functors j∗ and ι∗η induce an equivalence (j, ι•)∗ from

Cov(X) to the category C (X,U) of data

(7.2.12) E := (ϕ, (ψη, αη | η ∈ Smax))

where ϕ (resp. ψη) is an object of Cov(U) (resp. of Cov(f−1η), for every η ∈ Smax), andαη : ϕ×U Specκ(η)

∼→ ψη ×f−1η Uη is an isomorphism of U -schemes, for every η ∈ Smax (see(1.3.16)). On the basis of this description, it is easily seen that (i),(ii)⇒(iii). Furthermore, weremark :

Claim 7.2.13. (i) If j∗ is fully faithful, then the same holds for (j, ι•)∗.


(ii) For every open subset U ′ ⊂ X containing U , suppose that :(a) The pull-back functor Cov(U ′)→ Cov(U) is fully faithful.(b) If f−1Smax ⊂ U ′, the pull-back functor Cov(X)→ Cov(U ′) is an equivalence.

Then assertion (ii) holds.

Proof of the claim. (i): Since f−1η is a normal (even regular) scheme ([33, Ch.IV, Prop.17.5.7]),the pull-back functors ι∗η are fully faithful (lemma 7.1.7(ii.b)); the assertion is an immediateconsequence.

(ii): In light of (i), it remains only to check that (j, ι•)∗ is essentially surjective. Thus, let

ϕ : E → U be a finite etale morphism, such that i∗ηϕ extends to a finite etale morphismϕ′η : E ′η → f−1(η), for every maximal point η ∈ S. By claim 7.1.8, there is a largest opensubset Umax containing U , over which ϕ extends to a finite etale morphism ϕmax. To conclude,we have to show that Umax = X . However, for any maximal point η, let iη : f−1(η) → X(η)be the natural closed immersion. By lemma 7.1.7(i), i∗η is an equivalence, hence we may find afinite etale morphism ϕ′(η) : E ′(η) → X(η) such that i∗ηϕ

′(η) ' ϕ′η. By the same token, we also

see that E ′(η)×X(η) U(η) is U(η)-isomorphic to E ×U U(η).Next, S(η) is the limit of the filtered system V of all open subsets V ⊂ S with η ∈ V , hence

lemma 7.1.6 ensures that we may find V ∈ V and an object ϕ′V : E ′V → f−1V of Cov(f−1V )such that ϕ′V ×V S(η) ' ϕ′(η), and after shrinking V , we may also assume (again by lemma7.1.6) that E ′V ×X U is U -isomorphic to E ×S V . Hence we may glue E ′ and E along thecommon intersection, to deduce a finite etale morphism E ′ → U ′ := U ∪ f−1V that extendsϕ. It follows that f−1(η) ⊂ f−1V ⊂ Umax. Since η is arbitrary, (b) implies that the pull-backfunctor Cov(X) → Cov(Umax) is an equivalence, especially ϕ lies in the essential image ofj∗, as claimed. ♦

Claim 7.2.14. (i) Suppose that S is noetherian and normal, and X is separated. Then (i) holds.(ii) If furthermore, S is also excellent, then (ii) holds as well.

Proof of the claim. (i): Under the assumptions of the claim, X is normal and noetherian ([33,Ch.IV, Prop.17.5.7]), so (i) follows from lemma 7.1.7(ii.b), which also says – more generally –that assumption (a) of claim 7.2.13(ii) holds in this case, hence in order to show (ii) it sufficesto check that assumption (b) of claim 7.2.13(ii) holds whenever U ∪ f−1Smax ⊂ U ′ ⊂ X ,especially X\U ′ has codimension≥ 2 in X . Suppose first that S is regular; then the same holdsfor X ([33, Ch.IV, Prop.17.5.8]), and the contention follows from lemma 7.1.7(iii).

In the general case, let Sreg ⊂ S be the regular locus, which is open since S is excellent,and contains all the points of codimension ≤ 1, by Serre’s normality criterion ([31, Ch.IV,Th.5.8.6]). Consider the restriction f−1Sreg → Sreg of f , and the fibrewise dense open im-mersion jreg : U ′ ∩ f−1Sreg ⊂ f−1Sreg; by the foregoing, the functor j∗reg is an equivalence,hence we are easily reduced to showing that the functor Cov(X) → Cov(U ′ ∪ f−1Sreg) is anequivalence, i.e. we may assume that V := f−1Sreg ⊂ U ′. Moreover, since the full faithfulnessof j∗ is already known, we only need to show that any finite etale morphism ϕ : E → U ′

extends to an object of Cov(X). To this aim, by lemma 7.1.7(ii.b), it suffices to prove thatϕ ×U ′ (X(x) ×X U ′) extends to an object of Cov(X(x)), for every geometric point x of X .Let s := f(x), and denote by s ∈ S the support of s; by assumption, we may find a geomet-ric point ξ of f−1(s), whose support lies in U ′ ∩ f−1(s), and a strict specialization morphismX(ξ)→ X(x). There follows an essentially commutative diagram :

Cov(X(x))ρ //

δ

Cov(X(x)×X V )

γ

Cov(S(s)×S Sreg)αoo

βuujjjjjjjjjjjjjjj

Cov(X(ξ))τ // Cov(X(ξ)×X V )


where α and β are both equivalences, by theorem 7.2.2(ii); hence γ is an equivalence as well.Moreover, both Cov(X(x)) and Cov(X(ξ)) are equivalent to the category of finite sets, and δis obviously an equivalence. By construction, γ(ϕ×U ′ (X(ξ)×X V )) lies in the essential imageof τ , hence ϕ×U ′ (X(ξ)×X V ) lies in the essential image of ρ, so say it is isomorphic to ρ(ϕ′)for some object ϕ′ of Cov(X(x)). Using (i) (and [33, Ch.IV, Prop.18.8.12]) one checks easilythat ϕ′ ×X U ′ ' ϕ×U ′ (X(ξ)×X U ′), whence the contention. ♦

Claim 7.2.15. Let m ∈ N be any integer. Assertions (i) and (ii) hold if S and X are affineschemes of finite type over SpecZ, the fibres of f have pure dimension m, and furthermore :

(7.2.16) dim f−1(s)\U < m for every s ∈ S.

Proof of the claim. Indeed, in this situation, S admits finitely many maximal points, hence thenormalization morphism Sν → S is integral and surjective. Set :

S2 := Sν ×S Sν U1 := U ×S Sν U2 := U ×S S2.

Let β : X1 := X ×S Sν → X , f1 : X1 → Sν and j2 : U2 → X2 := X ×S S2 be theinduced morphisms; clearly f−1

1 (s′) has pure dimension m for every s′ ∈ Sν , and from (7.2.16)we deduce that dim f−1

1 (s′) \U1 < m, especially, U1 is dense in every fibre of f1; by thesame token, U2 is dense in X2. Then j∗2 is faithful (lemma 7.1.7(ii.a)), and lemma 7.1.2 andcorollary 1.5.35(ii) imply that j∗ is fully faithful, provided the same holds for the functor j∗1 :Cov(X1) → Cov(U1). In other words, in order to prove assertion (i), we may replace (f, U)by (f1, U1), which allows to assume that S is an affine normal scheme, and then it suffices toinvoke claim 7.2.14, to conclude.

Concerning assertion (ii) : by the foregoing, we already know that j∗ is fully faithful, hencethe same holds for (j, ι•)

∗ (claim 7.2.13(i)). To show that (j, ι•)∗ is essentially surjective, let

E be an object as in (7.2.12) of the category C (X,U); the normalization morphism induces abijection Sνmax

∼→ Smax : ην 7→ η, and clearly κ(ην) = κ(η) for every η ∈ Smax, whence adatum

Eν := (ϕ1 := ϕ×U U1, (ψη, αη | ην ∈ Sνmax))

of the analogous category C (X1, U1); by claim 7.2.14(ii), we may find an object ϕ′1 of Cov(X1)

and an isomorphism α : ϕ′1 ×X1 U1∼→ ϕ1. Let U3 := U2 ×U U1, and denote by j3 : U3 →

X3 := X2 ×X X1 the natural open immersion; by the foregoing, we know already that both j∗2and j∗3 are fully faithful; then corollary 1.5.35(iii) says that the natural essentially commutativediagram :

Desc(Cov, β) //

Desc(Cov, β ×X U)

Cov(X1) // Cov(U1)

is 2-cartesian. Thus, let ρ : Cov(U)→ Desc(Cov, β ×X U) be the functor defined in (1.5.27);it follows that the datum (ϕ′1, ρ(ϕ), α) comes from a descent datum (ϕ′1, ω) in Desc(Cov, β).By lemma 7.1.2, the latter descends to an object ϕ′ of Cov(X), and by construction we have(j, ι•)

∗ϕ′ = E, as required. ♦

Next, we consider assertions (i) and (ii) in case where both X and S are affine. We mayfind an affine open covering X = V0 ∪ · · · ∪ Vn such that the fibres of f|Vi : Vi → fVi areof pure dimension i, for every i = 0, . . . , n ([33, Ch.IV, Prop.17.10.2]). For i = 0, . . . , n, letji : Vi ∩ U → Vi be the induced open immersion; we have natural equivalences of categories :

Cov(X)∼→

n∏i=0

Cov(Vi) Cov(U)∼→

n∏i=0

Cov(Vi ∩ U)


which induce a natural identification : j∗ = j∗0 × · · · × j∗n. It follows j∗ is fully faithful ifand only if the same holds for every j∗i , and moreover D(S, f, U) decomposes as a productof n diagrams D(S, f|Vi , U ∩ Vi). Hence we may replace f by f|Vm , for any m ≤ n, afterwhich we may also assume that all the fibres f have the same pure dimension m. In that case,notice that the assumption on U is equivalent to (7.2.16). Next, say that X = SpecA, and letI ⊂ A be an ideal such that V (I) = X \U ; we may write I as the union of the filtered family(Iλ | λ ∈ Λ) of its finitely generated subideals. Set Uλ := X\V (Iλ) for every λ ∈ Λ; it followsthat U =

⋃λ∈Λ Uλ. For every λ ∈ Λ, set

Zλ := s ∈ S | dim f−1(s)\Uλ < m.

Claim 7.2.17. (i) Zλ is a constructible subset of S, for every λ ∈ Λ.

(ii) We have Zλ ⊂ Zµ whenever µ ≥ λ, and moreover S =⋃λ∈Λ Zλ.

Proof of the claim. (i): Let Vλ := x ∈ X | dimx f−1(f(x)) \ Uλ = m; according to [32,

Ch.IV, Prop.9.9.1], every Vλ is a constructible subset ofX , hence f(Vλ) is a constructible subsetof S ([30, Ch.IV, Th.1.8.4]), so the same holds for Zλ = S\f(Vλ).

(ii): Let µ, λ ∈ Λ, such that µ ≥ λ; then it is clear that f−1(s)\Uλ ⊂ f−1(s)\Uµ for everys ∈ S; using (7.2.16), the claim follows easily. ♦

Claim 7.2.17 and [30, Ch.IV, Cor.1.9.9] imply that Zλ = S for every sufficiently large λ ∈ Λ.Hence, after replacing Λ by a cofinal subset, we may assume that all the open subsets Uλ arefibrewise dense. We have a natural essentially commutative diagram :

Cov(U) //

2-limλ∈Λ

Cov(Uλ)

∏η∈Smax

Cov(Uη) //∏

η∈Smax2-limλ∈Λ

Cov(f−1(η) ∩ Uλ)

whose horizontal arrows are equivalences (notation of definition 1.3.12(i)); it follows formallythat j∗ is fully faithful, provided the same holds for all the pull-back functors Cov(X) →Cov(Uλ), and likewise, D(S, f, U) is 2-cartesian, provided the same holds for all the diagramsD(S, f, Uλ). Hence, we may replace U by Uλ, and assume that U is constructible, and (7.2.16)still holds.

Next, we may write S as the limit of a cofiltered family (Sλ | λ ∈ Λ) of affine schemes offinite type over SpecZ, and f as the limit of a cofiltered family f• := (fλ : Xλ → Sλ ∈ Λ) ofaffine finitely presented morphisms, such that :

• The natural morphism gλ : S → Sλ is dominant for every λ ∈ Λ.• fλ is smooth for every λ ∈ Λ ([33, Ch.IV, Prop.17.7.8(ii)]), and fµ = fλ ×Sλ Sµ

whenever µ ≥ λ.

Furthermore, we may find λ ∈ Λ such that U = Uλ ×Sλ S ([32, Ch.IV, Cor.8.2.11]), so that j isthe limit of the cofiltered system of open immersions (jµ : Uµ := Uλ ×Sλ Sµ → Xλ | µ ≥ λ),and after replacing Λ by a cofinal subset, we may assume that jµ is defined for every µ ∈ Λ.For every λ ∈ Λ and n ∈ N, let Xλ,n ⊂ Xλ be the open and closed subset consisting of allx ∈ Xλ such that dimx f

−1f(x) = n; clearly f• restricts to a cofiltered family f•,m := (fλ|Xλ,m :Xλ,m → Sλ | λ ∈ Λ), whose limit is again f . Hence we may replace Xλ by Xλ,m, and assumethat the fibres of fλ have pure dimension m, for every λ ∈ Λ. For every λ ∈ Λ, let :

Z ′λ := s ∈ Sλ | dim f−1λ (s)\Uλ = m.


and endow Z ′λ with its constructible topology Tλ; since Z ′λ is a constructible subset of Sλ ([32,Ch.IV, Prop.9.9.1]), (Z ′λ,Tλ) is a compact topological space, and due to (7.2.16), we have :

limλ∈Λ

Z ′λ = ∅.

Then [20, Ch.I, §9, n.6, Prop.8] implies that Z ′λ = ∅ for every sufficiently large λ ∈ Λ. Set :

Cov(X•) := 2-colimµ≥λ

Cov(Xµ) Cov(U•) := 2-colimµ≥λ

Cov(Uµ).

(See definition 1.3.12(ii).) There follows an essentially commutative diagram of categories:

(7.2.18)

Cov(X) //

j∗

Cov(X•)

j∗•

Cov(U) // Cov(U•)

where j∗• is the 2-colimit of the system of pull-back functors j∗µ : Cov(Xµ) → Cov(Uµ). Inlight of lemma 7.1.6, the horizontal arrows of (7.2.18) are equivalences, so j∗ will be fullyfaithful, provided the same holds for the functors j∗µ, for every large enough µ ∈ Λ.

Hence, in order to prove assertion (i) when X and S are affine, we may assume that S is offinite type over SpecZ, the fibres of f have pure dimension m, and (7.2.16) holds, which is thecase covered by claim 7.2.15.

Concerning assertion (ii), since the morphism gµ is dominant, for every η′ ∈ (Sµ)max we mayfind η ∈ Smax such that gµ(η) = η′. Denote by :

h : f−1η → f−1µ η′ and j′µ : (Uµ)η′ := Uµ ∩ f−1

µ η′ → f−1µ η′

the natural morphisms. With this notation, we have the following :

Claim 7.2.19. The induced essentially commutative diagram :

Cov(f−1µ η′)

j′∗µ //

h∗

Cov((Uµ)η′)

h∗U

Cov(f−1η)j∗η // Cov(Uη)

is 2-cartesian.

Proof of the claim. The pair (h∗, j′∗µ ) induces a functor (h, j′µ)∗ from Cov(f−1µ η′) to the category

of data of the form (ϕ, ϕ′, α), where ϕ′ (resp ϕ) is a finite etale covering of (Uµ)η′ (resp. off−1η) and α : ϕ×f−1η Uη

∼→ ϕ′×η′ η is an isomorphism in Cov(Uη), and the contention is that(h, j′µ)∗ is an equivalence. The full faithfulness of the functors j′∗µ and j∗η (lemma 7.1.7(ii.b))easily implies the full faithfulness of (h, j′µ)∗. To prove that (h, j′µ)∗ is essentially surjective,amounts to showing that if ϕ′ : E ′ → (Uµ)η′ is a finite etale morphism and

ϕ′′ := ϕ′ ×η′ η : E ′′ := E ′ ×η′ η → Uη

extends to a finite etale morphism ϕ : E → f−1η, then ϕ′ extends to a finite etale coveringof f−1

µ η′. Now, let L be the maximal purely inseparable extension of κ(η′) contained in κ(η).Since the induced morphism η′′ := SpecL→ Specκ(η′) is radicial, the base change functors

Cov(f−1µ η′)→ Cov((f−1

µ η′)×η′ η′′) Cov((Uµ)η′)→ Cov((Uµ)η′ ×η′ η′′)are equivalences (lemma 7.1.7(i)). Thus, we may replace η′ by η′′, and assume that the fieldextension κ(η′) ⊂ κ(η) is separable, hence the induced morphism Specκ(η) → Specκ(η′)is regular ([14, Ch.VIII, §7, no.3, Cor.1]), and then the same holds for the morphism h ([31,Ch.IV, Prop.6.8.3(iii)]). Given ϕ′ as above, set A := (j′µ ϕ′)∗OE′; then A is a quasi-coherent


Of−1µ η′-algebra, and we may define the quasi-coherent Of−1

µ η′-algebra B as the integral clo-sure of Of−1

µ η′ in A . By [31, Ch.IV, Prop.6.14.1], h∗B is the integral closure of Of−1η inh∗A = jη∗ h∗U(ϕ′′∗OE′′) = jη∗(ϕ∗OE). By [33, Ch.IV, Prop.17.5.7], it then follows thath∗B = ϕ∗OE , therefore B is a finite etale Of−1

µ η′-algebra ([33, Ch.IV, Prop.17.7.3(ii)] and [31,Ch.IV, Prop.2.7.1]). The claim follows. ♦

By the foregoing, we already know that j∗ is fully faithful, hence the same holds for (j, ι•)∗

(claim 7.2.13(i)). To show that (j, ι•)∗ is essentially surjective, consider any E as in (7.2.12);

we may find µ ∈ Λ, and a finite etale morphism ϕ′ : Eµ → Uµ such that ϕ = ϕ′×Uµ U , whenceobjects ϕ′η′ := ϕ′×Uµ (Uµ)η′ in Cov((Uµ)η′), for every η′ ∈ (Sµ)max. By construction, we haveϕ′η′ ×η′ η ' ψη×f−1η Uη for every η′ ∈ (Sµ)max and every η ∈ Smax such that gµ(η) = η′. Thenclaim 7.2.19 shows that, for every η′ ∈ (Sµ)max there exists an object ψ′η′ of Cov(f−1

µ η′) withisomorphisms :

ψ′η′ ×f−1µ η′ f

−1η ' ψη αη′ : ψ′η′ ×f−1µ η′ (Uµ)η′

∼→ ϕ′η′ .

Therefore, the datum Eµ := (ϕ′, (ψ′η′ , α′η′ | η′ ∈ (Sµ)max)) is an object of the 2-limit of the

diagram of categories

Cov(Uµ)j∗µ←− Cov(Xµ)

∏η′ ι∗η′−−−−→

∏η′∈(Sµ)max

Cov(f−1µ η′)

(where ιη′ : f−1µ η′ → Xµ is the natural immersion, for every η′ ∈ (Sµ)max). By claim 7.2.15,

the datum Eµ comes from an object ϕµ of Cov(Xµ). Let ϕ′′ be the image of ϕ′µ in Cov(X);by construction we have (j, ι•)

∗ϕ′′ = E, as required.This conclude the proof of (i) and (ii), in case X and S are affine. To deal with the general

case, let X =⋃i∈I Vi be a covering consisting of affine open subschemes, and for every i ∈ I ,

let fVi =⋃λ∈Λi

Siλ be an affine open covering of the open subscheme fVi ⊂ S; set alsoViλ := Vi ∩ f−1Siλ for every i ∈ I and λ ∈ Λi. The restrictions f|Viλ : Viλ → Siλ are smoothmorphisms; moreover, the image of the open immersion

jiλ := j|U∩Viλ : U ∩ Viλ → Viλ

is dense in every fibre of f|Vi . The induced morphism

(7.2.20) β : X ′ :=∐i∈I

∐λ∈Λi

Viλ → X

is faithfully flat, hence of universal 2-descent for (7.1.1); moreover, it is easily seen that X ′′ :=X ′ ×X X ′ is separated, and j′′ := j ×X X ′′ is a dense open immersion, hence j′′∗ is faithful(lemma 7.1.7(ii.a)). Then, by corollary 1.5.35(ii), j∗ is fully faithful, provided the pull-backfunctor Cov(X ′) → Cov(X ′ ×X U) is fully faithful, i.e. provided the same holds for thefunctors j∗iλ : Cov(Viλ)→ Cov(Viλ∩U). However, each Viλ is affine ([26, Ch.I, Prop.5.5.10]),hence assertion (i) is already known for the morphisms f|Viλ and the open subsets U ∩ Viλ; thisconcludes the proof of (i).

To show (ii), we use the criterion of claim 7.2.13(ii) : indeed, assumption (a) is alreadyknown, hence we are reduced to showing that assertion (iii) holds. To this aim, we consideragain the morphism β of (7.2.20), and denote by f ′′ : X ′′ → S the induced morphism. Clearlyf ′′ is smooth, and j′′ is an open immersion, such that f ′′−1(s) ×X U is dense in f ′′−1(s), forevery s ∈ S; then assertion (i) implies that j′′∗ is fully faithful. Moreover it is easily seen thatX ′′′ := X ′′ ×X X ′ is separated, and j ×X X ′′′ is a dense open immersion, so j′′′∗ is faithful(lemma 7.1.7(ii.a)), and therefore corollary 1.5.35(ii) reduces to showing that the pull-backfunctor Cov(X ′)→ Cov(X ′ ×X U) is an equivalence, or – what is the same – that this holdsfor the pull-back functors j∗iλ, which is already known.


7.2.21. We consider now the local counterpart of theorem 7.2.11. Namely, suppose that f :X → S is a smooth morphism, let x be any geometric point of X , set s := f(x), and lets ∈ S be the support of s. Then f induces the morphism fx : X(x) → S(s), and for everyopen immersion j : U → X(x), we may then consider the diagram D(S(x), fx, U) as in(7.2.10). Notice as well that, for every geometric point ξ of S, the fibre f−1

x (ξ) is normal, sinceit is a cofiltered limit of smooth |ξ|-schemes; on the other hand, f−1

x (ξ) is also connected, byproposition 7.1.31(ii), hence fx has geometrically irreducible fibres.

Theorem 7.2.22. In the situation of (7.2.21), suppose that U contains the generic point off−1x (s). Then :

(i) j∗ : Cov(X(x))→ Cov(U) is fully faithful.(ii) The diagram D(S(x), fx, U) is 2-cartesian.

Proof. (i): To begin with, since fx is generizing ([61, Th.9.5]), and S is local, every fibre of fxhas a point that specializes to the generic point ηs of f−1

x (s); since the fibres are irreducible, itfollows that the generic point of every fibre specializes to ηs. Therefore U is fibrewise dense inX(x), and moreover it is connected. Now, the category Cov(X) is equivalent to the categoryof finite sets, hence every object in the essential image of j∗ is (isomorphic to) a finite disjointunion of copies of U ; since U is connected, the morphisms of U -schemes between two suchobjects E and E ′ are in natural bijection with the set-theoretic mappings π0(E) → π0(E ′) oftheir sets of connected components, whence the assertion.

(ii): Let us write U as the union of a filtered family (Uλ | λ ∈ Λ) of constructible opensubsets of X(x); up to replacing Λ by a cofinal subset, we may assume that ηs ∈ Uλ for everyλ ∈ Λ. Arguing as in the proof of theorem 7.2.11, we see that D(S(x), fx, U) is the 2-limit ofthe system of diagrams D(S(x), fx, Uλ), hence it suffices to show the assertion with U = Uλ,for every λ ∈ Λ, which allows to assume that U is quasi-compact. Next, arguing as in the proofof proposition 7.1.31, we are reduced to the case where S = S(s). We may write X(x) asthe limit of a cofiltered system (Xλ | λ ∈ Λ) of affine schemes, etale over X , and for λ ∈ Λlarge enough, we may find an open subset Uλ ⊂ Xλ such that U = Uλ ×Xλ X(x) ([32, Ch.IV,Cor.8.2.11]). For every µ ≥ λ, set Uµ := Uλ ×Xλ Xµ, and denote by fµ : Xµ → S the naturalmorphism. Suppose first that S is irreducible; then, from lemma 7.1.6 it is easily seen thatD(S, fx, U) is the 2-colimit of the system of diagrams D(S, fµ, Uµ), so the assertion followsfrom theorem 7.2.11(ii) (more generally, this argument works whenever Smax is a finite set,since filtered 2-colimits of categories commute with finite products).

In the general case, let ϕ : E → U be a finite etale morphism, and suppose that ϕη :=ϕ×X(x) f

−1x (η) extends to an object ψη of Cov(f−1

x η), for every η ∈ Smax. The assertion boilsdown to showing that ϕ extends to an object ϕ′ of Cov(X(x)). To this aim, for every η ∈ Smax,let Zη → S be the closed immersion of the topological closure of η in S (which we endow withits reduced scheme structure); set also Yη := X ×S Zη. Then Zη is a strictly local scheme ([33,Ch.IV, Prop.18.5.6(i)]), and x factors through the closed immersion Y → X , which induces anisomorphism of Z-schemes :

Yη(x)∼→ X(x)×S Zη

(lemma 7.1.23(ii)). By the foregoing case, ϕ ×S Zη extends to an object ψη of Cov(Yη(x)).However, Zη is the limit of the cofiltered system (Zη,i | i ∈ I(η)) consisting of the constructibleclosed subschemes of S that contain Zη. By lemma 7.1.6, it follows that we may find i ∈ I(η)

and an object ψη,i of Cov(X(x) ×S Zη,i) whose image in Cov(Yη(x)) is isomorphic to ψη,and if i is large enough, ψη,i ×X(x) U agrees with ϕ ×S Zη,i in Cov(U ×S Zη,i). For eachη, η′ ∈ Smax, choose i ∈ I(η), i′ ∈ I(η′) with these properties, and to ease notation, set :

X ′η := X ×S Zη,i X ′′ηη′ := X ′η ×S Zη′,i′ ϕ′η := ψη,i


and denote by αη : ϕ′η ×X(x) U∼→ ϕ ×S Zη,i the given isomorphism. As in the foregoing, we

notice that x factors through X ′η, and the closed immersion X ′η → X induces an isomorphismX ′η(x)

∼→ X(x)×S Zη,i of Zη,i-schemes. According to [30, Ch.IV, Cor.1.9.9], we may then finda finite subset T ⊂ Smax such that the induced morphism :

β : X1 :=∐η∈T

X ′η(x)→ X(x)

is surjective. Set X2 := X1 ×X(x) X1 and X3 := X2 ×X(x) X1; notice that X2 is the disjointunion of schemes of the form X(x) ×S Zη,i ×S Zη′,i′ , for η, η′ ∈ T , and again, the latter isnaturally isomorphic to X ′′ηη′(x), for a unique lifting of the geometric point x to a geometricpoint of X ′′ηη′ . Similar considerations can be repeated for X3, and in light of (i), we deduce thatthe pull-back functors :

Cov(Xi ×X(x) U)→ Cov(Xi) i = 1, 2, 3

are fully faithful, in which case corollary 1.5.35(iii) says that the essentially commutative dia-gram of categories :

Desc(Cov, β) //

Desc(Cov, β ×X(x) U)

∏η∈T Cov(X ′η(x)) //

∏η∈T Cov(X ′η(x)×X(x) U)

is 2-cartesian. Let ρ : Cov(U) → Desc(Cov, β ×X(x) U) be the functor defined in (1.5.27);it follows that the datum ((ϕ′η, αη | η ∈ T ), ρ(ϕ)) comes from a descent datum (ϕ′1, ω) inDesc(Cov, β). By lemma 7.1.2, the latter descends to an object ϕ′ of Cov(X(x)), and byconstruction we have j∗ϕ′ = ϕ, as required.

7.3. Etale coverings of log schemes. We resume the general notation of (6.2.1), especially, wechoose implicitly τ to be either the Zariski or etale topology, and we shall omit further mentionof this choice, unless the omission might be a source of ambiguities.

7.3.1. Let Y := ((Y,N), T, ψ) be an object of the category Kint (see (6.6.10) : especially τ =Zar here), ϕ : T ′ → T an integral proper subdivision of the fan T (definition 3.5.22(ii),(iii)),and suppose that both T and T ′ are locally fine and saturated. Set ((Y ′, N ′), T ′, ψ′) := ϕ∗Y(proposition 6.6.14(iii)), and let f : Y ′ → Y be the morphism of schemes underlying thecartesian morphism ϕ∗Y → Y .

Proposition 7.3.2. In the situation of (7.3.1), the following holds :(i) For every geometric point ξ of Y , the fibre f−1(ξ) is non-empty and connected.

(ii) If Y is connected, the functor f ∗ : Cov(Y )→ Cov(Y ′) is an equivalence.

Proof. (i): The assertion is local on Y , hence we may assume that T = (SpecP )] for a fine,sharp and saturated monoid P . In this case, we may find a subdivision ϕ′ : T ′′ → T ′ such thatboth ϕ′ and ϕ ϕ′ are compositions of saturated blow up of ideals generated by at most twoelements of P (example 3.6.15(iii)). We are reduced to showing the assertion for the morphismsϕ′ and ϕ ϕ′, after which, we may further assume that ϕ is the saturated blow up of an idealgenerated by two elements of Γ(T,OT ), in which case the assertion follows from the moreprecise theorem 6.4.81.

(ii): First, we claim that the assertion is local on the Zariski topology of Y . Indeed, letY =

⋃i∈I Ui be a Zariski open covering, and set Uij := Ui ∩Uj for every i, j ∈ I; according to

corollary 1.5.35(ii) and lemma 7.1.2, it suffices to prove the contention for the objects (Ui ×Y


(Y,N), T, ψ|Ui) and (Uij ×Y (Y,N), T, ψ|Uij), and the morphism ϕ. Hence, we may againsuppose that T = SpecP , for a monoid P as in (i).

Arguing as in the proof of (i), we may next reduce to the case where ϕ is the saturated blowup of an ideal generated by two elements of Γ(T,OT ). We remark now that Y ′ is connected aswell. Indeed, suppose that Y ′ is the disjoint union of two open and closed subsets Z1 and Z2; itfollows easily from (i) that f(Z1) ∩ f(Z2) = ∅, and clearly Y = f(Z1) ∪ f(Z2). On the otherhand, proposition 6.6.31(ii) implies that f(Z1) and f(Z2) are closed in Y , which is impossible,since Y is connected.

Pick a geometric point ξ of Y ′; in this situation, assertion (ii) is equivalent to :

Claim 7.3.3. The continuous group homomorphism :

π1(Y ′et, ξ)→ π1(Yet, f(ξ))

induced by f , is an isomorphism of topological groups (see (7.1.12)).

Proof of the claim. In view of lemmata 7.1.13 and 1.6.2, it suffices to show that the inducedmap :

(7.3.4) H1(Yet, GX)→ H1(Y ′et, GX′)

is a bijection for every finite group G. However, the morphism of schemes f induces a mor-phism of topoi Y ′∼et → Y ∼et which we denote again by f ; let also g : Y ∼et → Set be the uniquemorphism of topoi (given by the global section functor on Y ∼et ). With this notation, theorem2.4.9 specializes to the exact sequence of pointed sets :

1 → H1(Yet, f∗GY ′)u−→ H1(Y ′et, GY ′)→ H0(Yet, R

1f∗GY ′).

On the other hand, assertion (i) and the proper base change theorem [5, Exp.XII, Th.5.1(i)] im-ply that the unit of adjunction GY → f∗f

∗GY = f∗GY ′ is an isomorphism, hence u is naturallyidentified to (7.3.4), and we are reduced to showing that the natural morphism τf,GY ′ : 1Y ∼et

→R1f∗GY ′ is an isomorphism (notation of (2.4.3)). The latter can be checked on the stalks, andin view of [5, Exp.XII, Cor.5.2(ii)] we are reduced to showing that H1(f−1(ξ)et, G) = 1 forevery finite group G, and every geometric point ξ of Y . However, according to theorem 6.4.81,the reduced geometric fibre f−1(ξ)red is either isomorphic to |ξ| (in which case the contentionis trivial), or else it is isomorphic to the projective line P1

κ(ξ), in which case – in view of lemma7.1.7(i) – it suffices to show that every finite etale morphism E → P1

κ(ξ) admits a section, whichis well known.

The class of etale morphisms of log schemes was introduced in section 6.3 : its definition andits main properties parallel those of the corresponding notion for schemes, found in [30, Ch.IV,§17]. In the present section this theme is further advanced : we will consider the logarithmicanalogue of the classical notion of etale covering of a scheme. To begin with, we make thefollowing :

Definition 7.3.5. (i) Let ϕ : T → S be a morphism of fans. We say that ϕ is of Kummer type,if the map (logϕ)t : OS,ϕ(t) → OT,t is of Kummer type, for every t ∈ T (see definition 3.4.40).

(ii) Let f : (Y,N)→ (X,M) be a morphism of log schemes.(a) We say that f is of Kummer type, if for every τ -point ξ of Y , the morphism of monoids

(log f)ξ : f ∗M ξ → N ξ is of Kummer type.(b) A Kummer chart for f is the datum of charts

ωP : PY → N ωQ : QX →M

and a morphism of monoids ϑ : Q → P such that (ωP , ωQ, ϑ) is a chart for f (seedefinition 6.1.15(iii)), and ϑ is of Kummer type.


Remark 7.3.6. (i) Let f : P → Q be a morphism of monoids of Kummer type, with P integraland saturated. It follows easily from lemma 3.4.41(iii,v), that the induced morphism of fans(Spec f)] : (SpecQ)] → (SpecP )] is of Kummer type.

(ii) In the situation of definition 7.3.5(ii), it is easily seen that f is of Kummer type, if andonly if the morphism of Y -monoids f ∗M ]

ξ → N ]ξ deduced from log f , is of Kummer type for

every τ -point ξ of Y .(iii) In the situation of definition 7.3.5(ii.b), suppose that the chart (ωP , ωQ, ϑ) is of Kummer

type, and Q is integral and saturated. It then follows easily from (i), (ii) and example 6.6.5(ii),that f is of Kummer type. In the same vein, we have the following :

Lemma 7.3.7. Let f : (Y,N) → (X,M) be a morphism of log schemes with coherent logstructures, ξ a τ -point of Y , and suppose that :

(a) M f(ξ) is fine and saturated.(b) The morphism (log f)ξ : M f(ξ) → N ξ is of Kummer type.

Then there exists a (Zariski) open neighborhood U of |ξ| in Y , such that the restriction f|U :(U,N |U)→ (X,M) of f is of Kummer type.

Proof. By corollary 6.1.34(i) and theorem 6.1.35(ii), we may find a neighborhood U ′ → Y ofξ in Yτ , and a finite chart (ωP , ωQ, ϑ) for the restriction f|U ′ , with Q fine and saturated. Set

SP := ω−1P,ξN

×ξ SQ := ω−1

Q,f(ξ)M×f(ξ) P ′ := S−1

P P Q′ := S−1Q Q.

According to claim 6.1.29(iii) we may find neighborhoods U ′′ → U ′ of ξ in Yτ , and V → X off(ξ) in Xτ , such that the charts ωP |U ′′ and ωQ|V extend to charts

ωP ′ : P ′U ′′ → N |U ′′ ωQ′ : Q′V →M |V .

Clearly ϑ extends as well to a unique morphism ϑ′ : Q′ → P ′, and after shrinking U ′′ wemay assume that the restriction f|U ′′ : (U ′′, N |U ′′) → (X,M) factors through a morphismf ′ : (U ′′, N |U ′′) → (V,M |V ), in which case it is easily seen that the datum (ωP ′ , ωQ′ , ϑ

′) is achart for f ′. Notice as well that Q′ is still fine and saturated (lemma 3.2.9(i)), and by claim6.1.29(iv) the maps ωP ′ and ωQ′ induce isomorphisms :

P ′]∼→ N ]

ξ Q′]∼→M ]

f(ξ).

Our assumption (b) then implies that the map Q′] → P ′] deduced from ϑ′ is of Kummer type,and then the morphism of fans Specϑ′ : SpecP ′ → SpecQ′ is of Kummer type as well (remark7.3.6(i)). However, let

ωP ′ : (U ′′, N ]|U ′′)→ (SpecP ′)] ωQ′ : (V,M ]

|V )→ (SpecQ′)]

be the morphisms of monoidal spaces deduced from ωP ′ and ωQ′; we obtain a morphism

(f ′, Specϑ′) : (U ′′, N |U ′′ , (SpecP ′)], ωP ′)→ (V,M |V , (SpecQ′)], ωQ′)

in the category K of (6.6.2), which – in view of remark 7.3.6(ii) – shows that f ′ is of Kummertype. This already concludes the proof in case τ = Zar, and for τ = et it suffices to remark thatthe image of U ′′ in Y is a Zariski open neighborhood U of ξ, such that the restriction f|U is ofKummer type.

We shall use the following criterion :

Proposition 7.3.8. Let k be a field, f : P → Q an injective morphism of fine monoids, withP sharp. Suppose that the scheme Spec k〈Q/mPQ〉 admits an irreducible component of Krulldimension 0. We have :

(i) f is of Kummer type, and k〈Q/mPQ〉 is a finite k-algebra.


(ii) If moreover, Q× is a torsion-free abelian group, then Q is sharp, and k〈Q/mPQ〉 is alocal k-algebra with k as residue field.

Proof. Notice that the assumption means especially that Q/mPQ 6= 1, so f is a local mor-phism. Set I := rad(mPQ) (notation of definition 3.1.8(ii)), and let p1, . . . , pn ⊂ Q be theminimal prime ideals containing I; then I = p1 ∩ · · · ∩ pn, by lemma 3.1.15. Clearly thenatural closed immersion Spec k〈Q/I〉 → Spec k〈Q/mPQ〉 is a homeomorphism; on the otherhand, say that q ⊂ k〈Q〉 is a prime ideal containing I; then q ∩ Q is a prime ideal of Qcontaining I , hence pi ⊂ q for some i ≤ n, i.e. Spec k〈Q/I〉 is the union of its closed sub-sets Spec k〈Q/piQ〉, for i = 1, . . . , n. The Krull dimension of each irreducible component ofSpec k〈Q/pi〉 = Spec k[Q\pi] equals

rkZ(Q\p)× + dimQ\p = rkZQ× + d− ht(pi) where d := dimQ

(claim 5.9.36(ii) and corollary 3.4.10(i,ii)). Our assumption then implies that Q× is a finitegroup, and ht(pi) = d, i.e. pi = mQ, for at least an index i ≤ n, and therefore I = mQ.Furthermore, since mQ is finitely generated, we have mn

Q ⊂ mPQ for a sufficiently large integern > 0, and then it follows easily that k〈Q/mPQ〉 is a finite k-algebra. If Q× is torsion-free,then we also deduce that Q is sharp, and moreover the maximal ideal of k〈Q/mPQ〉 generatedby mQ is nilpotent, hence the latter k-algebra is local.

Let now q ⊂ Q be any prime ideal, and pick x ∈ mQ\q; the foregoing implies that thereexists an integer r > 0 such that xr = f(pq)q for some pq ∈ mP and q ∈ Q, hence f(pq) /∈ q,and f(pq) is not invertible in Q, since f is local. If now q has height d− 1, it follows that f(pq)is a generator of (Q\q)R, which is an extremal ray of the polyhedral cone QR, and every suchextremal ray is of this form (proposition 3.4.7); furthermore, the latter cone is strongly convex,since Q× is finite. Hence the set S := (f(pq) | ht(q) = d − 1) is a system of generators ofthe polyhedral cone QR (see (3.3.15)). Let Q′ ⊂ Q be the submonoid generated by S; thenSQ = SR ∩QQ = QQ (proposition 3.3.22(iii)), i.e. f is of Kummer type.

7.3.9. Let f : (YZar, N) → (XZar,M) be a morphism of log schemes with Zariski log struc-tures; it follows easily from the isomorphism (6.1.8) and remark 7.3.6(ii) that f is of Kummertype if and only if u∗f : (Xet, u

∗M) → (Yet, u∗N) is a morphism of Kummer type between

schemes with etale log structures (notation of (6.2.2)). Suppose now that M is an integral andsaturated log structure on XZar, and denote by s.Kum(XZar,M) (resp. s.Kum(Xet, u

∗M))the full subcategory of sat.log/(XZar,M) (resp. of sat.log/(Xet, u

∗M) whose objects are allthe morphisms of Kummer type. In view of the foregoing (and of lemma 6.1.16(i)), we see thatu∗ restricts to a functor :

(7.3.10) s.Kum(XZar,M)→ s.Kum(Xet, u∗M).

Lemma 7.3.11. The functor (7.3.10) is an equivalence.

Proof. By virtue of proposition 6.2.3(ii) we know already that (7.3.10) is fully faithful, hencewe only need to show its essential surjectivity. Thus, let f : (Yet, N) → (Xet, u

∗M) be amorphism of Kummer type. By remark 7.3.6(ii), we know that log f induces an isomorphism

u∗f ∗M ]Q∼→ f ∗u∗M ]

Q∼→ N ]

Q

(notation of (3.3.20)). Since N is integral and saturated, the natural map N ] → N ]Q is a

monomorphism, so that the counit of adjunction u∗u∗N] → N ] is an isomorphism (lemma

2.4.26(ii)), and then the same holds for the counit of adjunction u∗u∗(Y,N) → (Y,N) (propo-sition 6.2.3(iii)).

Proposition 7.3.12. Let f : (Y,N)→ (X,M) be a morphism of fs log schemes. The followingconditions are equivalent:


(a) Every geometric point of X admits an etale neighborhood U → X such that YU :=U ×X Y decomposes as a disjoint union YU =

⋃ni=1 Yi of open and closed subschemes

(for some n ∈ N), and we have :(i) Each restriction Yi ×Y (Y,N) → U ×X (X,M) of f ×X 1U admits a fine, satu-

rated Kummer chart (ωPi , ωQi , ϑi) such that Pi and Qi are sharp, and the order ofCokerϑgp

i is invertible in OU .(ii) The induced morphism of U -schemes Yi → U ×SpecZ[Pi] SpecZ[Qi] is an isomor-

phism, for every i = 1, . . . , n.(b) f is etale, and the morphism of schemes underlying f is finite.

Proof. (a)⇒ (b): Indeed, it is easily seen that the morphism SpecZ[ϑi] is finite, hence the sameholds for the restriction Yi → U of f , in view of (a.ii), and then the same holds for f ×X 1U ,so finally f is finite on the underlying schemes ([31, Ch.IV, Prop.2.7.1]), and it is etale by thecriterion of theorem 6.3.37.

(b) ⇒ (a): Arguing as in the proof of theorem 6.3.37, we may reduce to the case whereτ = et. Suppose first that both X and Y are strictly local. In this case, M admits a fine andsaturated chart ωP : PX → M , sharp at the closed point (corollary 6.1.34(i)). Moreover, fadmits a chart (ωP , ωQ : QY → N, ϑ : P → Q), for some fine monoid Q such that Q× istorsion-free; also ϑ is injective, the induced morphism of X-schemes

g : Y → X ′ := X ×SpecZ[P ] SpecZ[Q]

is etale, and the order of Cokerϑgp is invertible in OX (corollary 6.3.42). Since f is finite, gis also closed ([27, Ch.II, Prop.6.1.10]), hence its image Z is an open and closed local sub-scheme, finite over X . Let k be the residue field of the closed point of X; it follows thatX ′ ×X Spec k ' Spec k〈Q/mPQ〉 admits an irreducible component of Krull dimension zero(namely, the intersection of Z with the fibre of X ′ over the closed point of X), in which casethe criterion of proposition 7.3.8(ii) ensures that ϑ is of Kummer type and Q is sharp, hence X ′

is finite over X , and moreover X ′ ×X Spec k is a local scheme with k as residue field. Since Xis strictly henselian, it follows that X ′ itself is strictly local, and therefore g is an isomorphism,so the proposition is proved in this case.

Let now X be a general scheme, and ξ a geometric point of X; denote by X(ξ) the stricthenselization of X at the point ξ, and set Y (ξ) := X(ξ) ×X Y . Since f is finite, Y (ξ)decomposes as the disjoint union of finitely many open and closed strictly local subschemesY1(ξ), . . . , Yn(ξ). Then we may find an etale neighborhood U → X of ξ, and open and closedsubschemes Y1, . . . , Yn of Y ×X U , with isomorphisms of X(ξ)-schemes Yi(ξ)

∼→ Yi×U X(ξ),for every i = 1, . . . , n ([32, Ch.IV, Cor.8.3.12]). We may then replaceX by U , and we reduce toproving the proposition for each of the restrictions Yi → U of f ×X 1U ; hence we may assumethat Y (ξ) is strictly local. By the foregoing case, we may find a chart

(7.3.13) PX(ξ) →M(ξ) QY (ξ) → N(ξ) ϑ : P → Q

of f ×X 1X(ξ), with ϑ of Kummer type, such that P and Q are sharp, the order d of Cokerϑgp

is invertible in OX(ξ), and the induced morphism of X(ξ)-schemes Y (ξ) → X(ξ) ×SpecZ[P ]

SpecZ[Q] is an isomorphism. By corollary 6.2.34 we may find an etale neighborhood U → Xof ξ such that (7.3.13) extends to a chart for f ×X 1U . After shrinking U , we may assumethat d is invertible in OU . Lastly, after further shrinking of U , we may ensure that the inducedmorphism of U -schemes U ×X Y → U ×SpecZ[P ] SpecZ[Q] is an isomorphism ([32, Ch.IV,Cor.8.8.2.4]).

Definition 7.3.14. Let f : (Y,N) → (X,M) be a morphism of fs log schemes. We say that fis an etale covering of (X,M), if f fulfills the equivalent conditions (a) and (b) of proposition7.3.12. We denote by

Cov(X,M)


the full subcategory of the category of (X,M)-schemes, whose objects are the etale coveringsof (X,M).

Remark 7.3.15. (i) Notice that all morphisms in Cov(X,M) are etale coverings, in light ofcorollary 6.3.25(ii).

(ii) Moreover, Cov(X,M) is a Galois category (see [42, Exp.V, Def.5.1]), and if ξ is anygeometric point of (X,M)tr, we obtain a fibre functor for this category, by the rule : f 7→f−1(ξ), for every etale covering f of (X,M). (Details left to the reader.) We shall denote by

π1((X,M)et, ξ)

the corresponding fundamental group.

Example 7.3.16. Let (f, log f) : (Y,N) → (X,M) be an etale covering of a regular logscheme (X,M), and suppose that X is strictly local of dimension 1 and Y is connected (hencestrictly local as well). Let x ∈ X (resp. y ∈ Y ) be the closed point; it follows that OX,xis a strictly henselian discrete valuation ring (corollary 6.5.29). The same holds for OY,y inview of theorem 6.5.44 and [33, Ch.IV, Prop.18.5.10]. In case dimMx = 0, the log structuresM and N are trivial, so f : Y → X is an etale morphism of schemes. Otherwise we havedimMx = 1 = dimNy (lemma 3.4.41(i)), and then M ]

x ' N ' N ]y (theorem 3.4.16(ii)); also,

the choice of a chart for f as in proposition 7.3.12, induces an isomorphism

OX,x ⊗M]xN ]y∼→ OY,y

where the map M ]x → N ]

y is the N -Frobenius map of N, where N > 0 is an integer invertiblein OX,x. Moreover, notice that the image of the maximal ideal of Mx generates the maximalideal of OX,x (and likewise for the image of Ny in OY,y), so the structure map of M induces anisomorphism M ]

x∼→ Γ+, onto the submonoid of the value group (Γ,≤) of OX,x consisting of

all elements ≤ 1 (and likewise for N ]y). In other words, OY,y is obtained from OX,x by adding

the N -th root of a uniformizer. It then follows that the ring homomorphism OX,x → OY,y is analgebraic tamely ramified extension of discrete valuation rings (see e.g. [36, Cor.6.2.14]).

Definition 7.3.17. Let X be a normal scheme, and Z → X a closed immersion such that :(a) Z is a union of irreducible closed subsets of codimension one in X .(b) For every maximal point z ∈ Z, the stalk OX,z is a discrete valuation ring.

Set U := X\Z, let f : U ′ → U be an etale covering, and g : X ′ → X the normalization of Xin U ′. Notice that g−1Z is a union of irreducible closed subsets of codimension one in X ′ (bythe going down theorem [61, Th.9.4(ii)]).

(i) We say that f is tamely ramified along Z if, for every geometric point ξ ∈ X ′ localizedat a maximal point of g−1Z, the induced finite extension

OshX,g(ξ) → Osh

X′,ξ

of strictly henselian discrete valuation rings ([33, Ch.IV, Cor.18.8.13]), is tamely rami-fied. Recall that the latter condition means the following : let Γg(ξ) → Γξ be the map ofvaluation groups induced by the above extension; then the ramification index of f at ξ

eξ(f) := (Γξ : Γf(ξ))

is invertible in the residue field κ(ξ), and g induces an isomorphism κ(f(ξ))∼→ κ(ξ).

(ii) Suppose that f is tamely ramified along Z, and that the set of maximal points of Z isfinite. Then the ramification index of f along Z is the least common multiple eZ(f)of the ramification indices eξ(f), where ξ ranges over the set of all geometric points ofX ′ supported at a maximal point of g−1Z.

(iii) We denote by Tame(X,U) the full subcategory of Cov(U) whose objects are theetale coverings of U that are tamely ramified along Z.


Lemma 7.3.18. Let ϕ : X ′ → X be a dominant morphism of normal schemes, Z ⊂ X a closedsubset, and set Z ′ := ϕ−1Z. Suppose that :

(a) The closed immersions Z → X and Z ′ → X ′ satisfy conditions (a) and (b) of definition7.3.17.

(b) ϕ restricts to a map MaxZ ′ → MaxZ on the subsets of maximal points of Z and Z ′.We have :

(i) ϕ∗ : Cov(U)→ Cov(U ′) restricts to a functor

Tame(X,U)→ Tame(X ′, U ′).

(ii) Suppose that MaxZ and MaxZ ′ are finite sets. Then, for any object f : V → U ofTame(X,U), the index eZ′(ϕ∗f) divides eZ(f).

(iii) Suppose that X is regular, and let j : U → X be the open immersion. Then theessential image of the pull-back functor j∗ : Cov(X)→ Tame(X,U) consists of theobjects f : V → U such that eZ(f) = 1.

Proof. Let V → U be an object of Tame(X,U), and denote by g : W → X (resp. g′ : W ′ →X ′) the normalization of X in V (resp. of X ′ in V ×U U ′). Let w′ be a geometric point ofW ′ localized at a maximal point of g′−1Z ′, and denote by w (resp. x′, resp. x) the image ofw′ in W (resp. in X ′, resp. in X); in order to check (i) and (ii), we have to show that theramification index of g′ at the geometric point w′ is invertible in κ(w′), and the residue fieldextension κ(x′) ⊂ κ(w′) is trivial. To this aim, in view of [33, Ch.IV, Prop.17.5.7], we mayreplace X ′ by X ′(x′), X by X(x), W by W (w), and assume from start that X , X ′ and W arestrictly local, sayX = SpecA, X ′ = SpecB and C = SpecW for some strictly henselian localrings A,B,C. In this case, D := B ⊗A C is a direct product of finite and local B-algebras, andx′ is localized at a closed point ofX ′, so w′ is localized at a closed point ofW ′, therefore OW ′,w′

is a direct factor of D, and κ(w′) is a quotient of κ(x′)⊗κ(x) κ(w); but we have κ(w) = κ(x) byassumption, so this already yields κ(w′) = κ(x′). Moreover, by construction, OW ′,w′⊗AFracAis a finite separable extension of FracB, and the diagram

(7.3.19)

FracA //

FracC

FracB // Frac OW ′,w′

is cocartesian (in the category of fields). Since V is tamely ramified along Z, the top horizontalarrow of (7.3.19) is a finite (abelian) extension whose degree ew is invertible in κ(x) ([36,Cor.6.2.14]); it follows that the bottom horizontal arrows is a Galois extension as well, and itsdegree divides ew, as required.

(iii): Let f : V → U be an object of Tame(X,U). By claim 7.1.8, there exists a largestopen subset Umax ⊂ X containing U , and such that f is the restriction of an etale coveringfmax : V ′ → Umax; now, if eZ(f) = 1, every point of codimension one of X \U lies in Umax,by claim 7.1.9. Hence X\U has codimension ≥ 2 in X , in which case lemma 7.1.7(iii) impliesthat X = Umax, as required.

Remark 7.3.20. The assumptions (a) and (b) of lemma 7.3.18 are fulfilled, notably, when f isfinite and dominant, or when f is flat ([61, Th.9.4(ii), Th.9.5]).

7.3.21. Let now X := (Xi | i ∈ I) be a cofiltered system of normal schemes, such that, forevery morphism ϕ : i → j in the indexing category I , the corresponding transition morphismfϕ : Xi → Xj is dominant and affine. Suppose also, that for every i ∈ I , there exists a closedimmersion Zi → Xi, fulfilling conditions (a) and (b) of definition 7.3.17, such that MaxZi is afinite set, and for every morphism ϕ : i→ j of I , we have :


• Zi = f−1ϕ Zj .

• The corresponding morphism fϕ restricts to a map MaxZi → MaxZj .Let alsoX be the limit ofX , and Z the limit of the system (Zi | i ∈ I), and suppose furthermorethat the closed immersion Z → X fulfills as well conditions (a) and (b) of definition 7.3.17.

Proposition 7.3.22. In the situation of (7.3.21), we have :(i) The induced morphism πi : X → Xi restricts to a map

MaxZ → MaxZi for every i ∈ I .

(ii) The morphisms πi induce an equivalence of categories :

2-colimI

Tame(Xi, Xi\Zi)→ Tame(X,X\Z).

Proof. (i): More precisely, we shall prove that there is a natural homeomorphism :

MaxZ∼→ lim

i∈IMaxZi.

(Notice the each MaxZi is a discrete finite set, hence this will show that MaxZ is a profinitetopological space.) Indeed, suppose that z := (zi | i ∈ I) is a maximal point of Z. For everyi ∈ I , let Ti ⊂ MaxZi be the subset of maximal generizations of zi in Zi. It is easily seen thatfϕTi ⊂ Tj for every morphism ϕ : i → j in I . Clearly Ti is a finite non-empty set for everyi ∈ I , hence the limit T of the cofiltered system (Ti | i ∈ I) is non-empty. However, any pointof T is a generization of z in Z, hence it must coincide with z. The assertion follows easily.

(ii): To begin with, lemma 7.3.18(i) shows that, for every morphism ϕ : i → j in I , thetransition morphism fϕ induces a pull-back functor Tame(Xj, Xj\Zj)→ Tame(Xi, Xi\Zi),so the 2-colimit in (ii) is well-defined, and combining (i) with lemma 7.3.18(i) we obtain indeeda well-defined functor from this 2-colimit to Tame(X,X\Z).

The full faithfulness of the functor of (ii) follows from lemma 7.1.6. Next, let g : V → X\Zbe an object of Tame(X,X \Z); invoking again lemma 7.1.6, we may descend g to an etalecovering gj : Vj → Xj\Zj , for some j ∈ I; after replacing I by I/j, we may assume that j isthe final object of I and we may define gi := f ∗ϕ(gj) for every ϕ : i → j in I . To conclude theproof, it suffices to show that there exists i ∈ I such that gi is tamely ramified along Zi.

Now, let gi : V i → Xi be the normalization of Xi in Vi ×Xj Xi, for every i ∈ I , andg : V → X the normalization of X in V . Given a geometric point v localized at a maximalpoint of g−1Z, let vi (resp. zi) be the image of v in V i (resp. in Zi), for every i ∈ I . Let also zbe the image of v in Z. Then

OshX,z = colim

i∈IOshXi,zi

OshV ,v

= colimi∈I

OshVi,vi

.

([33, Ch.IV, Prop.18.8.18(ii)]), and it follows easily that there exists i ∈ I such that the finiteextension Osh

Xi,zi→ Osh

Vi,viis already tamely ramified. Since only finitely many points of V i lie

over the support of zi, and since I is cofiltered, it follows that there exists i ∈ I such that theinduced morphism V i ×Xi Xi(zi) → Xi(zi) is already tamely ramified. However, notice thatπ−1i (zi) is open in MaxZ, for every i ∈ I , and every zi ∈ MaxZi. Therefore, we may find a

finite subset I0 ⊂ I , and for every i ∈ I0 a subset Ti ⊂ MaxZi, such that :• MaxZ =

⋃i∈I0 π

−1i (Ti).

• For every geometric point zi localized in Ti, the morphism V i ×Xi Xi(zi)→ Xi(zi) istamely ramified.

Since I is cofiltered, we may find k ∈ I with morphisms ϕi : k → i for every i ∈ I0;after replacing Ti by f−1

ϕi(Ti) for every i ∈ I0, we may then assume that I0 = k, so that

MaxZ = π−1k (Tk). It follows that MaxZi = f−1

ϕ Tk for some i ∈ I and some ϕ : i→ k. Thenit is clear that gi is tamely ramified along Zi, as required.


7.3.23. Let X be a scheme, U ⊂ X a connected open subset, ξ any geometric point of U ,and N > 0 an integer which is invertible in OU ; then the N -Frobenius map N of O×U gives aKummer exact sequence of abelian sheaves on Xet :

0→ µN,U → O×UN−−→ O×U → 0

(where µN,U is the N -torsion subsheaf of O×Uet: see [5, Exp.IX, §3.2]). Suppose now that µN,U

is a constant sheaf on Uet; this means especially that µN := (µN,U)ξ is a cyclic group of orderN . Indeed, µN certainly contains such a subgroup (since N is invertible in OU ), so denote by ζone of its generators; then every u ∈ µN satisfies the identity 0 = uN − 1 =

∏Ni=1(u− ζ i), and

each factor uiζ of this decomposition vanishes on a closed subset Ui of U(ξ); clearly Ui∩Uj = ∅for i 6= j, so we get a decomposition of U(ξ) as a disjoint union of open and closed subsetsU(ξ) = U1 ∪ · · · ∪ UN such that u = ζ i on Ui, for every i ≤ N ; since U(ξ) is connected, itfollows that U(ξ) = Ui for some i, i.e. ζ generates µN . There follows a natural map :

∂N : Γ(Uet,O×U )→ H1(Uet,µN,U)

∼→ Homcont(π1(Uet, ξ),µN)

(lemma 7.1.13). Recall the geometric interpretation of ∂N : a given u ∈ O×U (U) is viewed asa morphism of schemes u : U → Gm, where Gm := SpecZ[Z] is the standard multiplicativegroup scheme; we let U ′ be the fibre product in the cartesian diagram of schemes :

(7.3.24)

U ′ //

ϕu

Gm

SpecZ[NZ]

U

u // Gm.

Then ϕu is a torsor under the Uet-group µN,U , and to such torsor, lemma 7.1.13 associates awell defined continuous group homomorphism as required.

If M > 0 is any integer dividing N , a simple inspection yields a commutative diagram :

(7.3.25)

Γ(Uet,O×X )

∂N //

∂M **VVVVVVVVVVVVVVVVVVHomcont(π1(Uet, ξ),µN)

πN,M

Homcont(π1(Uet, ξ),µM)

where πN,M is induced by the map µN → µM given by the rule : x 7→ xN/M for every x ∈ µN .

7.3.26. Let now X be a strictly local scheme, x the closed point of X , and denote by p thecharacteristic exponent of the residue field κ(x) (so p is either 1 or a positive prime integer).Let also β : M → OX be a log structure on Xet, take U := (X,M)tr, suppose that U 6= ∅, andfix a geometric point ξ of U ; for every integer N > 0 with (N, p) = 1, we get a morphism ofmonoids :

(7.3.27) Mxβx−−→ Γ(Uet,O

×X )

∂N−−→ Homcont(π1(Uet, ξ),µN)

whose kernel contains MNx , the image of the N -Frobenius endomorphism of Mx. Notice that

the N -Frobenius map of O×X,x is surjective : indeed, since the residue field κ(x) is separablyclosed, and (N, p) = 1, all polynomials in κ(x)[T ] of the form TN − u (for u 6= 0) split as aproduct of distinct monic polynomials of degree 1; since OX,x is henselian, the same holds forall polynomials in OX,x[T ] of the form TN − u, with u ∈ O×X,x. It follows that (7.3.27) factorsthrough a natural map:

M ]gpx → Homcont(π1(Uet, ξ),µN)

which is the same as a group homomorphism :

(7.3.28) M ]gpx × π1(Uet, ξ)→ µN .


In view of the commutative diagram (7.3.25), it is easily seen that the pairings (7.3.28), for Nranging over all the positive integers with (N, p) = 1, assemble into a single pairing :

Mgpx × π1((X,M)tr,et, ξ)→

∏`6=p

Z`(1)

(where ` ranges over the prime numbers different from p, and Z`(1) := limn∈N

µ`n). The latter is

the same as a group homomorphism :

(7.3.29) π1((X,M)tr,et, ξ)→Mgp∨x ⊗Z

∏`6=p

Z`(1).

7.3.30. Let f : (X,M) → (Y,N) be a morphism of log schemes, such that both X and Y arestrictly local, and f maps the closed point x of X to the closed point y of Y . Then f restrictsto a morphism of schemes ftr : (X,M)tr → (Y,N)tr (remark 6.2.8(i)). Fix again a geometricpoint ξ of (X,M)tr; we get a diagram of group homomorphisms :

π1((X,M)tr,et, ξ)π1(ftr,ξ) //

π1((Y,N)tr,et, f(ξ))

Ngp∨y ⊗Z

∏6=p Z`(1) // Mgp∨

x ⊗Z∏

`6=p Z`(1)

whose vertical arrows (resp. bottom horizontal arrow) are the maps (7.3.29) (resp. is induced by(log fx)

gp∨), and by inspecting the constructions, it is easily seen that this diagram commutes.

7.3.31. Suppose now that (X,M) as in (7.3.26) is a fs log scheme, so that P := M ]x is fine and

saturated; in this case, we wish to give a second construction of the pairing (7.3.28).• Namely, for every integer N > 0 set SP := SpecZ[P ] and consider the finite morphism

of schemes gN : SP → SP induced by the N -Frobenius endomorphism of P . Notice thatSP contains the dense open subset UP := SpecZ[1/N, P gp], and it is easily seen that therestriction gN |UP : UP → UP of gN is an etale morphism. Let τ ∈ UP be any geometricpoint; then τ corresponds to a ring homomorphism Z[1/N, P gp] → κ := κ(τ), which is thesame as a character χτ : P gp → κ×. Likewise, any τ ′ ∈ g−1

N (τ) is determined by a characterχτ ′ : P gp → κ× extending χτ , i.e. such that χτ ′(xN) = χτ (x) for every x ∈ P gp. Notice thatevery character χτ as above admits at least one such extension χτ ′ , since κ is separably closed,and N is invertible in κ. Let CN be the cokernel of the N -Frobenius endomorphism of P gp;there follows a short exact sequence of finite abelian groups :

0→ HomZ(CN , κ×)→ HomZ(P gp, κ×)

N−→ HomZ(P gp, κ×)→ 0.

Especially, we see that the fibre g−1N (τ) is naturally a torsor under the group HomZ(CN , κ

×) =HomZ(P gp,µN(κ)), where µN(κ) ⊂ κ× is the N -torsion subgroup. Hence gN |UP is an etaleGalois covering of UP , whose Galois group is naturally isomorphic to HomZ(P gp,µN(κ));therefore, gN |UP is classified by a well defined continuous representation :

(7.3.32) π1(UP,et, τ)→ HomZ(P gp,µN(κ))

(lemma 7.1.13). The corresponding action of HomZ(P gp,µN(κ)) on UP can be extracted fromthe construction : namely, let χ : P gp → µN(κ) be a given character; by definition, the actionof χ sends the geometric point τ to the geometric point τ ′ whose character χτ ′ : P gp → κ× isgiven by the rule : x 7→ χ(x) · χτ for every x ∈ P gp. Consider the automorphism ρχ of theZ-algebra Z[P gp] given by the rule x 7→ χ(x) · x for every x ∈ P gp, and notice that ρ(τ) = τ ′;since the fibre functors are faithful on etale coverings, we conclude that χ acts as Spec ρχ onUP .


• Moreover, if λ : Q → P is any morphism of fine and saturated monoids, clearly we havea commutative diagram

UPSpecZ[λ]|UP //

gN|UP

UQ

gN|UQ

UPSpecZ[λ]|UP // UQ

whence, by remark 7.1.15(iii.a), a well defined group homomorphism

(7.3.33) HomZ(P gp,µN)→ HomZ(Qgp,µN)

that makes commute the induced diagram :

π1(UP,et, ξ) //

π1(UQ, ξ′)

HomZ(P gp,µN) // HomZ(Qgp,µN)

whose vertical arrows are the maps (7.3.32). By inspecting the constructions, it is easily seenthat (7.3.33) is just the map HomZ(λgp,µN).• Next, by corollary 6.1.34(i) the projection Mx → P admits a splitting

α : P →Mx

(which defines a sharp chart for M ); if N is invertible in OX , this splitting induces a morphismof schemes X → SP , which restricts to a morphism U → UP . If we let τ be the image of ξ, wededuce a continuous group homomorphism π1(Uet, ξ) → π1(UP,et, τ) ([42, Exp.V, §7]), whosecomposition with (7.3.32) yields a continuous map :

(7.3.34) π1(Uet, ξ)→ HomZ(P gp,µN(κ)).

We claim that the pairing P gp×π1(Uet, ξ)→ µN(κ) arising from (7.3.34) agrees with (7.3.28),under the natural identification µN(κ)

∼→ µN . Indeed, for given s ∈ P , let js : Z→ P gp be themap such that js(n) := sn for every n ∈ Z; by composing (7.3.34) with HomZ(js,µN(κ)), weobtain a map π1(Uet, ξ) → HomZ(Z,µN(κ))

∼→ µN(κ); in view of the foregoing, it is easilyseen that the corresponding µN(κ)-torsor is precisely ϕα(s), in the notation of (7.3.24), whencethe contention.

Proposition 7.3.35. In the situation of (7.3.26), suppose that (X,M) is a regular log scheme.Then (7.3.29) is a surjection.

Proof. From the discussion of (7.3.31), we are reduced to showing that the morphism (7.3.34)is surjective, for every N > 0 such that (N, p) = 1. The latter comes down to showing thatthe corresponding torsor gN |UP ×UP U : U ′ → U is connected (remark 7.1.15(i)). However,the map α : P → Mx induces a morphism of log schemes ψ : (X,M) → Spec(Z, P ) (see(6.2.13)); we remark the following :

Claim 7.3.36. Slightly more generally, for any Kummer morphism ν : P → Q of monoids,with Q fine, sharp and saturated, define (Xν ,Mν) as the fibre product in the cartesian diagram:

(7.3.37)

(Xν ,Mν) //

fν

Spec(Z, Q)

Spec(Z,ν)

(X,M)

ψ // Spec(Z, P )

Then (Xν ,Mν) is regular, and Xν is strictly local.


Proof of the claim. By lemma 6.5.40, (Xν ,Mν) is regular at every point of the closed fibreXν ×X Specκ(x), and since fν is a finite morphism, every point of Xν specializes to a pointof the closed fibre, therefore (Xν ,Mν) is regular (theorem 6.5.46). Moreover, Xν is connected,provided the same holds for the closed fibre; in turn, this follows immediately from proposition7.3.8(ii). Then Xν is strictly local, by [33, Ch.IV, Prop.18.5.10]. ♦

From claim 7.3.36 we see that Xν is normal (corollary 6.5.29) for every such ν, especiallythis holds for the N -Frobenius endomorphism of P , in which case U ′ is an open subset of Xν ;since the latter is connected, the same then holds for U ′.

Example 7.3.38. (i) In the situation of example 6.6.20, let X → S be any morphism ofschemes, set X := X ×S S, denote by (X,M) the log scheme underlying X , and define(X(k),M (k)) as the fibre product in the cartesian diagram of log schemes :

(X(k),M (k))π(k) //

kX

Spec(R,P )

Spec(R,kP )

(X,M)

π // Spec(R,P )

where π is the natural projection. Set as well

X(k) := ((X(k),M (k)), TP , ψP π](k))

which is an object of Kint; by proposition 6.6.14(iii), diagram (3.5.28) (with T := TP ) underliesa commutative diagram in Kint :

ϕ∗X(k)//

gX

X(k)

(kX ,kTP )

ϕ∗X // X

whose horizontal arrows are cartesian. Clearly this diagram is isomorphic to X ×S (6.6.21); wededuce that the morphism gX of log schemes underlying gX is finite, and of Kummer type; byproposition 7.3.12, gX is even an etale covering, if k is invertible in OX .

(ii) Suppose furthermore, that (X,M) is regular; by claim 7.3.36 it follows that (X(k),M (k))is regular as well, and then the same holds for the log schemes (Xϕ,Mϕ) and (Y,N) underlyingrespectively ϕ∗X and ϕ∗X(k) (proposition 6.6.14(v) and theorem 6.5.44). Let now y ∈ Y be apoint of height one in Y , lying in the closed subset Y \(Y,N)tr, and set x := gX(y); arguingas in example 7.3.16, we see that N ]

y and M ]ϕ,x are both isomorphic to N, and the induced map

OXϕ,x → OY,y is an extension of discrete valuation rings, whose corresponding extension ofvalued fields is finite. Denote by Γx → Γy the associated extension of valuation groups; in viewof (6.6.22) we see that the ramification index (Γy : Γx) equals k.

7.3.39. Resume the situation of (7.3.31). Every (finite) discrete quotient map

ρ : Mgp∨x ⊗Z

∏`6=p

Z`(1)→ G

corresponds, via composition with (7.3.29), to a GU -torsor, which can be explicitly constructedas follows. Set P := M ]

x, pick a splitting α as in (7.3.31), choose an integer N > 0 largeenough, so that (N, p) = 1, and ρ factors through a group homomorphism

ρ : HomZ(P gp,µN(κ))→ G

and set H := Ker ρ. Now, via (7.3.29), the quotient G′ := HomZ(P gp,µN(κ)) corresponds tothe G′U -torsor on U obtained from gN : UP → UP via pull-back along the morphism U → UPgiven by the chart α (notation of (7.3.31)). According to remark 7.1.15(ii), the sought GU is


therefore isomorphic to the one obtained from the quotient gN : UP/H → UP , via pull-backalong the same morphism. To exhibit such quotient, consider the exact sequence:

0→ HomZ(G, κ×)→ P gp ⊗Z Z/NZ→ HomZ(H, κ×)→ 0.

Define Qgp ⊂ P gp as the kernel of the induced map P gp → HomZ(H, κ×), and set Q :=Qgp ∩ P . By construction, the N -Frobenius endomorphism of P factors through an injectivemap ν : P → Q and the inclusion map j : Q→ P , so gN factors as a composition :

UP → UQh−→ UP .

The maps on geometric points UP (κ) → UQ(κ) → UP (κ) induced by gN and h correspond tojgp∗ and respectively νgp∗ in the resulting commutative diagram

0

0 // H // HomZ(P gp,µN(κ)) //

G

// 0

0 // H // HomZ(P gp, κ×)jgp∗

// HomZ(Qgp, κ×)

νgp∗

// 0

HomZ(P gp, κ×)

0

whose rows and column are exact. Hence, let τ and χτ : P gp → κ× be as in (7.3.31); thefibre h−1(τ) corresponds to the set of all characters χτ ′ : Qgp → κ× whose restriction χτ ′ νgp

to P gp agrees with χτ . Then, with the notation of (7.3.37), we conclude that the restriction(fν)tr : (Xν ,Mν)tr → U is the sought GU -torsor. Notice as well that fν is an etale covering of(X,M).• Here is another handier description of the same submonoid Q of P . Notice that ρ is the

same as a group homomorphism

ρ† : P gp∨ → HomZ(∏

`6=pZ`(1), G)

and let L := Ker ρ†. Fix a generator ζN of µN(κ); by definition

Qgp = x ∈ P gp | t(x)⊗ ξ = 0 for all t ∈ P gp∨ and ξ ∈ µN such that ρ(t⊗ ξ) = 0= x ∈ P gp | t(x)⊗ ζN = 0 for all t ∈ P gp∨ such that ρ(t⊗ ζN) = 0= x ∈ P gp | t(x) ∈ NZ for all t ∈ L.

On the other hand, notice that theN -Frobenius of P gp∨ factors through a map β : P gp∨ → L andthe inclusion map i : L→ P gp∨, and βi is theN -Frobenius map of L. Let ω : P gp ∼→ (P gp∨)∨

be the natural isomorphism. We may then write

Qgp = x ∈ P gp | ω(x)(t) ∈ NZ for all t ∈ L= x ∈ P gp | ω(x) i ∈ N · L∨ = i∨ β∨(L∨)= ω−1(Im β∨).

In other words, ω induces a natural isomorphism Qgp ∼→ L∨, that identifies β∨ with the mapjgp : Qgp → P gp, and then necessarily also the map i∨ with ν. Now, the image of Q inside


L∨ can be recovered just as L∨ ∩ P∨∨Q (the intersection here takes place in L∨Q, which contains(P gp∨)∨, via the injective map i∨ : details left to the reader).

7.3.40. Our chief supply of tamely ramified coverings comes from the following source. Letf : (Y,N) → (X,M) be an etale morphism of log schemes, whose underlying morphism ofschemes is finite, and suppose that (X,M) is regular. Then (Y,N) is regular, and both X andY are normal schemes (theorem 6.5.44 and corollary 6.5.29). Moreover, lemma 3.4.41(ii) andproposition 7.3.12 imply that (Y,N)tr = f−1(X,M)tr, hence the restriction of f

ftr : (Y,N)tr → (X,M)tr

is a finite etale morphism of schemes (corollary 6.3.27(i)). Furthermore, it follows easily fromcorollary 6.5.36(i) that the closed subset X\(X,M)tr is a union of irreducible closed subsets ofcodimension 1 inX (and the union is locally finite on the Zariski topology ofX); the same holdsalso for Y \(Y,N)tr, especially Y is the normalization of (Y,N)tr over X . Finally, example7.3.16 implies that ftr is tamely ramified alongX\(X,M)tr. It is then clear that the rule f 7→ ftr

defines a functor :F(X,M) : Cov(X,M)→ Tame(X, (X,M)tr).

It follows easily from remark 6.5.54 that F(X,M) is fully faithful; therefore, any choice of ageometric point ξ of (X,M)tr determines a surjective group homomorphism :

(7.3.41) π1((X,M)tr,et, ξ)→ π1((X,M)et, ξ)

([42, Exp.V, Prop.6.9]).

Proposition 7.3.42. In the situation of (7.3.26), suppose that (X,M) is a regular log scheme.Then the map (7.3.29) factors through (7.3.41), and induces an isomorphism :

(7.3.43) π1((X,M)et, ξ)∼→Mgp∨

x ⊗Z∏`6=p

Z`(1).

Proof. The discussion of (7.3.39) shows that (7.3.29) factors through (7.3.41), and proposition7.3.35 implies that (7.3.43) is surjective. Next, let f : (Y,N) → (X,M) be an etale cover-ing; according to proposition 7.3.12, f admits a Kummer chart (ωP , ωQ, ϑ) with Q fine, sharpand saturated, such that the order k of Cokerϑgp is invertible in OY . Moreover, the inducedmorphism of X-schemes Y → X ×SpecZ[P ] SpecZ[Q] is an isomorphism. With this notation,denote by G the cokernel of the induced group homomorphism HomZ(ϑgp,kP gp); there followsa map ρ : P gp⊗Z

∏6=p Z`(1)→ G, and by inspecting the construction, one may check that the

correspondingGU -torsor, constructed as in (7.3.39), is isomorphic to f×X 1U (details left to thereader). This shows the injectivity of (7.3.43), and completes the proof of the proposition.

The following result is the logarithmic version of the classical Abhyankar’s lemma.

Theorem 7.3.44. The functor F(X,M) is an equivalence.

Proof. First we show how to reduce to the case where τ = et.

Claim 7.3.45. Let (XZar,M) be a regular log structure on the Zariski site of X , and supposethat the theorem holds for u∗(XZar,M). Then the theorem holds for (XZar,M) as well.

Proof of the claim. Let g : V → (XZar,M)tr be an etale covering whose normalization overX is tamely ramified along X \ (XZar,M)tr. By assumption, there exists an etale coveringf : (Yet, N)→ u∗(XZar,M) such that ftr = g; by lemma 7.3.11, we may then find a morphismof log schemes of Kummer type fZar : u∗(Yet, N)→ (XZar,M) such that u∗fZar = f , thereforefZar is an etale covering of (XZar,M) (corollary 6.3.27(iii)), and clearly (fZar)tr = g, so thefunctor F(XZar,M) is essentially surjective. Full faithfulness for the same functor is derived


formally from the full faithfulness of the functor Fu∗(XZar,M), and that of the functor (7.3.10) :details left to the reader. ♦

Henceforth we assume that τ = et.

Claim 7.3.46. Let g : V → (X,M)tr be an object of Tame(X, (X,M)tr). Then g lies in theessential image of F(X,M) if (and only if) there exists an etale covering (Uλ → X | λ ∈ Λ) ofX ,such that, for every λ ∈ Λ, the etale covering g ×X Uλ lies in the essential image of F(Uλ,M |Uλ

).

Proof of the claim. We have to exhibit a finite etale covering f : (Y,N) → (X,M) suchthat ftr = g. However, given such f , theorem 6.5.44 and corollary 6.5.29 imply that Y isthe normalization of V over X , and proposition 6.5.52 says that N = j∗O

×V ∩ OY , where

j : V → Y is the open immersion; then log f : f ∗M → N is completely determined by f ,hence by g. Thus, we come down to showing that :

(i) N := j∗O×V ∩ OY is a regular log structure on the normalization Y of V over X .

(ii) The unique morphism f : (Y,N)→ (X,M) is an etale covering.However, [61, §33, Lemma 1] implies that Y is finite over X . To show (ii), it then suffices toprove that each restriction fλ := f ×X 1Uλ is etale (proposition 6.3.24(iii)). Likewise, (i) holds,provided the restriction N |Yλ is a regular log structure on Yλ := Y ×X Uλ, for every λ ∈ Λ. Letjλ : Vλ := V ×XUλ → Yλ be the induced open immersion. It is clear thatN |Yλ = jλ∗O

×Vλ∩OYλ ,

and Yλ is the normalization of Vλ over Uλ ([33, Ch.IV, Prop.17.5.7]); moreover, (Uλ,M |Uλ) isagain regular, so our assumption implies that (Yλ, N |Yλ) is the unique regular log structure onYλ whose trivial locus is Vλ, and that fλ is indeed etale. ♦

Claim 7.3.47. If F(X(ξ),M(ξ)) is essentially surjective for every geometric point ξ of X , the sameholds for F(X,M).

Proof of the claim. Indeed, let g : V → (X,M)tr be an object of Tame(X, (X,M)tr), and ξany geometric point ofX . By claim 7.3.46 it suffices to find an etale neighborhood U → X of ξ,such that g×X 1U lies in the essential image of F(U,M |U ). Denote by Y the normalization ofX inV , which is a finite X-scheme ([61, §33, Lemma 1]). The scheme Y (ξ) := Y ×X X(ξ) decom-poses as a finite disjoint union of strictly local open and closed subschemes Y1(ξ), . . . , Yn(ξ),and we may find an etale neighborhood U → X of ξ, and a decomposition of Y ×X U by openand closed subschemes Y1, . . . , Yn, with isomorphisms of X(ξ)-schemes Yi(ξ)

∼→ Yi ×X X(ξ)for every i = 1, . . . , n ([32, Ch.IV, Cor.8.3.12]). We are then reduced to showing that all therestrictions Yi → U lie in the essential image of F(U,M |U ). Thus, we may replace X by U , andY by any Yi, after which we may assume that Y (ξ) is strictly local. Proposition 6.5.32 andtheorem 6.5.46 imply that (X(ξ),M(ξ)) is a regular log scheme. Clearly gξ := g ×X 1X(ξ) isan object of Tame(X(ξ), (X(ξ),M(ξ))tr), so by assumption there exists a finite etale coveringh : (Z,N) → (X(ξ),M(ξ)) such that htr = gξ. By theorem 6.5.44 and corollary 6.5.29 weknow that Z is a normal scheme, and we deduce that Z = Y (ξ) ([33, Ch.IV, Prop.17.5.7]).

According to proposition 7.3.12, the morphism h admits a fine and saturated Kummer chart(ωP , ωQ, ϑ : P → Q), such that the order d of Cokerϑgp is invertible in OX,ξ, and such thatthe induced map Y (ξ) → X(ξ) ×SpecP SpecQ is an isomorphism. By proposition 6.2.28,there exists an etale neighborhood U → X of ξ, and a coherent log structure N ′ on Y ′ :=Y ×X U with an isomorphism Y (ξ) ×Y ′ (Y ′, N ′)

∼→ (Y (ξ), N). Moreover, let h′ : Y ′ → Ube the projection; after shrinking U , the map log h descends to a morphism of log structuresh′∗M |U → N ′, whence a morphism (h′, log h′) : (Y ′, N ′) → (U,M |U) of log schemes, suchthat h′ ×U 1X(ξ) = h. After further shrinking U , we may also assume that h′ admits a Kummerchart (ω′P , ω

′Q, ϑ) (corollary 6.2.34), that d is invertible in OU , and that the induced morphism

Y ′ → U ×SpecP SpecQ is an isomorphism ([32, Ch.IV, Cor.8.8.24]). Then h′ is an etalecovering, by proposition 7.3.12, and by construction, F(U,M |U )(h

′) = g ×X 1U , as desired. ♦


Claim 7.3.48. Assume that X is strictly local, denote by x the closed point of X , set U :=(X,M)tr, and P := M ]

x. Let h : V → U be any connected etale covering, tamely ramifiedalong X\U ; then h lies in the essential image of F(X,M), provided there exists a fine, sharp andsaturated monoid Q, and a morphism ν : P → Q of Kummer type, such that

hν := h×X Xν : V ×X Xν → U ×X Xν

admits a section (notation of (7.3.37)).

Proof of the claim. Suppose first that the order of Coker νgp is invertible in OX ; in this case,fν is an etale covering of log schemes (proposition 7.3.12), hence (fν)tr is an etale coveringof the scheme U . Then, by composing a section of hν with the projection V ×X Xν → V wededuce a morphism U ×X Xν → V of etale coverings of U . Such morphism shall be open andclosed, hence surjective, since V is connected; hence the π1(U, ξ)-set h−1(ξ) will be a quotientof f−1

ν (ξ), on which π1(U, ξ) acts through (7.3.29), as required. Next, for a general morphism νof Kummer type, let L ⊂ Qgp be the largest subgroup such that νgp(P gp) ⊂ L, and (L : P gp) isinvertible in OX . Set Q′ := L∩Q, and notice that Q′gp = L. Indeed, every element of L can bewritten in the form x = b−1a, for some a, b ∈ Q; then choose n > 0 such that bn ∈ νP , writex = b−n · (bn−1a) and remark that b−n, bn−1a ∈ Q′. The morphism ν factors as the compositionof ν ′ : P → Q′ and ψ : Q′ → Q, and therefore fν factors through a morphism fψ : Xν → Xν′ .In view of the previous case, we are reduced to showing that the morphism hν′ already admitsa section, hence we may replace (X,M) by (Xν′ ,Mν′) (which is still regular and strictly local,by claim 7.3.36), h by hν′ , and ν by ψ, after which we may assume that the order of Coker νgp

is pm for some integer m > 0, where p is the characteristic of the residue field κ(x), and thenwe need to show that h already admits a section.

Next, using again claim 7.3.36 and an easy induction, we may likewise reduce to the casewhere m = 1. Say that X = SpecA, and suppose first that A is a Fp-algebra (where Fp is thefinite field with p elements), so that Xν = SpecA ⊗Fp[P ] Fp[Q]; it is easily seen that the ringhomomorphism Fp[ν] : Fp[P ] → Fp[Q] is invertible up to Φ, in the sense of [36, Def.3.5.8(i)].Especially, SpecFp[ν] is integral, surjective and radicial, hence the same holds for fν , andtherefore the morphism of sites :

f ∗ν : Xet → Xν,et

is an equivalence of categories (lemma 7.1.7(i)). It follows that in this case, h admits a sectionif and only if the same holds for hν .

Next, suppose that the field of fractions K of A has characteristic zero; we may write Xν ×XSpecK = SpecKν and V ×X SpecK = SpecE for two field extensions Kν and E of K, suchthat [Kν : K] = p, and the section of hν yields a map E → Kν of K-algebras. Therefore wehave either E = K (in which case V = U , and then we are done), or else E = Kν , in whichcase V = (Xν ,Mν)tr, since both these schemes are normal and finite over U .

Hence, let us assume that V = (Xν ,Mν), and pick any point η ∈ X of codimension one,whose residue field κ(η) has characteristic p. Then Xν ×X X(η) = SpecB, where B :=OX,η ⊗Z[P ] Z[Q], and B := B ⊗A κ(η) = κ(η) ⊗Fp[P ] Fp[Q]. Since the map Fp[P ] → Fp[Q] isinvertible up to Φ, we easily deduce that B is local, and its residue field is a purely inseparableextension of κ(η), say of degree d. Therefore f−1

ν η consists of a single point η′, and OXν ,η′ ' Bis a normal and finite OX,η-algebra, hence it is a discrete valuation ring. Let e denote theramification index of the extension OX,η → OXν ,η′; then ed = p ([14, Ch.VI, §8, n.5, Cor.1]).Suppose that e = p; in view of [36, Lemma 6.2.5], the ramification index of the inducedextension of strict henselizations Osh

X,η → OshXν ,η′

is still equal to p, which is impossible, since his tamely ramified along X\U . In case e = 1, we must have d = p, and since the residue fieldκ(η)s of Osh

X,η is a separable closure of κ(η), it follows easily that the residue field of OshXν ,η′

must


be a purely inseparable extension of κ(η)s of degree p, which again contradicts the tameness ofh. ♦

Claim 7.3.49. If (X,M) = (X,M)2, then F(X,M) is essentially surjective (notation of definition6.2.7(i)).

Proof of the claim. By claim 7.3.47, we may assume that X is strictly henselian. In this case,let U , P and h : V → U be as in claim 7.3.48. By assumption d := dimP ≤ 2, and we haveto show that h×X Xν admits a section, for a suitable Kummer morphism ν : P → Q (notationof (7.3.37)). If d = 0, then P = 1, in which case U = X is strictly local, so its fundamentalgroup is trivial, and there is nothing to prove. In case d = 1, then P ' N (theorem 3.4.16(ii)),in which case X is a regular scheme (corollary 6.5.35), and X \U is a regular divisor (remark6.5.31) and then the assertion follows from the classical Abhyankar’s lemma ([42, Exp.XIII,Prop.5.2]). For d = 2, we may find e1, e2 ∈ P , and an integer N > 0 such that

Ne1 ⊕ Ne2 ⊂ P ⊂ P ′ := Ne1

N⊕ N

e2

N(see example 3.4.17(i)). Especially, the inclusion ν : P → P ′ is a morphism of Kummer type.Since a composition of morphisms of Kummer type is obviously of Kummer type, claims 7.3.48and 7.3.36 imply that we may replace (X,M) by (Xν ,Mν) and h by hν (notation of (7.3.37)),after which we may assume that P is isomorphic to N⊕2. In this case, X is again regular andX \U is a strictly normal crossing divisor, so the assertion follows again from [42, Exp.XIII,Prop.5.2]. ♦

Claim 7.3.50. In the situation of (7.3.1), suppose that (Y,N) is regular. Then the functor f ∗tr :Cov(Y,N)tr → Cov(Y ′, N ′)tr restricts to an equivalence :

(7.3.51) Tame(Y, (Y,N)tr)∼→ Tame(Y ′, (Y ′, N ′)tr).

Proof of the claim. Arguing as in the proof of proposition 7.3.2, we may reduce to the casewhere ϕ : T ′ → T is the saturated blow up of an ideal generated by two elements of Γ(T,OT ).

Notice that (f, log f) is an etale morphism (proposition 6.6.14(v)), and it restricts to an iso-morphism ftr : (Y ′, N ′)tr

∼→ (Y,N)tr (remark 6.6.17). Especially, (Y ′, N ′) is regular, and f ∗tris trivially an equivalence from the etale coverings of (Y,N)tr to those of (Y ′, N ′)tr.

Let g : V → (Y,N)tr be an object of Tame(Y, (Y,N)tr). By claim 7.3.49 and lemma 6.5.37,the functor Fu∗(Y,N)2 is essentially surjective, and then the same holds for F(Y,N)2 , in view ofclaim 7.3.45; hence we may find an etale covering (W,Q) → (Y,N)2, and an isomorphismV∼→ (Y,N)tr ×Y W of (Y,N)tr-schemes. On the other hand, example 3.5.56(ii) shows that ϕ

restricts to a morphism T ′1 → T2 (notation of (3.5.16)), consequently f restricts to a morphism(Y ′, N ′)1 → (Y,N)2. There follows a well defined etale covering :

(Y ′, N ′)1 ×(Y,N)2 (W,Q)→ (Y ′, N ′)1 ×Y ′ (Y ′, N ′).whose image under F(Y ′,N ′) is f ∗tr(g). However, (Y ′, N ′)1 contains all the points of height oneof Y ′\(Y ′, N ′)tr (corollary 6.5.36(i)), hence f ∗tr(g) is tamely ramified along Y ′\(Y ′, N ′)tr (see(7.3.40)). This shows that (7.3.51) is well defined, and clearly this functor is fully faithful, sincethe same holds for f ∗tr. ♦

Suppose again that (X,M) is strictly henselian, define P , U and h : V → U as in claim7.3.48, and set TP := (SpecP )]; it is easily seen that the counit of adjunction u∗u∗(X,M) →(X,M) is an isomorphism (cp. (6.6.48)), hence u∗(X,M) is regular (lemma 6.5.37). Let

X := (u∗(X,M), TP , πX)

be the object of K arising from u∗(X,M), as in (6.6.7); by theorem 6.6.32 (and its proof),we may find an integral proper simplicial subdivision ϕ : F → TP , such that the morphism


of schemes underlying (f, ϕ) : ϕ∗X → X is a resolution of singularities for X . Let k be theramification index of the covering h; we define (X(k),M (k)), (Xϕ,Mϕ), (Y,N), gX and kX asin example 7.3.38 : this makes sense, since, in the current situation, X is isomorphic to X×S S(where S is defined as in example 6.6.5(i)). Moreover, by the same token, the morphism ofschemes Y → X(k) is a resolution of singularities.

Set as well Uϕ := (Xϕ,Mϕ)tr and U(k) := (X(k),M (k))tr; by claim 7.3.50 and lemma7.3.18(i), we have an essentially commutative diagram of functors :

Tame(X,U)f∗tr //

k∗X

Tame(Xϕ, Uϕ)

g∗X

Tame(X(k), U(k)) // Tame(Y, (Y,N)tr)

whose horizontal arrows are equivalences. In light of claim 7.3.48, we are reduced to showingthat k∗X(h) admits a section. Set hϕ := f ∗tr(h); then it suffices to show that g∗X(hϕ) admits asection, and notice that the ramification index of hϕ along Xϕ\Uϕ divides k (lemma 7.3.18(ii)).

Let V → Xϕ be the normalization of hϕ overXϕ, and w a geometric point of V ×XϕY whosesupport has height one; denote by y (resp. v, resp. x) the image of w in Y (resp. in V , resp.in Xϕ), and suppose that the support of x lies in Uϕ. Denote by Ksh(x) the field of fractionsof the strict henselization Osh

X,x of OX,x, and define likewise Ksh(w), Ksh(y) and Ksh(v). Therefollow a commutative diagram of inclusions of valued fields :

Ksh(x) //

Ksh(y)

Ksh(v) // Ksh(w)

which is cocartesian in the category of fields (i.e. Ksh(w) is the compositum of Ksh(y) andKsh(v) : cp. the proof of lemma 7.3.18(i)). By the foregoing, the ramification index of Osh

V ,v

over OshXϕ,x

divides k; on the other hand, example 7.3.38(ii) shows that the ramification indexof Osh

Y,y over OshXϕ,x

equals k. It then follows (e.g. from [36, Claim 6.2.15]) that Ksh(v) ⊂Ksh(y), and therefore Ksh(w) = Ksh(y); this shows that the ramification index of g∗X(hϕ)

along Y \ (Y,N)tr equals 1, therefore g∗X(hϕ) is the restriction of an etale covering h of Y(lemma 7.3.18(iii)). By proposition 7.3.2(ii), h is the pull-back of an etale covering of X(k).Since X(k) is strictly local (claim 7.3.36), it follows that h admits a section, hence the sameholds for g∗X(hϕ), as required.

7.3.52. As an application of theorem 7.3.44, we shall determine the fundamental group of the“punctured” scheme obtained by removing the closed point from a strictly local regular logscheme (X,M) of dimension≥ 2. Indeed, let x ∈ X be the closed point, and set r := dimMx.Also, let y be any geometric point of X , localized at a point y ∈ U := X \ x, and ξ ageometric point of X localized at the maximal point. Let p (resp. p′) denote the characteristicexponent of κ(x) (resp. of κ(y)). Pick any lifting of ξ to a geometric point ξy of X(y); since(X(y),M(y)) is regular (theorem 6.5.46), according to proposition 7.3.42, the vertical arrowsof the commutative diagram in (7.3.30) factor through the surjections (7.3.41), and we get a


commutative diagram of group homomorphisms :

π1((X(y),M(y))et, ξy)ϕy //

π1((X,M)et, ξ)

Mgp∨

y ⊗Z∏

` 6=p′ Z`(1) // Mgp∨x ⊗Z

∏`6=p Z`(1)

whose vertical arrows are the isomorphisms (7.3.43), and whose top (resp. bottom) horizontalarrow is induced by the natural morphism X(y) → X (resp. by the specialization map Mx →My). Now, by corollary 6.5.29, the scheme U is connected and normal, hence the restrictionfunctor Cov(U) → Tame(X, (X,M)tr) is fully faithful (lemma 7.1.7(ii.b)); by [42, Exp.V,Prop.6.9] and theorem 7.3.44, it follows that the induced group homomorphism

(7.3.53) π1((X,M)et, ξ)→ π1(Uet, ξ)

is surjective (see (7.3.40)). Clearly, the image of ϕy lies in the kernel of (7.3.53). Especially :

(U,M |U)r 6= ∅⇒ π1(Uet, ξ) = 1since, for any geometric point y of (X,M)r, the specialization map Mx → My is an isomor-phism (notation of definition 6.2.7(i)). Thus, suppose that (X,M)r = x, set P := M ]

x, anddenote by ψ : X → SpecP the natural continuous map; we have M ]

y = Pψ(y), and if we letFy := P \ ψ(y), we get a short exact sequence of free abelian groups of finite rank :

(7.3.54) 0→ F gpy →M ]gp

x →M ]gpy → 0.

It is easily seen that ψ is surjective; hence, for any p ∈ SpecP of height r−1, pick y ∈ ψ−1(p).With this choice, we have F gp

y = Z; considering (7.3.54)∨, we deduce that π1(Uet, ξ) is aquotient of Z′(1) :=

∏` 6=p Z`(1). More precisely, let (SpecP )r−1 be the set of prime ideals of

P of height r − 1; then we have :

Theorem 7.3.55. If (X,M)r = x, we have a natural identification :

(7.3.56)P∨∑

p∈(SpecP )r−1P∨p⊗Z Z′(1)

∼→ π1(Uet, ξ)

and these two abelian groups are cyclic of finite order prime to p.

Proof. The foregoing discussion already yields a natural surjective map as stated; it remainsonly to check the injectivity, and to verify that the source is a cyclic finite group.

However, since d ≥ 2, and (X,M)r = x, corollary 6.5.36(i) implies that r ≥ 2, in whichcase (SpecP )r−1 must contain at least two distinct elements, say p and q. Let F := P \ p; theimage H of (Pq)

]gp∨ in F gp∨ is clearly a non-trivial subgroup, hence F gp∨/H is a cyclic finitegroup, and then the same holds for the source of (7.3.56).

Next, it follows easily from lemma 7.1.7(ii.b) that (7.3.53) identifies π1(Uet, ξ) with the quo-tient of π1((X,M)et, ξ) by the sum of the images of the maps ϕy, for y ranging over all thepoints of U . However, it is clear that the same sum is already spanned by the sum of the imagesof the ϕy such that y ∈ (X,M)r−1; whence the theorem.

Example 7.3.57. Let P ⊂ N⊕2 be the submonoid of all pairs (a, b) such that a+b ∈ 2N. ClearlyP is fine and saturated of dimension 2. Let K be any field; the K-scheme X := SpecK[P ] isthe singular quadric in A3

K cut by the equation XY −Z2 = 0. It is easily seen that Spec(K,P )2

consists of a single point x (the vertex of the cone); let x be a geometric point of X localizedat x, and U ⊂ X(x) the complement of the closed point. Then U is a normal K-scheme ofdimension 1, and we have a natural isomorphism

π1(U, ξ)∼→ Z/2Z.


Indeed, a simple inspection shows that P admits exactly two prime ideals of height one, namelyp := P \ (2N ⊕ 0) and q := P \ (0 ⊕ 2N). Then, Pp = (a, b) ∈ Z ⊕ N | a + b ∈ 2N,and similarly Pq is a submonoid of N ⊕ Z. The quotients P ]

p and P ]q are both isomorphic to

N, and are both generated by the class of (1, 1). Let ϕ : P ]p → Z be a map of monoids; then

the image of ϕ in P gp∨ is the unique map of monoids P → Z given by the rule : (2, 0) 7→ 0,(1, 1) 7→ ϕ(1, 1), (0, 2) 7→ 2ϕ(1, 1). Likewise, a map ψ : P ]

q → Z gets sent to the morphismP → Z such that (2, 0) 7→ 2ψ(1, 1), (1, 1) 7→ ψ(1, 1), and (0, 2) 7→ 0. We see therefore that(Pp)

]gp∨+(Pq)]gp∨ is a subgroup of index two in P gp∨, and the contention follows from theorem

7.3.55.

7.4. Local acyclicity of smooth morphisms of log schemes. In this section we consider asmooth and saturated morphism f : (X,M)→ (Y,N) of fine log schemes. We fix a geometricpoint x of X , localized at a point x, and let y := f(x). We shall suppose as well that Y isstrictly local and normal, that (Y,N)tr is a dense open subset of Y , and that y is localized at theclosed point y of Y . Let η be a strict geometric point of Y , localized at the generic point η; toease notation, set :

U := f−1x (η) Utr := (X(x),M(x))tr ∩ U U := U ×|η| |η| U tr := Utr ×|η| |η|

and notice that Utr is a dense open subset of U , by virtue of proposition 6.7.17(ii,iv). Choosea geometric point ξ of U tr, and let ξ′ be the image of ξ in Utr. There follows a short exactsequence of topological groups :

1→ π1(U tr,et, ξ)→ π1(Utr,et, ξ′)

α−→ π1([η|et, η)→ 1.

On the other hand, let p be the characteristic exponent of κ := κ(y); for every integer e suchthat (e, p) = 1, the discussion of (7.3.26) yields a commutative diagram :

(7.4.1)

Ny//

log fx

κ(η)× //

Homcont(π1(|η|et, η),µe(κ))

Homcont(α,µe(κ))

Mx

// Γ(Utr,O×U ) // Homcont(π1(Utr,et, ξ

′),µe(κ))

such that the composition of the two top (rep. bottom) horizontal arrows factors through N ]y

(resp. M ]x). There follows a system of natural group homomorphisms :

(7.4.2) Coker (log f)gpx → Homcont(π1(U tr,et, ξ),µe(κ)) where (e, p) = 1

which assemble into a group homomorphism :

(7.4.3) π1(U tr,et, ξ)→ Coker (log f gpx )∨ ⊗Z

∏`6=p

Z`(1).

7.4.4. We wish to give a second description of the map (7.4.3), analogous to the discussion in(7.3.31). To this aim, let R be a ring; for any monoid P , and any integer e > 0, denote byeP : P → P the e-Frobenius map of P . Let λ : P → Q be a local morphism of finitelygenerated monoids. For any integer e > 0, we get a commutative diagram of monoids :

(7.4.5)P

eP //

λ

P

λe

λ

&&MMMMMMMMMMMMM

Qµ′ // Q′

eQ|P // Q


whose square subdiagram is cocartesian, and such that eQ|P µ′ = eQ. The latter induces acommutative diagram of log schemes :

Spec(R,Q)gQ|P //

g ))TTTTTTTTTTTTTTTSpec(R,Q′)

g′ //

ge

Spec(R,Q)

g

Spec(R,P )

gP // Spec(R,P )

whose square subdiagram is cartesian, and such that gQ := g′ gQ|P is the morphism inducedby eQ. Also, by example 6.6.5, we have a commutative diagram of monoidal spaces :

Spec(R,Q)gQ|P //

ψQ

Spec(R,Q′)

ψQ′

ge // Spec(R,P )

ψP

TQϕ // TQ′ // TP

(where ϕ := (Spec eQ|P )]) which determines morphisms in the category K :

(Spec(R,Q), TQ, ψQ)→ (Spec(R,Q′), TQ′ , ψQ′)→ (Spec(R,P ), TP , ψP ).

7.4.6. Now, suppose that the closed point mQ of TQ lies in the strict locus of (Specλ)] (whichjust means that λ] is an isomorphism). Notice that the functor M 7→ M ] commutes withcolimits (since it is a left adjoint); taking into account lemma 3.1.12, we deduce that the squaresubdiagram of (7.4.5)] is still cocartesian, and therefore e]Q|P is an isomorphism, so the sameholds for ϕ.

More generally, let q ∈ SpecQ be any prime ideal in the strict locus of (Specλ)]; set p :=λ−1q, r := ϕ(q), and recall that TQq := SpecQq is naturally an open subset of TQ (see (3.5.16)).A simple inspection shows that the restriction TQq → TQ′r of ϕ is naturally identified with themorphism of affine fans (Spec eQq|Pp)

]. Since q is the closed point of TQq , the foregoing showsthat TQq lies in the strict locus of ϕ; in other words, Str(ϕ) is an open subset of TQ, and we have

(7.4.7) Str((Specλ)]) ⊂ Str(ϕ).

Moreover, recall that Spec eQ : TQ → TQ is the identity on the underlying topological space(see example 3.5.10(i)), and by construction it factors through ϕ, so the latter is injective on theunderlying topological spaces. Especially, Str(ϕ) = ϕ−1ϕ(Str(ϕ)), and therefore

(7.4.8) Str(gQ|P ) = ψ−1Q (Str(ϕ)) = g−1

Q|P (ψ−1Q′ ϕ(Str(ϕ))).

Hence, set g := Spec(R, λ); from (7.4.7) and (6.6.3) we obtain :

Str(g) ⊂ Str(gQ|P )

and together with (7.4.8) we deduce that :

• Str(gQ|P ) is an open subset of SpecR[Q].• ψ−1

Q′ ϕ(Str(ϕ)) is a locally closed subscheme of SpecR[Q′].• The restriction Str(g)→ ψ−1

Q′ ϕ(Str(ϕ)) of gQ|P is a finite morphism.

Lastly, suppose that e is invertible in R; in this case, the morphisms gP and gQ are etale (propo-sition 6.3.34), so the same holds for gQ|P (corollary 6.3.25(iii)). From corollary 6.3.27(i) wededuce that the restriction Str(g)→ ψ−1

Q′ ϕ(Str(ϕ)) of gQ|P is a (finite) etale covering.


7.4.9. In the situation of (7.4.4), suppose additionally, that R is a Z[1/e,µe]-algebra (where µeis the e-torsion subgroup of C×), P and Q are fine monoids, and λ is integral, so that Q′ is alsofine. Set

GP := HomZ(P gp,µe) GQ := HomZ(Qgp,µe) GQ|P := HomZ(Cokerλgp,µe).

Notice that the trivial locus Spec(R,P )tr is the open subset SpecR[P gp], and likewise forSpec(R,Q)tr and Spec(R,Q′)tr; therefore, (7.4.5)gp induces a cartesian diagram of schemes

Spec(R,Q′)tr

g′tr //

ge,tr

Spec(R,Q)tr

gtr

Spec(R,P )tr

gP,tr // Spec(R,P )tr.

Fix a geometric point τ ′Q of Spec(R,Q′)tr, and let τQ := g′(τ ′Q), τP := g(τQ), τ ′P := ge(τ′Q). It

was shown in (7.3.31) that g−1P (τP ) is aGP -torsor, so gP,tr is a Galois etale covering, correspond-

ing to a continuous representation (7.3.32) into the same group. Hence g′−1(τQ) is a GP -torsor,and g′tr is a Galois etale covering, whose corresponding representation of π1(Spec(R,Q)tr,et, τQ)is obtained by composing (7.3.32) with the natural continuous group homomorphism

π1(gtr, τQ) : π1(Spec(R,Q)tr,et, τQ)→ π1(Spec(R,P )tr,et, τP ).

By the same token, g−1Q (τQ) is a GQ-torsor, so also gQ,tr is a Galois etale covering, and a simple

inspection shows that the surjection

g−1Q (τQ)→ g′−1(τQ)

induced by gQ|P is GQ-equivariant, for the GQ-action on the target obtained from the map

HomZ(λgp,µe) : GQ → GP .

The situation is summarized by the commutative diagram of continuous group homomorphisms

π1(Spec(R,Q′)tr,et, τ′Q) //

π1(ge,tr,τ ′Q)

π1(Spec(R,Q)tr,et, τQ) //

π1(gtr,τQ)

GQ

HomZ(λgp,µe)

π1(Spec(R,P )tr,et, τ

′P ) // π1(Spec(R,P )tr,et, τP ) // GP

whose horizontal right-most arrows are the maps (7.3.32). Consequently, g−1Q|P (τ ′Q) is a GQ|P -

torsor, and gQ|P,tr is a Galois etale covering, classified by a continuous group homomorphism

(7.4.10) Ker π1(ge,tr, τ′Q)→ Ker π1(gtr, τQ)→ GQ|P .

7.4.11. Let us return to the situation of (7.4), and assume additionally, that both (X,M) and(Y,N) are fs log schemes. Take R := OY,y in (7.4.4); by corollary 6.3.42 and theorem6.1.35(iii), we may assume that there exist

• a local and saturated morphism λ : P → Q of fine and saturated monoids, such thatP is sharp, Q× is a free abelian group of finite type, say of rank r, and AssZ Cokerλgp

does not contain the characteristic exponent of κ(y);• a morphism of schemes π : Y → SpecR[P ], which is a section of the projection

SpecR[P ]→ Y , such that

(Y,N) = Y ×SpecR[P ] Spec(R,P ) (X,M) = Y ×SpecR[P ] Spec(R,Q)

and f is obtained by base change from the morphism g := Spec(R, λ). Moreover, theinduced chart QX →M shall be local at the geometric point x.


By claim 6.1.37, we may further assume that the projection Q → Q] admits a section σ :Q] → Q, such that λ(P ) lies in the image of σ. In this case, g factors through the morphismSpec(R,P )→ Spec(R,Q]) induced by λ], and σ induces an isomorphism of log schemes :

Spec(R,Q) = G⊕rm,Y ×Y Spec(R,Q])

(where G⊕rm,Y denotes the standard torus of rank r over Y ). In this situation, X is smooth overY ×SpecR[P ] SpecR[Q]], and more precisely

(7.4.12) (X,M) = G⊕rm,Y ×SpecR[P ] Spec(R,Q]).

Summing up, after replacing Q by Q], we may assume that Q is also sharp, and (7.4.12) holdswith Spec(R,Q) instead of Spec(R,Q]). Moreover, the image of x in TQ is the closed pointmQ.

Let e > 0 be an integer which is invertible in R; by inspecting the definition, it is easily seenthat there exists a finite separable extension Ke of κ(η) = Frac(R), such that the normalizationYe of Y in SpecKe fits into a commutative diagram of schemes :

Ye //

πe

Y

π

SpecR[P ]gP // SpecR[P ]

(whose top horizontal arrow is the obvious morphism); namely, π is defined by some morphismof monoids β : P → R, and one takes for Ke any subfield of κ(η) containing κ(η) and thee-th roots of the elements of β(P ). Set (Ye, N e) := Ye ×SpecR[P ] Spec(R,P ), and define logschemes (Xe,M e), (X ′e,M

′e) so that the two square subdiagrams of the diagram of log schemes

(X ′e,M′e)

he //

(Xe,M e) //

G⊕rm,Y ×Y (Ye, N e)

π′e

Spec(R,Q)gQ|P // Spec(R,Q′)

ge // Spec(R,P )

are cartesian (here π′e is the composition of πe and the projection G⊕rm,Y ×Y Ye → Ye), whence acommutative diagram :

(7.4.13)

(X ′e,M′e)

he //

f ′e &&MMMMMMMMMM(Xe,M e)

fe

(Ye, N e).

Notice that both fe and f ′e are smooth and saturated morphisms of fine log schemes. Also, byconstruction Ye is strictly local, and (Ye, N e)tr is a dense subset of Ye. In other words, fe andf ′e are still of the type considered in (7.4). Moreover, he is etale, since the same holds for gQ|P ,and the discussion in (7.4.6) shows that the restriction

Str(f ′e)→ Xe

is an etale morphism of schemes. Furthermore, the discussion in (7.4.9) shows that

he,tr : (X ′e,M′e)tr → (Xe,M e)tr

is a Galois etale covering.


7.4.14. More precisely, notice that λ] = log f ]x; combining with (7.4.10), we deduce a continu-ous group homomorphism :

(7.4.15) Ker π1(fe,tr, ξ′e)→ HomZ(Coker (log f)gp

x ,µe(κ))

where ξ′e is the image in Xe of the geometric point ξ. The geometric point y lifts uniquely to ageometric point ye of Ye, localized at the closed point ye, and the pair (x, ye) determines a uniquegeometric point xe such that fe(xe) = ye. Also, since the field extension κ(y)→ κ(ye) is purelyinseparable, it is easily seen that the induced map f−1

e (ye)→ f−1(y) is a homeomorphism; therefollows an isomorphism of X(x)-schemes ([33, Ch.IV, Prop.18.8.10]) :

Xe(xe)∼→ X(x)×Y Ye.

Let ηe be the generic point of Ye; by construction, η lifts to a geometric point ηe of Ye, localizedat ηe, and we have continuous group homomorphisms :

(7.4.16) π1(U tr, ξ)∼→ Ker (π1(Ue,tr, ξ

′e)→ π1(ηe, |ηe|))→ Ker π1(fe,tr, ξ

′e).

The composition of (7.4.15) and (7.4.16) is a continuous group homomorphism

(7.4.17) π1(U tr, ξ)→ HomZ(Coker (log f)gpx ,µe(κ))

whence, finally, a pairing

Coker (log f)gpx × π1(U tr,et, ξ)→ µe(κ).

We claim that this pairing agrees with the one deduced from (7.4.2). Indeed, by tracing backthrough the constructions, we see that (7.4.17) is the homomorphism arising from the Galoiscovering of U tr, which is obtained from gQ|P , after base change along the composition

U tr → Xe → SpecR[Q′].

On the other hand, the discussion of (7.3.31) shows that the homomorphism π1(Utr,et, ξ′)→ GQ

arising from the bottom row of (7.4.1), classifies the Galois covering C → U tr obtained fromgQ, by base change along the same map. By the same token, the top row of (7.4.1) correspondsto the GP -Galois covering C ′ → |η| obtained by base change of gP along the composition|η| → Y → SP . The map log fx induces a morphism of schemes C → C ′ ×|η| Utr, and (7.4.2)corresponds to the GQ|P -torsor obtained from a fibre of this morphism. Evidently, this torsor isisomorphic to g−1

Q|P (τ ′Q), whence the contention.

7.4.18. In the situation of (7.4), recall that there is a natural bijection between the set of maximalpoints of f−1

x (y), and the set Σ of maximal points of the closed fibre of the induced map

(7.4.19) SpecMx → SpecNy

(proposition 6.7.14). For every q ∈ Σ, denote by ηq the corresponding maximal point of f−1x (y),

choose a geometric point ηq localized at ηq, and let X(ηq) be the strict henselization of X(x) atηq. Also, set

Uq := U ×X(x) X(ηq) U q := Uq ×|η| |η|and notice that U q is an irreducible normal scheme. Notice as well that f induces a strictmorphism (X(ηq),M(ηq)) → (Y,N) (theorem 6.7.8(iii.a)), and therefore the log structure ofUq ×X(ηq) (X(ηq),M(ηq)) is trivial.

Recall that Z := U \ U tr is a finite union of irreducible closed subsets of codimensionone in U , and OU,z is a discrete valuation ring, for each maximal point z ∈ Z (proposition6.7.17(iv,v)); especially, the category Tame(U,U tr) is well defined (definition 7.3.17(iii)). Wedenote

Tame(f, x)


the full subcategory of Tame(U,U tr) consisting of all the coverings C → U tr such that, forevery q ∈ Σ, the induced covering

C ×UtrU q → U q

is trivial (i.e. C ×UtrU q is a disjoint union of copies of U q). It is easily seen that Tame(f, x)

is a Galois category (see [42, Exp.V, Def.5.1]), and we obtain a fibre functor for this category,by restriction of the usual fibre functor ϕ 7→ ϕ−1(ξ) defined on all etale coverings ϕ of U tr;we denote by π1(U tr/Yet, ξ) the corresponding fundamental group. According to [42, Exp.V,Prop.6.9], the fully faithful inclusion Tame(f, x)→ Cov(U tr) induces a continuous surjectivegroup homomorphism

(7.4.20) π1(U tr,et, ξ)→ π1(U tr/Yet, ξ).

Proposition 7.4.21. The map (7.4.3) factors through (7.4.20), and the induced group homo-morphism :

(7.4.22) π1(U tr/Yet, ξ)→ Coker (log f gpx )∨ ⊗Z

∏`6=p

Z`(1).

is surjective.

Proof. Let σY : Y → Y qfs be the natural morphism of schemes exhibited in remark 6.2.36(iv),and set

(Y,N ′) := Y ×Y qfs (Y,N)qfs (X,M ′) := (Y,N ′)×(Y,N) (X,M).

Since f is saturated, both (X,M ′) and (Y,N ′) are fs log schemes (see remark 6.2.36(i)); also,the morphism of schemes underlying the induced morphism of log schemes f ′ : (X,M ′) →(Y,N ′), agrees with that underlying f . Moreover, by construction we have N ′]z = (N ]

z)sat

for every geometric point z of Y , especially (Y,N ′)tr = (Y,N)tr. Likewise, M ′]z = (M ]

z)sat

(lemma 3.2.12(iii,iv)), therefore Str(f ′) = Str(f), and especially, (X,M ′)tr = (X,M)tr. Fur-thermore, notice that the the natural map

(7.4.23) Coker (log f)gpx → Coker (log f ′)gp

x

is surjective, and its kernel is a quotient of (M satx )×/M×

x , especially it is a torsion subgroup.However, the cokernel of (log f)gp

x equals the cokernel of (log f ]x)gp, hence it is torsion-free

(corollary 3.2.32(ii)), so (7.4.23) is an isomorphism. Thus, we may replace N (resp. M ) by N ′

(resp. M ′), and assume from start that f is a smooth, saturated morphism of fs log schemes.In this case, in light of the discussion of (7.4.14), it suffices to prove that (7.4.17) is a surjec-

tion, and that it factors through π1(U tr/Yet, ξ). To prove the surjectivity comes down to showingthat U tr ×Xe X ′e is a connected scheme. However, let xe be the support of xe, and notice thatψP πe(ye) = mP , the closed point of TP . Since x maps to the closed point of TQ, we deduceeasily that the image of xe in TQ′ is the closed point mQ′ , i.e. xe lies in the closed subschemeXe ×S′Q Specκ〈Q′/mQ′〉. Notice as well that eQ|P is of Kummer type (see definition 3.4.40);by proposition 7.3.8(ii), it follows that there exists a unique geometric point x′e of X ′e lying overxe, whence an isomorphism of Xe(xe)-schemes ([33, Ch.IV, Prop.18.8.10])

X ′e(x′e)∼→ X ′e ×Xe Xe(xe).

Hence U tr ×Xe X ′e is an open subset of |ηe| ×Ye X ′e(xe), and the latter is an irreducible scheme,by proposition 6.7.17(ii). We also deduce that the induced morphism

he,x : (X ′e(x′e),M

′e(x′e))→ (Xe(xe),M e(xe))

is a finite etale covering of log schemes. From the discussion in (7.4.11), we see that therestriction of he,x

Str(he,x)→ Xe(xe)


is an etale morphism, and Str(he,x) contains the strict locus of the induced morphism

f ′e,x : (X ′e(x′e),M

′e(x′e))→ (Ye, N e).

Since the field extension κ(y) → κ(ye) is purely inseparable, ηq lifts uniquely to a geometricpoint ηe,q ∈ Xe(xe), and as usual we deduce that the strict henselization Xe(ηq) of Xe(xe)at ηe,q is isomorphic, as an Xe(xe)-scheme, to Xe(xe) ×X(x) X(ηq). Moreover, if ηe,q is thesupport of ηe,q, a simple inspection shows that the fibre h−1

e,x(ηe,q) consists of maximal pointsof f ′−1

e,x (ye). By theorem 6.7.8(iii.a), every point of h−1e,x(ηe,q) lies in Str(f ′e,x), therefore the

induced morphism X ′e ×Xe Xe(ηq) → Xe(ηq) is finite and etale. Taking into account (7.3.40),we conclude that the etale covering X ′e×Xe U tr → U tr is an object of Tame(f, x), whence theproposition.

7.4.24. Say that Y = SpecR; for every algebraic field extension K of κ(η) = Frac(R), let RK

be the normalization of R in K, set |ηK | := SpecK and

YK := SpecRK (YK , NK) := YK ×Y (Y,N) (XK ,MK) := YK ×Y (X,M).

Moreover, let fK : (XK ,MK) → (YK , NK) be the induced morphism, and yK any geomet-ric point localized at the closed point yK of the strictly local scheme YK ; since the extensionκ(y)→ κ(yK) is purely inseparable, there exists a unique geometric point xK of XK lifting x,and we have an isomorphism of (X(x),M(x))-schemes :

(XK(xK),MK(xK))∼→ XK ×X (X(x),M(x)) = YK ×Y (X(x),M(x)).

Clearly the morphism fK is again of the type considered in (7.4.18); especially, the maximalpoints of f−1

K,xK(yK) are in natural bijection with the elements of Σ, and it is natural to denote

UK := U ×Y |ηK | UK,tr := Utr ×Y |ηK | UK,q := UK,tr ×X(x) X(ηq)

for every q ∈ Σ. Then, let ηK,q be the unique geometric point of f−1K,xK

(yK) lying over ηq, andηK,q the support of ηK,q; as usual, we have

(7.4.25) XK(ηK,q) = X(ηq)×Y YKhence the above notation is consistent with the one introduced for the original morphism f .Furthermore, if z is any maximal point of U\U tr, the image zK of z in UK is a maximal point ofUK\UK,tr, and since the induced map OUK ,zK → OU,z is faithfully flat, proposition 6.7.17(iv,v)easily implies that OUK ,zK is a discrete valuation ring. We may then denote

Tame(f, x,K)

the full subcategory of Tame(UK , UK,tr), consisting of those objects C → UK,tr, such that theinduced covering C ×UK,tr UK,q → UK,q is trivial, for every q ∈ Σ. We have a natural functor

(7.4.26) 2-colimK

Tame(f, x,K)→ Tame(f, x)

where the 2-colimit ranges over the filtered family of all finite separable extensions K of κ(η).


Proof. Let h : C → U tr be an object of the category Tame(f, x). According to [33, Ch.IV,Prop.17.7.8(ii)] and [32, Ch.IV, Prop.8.10.5], we may find a finite extension K of κ(η), suchthat h descends to a finite etale morphism

hK : CK → UK,tr.

Let C ′K (resp. C ′) denote the normalization of CK (resp. of C) over UK (resp. over U ). Sincethe morphism |η| → |ηK | is ind-etale, we have C ′ = C ′K ×|ηK | |η| ([33, Ch.IV, Prop.17.5.7]),


and it follows easily that CK is tamely ramified along the divisor UK \UK,tr. From (7.4.25) weget a natural isomorphism :

U q∼→ UK,q ×|ηK | |η|.

Thus, after replacing K by a larger finite separable extension of κ(η), we may assume that theinduced morphism CK ×UK,tr UK,q → UK,q is a trivial etale covering, for every q ∈ Σ.

This shows that (7.4.26) is essentially surjective; likewise one shows the full faithfulness :the details shall be left to the reader.

7.4.28. In the situation of (7.4.24), let K be an algebraic extension of κ(η), and

h : C → UK,tr

any object of Tame(UK , UK,tr), and denote by C ′ the normalization of XK(xK) in C. Weclaim that there exists a largest non-empty open subset

E(h) ⊂ XK(xK)

such that the restriction h−1E(h) → E(h) of h is etale. Indeed, in any case, h restricts to anetale morphism on a dense open subset containing UK,tr, and there exists a largest open subsetE ′ ⊂ C ′ such that h|E′ is etale (claim 7.1.8 and lemma 7.1.7(ii.b)). Then it is easily seen thatE(h) := XK(xK)\h(C ′\E ′) will do.

Lemma 7.4.29. With the notation of (7.4.28), the category Tame(f, x,K) is the full subcate-gory of Tame(UK , UK,tr) consisting of those objects h : C → UK,tr such that E(h) containsthe maximal points of XK(xK)×YK |yK |.

Proof. In view of claim 7.1.9, this characterization is a rephrasing of the definition of the cate-gory Tame(f, x,K).

7.4.30. Keep the notation of (7.4.28), and suppose that h is an object of Tame(f, x,K). Fixq ∈ Σ; then lemma 7.4.29 says that ηK,q ∈ E(h). Thus, we obtain a functor

Tame(f, x,K)→ Cov(|ηK,q|) : C 7→ C ′ ×XK(xK) |ηK,q|.However, the natural morphism |ηK,q| → |ηq| is radicial, hence it induces an equivalence

Cov(|ηq|)∼→ Cov(|ηK,q|)

(lemma 7.1.7(i)). Combining these two functors in the special special case where K := κ(η),we get a functor

(7.4.31) Tame(f, x)→ Cov(|ηq|) : (C → U tr) 7→ (C|ηq → |ηq|).Now, the rule ϕ 7→ ϕ−1(ηq) yields a fibre functor for the Galois category Cov(|ηq|); by com-position with (7.4.31), we deduce a fibre functor for Tame(f, x), whose group of automor-phisms we denote π1(U tr/Yet, ηq). Also, set F (q) := Mx\q, and notice that the structure mapMx → OX(x),x induces a group homomorphism

(7.4.32) F (q)gp → κ(ηq)×.

Lemma 7.4.33. With the notation of (7.4.30), we have :(i) The natural map Coker (log fx)→ F (q) induces a commutative diagram of groups

π1(|ηq|et, ηq) //

α

π1(U tr/Yet, ηq) // π1(U tr/Yet, ξ)

β

F (q)gp∨ ⊗Z

∏6=p Z`(1)

γ // Coker (log f gpx )∨ ⊗Z

∏` 6=p Z`(1)

where β is (7.4.22), and α is deduced from (7.4.32), as in the discussion of (7.3.26).


(ii) α is surjective, and γ is an isomorphism.

Proof. (i): The proof amounts to unwinding the definitions, and shall be left as an exercicefor the reader. Notice that the second arrow on the top row is only well-defined up to innerautomorphisms, but since the groups on the bottom row are abelian, the ambiguity does notaffect the statement.

(ii): Notice that log fx restricts to a map of monoids O×Y,y → F (q), which induces an iso-morphism Coker(log f)]

∼→ F (q)]; we deduce that γ is an isomorphism. Next, let Z be thetopological closure of ηq in X(x), and endow Z with its reduced subscheme structure; set also(Z,M(Z)) := Z ×X(x) (X,M). The map α factors as a composition

π1(|ηq|et, ηq)→ π1((Z,M(Z))tr,et, ηq)→ F (q)gp∨ ⊗Z∏` 6=p

Z`(1)

where the first map is surjective, by lemma 7.1.7. Lastly, notice that M(Z)red,x = F (q); bypropositions 7.3.42 and 6.7.14(ii), it follows that the second map is surjective as well, so theproof of (ii) is complete.

7.4.34. In the situation of (6.3.44), suppose that Yi is a strictly local normal scheme for everyi ∈ I , and the transition morphisms Yj → Yi are local and dominant, for every morphismi → j in I . Let x be a geometric point of X , and denote by xi the image of x in Xi, for everyi ∈ I . Suppose that the image y of x in Y is localized at the closed point. Also, let η be a strictgeometric point of Y , localized at the generic point ηi, and denote by ηi (resp. yi) the strictimage of η (resp. y) in Yi (see definition 7.1.10(iii)).

Lemma 7.4.35. In the situation of (7.4.34), suppose that (g, log g) : (X,M) → (Y,N) is asmooth and saturated morphism of fine log schemes. Then there exists i ∈ I , and a smooth andsaturated morphism (gi, log gi) : (Xi,M i) → (Yi, N i) of fine log schemes, such that log g =π∗i log gi.

Proof. By corollary 6.3.45, we can descend (g, log g) to a smooth morphism (gi, log gi) of finelog schemes, and after replacing I by I/i, we may assume that i = 0. Then the contentionfollows from corollary 6.2.34(ii).

7.4.36. Keep the situation of (7.4.34), and suppose that (g0, log g0) : (X0,M0)→ (Y0, N0) is asmooth and saturated morphism of fine log schemes; set

(Xi,M i) := Xi ×X0 (X0,M0) (Yi, N i) := Yi ×Y0 (Y0, Y 0)

and denote (gi, log gi) : (Xi,M i) → (Yi, N i) the induced morphism of log schemes, for ev-ery i ∈ I . Also, let (g, log g) : (X,M) → (Y,N) be the limit of the system of morphisms((gi, log gi) | i ∈ I). These are morphisms of the type considered in (7.4), so we may defineUi := g−1

i,xi(ηi), and introduce likewise the schemes Ui,tr, U i and U i,tr as in (7.4). Moreover, set

Zi := U i\U i,tr for every i ∈ I; clearly Zi = Zj ×Xj(xj) Xi(xi) for every morphism i→ j in I .Also, each Zi is a finite union of irreducible subsets of codimension one, and for every i → jin I , the transition morphisms Xi → Xj restrict to maps

MaxZi → MaxZj MaxXi(xi)×Yi |yi| → MaxXj(xj)×Yj |yj|

Combining proposition 7.3.22(ii) and lemma 7.4.29, we deduce a fully faithful functor

(7.4.37) 2-colimi∈I

Tame(gi, xi)→ Tame(g, x).



Proof. It remains only to show the essential surjectivity. Hence, let h be a given object ofTame(g, x); by proposition 7.3.22(ii), we know that there exists j ∈ I such that h descendsto an etale covering hj : Vj → U j,tr, tamely ramified along Zj , and after replacing I by I/i,we may assume that j is the final object of I , and define hi := U i,tr ×Uj,tr hj for every i → j

in I . Now, let E ′ ⊂ E(h) be a constructible open subset containing the maximal points ofX(x) ×Y |y|. For every i ∈ I , let Y i be the normalization of Yi in Specκ(ηi), and set X i :=Xi(xi) ×Yi Y i; according to [32, Ch.IV, Th.8.3.11], there exists i ∈ I such that E ′ descends toa constructible open subset E ′i ⊂ X i, and then necessarily E ′i contains all the maximal pointsof Xi(xi)×Yi |yi|. As usual, we may assume that i is the final object, so E ′i is defined for everyi ∈ I . Lastly, since h extends to an etale covering on E ′, we see that hi extends to an etalecovering of E ′k, for some k ∈ I ([33, Ch.IV, Prop.17.7.8(i)]). In view of lemma 7.4.29, thecontention follows.

Theorem 7.4.39. The map (7.4.22) is an isomorphism.

Proof. Arguing as in the proof of proposition 7.4.21, we may assume that both (X,M) and(Y,N) are fs log schemes, and in view of proposition 7.4.21, we need only show that (7.4.22)is injective. This comes down to the following assertion. For every object h : C → U tr of thecategory Tame(f, x), the induced action of π1(U tr, ξ) on h−1(ξ) factors through the quotientCoker (log f gp

x )∨ ⊗Z∏6=p Z`(1).

• By lemma 7.4.27, there exists a finite separable extension K of κ(η), and an object h :

CK → UK,tr of Tame(f, x,K), with an isomorphism CK ×UK,tr U tr∼→ C of U tr-schemes.

Since log fx = log fK,xK , the theorem will hold for the morphism f and the point x, if and onlyif it holds for fK and the point xK .

Claim 7.4.40. The theorem holds, if Y is noetherian of dimension one.

Proof of the claim. In this case, Y is the spectrum of a strictly henselian discrete valuationring R, and the same then holds for YK . Hence, we may replace throughout f by fK , andassume from start that there exists an object h : C → Utr of Tame(f, x, κ(η)), with an iso-morphism C

∼→ C ×Utr U tr of U tr-schemes. Endow Y with the fine log structure N ′ suchthat Γ(Y,N ′) = R \0; since (Y,N)tr is dense in Y , we have a well defined morphism oflog schemes π : (Y,N ′) → (Y,N), which is the identity on the underlying schemes. Set(X,M ′) := (Y,N ′)×(Y,N) (X,M). Then (Y,N ′) is a regular log scheme, and consequently thesame holds for (X,M ′), by theorem 6.5.44.

Furthermore, π trivially restricts to a strict morphism on the open subset |η|, hence the in-duced morphism (X,M ′) ×Y |η| → (X,M) ×Y |η| is an isomorphism, especially Utr is thetrivial locus of (X(x),M ′(x))×Y |η|. However, it is easily seen that (X(x),M ′(x))tr does notintersect the closed fibre f−1

x (y), so finally Utr = (X(x),M ′(x))tr.From theorem 7.3.44, we deduce that h extends to an etale covering of (X(x),M ′(x)). Then,

arguing as in (7.4) we get a commutative diagram of groups :

π1(U tr,et, ξ) //

Coker (log f gpx )∨ ⊗Z

∏` 6=p Z`(1)

π1(Utr,et, ξ′)

α // Mgp∨x ⊗Z

∏` 6=p Z`(1)

whose top horizontal arrow is (7.4.3), and whose right vertical arrow is deduced from the pro-jection Mgp

x → Coker (log f gpx ). Lastly, proposition 7.3.42 shows that the natural action of

π1(Utr,et, ξ′) on h−1(ξ′) factors through α, so the claim follows. ♦

• Next, suppose that Y is an arbitrary normal, strictly local scheme. Ths discussion in(7.4.14) implies that, in order to prove the theorem, it suffices to find an integer e > 0, a


(X,M)-scheme (X ′e,M′e) as in (7.4.11), and a geometric point x′e of X ′e lying over x, such

that h ×X(x) X′e(x′e) is a trival covering. To this aim, we write Y as the limit of a cofiltered

system (Yi | i ∈ I) of strictly local excellent and normal schemes (lemma 7.1.30), and wedenote by ηi the generic point of Yi, for every i ∈ I . By lemma 7.4.35, we may then descend(f, log f) to a smooth and saturated morphism (fi, log fi) : (Xi,M i) → (Yi, N i), for somei ∈ I , and as usual, we may assume that i is the final object of I . Let xi be the image of x inXi; by lemmata 7.4.38 and 7.4.27, the object h of Tame(f, x, κ(η)) descends to an object hiof Tame(fi, xi, K), for some i ∈ I , and some finite separable extension K of κ(ηi). It sufficestherefore to find e > 0, a (Xi,M i)-scheme (X ′i,e,M

′i,e), and a geometric point x′i,e of X ′i,e lying

over xi, such that hi ×Xi(xi) X ′i,e(x′i,e) is a trivial covering. In other words, we may replacethroughout Y by Yi,K , and assume from start that Y is excellent, and h descends to an objecth : C → Utr of Tame(f, x, κ(η)).• By [27, Ch.II, Prop.7.1.7] we may find a discrete valuation ring V and a local injective

morphism R→ V inducing an isomorphism on the respective fields of fractions. Let Vsh be thestrict henselization of V (at a geometric point whose support is the closed point), and set

Y := Spec Vsh (Y,N) := Y ×Y (Y,N) (X,M) := Y ×Y (X,M).

Also, let f : (X,M)→ (Y,N) be the induced morphism. Denote by y a geometric point localizedat the closed point y of Y; also, pick any geometric point x of X, whose image in X is x; theinduced morphism

(7.4.41) (X(x),M(x))→ (X(x),M(x))

restricts to a flat morphism f−1x (y)→ f−1

x (y) and from proposition 6.7.14, we see that the latterinduces a bijection between the sets of maximal points of the two fibres. On the other hand, letηV denote a geometric point localized at the generic point ηV of Y; then (7.4.41) restricts to anind-etale morphism f−1

x (ηV)→ f−1x (η). Hence, set

Utr := (X(x),M(x))tr ×Y |ηV|.From lemma 7.4.29, it follows easily that the covering C ×Utr Utr → Utr is an object ofTame(f, x, κ(ηV)).

For any integer e > 0 invertible in R, pick a (Y,N)-scheme (Ye, N e) as in (7.4.11), so thatwe may define the etale morphism (X ′e,M

′e) → (Xe,M e) of (Ye, N e)-schemes as in (7.4.13).

Notice that the morphism (X ′e,M′e) → (Ye, N e) is again of the type considered in (7.4), and

there exists, up to isomorphism, a unique geometric point x′e of X ′e lifting x; moreover, for anygeometric point ye supported at ye, the induced map

SpecM ′e,x′e→ SpecN ′e,ye

is naturally identified with (7.4.19). Likewise, pick a (Y,N)-scheme (Ye,Ne) in the same fash-ion, and denote by ηe (resp. ηV,e) the generic point of Ye (resp. of Ye), and by ye ∈ Ye (resp.ye ∈ Ye) the closed point. We may choose Ye so that κ(ηV,e) contains κ(ηe), in which case wehave a strict morphism

(Ye,Ne)→ (Ye, N e)

of log schemes, and we may set (X′e,M′e) := Ye ×Ye (X ′e,M

′e). Again, there exists, up to

isomorphism, a unique geometric point x′e of X′e lifting x, and by claim 7.4.40, we may assumethat both e and κ(ηV,e) have been chosen large enough, so that the base change

h×X(x) X′e(x′e)

shall be a trivial etale covering. Hence, we may replace Y by Ye, Y by Ye, X by X ′e, and h byh×X(x) X

′e(x′e), and assume from start that C ×Utr Utr is a trivial covering of Utr. The theorem

will follow, once we show that – in this case – h is a trivial etale covering.


Let C ′ (resp. C′) be the normalization of X(x) in C (resp. of X(x) in C ×Utr Utr), and defineE(h) as in (7.4.28). Then C′ is a trivial etale covering of X(x), and since X(x) is excellent, theinduced morphism h′ : C ′ → X(x) is finite.

Claim 7.4.42. For every maximal point ηq of f−1x (y), the induced covering C|ηq → |ηq| is trivial

(notation of (7.4.31)).

Proof of the claim. Let Zq denote the topological closure of ηq in X(x), and endow Zq withits reduced subscheme structure; then Eq := E(h) ∩ Zq is non-empty (lemma 7.4.29), andgeometrically normal (proposition 6.7.14(ii) and corollary 6.5.29). Also, (7.4.41) induces anisomorphism of κ(y)-schemes

Eq ×X(x) X(x)∼→ Eq ×κ(y) κ(y).

Moreover, Zq is strictly local, and Zq ×X(x) X(x) is the strict henselization of Zq ×κ(y) κ(y)at the point x. Furthermore, the morphism h′′ := h′ ×X(x) Eq is an etale covering of Eq, andh′′ ×κ(y) κ(y) is naturally identified with the restriction of C′ to the subscheme Eq ×X(x) X(x)([33, Ch.IV, Prop.17.5.7]), hence it is a trivial covering. By example 7.2.6, it follows that h′′ istrivial as well. Since C|ηq is the restriction of h′′ to |ηq|, the claim follows. ♦

Clearly (7.4.41) maps each stratum Uq of the logarithmic stratification of X(x), to the cor-responding stratum Uq of the logarithmic stratification of X(x) (see (6.5.49)). More pre-cisely, since (7.4.41)×Y |η| is ind-etale, proposition 6.7.17(iii) implies that the generic point ofUq ×Y |ηV| gets mapped to the generic point of Uq ×Y |η|. We conclude that E(h) contains thegeneric point of every stratum Uq ×Y |η|.

Claim 7.4.43. X(x)×Y |η| ⊂ E(h).

Proof of the claim. Notice first that (X,M)×Y |η| is a regular log scheme (corollary 6.5.45).For any geometric point ξ of X(x), denote by (X(ξ),M(ξ)) the strict henselization of

(X(x),M(x)) at ξ, and set C ′(ξ) := C ′ ×X(x) X(ξ). Now, suppose that the support of ξlies in the stratum Uq ×Y |η|, and let ξq be a geometric point localized at the generic pointof Uq. By assumption, C ′(ξ) is a finite X(ξ)-scheme, tamely ramified along the non-triviallocus of (X(ξ),M(ξ)). Likewise, C ′(ξq) is tamely ramified along the non-trivial locus of(X(ξq),M(ξq)). Pick any strict specialization map (X(ξq),M(ξq)) → (X(ξ),M(ξ)) (see(6.7.11)); it induces a functor

(7.4.44) Cov(X(ξ),M(ξ))→ Cov(X(ξq),M(ξq))

and theorem 7.3.44 implies that C ′(ξ) is an object of the source of (7.4.44), which is mapped,under this functor, to the object C ′(ξq). By proposition 7.3.42, for any geometric point ξ ofX(x) ×Y |η|, the category Cov(X(ξ),M(ξ)) is equivalent to the category of finite sets witha continuous action of (M ξ)

gp∨ ⊗Z∏

`6=p Z`(1). On the other hand, clearly M(x)] restrictsto a constant sheaf of monoids on (Uq)τ . In view of (7.3.52), we deduce that (7.4.44) is anequivalence; lastly, we have seen that the induced morphism C ′(ξq) → X(ξq) is etale, i.e. is atrivial covering, therefore the same holds for the morphism C ′(ξ) → X(ξ), and consequentlythe support of ξ lies in E(h) (claim 7.1.9). Since ξ is arbitrary, the assertion follows. ♦

Claim 7.4.45. There exists a non-empty open subset UY ⊂ Y such that X(x)×Y UY ⊂ E(h).

Proof of the claim. Since X(x) is a noetherian scheme, E(h) is a constructible open subset,hence Z := X(x)\E(h) is a constructible closed subset of X(x). The subset fx(Z) is pro-constructible ([30, Ch.IV, Prop.1.9.5(vii)]) and does not contain η (by claim 7.4.43), henceneither does its topological closure W ([30, Ch.IV, Th.1.10.1]). It is easily seen that UY :=Y \W will do. ♦


Claim 7.4.46. Let UY be as in claim 7.4.45. We have :(i) There exists an irreducible closed subset Z of Y of dimension one, such that Z ∩

(Y,N)tr ∩ UY 6= ∅.(ii) For any Z as in (i), the induced functor Cov(E(h)) → Cov(Z ×Y E(h)) is fully

faithful.

Proof of the claim. (i): More generally, letA be any local noetherian domain of Krull dimensiond ≥ 1, and W ⊂ SpecA a proper closed subset. Then we show that there exists an irreducibleclosed subset Z ⊂ SpecA of dimension one, not contained in W . We may assume that d > 1and W 6= ∅, and arguing by induction on d, we are reduced to showing that there exists anirreducible closed subset Z ′ ⊂ SpecA with dimZ ′ < d, which is not contained in W . This isan easy exercise that we leave to the reader.

(ii): It suffices to check that conditions (i)–(iii) of proposition 7.1.36 hold for Z and theopen subset E(h). However, condition (i) is immediate, since the generic point ηZ of Z liesin UY . Likewise, condition (ii) holds trivially for the fibre over the point ηZ , so it suffices toconsider the fibre over the closed point y of Z, in which case the assertion is just lemma 7.4.29.Lastly, condition (iii) follows directly from theorem 6.7.8(iii.b) and [33, Ch.IV, Prop.18.8.10,18.8.12(i)]. ♦

Let Z be as in claim 7.4.46(i), and endow Z with its reduced subscheme structure. Let alsoZ ′ be the normalization of Z; then both Z and Z ′ are strictly local, and the morphism Z ′ → Zis radicial and surjective, hence the induced functor

Cov(Z ×Y E(h))→ Cov(Z ′ ×Y E(h))

is an equivalence (lemma 7.1.7(i)). Taking into account claim 7.4.46, we are thus reduced toshowing that the morphism

(Z ′ ×Y E(h))×X(x) C′ → Z ′ ×Y E(h)

is a trivial etale covering. However, let ηZ′ be the generic point of Z ′, and set

(Z ′, N ′) := Z ′ ×Y (Y,N) (X ′,M ′) := Z ′ ×Y (X,M).

The open subset (Z ′, N ′)tr is dense in Z ′, by virtue of claim 7.4.46(i), so the induced mor-phism f ′ : (X ′,M ′) → (Z ′, N ′) is still of the type considered in (7.4), the geometric pointx lifts uniquely (up to isomorphism) to a geometric point x′ of X ′, and h ×Y Z ′ is an ob-ject of Tame(f ′, x′, κ(ηZ′)) (lemma 7.3.18(i)). Hence, we may replace from start (X,M) by(X ′,M ′), (Y,N) by (Z ′, N ′), h by h ×Y Z ′, after which, we may assume that Y is noetherianand of dimension one. Moreover, taking into account claim 7.4.42, we may assume that theinduced covering C|ηq → |ηq| is trivial, for every maximal point ηq of f−1

x (y), and it remains toshow that h is trivial under these assumptions.

To this aim, we look at the corresponding commutative diagram of groups, provided bylemma 7.4.33(i) : with the notation of loc.cit., we see that in the current situation, β is anisomorphism as well, by claim 7.4.40, therefore lemma 7.4.33(ii) says that the group homomor-phism π1(|ηq|et, ηq) → π1(U tr/Yet, ξ) is surjective, for any maximal point ηq of f−1

x (y). Fromthis, we deduce that h is a trivial covering, and therefore there exists an etale covering CY → |η|with an isomorphism C

∼→ CY ×|η|Utr ([42, Exp.IX, Th.6.1]). Denote by CνY the normalization

of Y in CY . Also, set E ′ := E(h) ⊂ Str(fx). In light of theorem 6.7.8(iii.a) and lemma 7.4.29,it is easily seen that the restriction E ′ → Y of fx is surjective; the latter is also a smooth mor-phism of schemes (corollary 6.3.27(i)). It follows that Cν

Y ×Y E ′ is the normalization of E ′ inC ([33, Ch.IV, Prop.17.5.7]), especially, it is an etale covering of E ′. We then deduce that Cν

Y

is already a (trivial) etale covering of Y ([33, Ch.IV, Prop.17.7.1(ii)]), and then clearly h mustbe a trivial covering as well.


Remark 7.4.47. Theorem 7.4.39 is the local acyclicity result that gives the name to this section.However, the title is admittedly not self-explanatory, and its full justification would require theintroduction of a more advanced theory of the log-etale site, that lies beyond the bounds of thistreatise. In rough terms, we can try to describe the situation as follows. In lieu of the standardstrict henselization, one should consider a suitable notion of strict log henselization for pointsof the log-etale topoi associated to log schemes. Then, for f : X → Y as in (7.4) with saturatedlog structures on both X and Y , and log-etale points x of X with image y in Y , one shouldlook, not at our fx, but rather at the induced morphism fx of strict log henselizations (of X atx and of Y at y). The (suitably defined) log geometric fibres of fx will be the log Milnor fibresof f at the log-etale point x, and one can state for such fibres an acyclicity result : namely,the prime-to-p quotients of their (again, suitably defined) log fundamental groups vanish. Theproof proceeds by reduction to our theorem 7.4.39, which, with hindsight, is seen to supply theessential geometric information encoded in the more sophisticated log-etale language.

8. THE ALMOST PURITY TOOLBOX

The sections of this rather eterogeneous chapter are each devoted to a different subject, andare linked to each other only very loosely, if they are at all. They have been lumped heretogether, because they each contribute a distinct self-contained little theory, that will find appli-cation in chapter 9, in one step or other of the proof of the almost purity theorem. The exceptionis section 8.7 : it studies a class of rings more general than the measurable algebras introducedin section 8.3; the results of section 8.7 will not be used elsewhere in this treatise, but they maybe interesting for other purposes.

Section 8.2 develops the yoga of almost pure pairs (see definition 8.2.25(i)); the relevanceto the almost purity theorem is clear, since the latter establishes the almost purity of certainpairs (X,Z) consisting of a scheme X and a closed subscheme Z ⊂ X . This section providesthe means to perform various kinds of reductions in the proof of the almost purity theorem,allowing to replace the given pair (X,Z) by more tractable ones.

Section 8.3 introduces measurable (and more generally, ind-measurable)K+-algebras, whereK+ is a fixed valuation ring of rank one : see definitions 8.3.3(ii) and 8.3.51. For modulesover a measurable algebra, one can define a well-behaved real-valued normalized length. Thislength function is non-negative, and additive for short exact sequences of modules. Moreover,the length of an almost zero module vanishes (for the standard almost structure associated toK+). Conversely, under some suitable assumptions, a module of normalized length zero will bealmost zero. In the proof of the almost purity theorem we shall encounter certain cohomologymodules whose normalized length vanishes, and the results of section 8.3 will enable us to provethat these modules are almost zero.

Section 8.4 discusses a category of topologically locally ringed spaces that contains the for-mal schemes of [26]. We call ω-admissible our generalized formal schemes, and they areobtained by gluing formal spectra of suitable ω-admissible topological rings (see definition8.4.18). It turns out that the so-called Fontaine rings reviewed in section 4.6 carry a natu-ral ω-admissible linear topology; hence the theory of section 8.4 attaches to such rings an ω-admissible formal scheme, endowed with a natural Frobenius automorphism. This construction– detailed in section 8.5 – lies at the heart of Faltings’ proof of the almost purity theorem.

Lastly, section 8.6 studies some questions concerning the formation of quotients of affinealmost schemes under a finite group action. These results will be used in the proof of theorem9.4.34, the last conspicuous stepping stone on the path to almost purity.

8.1. Non-flat almost structures. This section contains some material that complements thegeneralities of [36, §2.4, §2.5] : indeed, whereas many of the preliminaries in loc.cit. makeno assumptions on the basic setup (V,m) that underlies the whole discussion, for the more


advanced results it is usually required that m := m ⊗V m is a flat V -module. We shall showhow, with more work, one can remove this condition (or at least, weaken it significantly) andstill recover most of the useful almost ring theory of [36]. This extension shall be applied insection 9.6, in order to state and prove the most general case of almost purity.

For future reference, we recall the following :

Lemma 8.1.1. Let C be any Grothendieck abelian category, X•, Y • ∈ Ob(D(C )) any twocomplexes.

(i) If X• ∈ Ob(D≤b(C )), the natural map

HomD≤b(C )(X•, τ≤bY

•)→ HomD(C )(X•, Y •)

is an isomorphism.(ii) If Y • ∈ Ob(D≥a(C )), the natural map

HomD(C )(X•, Y •)→ HomD≥a(C )(τ≥aX

•, Y •)

is an isomorphism.(iii) If X• ∈ Ob(D≤b(C )) and Y • ∈ Ob(D≥a(C )), then the complex RHom•C (X•, Y •) lies

in D≥a−b(C ).

Proof. (Here τ≥a denotes the usual truncation functor D(R-Mod) → D≥a(R-Mod), and re-call that a Grothendieck abelian category has enough injectives, and admits a derived categoryD(C ).) We show only assertion (i) : mutatis mutandi, the same argument applies to prove (ii).In light of the distinguished triangle

τ≤bY• → Y • → τ>bY

• → (τ≤bY•)[1] = τ<b(Y

•[1])

we reduce to checking that

HomD(C )(X•, τ>bY

•) = 0 = HomD(C )(X•, τ>b+1(Y •[−1])).

We show the first vanishing : the same argument applies also to the second one. We pick aninjective resolution τ>bY •

∼→ J• such that Jq = 0 for every q ≤ b, and consider the spectralsequence

Epq1 := HomC (HpX•, Jq)⇒ HomD(C )(X

•, (τ>bY•)[q − p]).

Under the current assumptions we have Ep,−p1 = 0 for every p ∈ Z, whence the contention.

(iii): In view of the natural isomorphism

RiHomC (X•, Y •)∼→ HomD(C )(X

•, Y •[i]) for every i ∈ Zthe assertion follows from both (i) and (ii).

8.1.2. Let (V,m) be any basic setup (as defined in [36, §2.1.1]), and R any V -algebra. Forevery interval I ⊂ N, we have a localization functor

(8.1.3) CI(R-Mod)→ CI(Ra-Mod) K• 7→ K•a

from complexes of R-modules, to complexes of Ra-modules, which is obviously exact, henceit induces a derived localization functor :

(8.1.4) DI(R-Mod)→ DI(Ra-Mod).

The functor (8.1.3) admits a left (resp. right) adjoint

(8.1.5) CI(Ra-Mod)→ CI(R-Mod) K• 7→ K•! ( resp. K• 7→ K•∗ )

defined by applying termwise to K• the functor M 7→ M! (resp. M 7→ M∗) for Ra-modulesgiven by [36, §2.2.10, §2.2.21]. However, if m is not flat, the functor M 7→M! is obviously notexact, and the localization functor R-Mod → Ra-Mod does not send injectives to injectives(cp. [36, Cor.2.2.24]). This makes it trickier to deal with constructions in the derived category;


for instance, if m is flat, we get a left adjoint to (8.1.4), simply by deriving trivially the exactfunctor M 7→ M!. This fails in the general case, but we shall see later that a suitable derivedversion of the construction of M! is still available.

8.1.6. Let I ⊂ N be any interval. As a first step, denote by ΣI the multiplicative set of mor-phisms ϕ in DI(R-Mod) such that ϕa is an isomorphism in DI(Ra-Mod); as already pointedout in [36, §2.4.9], ΣI is locally small, hence the localized category Σ−1

I DI(R-Mod) exists,and obviously the derived localization functor factors through a natural functor

(8.1.7) Σ−1I DI(R-Mod)→ DI(Ra-Mod).

Lemma 8.1.8. For every interval I , the functor (8.1.7) is an equivalence.

Proof. A proof is sketched in [36, §2.4.9], in case m is flat, but in fact this assumption is super-fluous. Indeed, since the unit of adjunction M →Ma

! is an isomorphism ([36, Prop.2.2.23(ii)]),it is clear that the functor K• 7→ K•! of (8.1.5) descends to a well-defined functor

(8.1.9) DI(Ra-Mod)→ Σ−1I DI(R-Mod)

such that the composition (8.1.7) (8.1.9) is naturally isomorphic to the identity automorphismof DI(Ra-Mod). A simple inspection shows that, likewise, (8.1.9) (8.1.7) is naturally isomor-phic to the identity of Σ−1

I DI(R-Mod), whence the contention.

8.1.10. We sketch a few generalities on derived tensor products, since we shall use these func-tors to construct useful objects in various derived categories. Recall first that the usual tensorproduct −⊗R − on R-modules, descends to a bifunctor −⊗Ra − ([36, §2.2.6, §2.2.12]), and ifM is a flatR-module, thenMa is a flatRa-module ([36, Lemma 2.4.7]). It follows that the cate-gory Ra-Mod has enough flat objects, so every bounded above complex of Ra-modules admitsa bounded above flat resolution. Now, given bounded above complexes K•, L• of Ra-modules,set

K•L⊗Ra L• := (K•!

L⊗R L•! )a

which is a well defined object of D(Ra-Mod).• We claim that this rule yields a well defined functor

−L⊗Ra − : D−(Ra-Mod)× D−(Ra-Mod)→ D−(Ra-Mod).

Indeed, suppose ϕ• : K•1 → K•2 is a morphism in C(Ra-Mod), inducing an isomorphism inD(Ra-Mod), and set C• := Cone (ϕ•! ); clearly, C•a = 0 in D(Ra-Mod). Now, pick any flat

resolution P • → L•! , so that K•i! ⊗R P • computes K•i!L⊗R L•! , for i = 1, 2. We get natural

isomorphism :

Cone(ϕ•L⊗Ra L•)

∼→ (C• ⊗R P •)a∼→ C•a ⊗Ra P •a = 0.

so the derived tensor product depends only on the image of K• in D−(Ra-Mod); likewise forthe argument L•, whence the contention.• Next, suppose that P • → K• is a bounded above resolution, with P • a complex of flat

Ra-modules; we claim that there is a natural isomorphism in D(Ra-Mod)

K•L⊗Ra L•

∼→ P • ⊗Ra L•.

Indeed, pick any bounded above flat resolution Q• → L•! ; we have natural isomorphisms

K•L⊗Ra L•

∼→ (K•! ⊗R Q•)a∼→ K• ⊗Ra Q•a

∼→ P • ⊗Ra Q•a


in D(Ra-Mod), where the last holds, since Q•a is a complex of flat Ra-modules. Finally, theinduced map

P • ⊗Ra Q•a → P • ⊗Ra L•

is also an isomorphism in D(Ra-Mod), since P • is a complex of flat Ra-modules, so the claimfollows.• Notice that, for anyK• ∈ Ob(D−(R-Mod)) and any flat resolution P • → K•, the induced

morphism P •a → K•a is a flat resolution; it follows that, for any L• ∈ Ob(D(R-Mod)) we geta natural isomorphism

(8.1.11) (K•L⊗R L•)a

∼→ K•aL⊗Ra L•a in D(Ra-Mod).

Remark 8.1.12. Clearly, for the derived tensor products of Ra-modules, one has the samecommutativity and associativity isomorphisms as the ones detailed in remark 4.1.13(i) for usualmodules, as well as the vanishing properties of remark 4.1.13(ii).

We are now ready to return to the question of the existence of adjoints to derived localization.The key point is the following :

Lemma 8.1.13. In the situation of (8.1.2), let K• be any complex of R-modules, i ∈ Z anyinteger and suppose that :

(a) K• ∈ Ob(D≤i(R-Mod))(b) K•a ∈ Ob(D≤i−1(Ra-Mod)).

Then mL⊗V K• ∈ Ob(D≤i−1(R-Mod)).

Proof. We apply the standard spectral sequence

E2pq := TorVp (m, HqK•)⇒ Hq−p(m

L⊗V K•).

Indeed, (a) says that HqK• = 0 for every q > i, and (b) says that (H iK•)a = 0, and thereforemV ⊗V H iK• = 0 ([36, Rem.2.1.4(i)]). In either case, we conclude that E2

pq = 0 wheneverq − p ≥ i, whence the lemma.

8.1.14. Now, let us define inductively :

M•0 := V [0] and M•

i+1 := mL⊗V M•

i for every i ∈ N.A simple induction shows that M•

i ∈ D≤0(V -Mod) for every i ∈ N, so all these derivedtensor product are well defined in D≤0(V -Mod). Moreover, from the short exact sequence ofV -modules

Σ : 0→ m→ V → V/m→ 0

we obtain a distiguished triangle

M•i

L⊗V Σ : M•

i+1 →M•i →M•

i

L⊗V (V/m)→M•

i+1[1] for every i ∈ N.Especially, we get an inverse system of morphisms in D≤0(V -Mod) :

· · · →M•i+1

π•i−−→M•i

π•i−1−−−→M•i−1 → · · · →M•

0 := V [0].

Also, from (8.1.11) we deduce natural isomorphisms in D≤0(V a-Mod) :

(8.1.15) M•ai∼→ V a[0] for every i ∈ N

and under these identifications, the morphism π•ai corresponds to the identity automorphism ofV a[0] (details left to the reader). Furthermore, we deduce the following derived version of [36,Rem.2.1.4(i)] :

Proposition 8.1.16. Let a, b ∈ Z be any integers with a ≤ b. We have :


(i) For every K• ∈ Ob(D[a,b](R-Mod)), the following conditions are equivalent :(a) K•a ' 0 in D[a,b](Ra-Mod).

(b) τ≥a(M•b−a+1

L⊗V K•) ' 0 in D[a,b](R-Mod).

(ii) For every morphism ϕ• : K• → L• in D≤b(R-Mod), the following conditions areequivalent :(a) ϕ•a is an isomorphism in D[a,b](Ra-Mod).

(b) τ≥a(M•b−a+2

L⊗V ϕ•) is an isomorphism in D[a,b](R-Mod).

Proof. (i): From (8.1.15), it is easily seen that (b)⇒(a). The other direction follows straightfor-wardly from lemma 8.1.13, via an easy descending induction on b.

(ii): Again, the direction (b)⇒(a) is immediate from (8.1.15). For the other direction, denoteby C• the cone of ϕ•; then C• ∈ Ob(D[a−1,b](R-Mod)), and C•a ' 0 in D[a−1,b](Ra-Mod).

From (i) we deduce that τ≥a−1(M•b−a+2

L⊗V C•) ' 0 in D[a−1,b](R-Mod). Since the derived

tensor product is a triangulated functor, the assertion follows easily : details left to the reader.

Corollary 8.1.17. With the notation of (8.1.14), the morphism

τ≥2−iπ•i : τ≥2−iM

•i+1 → τ≥2−iM

•i

is an isomorphism in D[2−i,0](V -Mod) for every integer i ≥ 2.

Proof. By construction, we have a natural isomorphism :

Cone(τ≥2−iπ•i )∼→ τ≥1−i(M

•i

L⊗V (V/m)) in D[1−i,0](V -Mod)

in light of which, the assertion is an immediate consequence of proposition 8.1.16(i).

Proposition 8.1.18. In the situation of (8.1.2), we have :

(i) The functor

(8.1.19) D+(Ra-Mod)→ D+(R-Mod) : K• 7→ K•[∗] := RHom•Ra(Ra[0], K•)

is right adjoint to the localization functor.(ii) Let a, b, i ∈ Z be any three integers with b ≥ a and i ≥ b− a+ 2. Then

(a) The functor

D[a,b](Ra-Mod)→ D[a,b](R-Mod) : K• 7→ K•[!] := τ≥a(M•i

L⊗V K•[∗])

is left adjoint to the localization functor.(b) The functor

D[a,b](Ra-Mod)→ D[a,b](R-Mod) : K• 7→ τ≤bK•[∗]

is right adjoint to the localization functor.(iii) For every K• ∈ Ob(D+(Ra-Mod)) (resp. L• ∈ Ob(D[a,b](Ra-Mod))) the counit of

adjunction is an isomorphism

(K•[∗])a ∼→ K• ( resp. (τ≤bL

•[∗])

a ∼→ L• ).

(iv) For every L• ∈ Ob(D[a,b](Ra-Mod)), the unit of adjunction is an isomorphism

L• → (L•[!])a.


Proof. (i): This is analogous to lemma 5.1.25(iii). Recall the construction : we know that thecategoryRa-Mod admits enough injectives ([36, 2.2.18]), hence (8.1.19) can be represented byI•∗ , where K• ∼→ I• is any injective resolution of K•, and I•∗ is obtained by applying term-wiseto I• the functor M 7→M∗ of [36, §2.2.10]. Indeed, taking into account [36, Cor.2.2.19] we getnatural isomorphisms :

HomD+(R-Mod)(L•, I•∗ )

∼→H0Hom•R(L•, I•∗ )∼→H0Hom•Ra(L

•a, I•)∼→ HomD(Ra-Mod)(L

•a, I•)∼→ HomD(Ra-Mod)(L

•a, K•)

for every bounded below complex L• of R-modules.(ii.b): Let K• ∈ D[a,b](Ra-Mod); we may find an injective resolution K• ∼→ I• such that

Ij = 0 for every j < a, in which case I•∗ ∈ D≥a(R-Mod), and τ≤bI•∗ represents τ≤bK•[∗] inD[a,b](R-Mod). Then, in view of (i), the assertion is reduced to lemma 8.1.1(i).

(iii) follows by direct inspection of the definitions, taking into account that, for every Ra-module M , the counit of adjunction (M∗)

a → M is an isomorphism ([36, Prop.2.2.14(iii)]) :details left to the reader.

(ii.a): To ease notation, set

ωL• := τ≥a(M•i

L⊗V L•) for every L• ∈ Ob(D[a,b](R-Mod)).

Then ωL• is naturally an object of D[a,b](R-Mod), as explained in remark 4.1.13(iii), and like-wise for K•[!], if K• is any object of D[a,b](Ra-Mod). We begin with the following :

Claim 8.1.20. Let K•, L• ∈ Ob(D[a,b](R-Mod)) be any two objects. Then the natural map

HomD[a,b](R-Mod)(ωK•, L•)→ HomD[a,b](Ra-Mod)(K

•a, L•a)

is an isomorphism.

Proof of the claim. Arguing as in [36, §2.2.2], and taking into account lemma 8.1.8, we reduceto showing that, for any K• ∈ D[a,b](R-Mod), the natural morphism

ϕK• : ωK• → K•

is initial in the full subcategory of D[a,b](R-Mod)/K• whose objects are the morphisms ψ :L• → K• that lie in Σ[a,b]. However, for any such ψ, we have a commutative diagram inD[a,b](R-Mod) :

ωL•ϕL• //

L•

ψ

ωK•ϕK• // K•

whose left vertical arrow is an isomorphism, by proposition 8.1.16(ii). There follows a mor-phism ϕK• → ψ, and we have to check that this is the unique morphism from ϕK• to ψ.However, say that α, β : ϕK• → ψ are two such morphisms; then their difference is a mor-phism γ := α − β : ωK• → L• such that ψ γ = 0, so γ factors through a morphismγ : ωK• → Coneψ[−1]. Set C• := τ≤bConeψ[−1]; then C• ∈ D[a,b](R-Mod), and accordingto lemma 8.1.1(i), γ lifts uniquely to a morphism ωK• → C• that we denote again γ. We


deduce a commutative diagram

ω ωK• //

ϕωK•

ωC•

ϕC•

ωK•

γ // C•.

Now, by construction C•a = 0, therefore ωC• = 0 (proposition 8.1.16(i)); on the other hand,ϕωK• is an isomorphism, by proposition 8.1.16(ii). We conclude that γ = 0, whence α = β, assought. ♦

Assertion (ii.a) is an immediate consequence of (iii) and claim 8.1.20; from this, also (iv) isimmediate : details left to the reader.

Remark 8.1.21. Let a, b, i ∈ Z be any three integers such that a ≤ b and i ≥ b− a + 2. Frompropositions 8.1.18(ii.a,iii) and 8.1.16(ii) we deduce a natural isomorphism

τ≥a(M•i

L⊗V K•)

∼→ (K•a)[!] for every K• ∈ Ob(D[a,b](R-Mod))

(details left to the reader), which allows to compute (K•a)[!] purely in terms of K• and opera-tions within D(R-Mod). It turns out that an analogous isomorphism is available also for K•[∗] :this is contained in the following

Lemma 8.1.22. Let a, b, i ∈ N be any integers such that a ≤ b and i ≥ b − a + 2. For everyK• ∈ Ob(D[a,b](R-Mod)), we have a natural isomorphism :

τ≤bRHom•V (M•i , K

•).∼→ τ≤bK

•a[∗] .

Proof. Theorem 5.1.27(ii) yields a natural isomorphism

RHom•V (M•i , K

•)∼→ RHom•R(M•

i

L⊗V R[0], K•).

To compute the right-hand side, we may fix an injective resolution K• ∼→ I•; the complex I•a

is not necessarily injective, but we can find an injective resolution ϕ : I•a∼→ J• (in the category

of bounded below complexes of Ra-modules). In view of (8.1.11) and (8.1.15), the morphismϕ induces a natural transformation

ψ : RHom•R(M•i

L⊗V R[0], K•)→ RHom•Ra(R

a[0], K•a) = (K•a)[∗]

and it suffices to show that, for every j ≤ b, the map

Hjψ : HomD(R-Mod)(M•i

L⊗V R[0], K•[j])→ Hom•D(Ra-Mod)(R

a[0], K•a[j])

is an isomorphism. However, by lemma 8.1.1(i), the latter is the same as a map

(8.1.23) HomD(R-Mod)(M•i

L⊗V R[0], τ≤0K

•[j])→ Hom•D(Ra-Mod)(Ra[0], τ≤0K

•a[j])

and a direct inspection shows that (8.1.23) agrees with the map arising in claim 8.1.20, for everyj ≤ b. Especially, for every such j, the map (8.1.23) is an isomorphism, as sought.

Proposition 8.1.24. Let a, b, c ∈ Z be any three integers such that a ≤ b, and K•, L• any twoobjects of D[a,b](R-Mod)). Suppose that

(a) HomD(R-Mod)(K•, X[−j]) = 0 for all j ∈ [c, b] and all R-modules X with Xa = 0.

(b) HomD(R-Mod)(Y [−j], L•) = 0 for all j ∈ [a, c] and all R-modules Y with Y a = 0.Then the natural map

(8.1.25) HomD(R-Mod)(K•, L•)→ HomD(Ra-Mod)(K

•a, L•a)

is an isomorphism.


Proof. We start out with the following observation :

Claim 8.1.26. Consider the following conditions :(a’) HomD(R-Mod)(K

•, X•) = 0 for every X• ∈ Ob(D[c,b](R-Mod)) such that X•a = 0.(b’) HomD(R-Mod)(Y

•, L•) = 0 for every Y • ∈ Ob(D[a,c](R-Mod)) such that Y •a = 0.Then (a)⇔(a’) and (b)⇔(b’).

Proof of the claim. Obviously (a’)⇒(a). For the converse, one argues by decreasing inductionon c ≤ b. Indeed, the case c = b is immediate. Then, suppose that the sought equivalence hasalready been established for some d ≤ b; if X• ∈ D[d−1,b] and X•a = 0, and if we know that (a)holds with c := d− 1, we set H• := HcX•[−c], and consider the distinguished triangle

H• → X• → τ≥dX• → H•[1].

By inductive assumption, we have HomD(R-Mod)(K•, τ≥dX

•) = 0, and (a) says that

HomD(R-Mod)(K•, H•) = 0.

It then follows that HomD(R-Mod)(K•, X•) = 0, which shows that the equivalence holds for c.

The proof of the equivalence (b)⇔(b’) is wholly analogous. ♦Fix an integer i ≥ b− a+ 2, and set

C• := Cone(π•0 · · · π•i : M•i → V [0])

(notation of (8.1.14)); notice that C• ∈ D≤1(R-Mod), and C•a = 0. By virtue of condition (b),claim 8.1.26 and lemma 8.1.1(ii), it follows that :

RjHom•R(C•, L•) = HomD(R-Mod)(τ≥a(C•[−j]), L•) = 0 for every j < c

In other words, D• := RHom•R(C•, L•) ∈ D≥c(R-Mod), and clearly D•a = 0; also notice theinduced distinguished triangle

Σ : D• → L• → RHom•R(M•i , L

•)→ D•[−1].

Now, condition (a), claim 8.1.26 and lemma 8.1.1(i) imply that

HomD(R-Mod)(K•, D•[j]) = HomD(R-Mod)(K

•, τ≤bD•[j]) = 0 for every j ≤ 0

whence, by considering the distinguished triangle RHom•R(K•,Σ), natural isomorphisms

HomD(R-Mod)(K•, L•)

∼→ HomD(R-Mod)(K•, RHom•R(M•

i , L))

∼→ HomD(R-Mod)(M•i

L⊗V K•, L•) (by [75, Th.10.8.7])

∼→ HomD(R-Mod)(τ≥aM•i

L⊗V K•, L•) (by lemma 8.1.1(ii))

∼→ HomD(Ra-Mod)(K•a, L•a) (by claim 8.1.20)

whose composition, after a simple inspection, is seen to agree with the map (8.1.25).

Remark 8.1.27. (i) For every interval I ⊂ N, denote by

ΦI : DI(R/mR-Mod)→ DI(R-Mod)

the forgetful functor. It follows easily from lemmata 5.1.25(iii) and 8.1.1(i), that, for everyinterval [a, b], the functor Φ[a,b] admits the right adjoint

D[a,b](R-Mod)→ D[a,b](R/mR-Mod) K• 7→ Ψr[a,b]K

• := τ≤bRHom•R(R/mR[0], K•).

(ii) Likewise, theorem 5.1.27 and lemma 8.1.1(ii) imply that Φ[a,b] admits the left adjoint

D[a,b](R-Mod)→ D[a,b](R/mR-Mod) K• 7→ Ψl[a,b]K

• := τ≥a(K• ⊗R R/mR[0]).

(iii) Moreover, arguing as in the proof of claim 8.1.26 it is easily seen that condition (a) ofproposition 8.1.24 is equivalent to


(a”) HomD(R-Mod)(K•,Φ[c,b]X

•) = 0 for every X• ∈ Ob(D[c,b](R/mR-Mod)).(iv) Likewise, condition (b) is equivalent to(b”) HomD(R-Mod)(Φ[a,c]Y

•, L•) = 0 for every Y • ∈ Ob(D[a,c](R/mR-Mod)).

Proposition 8.1.28. Let a, b ∈ Z be any two integers such that a ≤ b. For every object K• ofD[a,b](R-Mod), the following conditions are equivalent :

(a) K• lies in the essential image of the left adjoint functor X• 7→ X•[!].

(b) K•L⊗R R/mR[0] ∈ Ob(D<a−1(R/mR-Mod)).

Proof. We start out with the following :

Claim 8.1.29. We may assume that V = R.

Proof of the claim. Clearly condition (b) does not depend on the underlying ring V . It sufficesthen to remark that condition (a) depends only on the basic setup (R,mR) (as opposed to theoriginal basic setup (V,m)). Indeed, notice that there is a natural equivalence

Ω : Ra-Mod∼→ (R,mR)a-Mod

(where Ra denotes, as in the foregoing, the image of R in the category of (V,m)-algebras), andthe induced equivalence of the respective derived categories fits into an essentially commutativediagram

D(R-Mod)

wwnnnnnnnnnnnn

))RRRRRRRRRRRRRR

D(Ra-Mod)D(Ω)

// D((R,mR)a-Mod)

whose downward arrows are the forgetful functors. Especially, the left (resp. right) adjoints ofthese two forgetful functors share the same essential images. ♦

Henceforth, we assume that V = R (and therefore, m = mR). Fix an integer i ≥ b− a + 2,

set L• := τ≥a−1(M•i

L⊗V K•), and notice first that, taking into account proposition 8.1.18(iv)

and remark 8.1.21, condition (a) is equivalent to :(c) The morphism π•0 · · · π•i−1 : Mi → V [0] induces an isomorphism τ≥aL

• → K•.(c)⇒(b): Indeed, set H := Ha−1L•; if (c) holds, we have a distinguished triangle

H[a− 1]→ L• → K• → H[a]

whence a distinguished triangle in D(V/m-Mod)

(8.1.30) H[a− 1]L⊗V V/m[0]→ L•

L⊗V V/m[0]→ K•

L⊗V V/m[0]→ H[a]

L⊗V V/m[0].

However, we have natural isomorphisms

τ≥a−1(L•L⊗V V/m[0])

∼→ τ≥a−1((M•i

L⊗V K•)

L⊗V V/m[0]) (by remark 4.1.13(ii))

∼→ τ≥a−1(M•i

L⊗V (K•

L⊗V V/m[0])) (by remark 4.1.13(i))

∼→ τ≥a−1(M•i

L⊗V τ≥a−1(K•

L⊗V V/m[0])) (by remark 4.1.13(ii))

∼→ 0 (by proposition 8.1.16(i))

whence (b), after considering the distinguished triangle τ≥a−1(8.1.30).(b)⇒(a): We remark

Claim 8.1.31. Condition (b) is equivalent to condition (a”) of remark 8.1.27(iii), with c := a−1.


Proof of the claim. Indeed, (b) holds if and only if Ψl[a−1,b]K

• = 0 in D[a−1,b](R/mR-Mod)(notation of remark 8.1.27(ii)). If the latter condition holds, then clearly (a”) holds with c :=a− 1. Conversely, if (a”) holds for this value of c, then HomD(R/mR-Mod)(Ψl

[a−1,b]K•, X•) = 0

for every X• ∈ D[a−1,b](R/mR-Mod); especially, the identity automorphism of Ψl[a−1,b]K

•

factors through 0, so Ψl[a−1,b]K

• vanishes. ♦

From claim 8.1.31 and remark 8.1.27(iii) we deduce that, if (b) holds, condition (a) of propo-sition 8.1.24 holds for c := a − 1, and condition (b) of the same proposition holds trivially forthis value of c, for every L• ∈ D[a;b](R-Mod). We conclude that the natural map

HomD(R-Mod)(K•, L•)→ HomD(Ra-Mod)(K

•a, L•a)∼→ HomD(R-Mod)(K

•a[!] , L

•)

is an isomorphism, for every L• ∈ D[a;b](R-Mod), whence (a).

Proposition 8.1.32. Let a, b ∈ Z be any two integers such that a ≤ b. For every L• ∈Ob(D[a,b](R-Mod)), the following conditions are equivalent :

(a) L• lies in the essential image of the right adjoint functor X• 7→ τ≤bX•[∗].

(b) RHom•R(R/mR[0], L•) ∈ Ob(D>b+1(R/mR-Mod)).

Proof. By the same argument as in the proof of claim 8.1.29, we reduce to the case whereV = R. Next, fix i ∈ N such that i ≥ b− a+ 2, define

K• := RHom•V (M•i , L

•) Pi := Mi

L⊗V V/m[0]

and notice that

(8.1.33) τ≥a−b−1Pi = 0 in D[a−b−1,0](V/m-Mod)

due to proposition 8.1.16(i). Morever, in view of proposition 8.1.18(iii) and lemma 8.1.22,condition (a) is equivalent to :

(c) The morphism π•0 · · · π•i−1 : Mi → V [0] induces an isomorphism L•∼→ τ≤bK

•.

(c)⇒(b): We argue as in the proof of proposition 8.1.28; namely, set H := Hb+1K•; if (c)holds, we obtain a distinguished triangle

(8.1.34) H[−b− 2]→ L• → τ≤b+1K• → H[−b− 1]

and by considering the induced distinguished triangle τ≤b+1RHom•V (V/m[0], (8.1.34)), we re-duce to observing that

τ≤b+1RHom•V (V/m[0], τ≤b+1K•)∼→ τ≤b+1RHom•V (V/m[0], K•) (by lemma 8.1.1(iii))∼→ τ≤b+1RHom•V (P•i , L

•) (by [75, Th.10.8.7])∼→ τ≤b+1RHom•V (τ≥a−b−1P

•i , L

•) (by lemma 8.1.1(iii))∼→ 0 (by (8.1.33)).

(b)⇒(a): Again, we proceed as in the proof of the corresponding assertion in proposition8.1.28; namely, arguing as in the proof of claim 8.1.31, we see that condition (b) is equivalentto condition (b”) of remark 8.1.27(iv), with c := b + 1. Hence, if (b) holds, condition (b) ofproposition 8.1.24 holds for c := b + 1, and notice that condition (a) of loc.cit. holds triviallyfor this value of c, for every K• ∈ D[a;b](R-Mod). We conclude that the natural map

HomD(R-Mod)(K•, L•)→ HomD(Ra-Mod)(K

•a, L•a)∼→ HomD(R-Mod)(K

•, τ≤bL•a[∗])

is an isomorphism, for every K• ∈ D[a;b](R-Mod), whence (a).


8.1.35. Let A be any V a-algebra; recall that the localization functor V -Alg→ V a-Alg admitsa left adjoint R 7→ R!!, whose restriction to the subcategory of A!!-algebras yields a left adjointfor the localization functor A!!-Alg→ A-Alg ([36, Prop.2.2.29]). In [36], we have studied thedeformation theory of A-algebras by means of this left adjoint, under the assumption that m isV -flat; here we wish to show that the same can be repeated in the current setting, if one makesappeal instead to the results of the foregoing paragraphs. To begin with, we remark :

Proposition 8.1.36. LetA→ B be any morphism of V a-algebras, N anyB!!-module. We have:(i) If the unit of adjunction N → Na

∗ is injective, the natural map

(8.1.37) ExalA!!(B!!, N)→ ExalA(B,Na)

is a bijection (notation of [36, §2.5.7]).(ii) If Na = 0, then ExalA!!

(B!!, N) = 0.

Proof. (i): LetΣ : 0→ Na → E

ϕ−→ B → 0

be any square-zero extension of A-algebras; there follows a square-zero extension

Σ!! : 0→ Na! /Kerϕ!! → E!! → B!! → 0

ofA!!-algebras. Under the stated assumption, the counit of adjunctionNa! → N factors uniquely

through a B-linear map gϕ : Na! /Kerϕ!! → N ; then gϕ ∗ Σ!! (defined as in [36, §2.5.5]) yields

an element of ExalA!!(B!!, N) whose image under (8.1.37) equals the class of Σ. Conversely, if

Ω : 0→ N → Fψ−→ B!! → 0

is a square-zero extension of A!!-algebras, then by adjunction we get a natural map

Ωa!! → Ω

which in turns, by simple inspection, induces an isomorphism gψa ∗ Ωa!!∼→ Ω in the category of

square-zero A!!-algebra extensions of B!! (details left to the reader). The assertion follows.(ii): Suppose Na = 0, and let Ω be as in the foregoing; it follows that ψa : F a → B is

an isomorphism of A-algebras. By adjunction, the morphism (ψa)−1 corresponds to a map ofA!!-algebras ϕ : B!! → E, and it is easily seen that ψ ϕ is the identity automorphism of B!!,whence the assertion.

Definition 8.1.38. Let f : A→ B be a morphism of V a-algebras. We set

LaB/A := (LB!!/A!!)a

which is a simplicial complex of B-modules that we call the almost cotangent complex of f .

Remark 8.1.39. (i) In case m is a flat V -module, we have introduced in [36, Def.2.5.20] a sim-plicial B!!-module LB/A; now, notice that the notation of loc.cit. agrees with the current one:indeed, [36, Prop.8.1.7(ii)] shows that complex (LB/A)a obtained by applying the derived local-ization functor to LB/A is naturally isomorphic (in D(s.B-Mod)) to the complex of definition8.1.38.

(ii) Depending on the context, we will want to regard LB/A either as a simplicial object,or as a cochain complex, via the Dold-Kan isomorphism ([75, Th.8.4.1]). The resulting slightnotational ambiguity should not be a source of confusion.

Theorem 8.1.40. In the situation of definition 8.1.38, let N be any B-module. Then there arenatural isomorphisms

DerA(B,N)∼→ Ext0

B(LB/A, N)

ExalA(B,N)∼→ Ext1

B(LB/A, N).

(Notation of [36, Def.2.5.22(i)]; so, here we view LB/A as an object of D≤0(B-Mod).)


Proof. The first isomorphism follows easily from [56, II.1.2.4.2] and the natural isomorphism

(8.1.41) ΩB!!/A!!

∼→ (ΩB/A)!

proved in [36, Lemma 2.5.29] (the proof in loc.cit. does not use the assumption that m is V -flat).Clearly we have

(8.1.42) HomD(B!!-Mod)(Y [0], N∗[0]) = 0 for every B!!-module Y such that Y a = 0.

On the other hand, we have :

Claim 8.1.43. HomD(B!!-Mod)(LB!!/A!!, X•) = 0 for every X• ∈ Ob(D[0,1](B!!-Mod)) such that

X•a = 0.

Proof of the claim. For every X• ∈ Ob(D[0,1](B!!-Mod)) we have a distinguished triangle

H0X•[0]→ X• → H1X•[−1]→ (H0X•)[1]

which reduces to considering the cases where X• = M [j] for some almost zero B!!-module M ,and j = 0,−1. The case where j = −1 follows from [56, III.1.2.3] and proposition 8.1.36(ii).The case where j = 0 follows easily from [56, II.1.2.4.2] and (8.1.41) : details left to thereader. ♦

Now, (8.1.42) says that L• := N∗[0] fulfills condition (b) of proposition 8.1.24, and claim8.1.43 says that K• := LB!!/A!!

fulfills condition (a), so the natural map

Ext1B!!

(LB!!/A!!, N∗)→ Ext1

B(LaB/A, N)

is an isomorphism. Taking into account proposition 8.1.36(i) and [56, III.1.2.3], the theoremfollows.

8.1.44. For the further study the almost cotangent complex, we shall need some preliminariesconcerning the derived functors of certain non-additive functors. This material generalizes theresults of [36, §8.1], that were obtained under the assumption that m is V -flat.

Lemma 8.1.45. Let (V,m) be any basic setup, R a simplicial V -algebra, n ∈ N an integer, Mand N two R-modules such that HiM = HiN = 0 for every i ≥ n. The following holds :

(i) If Ma = 0 in D(Ra-Mod), then a · 1M = 0 in D(R-Mod), for every a ∈ m.(ii) If ϕ : M → N is a morphism of R-modules such that ϕa in an isomorphism in

D(Ra-Mod), then for every a ∈ m we may find a morphism ψ : N → M inD(R-Mod), such that ψ ϕ = a · 1M and ϕ ψ = a · 1N in D(R-Mod).

Proof. (i): For every R-module X , set

τ≤−1X := σ ωX(notation of remark 4.5.21(iii)). According to [56, I.3.2.1.9(ii)], there exists a natural sequenceof morphisms

(8.1.46) τ≤−1X → X → s.H0(X)→ σ(τ≤−1X) in D(R-Mod)

whose induced sequence of normalized complexes is a distinguished triangle in D(V -Mod)(i.e. a distinguished triangle of D(R-Mod), in the terminology of [56, I.3.2.2.4], and in viewof [56, I.3.2.2.5]). Let now n and M be as in the lemma; we argue by induction on n. The casewhere n = 0 is trivial, so suppose that n > 0, and that the assertion has already been provenfor all almost zero R-modules N such that HiN = 0 for every i ≥ n − 1. Especially, forN := τ≤1X and P := s.H0(M) we have a · 1ωN = 0 and a · 1P = 0 in D(R-Mod), for everya ∈ m. Since the adjunction σ ωN → N is an isomorphism in D(R-Mod) ([56, I.3.2.1.10]),we deduce that

HomD(R-Mod)(M,N)a = 0 = HomD(R-Mod)(M,P )a


whence EndD(R-Mod)(M)a = 0, by virtue of (8.1.46) (with X := M ) and [56, I.3.2.2.10]. Theassertion follows.

(ii): Set C := Coneϕ; according to [56, I.3.2.2], we have a distinguished triangle

(8.1.47) Mϕ−→ N → C → σM in D(R-Mod)

whence – by [56, I.3.2.2.10] – an exact sequence of V -modules

HomD(R-Mod)(N,M)α−→ EndD(R-Mod)(N)

β−→ HomD(R-Mod)(N,C).

Now, let us write a =∑n

i=1 aibi for some a1, b1, . . . , an, bn ∈ m; the assumption on ϕ impliesthat Ca = 0 in D(Ra-Mod), therefore (i) yields β(ai · 1N) = ai · β(1N) = 0, so there existsa morphism ψi : N → M in D(R-Mod) such that α(ψi) = ai · 1N , i.e. ϕ ψi = ai · 1Nfor every i = 1, . . . , n. Likewise, by considering the long exact sequence Ext•R((8.1.47),M)provided by [56, I.3.2.2.10] we find, for every i = 1, . . . , n, a morphism ψ′i : N →M such thatψ′i ϕ = bi · 1M . Thus,

ai · ψ′i = ψ′i ϕ ψi = bi · ψi for every i = 1, . . . , n

and a simple computation shows that ψ :=∑n

i=1 bi · ψi will do.

Remark 8.1.48. Before considering non-additive functors, let us see how to define derivedtensor products in D(Ra-Mod), for any simplicial V -algebra R. We proceed as in (8.1.10) :for given Ra-modules M,N , set

M`⊗Ra N := (M!

`⊗R N!)

a.

(i) We claim that this rule yields a well defined functor

−`⊗Ra − : D(Ra-Mod)× D(Ra-Mod)→ D(Ra-Mod).

Indeed, say that ϕ : M → M ′ is a quasi-isomorphism of Ra-modules, and set C := Cone(ϕ!).

We need to check that (ϕ!

`⊗R N)a is a quasi-isomorphism, for any R-module N ; in light of

remark 4.5.21(iv), it then suffices to show that (C`⊗R N)a = 0; but the latter Ra-module may

be computed as(C⊗R ⊥R• N)a = Ca ⊗Ra (⊥R• N)a

whence the claim, since Ca = 0 in Ra-Mod.(ii) Next, suppose that M is a flat Ra-module; then we claim that the natural morphism of

Ra-modules

M`⊗Ra N →M ⊗Ra N

is a quasi-isomorphism, for every Ra-module N . Indeed, by Eilenberg-Zilber’s theorem 4.2.48,the assertion comes down to checking that the augmented simplicial Ra-module

(M!⊗R ⊥R• N!)a →M ⊗Ra N

is aspherical. But for every k ∈ N, the k-th column of the latter is isomorphic to the augmentedRa-module

M [k]⊗Ra[k] (⊥R[k]• N![k])a →M [k]⊗Ra[k] N [k]

which is aspherical, since M [k] is a flat Ra[k]-module, whence the assertion.(iii) Just as in (8.1.10), the foregoing immediately implies that we have a natural isomor-

phism

(M`⊗R N)a

∼→Ma`⊗Ra Na in D(Ra-Mod)

for any two R-modules M and N (details left to the reader).


(iv) Furthermore, we get suspension and loop functors σ and ω for Ra-modules, by the rule :

σM := (σM!)a and ωM := (ωM∗)

a for every Ra-module M

from which it follows that σ is left adjoint to ω. Then, it is clear that the assertions of remark4.5.21(iii) hold as well for these functors.

(v) Likewise, we define the cone of a morphism ϕ : M → N of Ra-modules, by the rule

Coneϕ := (Coneϕ!)a

and then the assertion of remark 4.5.21(iv) holds also for morphisms of Ra-modules.(vi) In view of (iv) and (v), it is then easy to check that also lemma 4.5.22 holds verbatim for

A := V , and any two Ra-modules X , Y .

Remark 8.1.49. (i) In the same vein, we may define derived tensor products of Ra-algebras,for any simplicial V -algebra R. Namely, if S and S ′′ are any two Ra-algebras, we set

S`⊗Ra S ′ := (S!!

`⊗R S ′!!)a

(see (8.1.35)), where`⊗R denotes the derived tensor product for R-algebras, defined in example

4.5.15. In view of remarks 4.5.21(ii) and 8.1.48(i), it is easily seen that this rule defines a functor

−`⊗Ra − : D(Ra-Alg)× D(Ra-Alg)→ D(Ra-Alg)

and moreover, the formation of these tensor products commutes with the forgetful functorD(Ra-Alg)→ D(Ra-Mod).

(ii) Moreover, they are computed by arbitrary flat resloutions : if S (or S ′) is a flat Ra-algebras, then the natural morphism

S`⊗Ra S ′ → S ⊗Ra S ′

is an isomorphism in D(Ra-Alg).(iii) Furthermore, if Ra → S is a given morphism of simplicial V a-algebras, we obtain a

well defined functor

D(Ra-Alg)→ D(S-Alg) S ′ 7→ S`⊗Ra S ′.

Namely, given an Ra-algebra S ′, we pick a resolution P → S ′ with P a flat Ra-algebra, andendow S ⊗Ra P with its natural S-algebra structure, which is independent, up to natural iso-morphism, of the choice of P . All the verifications are exercises for the reader.

8.1.50. Resume the situation of (4.5.25), let R be any simplicial V -algebra, d ∈ N any integer,and suppose additionally that :

• T is homogeneous of degree d, i.e. we have T (a · 1M) = ad · 1TM , for every object(A,M) of V -Alg.Mod, and every a ∈ V .• The ideal m ·H0(R) of H0(R) satisfies condition (B) of [36, §2.1.6].

Remark 8.1.51. (i) Let T and T ′ be two functors as in (8.1.50), with T (resp. T ′) homogeneousof degree d (resp. d′), and consider the functor

T ⊗ T ′ : V -Alg.Mod→ V -Alg.Mod (A,M) 7→ T (A,M)⊗A T ′(A,M).

Then it easily seen that T ⊗ T ′ fulfills the conditions of (8.1.50), and it is homogeneous ofdegree d+ d′.

(ii) Likewise, suppose that f : T → T ′ is a natural transformation of functors fulfilling theconditions of (8.1.50), for the same degree d; then the same holds for the functors Ker f andCoker f .


(iii) In the situation of (8.1.50), let f : H0(R) → B be any morphism of V -algebras, andϕ : M → N any B-linear map such that ϕa is an isomorphism; arguing as in the proof oflemma 8.1.45(ii), it is easily seen that, for every a ∈ m there exists a B-linear map ψ : N →Msuch that ϕ ψ = a · 1N and ψ ϕ = a · 1M . Since TB(M) and TB(M ′) are two B-modulesand (B) holds for m ·H0(R) (hence also for mB), the homogeneity property of T implies thatTB(ϕ)a is an isomorphism as well (details left to the reader). Especially, the above holds withB := R[p], for any p ∈ N, and f the natural map given by the degeneracies of the simplicialV -algebra R. It is then clear that TR induces a well defined functor

TRa : Ra-Mod→ Ra-Mod.

The following result shows that, in this situation, the construction of left derived functors de-scends likewise to Ra-modules.

Theorem 8.1.52. In the situation of (8.1.50), the following holds :(i) Let ϕ : M → N be a morphism of R-modules, and n ∈ N an integer such that

Hi(ϕ)a : (HiM)a → (HiN)a is an isomorphism of V a-modules, for every i ≤ n. Then

Hi(LTϕ)a : Hi(LTM)a → Hi(LTN)a

is an isomorphism of V a-modules, for every i ≤ n.(ii) Especially, the functor TR induces a well defined left derived functor

LTRa : D(Ra-Mod)→ D(Ra-Mod).

Proof. Clearly (ii) follows from (i).(i): In light of corollary 4.5.24(i), we may replace M and N by respectively cosknM and

cosknN , after which, we may assume that HiM = HiN = 0 for every i > n, so ϕa is anisomorphism in D(Ra-Mod). Next, by proposition 4.5.18, we may replace M and N by theirrespective standard free resolutions, in which case we are reduced to checking that the inducedmap (TRϕ)a is an isomorphism in D(Ra-Mod). Since T is homogeneous and (B) holds form ·H0(R), the latter assertion follows straightforwardly from lemma 8.1.45(ii).

8.1.53. Next, we wish to decide how much of the foregoing theory can be salvaged, when wedrop condition (B). We will concentrate on the functors that are relevant to the later study ofthe cotangent complex. Thus, henceforth, for every integer d ∈ N, we shall denote by T d oneof the three standard functors

Symd,Λd,Γd : V -Alg.Mod→ V -Alg.Mod

(namely, the symmetric and antisymmetric d-th power functors, and the d-th divided powerfunctor). Notice that the functor T d fulfills the conditions of (8.1.50), for every d ∈ N. More-over, in all three cases we have natural identifications :

(8.1.54) T 1 ∼→ 1V -Alg.Mod.

In general, we can no longer expect that T d descends to almost modules; indeed, consider thefollowing :

Example 8.1.55. Let (V,m) be as in (8.1.53), and p ∈ N any prime integer. The Frobeniusmap ΦR : R/pR → R/pR for V -algebras R, can be seen as a homogeneous polynomial lawof degree p on the V -module V/pV . Denote by m(p) ⊂ V the ideal generated by (xp | x ∈ m).Clearly Φ descends to a polynomial law

Φ : V/(pV + m)→ Wp := V/(pV + m(p))

which is still homogeneous of degree p, so it factors through a unique V -linear map

ΓpA(V/(pV + m))→ Wp.


The latter is surjective, since its image contains the class of the unit element of V . Now, theproof of [36, Prop.2.1.7(ii)] shows that – if (B) does not hold for m – there exists some prime psuch that pV + m(p) does not contain m, therefore W a

p 6= 0. This shows that the functor

V -Mod→ V a-Mod M 7→ (ΓpAM)a

does not factor through V a-Mod.

However, we will see that example 8.1.55 is, in a sense, the worst that can happen.

Lemma 8.1.56. In the situation of (8.1.53), we have :(i) There are natural transformations

T i+j → T i ⊗ T j → T i+j for every i, j ∈ Nwhose composition equals

(i+ji

)· 1T i+j . (Notation of remark 8.1.51(i).)

(ii) For every simplicial V -algebra R, the maps of (i) induce natural transformations

LT i+jR → LT iR`⊗R LT jR → LT i+jR for every i, j ∈ N

whose composition equals(i+ji

)· 1LT i+jR

.

Proof. (i): For T = Λ, the sought morphism Λi+j → Λi ⊗ Λj is the one denoted ∆i,j in [36,§4.3.20], and the morphism Λi ⊗ Λj → Λi+j is given by the rule : x ⊗ y 7→ x ∧ y, for every(A,M) ∈ Ob(V -Alg.Mod), every x ∈ Λi

AM and every y ∈ ΛjAM . The sought identity

follows easily from the explicit formula given in [36, (4.3.21)] : details left to the reader.For T = Sym, the natural transformation Symi+j → Symi ⊗ Symj is given by a sim-

ilar formula; namely, for every object (A,M) of V -Alg.Mod, every sequence of elementsx1, . . . , xi+j ∈ M , and every subset I ⊂ 1, . . . , i + j, set xI :=

∏k∈I xk ∈ Symk

AM (wherethe multiplication is formed in the graded ring Sym•AM ); one checks easily that the rule

x1 · · ·xi+j 7→∑I,J

xI ⊗ xJ

defines a well defined A-linear map on Symi+jA M (where the sum ranges over all the parti-

tions (I, J) of 1, . . . , i + j such that the cardinality of I equals i). Then one defines a mapSymi

AM ⊗A SymjAM → Symi+j

A M by the rule : u ⊗ v 7→ u · v for every u ∈ SymiAM and

v ∈ SymjAM . Again, the sought identity is verified by direct computation.

If T = Γ, then for every object (A,M) of V -Alg.Mod we define a homogeneous polyno-mial law M ΓiAM ⊗A ΓjAM of degree i+ j, by the rule : x 7→ x[i]⊗ x[j] for every A-algebraB, and every x ∈ B ⊗AM . This law yields a transformation Γi+j → Γi ⊗ Γj as sought. Next,the multiplication of the graded ring functor Γ• gives a transformation Γi ⊗ Γj → Γi+j . Thecomposition of these two transformations is characterized as the unique transformation ψ ofΓi+j such that ψ(A,M)(x

[i+j]) = x[i] · x[j] for every object (A,M) of V -Alg.Mod, and everyx ∈M . Then, by corollary 4.6.81 we must have ψ(A,M) =

(i+ji

)1M , as required.

(ii): Recall that the three functors Λ, Γ and Sym transform free A-modules into free A-modules, for every V -algebra A. Therefore, the sought natural transformation is none else thanthe map

T i+jR (⊥R• M)→ T iR(⊥R• M)⊗R T jR(⊥R• M)→ T i+jR (⊥R• M)

given by (i), for any R-module M .

Proposition 8.1.57. In the situation of (8.1.53), let p ∈ N be a prime integer, R a simplicialV ⊗Z Z(p)-algebra, M an R-module such that Ma = 0 in D(Ra-Mod). Then

(LT dRM)a = 0 in D(Ra-Mod), for every d ∈ N such that (p, d) = 1.


Proof. In light of corollary 4.5.24(i), we may replace M by coskiM , in which case lemma8.1.45(i) implies that a · 1M = 0 for every a ∈ m. Now, we apply lemma 8.1.56(ii) withi = 1 and j = d − 1 (notice that j ≥ 0, since d > 0); in view of (8.1.54), there result naturaltransformations

LT dRM →M ⊗R LT d−1R M → LT dRM

whose composition is d · 1LT dRM . Since the image of d is invertible in R, it follows easily thata · 1LT dRM = 0 for every a ∈ m, whence the assertion.

Theorem 8.1.58. Let d, n, p ∈ N be any three integers, with d > 0 and p a prime. Let also Rbe a simplicial V ⊗Z Z(p)-algebra, M an R-module, and suppose that

(a) HiM = 0 for every i < n.(b) Ma = 0 in D(Ra-Mod).

Then we have :(i) Hi(LΓdRM)a = 0 for every i < n.

(ii) Hi(LΛdRM)a = 0 for every i < n+ p− 1.

(iii) Hi(LSymdRM)a = 0 for every i < n+ 2(p− 1).

Proof. (i) is just a special case of corollary 4.5.24(ii), and for d < p, assertions (ii) and (iii)follow from the more general proposition 8.1.57, hence we may assume that d ≥ p. For everyj = 0, . . . , d, set

F j := Γj ⊗ Λd−j Gj := Λj ⊗ Symd−j.

By remark 8.1.51(i), these functors are homogeneous of degree d, and fulfill the conditions of(8.1.50). Moreover, since Γ, Λ and Sym transform free modules into free modules, we havenatural isomorphisms of functors :

(8.1.59) LF jR

∼→ LΓjR`⊗R LΛd−j

R LGjR

∼→ LΛjR

`⊗R LSymd−j

R for every j = 0, . . . , d

(details left to the reader).(ii): According to [56, I.4.3.1.7], there is a natural complex of functors :

(8.1.60) 0→ F dR

∂d−1−−−→ F d−1R → · · · → F 1

R∂0−→ F 0

R → 0

which is exact on flatR-modules. SetZp := Ker ∂p−1; due to remark 8.1.51(ii) we may considerthe derived functor LZp, and from corollary 4.5.24(ii) and assumption (a), we get

(8.1.61) Hi(LZpM) = 0 for every i < n.

The evaluation of (8.1.60) on (⊥R• M)∆ gives an exact sequence of R-modules :

0→ LZpM → LF p−1R M → · · · → LF 1

RM → LF 0RM → 0.

Notice as well, that

(8.1.62) (LF jRM)a = 0 in D(Ra-Mod) for j = 1, . . . , p− 1

in view of (8.1.59), remark 8.1.48(iii), and proposition 8.1.57. The assertion now follows from(8.1.61) and claim 4.5.39.

(iii): According to [56, I.4.3.1.7], there is a natural complex of functors :

Σ : 0→ GdR

∂d−1−−−→ Gd−1R → · · · → G1

R∂0−→ G0

R → 0.

Set Z ′p := Ker ∂p−1; due to remark 8.1.51(ii) we may consider the derived functor LZ ′p, andsince Σ is exact on flat R-modules, we obtain two exact sequences of R-modules :

Σ′ : 0→ LGdRM → LGd−1

R M → · · · → LGpRM → LZ ′pM → 0

Σ′′ : 0→ LZ ′pM → LGp−1R M → · · · → LG1

RM → LG0RM → 0.


On the one hand, in light of (ii), remark 8.1.48(iii,vi) and (8.1.59), we see that

coskn+p−1(LGjRM)a = 0 in D(Ra-Mod) for every j = 1, . . . , d.

By applying claim 4.5.39 to the exact sequence coskn+p−1Σ′a, we deduce thatHi(LZ′pM)a =

0 for every i < n+ p− 1. On the other hand, (8.1.59) and proposition 8.1.57 imply that

(LGjRM)a = 0 in D(Ra-Mod) for j = 1, . . . , p− 1

so the assertion follows, after applying claim 4.5.39 to the exact sequence Σ′′.

8.1.63. Let now ϕ : R → S be any morphism of simplicial V a-algebras; pick any resolutionρ : P → S with P a flat R-algebra, and consider the composition

(8.1.64) P ⊗R Sρ⊗RS−−−−→ S ⊗R S

µS−−→ S

where µS is the multiplication law of S. Then (8.1.64) represents a morphism

∆(ϕ) : S`⊗R S → S in D(R-Alg)

which is independent (up to unique isomorphism) of the choice of P . Notice that, if ϕ is an

isomorphism in D(s.V a-Alg), then the same holds for ϕ`⊗R ϕ : R→ S

`⊗R S; also, we have

ϕ = ∆(ϕ) (ϕ`⊗R ϕ) in D(s.V a-Alg).

More generally, set C := Coneϕ, and endow S`⊗R S with the S-module structure deduced

from its left tensor factor; then, by considering the section S`⊗R ϕ of ∆(ϕ), we get a natural

decomposition

S`⊗R S

∼→ S ⊕ (S`⊗R C) in D(S-Mod)

which identifies ∆(ϕ) with the natural projection, whence a natural isomorphism

(8.1.65) Cone ∆(ϕ)∼→ σS

`⊗R C.

Thus, say that C = σkC ′ in D(R-Mod), for some R-module C ′ with H0C′ 6= 0; there follows

a distinguished triangle in D(R-Mod)

σkC ′ → σkS`⊗R C ′ → σ2kC ′

`⊗R C ′ → σk+1C ′.

Especially, if k ≥ 2, then HkC = Hk(S`⊗R C), and we see that, in this case, Cone ∆(ϕ) =

σk+1C ′′ in D(S-Mod), for some C ′′ such that H0C′′ 6= 0. However, it is clear from (8.1.65)

that Cone ∆2(ϕ) = σ2C ′′ for some object C ′′ of D(S-Mod). We conclude that, if ∆n(ϕ) is anisomorphism in D(S-Mod) for some n ≥ 2, then the same holds already for ∆2(ϕ).

Definition 8.1.66. In the situation of (8.1.63), we say that ϕ is a weakly etale morphism, if∆2(ϕ) is an isomorphism in D(S-Alg).

8.1.67. Let ϕ : R′ → R be any morphism of simplicial V a-algebras, and S, T two R-algebras.Since the standard resolution FR(S) := FR!!

(S!!)a of example 4.5.15 is clearly functorial in both

R and S, we have a natural map

FR′(S)∆ → FR(S)∆

of simplicial R′-algebras, whence a natural morphism

(8.1.68) S`⊗R′ T → FR(S)∆ ⊗R′ T → S

`⊗R T.

Lemma 8.1.69. In the situation of (8.1.67), suppose that ϕ is a quasi-isomorphism. Then thesame holds for (8.1.68).


Proof. Let ψ : FR′(S)∆ ⊗R′ R → FR(S)∆ be the natural morphism; by construction, (8.1.68)factors as the composition of ψ ⊗R 1T and the natural isomorphism

FR′(S)∆ ⊗R′ T∼→ (FR′(S)∆ ⊗R′ R)⊗R T.

Since both FR′(S)∆ ⊗R′ R and FR(S)∆ are flat R-algebras, remark 8.1.49(ii) then reducesto checking that ψ is a quasi-isomorphism. Since the natural maps β : FR(S)∆ → S andα : FR′(S)∆ → FR′(∆)⊗R′ R are quasi-isomorphisms, it suffices to show that the compositionβ ψ α is a quasi-isomorphism. But the latter is none else than the standard resolutionFR′(S)→ S, whence the lemma.

Proposition 8.1.70. Let ϕ : R → S and ϕ′ : R′ → S ′ be two morphisms of simplicial V -algebras, and suppose we have a commutative diagram in D(R-Alg)

(8.1.71)R

ϕ //

ψ

S

ψ′

R′

ϕ′ // S ′

where ψ and ψ′ are isomorphisms (in D(R-Alg)). Then ϕa is weakly etale if and only if thesame holds for ϕ′a.

Proof. Denote by Hot(V -Alg) the homotopy category of simplicial V -algebras, and recall thatthe multiplicative system of quasi-isomorphisms in Hot(V -Alg) admits a right calculus of frac-tions ([56, I.3.1.8(ii)]). We begin with the following special case :

Claim 8.1.72. Suppose that ψ and ψ′ are also morphisms of simplicial V -algebras, and that(8.1.71) commutes in the category Hot(V -Alg). Then the proposition holds.

Proof of the claim. Indeed, in this situation it follows easily from lemma 8.1.69 that ϕa (resp.ϕ′a) is weakly etale if and only if the same holds for ψ′a ϕa (resp. for ϕ′a ψa), so we arereduced to the case where R = R′, S = S ′, and ϕ, ϕ′ : R → S are two homotopic morphismsof simplicial V -algebras. In this case, there exist morphisms of simplicial V -algebras ϕ′′ : R→S ′′ and d0, d1 : S ′′ → S, such that d0 and d1 are quasi-isomorphisms, and ϕ = d0 ϕ′′, ϕ′ =d1 ϕ′′ (see [56, I.2.3.2]). Then the claim follows by applying repeatedly lemma 8.1.69. ♦

Next, there exists a simplicial V -algebra S ′′ and morphisms β : S ′′ → S and γ : S ′′ → S ′

of simplicial V -algebras, such that both β and γ are quasi-isomorphisms, and ψ′ = γ β−1 inD(s.V -Alg). Moreover, there exists morphisms ϕ′′ : R′′ → S ′′ and ψ′′ : R′′ → R such that ψ′′

is a quasi-isomorphism, and the resulting diagram

R′′ϕ′′ //

ψ′′

S ′′

β

R

ϕ // S

commutes in Hot(V -Alg). By claim 8.1.72, we may then replace ϕ by ϕ′′ and ψ′ by β, afterwhich we may assume that ψ′ is a morphism of simplicial V -algebras.

Likewise, we may find morphisms β′ : R′′ → R and γ′ : R′′ → R′ of simplicial V -algebrasthat are quasi-isomorphisms, and such that ψ = γ′ β′−1 in D(s.V -Alg); again by lemma8.1.69, we see that ϕa is weakly etale if and only if the same holds for ϕa βa, so we mayreplace ϕ by ϕ β, ψ by ψ γ, and assume from start that also ψ is a map of simplicialV -algebras.

In this situation, by applying once again lemma 8.1.69 we see that ϕa (resp. ϕ′a) is weaklyetale if and only if the same holds for ψ′a ϕa (resp. for ϕ′a ψa). Therefore, we may assumefrom start that R = R′, S = S ′, and ϕ, ϕ′ : R → S are two maps of simplicial V -algebras that


represent the same morphism in D(s.V -Alg). In this case, we may find a morphism ϑ : R′′′ →R of simplicial V -algebras, such that ϑ is a quasi-isomorphism, and ϕ ϑ, ϕ′ ϑ : R′′′ → S arehomotopic maps; then the assertion follows from claim 8.1.72 (and again, from lemma 8.1.69:details left to the reader).

Corollary 8.1.73. Let ϕ : R → S and ψ : S → T be any two morphisms of simplicial V a-algebras. We have :

(i) If ϕ and ψ are weakly etale, the same holds for ψ ϕ.(ii) If ϕ and ψ ϕ are weakly etale, the same holds for ψ.

(iii) If ϕ is weakly etale, and R → R′ is any morphism of simplicial V a-algebras, then

ϕ′ := R′`⊗R ϕ is weakly etale.

(iv) ϕ is weakly etale if and only if the same holds for ∆(ϕ).

Proof. (iv) is immediate from the discussion of (8.1.63).

(iii): Endow S ′ := R′`⊗R S with the R′-algebra structure deduced from its left tensor factor;

then the assertion is an immediate consequence of the following more general

Claim 8.1.74. There exists a commutative diagram in D(R′-Alg)

S ′`⊗R′ S ′

//

∆(ϕ′)##GGGGGGGGG

R′`⊗R (S

`⊗R S)

R′`⊗R∆(ϕ)xxrrrrrrrrrrr

S ′

whose horizontal arrow is an isomorphism.

Proof of the claim. Pick any resolution ρ : P → S with P a flat R-algebra; then R′ ⊗R Prepresents R′

`⊗R S, and we have a natural isomorphism of R′-algebras

(R′ ⊗R P )⊗R′ (R′ ⊗R P )∼→ R′ ⊗R (P ⊗R P ).

In view of remark 4.5.21(i), the source of this map represents S ′`⊗R′S ′, and the target represents

R′`⊗R (S

`⊗R S). Under these identifications, the morphism R′

`⊗R ∆(ϕ) becomes the map

1R′⊗R (µS (ρ⊗Rρ)) (notation of (8.1.63)), whereas ∆(ϕ′) is represented by the multiplicationmap µR′⊗RP = 1R′ ⊗R µP of R′ ⊗R P . The claim follows straightforwardly. ♦

(i): Pick a resolution P → S!! with P a flatR!!-algebra, and a resolutionQ→ T!! withQ a flatP -algebra; in light of proposition 8.1.70 it suffices to show that the resulting morphismR→ Qa

is weakly etale, so we may replace S by P a and T by Qa, and assume from start that both ϕand ψ are flat morphisms. Due to (iv), it then suffices to check that ∆(ψ ϕ) : T ⊗R T → T isweakly etale; however, the latter can be factored as the composition

(8.1.75) T ⊗R T∼→ T ⊗S (S ⊗R S)⊗S T

T⊗S∆(ϕ)⊗ST−−−−−−−−−→ T ⊗S T∆(ψ)−−−−→ T

and by (iv) the maps ∆(ψ) and ∆(ϕ) are weakly etale, so the same holds for T ⊗S ∆(ϕ)⊗S T ,in view of (iii). We may therefore replace ψ by ∆(ψ) and ϕ by T ⊗S ∆(ϕ)⊗S T , after which,we may assume that ∆(ψ) is an isomorphism in D(s.A-Alg). In this case, the factorization(8.1.75) makes it clear that ∆(ψ ϕ) is weakly etale, as stated.

(ii): Set S ′ := S`⊗R S and T ′ := S

`⊗R T ; we may factor ψ as the composition

SS`⊗R(ψϕ)−−−−−−−→ T ′

T ′`⊗S′∆(ϕ)−−−−−−−→ T

so the assertion follows from (i), (iii) and (iv).


Theorem 8.1.76. Let ϕ : R → S be a morphism of simplicial V -algebras, c, p ∈ N twointegers, with p a prime, and suppose that ϕa is weakly etale. We have :

(i) If the ideal m ·H0(S) of H0(S) satisfies condition (B) of [36, §2.1.6], then LaS/R = 0

in D(Sa-Mod).(ii) If S is a Z(p)-algebra and HiLS/R = 0 for every i < c, then HiLaS/R = 0 for every

i < c+ 2p− 1.


Claim 8.1.77. In order to prove the theorem, we may assume that both ∆(ϕa) and H0(ϕ) areisomorphisms.

Proof of the claim. To ease notation, set S ′ := S`⊗RS, and consider the sequence of morphisms

(8.1.78) S1S

`⊗Rϕ−−−−−→ S ′

µB−−→ S

where µS is the composition of the multiplication map S ⊗R S → S and the natural mapS ′ → S ⊗R S; the transitivity triangle ([56, III.2.1.2]) relative to (8.1.78) yields a naturalisomorphism

LS′/S∼→ σS ⊗S′ LS′/S

∼→ σS ⊗S′ (S ′ ⊗S LS/R) in D(S-Mod)

where the last isomorphism follows from the base change theorem of [56, III.2.2.1]. Thus,LS′/S ' σLS/R in D(S-Mod), so assertion (i) (resp. (ii)) holds for ϕ, provided it holds for∆(ϕ), and then the claim follows from corollary 8.1.73(iv). ♦

Henceforth, we assume that both ∆(ϕa) and H0(ϕ) are isomorphisms. Let P → S be aresolution, with P an R-algebra, such that P [n] is a free R[n]-algebra for every n ∈ N; setP ′ := S ⊗R P , and recall that there are natural isomorphisms

LS/R∼→ S ⊗P Ω1

P/R∼→ S ⊗P ′ Ω1

P ′/S

where Ω1P/R denotes the flat simplicial P -module such that Ω1

P/R[n] := ΩP [n]/R[n] (and likewisefor Ω1

P ′/S), with faces and degeneracies deduced from those of P and R (resp. those of P ′ andS), in the obvious way. In other words, set

J := Ker(µS : P ′ → S)

(where µS is as in the proof of claim 8.1.77); then LS/R ' J/J2 in D(S-Mod), and notice that

(8.1.79) Ja = 0 in D(Sa-Mod)

since ∆(ϕa) is an isomorphism.

Claim 8.1.80. J is a quasi-regular ideal, and H0J = 0.

Proof of the claim. Since H0(ϕ) is an isomorphism, it is clear that H0J = 0. Next, for everyn ∈ N, the S[n]-algebra P ′[n] is free, hence we reduce to showing the following. Let B be anyring, C := B[Xi | i ∈ I] any free B-algebra (for any set I), and f : C → B any morphism ofB-algebras; then Ker f is a quasi-regular ideal of C. However, for every i ∈ I , set bi := f(Xi),and let g : C

∼→ C be the isomorphism of B-algebras such that g(Xi) = Xi − ai for everyi ∈ I; clearly Ker f is a quasi-regular ideal if and only if the same holds for g−1Ker f . But thelatter is the ideal (Xi | i ∈ I), so the assertion follows from remark 4.5.33(ii). ♦

(i): We need to show that Hn(J/J2)a = 0 for every n ∈ N, and we shall argue by inductionon n. The assertion for n = 0 is clear from (8.1.79), in view of the exact sequence

HnJ → Hn(J/J2)→ Hn−1J2 for every n ∈ N.


Hence, suppose that n > 0, and the assertion is already known for every degree < n. The sameexact sequence reduces to checking that Hn−1(J2)a = 0. Now, from claim 8.1.80 and theorem4.5.37, we know that Hn−1J

n = 0. Therefore, by an easy induction, we are further reducedto showing that Hn−1(J i/J i+1)a = 0 for every i = 2, . . . , n − 1. However, on the one hand,proposition 4.5.35(ii.a) says that the natural map

LSymiS(J/J2)→ J i/J i+1

is an isomorphism in D(S-Mod), for every i ∈ N; on the other hand, the inductive assumptionand theorem 8.1.52(i) imply thatHj(LSymiJ/J2)a = 0 for every j ≤ n, whence the contention.

(ii): We show, by induction on n, that Hn(J/J2)a = 0 for every n < c+ 2p− 1. Arguing asin the foregoing, we may assume that c+ 2p− 1 > n > 0, and the sought vanishing is alreadyknown in degrees < n, in which case we reduce to checking that Hn−1(LSymi

SJ/J2)a = 0 for

every i = 2, . . . , n − 1. Set M := coskn(s.truncnJ/J2); then HjM = 0 for every j < c, and

the inductive assumption implies that Ma = 0 in D(Sa-Mod), so Hn−1(LSymiSM)a = 0, by

theorem 8.1.58(iii). Lastly, by corollary 4.5.24(i), we see that

Hn−1(LSymiSM) = Hn−1(LSymi

SJ/J2) for every i ∈ N


Remark 8.1.81. Let A → B be any morphism of simplicial rings such that the multiplicationmap of B is an isomorphism µB : B⊗AB

∼→ B. Pick a resolution ϕ : P → B such that P [n] isa freeA[n]-algebra for every n ∈ N, and set P ′ := B⊗AP , J := Ker (µB(B⊗Aϕ) : P ′ → B).The assumption on B implies that H0J = 0. On the other hand, arguing as in the proof of claim8.1.80 we see that J is a quasi-regular ideal. Taking into account proposition 4.5.35(ii) thespectral sequence associated to the J-adic filtration of P ′ can be written as

E2pq := Hp+q(LSymq

B(LB/A))⇒ TorAp+q(B ⊗A B).

Though the filtration is not finite, one can show that this spectral sequence is convergent. Wewill not use this result.

Corollary 8.1.82. Let ϕ : A → B be a morphism of V a-algebras, p ∈ N a prime integer, andsuppose that

(a) B is a V a ⊗Z Z(p)-algebra(b) the induced morphism s.ϕ : s.A→ s.B of simplicial V a-algebras is weakly etale.

Then HiLaB/A = 0 for every i ≤ 2p.

Proof. Letting c := 0 in theorem 8.1.76(ii), we see that HiLaB/A = 0 for i = 0, 1. But in viewof claim 8.1.43, the latter implies that HiLB!!/A!!

= 0 for i = 0, 1. So, actually the morphism ϕfulfills the conditions of theorem 8.1.76(ii), with c = 2, whence the assertion.

8.1.83. We conclude this section with a few words on the cohomology of sheaves of almostmodules. Namely, let (V,m) again be an arbitrary basic setup, which we view as a basic setuprelative to the one-point topos pt, in the sense of [36, §3.3]. Let (X,OX) be a ringed topos,mX ⊂ OX an ideal, such that (OX ,mX) is a basic setup relative to the topos X , and supposealso that we are given a morphism of ringed topoi :

π : (X,OX)→ (pt, V ) such that (π∗m) · OX ⊂ mX .

We deduce a natural morphism :

Rπ∗ : D+((OX ,mX)a-Mod)→ D+((V,m)a-Mod)

which can be constructed as usual, by taking injective resolutions. In case m is a flat V -module,this is the same as setting

Rπ∗(K•) := (Rπ∗K

•! )a.

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