Étale cohomology: starting points · 2018-09-16 · Étale cohomology: starting points P. Deligne (notes by J. F. Boutot) 1977 Abstract At the AMS Summer Institute in Algebraic Geometry

Étale cohomology: starting pointsP. Deligne (notes by J. F. Boutot)

1977

AbstractAt the AMS Summer Institute in Algebraic Geometry in 1974, Deligne

gave a series of lectures “Inputs of étale cohomology” intended to explain thebasic étale cohomology required for his recent proof of the Weil conjectures.The lectures were written up by Boutot (in French) and published in SGA4 1/2 (Lecture Notes in Math., Springer, 1977). The typewritten documentwas typeset into LATEX by D.K. Miller,1 and his LATEX file was translated intoEnglish by an anonymous source. The document is available at jmilne.org(under /math/Documents).

ContentsI Grothendieck topologies 2

1 Sieves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Faithfully flat descent . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A special case: Hilbert’s theorem 90 . . . . . . . . . . . . . . . . . . 86 Grothendieck topologies . . . . . . . . . . . . . . . . . . . . . . . . . 9

II The étale topology 111 The étale topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Examples of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Fibers, direct images . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

IIIThe cohomology of curves 171 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Tsen’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 The cohomology of smooth curves . . . . . . . . . . . . . . . . . . . 201The source code may be found at GitHub (https://www.github.com/dkmiller/sga4.5).

Changes to the mathematics are slight: blackboard bold has become bold and script is usedinstead of roman for sheaves; a superscript × instead of ∗ is ueed to denote the group of units in aring. The numbering is unchanged except for equations, which is idiosyncratic in the original.

1

https://www.github.com/dkmiller/sga4.5

I GROTHENDIECK TOPOLOGIES 2

4 Unwindings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

IVThe proper base change theorem 241 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Proof for q = 0 or 1 and F = Z/n . . . . . . . . . . . . . . . . . . . 253 Constructible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 264 End of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Cohomology with proper support . . . . . . . . . . . . . . . . . . . . 296 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

V Local acyclicity of smooth morphisms 321 Locally acyclic morphisms . . . . . . . . . . . . . . . . . . . . . . . . 332 Local acyclicity of a smooth morphism . . . . . . . . . . . . . . . . . 363 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

VIPoincaré duality 411 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 The case of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Variants and applications . . . . . . . . . . . . . . . . . . . . . . . . 45

Original prefaceThis work contains the notes for six lectures given by P. Deligne at Arcata in August1974 (AMS Summer School2 on algebraic geometry) under the title “Inputs of étalecohomology”. The seventh lecture became “Rapport sur la formule des traces”,published in [SGA 4 1/2]. The purpose of the lectures was to give the proofs ofthe fundamental theorems in étale cohomology, freed from the gangue3 of nonsensethat surrounds them in SGA 4. We have not sought to state the theorems in theirmost general form, nor followed the reductions, sometimes clever, that their proofrequires. We have on the contrary emphasized the “irreducible” case, which, oncethe reductions have been made, remains to be treated.

We hope that this text, which makes no claim to originality, will help the readerconsult with profit the three volumes of SGA 4.

Conventions. We consider only schemes that are quasi-compact (= finite unionof open affines) and quasi-separated (= such that the intersection of two open affinesis quasi-compact), and we simply call them schemes.

I Grothendieck topologiesGrothendieck’s topologies first appeared as the foundation of his theory of descent(cf. SGA 1 VI, VIII); their use in the corresponding cohomology theories came

2Actually, Institute. All footnotes are by the translator.3The commercially valueless material in which ore is found (Oxford English Dictionary).


later. The same path is followed here: by formalizing the classical notions oflocalization, local property, and patching (§1, 2, 3), we obtain the general conceptof a Grothendieck topology (§6). To justify its introduction into algebraic geometry,we prove a faithfully flat descent theorem (§4), generalizing the classical theorem 90of Hilbert (§5).

The reader will find a more complete, but concise, exposition of this formalism inGiraud [Gir64]. The notes of M. Artin: “Grothendieck topologies” [Art62] (ChaptersI to III) also remain useful. The 866 pages of Exposés I to V of SGA 4 are valuablewhen considering exotic topologies, such as the one that gives rise to crystallinecohomology. To use étale topology, so close to the classical intuition, it is not strictlynecessary to read them.

1 SievesLet X be a topological space and f : X → R a real-valued function on X. Thecontinuity of f is a local property. In other words, if f is continuous on everysufficiently small open in X, then f is continuous on the whole of X. To formalizethe notion of “local property” we introduce some definitions.

We say that a set U of opens of X is a sieve if, for all U ∈ U and V ⊂ U , wehave V ∈ U . We say that a sieve is covering if the union of the opens belonging toit equals X.

The sieve generated by a family Ui of opens of X is defined to be the set ofopen subsets U of X such that U is contained in some Ui.

We say that a property P (U), defined for all open U in X, is local if, for everyopen U of X and every covering sieve U of U , P (U) is true if and only if P (V )is true for all V ∈ U . For example, given a map f : X → R, the property “f iscontinuous on U” is local.

2 SheavesWe make precise the notion of a function given locally on X.

2.1 The sieve point of view

Let U be a sieve of opens of X. We call a function given U -locally on X the data of,for every U ∈ U , a function fU on U such that, if V ⊂ U , then fV = fU |V .

2.2 The Čech point of view

If the sieve U is generated by a family of opens Ui of X, then to give a functionU-locally on X amounts to giving a function fi on each Ui such that fi|Ui ∩ Uj =fj |Ui ∩ Uj for all i, j.

In other words, if we let Z =∐Ui, then to give a function U-locally amounts

to giving a function on Z that is constant on the fibers of the natural projectionZ → X.


2.3

The continuous functions form a sheaf. This means that for every covering sieveU of an open V of X and every function fU given U-locally on V such that eachfU is continuous on U , there exists a unique continuous function f on V such thatf |U = fU for all U ∈ U .

3 StacksWe make precise the notion of vector bundle given locally on X.

3.1 The sieve point of view

Let U be a sieve of opens of X. We call a vector bundle given U-locally on X thedata of

a) a vector bundle EU on each U ∈ U ,

b) if V ⊂ U , an isomorphism ρU,V : EV∼−−→ EU |V , these being such that

c) if W ⊂ V ⊂ U , then the diagram

EW EU |W

EV |W

ρU,W

ρV,W ρU,V |W

commutes, that is, ρU,W = (ρU,V restricted to W ) ρV,W .

3.2 The Čech point of view

If the sieve U is generated by a family of opens Ui of X, then to give a vector bundleU-locally on X amounts to giving:

a) a vector bundle Ei on each Ui,

b) if Uij = Ui ∩ Uj = Ui ×X Uj , an isomorphism ρji : Ei|Uij∼−−→ Ej |Uij , these

being such that

c) if Uijk = Ui ×X Uj ×X Uk, the diagram

Ei|Uijk Ek|Uijk

Ej |Uijk

ρki|Uijk

ρji|Uijk ρkj |Uijk

commutes, that is, ρki = ρkj ρji on Uijk.


In other words, if we let Z =∐Ui and we let π : Z → X be the natural projection,

then to give a vector bundle U-locally on X amounts to giving

a) a vector bundle E on Z,

b) if x and y are two points of Z such that π(x) = π(y), an isomorphismρyx : Ex

∼−−→ Ey between the fibers of E at x and at y depending continuouslyon (x, y), these being such that

c) if x, y, and z are three points of Z such that π(x) = π(y) = π(z), thenρzx = ρzy ρyx.

3.3

A vector bundle E on X defines a vector bundle given U-locally EU , namely, thesystem of restrictions EU of E to the objects of U . The fact that the notion ofvector bundle is local can be expressed as follows: for every covering sieve U of X,the functor E 7→ EU from vector bundles on X to vector bundles given U -locally isan equivalence of categories.

3.4

If in §1, we replace “open of X” by “subset of X”, we get the notion of a sieve ofsubspaces of X. In this context also there are patching theorems. For example, letX be a normal space and C a sieve of subspaces of X generated by a finite locallyclosed covering of X; then the functor E 7→ EC from vector bundles on X to vectorbundles given C-locally is an equivalence of categories.

In algebraic geometry, it is useful to also consider “sieves of spaces above X”;this is what we will see in the next paragraph.

4 Faithfully flat descent4.1

In the setting of schemes, the Zariski topology is not fine enough for the study ofnonlinear problems, and one is led to replace the open immersions in the precedingdefinitions with more general morphisms. From this point of view, descent techniquesappear as localization techniques. Thus the following statement of descent can beparaphrased by saying that the properties considered are local for the faithfullyflat topology. [A morphism of schemes is said to be faithfully flat if it is flat andsurjective.]

Proposition 4.2. Let A be a ring and B a faithfully flat A-algebra. Then

(i) A sequence Σ = (M ′ →M →M ′′) of A-modules is exact if the sequence Σ(B),deduced from Σ by extension of scalars to B, is exact.

(ii) An A-module M is of finite type (resp. finite presentation, flat, locally free offinite rank, invertible i.e. locally free of rank one) if the B-module M(B) is.


Proof. (i) As the functor M 7→ M(B) is exact (flatness of B), it suffices to showthat, if an A-module N is nonzero, then N(B) is nonzero. If N is nonzero, then itcontains a nonzero monogenous submodule A/a, and so N(B) contains a monogenoussubmodule (A/a)(B) = B/aB, which is nonzero by the surjectivity of the structuremorphism ϕ : Spec(B)→ Spec(A) (if V (a) is nonempty, then ϕ−1(V (a)) = V (aB)is nonempty).

(ii) For any finite family (xi) of elements ofM(B), there exists a finitely generatedsubmodule M ′ of M such that M ′(B) contains the xi. If M(B) is of finite type andthe xi generate M(B), then M ′(B) = M(B), so M ′ = M and M is of finite type.

If M(B) is of finite presentation, then, from the above, we can find a surjectionAn →M . If N is the kernel of this surjection, then the B-module N(B) is of finitetype, so N is, and M is of finite presentation. The assertion for “flat” followsimmediately from (i). The condition “locally free of finite rank” means “flat andof finite presentation”, and the rank can be tested after extension of scalars to afield.

4.3

Let X be a scheme and S a class of X-schemes stable under fiber products overX. A class U ⊂ S is a sieve on X (relative to S) if, for every morphism ϕ : V → Uof X-schemes with U, V ∈ S and U ∈ U , we have V ∈ U . The sieve generated bya family Ui of X-schemes in S is the class of V ∈ S such that there exists amorphism of X-schemes from V into one of the Ui.

4.4

Let U be a sieve on X. We define a quasi-coherent module given U -locally on X tobe the data of

a) a quasi-coherent module EU on each U ∈ U ,

b) for every U ∈ U and every morphism ϕ : V → U of X-schemes in S, an isomor-phism ρϕ : EV

∼−−→ ϕ∗EU , these being such that

c) if ψ : W → V is a morphism of X-schemes in S, then the diagram

EW ψ∗ϕ∗EU

ψ∗EV

ρϕψ

ρψ ψ∗ρψ

commutes, in other words, ρϕψ = (ψ∗ρϕ) ρψ.

A quasi-coherent module E on X defines a quasi-coherent module EU givenU-locally, namely, for ϕU : U → X take the quasi-coherent module ϕ∗UE, and fora morphism ψ : V → U take the restriction isomorphism ρψ to be the canonicalisomorphism EV = (ϕU ψ)∗E ∼−−→ ψ∗ϕ∗UE = ψ∗EU .


Theorem 4.5. Let Ui ∈ S be a finite family of X-schemas flat over X such thatX is the union of the images of the Ui, and let U be the sieve generated by Ui.Then the functor E 7→ EU is an equivalence from the category of quasi-coherentmodules on X to the category of quasi-coherent modules given U-locally.

Proof. We treat only the case where X is affine and U is generated by an affineX-scheme U faithfully flat over X. The reduction to this case is formal. LetX = Spec(A) and U = Spec(B).

If the morphism U → X has a section, then X belongs to the sieve U , and theassertion is obvious. We will reduce the general case to this case.

A quasi-coherent module given U-locally defines modules M ′, M ′′, and M ′′′ onU , U ×X U , and U ×X U ×X U , and isomorphisms ρ : p∗M• 'M• for any projectionmorphism p between these spaces. There is a cartesian diagram

M∗ : M ′ M ′′ M ′′′

overU∗ : U U ×X U U ×X U ×X U .

Conversely, M∗ determines the module given U-locally: for V ∈ U , there ex-ists ϕ : V → U , and we put MU = ϕ∗M ′; for ϕ1, ϕ2 : V → U , we have naturalidentifications ϕ∗1M ′ ' (ϕ1 × ϕ2)∗M ′′ ' ϕ∗2M

′, and we see using M ′′′ that theseidentifications are compatible, and so the definition is valid. In summary, it amountsto the same thing to give a module U-locally or a cartesian diagram M∗ over U∗.

Let us translate this into algebraic terms: to give M∗ amounts to giving acartesian diagram of modules

M ′ M ′′ M ′′′∂0

∂1

∂0

∂1

∂2

over the diagram of rings

B B ⊗A B B ⊗A B ⊗A B.∂0

∂1

∂0

∂1

∂2

[More precisely: we have ∂i(bm) = ∂i(b) · ∂i(m), the usual identities such as ∂0∂1 =∂0∂0 are true, and “cartesian” means that the morphisms ∂i : M ′ ⊗B,∂i (B ⊗A B)→M ′′ and M ′′ ⊗B⊗AB,∂i (B ⊗A B ⊗A B)→M ′′′ are isomorphisms.]

Now E 7→ EU becomes the functor sending an A-module M to

M∗ =(M ⊗A B M ⊗A B ⊗A B M ⊗A B ⊗A B ⊗A B

)


It admits the functor

(M ′ M ′′ M ′′′) 7→ Ker (M ′ M ′′)

as a right adjoint. We have to prove that the adjunction arrows

M → Ker(M ⊗A B ⇒M ⊗A B ⊗A B)

andKer(M ′ ⇒M ′′)⊗A B →M ′

are isomorphisms. According to (4.2)(i), it suffices to do this after a faithfully flatbase change A→ A′ (B becoming B′ = B ⊗A A′). Taking A′ = B brings us back tothe case where U → X admits a section.

5 A special case: Hilbert’s theorem 905.1

Let k be a field, k′ a Galois extension of k, and G = Gal(k′/k). Then the homomor-phism

k′ ⊗k k′ →∏σ∈G

k′

x⊗ y 7→ x · σ(y)σ∈G

is bijective.It follows that to give a module locally for the sieve generated by Spec(k′) over

Spec(k) is the same as giving a k′-vector space with a semi-linear action of G, i.e.,

a) a k′- vector space V ′,

b) for all σ ∈ G, an endomorphism ϕσ of the underlying group of V ′ such that

ϕσ(λv) = σ(λ)ϕσ(v), for all λ ∈ k′ and v ∈ V ′,

satisfying the condition

c) for all σ, τ ∈ G, we have ϕτσ = ϕτ ϕσ.

Let V = V ′G be the group of invariants for this action of G. It is a k-vector

space and, according to Theorem 4.5, we have

Proposition 5.2. The inclusion of V into V ′ defines an isomorphism V ⊗k k′∼−−→

V ′.

In particular, if V ′ has dimension 1 and v′ ∈ V is nonzero, then ϕσ is determinedby the constant c(σ) ∈ k′× such that ϕσ(v′) = c(σ)v′, and the condition c) becomes

c(τσ) = c(τ) · τ(c(σ)).


According to the proposition, there is a nonzero invariant vector v = µv′, µ ∈ k′×.Therefore, for all σ ∈ G,

c(σ) = µ · σ(µ−1).

In other words, every 1-cocycle of G with values in k′× is a coboundary:

Corollary 5.3. We have H1(G, k′×) = 0.

6 Grothendieck topologiesWe now rewrite the definitions of the preceding paragraphs in an abstract frameworkthat encompasses both the case of topological spaces and that of schemes.

6.1

Let S be a category and U an object of S. We call a sieve on U a subset U ofOb(S/U) such that, if ϕ : V → U belongs to U and ψ : W → V is a morphism in S,then ϕ ψ : W → U belongs to U .

If ϕi : Ui → U is a family of morphisms, then the sieve generated by the Ui isdefined to be the set of morphisms ϕ : V → U which factor through one of the ϕi.

If U is a sieve on U and ϕ : V → U is a morphism, then the restriction UV of Uto V is defined to be the sieve on V consisting of the morphisms ψ : W → V suchthat ϕ ψ : W → U belongs to U .

6.2

A Grothendieck topology on S is the data of a set C(U) of sieves for every object Uof S, called the covering sieves, such that the following axioms are satisfied:

a) The sieve generated by the identity morphism of U is covering.

b) If U is a covering sieve U and V → U is a morphism, then the sieve UV iscovering.

c) A locally covering sieve is covering. In other words, if U is a covering sieve onU and U ′ is a sieve on U such that, for all V → U belonging to U , the sieveU ′V on V is covering, then U ′ is covering.

A site is defined to be a category furnished with a Grothendieck topology.

6.3

Let S be a site. A presheaf F on S is a contravariant functor from S to the categoryof sets. A section of F over an object U is an element of F (U). For a morphismV → U and an s ∈ F (U), we let s|V (s restricted to V ) denote the image of s inF (V ).

Let U be a sieve on U . We call a section of F given U-locally the data of, forevery V → U belonging to U , a section sV ∈ F (V ) such that, for every morphism


W → V , we have sV |W = sW . We say that F is a sheaf if, for every object U ofS, every covering sieve U on U , and every section sV given U-locally, there is aunique section s ∈ F (U) such that s|V = sV for all V → U belonging to U .

We define in a similar way abelian sheaves by replacing the category of sets bythat of abelian groups. One shows that the category of abelian sheaves on S is aabelian category with enough injectives. A sequence F

f−→ Gg−→ H of sheafs is

exact if, for every object U of S and every s ∈ G (U) such that g(s) = 0, there existslocally a t such that f(t) = s, i.e., there exists a covering sieve U of U and a sectionstV ∈ F (V ) for all V ∈ U such that f(tV ) = s|V .

6.4 Examples

We have seen two above.

a) Let X be a topological space and S the category whose objects are the opensof X and whose morphisms are the natural inclusions. The Grothendiecktopology on S corresponding to the usual topology X is that for which a sieveon an open U of X is covering if the union of the opens belonging to the sieveequals U . It is clear that the category of sheaves on S is equivalent to thecategory of sheaves on X in the usual sense.

b) Let X be a scheme and S the category of schemes over X. We define thefpqc (faithfully flat quasi-compact) topology on S to be the Grothendiecktopology for which a sieve on an X-schema U is covering if it is generated bya finite family of flat morphisms whose images cover U .

6.5 Cohomology

We will always assume that the category S has a final object X. The group ofglobal sections of an abelian sheaf F , denoted ΓF or H0(X,F ), is defined to bethe group F (X). The functor F 7→ ΓF from the category of abelian sheaves on Sto the category of abelian groups is left exact, and we denote its derived functors(or satellites) by Hi(X,−). These cohomology groups represent the obstructions topassing from the local to the global. By definition, if

0→ F → G →H → 0

is an exact sequence of abelian sheafs, then there is a long exact cohomology sequence:

0→ H0(X,F )→ H0(X,G )→ H0(X,H )→ H1(X,F )→ · · ·· · · → Hn(X,F )→ Hn(X,G )→ Hn(X,H )→ Hn+1(X,F )→ · · ·

6.6 4

Given an abelian sheaf F on S, we define an F -torsor to be a sheaf G endowed withan action F × G → G of F such that locally (after restriction to all the objects of

4The original repeats the number 6.5.

II THE ÉTALE TOPOLOGY 11

a covering sieve of the final object X) G equipped with the F -action is isomorphicto F equipped with the canonical action F ×F → F by translations.

The set H1(X,F ) can be interpreted as the set of isomorphism classes of F -torsors.

II The étale topologyWe specialize the definitions of the preceding chapter to the case of the étale topologyof a scheme X (§1, 2, 3). The corresponding cohomology coincides in the case thatX is the spectrum of a field K with the Galois cohomology of K.

1 The étale topologyWe begin by reviewing the notion of an étale morphism.

Definition 1.1. Let A be a (commutative) ring. An A-algebra B is said to be étaleif it is of finite presentation and if the following equivalent conditions are satisfied:

a) For every A-algebra C and every ideal of zero square J in C, the canonicalmap

HomA-alg(B,C)→ HomA-alg(B,C/J)

is a bijection.

b) B is a flat A-module and ΩB/A = 0 (here ΩB/A denotes the module of relativedifferentials).

c) Let B = A[X1, . . . , Xn]/I be a presentation of B. Then, for every prime idealp of A[X1, . . . , Xn] containing I, there exist polynomials P1, . . . , Pn ∈ I suchthat Ip is generated by the images of P1, . . . , Pn and det(∂Pi/∂Xj) /∈ p.

[Cf. [SGA 1, Exposé I], or [Ray70, Chapitre V].We say that a morphism of schemes f : X → S is étale if, for all x ∈ X,

there exists an open affine neighborhood U = Spec(A) of f(x) and an affine openneighborhood V = Spec(B) of x in X ×S U such that B is an étale A-algebra.

1.2 Examples

a) When A is a field, an A-algebra B is étale if and only if it is a finite productof separable extensions of A.

b) When X and S are schemes of finite type over C, a morphism f : X → S isétale if and only if the associated analytic map fan : Xan → San is a localisomorphism.


1.3 Sorite

a) (base change) If f : X → S is an étale morphism, then so also is fS′ : X×SS′ →S′ for any morphism S′ → S.

b) (composition) The composite of two étale morphisms is an étale morphism.

c) If f : X → S and g : Y → S are two étale morphisms, then every S-morphismfrom X to Y is étale.

d) (descent) Let f : X → S be a morphism. If there exists a faithfully flatmorphism S′ → S such that fS′ : X ×S S′ → S′ is étale, then f is étale.

1.4

Let X be a scheme and S the category of X-schemes. According to (1.3.c) everymorphism in S is étale. We define the étale topology on S to be the topology forwhich a sieve over U is covering if it is generated by a finite family of morphismsϕi : Ui → U such the union of the images of the ϕi covers U . We define the étalesite of X, denoted Xet, to be S endowed with the étale topology.

2 Examples of sheaves2.1 Constant sheaf

Let C be an abelian group, and suppose for simplicity that X is noetherian. We letCX (or just C if there is no ambiguity) denote the sheaf defined by U 7→ Cπ0(U),where π0(U) is the (finite) set of connected components of U . The most importantcase will be C = Z/n. By definition

H0(X,Z/n) = (Z/n)π0(X) .

In addition, H1(X,Z/n) is the set of isomorphism classes of Z/n-torsors (I.6.6), i.e.,of Galois finite étale coverings of X with group Z/n. In particular, if X is connectedand π1(X) is its fundamental group for a chosen base point, then

H1(X,Z/n) = Hom(π1(X),Z/n).

2.2 Multiplicative group

We denote by Gm,X (or Gm if there is no ambiguity) the sheaf U 7→ Γ(U,O×U ). Itis indeed a sheaf thanks to the faithfully flat descent theorem (I.4.5). We have bydefinition

H0(X,Gm) = H0(X,OX)×.

In particular, if X is reduced, connected, and proper over an algebraically closedfield k, then

H0(X,Gm) = k×.


Proposition 2.3. There is an isomorphism

H1(X,Gm) = Pic(X),

where Pic(X) is the group of isomorphism classes of invertible sheaves on X.Proof. Let ∗ be the functor which, to an invertible sheaf L over X, attaches thefollowing presheaf L ∗ on Xet: for ϕ : U → X étale,

L ∗(U) = IsomU (OU , ϕ∗L ).

According to (I.4.2) (i) and (I.4.5)5 (full faithfulness), this presheaf is a sheaf; it iseven a Gm-torsor. We see immediately that

a) the functor ∗ is compatible with (étale) localization;

b) it induces an equivalence of the category of trivial invertible sheaves (i.e.,isomorphic to OX) with the category of trivial Gm-torsors: L is trivial ifand only if L ∗ is.

Moreover, according to (I.4.2) (ii) and (I.4.5),c) the notion of invertible sheaf is local for the étale topology.It follows formally from a), b), c) that ∗ is an equivalence from the category

of invertible sheaves on X to the category of Gm-torsors on Xet. It induces theisomorphism sought. The inverse equivalence can be constructed as follows: if Tis a Gm-torsor, then there exists a finite étale covering Ui of X such that thetorsors T |Ui are trivial; T is then trivial on every V étale over X belonging to thesieve U ⊂ Xet generated by Ui. On each V ∈ U , T |V corresponds to an invertiblesheaf LV (by b)) and the LV constitute an invertible sheaf given U -locally LU (bya)). By c), the latter comes from an invertible sheaf L (T ) on X, and T 7→ L (T )is the inverse of ∗ sought.

2.4 Roots of unity

For an integer n > 0, we define the sheaf of nth roots of unity, denoted µn, to bethe kernel of “raising to the nth power” on Gm . If X is a scheme over a separablyclosed field k and n is invertible in k, then the choice of a primitive nth root of unityζ ∈ k defines an isomorphism i 7→ ζi of Z/n with µn.

The relationship between cohomology with coefficients in µm and cohomologywith coefficients in Gm is given by the exact cohomology sequence deduced fromKummer Theory 2.5. If n is invertible on X, then raising to the nth power inGm is a sheaf epimorphism, so there is an exact sequence

0→ µn → Gm → Gm → 0.

Proof. Let U → X be an étale morphism and let a ∈ Gm(U) = Γ(U,O×U ).As n is invertible on U , the equation Tn − a = 0 is separable; that is, U ′ =Spec (OU [T ]/(Tn − a)) is étale over U . Moreover U ′ → U is surjective and a admitsan nth root on U ′, whence the result.

5The original had (4.2) and (4.5). Similar corrections are made elsewhere.


3 Fibers, direct images3.1

A geometric point of X is a morphism x→ X, where x is the spectrum of a separablyclosed field k(x). By an abuse of language, we denote it by x, the morphism x→ Xbeing understood. If x is the image of x in X, then we say that x is centered at x.When the field k(x) is an algebraic extension of the residue field k(x), we say that xis an algebraic geometric point of X.

We define an étale neighborhood of x to be a commutative diagram

U

x X,

where U → X is an étale morphism.The strict localization of X at x is the ring OX,x = lim−→Γ(U,OU ), where the

inductive limit is over the étale neighborhoods of x. It is a strictly henselian localring whose residue field is the separable closure of the residue field k(x) of X at xin k(x). It plays the role of the local ring for the étale topology.

3.2

Given a sheaf F on Xet, we define the fiber of F at x to be the set (resp. the group,. . . ) Fx = lim−→F (U), where the inductive limit is again over the étale neighborhoodsof x.

In order for a homomorphism F → G of sheaves to be a mono-/epi-/isomorphism,it is necessary and sufficient that this is so of the morphisms Fx → Gx inducedon the fibers at all the geometric points of X. When X is of finite type over analgebraically closed field, it suffices to check this for the rational points of X.

3.3

If f : X → Y is a morphism of schemes and F a sheaf on Xet, then the direct imagef∗F of F by f is the sheaf on Yet defined by f∗F (V ) = F (X ×Y V ) for all V étaleover Y .

The functor f∗ : Sh(Xet)→ Sh(Yet) is left exact. Its right derived functors Rqf∗are called its higher direct images. If y is a geometric point of Y , we have

(Rqf∗F )y = lim−→Hq(V ×Y X,F ),

the inductive limit being over the étale neighborhoods V of y.Let OY,y be the strict localization of Y at y, let Y = Spec(OY,y), and let

X = X×Y Y . We can extend F to Xet (this is a special case of the general notion ofan inverse image) as follows: if U is a scheme étale over X, then there exists a étale


neighborhood V of y and a scheme U étale over X ×Y V such that U = U ×V Y ;we put

F (U) = lim−→F (U ×V V ) ,the inductive limit being over the étale neighborhoods V ′ of y which dominate V .With this definition, we have

(Rqf∗F )y = Hq(X,F ).

The functor f∗ has a left adjoint f∗, the “inverse image” functor. If x is ageometric point of X and f(x) its image in Y , we have (f∗F )x = Ff(x). Thisformula shows that f∗ is an exact functor. The functor f∗ thus transforms injectivesheaves into injective sheaves, and the spectral sequence of the composite functorΓ f∗ (resp. g∗ f∗) gives the

Leray spectral sequence 3.4. Let F be an abelian sheaf on Xet, and let f : X →Y be a morphism of schemes (resp. let X f−→ Y

g−→ Z be morphisms of schemes). Wehave a spectral sequence

Epq2 = Hp(Y,Rqf∗F )⇒ Hp+q(X,F )(resp. Epq2 = Rpg∗Rqf∗F ⇒ Rp+q(gf)∗F ).

Corollary 3.5. If Rqf∗F = 0 for all q > 0, then Hp(Y, f∗F ) = Hp(X,F ) (resp.Rpg∗(F∗F ) = Rp(gf)∗F ) for all p > 0.

This applies in particular in the following case:

Proposition 3.6. Let f : X → Y be a finite morphism (or, by passage to the limit,an integral morphism) and F an abelian sheaf on X. Then Rqf∗F = 0, for allq > 0.

Indeed, let y be a geometric point of Y , Y the spectrum of the strict localization ofY at y, and X = X×Y Y . According to the above, it suffices show that Hq(X,F ) = 0for all q > 0. But X is the spectrum of a product of strictly Henselian local rings (cf.[Ray70], Chapter I), and the functor Γ(X,−) is exact because every étale surjectivemap to X admits a section, whence the assertion.

4 Galois cohomologyFor X = Spec(K) the spectrum of a field, we will see that étale cohomology can beidentified with Galois cohomology.

4.1

Let us begin with a topological analogy. If K is the field of functions of an integralaffine algebraic variety Y = Spec(A) over C, then

K = lim−→f∈A

A[1/f ].


In other words, X = lim←−U , where U runs over all open sets of Y . We know thatthere are arbitrarily small Zariski opens which are K(π, 1)’s for the classical topology.Therefore, we should not be surprised if we can consider Spec(K) itself to be aK(π, 1), π being the group fundamental (in the algebraic sense) of X = Spec(K),i.e., the Galois group of K/K, where K is a separable closure of K.

4.2

More precisely, let K be a field, K a separable closure of K, and G = Gal(K/K) theGalois group with its natural topology. To a finite étale K-algebra A (finite productof separable extensions of K), attach the finite set HomK(A, K). The Galois groupG acts on this set through a discrete (hence finite) quotient. If A = K[T ]/(F ), thenthis set can be identified with the set of roots of the polynomial F in K. Galoistheory, in the form given by Grothendieck says thatProposition 4.3. The functor

finite étaleK-algebras

→

finite sets on whichG acts continuously

,

which to an étale algebra A attaches HomK(A, K), is an anti-equivalence of cate-gories.

We can deduce from this an analogous description of the sheaves for the étaletopology on Spec(K)Proposition 4.4. The functor

étale sheaveson Spec(K)

→

sets on whichG acts continuously

,

which to a sheaf F attaches its fiber FK at the geometric point Spec(K), is anequivalence of categories.

The group G is said to act continuously on a set E if the stabilizer of every elementof E is an open subgroup of G. The functor in the reverse direction has an obviousdescription: let A a finite étale K-algebra, U = Spec(A), and U(K) = HomK(A, K)the G-set corresponding to A; then F (U) = HomG-sets(U(K),FK).

In particular, if X = Spec(K), then F (X) = FGK. When we consider only

abelian sheaves, we get, by passage to the derived functors, canonical isomorphisms

Hq(Xet,F ) = Hq(G,FK)

4.5 Examples

a) The constant sheaf Z/n corresponds to Z/n with the trivial action of G.

b) The sheaf of n-th roots of unity µn corresponds to the group µn(K) of n-throots of unity in K with the natural action of G.

c) The sheaf Gm corresponds to the group K× with the natural action of G.

III THE COHOMOLOGY OF CURVES 17

III The cohomology of curvesIn the case of topological spaces, unwindings6 using the Künneth formula andsimplicial decompositions allow one to reduce the calculation of the cohomology tothat of the interval I = [0, 1] for which we have H0(I,Z) = Z and Hq(I,Z) = 0 forq > 0.

In our case, the unwindings lead to more complicated objects, namely, to curvesover an algebraically closed field. We will calculate their cohomology in this chapter.The situation is more complex than in the topological case because the cohomologygroups are zero only for q > 2. The essential ingredient for the calculations is thevanishing of the Brauer group of the function field of such a curve (Tsen’s theorem,§2).

1 The Brauer groupFirst recall the classical definition:

Definition 1.1. Let K be a field and A a K-algebra of finite dimension. We saythat A is a central simple algebra over K if it satisfies the following equivalentconditions:

a) A has no nontrivial two-sided ideal and its center is K.

b) There exists a finite Galois extension K ′/K such that AK′ = A ⊗K K ′ isisomorphic to a matrix algebra over K ′.

c) A is K-isomorphic to a matrix algebra over a skew field with center K.

Two such algebras are said to be equivalent if the skew fields associated withthem by c) are K-isomorphic. If the algebras have the same dimension, this amountsto requiring that they be K-isomorphic. Tensor product defines by passage to thequotient an abelian group structure on the set of equivalence classes. It is this groupthat we classically call the Brauer group of K and that we denote Br(K).

1.2

We let Br(n,K) denote the set of K-isomorphism classes of K-algebras A such thatthere exists a finite Galois extension K ′ of K for which AK′ is isomorphic to thealgebra Mn(K ′) of n× n matrices over K ′. By definition Br(K) is the union of thesubsets Br(n,K) for n ∈ N. Let K be a separable closure of K and G = Gal(K/K).Then Br(n,K) is the set of “forms ” of Mn(K), and so it is canonically isomorphicto H1 (G,Aut(Mn(K))

).

6“dévissages” in the original. The word “dévissage”, literally “unscrewing”, is used by Frenchmathematicians to mean the reduction of a problem to a special case, often by fairly standardarguments.


Every automorphism of Mn(K) is inner. Therefore the group Aut(Mn(K)) canbe identified with the projective linear group PGL(n, K), and there is a canonicalbijection

θn : Br(n,K) ∼−−→ H1 (G,PGL(n, K)).

On the other hand, the exact sequence

1→ K× → GL(n, K)→ PGL(n, K)→ 1, (3.1.2.1)

defines a coboundary map

∆n : H1 (G,PGL(n, K))→ H2(G, K×).

On composing θn with ∆n, we get a map

δn : Br(n,K)→ H2(G, K×).

It is easily checked that the maps δn are compatible among themselves and define agroup homomorphism

δ : Br(K)→ H2(G, K×).

Proposition 1.3. The homomorphism δ : Br(K)→ H2(G, K×) is bijective.

This is a consequence of the next two lemmas.

Lemma 1.4. The map ∆n : H1 (G,PGL(n, K))→ H2(G, K×) is injective.

According to [Ser65], cor. to prop. I-44, it suffices to check that, if the exactsequence (3.1.2.1) is twistd by an element of H1 (G,PGL(n, K)

), then the H1 of

the middle group is trivial. This middle group is the group of K-points of themultiplicative group of a central simple algebra A of degree n2 over K. To prove thatH1(G,A×

K) = 0, we interpret A× as the group of automorphisms of a free A-module

L of rank 1 and H1 as the set of “forms” of L; these are A-modules of rank n2 overK, which are automatically free.

Lemma 1.5. Let α ∈ H2(G, K×), let K ′ be a finite extension of K contained in Kwith [K ′ : K] = n, and let G′ = Gal(K/K ′). If the image of α in H2(G′, K×) iszero, then α belongs to the image of ∆n.

Note first that we have

H2(G′, K×) ' H2 (G, (K ⊗K K ′)×).

(From a geometric point of view, if we let x = Spec(K) and x′ = Spec(K ′), and welet π : x′ → x denote the canonical morphism, then Rqπ∗(Gm,x′) = 0 for q > 0 andhence Hq(x′,Gm,X′) ' Hq(x, π∗Gm,X′) for q > 0.)

Furthermore, the choice of a basis for K ′ as a vector space over K allows us todefine a homomorphism

(K ⊗K K ′)× → GL(n, K),


which sends an element x to the endomorphism “multiplication by x” of K ⊗K K ′.We then have a commutative diagram with exact rows

1 K× (K ⊗K K ′)× (K ⊗K K ′)×/K× 1

1 K× GL(n, K) PGL(n, K) 1

The lemma now follows from the commutative diagram deduced from the abovediagram by passing to the cohomology

H1(G, (K ⊗K K ′)×/K×) H2(G, K×) H2(G, (K ⊗K K ′)×)

H1(G,PGL(n, K)) H2(G, K×).∆n

The knowledge of the Brauer group, in particular its vanishing, is extremelyimportant in Galois cohomology as the following proposition shows.

Proposition 1.6. Let K a field, K a separable closure of K, and G = Gal(K/K).Suppose that Br(K ′) = 0 for every finite extension K ′ of K. Then

(i) Hq(G, K×) = 0 for all q > 0.

(ii) Hq(G,F ) = 0 for all torsion G-modules F and all q > 2.

For the proof, cf. [Ser65] or [Ser68].

2 Tsen’s theoremDefinition 2.1. A field K is said to be C1 if every nonconstant homogeneouspolynomial f(x1, . . . , xn) of degree d < n has a non-trivial zero.

Proposition 2.2. If K is a C1 field, then Br(K) = 0.

We have to show that every skew field D, finite over K with center K, equalsK. Let r2 be the degree of D over K and Nrd: D → K the reduced norm. [Locallyfor the étale topology on K, D is isomorphic — non canonically — to a matrixalgebra Mr and the determinant map on Mr defines a reduced norm map on D.This is independent of the isomorphism chosen between D and Mr because allautomorphisms of Mr are inner and similar matrices have the same determinant.This map, defined locally for the étale topology, descends because of its localuniqueness, to a map Nrd: D → K.]

The only zero of Nrd is the zero element of D because, if x 6= 0, then Nrd(x) ·Nrd(x−1) = 1. On the other hand, if e1, . . . , er2 is a basis for D over K andx =

∑xiei, then the function Nrd(x) is a homogeneous polynomial Nrd(x1, . . . , xr2)

of degree r [this is clear locally for the étale topology]. As K is C1, we have r2 6 r,which implies that r = 1 and D = K.


Theorem 2.3 (Tsen). Let K be an extension of transcendence degree 1 of analgebraically closed field k. Then K is C1.

Suppose first that K = k(X). Let

f(T ) =∑

ai1,...inTi11 · · ·T inn

be a homogeneous polynomial of degree d < n with coefficients in k(X). Aftermultiplying the coefficients by a common denominator we may suppose that theyare in k[X]. Let δ = sup deg(ai1...in). We search by the method of undeterminedcoefficients for a non-trivial zero in k[X] by writing each Ti (i = 1, . . . , n) as apolynomial of degree N in X. Then the equation f(T ) = 0 becomes a system ofhomogeneous equations in the n × (N + 1) coefficients of the polynomials Ti(X)expressing the vanishing of the polynomial coefficients in X obtained by replacing Tiwith Ti(X). This polynomial is of degree at most δ +ND, so there are δ +Nd+ 1equations in n× (N + 1) variables. As k is algebraically closed this system has anon-trivial solution if n(N +1) > Nd+δ+1, which will be the case for N sufficientlylarge when d < n.

It is clear that, to prove the theorem in the general case, it suffices to proveit when K is a finite extension of a pure transcendental extension k(X) of k. Letf(T ) = f(T1, . . . , Tn) be a homogeneous polynomial of degree d < n with coefficientsin K. Let s = [K : k(X)] and let e1, . . . , es be a basis for K over k(X). Introducesn new variables Uij such that Ti =

∑Uijej . In order for the polynomial f(T ) to

have a non-trivial zero in K, it suffices that the polynomial g(Xij) = NK/k(f(T ))have a nontrivial zero in k(X). But g is a homogeneous polynomial of degree sd insn variables, whence the result.

Corollary 2.4. Let K be an extension of transcendence degree 1 of an algebraicallyclosed field k. Then the étale cohomology groups Hq(Spec(K),Gm) are zero forq > 0.

3 The cohomology of smooth curvesHenceforth, and unless expressly mentioned otherwise, the cohomology groupsconsidered are the étale cohomology groups.

Proposition 3.1. Let k be an algebraically closed field and X a connected nonsin-gular projective curve over k. Then

H0(X,Gm) = k×,H1(X,Gm) = Pic(X),Hq(X,Gm) = 0 for q > 2.

Let η be the generic point of X, j : η → X the canonical morphism, and Gm,η

the multiplicative group of the field of fractions k(X) of X. For a closed point xof X, let ix : x → X be the canonical immersion and Zx the constant sheaf with


value Z on x. Then j∗Gm,η is the sheaf of nonzero meromorphic functions on X and⊕x ix∗Zx is the sheaf of divisors. We therefore have an exact sequence of sheaves

0 Gm j∗Gm,η

⊕x ix∗Zx 0.div (3.3.1.1)

Lemma 3.2. We have Rqj∗Gm,η = 0 for all q > 0.

It suffices to show that the fiber of this sheaf at every closed point x of X is zero.If OX,x is the henselization of X at x and K the field of the fractions of OX,x, wehave

Spec(K) = η ×X Spec(OX,x),

and so (Rqj∗Gm,η)x = Hq(Spec(K),Gm). But K is an algebraic extension of k(X),and hence an extension of transcendence degree 1 of k. The lemma now follows from(2.4).

Lemma 3.3. We have Hq(X, j∗Gm,η) = 0 for all q > 0.

Indeed from (3.2) and the Leray spectral sequence for j, we deduce that

Hq(X, j∗Gm,η) = Hq(η,Gm,η)

for all q > 0, and the second term is zero for q > 0 by (2.4).

Lemma 3.4. We have Hq(X,⊕

x∈X ix∗Zx)

= 0 for all q > 0.

Indeed, for a closed point x of X, we have Rqix∗Zx = 0 for q > 0 because ix is afinite morphism (3.6), and so

Hq(X, ix∗Zx) = Hq(x,Zx).

The second term is zero for q > 0 because x is the spectrum of an algebraicallyclosed field. [The lemma is true more generally for all “skyscraper” sheaves on X.]

We deduce from the preceding lemmas and the exact sequence (3.3.1.1), equalities

Hq(X,Gm) = 0 for q > 2,

and an exact cohomology sequence in low degrees

1→ H0(X,Gm)→ H0(X, j∗Gm,η)→ H0(X,⊕

xix∗Zx)→ H1(X,Gm)→ 1,

which is none other than the exact sequence

1→ k× → k(X)× → Div(X)→ Pic(X)→ 1.

From Proposition 3.1 we deduce that the cohomology groups of X with values inZ/n, n prime to the characteristic of k, have the expected values.


Corollary 3.5. If X has genus g and n is invertible in k, then the groups Hq(X,Z/n)are zero for q > 2, and free over Z/n of rank 1, 2g, 1 for q = 0, 1, 2. Replacing Z/nwith the isomorphic group µn, we get canonical isomorphisms

H0(X,µn) = µn

H1(X,µn) = Pic0(X)nH2(X,µn) = Z/n.

As the field k is algebraically closed, Z/n is isomorphic (noncanonically) to µn.From the Kummer exact sequence

0→ µn → Gm → Gm → 0,

and Proposition 3.1, we deduce the equalities,

Hq(X,Z/n) = 0 for q > 2,

and, in low degrees, exact sequences

0 H0(X,µn) k× k× 0n

0 H1(X,µn) Pic(X) h2(X,µn) 0.

We also have an exact sequence

0 Pic0(X) Pic(X) Z 0,deg

and Pic0(X) can be identified with the group of k-rational points of an abelianvariety of dimension g, the Jacobian X. In such a group, multiplication by n issurjective with kernel a Z/nZ-free module of rank 2g (because n is invertible in k);whence the corollary.

A clever unwinding, using the “trace method,” allows one to obtain the followingcorollary.

Proposition 3.6 ([SGA 4, IX 5.7]). Let k be an algebraically closed field, X analgebraic curve over k, and F a torsion sheaf on X. Then

(i) Hq(X,F ) = 0 for q > 2.

(ii) If X is affine, then we even have Hq(X,F ) = 0 for q > 1.

For the proof, as well as for an exposition of the “trace method”, we refer to[SGA 4, IX 5].

4 UnwindingsTo calculate the cohomology of varieties of dimension > 1 we fiber by curves, whichreduces the problem to the study of morphisms with fibers of dimension 6 1. Thisprinciple has several variants. We mention some of them.


4.1

Let A be a k-algebra of finite type with generators a1, . . . , an. If we put

X0 = Spec(k), . . . , Xi = Spec(k[a1, . . . , ai]), . . . , Xn = Spec(A),

then the canonical inclusions k[a1, . . . , ai]→ k[a1, . . . , ai, ai+1] define morphisms

Xn → Xn−1 → · · · → X1 → X0

whose fibers are of dimension 6 1.

4.2

In the case of a smooth morphism, we can be more precise. We define an elementaryfibration to be a morphism of schemes f : X → S that can be embedded in acommutative diagram

X X Y

S

j

ff

i

g

satisfying the following conditions:

(i) j is an open immersion dense in each fiber and X = X r Y ;

(ii) f is smooth and projective with irreducible geometric fibers of dimension 1;

(iii) g is a finite étale covering with no empty fibers.

We define a good neighborhood relative to S to be an S-scheme X such that thereexist S-schemes X = Xn, . . . , X0 = S and elementary fibrations fi : Xi → Xi−1,i = 1, . . . , n. One can show [SGA 4, XI 3.3] that, if X is a smooth scheme overan algebraically closed field k, then every rational point of X admits an openneighborhood that is a good neighborhood (relative to Spec(k)).

4.3

We can unwind a proper morphism f : X → S as follows. According to Chow’slemma, there exists a commutative diagram

X X

Sf

π

f

where π and f are projective morphisms, π being moreover an isomorphism over adense open of X. Locally on S, X is a closed subscheme of a projective space oftype Pn

S .

IV THE PROPER BASE CHANGE THEOREM 24

We unwind this last statement by considering the projection ϕ : PnS → P1

S thatsends the point with homogeneous coordinates (x0, x1, . . . , xn) to (x0, x1). It is arational map defined outside the closed subset Y ' Pn−2

S of PnS with homogeneous

equations x0 = x1 = 0. Let u : P → PnS be the blow up with center Y ; the fibers

of u have dimension 6 1. Moreover, there exists a natural morphism v : P → P1S

extending the rational map ϕ, and v makes P a P1S-scheme locally isomorphic to

the projective space of type Pn−1, which can in turn be projected onto P1, etc.

4.4

A smooth projective variety X can be fibered by a Lefschetz pencil. The blow up Xof the intersection of the axis of the pencil with X projects onto P1, and the fibersof this projection are the hyperplane sections of X by the hyperplanes of the pencil.

IV The proper base change theorem1 IntroductionThis chapter is devoted to the proof and applications of

Theorem 1.1. Let f : X → S be a proper morphism of schemes and F a torsionabelian sheaf on X. For every q > 0, the fiber of Rqf∗F at a geometric point s of Sis isomorphic to the cohomology Hq(Xs,F ) of the fiber Xs = X ⊗S Spec k(s) of fat s.

For f : X → S a separated proper continuous map (separated means that thediagonal of X ×S X is closed) between topological spaces and F an abelian sheafon X, the analogous result is well known and elementary: as f is closed, the setsf−1(V ) for V a neighborhood of s form a fundamental system of neighborhoodsof Xs, and it can be shown that H•(Xs,F ) = lim−→U

H•(U,F ) with U running overthe neighborhoods of Xs. In practice, Xs even has a fundamental system U ofneighborhoods U for which it is a deformation retract, and, for F constant, wetherefore have H•(Xs,F ) = H•(U,F ). In pictorial terms: the special fiber swallowsthe general fiber.

In the case of schemes the proof is more delicate and it is essential to assumethat F is torsion [SGA 4, XII.2]. Taking account of the description of the fibers ofRqf∗F (II.3.3), we see that the Theorem 1.1 is essentially equivalent to

Theorem 1.2. Let A be a strictly henselian local ring and S = Spec(A). Letf : X → S be a proper morphism with closd fiber X0. Then, for every torsion abeliansheaf F on X and q > 0,

Hq(X,F ) ∼−−→ Hq(X0,F ).

By passage to the limit, we see that it suffices to prove the theorem when Ais the strict henselization of a Z-algebra of finite type at a prime ideal. We first


treat the case q = 0 or 1 and F = Z/n (§2). An argument based on the notionof a constructible sheaf (§3) shows that it suffices to consider the case that F isconstant. On the other hand, the unwinding (III.4.3) allows us to assume that X0 isa curve. In this case it only remains to prove the theorem for q = 2 (§4).

Among other applications (§6), the theorem makes it possible to define the notionof cohomology with proper support (§5).

2 Proof for q = 0 or 1 and F = Z/n

The result for q = 0 and F constant is equivalent to the following proposition[Zariski connectedness theorem].

Proposition 2.1. Let A be a noetherian henselian local ring and S = Spec(A). Letf : X → S be a proper morphism and X0 the closed fiber of f . Then the sets ofconnected components π0(X) and π0(X0) are in natural bijection.

Proving this amounts to showing that the sets of subsets of X and X0 that areboth open and closed, Of(X) and Of(X0), are in natural bijection. We know thatthe set Of(X) is in natural bijection with the set of idempotents of Γ(X,OX), andsimilarly Of(X0) is in natural bijection with the set of idempotents of Γ(X0,OX0).It is therefore a question of showing that the canonical map

Idem Γ(X,OX)→ Idem Γ(X0,OX0)

is bijective.Let m denote the maximal ideal of A, Γ(X,OX)∧ the completion of Γ(X,OX) for

the m-adic topology, and, for every integer n > 0, Xn = X ⊗A A/mn+1. Accordingto the finiteness theorem for proper morphisms [EGA, III, 3.2], Γ(X,OX) is a finiteA-algebra; as A is henselian, it follows that the canonical map

Idem Γ(X,OX)→ Idem Γ(X,OX)∧

is bijective.According to the comparison theorem for proper morphisms [EGA, III, 4.1], the

canonical mapΓ(X,OX)∧ → lim←−Γ(Xn,OXn)

is bijective. In particular, the canonical map

Idem Γ(X,OX)∧ → lim←− Idem Γ(Xn,OXn)

is bijective. But, since Xn and X0 have the same underlying topological space, thecanonical map

Idem Γ(Xn,OXn)→ Idem Γ(X0,OX0)

is bijective for all n, which completes the proof.As H1(X,Z/n) is in natural bijection with the set of isomorphism classes of Galois

finite étale coverings of X with group Z/n, the theorem for q = 1 and F = Z/nresults from the following proposition.


Proposition 2.2. Let A be a noetherian henselian local ring and S = Spec(A). Letf : X → S be a proper morphism and X0 the closed fiber of f . Then the restrictionfunctor7

Rev. et(X)→ Rev. et(X0)

is an equivalence of categories.

[If X0 is connected and we have chosen a geometric point of X0 as base point,this amounts to saying that the canonical map π1(X0)→ π1(X) on the (profinite)fundamental groups is bijective.]

Proposition 2.1 shows that this functor is fully faithful. Indeed, if X ′ and X ′′ aretwo finite étale coverings of X, then an X-morphism from X ′ to X ′′ is determinedby its graph, which is an open and closed subset of X ′ ×X X ′′.

It remains to show that every finite étale covering X ′0 of X0 extends to X. Thefinite étale coverings do not depend on nilpotent elements [SGA 1, Chapt.1], andso X ′0 lifts uniquely to a finite étale coverng X ′n of Xn for all n > 0, i.e., to afinite étale covering X′ of the formal scheme X obtained by completing X along X0.According to the Grothendieck’s theorem on the algebraization of formal coherentsheaves [existence theorem, [EGA, III.5]], X′ is the formal completion of a finiteétale covering X ′ of X = X ⊗A A.

By passage to the limit, it suffices to prove the proposition in the case thatA is the henselization of a Z-algebra of finite type. We can then apply Artin’sapproximation theorem to the functor F : (A-algebras) → (sets) which, to an A-algebra B, makes correspond the set of isomorphism classes of finite étale coveringsof X ⊗A B. Indeed, this functor is locally of finite presentation: if Bi is an filteredinductive system of A-algebras and B = lim−→Bi, then F (B) = lim−→F (Bi). Accordingto Artin’s theorem, given an element ξ ∈ F (A), there exists a ξ ∈ F (A) having thesame image as ξ in F (A/m). When we take ξ to be the isomorphism class of X ′,this gives us a finite étale covering X ′ of X whose restriction to X0 is isomorphic toX ′0.

3 Constructible sheavesIn this paragraph, X is a noetherian scheme, and “sheaf on X” means an abeliansheaf on Xet.

Definition 3.1. We say that a sheaf F on X is locally constant constructible(abbreviated l.c.c) if it is represented by a finite étale covering of X.

Definition 3.2. We say that a F sheaf on X is constructible if it satisfies thefollowing equivalent conditions:

(i) There exists a finite surjective family of subschemes Xi of X such that therestriction of F to Xi is l.c.c..

7A “revêtement étale” is a “finite étale covering”, and so these are the categories of finite étalecoverings.


(ii) There exists a finite family of finite morphisms pi : X ′i → X, for each i aconstant constructible sheaf (= defined by a finite abelian group) Ci overX ′i, and a monomorphism F →

∏pi∗Ci.

It is easily checked that the category of constructible sheaves on X is abelian.Moreover, if u : F → G is a homomorphism of sheaves and F is constructible, thenthe sheaf im(u) is constructible.

Lemma 3.3. Every torsion sheaf F is a filtered inductive limit of constructiblesheaves.

Indeed, if j : U → X is étale of finite type, then an element ξ ∈ F (U) such thatnξ = 0 defines a homomorphism of sheaves j!Z/n

U→ F whose image (the smallest

subsheaf of F having ξ as a local section) is a constructible subsheaf of F . It isclear that F is the inductive limit of such subsheaves.

Definition 3.4. Let C be an abelian category and T a functor from C to the categoryof abelian groups. We say that T is effaceable in C if, for every object A of C andevery α ∈ T (A), there exists a monomorphism u : A→M in C such that T (u)α = 0.

Lemma 3.5. The functors Hq(X,−) for q > 0 are effaceable in the category ofconstructible sheaves on X.

It suffices to remark that, if F is a constructible sheaf, then there exists an integern > 0 such that F is a sheaf of Z/n-modules. Then there exists a monomorphismF → G , where G is a sheaf of Z/n-modules such that Hq(X,G ) = 0 for all q > 0.We can, for example, take G to be the Godement resolution

∏x∈X ix∗Fx, where

x runs through the points of X and ix : x → X is a geometric point centered atx. According to (3.3) G is an inductive limit of constructible sheaves, whence thelemma, because the functors Hq(X,−) commute with inductive limits.

Lemma 3.6. Let ϕ• : T • → T ′• be a morphism of cohomological functors defined

on an abelian category C with values in the category of abelian groups. Suppose thatT q is effaceable for q > 0 and let E be a subset of objects of C such that every objectof C is contained in an object belonging to E. Then the following conditions areequivalent:

(i) ϕq(A) is bijective for all q > 0 and all A ∈ Ob C.

(ii) ϕ0(M) is bijective and ϕq(M) is surjective for all q > 0 and all M ∈ E.

(iii) ϕ0(A) is bijective for all A ∈ Ob C and T ′q is effaceable for all q > 0.

The proof is by induction on q and does not present difficulties.

Proposition 3.7. Let X0 be a subscheme of X. Suppose that, for all n > 0 andevery X ′ scheme finite over X, the canonical map

Hq(X ′,Z/n)→ Hq(X ′0,Z/n), X ′0 = X ′ ×X X0,


is bijective for q = 0 and surjective for q > 0. Then, for every torsion sheaf F onX and all q > 0, the canonical map

Hq(X,F )→ Hq(X0,F )

is bijective.

By passage to the limit, it suffices to prove the assertion for a constructible F .We apply Lemma 3.6 taking C to be the category of constructible sheaves on X, T qto be Hq(X,−), T ′q to be Hq(X0,−), and E to be the set of constructible sheaves ofthe form

∏pi∗Ci, where pi : X ′i → X is a finite morphism and Ci is a finite constant

sheaf on X ′i.

4 End of the proofBy the method of fibering by curves (III.4.3), it suffices to prove the theorem inrelative dimension 6 1. According to the preceding paragraph, it suffices to showthat, if S is the spectrum of a strictly henselian local noetherian ring, f : X → S aproper morphism whose closed fiber X0 is of dimension 6 1, and n an integer > 0,then the canonical homomorphism

Hq(X,Z/n)→ Hq(X0,Z/n)

is bijective for q = 0 and surjective for q > 0.We saw the cases q = 0 and 1 earlier, and we know that Hq(X0,Z/n) = 0 for

q > 3; it therefore suffices to treat the case q = 2. We can obviously assume that nis a power of a prime number. If n = pr, where p is the residue field characteristicof S, then Artin-Schreier theory shows that we have H2(X0,Z/pr) = 0. If n = `r,` 6= p, we deduce from Kummer theory a commutative diagram

Pic(X) H2(X,Z/`r)

Pic(X0) H2(X0,Z/`r)

α

β

where the map β is surjective. [We saw this in Chapter III for a smooth curveover an algebraically closed field, but similar arguments apply to any curve over aseparably closed field.]

To conclude, it suffices to show

Proposition 4.1. Let S be the spectrum of a henselian noetherian local ring andf : X → S a proper morphism whose closed fiber X0 is of dimension 6 1. Then thecanonical restriction map

Pic(X)→ Pic(X0)

is surjective. [In fact, this holds with f any separated morphism of finite type.]


To simplify the proof, we will assume that X is integral, although this is notnecessary. As every invertible sheaf on X0 is associated with a Cartier divisor(because X0 is a curve, and so quasi-projective), it suffices to show that the canonicalmap Div(X)→ Div(X0) is surjective.

Every divisor on X0 is a linear combination of divisors whose support is concen-trated in a single non-isolated closed point of X0. Let x be such a point, t0 ∈ OX0,x

a non-invertible regular element of OX0,x, and D0 the divisor with support in xand local equation t0. Let U be an open neighborhood of x in X such that thereexists a section t ∈ Γ(U,OU ) lifting t0. Let Y be the closed subscheme of U withequation t = 0; by taking U sufficiently small, we can assume that x is the onlypoint of Y ∩X0. Then Y is quasi-finite above S at x. As S is the spectrum of alocal henselian ring, we deduce that Y = Y1 q Y2, where Y1 is finite over S and Y2does not meet X0. In addition, as X is separated over S, Y1 is closed in X.

After replacing U with a smaller open neighborhood of x, we can assume thatY = Y1, that is, that Y is closed in X. We then define a divisor D on X liftingD0 by putting D|X r Y = 0 and D|U = div(t), which makes sense because t isinvertible on U r Y .

4.2 Remark

In the case that f is proper, we could also make a proof in the same style as that ofthe Proposition 2.2. Indeed, as X0 is a curve, there is no obstruction to lifting aninvertible sheaf on X0 to the infinitesimal neighborhoods Xn of X0, and so to theformal completion X of X along X0. We can then conclude by applying successivelyGrothendieck’s existence theorem and Artin’s approximation theorem.

5 Cohomology with proper supportDefinition 5.1. Let X be a separated scheme of finite type over a field k. Accordingto a theorem of Nagata, there exists a scheme X over k and an open immersionj : X → X. For a torsion sheaf F on X, we let j!F denote the extension by 0 ofF to X, and we define the cohomology groups with proper support Hq

c(X,F ) bysetting

Hqc(X,F ) = Hq(X, j!F ).

We show that this definition is independent of compactification j : X → X chosen.Let j1 : X → X1 and j2 : X → X2 be two compactifications. Then X maps intoX1 × X2 by x 7→ (j1(x), j2(x)), and the closed image X3 of X by this map is a


compactification of X. We have a commutative diagram

X1

X X3 ,

X2

j1

(j1,j2)

j2

p1

p2

where p1 and p2, the restrictions of the natural projections to X3, are propermorphisms.

It suffices therefore to treat the case where we have a commutative diagram

X2

X

X1

p

j2

j1

with p a proper morphism.

Lemma 5.2. We have p∗(j2!F ) = j1!F and Rqp∗(j2!F ) = 0, for q > 0 .

Note immediately that the lemma completes the proof because, from the Lerayspectral sequence of the morphism p, we can deduce that, for all q > 0,

Hq(X2, j2!F ) = Hq(X1, j1!F ).

To prove the lemma, we argue fiber by fiber using the base change theorem (1.1)for p. The result is immediate because, over a point of X, p is an isomorphism, andover a point of X1 rX, the sheaf j2!F is zero on the fiber of p.

5.3

Similarly, for a separated morphism of finite type of noetherian schemes f : X → S,there exists a proper morphism f : X → S and an open immersion j : X → X overS. We then define the higher direct images with proper support of a torsion sheafF on X by setting

Rqf!F = Rqf∗(j!F ).One can check as before that this definition is independent of the chosen com-

pactification.

Theorem 5.4. Let f : X → S be a separated morphism of finite type of noetherianschemes and F a torsion sheaf on X. Then the fiber of Rqf!F at a geometric points of S is canonically isomorphic to the cohomology with proper support Hq

c(Xs,F )of the fiber Xs of f at s.


This is a simple variant of the base change theorem for a proper morphism (1.1).More generally, if

X X ′

S S′

f

g′

f ′

g

is a cartesian diagram, we have a canonical isomorphism

g∗(Rqf!F ) ' Rqf ′! (g′∗F ). (4.5.4.1)

6 ApplicationsVanishing Theorem 6.1. Let f : X → S be a separated morphism of finite typewhose fibers are of dimension 6 n and F a torsion sheaf on X. Then Rqf!F = 0for q > 2n.

Thanks to the base change theorem, we can assume that S is the spectrum of aseparably closed field. If dimX = n, then there is an open affine U of X such thatdim(X r U) < n. We then have an exact sequence 0 → FU → F → FXrU → 0and, by induction on n, it suffices to prove the theorem for X = U affine. Thenthe method of fibering by curves (III.4.1) and the base change theorem reduce theproblem to the case of a curve over a separably closed field for which we can deducethe desired result from Tsen’s theorem (III.3.6).

Finiteness Theorem 6.2. Let f : X → S be a separated morphism of finite typeand F a constructible sheaf on X. Then the sheaves Rqf!F are constructible.

We consider only the case where F is killed by an integer invertible on X.Proving the theorem comes down to the case where F is a constant sheaf Z/n and

f : X → S is a smooth proper morphism whose fibers are geometrically connectedcurves of genus g. For n invertible on X, the sheaves Rqf∗F are then locally freeof finite rank and zero for q > 2 (6.1). Replacing Z/n with the locally isomorphicsheaf (on S) µn, we have canonically

R0f∗µn = µn

R1f∗µn = Pic(X/S)nR2f∗µn = Z/n.

(4.6.2.1)

Theorem 6.3 (comparison with the classical cohomology). Let f : X → S be aseparated morphism of schemes of finite type over C and F a torsion sheaf on X.We use the exponent an to denote the functor of passing to the usual topological spaces,and Rqfan

! the derived functors of the direct-image functor with proper support fan.Then

(Rqf!F )an ' Rqfan! F an.

In particular, for S = a point and F the constant sheaf Z/n,

Hqc(X,Z/n) ' Hq

c(Xan,Z/n).

V LOCAL ACYCLICITY OF SMOOTH MORPHISMS 32

Unwinding using the base change theorem brings us back to case that X is asmooth proper curve with S = a point and F = Z/n. The relevant cohomologygroups are then zero for q 6= 0, 1, 2, and we can invoke GAGA ([GAGA]): indeed, if Xis proper over C, we have π0(X) = π0(Xan) and π1(X) = profinite completion of π1(Xan),whence the assertion for q = 0, 1. For q = 2, we use the Kummer exact sequenceand the fact that, by GAGA again, Pic(X) = Pic(Xan).

Theorem 6.4 (cohomological dimension of affine schemes). Let X be an affinescheme of finite type over a separably closed field and F a torsion sheaf on X. ThenH1(X,F ) = 0 for q > dim(X).

For the very pretty proof, we refer to [SGA 4, XIV §2 and 3].

6.5 Remark

This theorem is in a way a substitute for Morse theory. Indeed, consider the classicalcase, where X is affine and smooth over C and is embedded in an affine space oftype CN . Then, for almost all points p ∈ CN , the function “distance to p” on X isa Morse function and the indices of its critical points are less than dim(X). Thus Xis obtained by gluing handles of index smaller than dim(X), whence the classicalanalog of (6.4).

V Local acyclicity of smooth morphismsLet X be a complex analytic variety and f : X → D a morphism from X to the disk.We denote by [0, t] the closed line segment with endpoints 0 and t in D and by (0, t]the semi-open segment. If f is smooth, then the inclusion

j : f−1 ((0, t]) → f−1 ([0, t])

is a homotopy equivalence: we can push the special fiber X0 = f−1(0) into f−1([0, t]).In practice, for t small enough, f−1((0, t]) will be a fiber bundle on (0, t] so that

the inclusionXt = f−1(t) → f−1 ((0, t])

will also be a homotopy equivalence. We then define the cospecialization morphismto be the homotopy class of maps

cosp: X0 → f−1 ([0, t])∼≈←− f−1 ((0, t])

∼≈←− Xt.

This construction can be expressed in pictorial terms by saying that for a smoothmorphism, the general fiber swallows the special fiber.

Let us assume no longer that f is necessarily smooth (but assume that f−1((0, t])is a fiber bundle over (0, t]). We can still define a morphism cosp• on cohomology


provided j∗Z = Z and Rqj∗Z = 0 for q > 0. Under these assumptions, the Lerayspectral sequence for j shows that we have

H•(f−1 ([0, t]) ,Z

) ∼−−→ H•(f−1 ((0, t]) ,Z

)and cosp• is the composite morphism

cosp• : H•(Xt,Z) ∼←− H•(f−1 ((0, t]) ,Z

) ∼←− H•(f−1 ([0, t]) ,Z

)→ H•(X0,Z).

The fiber of Rqj∗Z at a point x ∈ X0 is computed as follows. We take in theambient space a ball Bε with center x and sufficiently small radius ε, and for ηsufficiently small, we put E = X ∩Bε ∩ f−1(ηt); it is the variety of vanishing cyclesat x. We have

(Rqj∗Z)x∼←− Hq

(X ∩Bε ∩ f−1 ((0, ηt]) ,Z

) ∼−→ Hq(E,Z),

and the cospecialization morphism is defined in cohomology when the varieties ofvanishing cycles are acyclic [H0(E,Z) = Z and Hq(E,Z) = 0 for q > 0], which canbe expressed by saying that f is locally acyclic.

This chapter is devoted to the analogue of this situation for a smooth morphismof schemes and the étale cohomology. However it is essential in this context to restrictthe coefficients to be torsion of order prime to the residue characteristics. Paragraph1 is devoted to generalities on locally acyclic morphisms and cospecialization maps.In paragraph 2, we show that a smooth morphism is locally acyclic. In paragraph 3,we combine this result with those of the preceding chapter to deduce two applications:a specialization theorem for cohomology groups (the cohomology of the geometricfibers of a proper smooth morphism is locally constant) and a base change theoremfor a smooth morphism.

In the following, we fix an integer n, and “scheme” means “scheme on whichn is invertible”. “Geometric point” will always mean “algebraic geometric point”x : Spec(k)→ X with k algebraically closed (II.3.1).

1 Locally acyclic morphisms1.1 Notation

Given a S scheme and a geometric point s of S, we let Ss denote the spectrum ofthe strict localization of S at s.

Definition 1.2. We say that a geometric point t of S is a generization of s if itis defined by an algebraic closure of the residue field of a point of Ss. We also saythat s is a specialization of t, and we call the S-morphism t→ Ss the specializationarrow.

Definition 1.3. Let f : X → S be a morphism of schemes. Let s be a geometric pointof S, t a generization of s, x a geometric point of X above s, and Xx

t = Xx ×Sst.

Then we say that Xxt is a variety of vanishing cycles of f at the point x.


We say that f is locally acyclic if the reduced cohomology of every variety ofvanishing cycles Xx

t is zero:

H•(Xx

t ,Z/n) = 0, (5.1.3.1)

i.e., H0(Xxt ,Z/n) = Z/n and Hq(Xx

t ,Z/n) = 0 for q > 0.

Lemma 1.4. Let f : X → S be a locally acyclic morphism and g : S′ → S a quasi-finite morphism (or projective limit of quasi-finite morphisms). Then the morphismf ′ : X ′ → S′ deduced from f by base change is locally acyclic.

One can show that, in fact, every variety of vanishing cycles of f ′ is a variety ofvanishing cycles of f .

Lemma 1.5. Let f : X → S be a locally acyclic morphism. For every geometricpoint t of S and corresponding cartesian diagram

Xt X

t S,

ε′

f

ε

we have ε′∗Z/n = f∗ε∗Z/n and Rqε′∗Z/n = 0 for q > 0.

Let S be the closure ε(t) and S′ the normalization of S in k(t). Consider thecartesian diagram

Xt X ′ X

t S′ S.

i′ α′

f ′ f

i α

The local rings of S′ are normal with separably closed fields of fractions. They aretherefore strictly henselian, and the local acyclicity of f ′ (1.4) gives i′∗Z/n = Z/n,Rqi′∗Z/n = 0 for q > 0. As α is integral, we then have

Rqε′∗Z/n = α′∗Rqi′∗Z/n = α′∗f′∗Rqi∗Z/n = f∗α∗Rqi∗Z/n = f∗Rqε∗Z/n,

and the lemma follows

1.6

Given a locally acyclic morphism f : X → S and a specialization arrow t→ Ss, wewill define canonical homomorphisms, called cospecialization maps

cosp• : H•(Xt,Z/n)→ H•(Xs,Z/n)

relating the cohomology of the general fiber Xt = X ×S t to that of the special fiberXs = X ×X s.


Consider the cartesian diagram

Xt X Xs

t Ss s.

ε′

f ′

ε

deduced from f by base change. According to (1.4), f ′ is still locally acyclic. Fromthe definition of local acyclicity, we deduce immediately that the restriction to Xs ofthe sheaf Rqε′∗Z/n is Z/n for q = 0, and 0 for q > 0. By (1.5), we even know thatRqε′∗Z/n = 0 for q > 0. We define cosp• to be the composite

H•(Xt,Z/n) ' H•(X, ε′∗Z/n)→ H•(Xs,Z/n) . (5.1.6.1)

Variant Let S be the closure of ε(t) in Ss, let S′ be the normalization of S ink(t), and let X ′/S′ be deduced from X/S by base change. The display (5.1.6.1) canstill be written

H•(Xt,Z/n) ' H•(X ′,Z/n)→ H•(Xs,Z/n).

Theorem 1.7. Let S be a locally noetherian scheme, s a geometric point of S, andf : X → S a morphism. We assume that

a) the morphism f is locally acyclic,

b) for every specialization arrow t→ Ss and for all q > 0, the cospecializationmaps Hq(Xt,Z/n)→ Hq(Xs,Z/n) are bijective.

Then the canonical homomorphism (Rqf∗Z/n)s → Hq(Xs,Z/n) is bijective for allq > 0.

In proving the theorem, we can clearly assume that S = Ss. We will in factshow that, for every sheaf of Z/nZ-modules F on S, the canonical homomorphismϕq(F ) : (Rqf∗f

∗F )s → Hq(Xs, f∗F ) is bijective.

Every sheaf of Z/nZ-modules is a filtered inductive limit of constructible sheavesof Z/nZ-modules (IV.3.3). Moreover, every constructible sheaf of Z/nZ-modulesembeds into a sheaf of the form

∏iλ∗Cλ, where (iλ : tλ → S) is a finite family of

generizations of s and Cλ is a free Z/nZ-module of finite rank over tλ. Accord-ing to the definition of the cospecialization maps, condition b) means that thehomomorphisms ϕq(F ) are bijective if F is of this form.

We conclude with the help of a variant of lemma (IV.3.6):

Lemma 1.8. Let C be an abelian category in which filtered inductive limits exist.Let ϕ• : T • → T ′

• be a morphism of cohomological functors commuting with filteredinductive limits, defined on C and with values in the category of abelian groups.Suppose that there exist two subsets D and E of objects of C such that


a) every object of C is a filtered inductive limit of objects belonging at D,

b) every object belonging to D is contained in an object belonging to E.

Then the following conditions are equivalent:

(i) ϕq(A) is bijective for all q > 0 and all A ∈ Ob C.

(ii) ϕq(M) is bijective for all q > 0 and all M ∈ E.

The lemma is proved by passing to the inductive limit, using induction on q,and by repeated application of the five-lemma to the diagram of exact cohomologysequences deduced from an exact sequence 0 → A → M → A′ → 0, with A ∈ D,M ∈ E , A′ ∈ Ob C.

Corollary 1.9. Let S be the spectrum of a strictly henselian local noetherian ringand f : X → S a locally acyclic morphism. Suppose that, for every geometric point tof S we have H0(Xt,Z/n) = Z/n and Hq(Xt,Z/n) = 0 for q > 0 (i.e., the geometricfibers of f are acyclic). Then f∗Z/n = Z/n and Rqf∗Z/n = 0 for q > 0.

Corollary 1.10. Let f : X → Y and g : Y → Z be morphisms of locally noetherianschemes. If f and g are locally acyclic, then so also is g f .

We may suppose that X, Y , and Z are strictly local and that f and g are localmorphisms. We have to show that, if z is an algebraic geometric point of Z, thenH0(Xz,Z/n) = Z/n and Hq(Xz,Z/n) = 0 for q > 0.

As g is locally acyclic, we have H0(Yz,Z/n) = Z/n and Hq(Yz,Z/n) = 0 forq > 0. Moreover, the morphism fz : Xz → Yz is locally acyclic (1.4) and its geometricfibers are acyclic because they are varieties of vanishing cycles of f . From (1.9), wehave Rqfz∗Z/n = 0 for q > 0. Moreover, fz∗Z/n is constant with fibre Z/n over Yz.We conclude with the help of the Leray spectral sequence of fz.

2 Local acyclicity of a smooth morphismTheorem 2.1. Smooth morphisms are locally acyclic.

Let f : X → S be a smooth morphism. The assertion is local for the étaletopology on X and S, so we can assume that X is affine space of dimension d overS. By passage to the limit, we can assume that S is noetherian, and the transitivityof local acyclicity (1.10) shows that it suffices to treat the case d = 1.

Let s be a geometric point of S and x a geometric point of X centered at a closedpoint of Xs. We have to show that the geometric fibers of the morphism Xx → Ss

are acyclic. We now put S = Ss = Spec(A) and X = Xx. We have X ' SpecAT,where AT is the henselization of A[T ] at the point T = 0 above s.

If t is a geometric point of S, then the fiber Xt is a projective limit of smoothaffine curves over t. Therefore Hq(Xt,Z/n) = 0 for q > 2 and it suffices to show thatH0(Xt,Z/n) = Z/n and H1(Xt,Z/n) = 0 for n prime to the residue characteristicof S. This follows from the next two propositions.


Proposition 2.2. Let A be a strictly henselian local ring, S = Spec(A), andX = SpecAT. Then the geometric fibers of X → S are connected.

By passage to the limit, we need only consider the case that A is a stricthenselization of a Z-algebra of finite type.

Let t be a geometric point of S, localized at t, and k′ a finite separable extensionof k(t) in k(t). We put t′ = Spec(k′) and Xt′ = X ×S Spec(k′). We have to showthat, for all t and t′, Xt′ is connected (by which we mean connected and nonempty).Let A′ be the normalization of A in k′, i.e., the ring of elements of k′ integral overthe image of A in k(t). We have AT ⊗A A′

∼−−→ A′T: the ring on the left isindeed henselian local (because A′ is finite over A and local) and a limit of étalelocal algebras over A′[T ] = A[T ]⊗A A′. The scheme Xt′ is again the fiber at t′ ofX ′ = Spec (A′T) on S′ = Spec(A′). The local scheme X ′ is normal, thereforeintegral; its localization Xt′ is still integral, a fortiori, connected.

Proposition 2.3. Let A be a strictly henselian local ring, S = Spec(A), andX = Spec (AT). Let t be a geometric point of S and Xt the correspondinggeometric fiber. Then every finite étale Galois covering of Xt of order prime to thecharacteristic of the residue field of A is trivial.

Lemma 2.3.1 (Zariski-Nagata purity theorem in dimension 2). Let C be a regularlocal ring of dimension 2 and C ′ a finite normal C-algebra, étale over the opencomplement of a closed point of Spec(C). Then C ′ is étale over C.

Indeed C ′ is normal of dimension 2, so prof(C ′) = 2. As prof(C ′)+dim proj(C ′) =dim(C) = 2, we conclude that C ′ is free over C. Then the set of points of C whereC ′ is ramified is defined by a single equation, namely, the discriminant. As it doesnot contain a point of height 1, it is empty.

Lemma 2.3.2 (special case of the Abhyankar lemma). Let V be a discrete valuationring, S = Spec(V ), π a uniformizer, η the generic point of S, X irreducible andsmooth over S of relative dimension 1, X1 a finite étale Galois covering Xη of degreen invertible on S, and S1 = Spec

(V [π1/n]

). Denote by a subscript 1 base change

from S to S1. Then, X1η extends to a finite étale covering of X1.

Let X1 be the normalization of X1 in X1η. In view of the structure of the tameinertia groups of the discrete valuation rings that are localizations of X at genericpoints of the special fiber Xs, we see that X1 is étale over X1 over the generic fiberand at generic points of the special fiber. By (2.3.1), it is étale everywhere.

2.3.3.

We now prove Proposition 2.3. Let t denote the point at which t is localized.We are free to replace A with the normalization of A in a finite separable extensionk(t′) of k(t) in k(t) (cf. 2.2). This, and a preliminary passage to the limit, allow usto assume that

a) A is noetherian and normal, and t is the generic point of S.


b) The finite étale covering of Xt in question comes from a finite étale coveringof Xt.

c) In fact, it comes from a finite étale covering XU of the inverse image XU of anonempty open U of S (consequence of b): t is the limit of the U).

d) The complement of U is of codimension > 2 (this at the cost of enlarging k(t);apply (2.3.2) to the discrete valuation rings obtained by localizing S at pointsof S r U of codimension 1 in S; there are only finitely many such points).

e) The finite étale covering XU is trivial above the subscheme T = 0 (thisrequires that k(t) be further enlarged).

When these conditions are met, we will see that the finite étale covering XU istrivial.

Lemma 2.3.4. Let A be a noetherian normal strictly local henselian ring, U anopen of Spec(A) whose complement has codimension > 2, V its inverse image inX = Spec (AT), and V ′ a finite étale covering of V . If V ′ is trivial over T = 0,then V ′ is trivial.

Let B = Γ(V ′,O). As V ′ is the inverse image of V in Spec(B), it suffices toshow that B is finite and étale over AT (hence decomposed as AT is strictlyhenselian). Let X = AJT K, and denote by (−)∧ base change from X to X. Thescheme X is faithfully flat over X. Therefore Γ(V ′,O) = B ⊗AT AJT K, and itsuffices to show that this ring B is finite and étale over AJT K.

Let Vm (resp. V ′m) be the subscheme of V (resp. V ′) defined by the equationTm+1 = 0. By hypothesis, V ′0 is a trivial finite étale covering of V0, i.e., a sum of ncopies V0. Likewise V ′m/Vm is trivial because finite étale coverings are insensitive tonilpotents. We deduce a map

ϕ : Γ(V ′,O)→ lim←−m

Γ(V ′m,O) = (lim←−m

Γ(V,O))n.

By hypothesis, the complement of U is of depth > 2: we have Γ(Vm,O) =A[T ]/(Tm+1), and ϕ is a homomorphism from B to AJT Kn. Over U , it provides ndistinct sections of V ′/V ; it follows that V ′ is trivial, i.e., a sum of n copies of V .The complement of V in X still being of codimension > 2 (therefore of depth > 2),we deduce that B = AJT Kn, whence the lemma.

3 ApplicationsTheorem 3.1 (specialization of cohomology groups). Let f : X → S be a properlocally acyclic morphism, for example, a proper smooth morphism. Then the sheavesRqf∗Z/n are constant and locally constructible and for every specialization arrowt→ Ss, the cospecialization arrows Hq(Xt,Z/n)→ Hq(Xs,Z/n) are bijective.


This follows immediately from the definition of the cospecialization morphismsand the finiteness and base change theorems for proper morphisms.

Theorem 3.2 (smooth base change). Let

X ′ X

S′ S

g′

f ′ f

g

be a cartesian diagram with g smooth. For every torsion sheaf F on X whose torsionis prime to the residue characteristics of S,

g∗Rqf∗F∼−−→ Rqf ′∗(g′∗F )

By passing to an open covering of X, we may suppose that X is affine, and thenby passing to the limit, that X is of finite type on S. Then f factors into an openimmersion j : X → X and a proper morphism f : X → S. From the Leray spectralsequence for f j and the base change theorem for proper morphisma, we deducethat it suffices to prove the theorem in the case where X → S is an open immersion.

In this case, if F is of the form ε∗C, where ε : t→ X is a geometric point of X,then the theorem is a corollary of (1.5). The general case follows from Lemma 1.8.

Corollary 3.3. Let K/k be an extension of separably closed fields, X a k-scheme,and n an integer prime to the characteristic of k. Then the canonical map Hq(X,Z/n)→Hq(XK ,Z/n) is bijective for all q > 0.

It suffices to remark that K is an inductive limit of smooth k-algebras.

Theorem 3.4 (relative purity). Consider a diagram

U X Y

S

j

f

i

h(5.3.4.1)

with f smooth of pure relative dimension N , h smooth of pure relative dimension N−1, i a closed immersion, and U = X r Y . For n prime to the residue characteristicsof S, we have

j∗Z/n = Z/nR1j∗Z/n = Z/n(−1)YRqj∗Z/n = 0 for q > 2.

In these formulas, Z/n(−1) denotes the Z/n-dual of µn. If t is a local equationfor Y , then the isomorphism R1j∗Z/n ' Z/n(−1)Y is defined by the map

a : Z/n→ R1j∗µn


sending 1 to the class of the µn-torsor of nth roots of t.The question is local. This allows us to replace (X,Y ) with a locally isomorphic

pair, for example,A1T P1

T T

T

j

gf

i

with T = An−1S and i the section at infinity. Corollary 1.9 applies to g, and shows

that Rqg∗Z/n = Z/n for q = 0 and 0 for q > 0. For f , we have moreover (4.6.2.1)

Rqf∗Z/n = Z/n, 0, Z/n(−1), 0 for q = 0, 1, 2, > 2.

It is easily checked that j∗Z/n = Z/n, and that the Rqj∗Z/n are concentratedon i(T ) for q > 0. The Leray spectral sequence

Epq2 = Rpf∗Rqj∗Z/n =⇒ Rp+qg∗Z/n

therefore simplifies to

i∗Rqj∗Z/n 0 . . .

i∗R1j∗Z/n 0 . . .

Z/n 0 Z/n(−1) 0 . . . .

d2

Rqj∗Z/n = 0 for q > 2, and R1j∗Z/n is the extension by zero of a locally free sheafof rank one over T (isomorphic, via d2, to Z/n(−1)). The map a defined above,being injective (as can be checked fiber by fiber), is an isomorphism, which completesthe proof of Theorem 3.4.

3.5

We refer to [SGA 4, XVI §4, §5] for the proofs of the following applications of theacyclicity theorem (2.1).

3.5.1. Let f : X → S be a morphism of schemes finite type over C and F aconstructible sheaf on X. Then

(Rqf∗F )an ∼ Rqfan∗ (F an)

(cf. IV.6.3; in ordinary cohomology, it is necessary to assume that F is constructibleand not just torsion).

3.5.2. Let f : X → S be a morphism of schemes of finite type over a field k ofcharacteristic 0 and F a sheaf on X. If F is constructible, then so also are thesheaves Rqf∗F .

VI POINCARÉ DUALITY 41

The proof uses resolution of singularities and Theorem 3.4. It is generalized tothe case of a morphism of finite type of excellent schemes of characteristic 0 in [SGA4, XIX.5]. Another proof, independent of resolution, is given in [SGA 4 1/2, Th.Finitude, 1.1]. It applies to a morphism of schemes of finite type over a field or overa Dedekind ring.

VI Poincaré duality1 IntroductionLet X be an oriented topological manifold of pure dimension N , and assume that Xadmits a finite open covering U = (Ui)16i6K such that all nonempty intersections ofthe opens Ui are homeomorphic to balls. For such a manifold, the Poincaré dualitytheorem can be described as follows.

A. The cohomology of X is the Čech cohomology of the covering U . This is thecohomology of the complex

0→ ZA0 → ZA1 → · · · (1)

whereAk = (i0, . . . , ik) : i0 < · · · < ik and Ui0 ∩ · · · ∩ Uik 6= ∅.

B. For a = (i0, . . . , ik) ∈ Ak, let Ua = Ui0 ∩ · · · ∩ Uik and let ja be the inclusionof Ua into X. The constant sheaf Z on X admits the (left) resolution

· · ·⊕a∈A1

ja!Z⊕a∈A0

ja!Z 0

Z.

(2)

The cohomology with compact support H•c(X, ja!Z) is nothing but the cohomologywith proper support of the (oriented) ball Ua:

Hic(X, ja!Z) =

0 if i 6= N

Z if i = N .

The spectral sequence of hypercohomology for the complex (2) and cohomologywith proper support, show therefore that Hi

c(X) is the (i−N)th cohomology groupof the complex

· · · → ZA1 → ZA0 → 0. (3)

This complex is the dual of the complex (1), whence Poincaré duality.The essential points of this construction are

a) the existence of a cohomology theory with proper support:


b) the fact that every point x of a manifold X of pure dimension N has afundamental system of open neighborhoods U for which

Hic(U) =

0 for i 6= N ,Z for i = N .

(4)

Poincaré duality in étale cohomology can be constructed on this model. For Xsmooth of pure dimension N over an algebraically closed field k, n invertible on X,and x a closed point of X, the key point is to calculate the projective limit over theétale neighborhoods U of x,

lim←−Hic(U,Z/n) =

0 if i 6= 2NZ/n if i = 2N .

(5)

Just as when working with topological manifolds we have to directly treat firstthe case of an open ball (or simply the interval (0, 1]), here we must directly treatfirst the case of the curves (§2). The local acyclicity theorem for smooth morphismsthen allows us to reduce the general case (5) to this particular case (§3).

The isomorphisms (4) and (5) are not canonical: they depend on the choice of anorientation of X. For n invertible on a scheme X, µn is a sheaf of free Z/n-modulesof rank one. We denote by Z/n(N) its Nth tensor power (N ∈ Z). The intrinsicform of the second line of (5) is

lim←−H2Nc (U,Z/n(N)) = Z/n, (6)

and Z/n(N) is called the orientation sheaf of X. The sheaf Z/n(N) is constant andisomorphic to Z/n, and so we can move the sign N and write instead

lim←−H2Nc (U,Z/n) = Z/n(−N). (7)

Now Poincaré duality takes the form of a perfect duality, with values in Z/n(−N),between Hi(X,Z/n) and H2N−i

c (X,Z/n).

2 The case of curves2.1

Let X be a smooth projective curve over an algebraically closed field k, and let n beinvertible on X. When X is connected, the proof of (III.3.5) gives us a canonicalisomorphism

H2(X,µn) = Pic(X)/nPic(X) Z/n.deg∼

Let D be a reduced divisor on X and X = X rD,

X X D.j i


The exact sequence0→ j!µn → µn → i∗µn → 0

gives us an isomorphism

H2c(X,µn) = H2(X, j!µn) H2(X,µn) Z/n.∼ ∼

because Hi(X, i∗µn) = Hi(D,µn) = 0 for i > 0. When X is not necessarilyconnected, we have similarly

H2c(X,µn) ' (Z/n)π0(X),

and we define the trace morphism to be the sum

Tr: H2c(X,µn) ' (Z/n)π0(X) Z/n.Σ

Theorem 2.2. The form Tr(a∪ b) identifies each of the two groups Hi(X,Z/n) andH2−ic (X,µn) with the dual (with values in Z/n) of the other.

Transcendental proof. If X is a smooth projective curve over the spectrum S of adiscrete valuation ring and j : X → X is the inclusion of the complement of a divisorD étale over S the cohomologies (resp. the cohomologies with proper support) of thespecial and generic geometric fibers of X/S are “the same,” i.e., the fibers of locallyconstant sheaves on S. This can be deduced from the similar facts for X and D usingthe exact sequence 0→ j!Z/n→ Z/n→ (Z/n)D → 0 (for cohomology with propersupport) and the formulas j∗Z/n = Z/n, R1j∗Z/n ' (Z/n)D(−1), Rij∗Z/n = 0(i > 2) (for ordinary cohomology) (V.3.4).

This principle of specialization reduces the general case of (2.2) to the casewhere k has characteristic 0. By (V.3.3), we can then take k = C. Finally, fork = C, the groups H•(X,Z/n) and H•c(X,µn) coincide with the groups of the samename, calculated for the classical topological space Xcl and, via the isomorphismZ/n→ µn : x 7→ exp

( 2πixn

), the trace morphism becomes identified with “integration

over the fundamental class,” so that (2.2) results from Poincaré duality for Xcl.

2.3 Algebraic proof

For a very economical proof, see [SGA 4 1/2, Dualité §2]. Here is another, tied tothe autoduality of the Jacobian.

We return to the notation of (2.1). We may suppose — and we do suppose —that X is connected. The cases i = 0 and i = 2 being trivial, we suppose also thati = 1. Define DGm by the exact sequence

0→ DGm → Gm → i∗Gm → 0

(sections of Gm congruent to 1 mod D). The group H1(X,DGm) classifies theinvertible sheaves on X trivialized over D. It is the group of points of PicD(X),which is an extension of Z (the degree) by the group of points of Rosenlicht’s


generalized Jacobian Pic0D(X) (corresponding to the conductor 1 at each point of

D). This last is itself an extension of the abelian variety Pic0(X) by the torusGDm/(Gm diagonal).

a) The exact sequence 0→ j!µn → DGmx 7→xn−−−−→ DGm → 0 provides an isomor-

phismH1c(X,µn) = Pic0

D(X)n. (6.2.3.1)

b) The map sending x ∈ X(k) to the class of the sheaf invertible O(x) on Xtrivialized by 1 over D comes from a morphism

f : X → PicD(X).

For the rest, we fix a base point 0 and we put f0(x) = f(x)− f(0). For a homomor-phism v : Pic0

D(X)n → Z/n, let v ∈ H1 (Pic0D(X),Z/n

)denote the image by v of

the class in H1 (Pic0D(X),Pic0

D(X)n)of the torsor defined by the extension

0 Pic0D(X)n Pic0

D(X) Pic0D(X) 0n

Geometric class field theory (as explained in Serre [Ser59]) shows that the mapv 7→ f∗0 (v):

Hom(Pic0

D(X)n,Z/n)→ H1(X,Z/n) (6.2.3.2)

is an isomorphism. To deduce (2.2) from (6.2.3.1) and (6.2.3.2), it remains to showthat

Tr (u ∪ f∗0 (v)) = −v(u). (6.2.2.3)

This compatibility is proved in [SGA 4 1/2, Dualité, 3.2.4].

3 The general caseLet X be a smooth algebraic variety of pure dimension N over an algebraicallyclosed field k. To state the Poincaré duality theorem, we must first define the tracemorphism

Tr: H2Nc (X,Z/n(N))→ Z/n.

The definition is a painful unwinding starting from the case of curves [SGA 4, XVIII§2]. We then have

Theorem 3.1. The form Tr(a ∪ b) identifies each of the groups Hic (X,Z/n(N))

and H2N−i(X,Z/n) with the dual (with values in Z/n) of the other.

Let x ∈ X be a closed point and Xx the strict localization of X at x. We supposethat, for U running over the étale neighborhoods of x,

H•c(Xx,Z/n) = lim←−H•c(U,Z/n). (6.3.1.1)


It would be better to consider rather the pro-object “ lim←− ”H•c(U,Z/n) but, thegroups in play being finite, the difference is inessential. As we endeavored to explainin the introduction, (3.1) follows from

Hic(Xx,Z/n) = 0 for i 6= 2N and

Tr: H2Nc (Xx,Z/n(N)) ∼−−→ Z/n is an isomorphism.

(6.3.1.2)

The case N = 0 is trivial. When N > 0, let Yy denote the strict localization at aclosed point of a smooth scheme Y of pure dimension N − 1, and let f : Xx → Yybe an essentially smooth morphism (of relative dimension one). The proof uses theLeray spectral sequence for cohomology with proper support for f to reduce thequestion to the case of curves. The “cohomology with proper support” consideredbeing defined by limits (6.3.1.1), the existence of such a spectral sequence posesvarious problems of passage to the limit, treated with too much detail in [SGA 4,XVIII]. Here we are content to calculate. For every geometric point z of Yy, we have

(Rif!Z/n)z = Hic(f−1(z),Z/n).

The geometric fiber f−1(z) is a projective limit of smooth curves over an algebraicallyclosed field. It satisfies Poincaré duality. Its ordinary cohomology is given by thelocal acyclicity theorem for smooth morphisms,

Hi(f−1(z),Z/n) =

Z/n for i = 00 for i > 0.

By duality, we have

Hic(f−1(z),Z/n) =

Z/n(−1) for i = 20 for i 6= 2,

and the Leray spectral sequence becomes

Hic (Xx,Z/n(N)) = Hi−2

c (Yy,Z/n(N − 1)) .

We conclude by induction on N .

4 Variants and applicationsIt is possible to construct, in étale cohomology, a “duality formalism” (= functors Rf∗,Rf!, f∗, Rf !, satisfying various compatibilities and adjunction formulas) parallel tothat existing in coherent cohomology. In this language, the results of the precedingparagraph can be rewritten as follows: if f : X → S is smooth of pure relativedimension N and S = Spec(k) with k algebraically closed, then

Rf !Z/n = Z/n[2N ](N).

This statement is valid without hypothesis on S. It admits the

REFERENCES 46

Corollary 4.1. If f is smooth of pure relative dimension N , and the sheavesRif!Z/n are locally constant, then the sheaves Rif∗Z/n are also locally constant,and

Rif∗Z/n = Hom(R2N−if!Z/n(N),Z/n

).

In particular, under the hypotheses of the corollary, the sheaves Rif∗Z/n areconstructible. Starting from that, we can show that, if S is of finite type over thespectrum of a field or Dedekind ring, then, for every morphism of finite type f : X →S and every constructible sheaf F on X, the sheaves Rif∗F are constructible [SGA4 1/2, Th. finitude 1.1].

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[Ser59] Jean-Pierre Serre. Groupes algébriques et corps de classes. Publica-tions de l’institut de mathématique de l’université de Nancago, VII.Hermann, Paris, 1959, 202.

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September 16, 2018