arXiv:math/0002064v1 [math.AG] 9 Feb 2000 · We deﬁne a uniform structure on the set I of ideals of V as follows. For every ﬁnitely generated ideal m0 ⊂ mthe subset of I ×

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ALMOST RING THEORY

OFER GABBER AND LORENZO RAMERO

third release

CONTENTS

1. Introduction 12. Homological theory 22.1. Some ring-theoretic preliminaries 22.2. Almost categories 52.3. Almost homological algebra 92.4. Almost homotopical algebra 163. Almost ring theory 233.1. Flat, unramified and etale morphisms 233.2. Almost traces 253.3. Lifting theorems 293.4. Descent 343.5. Behaviour of etale morphisms under Frobenius 414. Appendix 464.1. 46References 50

1. INTRODUCTION

Broadly speaking, the aim of this work is to describe “how to do ring theory” within monoidalcategories that arise as localisations of categories of modules over certain rings. A reader look-ing for forerunners of our themes would be drawn inevitably to Gabriel’s “Des categories abeli-ennes” [8], and might even conclude that Gabriel’s memoir must have been the main instigationfor the present article. In truth, the initial motivation isto be found elsewhere, namely in thewant of adequately documented foundations for the method ofalmost etale extensions that un-derpins Faltings’ approach top-adic Hodge theory as presented in [6]. However, as is often thecase with healthy offsprings, our subject matter has eventually resolved to venture beyond itsoriginal boundaries and pursue an autonomous existence.

In any case, we are glad to report that our paper remains true to its first vocation, which isto serve as a comprehensive reference, paving the way to deeper aspects of almost etale theory,especially to the difficult purity theorem of [6]. The notions of almost unramified and almostetale morphism are defined and their main properties are established, including the analogues ofthe classical lifting theorems over nilpotent extensions,and invariance under Frobenius. Also,to any almost finitely presented almost flat morphism we attach an almost trace form, and wecharacterize almost etale morphisms in terms of this form.Finally, we study some cases ofnon-flat descent for almost etale maps.

A preliminary version of this manuscript was prepared in 1997, while the second author was supported by theIHES.

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http://arxiv.org/abs/math/0002064v1

2 OFER GABBER AND LORENZO RAMERO

Actually, our terminology is slightly different, in that wereplace usual modules and algebrasby their “almost” counterparts, which live in the category of almost modules, a localisation ofthe category of modules. So for instance, instead of almost ´etale morphisms of algebras, wehave etale morphisms of almost algebras.

The categories of almost modules (or almost algebras) and ofusual modules (resp. algebras)are linked in manifold ways. First of all we have of course thelocalisation functor. Then, as ithad already been observed by Gabriel, there is a right adjoint to localisation. Furthermore, weshow that there is aleft adjoint as well, that, to our knowledge, has not been exploited before, inspite of its several useful qualities which will establish it quickly as one of our main tools. Theensemble of localisation and right, left adjoints exhibitssome remarkable exactness properties,that are typically associated to open imbeddings of topoi, all of which seems to suggest theexistence of some deeper geometric structure, still to be unearthed. We may have encounteredhere an instance of a general principle, apparently evoked first by Deligne, according to whichone should try to do algebraic geometry on arbitrary abeliantensor categories (notice though,that our categories are more general than the tannakian categories of [4]).

A large part of the paper is devoted to the construction and study of the almost version ofIllusie’s cotangent complex, on which we base our deformation theory for almost algebras.Faltings’ original method was based on Hochschild cohomology rather than the cotangent com-plex. While Faltings’ approach has the advantage of being more explicit and elementary, italso has the drawback of involving a very large number of longand tedious manipulations withcocycles, and requires a painstaking tracking of the “epsilon book-keeping”. The method pre-sented here avoids (or at least removes from view!) these problems, and also leads to moregeneral results (especially, we can drop all finiteness assumptions from the statements of thelifting theorems).

Though we have strived throughout for the widest generality, in a few places one could havegone even further : for instance it would have been possible to globalise all definitions and mostresults to arbitrary schemes. However, the extension to schemes is completely straightforward,and in practice seems to be scarcely useful. Similarly, there is currently not much incentive tostudy a notion of “almost smooth morphism”.

2. HOMOLOGICAL THEORY

2.1. Some ring-theoretic preliminaries. Unless otherwise stated, every ring is commutativewith unit. Our basic setup consists of a fixed base ringV containing an idealm such thatm2 = m. Starting from section 2.4, we will also assume thatm = m⊗V m is a flatV -module.

Example 2.1.1. i) The main example is given by a non-discrete valuation ringV with valuationν : V − 0 → Γ of rank one (whereΓ is the totally ordered abelian group of values ofν).Then we can takem = 0 ∪ x ∈ V − 0 | ν(x) > ν(1).

ii) Suppose thatm = V . This is the “classical limit”. In this case almost ring theory reducesto usual ring theory. Thus, all the discussion that follows specialises to, and sometimes givesalternative proofs for, statements about rings and their modules.

We define a uniform structure on the setI of ideals ofV as follows. For every finitelygenerated idealm0 ⊂ m the subset ofI × I given by(I, J) | m0 · J ⊂ I andm0 · I ⊂ J isan entourage for the uniform structure, and the subsets of this kind form a fundamental systemof entourages (cp. [3] Ch.II§1). The uniform structure induces a topology onI and moreoverthe notion of convergent (resp. Cauchy) sequence of ideals is well defined. We will also needa notion of “Cauchy product” : let

∏∞n=0 In be a formal infinite product of ideals. We say that

the formal productsatisfies the Cauchy condition(or briefly : is a Cauchy product) if, for everyneighborhoodU of V ∈ I there existsn0 ≥ 0 such that

∏n+pm=n Im ∈ U for all n ≥ n0 and all

p ≥ 0.

ALMOST RING THEORY 3

Remark 2.1.2. Suppose thatJ0 ⊂ J1 ⊂ ... is an increasing infinite sequence of ideals ofI suchthat lim

k→∞Jk = V (convergence for the above uniform structure onI). Then one checks easily

that⋃∞k=0m · Jk = m.

Let M be a givenV -module. We say thatM is almost zeroif m · M = 0. A mapφ ofV -modules is analmost isomorphismif both Ker(φ) and Coker(φ) are almost zeroV -modules.

Remark 2.1.3. (i) It is easy to check that aV -moduleM is almost zero if and only ifm⊗VM =0. Similarly, a mapM → N of V -modules is an almost isomorphism if and only if the inducedmapm⊗V M → m⊗V N is an isomorphism. Notice also that, ifm is flat, thenm ≃ m.

(ii) Let V → W be a ring homomorphism. For aV -moduleM setMW = W ⊗V M . Wehave an exact sequence

0 → K → mW → m ·W → 0(2.1.4)

whereK = TorV1 (V/m,W ) is an almost zeroW -module. By (i) it follows thatm ⊗V K ≃(m ·W )⊗W K ≃ 0. Then, applyingmW ⊗W − and−⊗W (m ·W ) to (2.1.4) we derive

mW ⊗W mW ≃ mW ⊗W (m ·W ) ≃ (m ·W )⊗W (m ·W )

i.e. mW ≃ (m ·W ). In particular, ifm is a flatV -module, thenmW is a flatW -module. Thismeans that our basic assumptions on the pair(V,m) are stable under arbitrary base extension.Notice that the flatness ofm does not imply the flatness ofm ·W . This partly explains why weinsist thatm, rather thanm, be flat.

Before moving on, we want to analyze in some detail how our basic assumptions relate tocertain other natural conditions that can be postulated on the pair(V,m). Indeed, let us considerthe following two hypotheses :

(A) m = m2 andm is a filtered colimit of principal ideals.

(B) m = m2 and, for all integersk > 1, thek-th powers of elements ofm generatem.

Clearly (A) implies (B). Less obvious is the following result.

Proposition 2.1.5. (i) (A) implies thatm is flat.(ii) If m is flat then (B) holds.

Proof. Suppose that (A) holds, so thatm = colimα∈I

V xα, whereI is a directed set parametrizing

elementsxα ∈ m (andα ≤ β ⇔ V xα ⊂ V xβ). For anyα ∈ I we have natural isomorphisms

V xα ≃ V/AnnV (xα) ≃ (V xα)⊗V (V xα).(2.1.6)

Forα ≤ β, let jαβ : V xα → V xβ be the imbedding; we have a commutative diagram

Vµz2 //

πα

V

πβ

(V xα)⊗V (V xα)jαβ⊗jαβ // (V xβ)⊗V (V xβ)

wherez ∈ V is such thatxα = z ·xβ, µz2 is multiplication byz2 andπα is the projection inducedby (2.1.6) (and similarly forπβ). Sincem = m2, for all α ∈ I we can findβ such thatxα is amultiple ofx2β . Sayxα = y ·x2β; then we can takez = y ·xβ, soz2 is a multiple ofxα and in theabove diagram Ker(πα) ⊂ Ker(µz2). Hence one can define a mapλαβ : (V xα)⊗V (V xα) → Vsuch thatπβ λαβ = jαβ ⊗ jαβ andλαβ πα = µz2. It now follows that for everyV -moduleN , the induced morphism TorV1 (N, (V xα) ⊗V (V xα)) → TorV1 (N, (V xβ) ⊗V (V xβ)) is thezero map. Taking the colimit we derive thatm is flat. This shows (i). In order to show (ii) weconsider, for any prime numberp, the following condition


(∗p) m/p ·m is generated (as aV -module) by thep-th powers of its elements.

Clearly (B) implies (∗p) for all p. In fact we have :

Claim 2.1.7. (B) holds if and only if (∗p) holds for every primep.

Proof of the claim:Suppose that (∗p) holds for every primep. The polarization identity

k! · x1 · x2 · ... · xk =∑

I⊂1,2,...,k

(−1)k−|I| ·

(∑

i∈I

xi

)k

shows that ifN =∑

x∈mV xk thenk! · m ⊂ N . To prove thatN = m it then suffices to show

that for every primep dividing k! we havem = p · m + N . Let φ : V/p · V → V/p · V be theFrobenius (x 7→ xp); we can denote by(V/p · V )φ the ringV/p · V seen as aV/p · V -algebravia the homomorphismφ. Also setφ∗M = M ⊗V/p·V (V/p · V )φ for a V/p · V -moduleM .Then the mapφ∗(m/p ·m) → (m/p ·m) (defined by raising top-th power) is surjective by (∗p).Hence so is(φr)∗(m/p ·m) → (m/p ·m) for everyr > 0, which says thatm = p ·m+N whenk = pr, hence for everyk.

Next recall (see [11] Exp. XVII 5.5.2) that, ifM is aV -module, the module of symmetrictensors TSk(M) is defined as(⊗k

VM)Sk , the invariants under the natural action of the symmetricgroupSk on⊗k

VM . We have a natural mapΓk(M) → TSk(M) that is an isomorphism whenMis flat (seeloc. cit. 5.5.2.5; hereΓk denotes thek-th graded piece of the divided power algebra).

Claim 2.1.8. The groupSk acts trivially on⊗kVm and the mapm⊗V m → m (x⊗y⊗z 7→ x⊗yz)

is an isomorphism.

Proof of the claim:The first statement is reduced to the case of transpositions and tok = 2.There we can compute :x⊗ yz = xy ⊗ z = y ⊗ xz = yz ⊗ x. For the second statement notethat the imbeddingm → V is an almost isomorphism, and apply remark 2.1.3(i).

Suppose now thatm is flat and pick a primep. ThenSp acts trivially on⊗pV m. Hence

Γp(m) ≃ ⊗pV m ≃ m.(2.1.9)

But Γp(m) is spanned as aV -module by the productsγi1(x1) · ... · γik(xk) (wherexi ∈ m and∑j ij = p). Under the isomorphism (2.1.9) these elements map to

(p

i1,...,ik

)·xi11 · ... ·xikk ; but such

an element vanishes inm/p · m unlessik = p for somek. Thereforem/p · m is generated byp-th powers, so the same is true form/p ·m, and by the above, (B) holds, which shows (ii).

Proposition 2.1.10.Suppose thatm is countably generated as aV -module. Then we have :i) m is countably presented as aV -module;ii) if m is a flatV -module, then it is of homological dimension≤ 1.

Proof. Let (εi)i∈I be a countable generating family ofm. Thenεi ⊗ εj generatem andεi · εj ·(εk⊗εl) = εk ·εl · (εi⊗εj) for all i, j, k, l ∈ I. For everyi ∈ I, we can writeεi =

∑j xijεj, for

certainxij ∈ m. LetF be theV -module defined by generators(eij)i,j∈I , subject to the relations:

εi · εj · ekl = εk · εl · eij eik =∑

j

xijejk for all i, j, k, l ∈ I.

We get an epimorphismπ : F → m by eij 7→ εi⊗εj . The relations imply that, ifx =∑

k,l ykl ∈

Ker(π), thenεi · εj · x = 0, som · Ker(π) = 0. Whencem ⊗V Ker(π) = 0 and1m ⊗V π is anisomorphism. We consider the diagram

m⊗V F∼ //

φ

m⊗V m

ψ

F

π // m


whereφ andψ are induced by scalar multiplication. We already know thatψ is an isomorphism,and sinceF = m · F , we see thatφ is an epimorphism, soπ is an isomorphism, which shows(i). Now (ii) follows from (i), since it is well-known that a flat countably presented module is ofhomological dimension≤ 1 (see [16] (Ch.I, Th.3.2) and the discussion in [19] pp.49-50).

2.2. Almost categories. If C is a category, andX, Y two objects ofC, we will usually denoteby HomC(X, Y ) the set of morphisms inC from X to Y and by1X the identity morphism ofX. Moreover we denote byCo the opposite category ofC and bys.C the category of simplicialobjects overC, that is, functors∆o → C, where∆ is the category whose objects are the orderedsets[n] = 0, ..., n for each integern ≥ 0 and where a morphismφ : [p] → [q] is a non-decreasing map. A morphismf : X → Y in s.C is a sequence of morphismsf[n] : X [n] →Y [n], n ≥ 0 such that the obvious diagrams commute. We can imbedC in s.C by sending eachobjectX to the “constant” objects.X such thats.X [n] = X for all n ≥ 0 ands.X [φ] = 1X forall morphismsφ in ∆.

If C is an abelian category,D(C) will denote the derived category ofC. As usual we havealso the full subcategoriesD+(C),D−(C) of complexes of objects ofC that are exact for suf-ficiently large negative (resp. positive) degree. IfR is a ring, the category ofR-modules(resp. R-algebras) will be denoted byR-Mod (resp. R-Alg). Most of the times we willwrite HomR(M,N) instead of HomR-Mod(M,N).

We denote bySet the category of sets. The symbolN denotes the set of non-negative inte-gers; in particular0 ∈ N.

The full subcategoryΣ of V -Mod consisting of allV -modules that are almost isomorphicto 0 is clearly a Serre subcategory and hence we can form the quotient categoryV -Mod/Σ.There is a localization functor

V -Mod → V -Mod/Σ M 7→Ma

that takes aV -moduleM to the same module, seen as an object ofV -Mod/Σ. In particular,we have the objectV a associated toV ; it seems therefore natural to use the notationV a-Mod

for the categoryV -Mod/Σ, and an object ofV a-Mod will be indifferently referred to as “aV a-module” or “an almostV -module”. In case we need to stress the dependance on the idealm, we can write(V,m)a-Mod.

Since the almost isomorphisms form a multiplicative system(seee.g. [22] Exerc.10.3.2), itis possible to describe the morphisms inV a-Mod via a calculus of fractions, as follows. LetV -al.Iso be the category that has the same objects asV -Mod, but such that HomV -al.Iso(M,N)consists of all almost isomorphismsM → N . If M is any object ofV -al.Iso we write(V -al.Iso/M) for the category of objects ofV -al.Iso overM (i.e. morphismsφ : X → M).If φi : Xi → M (i = 1, 2) are two objects of(V -al.Iso/M) then Hom(V -al.Iso/M)(φ1, φ2)consists of all morphismsψ : X1 → X2 in V -al.Iso such thatφ1 = φ2 ψ. For any twoV -modulesM,N we define a functorFN : (V -al.Iso/M)o → V -Mod by associating to anobjectφ : P → M the V -module HomV (P,N) and to a morphismα : P → Q the mapHomV (Q,N) → HomV (P,N) : β 7→ β α. Then we have

HomV a-Mod(Ma, Na) = colim

(V -al.Iso/M)oFN .(2.2.1)

However, formula (2.2.1) can be simplified considerably, byremarking that, for anyV -moduleM , the natural morphismm ⊗V M → M is an initial object of(V -al.Iso/M). Indeed, letφ : N →M be an almost isomorphism; the diagram

m⊗V N∼ //

m⊗V M

N

φ // M


(cp. remark 2.1.3(i)) allows one to define a morphismψ : m ⊗V M → N overM . Weneed to show thatψ is unique. But ifψ1, ψ2 : m ⊗V M → N are two maps overM , thenIm(ψ1−ψ2) ⊂ Ker(φ) is almost zero, hence Im(ψ1−ψ2) = 0, sincem⊗V M = m · (m⊗V M).Consequently, (2.2.1) boils down to

HomV a-Mod(Ma, Na) = HomV (m⊗V M,N).(2.2.2)

In particular HomV a-Mod(M,N) has a natural structure ofV -module for any twoV a-modulesM,N , i.e. HomV a-Mod(−,−) is a bifunctor that takes values in the categoryV -Mod.

One checks easily (for instance using (2.2.2)) that the usual tensor product induces a bifunctor−⊗V − on almostV -modules, which, in the jargon of [4] makes ofV a-Mod anabelian tensorcategory. Then analmostV -algebrais just a commutative unitary monoid in the tensor categoryV a-Mod. Let us recall what this means. Quite generally, let(C,⊗, U) be any abelian tensorcategory, so that⊗ : C × C → C is a biadditive functor,U is the identity object ofC (see[4] p.105) and for any two objectsM andN in C we have a “commutativity constraint” (i.e.a functorial isomorphismηM |N : M ⊗ N → N ⊗ M that “switches the two factors”) and afunctorial isomorphismνM : U ⊗M →M . Then aC-monoidA is an object ofC endowed witha morphismµA : A⊗ A→ A (the “multiplication” ofA) satisfying the associativity condition

µA (1A ⊗ µA) = µA (µA ⊗ 1A).

We say thatA is unitary if additionallyA is endowed with a “unit morphism”1A : U → Asatisfying the (left and right) unit property :

µA (1A ⊗ 1A) = νA µA (1A ⊗ 1A) ηA|U = µA (1A ⊗ 1A).

Finally A is commutativeif µA = µA ηA|A (to be rigorous, in all of the above one shouldindicate the associativity constraints, which we have omitted : see [4]). A commutative unitarymonoid will also be simply called analgebra. With the morphisms defined in the obviousway, theC-monoids form a category; furthermore, given aC-monoidA, a left A-moduleis anobjectM of C endowed with a morphismσM : A ⊗M → M such thatσM (1A ⊗ σM ) =σM (µA ⊗ 1M). Similarly one defines rightA-modules andA-bimodules. In the case ofbimodules we have left and right morphismsσM,l : A⊗M →M , σM,r :M ⊗A →M and oneimposes that they “commute”,i.e. that

σM,r (σM,l ⊗ 1A) = σM,l (1A ⊗ σM,r).

Clearly the (left resp. right)A-modules (and theA-bimodules) form an additive category withA-linear morphismsdefined as one expects. One defines the notion of a submodule asanequivalence class of monomorphismsN → M such that the compositionA ⊗ N → A ⊗M → M factors throughN . Now, if f : M → N is a morphism of leftA-modules, thenKer(f) exists in the underlying abelian categoryC and one checks easily that it has a uniquestructure of leftA-module which makes it a submodule ofM . If moreover⊗ is right exactwhen either argument is fixed, then also Coker(f) has a uniqueA-module structure for whichN → Coker(f) isA-linear. In this case the category of leftA-modules is abelian. Similarly, ifA is a unitaryC-monoid, then one defines the notion ofunitary left A-module by requiring thatσM (1A ⊗ 1M) = νM and these form an abelian category when⊗ is right exact.

Specialising to our case we obtain the categoryV a-Alg of almostV -algebras and, for everyalmostV -algebraA, the categoryA-Mod of unitary leftA-modules. Clearly the localizationfunctor restricts to a functorV -Alg → V a-Alg and for anyV -algebraR we have a localizationfunctorR-Mod → Ra-Mod.

Next, if A is an almostV -algebra, we can define the categoryA-Alg of A-algebras. Itconsists of all the morphismsA→ B of almostV -algebras.

Let again(C,⊗, U) be any abelian tensor category. By [4] p.119, the endomorphism ringEndC(U) of U is commutative. For any objectM of C, denoteM∗ = HomC(U,M); thenM 7→


M∗ defines a functorC → EndC(U)-Mod. Moreover, ifA is aC-monoid,A∗ is an associativeEndC(U)-algebra, with multiplication given as follows. Fora, b ∈ A∗ leta·b = µA(a⊗b)ν

−1U .

Similarly, if M is anA-module,M∗ is anA∗-module in a natural way, and in this way we obtaina functor fromA-modules andA-linear morphisms toA∗-modules andA∗-linear maps. Using[4] (Prop. 1.3), one can also check that EndC(U) = U∗ as EndC(U)-algebras, whereU isviewed as aC-monoid usingνU .

All this applies especially to our categories of almost modules and almost algebras : in thiscase we callM 7→M∗ thefunctor of almost elements. So, ifM is an almost module, an almostelement ofM is just an honest element ofM∗. Using (2.2.2) one can show easily that for everyV -moduleM the natural mapM → (Ma)∗ is an almost isomorphism.

LetA be an almostV -algebra. For any twoA-modulesM,N , the set HomA-Mod(M,N) hasa natural structure ofA∗-module and we obtain an internal Hom functor by letting

alHomA(M,N) = HomA-Mod(M,N)a.

This is the functor ofalmost homomorphismsfromM toN .For anyA-moduleM we have also a functor of tensor productM ⊗A − on A-modules

which, in view of the following proposition 2.2.4 can be shown to be a left adjoint to the functoralHomA(M,−). It can be defined asM ⊗A N = (M∗ ⊗A∗

N∗)a but an appropriate almost

version of the usual construction would also work.With this tensor product,A-Mod is an abelian tensor category as well, andA-Alg could

also be described as the category of (A-Mod)-algebras. Under this equivalence, a morphismφ : A→ B of almostV -algebras becomes the unit morphism1B : A→ B of the correspondingmonoid. We will sometimes drop the subscript and write simply 1.

Remark 2.2.3. Let V → W be a map of base rings,W taken with the extended idealm ·W .ThenW a is an almostV -algebra so we have defined the categoryW a-Mod using base ringVand the category(W,m ·W )a-Mod using baseW . One shows easily that they are equivalent:we have an obvious functor(W,m·W )a-Mod →W a-Mod and an essential inverse is providedbyM 7→M∗. Similar base comparison statements hold for the categories of almost algebras.

Proposition 2.2.4. i) There is a natural isomorphismA ≃ Aa∗ of almostV -algebras.ii) Let R be anyV -algebra. Then the functorM 7→ M∗ fromRa-Mod to R-Mod (resp.

fromRa-Alg toR-Alg) is right adjoint to the localization functorR-Mod → Ra-Mod (resp.R-Alg → Ra-Alg).

iii) The counit of the adjunctionMa∗ →M is a natural isomorphism from the composition of

the two functors to the identity functor1A-Mod (resp.1A-Alg).

Proof. (i) has already been remarked. We show (ii). In light of remark 2.2.3 (applied withW = R) we can assume thatV = R. LetM be aV -module andN an almostV -module; wehave natural bijections

HomV a-Mod(Ma, N)≃ HomV a-Mod(M

a, (N∗)a) ≃ HomV (m⊗V M,N∗)

≃ HomV (M,HomV (m, N∗)) ≃ HomV (M,HomV a-Mod(V, (N∗)a))

≃ HomV (M,N∗)

which proves (ii). Now (iii) follows by inspecting the proofof (ii), or by [8] (ch.III Prop.3).

Remark 2.2.5. The existence of the right adjoint follows also directly from [8] (chap.III §3Cor.1 or chap.V§2).

Corollary 2.2.6. The functorM 7→M∗ fromRa-Mod toR-Mod sends injectives to injectivesand injective envelopes to injective envelopes.

Proof. The functorM 7→ M∗ is right adjoint to an exact functor, hence it preserves injectives.Now, let J be an injective envelope ofM ; to show thatJ∗ is an injective envelope ofM∗, it


suffices to show thatJ∗ is an essential extension ofM∗. However, ifN ⊂ J∗ andN ∩M∗ = 0,thenNa ∩M = 0, hencem ·N = 0, butJ∗ does not containm-torsion, thusN = 0.

Corollary 2.2.7. The categoriesA-Mod andA-Alg are both complete and cocomplete.

Proof. We recall that the categoriesA∗-Mod andA∗-Alg are both complete and cocomplete.Now let I be any small indexing category andM : I → A-Mod be any functor. Denote byM∗ : I → A∗-Mod the composed functori 7→M(i)∗. We claim that colim

IM = (colim

IM∗)

a.

The proof is an easy application of proposition 2.2.4(iii).A similar argument also works forlimits and for the categoryA-Alg.

Note that the essential image ofM 7→M∗ is closed under limits. Next recall that the forgetfulfunctorA∗-Alg → Set (resp.A∗-Mod → Set) has a left adjointA∗[−] : Set → A∗-Alg

(resp. A(−) : Set → A∗-Mod) that assigns to a setS the freeA∗-algebraA∗[S] (resp. thefreeA∗-moduleA(S)

∗ ) generated byS. If S is any set, it is natural to writeA[S] (resp.A(S)) fortheA-algebra(A∗[S])

a (resp. for theA-module(A(S)∗ )a. This yields a left adjoint, called the

freeA-algebrafunctorSet → A-Alg (resp. thefreeA-modulefunctorSet → A-Mod) to the“forgetful” functorA-Alg → Set (resp.A-Mod → Set) B 7→ B∗.

Now letR be anyV -algebra; we want to construct a left adjoint to the localisation functorR-Mod → Ra-Mod. For a givenRa-moduleM , let

M! = m⊗V (M∗).(2.2.8)

We have the natural map (unit of adjunction)R → Ra∗, so that we can viewM! as anR-module.

Proposition 2.2.9. i) The functorRa-Mod → R-Mod defined by(2.2.8) is left adjoint tolocalisation.

ii) The unit of the adjunctionM → Ma! is a natural isomorphism from the identity functor

1Ra-Mod to the composition of the two functors.

Proof. (i) follows easily from (2.2.2) and (ii) follows easily from(i).

Corollary 2.2.10. Suppose thatm is a flatV -module. Then we have :i) the functorM 7→M! is exact;ii) the localisation functorR-Mod → Ra-Mod sends injectives to injectives.

Proof. By proposition 2.2.9 it follows thatM 7→ M! is right exact. To show that it is also leftexact whenm is a flatV -module, it suffices to remark thatM 7→ M∗ is left exact. Now, by (i),the functorM 7→Ma is right adjoint to an exact functor, so (ii) is clear.

Next, letB be anyA-algebra. The multiplication onB∗ is inherited byB!, which is thereforea non-unital ring in a natural way. We endow theV -moduleV ⊕ B! with the ring structuredetermined by the rule:(v, b) · (v′, b′) = (v · v′, v · b′ + v′ · b + b · b′) for all v, v′ ∈ V andb, b′ ∈ B!. ThenV ⊕ B! is a (unital) ring. Letµm : m → m be the map defined byx⊗ y 7→ xyfor all x, y ∈ m; we notice that the subset of all elements of the form(µ(s),−s ⊗ 1) (forarbitrarys ∈ m) forms an idealI of V ⊕ B!. SetB!! = (V ⊕ B!)/I. Thus we have a sequenceof V -modules

0 → m → V ⊕ B! → B!! → 0(2.2.11)

which in general is only right exact.

Definition 2.2.12. We say thatB is anexactA-algebraif the sequence (2.2.11) is exact.

Remark 2.2.13. Notice that if m∼→ m (e.g. whenm is flat), then allA-algebras are exact.

In the general case, ifB is anyA-algebra, thenV a × B is always exact. Indeed, we have(V a ×B)∗ ≃ V a

∗ × B∗ and, by remark 2.1.3(i),m⊗V Va∗ ≃ m.


Clearly we have a natural isomorphismB ≃ Ba!!.

Proposition 2.2.14.The functorB 7→ B!! is left adjoint to the localisation functorA!!-Alg →A-Alg.

Proof. Let B be anA-algebra,C anA!!-algebra andφ : B → Ca a morphism ofA-algebras.By proposition 2.2.9 we obtain a naturalA∗-linear morphismB! → C. Together with thestructure morphismV → C this yields a mapφ : V ⊕ B! → C which is easily seen to be aring homomorphism. It is equally clear that the idealI defined above is mapped to zero byφ,hence the latter factors through a map ofA!!-algebrasB!! → C. Conversely, such a map inducesa morhism ofA-algebrasB → Ca just by taking localisation. It is easy to check that the twoprocedures are inverse to each other, which shows the assertion.

Remark 2.2.15. The functor of almost elements commutes with arbitrary limits, because allright adjoints do. It does not in general commute with colimits, not even with arbitrary infinitedirect sums. Dually, the functorsM 7→ M! andB 7→ B!! commute with all colimits. Inparticular, the latter commutes with tensor products.

2.3. Almost homological algebra. In this section we fix an almostV -algebraA and we con-sider various constructions in the category ofA-modules.

Remark 2.3.1. i) Let M1,M2 be twoA-modules. By proposition 2.2.4 it is clear that a mor-phismφ : M1 → M2 of A-modules is uniquely determined by the induced morphismM1∗ →M2∗.

ii) It is a bit tricky to deal with preimages of almost elements under morphisms: for instance,if φ :M1 →M2 is an epimorphism (by which we mean that Coker(φ) ≃ 0) andm2 ∈ M2∗, thenit is not true in general that we can find an almost elementm1 ∈ M1∗ such thatφ∗(m1) = m2.What remains true is that for arbitraryε ∈ m we can findm1 such thatφ∗(m1) = ε ·m2.

The abelian categoryA-Mod satisfies axiom (AB5) (seee.g. [22] (§A.4)) and it has a gen-erator, namely the objectA itself. It then follows by a general result thatA-Mod has enoughinjectives. By corollary 2.2.7 any inverse systemMn | n ∈ N of A-modules has an (inverse)limit lim

n∈NM . As usual, we denote by lim1 the right derived functor of the inverse limit functor.

Notice that [22] (Cor. 3.5.4) holds in the almost case since axiom (AB4*) holds inA-Mod (onthe other hand, it is not clear whether [22] (Lemma 3.5.3) holds under (AB4*), since the proofuses elements).

Lemma 2.3.2. LetMn ; φn :Mn →Mn+1 | n ∈ N (resp.Nn ; ψn : Nn+1 → Nn | n ∈ N)be a direct (resp. inverse) system ofA-modules and morphisms andεn | n ∈ N a sequence ofideals ofV converging toV (for the uniform structure introduced in section 2.1).

i) If εn ·Mn = 0 for all n ∈ N thencolimn∈N

Mn ≃ 0.

ii) If εn ·Nn = 0 for all n ∈ N thenlimn∈N

Nn ≃ 0 ≃ limn∈N

1Nn.

iii) If εn · Coker(ψn) = 0 for all n ∈ N and∏∞

j=0 εj is a Cauchy product, thenlimn∈N

1Nn ≃ 0.

Proof. (i) and (ii) : we remark only that limn∈N

1Nn ≃ limn∈N

1Nn+p for all p ∈ N and leave the

details to the reader. We prove (iii). From [22] (Cor. 3.5.4)it follows easily that(limn∈N

1Nn∗)a ≃

limn∈N

1Nn. It then suffices to show that limn∈N

1Nn∗ is almost zero. We haveε2n · Coker(ψn∗) = 0

and the product∏∞

j=0(ε2j) is again a Cauchy product. Next letN ′

n =⋂p≥0 Im(Nn+p∗ → Nn∗).

If Jn =⋂p≥0(εn · εn+1 · ... · εn+p)

2 thenJn · Nn∗ ⊂ N ′n and lim

n→∞Jn = V . In view of (ii),

limn∈N

1Nn∗/N′n is almost zero, hence we reduce to showing that lim

n∈N

1N ′n is almost zero. But

Jn+p+q ·N′n ⊂ Im(N ′

n+p+q → N ′n) ⊂ Im(N ′

n+p → N ′n)


for all n, p, q ∈ N. On the other hand, by remark 2.1.2 we get⋃∞q=0m · Jn+p+q = m, hence

m·N ′n ⊂ Im(N ′

n+p → N ′n) and finallym·N ′

n = m2 ·N ′n ⊂ Im(m·N ′

n+p → m·N ′n) which means

thatm ·N ′n is a surjective inverse system, so its lim1 vanishes and the result follows.

Example 2.3.3.Let (V,m) be as in example 2.1.1. Then every ideal inV is principal, so in thesituation of the lemma we can writeεj = (xj) for somexj ∈ V . Then the hypothesis in (iii)can be stated by saying that there existsc ∈ N such thatxj 6= 0 for all j ≥ c and the sequencen 7→

∑nj=c ν(xj) is Cauchy inΓ.

Definition 2.3.4. LetM be anA-module.i) We say thatM is flat (resp.faithfully flat) if the functorN 7→M ⊗A N , from the category

of A-modules to itself is exact (resp. exact and faithful).M is almost projectiveif the functorN 7→ alHomA(M,N) is exact.

ii) We say thatM is finitely generatedif there exists a positive integern and an epimorphismAn → M . We say thatM is almost finitely generatedif, for arbitrary ε ∈ m, there exists afinitely generated submoduleMε ⊂M such thatε ·M ⊂Mε.

iii) We say thatM is almost finitely presentedif, for arbitrary ε, δ ∈ m there exist positive

integersn = n(ε),m = m(ε) and a three term complexAmψε→ An

φε→ M with ε·Coker(φε) = 0

andδ · Ker(φε) ⊂ Im(ψε).

Proposition 2.3.5. (i) An A-moduleM is almost finitely generated if and only if for everyfinitely generated idealm0 ⊂ m there exists a finitely generated submoduleM0 ⊂ M suchthatm0 ·M ⊂M0.

(ii) An A-module is almost finitely presented if and only if, for everyfinitely generated ideal

m0 ⊂ m there is a complexAmψ→ An

φ→M withm0 ·Coker(φ) = 0 andm0 ·Ker(φ) ⊂ Im(ψ).

Proof. (i) is easy and we leave it to the reader. To prove (ii), take a finitely generated idealm1 ⊂ m such thatm0 ⊂ m · m1, pick a morphismφ : An → M whose cokernel is annihilatedby m1, and apply the following lemma 2.3.6.

Lemma 2.3.6. If M is almost finitely presented andφ : F → M is a morphism withF ≃ An,then for every finitely generated idealm1 ⊂ m · AnnV (Coker(φ)) there is a finitely generatedsubmodule ofKer(φ) containingm1 · Ker(φ).

Proof. We need the following

Claim 2.3.7. Let F1 be a finitely generatedA-module and suppose that we are givena, b ∈ Vand a (not necessarily commutative) diagram

F1

p //

φ

M

F2

ψ

OO

q

>>

such thatq φ = a · p, p ψ = b · q. Let I ⊂ V be an ideal such that Ker(q) has a finitelygenerated submodule containingI · Ker(q). Then Ker(p) has a finitely generated submodulecontaininga · b · I · Ker(p).

Proof of the claim: Let R be the submodule of Ker(q) given by the assumption. We haveIm(ψφ−a ·b ·1F1) ⊂ Ker(p) andψ(R) ⊂ Ker(p). We takeR1 = Im(ψφ−a ·b ·1F1)+ψ(R).Clearlyφ(Ker(p)) ⊂ Ker(q), soI · φ(Ker(p)) ⊂ R, henceI ·ψ φ(Ker(p)) ⊂ ψ(R) and finallya · b · I · Ker(p) ⊂ R1.

Now, let δ ∈ AnnV (Coker(φ)) andε1, ε2, ε3, ε4 ∈ m. By assumption there is a complex

Art→ As

q→ M with ε1 · Coker(q) = 0, ε2 · Ker(q) ⊂ Im(t). LettingF1 = F , F2 = As,


a = ε1 · ε3, b = ε4 · δ, one checks easily thatψ andφ can be given such that all the assumptionsof the above claim are fulfilled. So, withI = ε2 · V we get thatε1 · ε2 · ε3 · ε4 · δ ·Ker(φ) lies ina finitely generated submodule of Ker(φ). Butm1 is contained in an ideal generated by finitelymany such productsε1 · ε2 · ε3 · ε4 · δ.

The following proposition generalises a well-known characterization of finitely presentedmodules over usual rings.

Proposition 2.3.8. LetM be anA-module.i) M is almost finitely generated if and only if, for every filteredinductive system(Nλ, φλµ)

(indexed by a directed setΛ) the natural morphism

ν : colimΛ

alHomA(M,Nλ) → alHomA(M, colimΛ

Nλ)

is a monomorphism.ii) M is almost finitely presented if and only if for every filtered inductive sytem as above,ν

is an isomorphism.

Proof. The “only if” part in (i) (resp. (ii)) is first checked whenM is finitely generated (resp.finitely presented) and then extended to the general case. Weleave the details to the reader andwe proceed to verify the “if” part. For (i), choose a setI and an epimorphismp : A(I) → M .Let Λ be the directed set of finite subsets ofI, ordered by inclusion. ForS ∈ Λ, let MS =p(AS). Then colim

Λ(M/MS) = 0, so the assumption gives colim

ΛalHomA(M,M/MS) = 0,

i.e. colimΛ

HomA(M,M/MS) = 0 is almost zero, so, for everyε ∈ m, the image ofε · 1Min the above colimit is0, i.e. there existsS ∈ Λ such thatε · M ⊂ MS, which proves thecontention. For (ii), we presentM as a filtered colimit colim

ΛMλ, where eachMλ is finitely

presented (this can be donee.g.by taking such a presentation of theA∗-moduleM∗ and applyingN 7→ Na). The assumption of (ii) gives that colim

ΛHomA(M,Mλ) → HomA(M,M) is an

almost isomorphism, hence, for everyε ∈ m there isλ ∈ Λ andφε : M → Mλ such thatpλ φε = ε · 1M , wherepλ : Mλ → M is the natural morphism to the colimit. If such aφεexists forλ, then it exists for everyµ ≥ λ. Hence, ifm0 ⊂ m is a finitely generated subideal,saym0 =

∑kj V εj , then there existλ ∈ Λ andφi : M → Mλ such thatpλ φi = εi · 1M for

i = 1, ..., k. Hence Im(φipλ−εi ·1Mλ) is contained in Ker(pλ) and containsεi ·Ker(pλ). Hence

Ker(pλ) has a finitely generated submoduleL containingm0 · Ker(pλ). Choose a presentationAm → An

π→ Mλ. Then one can liftm0 · L to a finitely generated submoduleL′ of An. Then

Ker(π)+L′ is a finitely generated submodule of Ker(pλ π) containingm20 ·Ker(pλ π). Since

we also havem0 ·Coker(pλπ) = 0 andm0 is arbitrary, the conclusion follows from proposition2.3.5.

Lemma 2.3.9. Let0 →M ′ →M → M ′′ → 0 be an exact sequence ofA-modules. Then:i) If M ′,M ′′ are almost finitely generated (resp. presented) then so isM .ii) If M is almost finitely presented, thenM ′′ is almost finitely presented if and only ifM ′ is

almost finitely generated.

Proof. These facts can be deduced from proposition 2.3.8 and remark2.3.11(iii), or proveddirectly.

Lemma 2.3.10.LetP be one of the properties : “flat”, “almost projective”, “almost finitelygenerated”, “almost finitely presented”. IfB is aP A-algebra, andM is aP B-module, thenM is P as anA-module.

Proof. Left to the reader.


Let R be aV -algebra andM a flat (resp. faithfully flat)R-module (in the usual sense, see[18] p.45). ThenMa is a flat (resp. faithfully flat)Ra-module. Indeed, the functorM ⊗R −preserves the Serre subcategory of almost zero modules, so by general facts it induces an exactfunctor on the localized categories (cp. [8] p.369). For thefaithfullness we have to show thatanR-moduleN is almost zero wheneverM ⊗RN is almost zero. However,M ⊗RN is almostzero⇔ M ⊗R (m ⊗V N) = 0 ⇔ m ⊗V N = 0 ⇔ N is almost zero. It is clear thatA-Mod

has enough almost projective (resp. flat) objects. LetR be aV -algebra. The localisationfunctor induces a functorG : D(R) → D(Ra) and, in view of corollary 2.2.10,M 7→ M!

induces a functorF : D(Ra) → D(R). We have a natural isomorphismG F ≃ 1D(Ra)

and a natural transformationF G → 1D(R). These satisfy the triangular identities of [17](p.83) soF is a left adjoint toG. If Σ denotes the multiplicative set of morphisms inD(R)which induce almost isomorphisms on the cohomology modules, then the localised categoryΣ−1D(R) exists (seee.g.[22] (Th.10.3.7)) and by the same argument we get an equivalence ofcategoriesΣ−1D(R) ≃ D(Ra).

Given anA-moduleM , we can derive the functorsM ⊗A − (resp. alHomA(M,−), resp.alHomA(−,M)) by taking flat (resp. injective, resp. almost projective) resolutions : one re-marks that bounded above exact complexes of flat (resp. almost projective)A-modules areacyclic for the functorM ⊗A− (resp. alHomA(−,M)) (recall the standard argument: ifF• is acomplex of flatA-modules, letΦ• be a flat resolution ofM ; then Tot(Φ• ⊗A F•) → M ⊗A F•

is a quasi-isomorphism since it is so on rows, and Tot(Φ• ⊗A F•) is acyclic since its columsare; similarly, ifP• is a complex of almost projective objects, one considers thedouble com-plex alHomA(P•, J

•) whereJ• is an injective resolution ofM ; cp. [22] §2.7); then one usesthe construction detailed in [22] (Th.10.5.9). We denote byTorAi (M,−) (resp. alExtiA(M,−),resp. alExtiA(−,M)) the corresponding derived functors. IfA = Ra for someV -algebraR, weobtain easily natural isomorphisms TorR

i (M,N)a ≃ TorAi (Ma, Na) for allR-modulesM,N . A

similar result holds for ExtiR(M,N).

Remark 2.3.11. i) Clearly, anA-moduleM is flat (resp. almost projective) if and only ifTorAi (M,N) = 0 (resp. alExtiA(M,N) = 0) for all A-modulesN and alli > 0.

ii) Let M,N be two flat (resp. almost projective)A-modules. ThenM ⊗A N is a flat (resp.almost projective)A-module and for anyA-algebraB, theB-moduleB ⊗A M is flat (resp.almost projective).

iii) Resume the notation of proposition 2.3.8. IfM is almost finitely presented, then one hasalso that the natural morphism colim

ΛalExt1A(M,Nλ) → alExt1A(M, colim

ΛNλ) is a monomor-

phism. This is deduced from proposition 2.3.8(ii), using the fact that(Nλ) can be injected intoan inductive system(Jλ) of injective almost modules (e.g. Jλ = EHomA(Nλ,E), whereE is aninjective cogenerator forA-Mod), and by applying alExt sequences.

Lemma 2.3.12.LetM be an almost finitely generatedA-module. Consider the following prop-erties:

i) M is almost projective.ii) For arbitrary ε ∈ m there existn(ε) ∈ N andA-linear morphisms

Muε // An(ε)

vε // M(2.3.13)

such thatvε uε = ε · 1M .iii) M is flat.Then (i)⇔ (ii) ⇒ (iii).

Proof. (ii)⇒(i): for given ε ∈ m, we consider anyA-moduleN and we apply the functoralExtiA(−, N) to (2.3.13) :

alExtiA(M,N) // alExtiA(An(ε), N) ≃ 0 // alExtiA(M,N)


which impliesε · alExti(M,N) = 0 for all i > 0. Sinceε is arbitrary, (i) follows from remark2.3.11(i).

(i)⇒(ii): by hypothesis, for arbitraryε ∈ m we can findn = n(ε) and a morphismφε :An → M such thatε · Coker(φε) = 0. Let Mε be the image ofφε, so thatφε factors as

An(ε)ψε−→ Mε

jε−→ M . Also ε · 1M : M → M factors asM

γε−→ Mε

jε−→ M. Since by

hypothesisM is almost projective, the natural morphism induced byψε

alHomA(M,An)ψ∗ε // alHomA(M,Mε)

is an epimorphism. Then for arbitraryδ ∈ m the morphismδ · γε is in the image ofψ∗ε , in other

words, there exists anA-linear morphismuεδ :M → An such thatψε uεδ = δ · γε. If now wetakevεδ = φε, it is clear thatvεδ uεδ = ε · δ · 1M . This proves (ii), since theε ∈ m satisfyingthe assertion of (ii) form an ideal.

(ii)⇒(iii): for a givenA-moduleN , apply the functor TorAi (−, N) to the sequence (2.3.13).This yieldsε · TorAi (M,N) = 0. Now the claim follows from remark 2.3.11(i).

There is a converse to lemma 2.3.12 in caseM is almost finitely presented. Before stating it,we need the following lemma.

Lemma 2.3.14.Let R be any ring,M anyR-module andC = Coker(φ : Rn → Rm) anyfinitely presented (left)R-module. LetC ′ = Coker(φ∗ : Rm → Rn) be the cokernel of thetranspose of the mapφ. Then there is a natural isomorphism

TorR1 (C′,M) ≃ HomR(C,M)/Im(HomR(C,R)⊗RM).

Proof. We have a spectral sequence :

E2ij = TorRi (Hj(Cone(φ∗)),M) ⇒ Hi+j(Cone(φ∗)⊗R M).

On the other hand we have also natural isomorphisms

Cone(φ∗)⊗R M ≃ HomR(Cone(φ), R)[1]⊗RM ≃ HomR(Cone(φ),M)[1].

Hence :E2

10 ≃ E∞10 ≃ H1(Cone(φ∗)⊗R M)/E∞

01 ≃H0(HomR(Cone(φ),M))/Im(E201)

≃ HomR(C,M)/Im(HomR(C,R)⊗R M)

which is the claim.

Proposition 2.3.15. (i) Every almost finitely generated almost projectiveA-module is almostfinitely presented.

(ii) Every almost finitely presented flatA-module is almost projective.

Proof. (ii) : let M be such anA-module. Letε, δ ∈ m and pick a three term complex

Amψ // An

φ // M

such thatε · Coker(φ) = δ · Ker(φ)/Im(ψ) = 0. SetP = Coker(ψ∗); this is a finitely presentedA∗-module andφ∗ factors through a morphismφ∗ : P → M∗. Let γ ∈ m; from lemma 2.3.14we see thatγ · φ is the image of some element

∑nj=1 φj ⊗ mj ∈ HomA∗

(P,A∗) ⊗A∗M∗.

If we defineL = An∗ and v : P → L, w : L → M∗ by v(x) = (φ1(x), ..., φn(x)) andw(y1, ..., yn) =

∑nj=1 yj ·mj, then clearlyγ ·φ = w v. LetK = Ker(φ∗). Thenδ ·Ka = 0 and

the mapδ ·1P a factors through a morphismσ : (P/K)a → P a. Similarly the mapε ·1M factorsthrough a morphismλ :M → (P/K)a. Letα = va σ λ :M → La andβ = wa : La → M .The reader can check thatβ α = ε · δ · γ · 1M . By lemma 2.3.12 the claim follows.

(i) : let P be such an almost finitely generated almost projectiveA-module. For any finitelygenerated idealm0 ⊂ m pick a morphismφ : Ar → P such thatm0 · Coker(φ) = 0. If


ε1, ..., εk is a set of generators form0, a standard argument shows that, for anyi ≤ k, εi · 1Plifts to a morphismψi : P → Ar/Ker(φ); then, sinceP is almost projective,εjψi lifts to amorphismψij : P → Ar. Now claim 2.3.7 applies withF1 = Ar, F2 = M = P , p = φ,q = 1P andψ = ψij and shows that Ker(φ) has a finitely generated submoduleMij containingεi · εj · Ker(φ). Then the span of all suchMij is a finitely generated submodule of Ker(φ)containingm2

0 · Ker(φ). By proposition 2.3.5(ii), the claim follows.

Definition 2.3.16. For anA-moduleM , the dual A-moduleof M is theA-moduleM∗ =alHomA(M,A). M is reflexiveif the natural morphism

M → (M∗)∗ m 7→ (f 7→ f(m))(2.3.17)

is an isomorphism ofA-modules.

Remark 2.3.18. Notice that ifB is anA-algebra andM anyB-module, then by “restrictionof scalars”M is also anA-module and the dualA-moduleM∗ has a natural structure ofB-module. This is defined by the rule(b · f)(m) = f(b · m) (b ∈ B∗, m ∈ M∗ andf ∈ M∗

∗ ).With respect to this structure (2.3.17) becomes aB-linear morphism. Incidentally, notice thatthe two meanings of “M∗

∗ ” coincide,i.e. (M∗)∗ ≃ (M∗)∗.

Proposition 2.3.19.LetP be an almost projectiveA-module and denote byIP the image of thenatural “evaluation morphism”P ⊗A P

∗ → A.i) For every morphism of algebrasA→ B we haveIB⊗AP = IP · B.ii) IP = I2P .iii) P = 0 if and only ifIP = 0.iv) P is faithfully flat if and only ifIP = A.

Proof. Pick an indexing setI large enough, and an epimorphismφ : F = A(I) → P . Foreveryi ∈ I we have the standard morphismsA

ei→ Fπi→ A such thatπi ej = δij · 1A and∑

i∈I ei πi = 1F . For everyx ∈ m chooseψx ∈ HomA(P, F ) such thatφ ψx = x · 1P . It iseasy to check thatIP is generated by the almost elementsπi ψx φ ej (i, j ∈ I, x ∈ m). (i)follows already. For (iii), the “only if” is clear; ifIP = 0, thenψx φ = 0 for all x ∈ m, henceψx = 0 and thereforex · 1P = 0. Next, notice that, from (i) and (iii) we deriveP/(IP · P ) = 0,i.e. P = IP · P , so (ii) follows directly from the definition ofIP . SinceP is flat, to show (iv)we have only to verify that the functorM 7→ P ⊗A M is faithful. To this purpose, it sufficesto check thatP ⊗A (A/J) 6= 0 for every proper idealJ of A. This follows easily from (i) and(iii).

If E, F andN areA-modules, there is a natural morphism :

E ⊗A alHomA(F,N) → alHomA(F,E ⊗A N).(2.3.20)

Lemma 2.3.21. (i) The morphism(2.3.20)is an isomorphism in the following cases :a) whenE is flat andF is almost finitely presented;b) when eitherE or F is almost finitely generated and almost projective;c) whenF is almost projective andE is almost finitely presented;d) whenE is almost projective andF is almost finitely generated.

(ii) The morphism(2.3.20)is a monomorphism in the following cases :a) whenE is flat andF is almost finitely generated;b) whenE is almost projective.

(iii) The morphism(2.3.20)is an epimorphism whenF is almost projective andE is almostfinitely generated.

Proof. If F ≃ A(I) for some finite setI, then alHomA(F,N) ≃ N (I) and the claims are obvious.More generally, ifF is almost projective and almost finitely generated, for anyε ∈ m there


exists a finite setI = I(ε) and morphisms

Fuε // A(I)

vε // F(2.3.22)

such thatvε uε = ε · 1F . We apply the natural transformation

E ⊗A alHomA(−, N) → alHomA(−, E ⊗A N)

to (2.3.22) : an easy diagram chase allows then to conclude that the kernel and cokernel of(2.3.20) are killed byε. As ε is arbitrary, it follows that (2.3.20) is an isomorphism in this case.An analogous argument works whenE is almost finitely generated almost projective, so we get(i.b). If F is only almost projective, then we still have morphisms of the type (2.3.22), but nowI(ε) is no longer necessarily finite. However, the cokernels of the induced morphisms1E ⊗ uεand alHomA(vε, E ⊗A N) are still annihilated byε. Hence, to show (iii) (resp. (i.c)) it sufficesto consider the case whenF is free andE is almost finitely generated (resp. presented). Bypassing to almost elements, we can further reduce to the analogous question for usual ringsand modules, and by the usual juggling we can even replaceE by a finitely generated (resp.presented)A∗-module andF by a freeA∗-module. This case is easily dealt with, and (iii) and(i.c) follow. Case (i.d) (resp. (ii.b)) is similar : one considers almost elements and replacesE∗ by a freeA∗-module (resp. andF∗ by a finitely generatedA∗-module). In case (ii.a) (resp.(i.a)), for every finitely generated submodulem0 of m we can find, by proposition 2.3.5, afinitely generated (resp. presented)A-moduleF0 and a morphismF0 → F whose kernel andcokernel are annihilated bym0. It follows easily that we can replaceF by F0 and suppose thatF is finitely generated (resp. presented). Then the argument in [2] (Ch.I §2 Prop.10) can betaken oververbatimto show (ii.a) (resp. (i.a)).

Lemma 2.3.23. i) Let P be anA-module andB an A-algebra. If eitherP or B is almostfinitely generated almost projective as anA-module, then the natural morphism

B ⊗A alHomA(P,N) → alHomB(B ⊗A P,B ⊗A N)(2.3.24)

is an isomorphism for allA-modulesN .ii) Every almost projective almost finitely generatedA-module is reflexive.

Proof. (i) is an easy consequence of lemma 2.3.21(i.b). To prove (ii), we we apply the naturaltransformation (2.3.17) to (2.3.22) : by diagram chase one sees that the kernel and cokernel ofthe morphismF → (F ∗)∗ are killed byε.

Lemma 2.3.25.Let Mn ; φn : Mn → Mn+1 | n ∈ N be a direct system ofA-modules andsuppose there exist sequencesεn | n ∈ N andδn | n ∈ N of ideals ofV such that

i) limn→∞

εn = V (convergence for the uniform structure on ideals ofV ) and∏∞

j=0 δj is a

Cauchy product;ii) for all n ∈ N there exist integersN(n) and morphisms ofA-modulesψn : AN(n) → Mn

such thatεn · Coker(ψn) = 0;iii) δn · Coker(φn) = 0 for all n ∈ N.Thencolim

n∈NMn is an almost finitely generatedA-module.

Proof. LetM = colimn∈N

Mn. For anyn ∈ N let an =⋂m≥0(

∏n+mj=n δj). Then lim

n→∞an = V . For

m > n setφn,m = φm ... φn+1 φn : Mn → Mm+1 and letφn,∞ : Mn → M be the naturalmorphism. An easy induction shows that

∏mj=n δj · Coker(φn,m) = 0 for all m > n ∈ N. Since

Coker(φn,∞) = colimm∈N

Coker(φn,m) we obtainan · Coker(φn,∞) = 0 for all n ∈ N. Therefore

εn · an · Coker(φn,∞ ψn) = 0 for all n ∈ N. Since limn→∞

εn · an = V , the claim follows.


Lemma 2.3.26.Let Mn ; φn : Mn → Mn+1 | n ∈ N be a direct system ofA-modules andsuppose there exist sequencesεn | n ∈ N andδn | n ∈ N of ideals ofV such that

i) limn→∞

εn = V and∏∞

j=0 δj is a Cauchy product;

ii) εn · alExtiA(Mn, N) = δn · alExtiA(Coker(φn), N) = 0 for all A-modulesN , all i > 0 andall n ∈ N;

iii) δn · Ker(φn) = 0 for all n ∈ N.Thencolim

n∈NMn is an almost projectiveA-module.

Proof. LetM = colimn∈N

Mn. By the above remark 2.3.11(i) it suffices to show that alExtiA(M,N)

vanishes for alli > 0 and allA-modulesN . The mapsφn define a mapφ : ⊕nMn → ⊕nMn

such that we have a short exact sequence0 → ⊕nMn1−φ−→ ⊕nMn −→ M → 0. Applying the

long exact alExt sequence one obtains a short exact sequence(cp. [22] (3.5.10))

0 → limn∈N

1alExti−1A (Mn, N) → alExtiA(M,N) → lim

n∈NalExtiA(Mn, N) → 0.

Then lemma 2.3.2(ii) implies that alExtiA(M,N) ≃ 0 for all i > 1 and moreover alExt1A(M,N)

is isomorphic to limn∈N

1alHomA(Mn, N). Let

φ∗n : alHomA(Mn+1, N) → alHomA(Mn, N) f 7→ f φn

be the transpose ofφn and writeφn as a compositionMnpn−→ Im(φn)

qn→ Mn+1, so that

φ∗n = q∗np

∗n, the composition of the respective transposed morphims. Wehave monomorphisms

Coker(p∗n) → alHomA(Ker(φn), N)Coker(q∗n) → alExt1A(Coker(φn), N)

for all n ∈ N. Henceδ2n · Coker(φ∗n) = 0 for all n ∈ N. Since

∏∞n=0 δ

2n is a Cauchy product,

lemma 2.3.2(iii) shows that limn∈N

1alHomA(Mn, N) ≃ 0 and the assertion follows.

Proposition 2.3.27.Suppose thatm is a flatV -module. Then for anyV -algebraR the functorM 7→M! commutes with tensor products and takes flatRa-modules to flatR-modules.

Proof. Let M be a flatRa-module andN → N ′ an injective map ofR-modules. Denote byK the kernel of the induced mapM! ⊗R N → M! ⊗R N

′; we haveKa ≃ 0. We obtain anexact sequence0 → m ⊗V K → m ⊗V M! ⊗R N → m ⊗V M! ⊗R N

′. But one sees easilythat m ⊗V K = 0 andm ⊗V M! ≃ M!, which shows thatM! is a flatR-module. Similarly,let M,N be twoRa-modules. Then the natural mapM∗ ⊗R N∗ → (M ⊗Ra N)∗ is an almostisomorphism and the assertion follows from remark 2.1.3(i).

2.4. Almost homotopical algebra. The formalism of abelian tensor categories provides a min-imal framework wherein the rudiments of deformation theorycan be developed.

Let (C,⊗, U) be an abelian tensor category; we assume henceforth that⊗ is a right exactfunctor. LetA be a givenC-monoid. A two-sided idealof A is anA-sub-bimoduleI → A.The quotientA/I in the underlying abelian categoryC has a uniqueC-monoid structure suchthatA → A/I is a morphism of monoids.A/I is unitary ifA is. ForI, J subobjects ofA onedenotesIJ = Im(I ⊗ J → A⊗ A

µA→ A). If I is a two-sided ideal ofA such thatI2 = 0, then,using the right exactness of⊗ one checks thatI has a natural structure of anA/I-bimodule,unitary whenA is.

Definition 2.4.1. A C-extensionof aC-monoidB by aB-bimoduleI is a short exact sequenceof objects ofC

X : 0 // I // Cp // B // 0(2.4.2)


such thatC is aC-monoid,p is a morphism ofC-monoids,I is a square zero two-sided ideal inC and theE/I-bimodule structure onI coincides with the given bimodule structure onI. TheC-extensions form a categoryExmonC. The morphisms are commutative diagrams with exactrows

X :

0 // I //

f

Ep //

g

B //

h

0

X ′ : 0 // I ′ // E ′p′ // B′ // 0

such thatg andh are morphisms ofC-monoids. We letExmonC(B, I) be the subcategory ofExmonC consisting of allC-extensions ofB by I, where the morphisms are all short exactsequences as above such thatf = 1I andh = 1B.

We have also the variant in which all theC-monoids in (2.4.2) are required to be unitary(resp. to be algebras) andI is a unitaryB-bimodule (resp. whose left and rightB-moduleactions coincide,i.e. are switched by composition with the “commutativity constraints” ηB|I

and ηI|B, see 2.2); we will callExunC (resp. ExalC) the corresponding category. For amorphismφ : C → B of C-monoids, and aC-extensionX in ExmonC(B, I), we can pullbackX via φ to obtain an exact sequenceX ∗ φ with a morphismφ∗ : X ∗ φ → X; one checkseasily that there exists a unique structure ofC-extension onX ∗φ such thatφ∗ is a morphism ofC-extension; thenX ∗ φ is an object inExmonC(C, I). Similarly, given aB-linear morphismψ : I → J , we can push outX and obtain a well defined objectψ ∗ X in ExmonC(B, J)with a morphismX → ψ ∗ X of ExmonC. In particular, ifI1 andI2 are twoB-bimodules,the functorspi∗ (i = 1, 2) associated to the natural projectionspi : I1 ⊕ I2 → Ii establish anequivalence of categories

ExmonC(B, I1 ⊕ I2)∼→ ExmonC(B, I1)×ExmonC(B, I2)(2.4.3)

whose essential inverse is given by(E1, E2) 7→ (E1 ⊕ E2) ∗ δ, whereδ : B → B ⊕ B is thediagonal morphism. A similar statement holds forExal andExun. These operations can beused to induce an abelian group structure on the set ExmonC(B, I) of isomorphism classes ofobjects ofExmonC(B, I) as follows. For any two objectsX, Y of ExmonC(B, I) we canformX ⊕ Y which is an object ofExmonC(B⊕B, I ⊕ I). Letα : I ⊕ I → I be the additionmorphism ofI. Then we setX+Y = α∗ (X⊕Y )∗ δ. One can check thatX+Y ≃ Y +X foranyX, Y and that the trivial splitC-extensionB⊕ I is a neutral element for+. Moreover everyisomorphism class has an inverse−X. The functorsX 7→ X ∗ φ andX 7→ ψ ∗ X commutewith the operation thus defined, and induce group homomorphisms

∗φ : ExmonC(B, I) → ExmonC(C, I)ψ∗ : ExmonC(B, I) → ExmonC(B, J).

We will need the variant ExalC(B, I) defined in the same way, starting fromExalC(B, I). Forinstance, ifA is an almost algebra (resp. a commutative ring), we can consider the abelian tensorcategoryC = A-Mod. In this case theC-extensions will be called simplyA-extensions, andwe will write ExalA rather thanExalC. In fact the commutative unitary case will soon becomeprominent in our work, and the more general setup is only required for technical reasons, in theproof of proposition 2.4.6 below, which is the abstract version of a well-known result on thelifting of idempotents over nilpotent ring extensions.

Let A be aC-monoid. We form the biproductA† = U ⊕ A in C. We denote byp1, p2the associated projections fromA† ontoU and respectivelyA. Also, let i1, i2 be the naturalmonomorphisms fromU , resp.A toA†. A† is equipped with a unitary monoid structure

µ† = i2 µ (p2 ⊗ p2) + i2 ℓ−1A (p1 ⊗ p2) + i2 r

−1A (p2 ⊗ p1) + i1 u

−1 (p1 ⊗ p1)


whereℓA, rA are the natural isomorphisms provided by [4] (Prop. 1.3) andu : U → U⊗U is asin loc. cit. §1. In terms of the ringA†

∗ ≃ U∗ ⊕ A∗ this is the multiplication(u1, b1) · (u2, b2) =(u1 · u2, b1 · b2 + b1 · u2 + u1 · b2). Theni2 is a morphism of monoids and one verifies thatthe “restriction of scalars” functori∗2 defines an equivalence from the categoryA†-Uni.Mod ofunitaryA†-modules to the categoryA-Mod of all A-modules; letj denote the inverse functor.A similar discussion applies to bimodules. Similarly, we derive equivalences of categories

ExunC(A†, j(M))

∗i2 //ExmonC(A,M)

(−)†oo

for all A-bimodulesM .Next we specialise toA = U : for a givenU-moduleM let eM = σM ℓM : M →

M ; working out the definitions one finds that the condition thatthis is a module structure isequivalent toe2M = eM . LetU × U be the product ofU by itself in the category ofC-monoids.There is an isomorphism of unitaryC-monoidsζ : U † → U × U given byζ = i1 p1 + i2 p1 + i2 p2. Another isomorphism isτ ζ , whereτ is the flip i1 p2 + i2 p1. Hence we getequivalences of categories

U-Modj //

U †-Uni.Mod(ζ−1)∗

//

i∗2

oo (U × U)-Uni.Mod(τζ)∗

oo

The compositioni∗2 (ζ−1 τ ζ)∗ j defines a self-equivalence ofU-Mod which associates

to a givenU-moduleM the newU-moduleMflip whose underlying object inC isM and suchthat eMflip = 1M − eM . The same construction applies toU-bimodules and finally we getequivalences

ExmonC(U,M)∼ // ExmonC(U,M

flip) X 7→ Xflip(2.4.4)

for all U-bimodulesM . If X = (0 → M → Eπ→ U → 0) is an extension andXflip = (0 →

Mflip → Eflip → U → 0), then one verifies that there is a natural isomorphismXflip → Xof complexes inC inducing−1M on M , the identity onU and carrying the multiplicationmorphism onEflip to

−µE + ℓ−1E (π ⊗ 1E) + r−1

E (1E ⊗ π) : E ⊗E → E.

In terms of the associated rings, this corresponds to replacing the given multiplication(x, y) 7→x · y of E∗ by the new operation(x, y) 7→ π∗(x) · y + π∗(y) · x− x · y.

Lemma 2.4.5. If M is aU-bimodule whose left and right actions coincide, then everyextensionofU byM splits uniquely.

Proof. Using the idempotenteM we get aU-linear decompositionM ≃ M1 ⊕ M2 wherethe bimodule structure onM1 is given by the zero morphisms and the bimodule structure onM2 is given byℓ−1

M andr−1M . We have to prove thatExmonC(U,M) is equivalent to a one-

point category. By (2.4.3) we can assume thatM = M1 or M = M2. By (2.4.4) we haveExmonC(U,M2) ≃ ExmonC(U,M

flip2 ) and onMflip

2 the bimodule actions are the zero mor-phisms. So it is enough to considerM =M1. In this case, ifX = (0 →M → E → U → 0) isany extension,µE : E ⊗E → E factors through a morphismU ⊗U → E and composing withu : U → U ⊗ U we get a right inverse ofE → U , which shows thatX is the split extension.Then it is easy to see thatX does not have any non-trivial automorphisms, which proves theassertion.

Proposition 2.4.6. i) Let X = (0 → I → Ap→ A′ → 0) be aC-extension and suppose that

e′ ∈ A′∗ is an idempotent element whose left action on theA′-bimoduleI coincides with its right

action. Then there exists a unique idempotente ∈ A∗ such thatp∗(e) = e′.


ii) Especially, ifA′ is unitary andI is a unitaryA′-bimodule, then every extension ofA′ byIis unitary.

Proof. (i) : the hypothesise′2 = e′ implies thate′ : U → A′ is a morphism of (non-unitary)C-monoids. We can then replaceX byX ∗e′ and thereby assume thatA′ = U , p : A→ U andI isa (non-unitary)U-bimodule and the right and left actions onI coincide. The assertion to proveis that1U lifts to a unique idempotente ∈ A∗. However, this follows easily from lemma 2.4.5.To show (ii), we observe that, by (i), the unit1A′ of A′

∗ lifts uniquely to an idempotente ∈ A∗.We have to show thate is a unit forA∗. Let us show the left unit property. Viae : U → Awe can view the extensionX as an exact sequence of leftU-modules. We can then splitX asthe direct sumX1 ⊕ X2 whereX1 is a sequence of unitaryU-modules andX2 is a sequenceof U-modules with trivial actions. But by hypothesis, onI and onA theU-module structure isunitary, soX = X1 and this is the left unit property.

So much for the general nonsense; we now return to almost algebras. As already announced,from here on, we assume throughout thatm is a flatV -module. As an immediate consequenceof proposition 2.4.6 we get natural equivalences of categories

ExalA1(B1,M1)×ExalA2(B2,M2)∼ // ExalA1×A2(B1 ×B2,M1 ⊕M2)(2.4.7)

wheneverA1,A2 areV a-algebras,Bi is aAi-algebra andMi is a (unitary)Bi-module,i = 1, 2.Notice that, ifA = Ra for someV -algebraR, S (resp.J) is aR-algebra (resp. anS-module)

andX is any object ofExalR(S, J), then by applying termwise the localisation functor we getan objectXa of ExalA(Sa, Ja). With this notation we have the following lemma.

Lemma 2.4.8. i) LetB be anyA-algebra andI aB-module. The natural functor

ExalA!!(B!!, I∗) → ExalA(B, I) X 7→ Xa(2.4.9)

is an equivalence of categories.ii) The equivalence(2.4.9) induces a group isomorphismExalA!!

(B!!, I∗)∼→ ExalA(B, I)

functorial in all arguments.

Proof. Of course (ii) is an immediate consequence of (i). To show (i), letX = (0 → I → E →B → 0) be any object ofExalA(B, I). Using corollary 2.2.10 one sees easily that the sequenceX! = (0 → I! → E!! → B!! → 0) is right exact;X! won’t be exact in general, unlessB (andthereforeE) is an exact algebra. In any case, the kernel ofI! → E!! is almost zero, so we getan extension ofB!! by a quotient ofI! which maps toI∗. In particular we get by pushout anextensionX!∗ by I∗, i.e. an object ofExalA!!

(B!!, I∗) and in fact the assignmentX 7→ X!∗ is anessential inverse for the functor (2.4.9).

Remark 2.4.10. By inspecting the proof, we see that one can replaceI∗ by I!∗ = Im(I! → I∗)in (i) and (ii) above. WhenB is exact, alsoI! will do.

In [14] (II.1.2) it is shown how to associate to any ring homomorphismR → S a naturalsimplicial complex ofS-modules denotedLS/R and called the cotangent complex ofS overR.

Definition 2.4.11. Let A → B be a morphism of almostV -algebras. Thealmost cotangentcomplexof B overA is the simplicialB!!-module

LB/A = B!! ⊗(V a×B)!! L(V a×B)!!/(V a×A)!! .

Usually we will want to viewLB/A as an object of the derived categoryD•(s.B!!) of simpli-cialB!!-modules. Indeed, the hyperext functors computed in this category relate the cotangentcomplex to a number of important invariants. Recall that, for any simplicial ringR and any twoR-modulesE, F the hyperext ofE andF is the abelian group defined as

ExtpR(E, F ) = colimn≥−p

HomD•(R)(σnE, σn+pF )


(whereσ is the suspension functor of [14] (I.3.2.1.4)).Let us fix an almost algebraA. First we want to establish the relationship with differentials.

Definition 2.4.12. LetB be anyA-algebra,M anyB-module.i) An A-derivationof B with values inM is anA-linear morphism∂ : B → M such that

∂(b1 · b2) = b1 · ∂(b2) + b2 · ∂(b1) for b1, b2 ∈ B∗. The set of allM-valuedA-derivations ofBforms aV -module DerA(B,M) and the almostV -module DerA(B,M)a has a natural structureof B-module.

ii) We reserve the notationIB/A for the ideal Ker(µB/A : B ⊗A B → B). Themodule ofrelative differentialsof φ is defined as the (left)B-moduleΩB/A = IB/A/I

2B/A. It is endowed

with a naturalA-derivationδ : B → ΩB/A defined byb 7→ 1 ⊗ b − b ⊗ 1 for all b ∈ B∗. Theassignment(A→ B) 7→ ΩB/A defines a functor

Ω : V a-Alg.Morph → V a-Alg.Mod

from the category of morphismsA → B of almostV -algebras to the categoryV a-Alg.Mod

consisting of all pairs(B,M) whereB is an almostV -algebra andM is aB-module. The mor-phisms inV a-Alg.Morph are the commutative squares; the morphisms(B,M) → (B′,M ′)in V a-Alg.Mod are all pairs(φ, f) whereφ : B → B′ is a morphism of almostV -algebrasandf : B′ ⊗B M →M ′ is a morphism ofB′-modules.

The module of relative differentials enjoys the familiar universal properties that one expects.In particularΩB/A represents the functor DerA(B,−), i.e. for any (left)B-moduleM the mor-phism

HomB(ΩB/A,M) → DerA(B,M) f 7→ f δ(2.4.13)

is an isomorphism. As an exercise, the reader can supply the proof for this claim and for thefollowing standard proposition.

Proposition 2.4.14. i) LetB andC be twoA-algebras. Then there is a natural isomorphism:

ΩC⊗AB/C ≃ C ⊗A ΩB/A.

ii) Let B be anA-algebra,C aB-algebra. There is a natural exact sequence ofC-modules:

C ⊗B ΩB/A → ΩC/A → ΩC/B → 0.

iii) Let I be an ideal of theA-algebraB and letC = B/I be the quotientA-algebra. Thenthere is a natural exact sequence:I/I2 → C ⊗B ΩB/A → ΩC/A → 0.

iv) The functorΩ : V a-Alg.Morph → V a-Alg.Mod commutes with all colimits.

Lemma 2.4.15.For anyA-algebraB there is a natural isomorphism ofB!!-modules

(ΩB/A)! ≃ ΩB!!/A!!.

Proof. Using the adjunction (2.4.13) we are reduced to showing thatthe natural map

φM : DerA!!(B!!,M) → DerA(B,M

a)

is a bijection for allB!!-modulesM . Given∂ : B →Ma we construct∂! : B! →Ma! →M . We

extend∂! to V ⊕ B! by setting it equal to zero onV . Then it is easy to check that the resultingmap descends toB!!, hence giving anA-derivationB!! → M . This procedure yields a rightinverseψM to φM . To show thatφM is injective, suppose that∂ : B!! → M is an almost zeroA-derivation. Composing with the naturalA-linear mapB! → B!! we obtain an almost zeromap∂′ : B! → M . But m · B! = B!, hence∂′ = 0. This implies that in fact∂ = 0, and theassertion follows.

Proposition 2.4.16.LetM be aB-module. There exists a natural isomorphism ofB!!-modules

Ext0B!!(LB/A,M!) ≃ DerA(B,M).


Proof. To ease notation, setA = V a ×A andB = V a ×B. We have natural isomorphisms :

Ext0B!!(LB/A,M!)≃ Ext0

B!!(LB!!/A!!

,M!) by [14] (I.3.3.4.4)

≃ DerA!!(B!!,M!) by [14] (II.1.2.4.2)

≃ DerA(B,M) by lemma 2.4.15.

But it is easy to see that the natural map DerA(B,M) → DerA(B,M) is an isomorphism.

Theorem 2.4.17.There is a natural isomorphism

ExalA(B,M) → Ext1B!!(LB/A,M!).

Proof. With the notation of the proof of proposition 2.4.16 we have natural isomorphisms

Ext1B!!(LB/A,M!)≃ Ext1

B!!(LB!!/A!!

,M!) by [14] (I.3.3.4.4)

≃ ExalA!!(B!!,M!) by [14] (III.1.2.3)

≃ ExalA(B,M)

where the last isomorphism follows directly from lemma 2.4.8(ii) and the subsequent remark2.4.10. Finally, (2.4.7) shows that ExalA(B,M) ≃ ExalA(B,M), as required.

Moreover we have the following transitivity theorem as in [14] (II.2.1.2).

Theorem 2.4.18.LetA → B → C be a sequence of morphisms of almostV -algebras. Thereexists a natural distinguished triangle ofD•(s.C!!)

C!! ⊗B!!LB/A

u // LC/Av // LC/B // C!! ⊗B!!

σLB/A

where the morphismsu andv are obtained by functoriality ofL.

Proof. It follows directly from loc. cit.

Proposition 2.4.19.Let (Aλ → Bλ)λ∈I be a system of almostV -algebra morphisms indexedby a small filtered categoryI. Then there is a natural isomorphism inD•(s.colim

λ∈IBλ!!)

colimλ∈I

LBλ/Aλ≃ Lcolim

λ∈IBλ/colim

λ∈IAλ.

Proof. Remark 2.2.15 gives an isomorphism : colimλ∈I

Aλ!!∼→ (colim

λ∈IAλ)!! (and likewise for

colimλ∈I

Bλ). Then the claim follows from [14] (II.1.2.3.4).

Next we want to prove the almost version of the flat base changetheorem [14] (II.2.2.1). Tothis purpose we need some preparation.

Proposition 2.4.20.LetB andC be twoA-algebras and setTi = TorA!!i (B!!, C!!). If A, B, C

andB ⊗A C are all exact, then for everyi > 0 the natural morphismm ⊗V Ti → Ti is anisomorphism.

Proof. For any almostV -algebraD we letkD denote the complex ofD!!-modules[m⊗V D!! →D!!] placed in degrees−1, 0; we have a distiguished triangle

T(D) : m⊗V D!!// D!!

// kD // m⊗V D!![1].

By the assumption, the natural mapkA → kB is a quasi-isomorphism andm ⊗V B!! ≃ B!. Onthe other hand, for alli ∈ N we have

TorA!!i (kB, C!!) ≃ TorA!!

i (kA, C!!) ≃ H−i(kA ⊗A!!C!!) = H−i(kC).

In particular TorA!!i (kB, C!!) = 0 for all i > 1. As m is flat overV , we havem ⊗V Ti ≃

TorA!!i (m ⊗V B!!, C!!). Then by the long exact Tor sequence associated toT(B)

L⊗A!!

C!! we


get the assertion for alli > 1. Next we consider the natural map of distinguished triangles

T(A)L⊗A!!

A!! → T(B)L⊗A!!

C!!; writing down the associated morphism of long exact Tor se-quences, we obtain a diagram with exact rows :

0 // TorA!!1 (kA, A!!)

∂ //

(m⊗V A!!)⊗A!!A!!

i //

A!! ⊗A!!A!!

TorA!!1 (kB, C!!)

∂′ // (m⊗V B!!)⊗A!!C!!

i′ // B!! ⊗A!!C!!

By the above, the leftmost vertical map is an isomorphism; moreover, the assumption givesKer(i) ≃ Ker(m → V ) ≃ Ker(i′). Then, since∂ is injective, also∂′ must be injective, whichimplies our assertion for the remaining casei = 1.

Corollary 2.4.21. Keep the notation of proposition 2.4.20 and suppose thatTorAi (B,C) ≃ 0for somei > 0. Then the correspondingTi vanishes.

Theorem 2.4.22.LetB,A′ be twoA-algebras. Suppose that the natural morphismBL⊗AA

′ →B′ = B ⊗A A

′ is an isomorphism inD•(s.A). Then the natural morphisms

B′!! ⊗B!!

LB/A −→ LB′/A′

(B′!! ⊗B!!

LB/A)⊕ (B′!! ⊗A′

!!LA′/A) −→ LB′/A

are quasi-isomorphisms.

Proof. Let us remark that the functorD 7→ V a×D : A-Alg → (V a×A)-Alg commutes withtensor products; hence the same holds for the functorD 7→ (V a × D)!! (see remark 2.2.15).Then, in view of corollary 2.4.21, the theorem is reduced immediately to [14] (II.2.2.1).

As an application we obtain the vanishing of the almost cotangent complex for a certain classof morphisms.

Theorem 2.4.23.LetR → S be a morphism of almost algebras such that

TorRi (S, S) ≃ 0 ≃ TorS⊗RSi (S, S) for all i > 0

(for the naturalS ⊗R S-module structure induced byµS/R). ThenLS/R ≃ 0 in D•(S!!).

Proof. Since TorRi (S, S) = 0 for all i > 0, theorem 2.4.22 applies (withA = R andB = A′ =S), giving the natural isomorphisms

(S ⊗R S)!! ⊗S!!LS/R ≃ LS⊗RS/S

((S ⊗R S)!! ⊗S!!LS/R)⊕ ((S ⊗R S)!! ⊗S!!

LS/R) ≃ LS⊗RS/R(2.4.24)

Since TorS⊗RSi (S, S) = 0, the same theorem also applies withA = S ⊗R S, B = S, A′ = S,

and we notice that in this caseB′ ≃ S; hence we have

LS/S⊗RS ≃ S!! ⊗S!!LS/S⊗RS ≃ LS/S ≃ 0.(2.4.25)

Next we apply transitivity to the sequenceR → S ⊗R S → S, to obtain (thanks to (2.4.25))

S!! ⊗S⊗RS!!LS⊗RS/R ≃ LS/R.(2.4.26)

Applying S!! ⊗S⊗RS!!− to the second isomorphism (2.4.24) we obtain

LS/R ⊕ LS/R ≃ S!! ⊗S⊗RS!!LS⊗RS/R.(2.4.27)

Finally, composing (2.4.26) and (2.4.27) we derive

LS/R ⊕ LS/R∼→ LS/R.(2.4.28)

However, by inspection, the isomorphism (2.4.28) is the summap. ConsequentlyLS/R ≃ 0, asclaimed.


Finally we have a fundamental spectral sequence as in [14] (III.3.3.2).

Theorem 2.4.29.Let φ : A → B be a morphism of almost algebras such thatB ⊗A B ≃ B(e.g.such thatB is a quotient ofA). Then there is a first quadrant homology spectral sequenceof bigraded almost algebras

E2pq = Hp+q(Symq

B(LaB/A)) ⇒ TorAp+q(B,B).

Proof. We replaceφ by 1V a × φ and apply the functorB 7→ B!! (which commutes with ten-sor products by remark 2.2.15) thereby reducing the assertion to the above mentioned [14](III.3.3.2).

3. ALMOST RING THEORY

3.1. Flat, unramified and etale morphisms. LetA→ B be a morphism of almostV -algebras.Using the natural “multiplication” morphism ofA-algebrasµB/A : B ⊗A B → B we can viewB as aB ⊗A B-algebra.

Definition 3.1.1. Let φ : A→ B be a morphism of almostV -algebras.i) We say thatφ is aflat (resp.faithfully flat, resp.almost projective) morphismif B is a flat

(resp. faithfully flat, resp. almost projective)A-module.ii) We say thatφ is almost finite(resp.finite) if B is an almost finitely generated (resp. finitely

generated)A-module.iii) We say thatφ is weakly unramified(resp.unramified) if B is a flat (resp. almost projec-

tive)B ⊗A B-module (via the morphismµB/A defined above).iv) φ is weaklyetale(resp.etale) if it is flat and weakly unramified (resp. unramified).

Lemma 3.1.2. Letφ : A→ B andψ : B → C be morphisms of almostV -algebras.i) Any base change of a flat (resp. almost projective, resp. faithfully flat, resp. almost finite,

resp. weakly unramified, resp. unramified, resp. weaklyetale, resp.etale) morphism is flat(resp. almost projective, resp. faithfully flat, resp. almost finite, resp. weakly unramified, resp.unramified, resp. weaklyetale, resp.etale);

ii) if both φ andψ are flat (resp. almost projective, resp. faithfully flat, resp. almost finite,resp. weakly unramified, resp. unramified, resp. weaklyetale, resp.etale), then so isψ φ;

iii) if φ is flat andψ φ is faithfully flat, thenφ is faithfully flat;iv) if φ is weakly unramified andψ φ is flat (resp. weaklyetale), thenψ is flat (resp. weakly

etale);v) If φ is unramified andψ φ is etale, thenψ is etale;vi) φ is faithfully flat if and only if it is a monomorphism andB/A is a flatA-module;vii) if φ is almost finite and weakly unramified, thenφ is unramified.

Proof. For (vi) use the Tor sequences. In view of proposition 2.3.15(ii), to show (vii) it sufficesto know thatB is an almost finitely presentedB ⊗A B-module; but this follows from theexistence of an epimorphism ofB ⊗A B-modules(B ⊗A B) ⊗A B → Ker(µB/A) defined byx ⊗ b 7→ x · (1 ⊗ b − b ⊗ 1). Of the remaining assertions, only (iv) and (v) are not obvious,but the proof is just the “almost version” of a well-known argument. Let us show (v); the sameargument applies to (iv). We remark thatµB/A is an etale morphism, sinceφ is unramified.DefineΓψ = 1C ⊗B µB/A. By (i), Γψ is etale. Define alsop = (ψ φ) ⊗A 1B. By (i), p is flat(resp. etale). The claim follows by remarking thatψ = Γψ p and applying (ii).

Remark 3.1.3. i) Suppose we work in the classical limit case, that is,V = m (cp. example2.1.1(ii)). Then we caution the reader that our notion of “etale morphism” is more generalthan the usual one, as defined in [10]. The relationship between the usual notion and ours isdiscussed in the digression at the end of section 3.4.


ii) The naive hope that the functorA 7→ A!! might preserve flatness is crushed by the fol-lowing counterexample. Let(V,m) be as in example 2.1.1(i) and letk be the residue fieldof V . Consider the flat mapV × V → V defined as(x, y) 7→ x. We get a flat morphismV a × V a → V a in V a-Alg; applying the left adjoint to localisation yields a mapV ×k V → Vthat is not flat. On the other hand, faithful flatnessis preserved. Indeed, letφ : A→ B be a mor-phism of almost algebras. Thenφ is a monomorphism if and only ifφ!! is injective; moreover,B!!/Im(A!!) ≃ B!/A!, which is flat overA!! if and only if B/A is flat overA, by proposition2.3.27.

We will find useful to study certain “almost idempotents”, asin the following proposition.

Proposition 3.1.4. A morphismφ : A → B is unramified if and only if there exists an almostelementeB/A ∈ B ⊗A B∗ such that

i) e2B/A = eB/A;ii) µB/A(eB/A) = 1;iii) x · eB/A = 0 for all x ∈ IB/A∗.

Proof. Suppose thatφ is unramified. We start by showing that for everyε ∈ m there existalmost elementseε of B ⊗A B such that

e2ε = ε · eε µB/A(eε) = ε · 1 IB/A∗ · eε = 0.(3.1.5)

SinceB is an almost projectiveB ⊗A B-module, for everyε ∈ m there exists an “approximatesplitting” for the epimorphismµB/A : B ⊗A B → B, i.e. a B ⊗A B-linear morphismuε :B → B ⊗A B such thatµB/A uε = ε · 1B. Seteε = uε 1 : A → B ⊗A B. We see thatµB/A(eε) = ε · 1. To show thate2ε = ε · eε we use theB ⊗A B-linearity ofuε to compute

e2ε = eε · uε(1) = uε(µB/A(eε) · 1) = uε(µB/A(eε)) = ε · eε.

Next take any almost elementx of IB/A and compute

x · eε = x · uε(1) = uε(µB/A(x) · 1) = 0.

This establishes (3.1.5). Next let us take any otherδ ∈ m and a corresponding almost elementeδ.Bothε·1−eε andδ·1−eδ are elements ofIB/A∗, hence we have(δ·1−eδ)·eε = 0 = (ε·1−eε)·eδwhich implies

δ · eε = ε · eδ for all ε, δ ∈ m.(3.1.6)

Let us define a mapeB/A : m⊗V m → B ⊗A B∗ by the rule

ε⊗ δ 7→ δ · eε for all ε, δ ∈ m.(3.1.7)

To show that (3.1.7) does indeed determine a well defined morphism, we need to check thatδ · v · eε = δ · ev·ε andδ · eε+ε′ = δ · (eε + eε′) for all ε, ε′, δ ∈ m and allv ∈ V . However, bothidentities follow easily by a repeated application of (3.1.6). It is easy to see thateB/A defines analmost element with the required properties.

Conversely, suppose an almost elementeB/A of B ⊗A B is given with the stated properties.We defineu : B → B ⊗A B by b 7→ eB/A · (1⊗ b) (b ∈ B∗) andv = µB/A. Then (iii) says thatu is aB ⊗A B-linear morphism and (ii) shows thatv u = 1B. Hence, by lemma 2.3.12,φ isunramified.

Corollary 3.1.8. Under the hypotheses and notation of the proposition, the ideal I = IB/A hasa natural structure ofA-algebra, with unit morphism given by1I/A = 1B⊗AB/A

− eB/A andwhose multiplication is the restriction ofµB⊗AB/A to I. Moreover the natural morphism

B ⊗A B → IB/A ⊕B x 7→ (x · 1I/A ⊕ µB/A(x))

is an isomorphism ofA-algebras.

Proof. Left to the reader as an exercise.


3.2. Almost traces. LetA be an almostV -algebra. For any integern > 0, the standard direct

sum decomposition ofAn determines uniquelyA-linear morphismsAeAi−→ An

πAj

−→ A (fori, j = 1, ..., n) such thatπAj eAi = δij · 1A for all i, j and

∑ni=1 e

Ai πAi = 1An. We can then

define a naturaltrace homomorphism

Tr : alHomA(An, An) → A φ 7→

n∑

i=1

πAi φ eAi(3.2.1)

which is anA-linear morphism. For anyφ, ψ ∈ alHomA(An, An)∗ we have Tr(φ ψ) =

Tr(ψ φ). It follows easily that Tr is independent of the given directsum decomposition ofAn.More generally, suppose thatM is an almost projective almost finitely generatedA-module.Then for anyε ∈ m we can findn = n(ε) and morphisms

Muε // An

vε // M(3.2.2)

such thatvε uε = ε · 1M . Let E(M) = alHomA(M,M); notice thatE(M)∗ is naturallyisomorphic to HomA(M,M). We consider theA-linear morphism

tε : E(M) → A φ 7→ Tr(uε φ vε) (φ ∈ E(M)∗).(3.2.3)

Now, pick any otherδ ∈ m. We compute

ε · tδ(φ) = ε · Tr(uδ φ vδ) = Tr(uδ vε uε φ vδ)= Tr(uε φ vδ uδ vε) = δ · Tr(uε φ vε) = δ · tε(φ).

This allows us to define a map

tM/A : m⊗V m⊗V E(M)∗ → A∗

by settingε⊗ δ⊗φ 7→ ε · tδ(φ). We leave to the reader the verification thattM/A is well definedand does not depend on the choice oftδ. It induces a morphismE(M) → A that we denoteagain bytM/A and we call thealmost trace morphismfor the almostA-moduleM .

Let f ∈ M∗∗ , m ∈ M∗ and defineφf,m ∈ E(M)∗ by the formulaφf,m(m′) = f(m′) ·m for

all m′ ∈ M∗. We have the following :

Lemma 3.2.4. With the above notation :tM/A(φf,m) = f(m).

Proof. Let f : M → A andm : A → M be given. Obviously we haveφf,m = m f andf(m) = f m. Pick morphismsuε andvε as in (3.2.2). Using the foregoing notation, we canwrite :

tε(φf,m) =∑n

i=1(πAi uε m) (f vε e

Ai )

=∑n

i=1(f vε eAi ) (π

Ai uε m)

= f vε uε m = ε · f m

from which the claim follows directly.

Lemma 3.2.5. Let M andN be almost finitely generated almost projectiveA-modules, andφ :M → N , ψ : N →M twoA-linear morphisms. Then :

i) tM/A(ψ φ) = tN/A(φ ψ).ii) If ψ φ = a · 1M andφ ψ = a · 1N for somea ∈ A∗, and if, furthermore, there exist

u ∈ EndA(M), v ∈ EndA(N) such thatv φ = φ u, thena · (tM/A(u)− tN/A(v)) = 0.

Proof. (i) is left to the reader as an exercise. For (ii) we compute using (i) : a · tM/A(u) =tM/A(ψ φ u) = tM/A(ψ v φ) = tN/A(v φ ψ) = a · tN/A(v).

Proposition 3.2.6. LetM = (0 → M1i→ M2

p→ M3 → 0) be an exact sequence of almost

finitely generated almost projectiveA-modules, and letu = (u1, u2, u3) : M → M be anendomorphism ofM , given by endomorphismsui : Mi → Mi (i = 1, 2, 3). Then we havetM2/A(u2) = tM1/A(u1) + tM3/A(u3).


Proof. Suppose first that there exists a splittings : M3 → M2 for p, so that we can viewu2

as a matrix

(u1 v0 u3

), wherev ∈ HomA(M3,M1). By additivity of the trace, we are then

reduced to show thattM2/A(iv p) = 0. By lemma 3.2.5(i), this is the same astM3/A(p iv),which obviously vanishes. In general, for anya ∈ m we consider the morphismµa = a · 1M3

and the pull back morphismM ∗ µa →M :

0 // M1i // M2

p // M3// 0

0 // M1// P //

φ

OO

M3//

µa

OO

0

ThenM ∗ µa is a split exact sequence with the endomorphismu ∗ µa = (u1, v, u3), for acertainv ∈ EndA(P ). The pair of morphisms(a · 1M2, p) induces a morphismψ : M2 → P ,and it is easy to check thatφ ψ = a · 1M2 andψ φ = a · 1P . We can therefore applylemma 3.2.5 to deduce thata · (tP/A(v) − tM/A(u)) = 0. By the foregoing we know thattP/A(v) = tM1/A(u1) + tM3/A(u3), so the claim follows.

Suppose now thatB is an almost finitely generated almost projectiveA-algebra. For anyb ∈ B∗, denote byµb : B → B theB-linear morphismb′ 7→ b · b′. The mapb 7→ µb defines aB-linear monomorphismµ : B → E(B). The composition

TrB/A = tB/A µ : B → A

will also be called the almost trace morphism of theA-algebraB.

Proposition 3.2.7. LetA andB be as in the above discussion.i) If φ : A→ B is an isomorphism, thenTrB/A = φ−1.ii) If C any otherA-algebra, thenTrC⊗AB/C = 1C ⊗A TrB/A.iii) If C is an almost projective almost finiteB-algebra, thenTrC/A = TrB/A TrC/B.

Proof. (i) and (ii) are left as exercises for the reader. We verify (iii). For givenε, δ ∈ m pick

morphismsBuε−→ An

vε−→ B andCu′δ−→ Bm

v′δ−→ C such thatvεuε = ε·1B andv′δu′δ = δ·1C .

If we setu⊕mε = uε ⊗A 1Am, u′′δε = u⊕mε u′δ : C → An ⊗A Am, v⊕mε = vε ⊗A 1Am and

v′′δε = v′δ v⊕mε : An ⊗A A

m → C then we havev′′δε u′′δε = ε · δ · 1C . Define

tε,B/A : B → A b 7→ Tr(uε µb vε)tδ,C/B : C → B c 7→ Tr(u′δ µc v

′δ)

tδε,C/A : C → A c 7→ Tr(u′′δε µc v′′δε).

Using (3.2.1) we can write

tδε,C/A(c) =

n,m∑

i,j=1

(πAi ⊗A πAj ) u

′′δε µc v

′′δε (e

Ai ⊗A e

Aj )

=

n,m∑

i,j=1

(πAi ⊗ πAj ) u⊕mε u′δ µc v

′δ v

⊕mε (eAi ⊗ eAj )

=

n,m∑

i,j=1

πAi uε πBj u′δ µc v

′δ e

Bj vε e

Ai

=n∑

i=1

πAi uε tδ,C/B(c) vε eAi

= tε,B/A tδ,C/B(c)

which implies immediately the claim.


Corollary 3.2.8. LetA → B be a faithfully flat almost finitely presented andetale morphismof almostV -algebras. ThenTrB/A : B → A is an epimorphism.

Proof. Under the stated hypotheses,B is an almost projectiveA-module (by proposition 2.3.15).Let C = Coker(TrB/A) and TrB/B⊗AB the trace morphism for the morphism of almostV -algebrasµB/A. By faithful flatness, the natural morphismC → C ⊗A B = Coker(TrB⊗AB/B)is a monomorphism, hence it suffices to show that TrB⊗AB/B is an epimorphism (hereB ⊗A Bis considered as aB-algebra via the second factor). However, from proposition3.2.7(i) and (iii)we see that TrB/B⊗AB is a right inverse for TrB⊗AB/B. The claim follows.

It is useful to introduce theA-linear morphism

trB/A = TrB/A µB/A : B ⊗A B → A.

We can view trB/A as a bilinear form; it induces anA-linear morphism

τB/A : B → B∗ = alHomA(B,A)

characterized by the equality trB/A(b1 ⊗ b2) = τB/A(b1)(b2) for all b1, b2 ∈ B∗. We say thattrB/A is a perfect pairingif τB/A is an isomorphism.

Theorem 3.2.9.An almost projective and almost finite morphismφ : A → B of almostV -algebras isetale if and only if the trace formtrB/A is a perfect pairing.

Proof. Suppose thatφ is etale. LeteB/A be the idempotent almost element ofB⊗AB providedby proposition 3.1.4. We define a morphismσ : B∗ → B by f 7→ (f ⊗A 1B)(eB/A). Tostart with, we remark that bothτB/A andσ areB-linear morphisms (for the naturalB-modulestructure ofB∗ defined in remark 2.3.18). Indeed, let us pick anyb, b′, b′′ ∈ B∗, f ∈ B∗

∗ andcompute directly

(b · τB/A(b′))(b′′) = τB/A(b

′)(bb′′) = TrB/A(bb′b′′) = (τB/A(bb′))(b′′).

b · σ(f) = b · (f ⊗A 1B)(eB/A) = (f ⊗A 1B)((1B/A ⊗ b) · eB/A)= (f ⊗A 1B)((b⊗ 1B/A) · eB/A) = ((b · f)⊗A 1B)(eB/A)= σ(b · f).

Next we show thatσ is a left inverse forτB/A. In fact, letb ∈ B∗. We have

σ τB/A(b) = (τB/A(b)⊗A 1B)(eB/A) = (TrB/A ⊗A 1B)((b⊗ 1B/A) · eB/A)= TrB⊗AB/B((1B/A ⊗ b) · eB/A) = b · TrB⊗AB/B(eB/A).

Therefore it suffices to show that TrB⊗AB/B(eB/A) = 1. However, by hypothesisφ is unrami-fied, hence corollary 3.1.8 gives a decompositionB ⊗A B ≃ B ⊕ IB/A such thateB/A acts asthe identity on the first factor and as the zero morphism on thesecond factor.

Now, letX = Ker(σ). From the above we derive aB-linear isomorphismB∗ ≃ B ⊕X. Wedualize and apply lemma 2.3.23(ii) to obtain anotherB-linear isomorphism

B ≃ (B∗)∗ ≃ (B ⊕X)∗ ≃ B∗ ⊕X∗ ≃ B ⊕X ⊕X∗.(3.2.10)

Finally, composing the isomorphism (3.2.10) with the projection on the first factor, we get asplitB-linear epimorphismB → B, hence asurjectiveB∗-linear morphismB∗ → B∗. Such amorphism is necessarily an isomorphism, and, tracing backward, the same must hold forτB/A.

Conversely, suppose that the trace form is a perfect pairing. By lemma 2.3.23(i) the naturalmorphismα : B∗ ⊗AB → alHomB(B⊗AB,B) is an isomorphism and one verifies easily thatα(τB/A⊗A1B) = τB⊗AB/B . In particularτB⊗AB/B is also an isomorphism. The multiplicationgives an almost elementµB/A ∈ alHomB(B ⊗A B,B)∗; let e = τ−1

B⊗AB/B(µB/A). We derive

TrB⊗AB/B(e) = τB⊗AB/B(e)(1B⊗AB) = µB/A(1B⊗AB

) = 1B/A.(3.2.11)


Furthermore, we have already remarked thatτB/A is aB-linear morphism, henceτB⊗AB/B is aB ⊗A B-linear morphism. Consequently, for any almost elementx of B ⊗A B we have

τB⊗AB/B(x · e) = x · τB⊗AB/B(e) = x · µB/A = µB/A(x) · µB/A = µB/A(x) · τB⊗AB/B(e).

Since by hypothesisτB/A is an isomorphism, this implies

x · e = µB/A(x) · e.(3.2.12)

Consider the morphismµe : B ⊗A B → B ⊗A B defined byx 7→ e · x; thenµe isB-linear (fortheB-module structure defined by the second factor). Applying (3.2.12) and lemma 3.2.4 weconclude thattB⊗AB/B(µe) = µB/A(e). On the other hand, (3.2.11) says that this trace is equalto 1B/A, hence

µB/A(e) = 1B/A.(3.2.13)

Let β : B → B ⊗A B be defined asb 7→ b · e. From (3.2.12) we see that bothβ andµB/AareB ⊗A B-linear morphisms and from (3.2.13) we know moreover thatµB/A β = 1B. Bylemma 2.3.12 we deduce thatB is an almost projectiveB ⊗A B-module,i.e. φ is unramified,as claimed.

Definition 3.2.14. Thenilradical of an almost algebraA is the ideal nil(A) = nil(A∗)a (where,

for a ringR, we denote by nil(R) the ideal of nilpotent elements inR). We say thatA is reducedif nil (A) ≃ 0.

Notice that, ifR is aV -algebra, then every nilpotent ideal inRa is of the formIa, whereIis a nilpotent ideal inR (indeed, it is of the formIa whereI is an ideal, andm · I is seen tobe nilpotent). It follows easily that nil(A) is the colimit of the nilpotent ideals inA; moreovernil(R)a = nil(Ra). Using this one sees thatA/nil(A) is reduced.

Proposition 3.2.15.LetA → B be anetale almost finitely presented morphism of almost al-gebras. IfA is reduced thenB is reduced as well.

Proof. Under the stated hypothesis,B is an almost projectiveA-module (by virtue of proposi-tion 2.3.15(ii)). Hence, for givenε ∈ m, pick a sequence of morphismsB

uε→ Anvε→ B such

thatvε uε = ε · 1B; with the notation of (3.2.3), defineνb : An → An by νb = vε µb uε, sothattε(b) = Tr(νb). One verifies easily thatνmb = εm−1 · νbm for all integersm > 0.

Now, suppose thatb ∈ nil(B∗). It follows thatbm = 0 for m sufficiently large, henceνmb = 0for m sufficiently large. Letp be any prime ideal ofA∗; let π : A∗ → A∗/p be the naturalprojection andF the fraction field ofA∗/p. TheF -linear morphismνb∗ ⊗A∗

1F is nilpotent onthe vector spaceF n, henceπTr(νb∗) = Tr(νb∗⊗A∗

1F ) = 0. This shows that Tr(νb∗) lies in theintersection of all prime ideals ofA∗, hence it is nilpotent. Since by hypothesisA is reduced, weget Tr(νb∗) = 0. Finally, this implies TrB/A(b) = 0. Now, for anyb′ ∈ B∗, the almost elementbb′ will be nilpotent as well, so the same conclusion applies to it. This shows thatτB/A(b) = 0.But by hypothesisB is etale overA, hence theorem 3.2.9 yieldsb = 0, as required.

Remark 3.2.16. LetM be anA-module. We say that an almost elementa of A isM-regularif the multiplication morphismm 7→ am : M → M is a monomorphism. Assume (A) (cf.section 2.1) and suppose furthermore thatm is generated by a multiplicative systemS which isa cofiltered semigroup under the preorder structure(S,≻) induced by the divisibility relationin V . We say thatS is archimedean if, for alls, t ∈ S there existsn > 0 such thatsn ≻ t.Suppose thatS is archimedean and thatA is a reduced almost algebra. ThenS consists ofA-regular elements. Indeed, by hypothesis nil(A∗)

a = 0; since the annihilator ofS in A∗ is 0we get nil(A∗) = 0. Suppose thats · a = 0 for somes ∈ S anda ∈ A∗. Let t ∈ S be arbitraryand pickn > 0 such thattn ≻ s. Then(ta)n = 0 henceta = 0 for all t ∈ S, hencea = 0.


3.3. Lifting theorems. Throughout the following, the terminology “epimorphism ofV a-alge-bras” will refer to a morphism ofV a-algebras that induces an epimorphism on the underlyingV a-modules.

Lemma 3.3.1. LetA → B be an epimorphism of almostV -algebras with kernelI. LetU betheA-extension0 → I/I2 → A/I2 → B → 0. Then the assignmentf 7→ f ∗ U defines anatural isomorphism

HomB(I/I2,M)

∼ // ExalA(B,M).(3.3.2)

Proof. LetX = (0 → M → Ep→ B → 0) be anyA-extension ofB byM . The composition

g : A → Ep→ B of the structural morphism forE followed byp coincides with the projection

A→ B. Thereforeg(I) ⊂M andg(I2) = 0. Henceg factors throughA/I2; the restriction ofgto I/I2 defines a morphismf ∈ HomB(I/I

2,M) and a morphism ofA-extensionsf ∗U → X.In this way we obtain an inverse for (3.3.2).

Now consider any morphism ofA-extensions

B :

f

0 // I //

u

B //

f

B0//

f0

0

C : 0 // J // C // C0// 0.

(3.3.3)

The morphismu induces by adjunction a morphism ofC0-modules

C0 ⊗B0 I → J(3.3.4)

whose image is the idealI · C, so that the square diagram of almost algebras defined byf iscofibred (i.e.C0 ≃ C ⊗B B0) if and only if (3.3.4) is an epimorphism.

Lemma 3.3.5. Let f : B → C be a morphism ofA-extensions as above, such that the corre-sponding square diagram of almost algebras is cofibred. Thenthe morphismf : B → C is flatif and only iff0 : B0 → C0 is flat and(3.3.4)is an isomorphism.

Proof. It follows directly from the (almost version of the) local flatness criterion (see [18] Th.22.3).

We are now ready to put together all the work done so far and begin the study of deformationsof almost algebras.

The morphismu : I → J is an element in HomB0(I, J); by lemma 3.3.1 the latter group isnaturally isomorphic to ExalB(B0, J). By applying transitivity (theorem 2.4.18) to the sequence

of morphismsB → B0f0→ C0 we obtain an exact sequence of abelian groups

ExalB0(C0, J) → ExalB(C0, J) → HomB0(I, J)∂→ Ext2C0!!

(LC0/B0, J!).

Hence we can form the elementω(B, f0, u) = ∂(u) ∈ Ext2C0!!(LC0/B0 , J!). The proof of the

next result goes exactly as in [14] (III.2.1.2.3).

Proposition 3.3.6. i) Let theA-extensionB, theB0-linear morphismu : I → J and themorphism ofA-algebrasf0 : B0 → C0 be given as above. Then there exists anA-extensionCand a morphismf : B → C completing diagram(3.3.3) if and only ifω(B, f0, u) = 0. (i.e.ω(B, f0, u) is the obstruction to the lifting ofB overf0.)

ii) Assume that the obstructionω(B, f0, u) vanishes. Then the set of isomorphism classes ofA-extensionsC as in (i) forms a torsor under the groupExalB0(C0, J) (≃ Ext1C0!!

(LC0/B0 , J!)).

iii) The group of automorphisms of anA-extensionC as in (i) is naturally isomorphic toDerB0(C0, J) (≃ Ext0C0!!

(LC0/B0 , J!)).


The obstructionω(B, f0, u) depends functorially onu. More exactly, if we denote by

ω(B, f0) ∈ Ext2C0!!(LC0/B0 , (C0 ⊗B0 I)!)

the obstruction corresponding to the natural morphismI → C0 ⊗B0 I, then for any othermorphismu : I → J we have

ω(B, f0, u) = v! ω(B, f0)

wherev is the morphism (3.3.4). Taking lemma 3.3.5 into account we deduce

Corollary 3.3.7. Suppose thatB0 → C0 is flat. Theni) The classω(B, f0) is the obstruction to the existence of a flat deformation ofC0 overB,

i.e. of a B-extensionC as in (3.3.3)such thatC is flat overB andC ⊗B B0 → C0 is anisomorphism.

ii) If the obstructionω(B, f0) vanishes, then the set of isomorphism classes of flat deforma-tions ofC0 overB forms a torsor under the groupExalB0(C0, C0 ⊗B0 I).

iii) The group of automorphisms of a given flat deformation ofC0 overB is naturally iso-morphic toDerB0(C0, C0 ⊗B0 I).

Now, suppose we are given twoA-extensionsC1, C2 with morphisms ofA-extensions

B :

f i

0 // I //

ui

B //

f i

B0//

f i0

0

C i : 0 // J i // C i // C i0

// 0

and morphismsv : J1 → J2, g0 : C10 → C2

0 such that

u2 = v u1 and f 20 = g0 f

10 .(3.3.8)

We consider the problem of finding a morphism ofA-extensions

C1 :

g

0 // J1 //

v

C1 //

g

C10

//

g0

0

C2 : 0 // J2 // C2 // C20

// 0

(3.3.9)

such thatf 2 = g f 1. Let us denote bye(C i) ∈ Ext1Ci0!!(LCi

0/B, J i! ) the classes defined by the

B-extensionsC1, C2 via the isomorphism of theorem 2.4.17 and by

v∗ : Ext1C10!!(LC1

0/B, J1

! ) → Ext1C10!!(LC1

0/B, J2

! )

∗g0 : Ext1C20!!(LC2

0/B, J2

! ) → Ext1C20!!(C2

0!! ⊗C10!!LC1

0/B, J2

! )

the canonical morphisms defined byv andg0. Using the natural isomorphism

Ext1C10!!(LC1

0/B, J2

! ) ≃ Ext1C20!!(C2

0!! ⊗C10!!LC1

0/B, J2

! )

we can identify the target of bothv∗ and∗g with Ext1C10!!(LC1

0/B, J2

! ). It is clear that the problem

admits a solution if and only if theA-extensionsv ∗ C1 and C2 ∗ g0 coincide, i.e. if andonly if v ∗ e(C1) − e(C2) ∗ g0 = 0. By applying transitivity to the sequence of morphismsB → B0 → C1

0 we obtain an exact sequence

Ext1C10!!(LC1

0/B0, J2

! ) → Ext1C10!!(LC1

0/B, J2

! ) → HomC10(C1

0 ⊗B0 I, J2)


It follows from (3.3.8) that the image ofv∗e(C1)−e(C2)∗g0 in the group HomC10(C1

0⊗B0 I, J2)

vanishes, therefore

v ∗ e(C1)− e(C2) ∗ g0 ∈ Ext1C10!!(LC1

0/B0, J2

! ).(3.3.10)

In conclusion, we derive the following result as in [14] (III.2.2.2).

Proposition 3.3.11.With the above notations, the class(3.3.10)is the obstruction to the exis-tence of a morphism ofA-extensionsg : C1 → C2 as in(3.3.9)such thatf 2 = g f 1. When theobstruction vanishes, the set of such morphisms forms a torsor under the groupDerB0(C

10 , J

2)(the latter being identified withExt0C2

0!!(C2

0!! ⊗C10!!LC1

0/B0, J2

! )).

For a given almostV -algebraA, we define the categoryw.Et(A) as the full subcategory ofA-Alg consisting of all weakly etaleA-algebras. Notice that, by lemma 3.1.2(iv) all morphismsin w.Et(A) are weakly etale.

Theorem 3.3.12. i) LetA → B be a weaklyetale morphism of almost algebras. LetC be anyA-algebra andI ⊂ C a nilpotent ideal. Then the natural morphism

HomA-Alg(B,C) → HomA-Alg(B,C/I)

is bijective.ii) Let A be aV a-algebra,I ⊂ A a nilpotent ideal andA′ = A/I. Then the natural functor

w.Et(A) → w.Et(A′) (φ : A→ B) 7→ (1A′ ⊗A φ : A′ → A′ ⊗A B)

is an equivalence of categories.iii) The equivalence of (ii) restricts to an equivalenceEt(A) → Et(A′).

Proof. By induction we can assumeI2 = 0. Then (i) follows directly from proposition 3.3.11and theorem 2.4.23. We show (ii) : by corollary 3.3.7 (and again theorem 2.4.23) a givenweakly etale morphismφ′ : A′ → B′ can be lifted to auniqueflat morphismφ : A → B.We need to prove thatφ is weakly etale,i.e. thatB is B ⊗A B-flat. However, it is clear thatµB′/A′ : B′ ⊗A′ B′ → B′ is weakly etale, hence it has a flat liftingµ : B ⊗A B → C. Thenthe compositionA → B ⊗A B → C is flat and it is a lifting ofφ′. We deduce that there is anisomorphism ofA-algebrasα : B → C lifting 1B′ and moreover the morphismsb 7→ µ(b⊗ 1)andb 7→ µ(1 ⊗ b) coincide withα. Claim (ii) follows. To show (iii), suppose thatA′ → B′

is etale and letIB′/A′ denote as usual the kernel ofµB′/A′ . By corollary 3.1.8 there is a naturalmorphism of almost algebrasB′ ⊗A′ B′ → IB′/A′ which is clearly etale. HenceIB′/A′ lifts toa weakly etaleB ⊗A B-algebraC, and the isomorphismB′ ⊗A′ B′ ≃ IB′/A′ ⊕ B′ lifts to anisomorphismB ⊗A B ≃ C ⊕ B of B ⊗A B-algebras. It follows thatB is an almost projectiveB ⊗A B-module,i.e.A→ B is etale, as claimed.

We conclude with some results on deformations of almost modules. These can be establishedindependently of the theory of the cotangent complex, alongthe lines of [14] (IV.3.1.12).

We begin by recalling some notation fromloc. cit. LetR be a ring andJ ⊂ R an ideal withJ2 = 0. SetR′ = R/J ; an extension ofR-modulesM = (0 → K →M

p→ M ′ → 0) whereK

andM ′ are killed byJ , defines a natural morphism ofR′-modulesu(M) : J⊗R′M ′ → K suchthatu(M)(x ⊗m′) = xm for x ∈ J , m ∈ M andp(m) = m′. By the local flatness criterion([18] Th.22.3)M is flat overR if and only ifM ′ is flat overR′ andu(M) is an isomorphism.One can then show the following.

Proposition 3.3.13. (cp. [14] (IV.3.1.5))i) GivenR′-modulesM ′ andK and a morphismu′ : J ⊗R′ M ′ → K there exists an obstruc-

tion ω(R, u′) ∈ Ext2R′(M ′, K) whose vanishing is necessary and sufficient for the existence ofan extension ofR-modulesM ofM ′ byK such thatu(M) = u′.


ii) Whenω(R, u′) = 0, the set of isomorphism classes of such extensionsM forms a torsorunderExt1R′(M ′, K); the group of automorphisms of such an extension isHomR′(M ′, K).

Lemma 3.3.14.LetA → B be a finite morphism of almost algebras with nilpotent kernel. Letφ : M → N be anA-linear morphism and setφB = φ ⊗A 1B : M ⊗A B → N ⊗A B. Thenthere existsm ≥ 0 such that

i) AnnA(Coker(φB))m ⊂ AnnA(Coker(φ)).ii) (AnnV (Ker(φB)) · AnnV (TorA1 (B,N)) · AnnV (Coker(φ)))m ⊂ AnnA(Ker(φ)).

If B = A/I for some nilpotent idealI, andIn = 0, then we can takem = n in (i) and (ii).

Proof. Under the assumptions, we can find a finitely generatedA∗-moduleQ such thatm ·B∗ ⊂Q ⊂ B∗. By [12] (1.1.5), there exists a finite filtration0 = Jm ⊂ ... ⊂ J1 ⊂ J0 = A∗ such thateachJi/Ji+1 is a quotient of a direct sum of copies ofQ. This implies that, for everyA-moduleM , we have

AnnA(M ⊗A B)m ⊂ AnnA(M).(3.3.15)

(i) follows easily. Notice that ifB = A/I andIn = 0, then we can takem = n in (3.3.15).

For (ii) letC• = Cone(φ). We estimateH = H−1(C•L⊗AB) in two ways. By the first spectral

sequence of hyperhomology we have an exact sequence TorA1 (N,B) → H → Ker(φB). By the

second spectral sequence for hyperhomology we have an exactsequence TorA2 (Coker(φ), B) →Ker(φ)⊗A B → H. Hence Ker(φ)⊗A B is annihilated by the product of the three annihilatorsin (ii) and the result follows by applying (3.3.15) withM = Ker(φ).

Lemma 3.3.16.Keep the assumptions of lemma 3.3.14, letM be anA-module and setMB =B ⊗AM .

i) If A → B is an epimorphism,M is flat andMB is almost projective overB, thenM isalmost projective overA.

ii) If MB is an almost finitely generatedB-module thenM is an almost finitely generatedA-module.

iii) If TorA1 (B,M) = 0 andMB is almost finitely presented overB, thenM is almost finitelypresented overA.

Proof. (i) : we have to show that Ext1A(M,N) is almost zero for everyA-moduleN . Let

I = Ker(A → B); by assumptionI is nilpotent, so by the usual devissage we may assumethatI · N = 0. If χ ∈ Ext1A(M,N) is represented by an extension0 → N → Q → M → 0then after tensoring byB and using the flatness ofM we get an exact sequence ofB-modules0 → N → B ⊗A Q → MB → 0. Thusχ comes from an element of Ext1

B(MB, N) which isalmost zero by assumption.

(ii) : let m0 = (ε1, ..., εm) be a finitely generated subideal ofm. By assumption there is amapφ′ : Br → MB such thatm0 · Coker(φ′) = 0. For allj ≤ m the morphismεj · φ′ lifts to amorphismφj : Ar → M . Thenφ = φ1⊕...⊕φm : Arm →M satisfiesm2

0 ·Coker(φ⊗A1B) = 0.By lemma 3.3.14(i) it followsm2n

0 · Coker(φ) = 0 for somen ≥ 0. As m0 was arbitrary, theresult follows.

(iii) Let m0 be as above. By (ii),M is almost finitely generated overA, so we can choose amorphismφ : Ar →M such thatm0 ·Coker(φ) = 0. ConsiderφB = φ⊗A 1B : Br →MB. Bylemma 2.3.6, there is a finitely generated submoduleN of Ker(φB) containingm2

0 · Ker(φB).Notice that Ker(φ)⊗A B maps onto Ker(Br → Im(φ)⊗A B) and Ker(Im(φ)⊗A B →MB) ≃TorA1 (B,Coker(φ)) is annihilated bym0. Hencem0 · Ker(φB) is contained in the image ofKer(φ) and therefore we can lift a finite generating setx′1, ..., x

′n for m2

0 ·N to almost elementsx1, ..., xn of Ker(φ). If we quotientAr by the span of thesexi, we get a finitely presentedA-moduleF with a morphismφ : F → M such that Ker(φ ⊗A B) is annihilated bym4

0 and


Coker(φ) is annihilated bym0. By lemma 3.3.14(ii) we derivem5m0 · Ker(φ) = 0 for some

m ≥ 0. Sincem0 is arbitrary, this proves the result.

Remark 3.3.17. (i) Inspecting the proof, one sees that parts (ii) and (iii) of lemma 3.3.16 holdwhenever (3.3.15) holds. For instance, ifA → B is any faithfully flat morphism, then (3.3.15)holds withm = 1.

ii) Consequently, ifA→ B is faithfully flat andM is anA-module such thatMB is flat (resp.almost finitely generated, resp. almost finitely presented)overB, thenM is flat (resp. almostfinitely generated, resp. almost finitely presented) overA.

iii) On the other hand, we do not know whether a general faithfully flat morphismA → Bdescends almost projectivity. However, using (ii) and proposition 2.3.15 we see that if theB-moduleMB is almost finitely generated almost projective, thenM has the same property.

iv) However, ifB is faithfully flat and almost finitely presented as anA-module, thenA→ Bdoes descend almost projectivity, as can be easily deduced from lemma 2.3.23(i) and proposition2.3.15(ii).

Theorem 3.3.18.LetI be a nilpotent ideal of the almost algebraA and setA′ = A/I. Supposethat m is a (flat)V -module of homological dimension≤ 1. LetP ′ be an almost projectiveA′-module.

i) There is an almost projectiveA-moduleP withA′ ⊗A P ≃ P ′.ii) If P ′ is almost finitely presented, thenP is almost finitely presented.

Proof. As usual we reduce toI2 = 0. Then proposition 3.3.13(i) applies withR = A∗, J =I∗, R′ = A∗/I∗, M ′ = P ′

! , K = I∗ ⊗R′ P ′! andu′ = 1K . We obtain a classω(A∗, u

′) ∈Ext2R′(P ′

! , I∗ ⊗R′ P ′! ) which gives the obstruction to the existence of a flatA∗-moduleF lifting

P ′! . SinceP ′

! is almost projective, we know thatm · Ext2R′(P ′! , I∗ ⊗R′ P ′

! ) = 0, which says that0 = ε · ω(A∗, u

′) = ω(A∗, ε · u′) for all ε ∈ m. In other words, for everyε ∈ m we can

find an extension ofA∗-modulesPε of P ′! by I∗ ⊗R′ P ′

! such thatu(Pε) = ε · 1I∗⊗R′P ′!. Let

χε ∈ Ext1A∗(P ′

! , I∗ ⊗R′ P ′! ) be the class ofPε. Notice that, for anyδ ∈ m, δ · χε is the class of

an extensionX such thatu(X) = δ · u(Pε) = δ · ε · 1I∗⊗R′P ′!, hence, by proposition 3.3.13(ii),

γ · (δ · χε − χδ·ε) = 0 for all γ ∈ m. Hence we can define a morphism

χ : m⊗V m⊗V m → Ext1A∗(P ′

! , I∗ ⊗R′ P ′! ) ε⊗ δ ⊗ γ 7→ δ · γ · χε.

However, one sees easily thatm⊗V m⊗V m ≃ m andm⊗V P′! ≃ P ′

! , hence we can viewχ asan element of HomV (m,Ext1A∗

(P ′! , I∗ ⊗R′ P ′

! )) and moreover we have a spectral sequence

Epq2 = ExtpV (m,ExtqA∗

(P ′! , I∗ ⊗R′ P ′

! )) ⇒ Extp+qA∗(P ′

! , I∗ ⊗R′ P ′! )

with Epq2 = 0 for all p ≥ 2 (this spectral sequence is constructede.g. from the double complex

HomV (Fp,HomA∗(F ′

q, I∗⊗R′P ′! ))whereF• (resp.F ′

•) is a projective resolution ofm (resp.P ′! )).

In particular, ourχ is an element inE012 which therefore survives in the abutment as a class of

E01∞ . The latter can be lifted to an elementχ via the surjection Ext1A∗

(P ′! , I∗⊗R′ P ′

! ) → E01∞ . Let

0 → I∗ ⊗R′ P ′! → Q → P ′

! → 0 be an extension representingχ. Checking compatibilities, wesee thatδ · ε · χ = δ ·χε for everyε, δ ∈ m. Henceu(χ) : I∗⊗R′ P ′

! → I∗⊗R′ P ′! coincides with

the identity map on the submodulem · I∗ ⊗R′ P ′! . Sincem ·P! = P!, we see thatu(χ) is actually

the identity map. By the local flatness criterion, it then follows thatQ is flat overR, hence theA-moduleP = Qa is a flat lifting ofP ′, so it is almost projective, by lemma 3.3.16(i). Now (ii)follows from (i), lemma 3.3.16(ii) and proposition 2.3.15(i).

Remark 3.3.19. (i) According to proposition 2.1.10(ii), theorem 3.3.18 applies especially whenm is countably generated as aV -module.

(ii) For P andP ′ as in theorem 3.3.18(ii) letσP : P → P ′ be the projection. It is natural to askwhether the pair(P, σP ) is uniquely determined up to isomorphism,i.e. whether, for any other


pair (Q, σQ : Q → P ′) for which theorem 3.3.18 holds, there exists anA-linear isomorphismφ : P → Q such thatσQ φ = σP . The answer is negative in general. Consider the caseP ′ = A′. TakeP = Q = A and letσP be the natural projection, whileσQ = (u′ · 1A′) σP ,whereu′ is a unit inA′

∗. Then the uniqueness question amounts to whether every unitin A′∗

lifts to a unit ofA∗. The following counterexample is related to the fact that the completion ofthe algebraic closureQp of Qp is not maximally complete. LetV = Zp, the integral closure ofZp in Qp. ThenV is a non-discrete valuation ring of rank one, and we takem as in example2.1.1(i),A = (V/p2V )a andA′ = A/pA. Choose a compatible system of roots ofp. Analmost element ofA′ is just aV -linear morphismφ : colim

n>0p1/n!V → V/pV . Such aφ can be

represented (in a non-unique way) by an infinite series of theform∑∞

n=1 anp1−1/n! (an ∈ V ).

The meaning of this expression is as follows. For everym > 0, scalar multiplication by theelement

∑mn=1 anp

1−1/n! ∈ V defines a morphismφm : p1/m!V → V/pV . Form′ > m, letjm,m′ : p1/m!V → p1/m

′!V be the imbedding. Then we haveφm′ jm,m′ = φm, so that we candefineφ = colim

m>0φm. Similarly, every almost element ofA can be represented by an expression

of the forma0+∑∞

n=1 anp2−1/n!. Now, if σ : A→ A′ is the natural projection, the induced map

σ∗ : A∗ → A′∗ is given by:a0 +

∑∞n=1 anp

2−1/n! 7→ a0. In particular, its image is the subringV/p ⊂ (V/p)∗ = A′

∗. For instance, the unit∑∞

n=1 p1−1/n! of A′

∗ does not lie in the image of thismap.

In the light of the above remark, the best one can achieve in general is the following result.

Proposition 3.3.20.Assume (A) (see section 2.1) and keep the notation of theorem 3.3.18.Suppose moreover that(Q, σQ : Q → P ′) and (P, σP : P → P ′) are two pairs as in remark3.3.19. Then for allε ∈ m there existA-linear morphismstε : P → Q andsε : Q → P suchthat

PQ(ε)σQ tε = ε · σP σP sε = ε · σQsε tε = ε2 · 1P tε sε = ε2 · 1Q.

Proof. Since bothQ andP are almost projective andσP , σQ are epimorphisms, there existmorphismstε : P → Q andsε : Q → P such thatσQ tε = ε · σP andσP sε = ε · σQ.Then we haveσP (sε tε − ε2 · 1P ) = 0 andσQ (tε sε − ε2 1Q) = 0, i.e. the morphismuε = ε2 ·1P − sε tε (resp.vε = ε2 ·1Q− tε sε) has image contained in the almost submoduleIP (resp.IQ). SinceIm = 0 this impliesumε = 0 andvmε = 0. Hence

ε2m · 1P = (ε21P )m − umε = (m−1∑

a=0

ε2aum−1−aε ) sε tε.

Defines(2m−1)ε = (∑m−1

a=0 ε2aum−1−a

ε ) sε. Notice thats(2m−1)ε = sε (∑m−1

a=0 ε2avm−1−a

ε ).This implies the equalitiess(2m−1)ε tε = ε2m · 1P andtε s(2m−1)ε = ε2m · 1Q. Then the pair(s(2m−1)ε, ε

2(m−1) · tε) satisfiesPQ(ε2m−1). Under (A), every element ofm is a multiple of anelement of the formε2m−1, therefore the claim follows for arbitraryε ∈ m.

3.4. Descent.Faithfully flat descent in the almost setting presents no particular surprises:since the functorA 7→ A!! preserves faithful flatness of morphisms (see remark 3.1.3)manywell-known results for usual rings and modules extendverbatimto almost algebras. So forinstance, faithfully flat morphisms are of universal effective descent for the fibred categoriesF : V a-Alg.Modo → V a-Algo andG : V a-Alg.Morpho → V a-Algo (see definition 2.4.12:for an almostV -algebraB, the fibreFB (resp.GB) is the opposite of the category ofB-modules(resp.B-algebras)). Then, using remark 3.3.17, we deduce also universal effective descent forthe fibred subcategories of flat (resp. almost finitely generated, resp. almost finitely presented,resp. almost projective almost finitely generated) modules. Likewise, a faithfully flat morphism


is of universal effective descent for the fibred subcategoriesEto→ V a-Algo of etale (resp.

w.Eto→ V a-Algo of weakly etale) algebras.

More generally, since the functorA 7→ A!! preserves pure morphisms in the sense of [20],and since, by a theorem of Olivier (loc. cit.), pure morphisms are of universal effective descentfor modules, the same holds for pure morphisms of almost algebras.

Non-flat descent is more delicate. Our results are not as complete here as it could be wished,but nevertheless, they suffice for current applications (namely, for the cases needed in [6]).

Our first statement is the almost version of a theorem of Gruson and Raynaud (cp. [13] (PartII, Th.1.2.4)).

Proposition 3.4.1. A finite monomorphism of almost algebras descends flatness.

Proof. Let φ : A → B be such a morphism. Under the assumption, we can find a finiteA∗-moduleQ such thatm · B∗ ⊂ Q ⊂ B∗. One sees easily thatQ is a faithfulA∗-module, so by[13] (Part II, Th.1.2.4 and lemma 1.2.2),Q satisfies the following condition :

If (0 → N → L → P → 0) is an exact sequence ofA∗-modules withL flat, suchthat Im(N ⊗A∗

Q) is a pure submodule ofL⊗A∗Q, thenP is flat.(3.4.2)

Now letM be anA-module such thatM ⊗AB is flat. Pick an epimorphismp : F →M with Ffree overA. ThenY = (0 → Ker(p⊗A 1B) → F ⊗A B →M ⊗A B → 0) is universally exactoverB, hence overA. Consider the sequenceX = (0 → Im(Ker(p)! ⊗A∗

Q) → F! ⊗A∗Q →

M!⊗A∗Q→ 0). ClearlyXa ≃ Y . However, it is easy to check that a sequenceE ofA-modules

is universally exact if and only if the sequenceE! is universally exact overA∗. We concludethatX = (Xa)! is a universally exact sequence ofA∗-modules, hence, by condition (3.4.2),M!

is flat overA∗, i.e.M is flat overA as required.

Corollary 3.4.3. LetA → B be a finite morphism of almost algebras, with nilpotent kernel. IfC is a flatA-algebras such thatC ⊗A B is weaklyetale (resp.etale) overB, thenC is weaklyetale (resp.etale) overA.

Proof. In the weakly etale case, we have to show that the multiplication morphismµ : C ⊗A

C → C is flat. AsN = Ker(A → B) is nilpotent, the local flatness criterion reduces thequestion to the situation overA/N . So we may assume thatA→ B is a monomorphism. ThenC⊗AC → (C⊗AC)⊗AB is a monomorphism, butµ⊗C⊗AC 1(C⊗AC)⊗AB is the multiplicationmorphism ofC ⊗A B, which is flat by assumption. Therefore, by proposition 3.4.1,µ is flat.

For the etale case, we have to show thatC is almost finitely presented as aC⊗AC-module. ByhypothesisC⊗AB is almost finitely presented as aC⊗AC⊗AB-module and we know alreadythatC is flat as aC ⊗A C-module, so by lemma 3.3.16(iii) (applied to the finite morphismC ⊗A C → C ⊗A C ⊗A B) the claim follows.

Next we consider the following situation. We are given a cartesian diagram of almost algebras

A0f2 //

f1

A2

g2

A1

g1 // A3

(3.4.4)

such that one of the morphismsAi → A3 (i = 1, 2) is an epimorphism. We denote byMi

(resp.Mi,fl, resp.Mi,proj) the category of all (resp. flat, resp. almost projective)Ai-modules, fori = 0, ..., 3. Diagram (3.4.4) induces an essentially commutative diagram for the correspondingcategoriesMi, where the arrows are given by the “extension of scalars” functors. There followsa natural functor

π : M0 → M1 ×M3M2


from M0 to the 2-fibred products ofM1 andM2 over M3. Recall (cp. [1] (Ch.VII§3)) thatM1×M3

M2 is the category whose objects are the triples(M1,M2, ξ), whereMi is anAi-module(i = 1, 2) andξ : A3 ⊗A1 M1

∼→ A3⊗A2 M2 is anA3-linear isomorphism. Given such an object

(M1,M2, ξ), let us denoteM3 = A3 ⊗A2 M2; we have a natural morphismM2 → M3, andξgives a morphismM1 →M3, so we can form the fibre productT (M1,M2, ξ) =M1×M3M2. Inthis way we obtain a functorT : M1×M3

M2 → M0, and we leave to the reader the verificationthatT is right adjoint toπ. Let us denote byε : 1M0

→ T π andη : π T → 1M1×M3M2

the

unit and counit of the adjunction.

Lemma 3.4.5. The functorπ induces an equivalence of full subcategories :

X ∈ Ob(M0)|εX is an isomorphism π // Y ∈ Ob(M1 ×M3M2)|ηY is an isomorphism

havingT as essential inverse.

Proof. General nonsense.

Lemma 3.4.6. LetM be anyA0-module. ThenεM is an epimorphism. IfM is flat overA0, εMis an isomorphism.

Proof. Indeed,εM :M → (A1 ⊗A0 M)×A3⊗A0M (A2 ⊗A0 M) is the natural morphism. So, the

assertions follow by applying−⊗A0 M to the short exact sequence ofA0-modules

0 → A0f→ A1 ⊕ A2

g→ A3 → 0(3.4.7)

wheref(a) = (f1(a), f2(a)) andg(a, b) = g1(a)− g2(b).

There is another case of interest, in whichεM is an isomorphism. Namely, suppose that oneof the morphismsAi → A3 (i = 1, 2), sayA1 → A3, has a section. Then also the morphismA0 → A2 gains a sections : A2 → A0 and we have the following :

Lemma 3.4.8. In the above situation, suppose that theA0-moduleM arises by extension ofscalars from anA2-moduleM ′, via the sections : A2 → A0. ThenεM is an isomorphism.

Proof. Indeed, in this case, (3.4.7) is split exact as a sequence ofA2-modules, and it remainssuch after tensoring byM ′.

Lemma 3.4.9. η(M1,M2,ξ) is an isomorphism for all objects(M1,M2, ξ).

Proof. To fix ideas, suppose thatA1 → A3 is an epimorphism. Consider any object(M1,M2, ξ)of M1 ×M3

M2. LetM = T (M1,M2, ξ); we deduce a natural morphism

φ : (M ⊗A0 A1)×M⊗A0A3 (M ⊗A0 A2) →M1 ×M3 M2

such thatφ εM = 1M . It follows thatεM is injective, hence it is an isomorphism, by lemma3.4.6. We derive a commutative diagram with exact rows :

0 // M // (M ⊗A0 A1)⊕ (M ⊗A0 A2)

φ1⊕φ2

// M ⊗A0 A3//

φ3

0

0 // M // M1 ⊕M2// M3

// 0.

From the snake lemma we deduce(∗) Ker(φ1)⊕ Ker(φ2) ≃ Ker(φ3)(∗∗) Coker(φ1)⊕ Coker(φ2) ≃ Coker(φ3).

SinceM3 ≃M1⊗A1A3 we haveA3⊗A1 Coker(φ1) ≃ Coker(φ3). But by assumptionA1 → A3

is an epimorphism, so also Coker(φ1) → Coker(φ3) is an epimorphism. Then(∗∗) impliesthat Coker(φ2) = 0. But φ3 = 1A3 ⊗A2 φ2, thus Coker(φ3) = 0 as well. We look at the


exact sequence0 → Ker(φ1) → M ⊗A0 A1φ1→ M1 → 0 : applyingA3 ⊗A1 − we obtain an

epimorphismA3⊗A1 Ker(φ1) → Ker(φ3). From(∗) it follows that Ker(φ2) = 0. In conclusion,φ2 is an isomorphism. Hence the same is true forφ3 = 1A3 ⊗A2 φ2, and again(∗), (∗∗) showthatφ1 is an isomorphism as well, which implies the claim.

Lemma 3.4.10. If (A1 × A2)⊗A0 M is flat overA1 ×A2, thenM is flat overA0.

Proof. Suppose thatA1 → A3 is an epimorphism and letI be its kernel. LetA = A1!!×A3!!A2!!;

it suffices to show thatM! is a flat A-module. However, in view of proposition 2.3.27, theassumption implies that(A1!! × A2!!)⊗AM! is a flatA1!! × A2!!-module.I! is the kernel of theepimorphismA1!! → A3!!. Moreover,I! identifies naturally with an ideal ofA andA/I! ≃ A2!!.Then the desired conclusion follows from [7] (lemma inloc. cit.).

Proposition 3.4.11.The functorπ restricts to equivalences :

M0,fl∼→ M1,fl ×M3,fl

M2,fl

M0,proj∼→ M1,proj ×M3,proj

M2,proj.

Proof. The assertion for flat almost modules follows directly from lemmata 3.4.5, 3.4.6, 3.4.9and 3.4.10. SetB = A1 × A2. To establish the second equivalence, it suffices to show that, ifP is anA0-module such thatB ⊗A0 P is almost projective overB, thenP is almost projectiveoverA0, or which is the same, that alExti

A0(P,N) ≃ 0 for all i > 0 and anyA0-module

N . We know already thatP is flat. LetM be anyA0-module andN anyB-module. The

standard isomorphismRHomB(BL⊗A0M,N) ≃ RHomA0(M,N) yields a natural isomorphism

alExtiB(B ⊗A0 M,N) ≃ alExtiA0(M,N), whenever TorA0

j (B,M) = 0 for every j > 0. Inparticular, we have alExtiA0

(P,N) ≃ 0 wheneverN comes from either anA1-module, or anA2-module. For a generalA0-moduleN there is a 3-step filtration such that Fil0(N) = 0,gr1(N) = Fil1(N) = Ker(εN), gr2(N) = Ker(A1⊗A0N → A3⊗A0N) and gr3(N) = A2⊗A0N .By an easy devissage, we reduce to verify that alExti

A0(P, grj(N)) = 0 for every i > 0 and

j = 1, 2, 3. However, gr2(N) is anA1-module and gr3(N) is anA2-module, so the requiredvanishing follows forj = 2, 3. Moreover, applying−⊗A0 N to (3.4.7), we derive a short exactsequence :

0 → TorA01 (N,A2) →

TorA01 (N,A3)

TorA01 (N,A1)

→ gr1(N) → 0.(3.4.12)

Here again, the leftmost term of (3.4.12) is anA2-module, and the middle term is anA1-module,so the same devissage yields the sought vanishing also forj = 1.

Corollary 3.4.13. In the situation of(3.4.4), denote byAi,fl (resp. Eti, resp.w.Eti) the cate-gory of flat (resp.etale, resp. weaklyetale)Ai-algebras. The functorπ induces equivalences

A0,fl∼→ A1,fl ×A3,fl

A2,fl Et0∼→ Et1 ×Et3

Et2 w.Et0∼→ w.Et1 ×w.Et3

w.Et2.

Next we want to reinterpret the equivalences of proposition3.4.11 in terms of descent data.If F : C → V a-Algo is a fibred category over the opposite of the category of almost algebras,and if X → Y is a given morphism of almost algebras, we shall denote byDesc(C, Y/X)the category of objects of the fibre categoryFY , endowed with a descent datum relative to themorphismX → Y (cp. [9] (Ch.II §1)). In the arguments hereafter, we consider morphismsof almost algebras and modules, and one has to reverse the direction of the arrows to pass tomorphisms in the considered fibred category. Denote bypi : Y → Y ⊗X Y (i = 1, 2), resp.pij : Y ⊗X Y → Y ⊗X Y ⊗X Y (1 ≤ i < j ≤ 3) the usual morphisms. As an example,Desc(V a-Alg.Modo, Y/X) consists of the pairs(M,β) whereM is aY -module andβ is a


Y ⊗X Y -linear isomorphismβ : p∗2(M)∼→ p∗1(M) such that

p∗12(β) p∗23(β) = p∗13(β).(3.4.14)

Let nowI ⊂ X be an ideal, and setX = X/I, Y = Y/I · Y . For anyF : C → V a-Algo asabove, one has an essentially commutative diagram:

Desc(C, Y/X) //

Desc(C, Y /X)

FY // FY .

This induces a functor :

Desc(C, Y/X) → Desc(C, Y /X)×FYFY .(3.4.15)

Lemma 3.4.16.With the above notation, suppose moreover that the natural morphismI →I · Y is an isomorphism. Then the functor(3.4.15)is an equivalence wheneverC is one of thefibred categoriesV a-Alg.Modo, V a-Alg.Morpho, Et

o, w.Et

o.

Proof. For anyn > 0, denote byY ⊗n (resp.Y⊗n

) then-fold tensor product ofY (resp.Y ) withitself overX (resp.X), and byρn : Y ⊗n → Y

⊗nthe natural morphism. First of all we claim

that, for everyn > 0, the natural diagram of almost algebras

Y ⊗nρn //

µn

Y⊗n

µn

Yρ1 // Y

(3.4.17)

is cartesian (whereµn andµn aren-fold multiplication morphisms). For this, we need to verifythat, for everyn > 0, the induced morphism Ker(ρn) → Ker(ρ1) (defined by multiplicationof the first two factors) is an isomorphism. It then suffices tocheck that the natural morphismKer(ρn) → Ker(ρn−1) is an isomorphism for alln > 1. Indeed, consider the commutativediagram

I ⊗X Y⊗n−1

p // I · Y ⊗n−1 i //

ψ

Y ⊗n−1

φ⊗1Y ⊗n−1

1Y ⊗n−1

((QQQQQQQQQQQQQ

I ⊗X Y⊗n−1

p′// Ker(ρn)

i′// Y ⊗n

µY/X⊗1Y ⊗n−2

// Y ⊗n−1

FromI ·Y = φ(Y ), it follows thatp′ is an epimorphism. Hence alsoψ is an epimorphism. Sincei is a monomorphism, it follows thatψ is also a monomorphism, henceψ is an isomorphismand the claim follows easily.

We consider first the caseC = V a-Alg.Modo; we see that (3.4.17) is a diagram of the kindconsidered in (3.4.4), hence, for everyn > 0, we have the associated functorπn : Y ⊗n-Mod →

Y⊗n

-Mod ×Y -Mod Y -Mod and also its right adjointTn. Denote bypi : Y → Y⊗2

(i = 1, 2)

the usual morphisms, and similarly definepij : Y⊗2

→ Y⊗3

. Suppose there is given a de-scent datum(M,β) for M , relative toX → Y . The cocycle condition (3.4.14) implies eas-ily that µ∗

2(β) is the identity onµ∗2(p

∗i (M)) = M . It follows that the pair(β, 1M) defines

an isomorphismπ2(p∗1M)∼→ π2(p

∗2M) in the categoryY

⊗2-Mod ×Y -Mod Y -Mod. Hence

T2(β, 1M) : T2 π2(p∗1M) → T2 π2(p

∗2M) is an isomorphism. However, we remark that

either morphismpi yields a section forµ2, hence we are in the situation contemplated in lemma3.4.8, and we derive an isomorphismβ : p∗2(M)

∼→ p∗1(M). We claim that(M,β) is an ob-

ject of Desc(C, Y/X), i.e. that β verifies the cocycle condition (3.4.14). Indeed, we can


compute: π3(p∗ijβ) = (ρ∗3(p∗ijβ), µ

∗3(p

∗ijβ)) and by construction we haveρ∗3(p

∗ijβ) = p∗ij(β)

and µ∗3(p

∗ijβ) = µ∗

2(β) = 1M . Therefore, the cocycle identity forβ implies the equalityπ3(p

∗12(β)) π3(p

∗23(β)) = π3(p

∗13(β)). If we now apply the functorT3 to this equality, and

then invoke again lemma 3.4.8, the required cocycle identity for β will ensue. Clearlyβ isthe only descent datum onM lifting β. This proves that (3.4.15) is essentially surjective. Thesame sort of argument also shows that the functor (3.4.15) induces bijections on morphisms,so the lemma follows in this case. Next, the caseC = V a-Alg.Morpho can be deducedformally from the previous case, by applying repeatedly natural isomorphisms of the kindp∗i (M ⊗Y N) ≃ p∗i (M) ⊗Y ⊗XY p∗i (N) (i = 1, 2). Finally, the “etaleness” of an object ofDesc(V a-Alg.Morpho, Y/X) can be checked on its projection ontoY -Algo, hence also thecasesC = w.Et

oandC = Et

ofollow directly.

Now, letB = A1 × A2; to an objet(M,β) in Desc(V a-Alg.Modo, B/A) we assign anobject (M1,M2, ξ) of M1 ×M3

M2, as follows. SetMi = Ai ⊗B M (i = 1, 2) andAij =

Ai⊗A0Aj . We can writeB⊗A0B =∏2

i,j=1Aij andβ gives rise to theAij-linear isomorphisms

βij : Aij ⊗B⊗A0B p∗2(M)

∼→ Aij ⊗B⊗A0

B p∗1(M). In other words, we obtain isomorphismsβij : Ai ⊗A0 Mj → Mi ⊗A0 Aj. However, we have a natural isomorphismA12 ≃ A3 (indeed,suppose thatA1 → A3 is an epimorphism with kernelI; then I is also an ideal ofA0 andA0/I ≃ A2; now the claim follows by remarking thatI · A1 = I). Hence we can chooseξ = β12. In this way we obtain a functor :

Desc(V a-Alg.Modo, B/A0) → (M1 ×M3M2)

o.(3.4.18)

Proposition 3.4.19.The functor(3.4.18)is an equivalence of categories.

Proof. Let us say thatA1 → A3 is an epimorphism with kernelI. ThenI is also an idealof B and we haveB/I ≃ A3 × A2 andA0/I ≃ A2. We intend to apply lemma 3.4.16to the morphismA0 → B. However, the induced morphismB = B/I → A0 = A0/I inV a-Algo has a section, and hence it is of universal effective descentfor every fibred category.Thus, we can replace in (3.4.15) the categoryDesc(V a-Alg.Modo, B/A0) byA0-Modo, andthereby, identify (up to equivalence) the target of (3.4.15) with the 2-fibred product(M1 ×M2)

o ×(M3×M2)o Mo2. The latter is equivalent to the categoryM

o1 ×M

o3

Mo2 and the resulting

functorDesc(V a-Alg.Modo, B/A0) → Mo1 ×M

o3

Mo2 is canonically isomorphic to (3.4.18),

which gives the claim.

Putting together propositions 3.4.11 and 3.4.19 we obtain the following :

Corollary 3.4.20. In the situation of(3.4.4), the morphismA0 → A1×A2 is of effective descentfor the fibred categories of flat almost modules and of almost projective almost modules.

Next we would like to give sufficient conditions to ensure that a morphism of almost algebrasis of effective descent for the fibred categoryw.Et

o→ V a-Algo of weakly etale algebras (resp.

for etale algebras). To this aim we are led to the following :

Definition 3.4.21. A morphismφ : A → B of almost algebras is said to bestrictly finite ifKer(φ) is nilpotent andB ≃ Ra, whereR is a finiteA∗-algebra.

Theorem 3.4.22.Letφ : A→ B be a strictly finite morphism of almost algebras. Then :i) For everyA-algebraC, the induced morphismC → C ⊗A B is again strictly finite.ii) If M is a flatA-module andB ⊗A M is almost projective overB, thenM is almost

projective overA.iii) A → B is of universal effective descent for the fibred categories of weaklyetale (resp.

etale) almost algebras.


Proof. (i): suppose thatB = Ra for a finiteA∗-algebraR; thenS = C∗ ⊗A∗R is a finiteC∗-

algebra andSa ≃ C. It remains to show that Ker(C → C ⊗A B) is nilpotent. Suppose thatRis generated byn elements as anA∗-module and letFA∗

(R) (resp.FC∗(S)) be the Fitting ideal

of R (resp.ofS); we have AnnC∗(S)n ⊂ FC∗

(S) ⊂ AnnC∗(S) (see [15] (Chap.XIX Prop.2.5));

on the other handFC∗(S) = FA∗

(R) · C∗, so the claim is clear.(iii): we shall consider the fibred categoryF : w.Et

o→ V a-Algo; the same argument

applies also to etale almost algebras. We begin by establishing a very special case :

Claim 3.4.23. Assertion (iii) holds whenB = (A/I1) × (A/I2), whereI1 andI2 are ideals inA such thatI1 ∩ I2 is nilpotent.

Proof of the claim:First of all we remark that the situation considered in the claim is stableunder arbitrary base change, therefore it suffices to show thatφ is of F -2-descent in this case.Then we factorφ as a compositionA→ A/Ker(φ) → B and we remark thatA→ A/Ker(φ) isof F -2-descent by theorem 3.3.12; since a composition of morphisms ofF -2-descent is againof F -2-descent, we are reduced to show thatA/Ker(φ) → B is of F -2-descent,i.e. we canassume that Ker(φ) ≃ 0. However, in this case the claim follows easily from corollary 3.4.20.

Claim 3.4.24. More generally, assertion (iii) holds whenB =∏n

i=1A/Ii, whereI1, ..., In areideals ofA, such that

⋂ni=1 Ii is nilpotent.

Proof of the claim:We prove this by induction onn, the casen = 2 being covered by claim3.4.23. Therefore, suppose thatn > 2, and setB′ = A/(

⋂n−1i=1 Ij). By induction, the morphism

B′ →∏n−1

i=1 A/Ii is of universalF -2-descent. However, according to [9] (Chap.II Prop.1.1.3),the sieves of universalF -2-descent form a topology onV a-Algo; for this topology,A,B is acovering family ofA×B and(A→ B′ × (A/In))

o is a covering morphism, henceB′, A/In

is a covering family ofA, and then, by composition of covering families,∏n−1

i=1 A/Ii, A/Inis a covering family ofA, which is equivalent to the claim.

Now, let A → B be a general strictly finite morphism, so thatB = Ra for some fi-nite A∗-algebraR. Pick generatorsf1, ..., fm of theA∗-moduleR, and monic polynomialsp1(X), ..., pm(X) such thatpi(fi) = 0 for i = 1, ..., m.

Claim 3.4.25. There exists a finite and faithfully flat extensionC of A∗ such that the images inC[X ] of p1(X),...,pm(X) split as products of monic linear factors.

Proof of the claim:This extensionC can be obtained as follows. It suffices to find, for eachi = 1, ..., m, an extensionCi that splitspi(X), because thenC = C1 ⊗A∗

... ⊗A∗Cm will split

them all, so we can assume thatm = 1 andp1(X) = p(X); moreover, by induction on thedegree ofp(X), it suffices to find an extensionC ′ such thatp(X) factors inC ′[X ] as a productof the formp(X) = (X − α) · q(X), whereq(X) is a monic polynomial of degreedeg(p)− 1.Clearly we can takeC ′ = A∗[T ]/(p(T )).

Given aC as in claim 3.4.25, we remark that the morphismA → Ca is of universalF -2-descent. Considering again the topology of universalF -2-descent, it follows thatA → B is ofuniversalF -2-descent if and only if the same holds for the induced morphismCa → Ca ⊗A B.Therefore, in proving assertion (iii) we can replaceφ by 1C ⊗A φ and assume from start thatthe polynomialspi(X) factor inA∗[X ] as product of linear factors. Now, letdeg(pi) = di andpi(X) =

∏dij (X − αij) (for i = 1, ..., m). We get a surjective homomorphism ofA∗-algebras

D = A∗[X1, ..., Xm]/(p1(X1), ..., pm(Xm)) → R by the ruleXi 7→ fi (i = 1, ..., m). Moreover,any sequenceα = (α1,j1, α2,j2, ..., αm,jm) yields a homomorphismψα : D → A∗, determinedby the assignmentXi 7→ αi,ji. A simple combinatorial argument shows that

∏α Ker(ψα) = 0,

whereα runs over all the sequences as above. Hence the product map∏

α ψα : D →∏

αA∗ hasnilpotent kernel. We notice that theA∗-algebra(

∏αA∗)⊗D R is a quotient of

∏αA∗, hence it


can be written as a product of rings of the formA∗/Iα, for various idealsIα. By (i), the kernelof the induced homomorphismR →

∏αA∗/Iα is nilpotent, hence the same holds for the kernel

of the compositionA →∏

αA/Iaα, which is therefore of the kind considered in claim 3.4.24.

HenceA →∏

αA/Iaα is of universalF -2-descent. Since such morphisms form a topology, it

follows that alsoA→ B is of universalF -2-descent, which concludes the proof of (iii).Finally, letM be as in (ii) and pick againC as in the proof of claim 3.4.25. By remark

3.3.17(iv),M is almost projective overA if and only ifCa ⊗AM is almost projective overCa;hence we can replaceφ by 1Ca ⊗A φ, and by arguing as in the proof of (iii), we can assumefrom start thatB =

∏nj=1(A/Ij) for idealsIj ⊂ A, j = 1, ..., n such thatI =

⋂nj=1 Ij is

nilpotent. By an easy induction, we can furthermore reduce to the casen = 2. We factorφ asA → A/I → B; by proposition 3.4.11 it follows that(A/I) ⊗A M is almost projective overA/I, and then lemma 3.3.16(i) says thatM itself is almost projective.

Remark 3.4.26. It is natural to ask whether theorem 3.4.22 holds if we replace everywhere“strictly finite” by “finite with nilpotent kernel” (or even by “almost finite with nilpotent ker-nel”). We do not know the answer to this question.

We conclude with a digression to explain the relationship between our results and relatedfacts that can be extracted from the literature. So, we now place ourselves in the “classicallimit” V = m (cp. example 2.1.1(ii)). In this case, weakly etale morphisms had already beenconsidered in some earlier work, and they were called “absolutely flat” morphisms. A ringhomomorphismA → B is etale in the usual sense of [10] if and only if it is absolutely flat andof finite presentation. Let us denote byu.Et

othe fibred category overV -Algo, whose fibre

over aV -algebraA is the opposite of the category of etaleA-algebras in the usual sense. Weclaim that, if a morphismA → B of V -algebras is of universal effective descent for the fibredcategoryw.Et

o(resp. Et

o), then it is a morphism of universal effective descent foru.Et

o.

Indeed, letC be an etaleA-algebra (in the sense of definition 3.1.1) and such thatC ⊗A B isetale overB in the usual sense. We have to show thatC is etale in the usual sense,i.e. that it isof finite presentation overA. This amounts to showing that, for every filtered inductive system(Aλ)λ∈Λ of A-algebras, we have colim

λ∈ΛHomA-Alg(C,Aλ) ≃ HomA-Alg(C, colim

λ∈ΛAλ). Since, by

assumption, this is known after extending scalars toB and toB ⊗A B, it suffices to show that,for anyA-algebraD, the natural sequence

HomA-Alg(C,D) // HomB-Alg(CB, DB)//// HomB⊗AB-Alg(CB⊗AB, DB⊗AB)

is exact. For this, note that HomA-Alg(C,D) = HomD-Alg(CD, D) (and similarly for the otherterms) and by hypothesis(D → D ⊗A B)o is a morphism of 1-descent for the fibred categoryw.Et

o(resp.Et

o).

As a consequence of these observations and of theorem 3.4.22, we see that any finite ringhomomorphismφ : A→ B with nilpotent kernel is of universal effective descent forthe fibredcategory of etale algebras. This fact was known as follows.By [10] (Exp.IX, 4.7), Spec(φ) isof universal effective descent for the fibred category of separated etale morphisms of finite type.One has to show that ifX is such a scheme overA, such thatX⊗AB is affine, thenX is affine.This follows by reduction to the noetherian case and [5] (Chap.II, 6.7.1).

3.5. Behaviour of etale morphisms under Frobenius.We consider the following categoryBof base rings. The objects ofB are the pairs(V,m), whereV is a ring andm is an ideal ofVwith m = m2 andm is flat. The morphisms(V,mV ) → (W,mW ) between two objects ofB arethe ring homomorphismsf : V →W such thatmW = f(mV ) ·W .

We have a fibred and cofibred categoryB-Mod → B (see [10] (Exp.VI§5,6,10) for gener-alities on fibred categories). An object ofB-Mod (which we may call a “B-module”) consistsof a pair((V,m),M), where(V,m) is an object ofB andM is aV -module. Given two objects


X = ((V,mV ),M) andY = ((W,mW ), N), the morphismsX → Y are the pairs(f, g), wheref : (V,mV ) → (W,mW ) is a morphism inB andg :M → N is anf -linear map.

Similarly one has a fibred and cofibred categoryB-Alg → B of B-algebras. We will alsoneed to consider the fibred and cofibred categoryB-Mon → B of non-unitary commutativeB-monoids: an object ofB-Mon is a pair((V,m), A) whereA is aV -module endowed witha morphismA ⊗V A → A subject to associativity and commutativity conditions, asdiscussedin section 2.2. The fibre over an object(V,m) of B, is the category ofV -monoids denoted(V,m)-Mon or simplyV -Mon.

The almost isomorphisms in the fibres ofB-Mod → B give a multiplicative systemΣin B-Mod, admitting a calculus of both left and right fractions. The “locally small” con-ditions are satisfied (see [22] p.381), so that one can form the localised categoryBa-Mod =Σ−1(B-Mod). The fibres of the localised category over the objects ofB are the previously con-sidered categories of almost modules. Similar considerations hold forB-Alg andB-Mon, andwe get the fibred and cofibred categoriesB

a-Mod → B, Ba-Alg → B andB

a-Mon → B.In particular, for every object(V,m) of B, we have an obvious notion of almostV -monoid andthe category consisting of these is denotedV a-Mon. The localisation functors

B-Mod → Ba-Mod : M 7→Ma

B-Alg → Ba-Alg : B 7→ Ba

have left and right adjoints. These adjoints can be chosen asfunctors of categories overBsuch that the adjunction units and counits are morphisms over identity arrows inB. On the fibresthese induce the previously considered left and right adjointsM 7→ M!, M 7→ M∗, B 7→ B!!,B 7→ B∗. We will use the same notation for the corresponding functors on the larger categories.Then it is easy to check that the functorM 7→ M! is cartesian and cocartesian (i.e. it sendscartesian arrows to cartesian arrows and cocartesian arrows to cocartesian arrows),M 7→ M∗

andB 7→ B∗ are cartesian, andB 7→ B!! is cocartesian.Let B/Fp be the full subcategory ofB consisting of all objects(V,m) whereV is anFp-

algebra. Define similarlyB-Alg/Fp, B-Mon/Fp andBa-Alg/Fp, B

a-Mon/Fp, so that wehave again fibred and cofibred categoriesB

a-Alg/Fp → B/Fp andBa-Alg/Fp → B/Fp (resp.

the same for non-unitary monoids). We remark that the categoriesBa-Alg/Fp andB

a-Mon/Fphave small limits and colimits, and these are preserved by the projection toB/Fp. Especially, ifA→ B andA→ C are two morphisms inBa-Alg/Fp or B

a-Mon/Fp, we can defineB⊗A Cas such a colimit.

If A is a (unitary or non-unitary)B-monoid overFp, we denote byφA : A → A the Frobe-nius endomorphismx 7→ xp. If (V,m) is an object ofB/Fp, it follows from proposition2.1.5(ii) thatφV : (V,m) → (V,m) is a morphism inB. If B is an object ofB-Alg/Fp(resp. B-Mon/Fp) over V , then the Frobenius map induces a morphismφB : B → B inB-Alg/Fp (resp.B-Mon/Fp) overφV . In this way we get a natural transformation from theidentity functor ofB-Alg/Fp (resp.B-Mon/Fp) to itself that induces a natural transformationon the identity functor ofBa-Alg/Fp (resp.Ba-Mon/Fp).

Using the pull-back functors, any objectB of B-Alg overV defines new objectsB(m) ofB-Alg (m ∈ N) overV , whereB(m) = (φmV )

∗(B), which is justB considered as aV -algebra

via the homomorphismVφm

−→ V → B. These operations also induce functorsB 7→ B(m) onalmostB-algebras.

Definition 3.5.1. i) Let (V,m) be an object ofB/Fp; we say that a morphismf : A → B ofalmostV -algebras (resp. almostV -monoids) isinvertible up toφm if there exists a morphismf ′ : B → A in B

a-Alg (resp.Ba-Mon) overφmV , such thatf ′ f = φmA andf f ′ = φmB .ii) We say that an almostV -monoidI (e.g.an ideal in aV a-algebra) isFrobenius nilpotentif

φI is nilpotent.

Notice that a morphismf of V a-Alg (or V a-Mon) is invertible up toφm if and only iff∗ : A∗ → B∗ is so as a morphism ofFp-algebras.


Lemma 3.5.2. Let (V,m) be an object ofB/Fp and letf : A → B, g : B → C be morphismsof almostV -algebras or almostV -monoids.

i) If f is invertible up toφn andg is invertible up toφm, theng f is invertible up toφm+n.ii) If f is invertible up toφn andg f is invertible up toφm, theng is invertible up toφm+n.iii) If g is invertible up toφn andg f is invertible up toφm, thenf is invertible up toφm+n.iv) The Frobenius morphisms induceφV -linear morphisms (i.e. morphisms inBa-Mod over

φV ) φ′ : Ker(f) → Ker(f) andφ′′ : Coker(f) → Coker(f), andf is invertible up to somepower ofφ if and only if bothφ′ andφ′′ are nilpotent.

v) Consider a map of short exact sequences of almostV -monoids :

0 // A′ //

f ′

A //

f

A′′ //

f ′′

0

0 // B′ // B // B′′ // 0

and suppose that two of the morphismsf ′, f, f ′′ are invertible up to a power ofφ. Then also thethird morphism has this property.

Proof. (i): if f ′ is an inverse off up toφn andg′ is an inverse ofg up toφm, thenf ′ g′ is aninverse ofg f up toφm+n. (ii): given an inversef ′ of f up toφn and an inverseh′ of h = g fup toφm, let g′ = φnB f h′. We compute :

g g′ = g φnB f h′ = φnC g f h = φnC φmCg′ g = φnB f h′ g = f h′ g φnB = f h′ g f f ′

= f φmA f ′ = φmB f f ′ = φmB φnB.

(iii) is similar and (iv) is an easy diagram chasing left to the reader. (v) follows from (iv) andthe snake lemma.

Lemma 3.5.3. Let (V,m) be an object ofB/Fp.(i) If f : A → B is a morphism of almostV -algebras, invertible up toφn, then so isA′ →

A′ ⊗A B for every morphismA→ A′ of almost algebras.(ii) If f : (V,mV ) → (W,mW ) is a morphism inB/Fp, the functorsf∗ : (V,mV )

a-Alg →(W,mW )a-Alg and f ∗ : (W,mW )a-Alg → (V,mV )

a-Alg preserve the class of morphismsinvertible up toφn.

Proof. (i): givenf ′ : B → A(m), construct a morphismA′ ⊗A B → A′(m) using the morphism

A′ → A′(m) coming fromφmA′ andf ′. (ii): the assertion forf ∗ is clear, and the assertion forf∗

follows from (i).

Remark 3.5.4. Statements like those of lemma 3.5.3 hold for the classes of flat, (weakly) un-ramified, (weakly) etale morphisms.

Theorem 3.5.5.Let (V,m) be an object ofB/Fp andf : A → B a weaklyetale morphism ofalmostV -algebras.

(i) If f is invertible up toφn (n ≥ 0), then it is an isomorphism.(ii) For every integerm ≥ 0 the natural square diagram

Af //

φmA

B

φmB

A(m)

f(m) // B(m)

(3.5.6)

is cocartesian.


Proof. (i): we first show thatf is faithfully flat. Sincef is flat, it remains to show that ifMis anA-module such thatM ⊗A B = 0, thenM = 0. It suffice to do this forM = A/I, foran arbitrary idealI of A. After base change byA → A/I, we reduce to show thatB = 0impliesA = 0. However,A∗ → B∗ is invertible up toφn, soφnA∗

= 0 which meansA∗ = 0. Inparticular,f is a monomorphism, hence the proof is complete in case thatf is an epimorphism.

In general, consider the compositionB1B⊗f−→ B ⊗A B

µB/A−→ B. From lemma 3.5.3(i) it follows

that1B⊗f is invertible up toφn; then lemma 3.5.2(ii) says thatµB/A is invertible up toφn. Thelatter is also weakly etale; by the foregoing we derive thatit is an isomorphism. Consequently1B ⊗ f is an isomorphism, and finally, by faithful flatness,f itself is an isomorphism.

(ii): the morphismsφmA andφmB are invertible up toφm. By lemma 3.5.3(i) it follows that1B ⊗ φmA : B → B ⊗A A(m) is invertible up toφm; hence, by lemma 3.5.2(ii), the morphismh : B ⊗A A(m) → B(m) induced byφmB andf(m) is invertible up toφ2m (in fact one verifies thatit is invertible up toφm). But h is a morphism of weakly etaleA(m)-algebras, so it is weaklyetale, so it is an isomorphism by (i).

Remark 3.5.7. Theorem 3.5.5(ii) extends a statement of Faltings ([6] p.10) for his notion ofalmost etale extensions.

We recall (cp. [9] (Chap.0, 3.5)) that a morphismf : X → Y of objects in a site is calledbicoveringif the induced map of associated sheaves of sets is an isomorphism; if f is squarable(“quarrable” in French), this is equivalent to the condition that bothf and the diagonal mor-phismX → X ×Y X are covering morphisms.

Let F → E be a fibered category andf : P → Q a squarable morphism ofE. Consider thefollowing condition:

for every base changeP×QQ′ → Q′ of f , the inverse image functorFQ′ → FP×QQ′

is an equivalence of categories.(3.5.8)

Inspecting the arguments in [9] (Chap.II,§1.1) one can show:

Lemma 3.5.9. With the above notation, letτ be the topology of universal effective descentrelative toF → E. Then we have :

i) if (3.5.8)holds, thenf is a covering morphism for the topologyτ ;ii) f is bicovering forτ if and only if (3.5.8)holds both forf and for the diagonal morphism

P → P ×Q P .

Remark 3.5.10. In [9] (Chap.II, 1.1.3(iv)) it is stated that “la reciproque est vraie sii = 2”,meaning that (3.5.8) is equivalent to the condition thatf is bicovering forτ . (Actually the citedstatement is given in terms of presheaves, but one can show that (3.5.8) is equivalent to thecorresponding condition for the fibered categoryF+ → EU considered inop.cit.) However,this fails in general : as a counterexample we can give the following. LetE be the category ofschemes of finite type over a fieldk; setP = A1

k, Q = Spec(k). Finally letF → E be thediscretely fibered category defined by the presheafX 7→ H0(X,Z). Then it is easy to showthatf satisfies (3.5.8) but the diagonal map does not, sof is not bicovering. The mistake in theproof is in [9] (Chap.II, 1.1.3.5), where one knows thatF+(d) is an equivalence of categories(notation ofloc.cit.) but one needs it also after base changes ofd.

Lemma 3.5.11. (i) If f : A → B is a morphism ofV a-algebras which is invertible up toφm,then the induced functorsEt(A) → Et(B) andw.Et(A) → w.Et(B) are equivalences ofcategories.

ii) If A → B is weaklyetale andC → D is a morphism ofA-algebras invertible up toφm,then the induced mapHomA-Alg(B,C) → HomA-Alg(B,D) is bijective.

Proof. We first consider (i) for the special case wheref = φmA : A → A(m). The functor(φmV )

∗ : V a-Alg → V a-Alg induces a functor(−)(m) : A-Alg → A(m)-Alg, and by restriction


(see remark 3.5.3) we obtain a functor(−)(m) : Et(A) → Et(A(m)); by theorem 3.5.5(ii), thelatter is isomorphic to the functor(φm)∗ : Et(A) → Et(A(m)) of the lemma. Furthermore,from remark 2.1.3(ii) and (2.2.2) we derive a natural ring isomorphismω : A(m)∗ ≃ A∗, hencean essentially commutative diagram

Et(A) //

(φm)∗

A-Algα //

(−)(m)

(A∗,m ·A∗)a-Alg

ω∗

Et(A(m)) // A(m)-Algβ // (A(m)∗,m ·A(m)∗)

a-Alg

whereα andβ are the equivalences of remark 2.2.3. Clearlyα andβ restrict to equivalences onthe corresponding categories of etale algebras, hence thelemma follows in this case.

For the general case of (i), letf ′ : B → A(m) be a morphism as in definition 3.5.1. Diagram(3.5.6) induces an essentially commutative diagram of the corresponding categories of algebras,so by the previous case, the functor(f ′)∗ : Et(B) → Et(A(m)) has both a left essential inverseand a right essential inverse; these essential inverses must be isomorphic, sof∗ has an essentialinverse as desired. Finally, we remark that the map in (ii) isthe same as the map HomC-Alg(B⊗A

C,C) → HomD-Alg(B ⊗A D,D), and the latter is a bijection in view of (i).

Remark 3.5.12. Notice that lemma 3.5.11(ii) generalises the lifting theorem 3.3.12(i) (in caseV is anFp-algebra). Similarly, it follows from lemmata 3.5.11(i) and 3.5.2(iv) that, in caseVis anFp-algebra, one can replace “nilpotent” in theorem 3.3.12 parts (ii) and (iii) by “Frobeniusnilpotent”.

In the followingτ will denote indifferently the topology of universal effective descent definedby either of the fibered categoriesw.Et

o→ V a-Algo or Et

o→ V a-Algo.

Proposition 3.5.13. If f : A → B is a morphism of almostV -algebras which is invertible upto φm, thenf o is bicovering for the topologyτ .

Proof. In light of lemmata 3.5.9(ii) and 3.5.11(i), it suffices to show thatµB/A is invertible up

to a power ofφ. For this, factor the identity morphism ofB asB1B⊗f−→ B ⊗A B

µB/A−→ B and

argue as in the proof of theorem 3.5.5.

Proposition 3.5.14.LetA→ B be a morphism of almostV -algebras andI ⊂ A an ideal. SetA = A/I andB = B/I · B. Suppose that either

a) I → I · B is an epimorphism with nilpotent kernel, orb) V is anFp-algebra andI → I ·B is invertible up to a power ofφ.Then we have :i) conditions (a) and (b) are stable under any base changeA→ C.ii) (A→ B)o is covering (resp. bicovering) forτ if and only if(A→ B)o is.

Proof. Suppose first thatI → I · B is an isomorphism; in this case we claim thatI · C →I · (C ⊗A B) is an epimorphism and Ker(I · C → I · (C ⊗A B))2 = 0 for anyA-algebraC.Indeed, since by assumptionI ≃ I · B, C ⊗A B acts onC ⊗A I, hence Ker(C → C ⊗A B)annihilatesC ⊗A I, hence annihilates its imageI · C, whence the claim. If, moreover,V is anFp-algebra, lemma 3.5.2(iv) implies thatI ·C → I · (C ⊗A B) is invertible up to a power ofφ.

In the general case, consider the intermediate almostV -algebraA1 = A×B B equipped withthe idealI1 = 0 ×B (I · B). In case (a),I1 = I · A1 andA → A1 is an epimorphism withnilpotent kernel, hence it remains such after any base changeA→ C. To prove (i) in case (a), itsuffices then to consider the morphismA1 → B, hence we can assume from start thatI → I ·Bis an isomorphism, which is the case already dealt with. To prove (i) in case (b), it suffices toconsider the cases of(A, I) → (A1, I1) and(A1, I1) → (B, I · B). The second case is treated


above. In the first case, we do not necessarily haveI1 = I · A1 and the assertion to be checkedis that, for everyA-algebraC, the morphismI ·C → I1 · (A1 ⊗A C) is invertible up to a powerof φ. We apply lemma 3.5.2(v) to the commutative diagram with exact rows:

0 // I //

A //

A/I // 0

0 // I · B // A1// A/I // 0

to deduce thatA → A1 is invertible up to some power ofφ, hence so isC → A1 ⊗A C, whichimplies the assertion.

As for (ii), we remark that the “only if” part is trivial; and we assume therefore that(A→ B)o

is τ -covering (resp.τ -bicovering). Consider first the assertion for “covering”.We need to showthat(A → B)o is of universal effective descent forF , whereF is either one of our two fiberedcategories. In light of (i), this is reduced to the assertionthat(A → B)o is of effective descentfor F . We notice that(A → A1)

o is bicovering forτ (in case (a) by theorem 3.3.12 andlemma 3.5.9(ii), in case (b) by proposition 3.5.13). As(A → A1/I1)

o is an isomorphism, theassertion is reduced to the case whereI → I · B is an isomorphism. In this case, by lemma3.4.16, there is a natural equivalence:Desc(F,B/A)

∼→ Desc(F,B/A) ×FB

FB. Then theassertion follows easily from corollary 3.4.13. Finally suppose that(A → B)o is bicovering.The foregoing already says that(A → B)o is covering, so it remains to show that(B ⊗A B →B)o is also covering. The above argument again reduces to the case whereI → I · B is anisomorphism. Then, as in the proof of lemma 3.4.16, the induced morphismI ·(B⊗AB) → I ·Bis an isomorphism as well. Thus the assertion for “bicovering” is reduced to the assertion for“covering”.

4. APPENDIX

4.1. In this appendix we have gathered a few miscellaneous results that were found in thecourse of our investigation, and which may be useful for other applications.

We need some preliminaries on simplicial objects : first of all, a simplicial almost algebrais just an object in the categorys.(V a-Alg). Then for a given simplicial almost algebraAwe have the categoryA-Mod of A-modules : it consists of all simplicial almostV -modulesM such thatM [n] is anA[n]-module and such that the face and degeneracy morphismsdi :M [n] → M [n − 1] andsi : M [n] → M [n + 1] (i = 0, 1, ..., n) areA[n]-linear. We will needalso the derived category ofA-modules; it is defined as follows.

A bit more generally, letC be any abelian category. For an objectX of s.C let N(X) bethe normalized chain complex (defined as in [14] (I.1.3)). Bythe theorem of Dold-Kan ([22]th.8.4.1)X 7→ N(X) induces an equivalenceN : s.C → C•(C). Now we say that a morphismX → Y in s.C is a quasi-isomorphismif the induced morphismN(X) → N(Y ) is a quasi-isomorphism of chain complexes.

In the following we fix a simplicial almost algebraA.

Definition 4.1.1. We say thatA is exactif the almost algebrasA[n] are exact for alln ∈ N.A morphismφ : M → N of A-modules (orA-algebras) is aquasi-isomorphismif the mor-phismφ of underlying simplicial almostV -modules is a quasi-isomorphism. We define thecategoryD•(A) (resp. the categoryD•(A-Alg)) as the localization of the categoryA-Mod

(resp.A-Alg) with respect to the class of quasi-isomorphisms.

As usual, the morphisms inD•(A) can be computed via a calculus of fraction on the categoryHot•(A) of simplicial complexes up to homotopy. Moreover, ifA1 andA2 are two simplicial


almost algebras, then the “extension of scalars” functors define equivalences of categories

D•(A1 × A2)∼

−→ D•(A1)× D•(A2)

D•(A1 × A2-Alg)∼

−→ D•(A1-Alg)× D•(A2-Alg).

Proposition 4.1.2. (i) The functor onA-algebras given byB 7→ (s.V a ×B)!! preserves quasi-isomorphisms and therefore induces a functorD•(A-Alg) → D•((s.V

a × A)!!-Alg).(ii) The localisation functorR 7→ Ra followed by “extension of scalars” vias.V a × A→ A

induces a functorD•((s.Va × A)!!-Alg) → D•(A-Alg) and the composition of this and the

above functor is naturally isomorphic to the identity functor on D•(A-Alg).

Proof. (i) : let B → C be a quasi-isomorphism ofA-algebras. Clearly the induced morphisms.V a × B → s.V a × C is still a quasi-isomorphism ofV -algebras. But by remark 2.2.13,s.V a × B and s.V a × C are exact simplicial almostV -algebras; moreover, it follows fromcorollary 2.2.10 that(s.V a × B)! → (s.V a × C)! is a quasi-isomorphism ofV -modules. Thenthe claim follows easily from the exactness of the sequence (2.2.11). Now (ii) is clear.

Remark 4.1.3. In casem is flat, then allA-algebras are exact, and the same argument showsthat the functorB 7→ B!! induces a functorD•(A-Alg) → D•(A!!-Alg). In this case, composi-tion with localisation is naturally isomorphic to the identity functor onD•(A-Alg).

Proposition 4.1.4. Let f : R → S be a map ofV -algebras such thatfa : Ra → Sa is anisomorphism. ThenLaS/R ≃ 0 in D•(s.S

a).

Proof. We show by induction onq that

VAN (q;S/R) Hq(LaS/R) = 0.

Forq = 0 the claim follows immediately from [14] (II.1.2.4.2). Therefore suppose thatq > 0and thatVAN (j;D/C) is known for all almost isomorphisms ofV -algebrasC → D and allj < q. LetR = f(R). Then by transitivity ([14] (II.2.1.2)) we have a distinguished triangle inD•(s.S

a)

(S ⊗R LR/R)a u // LaS/R

v // LaS/R

// σ(S ⊗R LR/R)a.

We deduce thatVAN (q;R/R) andVAN (q;S/R) imply VAN (q;S/R), thus we can assume thatf is either injective or surjective. LetS• → S be the simplicialV -algebra augmented overS defined byS• = PV (S). It is a simplicial resolution ofS by freeV -algebras, in particularthe augmentation is a quasi-isomorphism of simplicialV -algebras. SetR• = S• ×S R. Thisis a simplicialV -algebra augmented overR via a quasi-isomorphism. Moreover, the inducedmorphismsR[n]a → S[n]a are isomorphisms. By [14] (II.1.2.6.2) there is a quasi-isomorphismLS/R ≃ L∆

S•/R•. On the other hand we have a spectral sequence

E1ij = Hj(LS[i]/R[i]) ⇒ Hi+j(L

∆S•/R•

).

It follows easily thatVAN(j;S[i]/R[i]) for all i ≥ 0, j ≤ q impliesVAN(q;S/R). Thereforewe are reduced to the case whereS is a freeV -algebra andf is either injective or surjective.We examine separately these two cases. Iff : R → V [T ] is surjective, then we can find a rightinverses : V [T ] → R for f . By applying transitivity to the sequenceV [T ] → R → V [T ] weget a distinguished triangle

(V [T ]⊗R LR/V [T ])a u→ LaV [T ]/V [T ]

v→ LaV [T ]/R → σ(V [T ]⊗R LR/V [T ])

a.

SinceLaV [T ]/V [T ] ≃ 0 there follows an isomorphism :Hq(LV [T ]/R)a ≃ Hq−1(V [T ]⊗RLR/V [T ])

a.Furthermore, sincefa is an isomorphism,sa is an isomorphism as well, hence by induction (andby a spectral sequence of the type [14] (I.3.3.3.2))Hq−1(V [T ] ⊗R LR/V [T ])

a ≃ 0. The claimfollows in this case.


Finally suppose thatf : R → V [T ] is injective. WriteV [T ] = Sym(F ), for a freeV -moduleF and setF = m⊗V F ; sincefa is an isomorphism, Im(Sym(F ) → Sym(F )) ⊂ R. We applytransitivity to the sequence Sym(F ) → R → Sym(F ). By arguing as above we are reduced toshowing thatLa

Sym(F )/Sym(F )≃ 0. We know thatH0(L

aSym(F )/Sym(F )

) ≃ 0 and we will show that

Hq(LaSym(F )/Sym(F )

) ≃ 0 for q > 0. To this purpose we apply transitivity to the sequenceV →

Sym(F ) → Sym(F ). AsF andF are flatV -modules, [14] (II.1.2.4.4) yieldsHq(LSym(F )/V ) ≃

Hq(LSym(F )/V ) ≃ 0 for q > 0 andH0(LSym(F )/V ) is a flat Sym(F )-module. In particularHj(Sym(F )⊗Sym(F ) LSym(F )/V ) ≃ 0 for all j > 0. ConsequentlyHj+1(LSym(F )/Sym(F )) ≃ 0 forall j > 0 andH1(LSym(F )/Sym(F )) ≃ Ker(Sym(F )⊗Sym(F ) ΩSym(F )/V → ΩSym(F )/V ). The lattermodule is easily seen to be almost zero.

Theorem 4.1.5.Let φ : R → S be a map of simplicialV -algebras inducing an isomorphismRa ∼

→ Sa in D•(Ra). Then(L∆

S/R)a ≃ 0 in D•(S

a).

Proof. Apply the base change theorem ([14] II.2.2.1) to the (flat) projections ofs.V × R ontoR and respectivelys.V to deduce that the natural mapL∆

s.V×S/s.V×R → L∆S/R ⊕ L∆

s.V/s.V →

L∆S/R is a quasi-isomorphism inD•(s.V × S). By proposition 4.1.2 the induced morphism

(s.V × R)a!! → (s.V × S)a!! is still a quasi-isomorphism. There are spectral sequences

E1ij = Hj(L(V×R[i])/(V ×R[i])a!!

) ⇒ Hi+j(L∆(s.V×R)/(s.V ×R)a!!

)

F 1ij = Hj(L(V ×S[i])/(V×S[i])a!!

) ⇒ Hi+j(L∆(s.V×S)/(s.V×S)a!!

).

On the other hand, by proposition 4.1.4 we haveLa(V×R[i])/(V ×R[i])a!!≃ 0 ≃ La(V ×S[i])/(V×S[i])a!!

for all i ∈ N. Then the theorem follows directly from [14] (II.1.2.6.2(b)) and transitivity.

Proposition 4.1.6. LetA → B be a morphism of exact almostV -algebras. Then the naturalmapm⊗V LB!!/A!!

→ LB!!/A!!is a quasi-isomorphism.

Proof. By transitivity we may assumeA = V a. LetP• = PV (B!!) be the standard resolution ofB!! (see [14] II.1.2.1). EachP [n]a containsV as a direct summand, hence it is exact, so that wehave an exact sequence of simplicialV -modules0 → s.m → s.V ⊕ (P a

• )! → (P a• )!! → 0. The

augmentation(P a• )! → (Ba

!!)! ≃ B! is a quasi-isomorphism and we deduce that(P a• )!! → B!!

is a quasi-isomorphism; hence(P a• )!! → P• is a quasi-isomorphism as well. We haveP [n] ≃

Sym(Fn) for a freeV -moduleFn and the map(P [n]a)!! → P [n] is identified with Sym(m ⊗V

Fn) → Sym(Fn), whenceΩP [n]a!!/V⊗P [n]a!!

P [n] → ΩP [n]/V is identified withm ⊗V ΩP [n]/V →

ΩP [n]/V . By [14] (II.1.2.6.2) the mapL∆(P a

• )!!/V→ L∆

P•/Vis a quasi-isomorphism. In view of [14]

(II.1.2.4.4) we derive thatΩ(P a• )!!/V → ΩP•/V is a quasi-isomorphism,i.e. m⊗V ΩP•/V → ΩP•/V

is a quasi-isomorphism. Sincem is flat andΩP•/V → ΩP•/V ⊗P•B!! = LB!!/V is a quasi-

isomorphism, we get the desired conclusion.

In view of proposition 4.1.4 we haveLa(V a×A)!!/V×A!!≃ 0 in D•(V

a×A). By this, transitivityand localisation ([14] II.2.3.1.1) we derive thatLaB/A → LaB!!/A!!

is a quasi-isomorphism for allA-algebrasB. If A andB are exact (e.g. if m is flat), we conclude from proposition 4.1.6 thatthe natural mapLB/A → LB!!/A!!

is a quasi-isomorphism.Finally we want to discuss left derived functors of (the almost version of) some notable non-

additive functors that play a role in deformation theory. Let R be a simplicialV -algebra. Thenwe have an obvious functorG : D•(R) → D•(R

a) obtained by applying dimension-wise thelocalisation functor. LetΣ be the multiplicative set of morphisms ofD•(R) that induce almostisomorphisms on the cohomology modules. An argument as in section 2.3 shows thatG inducesan equivalence of categoriesΣ−1D•(R) → D•(R

a).Now let R be aV -algebra andFp one of the functors⊗p, Λp, Symp, Γp defined in [14]

(I.4.2.2.6).


Lemma 4.1.7. Let φ : M → N be an almost isomorphism ofR-modules. ThenFp(φ) :Fp(M) → Fp(N) is an almost isomorphism.

Proof. Let ψ : m ⊗V N → M be the map corresponding to(φa)−1 under the bijection (2.2.2).By inspection, the compositionsφ ψ : m ⊗V N → N andψ (1

m⊗ φ) : m ⊗V M → M

are induced by scalar multiplication. Pick anys ∈ m and lift it to an elements ∈ m; defineψs : N →M by n 7→ ψ(s⊗ n) for all n ∈ N . Thenφ ψs = s · 1N andψs φ = s · 1M . Thiseasily implies thatsp annihilates KerFp(φ) and CokerFp(φ). In light of proposition 2.1.5(ii),the claim follows.

Let B be an almostV -algebra. We define a functorFap on B-Mod by M 7→ (Fp(M!))

a,whereM! is viewed as aB!!-module or aB∗-module (to show that these choices define the samefunctor it suffices to observe thatB∗ ⊗B!!

N ≃ N for all B∗-modulesN such thatN = m ·N).For allp > 0 we have diagrams :

R-ModFp //

R-Mod

Ra-Mod

Fap //

OO

Ra-Mod

OO(4.1.8)

where the downward arrows are localisation and the upward arrows are the functorsM 7→M!. Lemma 4.1.7 implies that the downward arrows in the diagramcommute (up to a naturalisomorphism) with the horizontal ones. It will follow from the following proposition 4.1.9 thatthe diagram commutes also going upward.

For anyV -moduleN we have an exact sequenceΓ2N → ⊗2N → Λ2N → 0. As observed inthe proof of proposition 2.1.5, the symmetric groupS2 acts trivially on⊗2m andΓ2m ≃ ⊗2m,soΛ2m = 0. Also we have natural isomorphismsΓpm ≃ m for all p > 0.

Proposition 4.1.9. LetR be a commutative ring andL a flatR-module withΛ2L = 0. Thenfor p > 0 and for allR-modulesN we have natural isomorphisms

Γp(L)⊗R Fp(N)∼→ Fp(L⊗R N).

Proof. Fix an elementx ∈ Fp(N). For eachR-algebraR′ and each elementl ∈ R′ ⊗R L weget a mapφl : R′⊗RN → R′⊗RL⊗RN by y 7→ l⊗ y, hence a mapFp(φl) : R

′⊗R Fp(N) ≃Fp(R

′ ⊗R N) → Fp(R′ ⊗R L ⊗R N) ≃ R′ ⊗R Fp(L ⊗R N). For varyingl we obtain a

map of setsψR′,x : R′ ⊗R L → R′ ⊗R Fp(L ⊗R N) : l 7→ Fp(φl)(1 ⊗ x). According tothe terminology of [21], the system of mapsψR′,x for R′ ranging over allR-algebras forms ahomogeneous polynomial law of degreep from L to Fp(L ⊗R N), so it factors through theuniversal homogeneous degreep polynomial lawγp : L → Γp(L) . The resultingR-linearmapψx : Γp(L) → Fp(L ⊗R N) dependsR-linearly onx, hence we derive anR-linear mapψ : Γp(L)⊗RFp(N) → Fp(L⊗RN). Next notice that by hypothesisS2 acts trivially on⊗2L soSp acts trivially on⊗pL and we get an isomorphismβ : Γp(L)

∼−→ ⊗pL. We deduce a natural

map(⊗pL) ⊗R Fp(N) → Fp(L ⊗R N). Now, in order to prove the proposition for the caseFp = ⊗p, it suffices to show that this last map is just the natural isomorphism that “reorders thefactors”. Indeed, letx1, ..., xn ∈ L andq = (q1, ..., qn) ∈ Nn such that|q| =

∑i qi = p; then

β sends the generatorx[q1]1 · ... · x[qn]n to

(p

q1,...,qn

)· x⊗q11 ⊗ ... ⊗ x⊗qnn . On the other hand, pick

anyy ∈ ⊗pN and letR[T ] = R[T1, ..., Tr] be the polynomialR-algebra inn variables; write(T1 ⊗ x1 + ... + Tn ⊗ xn)

⊗p ⊗ y = ψR[T ],y(T1 ⊗ x1 + ... + Tn ⊗ xn) =∑

r∈Nn T r ⊗ wr with

wr ∈ ⊗p(L ⊗R N). Thenψ((x[q1]1 · ... · x[qn]n ) ⊗ y) = wq (see [21] pp.266-267) and the claim

follows easily. Next notice thatΓp(L) is flat, so that tensoring withΓp(L) commutes with takingcoinvariants (resp. invariants) under the action of the symmetric group; this implies the assertionfor Fp = Symp (resp.Fp = TSp). To deal withFp = Λp recall that for anyV -moduleM and


p > 0 we have the antisymmetrization operatoraM =∑

σ∈Spsgn(σ) · σ : ⊗pM → ⊗pM and a

surjectionΛp(M) → Im(aM) which is an isomorphism forM free, hence forM flat. The resultfor Fp = ⊗p (and again the flatness ofΓp(L)) then givesΓp(L)⊗ Im(aN ) ≃ Im(aL⊗RN ), hence

the assertion forFp = Λp andN flat. For generalN let F1∂→ F0

ε→ N → 0 be a presentation

with Fi free. Definej0, j1 : F0 ⊕ F1 → F0 by j0(x, y) = x + ∂(y) andj1(x, y) = x. Byfunctoriality we derive an exact sequence

Λp(F0 ⊕ F1)//// Λp(F0) // Λp(N) // 0

which reduces the assertion to the flat case. ForFp = Γp the same reduction argument works aswell (cf. [21] p.284) and for flat modules the assertion forΓp follows from the correspondingassertion for TSp.

Lemma 4.1.10.LetA be a simplicial almost algebra,L,E andF threeA-modules,f : E → Fa quasi-isomorphism. IfL is flat orE, F are flat, thenL⊗A f : L⊗A E → L⊗A F is a quasi-isomorphism.

Proof. It is deduced directly from [14] (I.3.3.2.1) by applyingM 7→ M!.

As usual, this allows one to show that⊗ : Hot•(A)×Hot•(A) → Hot•(A) admits a left derived

functorL⊗ : D•(A)× D•(A) → D•(A). If R is a simplicialV -algebra then we have essentially

commutative diagrams

D•(R)× D•(R)

L⊗ //

D•(R)

D•(R

a)× D•(Ra)

L⊗ //

OO

D•(Ra)

OO

where again the downward (resp. upward) functors are induced by localisation (resp. byM 7→M!).

We mention the derived functors of the non-additive functorFp defined above in the simplestcase of modules over a constant simplicial ring. LetA be a (commutative) almost algebra.

Lemma 4.1.11. If u : X → Y is a quasi-isomorphism of flats.A-modules thenFap(u) :

Fap(X) → F

ap(Y ) is a quasi-isomorphism.

Proof. This is deduced from [14] (I.4.2.2.1) applied toN(X!) → N(Y!) which is a quasi-isomorphism of chain complexes of flatA!!-modules. We note thatloc. cit. deals with a moregeneral mixed simplicial construction ofFp which applies to bounded above complexes, butone can check that it reduces to the simplicial definition forcomplexes inC•(A!!).

Using the lemma one can constructLFap : D•(s.A) → D•(s.A). If R is aV -algebra we have

the derived category version of the essentially commutative squares (4.1.8), relatingLFp :D•(s.R) → D•(s.R) andLF

ap : D•(s.R

a) → D•(s.Ra).

AcknowledgementsThe second author is very much indebted to Gerd Faltings for many patient explanations onthe method of almost etale extensions. Next he would like toacknowledge several interesting discussions withIoannis Emmanouil. He is also much obliged to Pierre Deligne, for a useful list of critical remarks. Finally, heowes a special thank to Roberto Ferretti, who has read the first tentative versions of this article, has corrected manyslips and has made several valuable suggestions.

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Ofer Gabber Lorenzo RameroI.H.E.S. Universite de Bordeaux ILe Bois-Marie Laboratoire de Mathematiques Pures35, route de Chartres 351, cours de la LiberationF-91440 Bures-sur-Yvette F-33405 Talence CedexE-mail address:[email protected] E-mail address:[email protected]

web:http://www.math.u-bordeaux.fr/∼ramero

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arXiv:math/0002064v1 [math.AG] 9 Feb 2000 · We deﬁne a uniform structure on the set I of ideals of V as follows. For every ﬁnitely generated ideal m0 ⊂ mthe subset of I ×