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basics of etale cohomology

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This page goes through some basics of étale cohomology.

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Étale topos

To every scheme XX is assigned a site which is a geometric analog of the collection of étale spaces over a topological space. This is called the étale site X etX_{et} of the scheme. The category of sheaves on that site is called the étale topos of the scheme. The intrinsic cohomology of that topos, hence the abelian sheaf cohomology over the étale site, is the étale cohomology of XX.

This section starts with looking at some basic aspects of the étale topos as such, the basic definitions and the central descent theorem for characterizing its sheaves. The next section then genuinely considers the corresponding abelian sheaf cohomology.

Étale cohomology is traditionally motivated by the route by which it was historically discovered, namely as a fix for technical problems encountered with the Zariski topology. You can find this historical motivation in all textbooks and lectures, see the References below.

But étale cohomology has a more fundamental raison d’être than this. As discussed at étale topos it is induced in any context in which one has a “reduction modality”. While fundamental, this is actually a simple point of view which leads to a simple characterization of étale morphisms, and this is what we start with now.

Étale morphisms

Of the many equivalent characterizations of étale morphisms, here we will have use of the following incarnation:

Definition

A morphisms of schemes is an étale morphism of schemes if it is

  1. formally étale– recalled in a moment;

  2. locally of finite presentation.

Remark

The first condition makes an étale morphism of schemes be like an étale space over its codomain. The second essentially just says demands this has finite fibers.

Definition

For XX a scheme, its étale site has a objects the étale morphisms of schemes into XX, as morphisms the morphisms of schemes over XX, and as coverings the jointly surjective étale morphisms over XX.

The category of sheaves on X etX_{et} is the étale topos of XX. The corresponding abelian sheaf cohomology is its étale cohomology.

The definition of formally étale in components goes like this.

Definition

A morphism of commutative rings RAR \longrightarrow A is called formally étale if for every ring BB and for every nilpotent ideal IBI \subset B and for every commuting diagram of the form

B/I A B R \array{ B/I &\leftarrow& A \\ \uparrow && \uparrow \\ B &\leftarrow& R }

there is a unique diagonal morphism

B/I A B R \array{ B/I &\leftarrow& A \\ \uparrow &\swarrow& \uparrow \\ B &\leftarrow& R }

that makes both triangles commute.

(e.g. Stacks Project 57.9, 57.12)

Remark

So dually this means that Spec(A)Spec(R)Spec(A) \to Spec(R) is formally étale if it has the unique right lifting property against all infinitesimal extensions

Spec(B red) Spec(A) Spec(B) Spec(R). \array{ Spec(B_{red}) &\longrightarrow& Spec(A) \\ \downarrow &\nearrow& \downarrow \\ Spec(B) &\longrightarrow & Spec(R) } \,.

and locality this yields a notion of formally étale morphisms of affine varieties and of schemes.

It is useful to realize this equivalently but a bit more naturally as follows.

Definition

Write CRing finCRing_{fin} for the category of finitely generated commutative rings and write CRing fin extCRing_{fin}^{ext} for the category of infinitesimal ring extensions. Write

Red:CRing fin extCRing fin Red \;\colon\; CRing_{fin}^{ext} \longrightarrow CRing_{fin}

for the functor which sends an infinitesimal ring extension to the underlying commutative ring (in the maximal case this sends a commutative ring to its reduced ring, whence the name of the functor), and write

i:CRing finCRing fin ext i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext}

for the full subcategory inclusion that regards a ring as the trivial infinitesimal extension over itself.

Proposition

There is an adjoint triple of idempotent (co-)monads

(Red inf inf):PSh((CRing fin ext) op)PSh((CRing fin ext) op) (Red \dashv \int_{inf} \dashv \flat_{inf}) \;\colon\; PSh((CRing_{fin}^{ext})^{op}) \longrightarrow PSh((CRing_{fin}^{ext})^{op})

where the left adjoint comonad RedRed is given on representables by the reduction functor of def. (followed by the inclusion).

This statement and the following prop. is a slight paraphrase of an observation due to (Kontsevich-Rosenberg 04). A closely related adjunction appeared in (Simpson-Teleman 13) in the discussion of de Rham spaces. The general abstract situation of “differential cohesion” has been discussed in (Schreiber 13).

Proof

The functors from def. form an adjoint pair (Redi)(Red \dashv i) because an extension element can only map to an extension element; so for R^R\widehat R \to R an infinitesimal ring extension of R=Red(R^)R = Red(\widehat R), and for SS a commutative ring with i(S)=(SS)i(S) = (S \to S) its trivial extension, there is a natural isomorphism

Hom CRing fin ext(R^,i(S))Hom CRing fin(R,S). Hom_{CRing_{fin}^{ext}}(\widehat R, i(S)) \simeq Hom_{CRing_{fin}}(R,S) \,.

This exhibits CRing finCRing_{fin} as a reflective subcategory of CRing fin extCRing_{fin}^{ext}.

(Redi):CRing finiRedCRing fin ext. (Red \dashv i) \;\colon\; CRing_{fin} \stackrel{\overset{Red}{\leftarrow}}{\underset{i}{\hookrightarrow}} CRing_{fin}^{ext} \,.

Via Kan extension this adjoint pair induces an adjoint quadruple of functors on categories of presheaves

PSh(CRing fin op)Red *Red *i *=Red !i !PSh((CRing fin ext) op). PSh(CRing_{fin}^{op}) \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast = Red_!}{\leftarrow}}{\stackrel{\overset{Red^\ast}{\hookrightarrow}}{\underset{Red_\ast}{\leftarrow}}}} PSh((CRing_{fin}^{ext})^{op}) \,.

The adjoint triple to be shown is obtained from composing these adjoints pairwise.

That RedRed coincides with the reduction functor on representables is a standard property of left Kan extension (see here for details).

Remark

These considerations make sense in the general abstract context of “differential cohesion” where the adjoint triple of prop. would be called:

(reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality).

Due to the full subcategory inclusion i !i_! in the proof of prop. we may equivalently regard presheaves on (CRing fin) op(CRing_{fin})^{op} (e.g. schemes) as presheaves on (CRing fin ext) op(CRing_{fin}^{ext})^{op} (e.g. formal schemes). This is what we do implicitly in the following.

Proposition

A morphism f:SpecASpecRf \;\colon\; Spec A \to Spec R in CRing fin opPSh(CRing fin op)CRing_{fin}^{op} \hookrightarrow PSh(CRing_{fin}^{op}) is formally étale, def. , precisely if it is inf\int_{inf}-modal relative SpecRSpec R, hence if the naturality square of the infinitesimal shape modality-unit

SpecA infSpecA SpecR infSpecR \array{ Spec A &\longrightarrow& \int_{inf} Spec A \\ \downarrow && \downarrow \\ Spec R &\longrightarrow& \int_{inf} Spec R }

is a pullback square.

Proof

Evaluated on IRR/ICRing fin extI \hookrightarrow R \to R/I \in CRing_{fin}^{ext} any object, by the Yoneda lemma and the (Red inf)(Red \dashv \int_{inf})-adjunction the naturality square becomes

CRing(A,B) CRing(A,B/I) CRing(R,B) CRing(R,B/I). \array{ CRing(A,B) &\longrightarrow& CRing(A,B/I) \\ \downarrow && \downarrow \\ CRing(R,B) &\longrightarrow& CRing(R,B/I) } \,.

in Set. Chasing elements through this shows that this is a pullback precisely if the condition in def. holds.

The basic stability property of étale morphisms, which we need in the following, immediately follows from this characterization:

Proposition

For fg\stackrel{f}{\to} \stackrel{g}{\to} two composable morphisms, then

  1. if ff and gg are both (formally) étale, then so is their composite gfg \circ f;

  2. if gg and gf g\circ f are (formally) étale, then so is ff;

  3. the pullback of a (formally) étale morphism along any morphism is again (formally) étale.

Proof

With prop. this is equivalently the statement of the pasting law for pullback diagrams.

Apart from that, for the proofs in the following we need the following basic facts

Proposition
  • Every etale morphism is a flat morphism.

  • Flat morphism between affines Spec(B)Spec(A)Spec(B) \to Spec(A) is faithfully flat precisely if it is surjective

We repeatedly use the following example of étale morphisms.

Proposition

Every open immersion of schemes is an étale morphism of schemes. In particular a standard open inclusion (a cover in the Zariski topology) induced by the localization of a commutative ring

Spec(R[S 1])Spec(R) Spec(R[S^{-1}]) \longrightarrow Spec(R)

is étale.

(e.g. Stacks Project, lemma 28.37.9)

Proof

By def. we need to check that the map Spec(R[S 1])Spec(R)Spec(R[S^{-1}]) \longrightarrow Spec(R) is a formally étale morphism and locally of finite presentation.

The latter is clear, since the very definition of localization of a commutative ring

R[S 1]=R[s 1 1,,s n 1](s 1s 1 11,,s ns n 11) R[S^{-1}] = R[s_1^{-1}, \cdots, s_n^{-1}](s_1 s_1^{-1} - 1, \cdots , s_n s_n^{-1} - 1)

exhibits a finitely presented algebra over RR.

To see that it is formally étale we need to check that for every commutative ring TT with nilpotent ideal JJ we have a pullback diagram

Hom(R[S 1],T) Hom(R[S 1],T/J) Hom(R,T) Hom(R,T/J). \array{ Hom(R[S^{-1}], T) &\longrightarrow& Hom(R[S^{-1}],T/J) \\ \downarrow && \downarrow \\ Hom(R, T) &\longrightarrow& Hom(R, T/J) } \,.

Now by the universal property of the localization, a homomorphism R[S 1]TR[S^{-1}] \longrightarrow T is a homomorphism RTR \longrightarrow T which sends all elements in SRS \hookrightarrow R to invertible elements in TT. But no element in a nilpotent ideal can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from RR to TT such that SS is taken to invertibles, which is indeed the top left set.

Descent theorem and examples of étale sheaves

Since there are “many more” étale morphisms of schemes than there are open immersions of schemes, a priori the discussion of descent over the étale site is more intricate than that in, say, the Zariski topology. However, the following proposition drastically reduces the types of étale covers over which descent has to be checked in addition to the open immersions. Then the following descent theorem effectively solves the descent problem over these remaining covers.

Proposition

For XX a scheme, and APSh(X et)A \in PSh(X_{et}) a presheaf on its étale site, def. , for checking the sheaf condition it is sufficient to check descent on the following two kinds of covers in the étale site

  1. jointly surjective collections of open immersions of schemes;

  2. single faithfully flat morphisms between affine schemes

(all over XX).

(Tamme, II Lemma (3.1.1), Milne, prop. 6.6)

Proof

Suppose given an arbitrary étale covering {X iX}\{X'_i \to X'\} over XX. We show how to refine it to a more special cover which itslf is the composition of covers of the form as in the statement.

To that end, first choose a cover {U jX}\{U'_j \to X'\} of X iX_i by affine open immersions of schemes. Then pulling back the original cover along that one yields covers

{X i× XU jU j} \{X'_i \times_{X'} U'_j \to U'_j\}

of each of the open affines. By pullback stability, prop. , these are still étale maps. Now these patches in turn we cover by open affines

{{U ijkX i× XU j}} \{ \{U'_{i j k} \to X'_i \times_{X'} U'_j \} \}

leading to covers

{U ijkU j} \{ U'_{i j k} \to U'_j \}

by affines.

(Notice here crucially that while the U ijkU'_{i j k} are affine open immersions in X i× XU jX'_i \times_{X'} U'_j, after this composition with an étale morphism they no longer need to be open immersions in U jU'_j, all we know is that the map is étale. This is the source of the second condition in the proposition to be shown, as discussed now. )

Since each U jU'_j, being affine, is a quasi-compact scheme, we may find a finite subcover

{U jlU j}. \{ U'_{j l} \to U'_j \} \,.

Composed with the original {U jX}\{U'_j \to X'\} this yields a refinement of the original cover by open affines.

Hence for checking descent it is sufficient to check it for these two kinds of overs. The latter is by open immersions. For the former, we may factor

{U jlU j}\{U'_{j l} \to U'_j\} as a collection of open immersions

{U jiU ji} \{U'_{j i} \to \coprod U'_{j i}\}

followed by the epimorphism of affines of the form

{U jiU j}. \{ \coprod U'_{j i} \to U'_j \} \,.

Now this is morphism is etale, hence flat, but also surjective. That makes it a faithfully flat morphism.

Therefore we are led to consider descent along faithfully flat morphisms of affines. For these the descent theorem says that they are effective epimorphisms:

Definition

Given a commutative ring RR and an RR-associative algebra AA, hence a ring homomorphism f:RAf \colon R \longrightarrow A, the Amitsur complex is the Moore complex of the dual Cech nerve of Spec(A)Spec(R)Spec(A) \to Spec(R), hence the chain complex

0RfA1idid1A RAA RA RA. 0 \to R \stackrel{f}{\to} A \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots \,.

(See also at Sweedler coring and at commutative Hopf algebroid for the same or similar constructions.)

Proposition

(descent theorem)

If ABA \to B is faithfully flat then its Amitsur complex is exact.

This is due to (Grothendieck, FGA1). The following reproduces the proof in low degree following (Milne, prop. 6.8).

Proof

We show that

0AfB1idid1B AB 0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B

is an exact sequence if f:ABf \colon A \longrightarrow B is faithfully flat.

First observe that the statement follows if ABA \to B admits a section s:BAs \colon B \to A. Because then we can define a map

k:B ABB k \colon B \otimes_A B \longrightarrow B
k:b 1b 2b 1f(s(b 2)). k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2)) \,.

This is such that applied to a coboundary it yields

k(1bb1)=f(s(b))b k(1 \otimes b - b \otimes 1) = f(s(b)) - b

and hence it exhibits every cocycle bb as a coboundary b=f(s(b))b = f(s(b)).

So the statement is true for the special morphism

BB AB B \to B \otimes_A B
bb1 b \mapsto b \otimes 1

because that has a section given by the multiplication map.

But now observe that the morphism BB ABB \to B \otimes_A B is the tensor product of the morphism ff with BB over AA, hence the Amitsur complex of this morphism is exact.

Finally, the fact that ABA \to B is faithfully flat by assumption, hence that it exhibits BB as a faithfully flat module over AA, means by definition that the Amitsur complex for (AB) AB(A \to B)\otimes_A B is exact precisely if that for ABA \to B is exact.

Proposition

For ZXZ \to X any scheme over a scheme XX, the induced presheaf on the étale site

(U YX)Hom X(U Y,Z) (U_Y \to X) \mapsto Hom_X(U_Y, Z)

is a sheaf.

This is due to (Grothendieck, SGA1 exp. XIII 5.3) A review is in (Tamme, II theorem (3.1.2), Milne, 6.2).

Proof

By prop. we are reduced to showing that the represented presheaf satisfies descent along collections of open immersions and along surjective maps of affines. For the first this is clear (it is Zariski topology-descent).

For the second case of a faithfully flat cover of affines Spec(B)Spec(A)Spec(B) \to Spec(A) it follows with the exactness of the corresponding Amitsur complex, by the descent theorem, prop. .

Remark

This map from XX-schemes to sheaves on X etX_{et} is not injective, different XX-schemes may represent the same sheaf on X etX_{et}. Unique representatives are given by étale schemess over XX.

(e.g. Tamme, II theorem 3.1)

We consider some examples of sheaves of abelian groups induced by prop. from group schemes over XX.

Example

The additive group over XX is the group scheme

𝔾 aSpec([t])× Spec()X. \mathbb{G}_a \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X \,.

By the universal property of the pullback, the corresponding sheaf (𝔾 a) X(\mathbb{G}_a)_X is given by the assignment

(𝔾 a) X(U XX) =Hom X(U X,Spec([t])× Spec()X) =Hom(U X,Spec([t])) =Hom([t],Γ(U X,𝒪 U X)) =Γ(U X,𝒪 U X). \begin{aligned} (\mathbb{G}_a)_X(U_X \to X) & = Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X) \\ & = Hom(U_X, Spec(\mathbb{Z}[t])) \\ & = Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X})) \\ & = \Gamma(U_X, \mathcal{O}_{U_X}) \end{aligned} \,.
Remark

In other words, the sheaf represented by the additive group is the abelian sheaf underlying the structure sheaf of XX, and in particular the structure sheaf is indeed an étale sheaf.

Similarly one finds:

Example

The multiplicative group over XX

𝔾 mSpec([t,t 1])× Spec()X \mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X

represents the sheaf (𝔾 m) X(\mathbb{G}_m)_X given by

(𝔾 m) X(U X)Γ(U X,𝒪 U X) ×. (\mathbb{G}_m)_X(U_X) \mapsto \Gamma(U_X, \mathcal{O}_{U_X})^\times \,.

(e.g. Tamme, II, 3)

Base change and sheaf cohomology

Definition

For f:XYf \colon X \longrightarrow Y a homomorphism of schemes, there is induced a functor on the categories underlying the étale site

f 1:Y etX et f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}

given by sending an object U YYU_Y \to Y to the fiber product/pullback along ff

f 1:(U YY)(X× YU YX). f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X) \,.
Proposition

The morphism in def. is a morphism of sites and hence induces a geometric morphism between the étale toposes

(f *f *):Sh(X et)f *f *Sh(Y et). (f^\ast \dashv f_\ast) \;\colon\; Sh(X_{et}) \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}} Sh(Y_{et}) \,.

Here the direct image is given on a sheaf Sh(X et)\mathcal{F} \in Sh(X_{et}) by

f *:(U YY)(f 1(U Y))=(X× XU Y) f_\ast \mathcal{F} \;\colon\; (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)

while the inverse image is given on a sheaf Sh (Y et)\mathcal{F} \in Sh_(Y_{et}) by

f *:(U XX)limU Xf 1(U Y)(U Y). f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y) \,.

By the discussion at morphisms of sites – Relation to geometric morphisms. See also for instance (Tamme I 1.4).

Proposition

The qqth derived functor R qf *R^q f_\ast of the direct image functor of def. sends Ab(Sh(X et))\mathcal{F} \in Ab(Sh(X_{et})) to the sheafification of the presheaf

(U YY)H q(X× YU Y,), (U_Y \to Y) \mapsto H^q(X \times_Y U_Y, \mathcal{F}) \,,

where on the right we have the degree qq abelian sheaf cohomology group with coefficients in the given \mathcal{F} (étale cohomology).

(e.g. Tamme, I (3.7.1), II (1.3.4), Milne, 12.1).

Proof

We have a commuting diagram

Ab(PSh(X)) ()f 1 Ab(PSh(Y)) inc L Ab(Sh(X)) f * Ab(Sh(Y)), \array{ Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y)) \\ \uparrow^{\mathrlap{inc}} && \downarrow^{L} \\ Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y)) } \,,

where the right vertical morphism is sheafification. Because ()f 1(-) \circ f^{-1} and LL are both exact functors it follows that for I \mathcal{F} \to I^\bullet an injective resolution that

R pf *() :H p(f *I) =H p(LI (f 1())) =L(H p(I )(f 1())) \begin{aligned} R^p f_\ast(\mathcal{F}) & :\simeq H^p( f_\ast I) \\ & = H^p(L I^\bullet(f^{-1}(-))) \\ & = L (H^p(I^\bullet)(f^{-1}(-))) \end{aligned}
Remark

For O Xf 1O Yg 1O ZO_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z two composable morphisms of sites, the Grothendieck spectral sequence for the corresponding direct images is of the form

E 2 p,q=R pg *(R qf *())E p+q=R p+q(gf) *(). E^{p,q}_2 = R^p g_\ast(R^q f_\ast(\mathcal{F})) \Rightarrow E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F}) \,.

For the special case that S Z=*S_Z = \ast and g 1g^{-1} includes an étale morphism U YYU_Y \to Y this yields the Leray spectral sequence

E 2 p,q=H p(U Y,R qf *)E p+q=H p+q(U Y× YX,). E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F}) \Rightarrow E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F}) \,.

Étale cohomology

With some basic facts about sheaves on the étale site in hand, we now consider basics of abelian sheaf cohomology with coefficients in some such sheaves.

  1. With coefficients in coherent modules

  2. With coefficients in cyclic groups

  3. With coefficients in the multiplicative group

This may serve to give a first idea of the nature of étale cohomology. An outlook on the deep structurual theorems about étale cohomology is in the next section below.

With coefficients in coherent modules

Proposition

For XX a scheme and NN a (flat) quasicoherent module over its structure sheaf 𝒪 X\mathcal{O}_X, then this induces an abelian sheaf on the étale site by

N et:(U XX)Γ(U Y,N 𝒪 X𝒪 U Y). N_{et} \;\colon\; (U_X \to X) \mapsto \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_Y}) \,.

(e.g. Tamme, II 3.2.1)

Proof

By prop. it is sufficient to test the sheaf condition on open affine covers and on singleton covers by faithfully flat morphisms of affines. For the first case we have a sheaf since this is just the sheaf condition in the Zariski topology. For the second case the corresponding Cech complexes are the Amitsur complexes of a faithfully flat ABA \to B tensored with NN. By the descent theorem, prop. this is exact, hence verifies the sheaf condition.

We consider now the étale abelian sheaf cohomology with coefficients in such coherent modules.

Remark

A cover in the Zariski topology on schemes is an open immersion of schemes and hence is in particular an étale morphism of schemes. Hence the étale site is finer than the Zariski site and so every étale sheaf is a Zarsiki sheaf, but not necessarily conversely.

Remark

For XX a scheme, the inclusion

ϵ:X ZarX et \epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}

of the Zariski site into the étale site is indeed a morphism of sites. Hence there is a Leray spectral sequence, remark , which computes étale cohomology in terms of Zarsiki cohomology

E 2 p,q=H p(X Zar,R qϵ *)E p+q=H p+q(X et,). E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F}) \Rightarrow E^{p+q} = H^{p+q}(X_{et}, \mathcal{F}) \,.

This is originally due to (Grothendieck, SGA 4 (Chapter VII, p355)). Reviews include (Tamme, II 1.3).

Proposition

For NN a quasi-coherent sheaf of 𝒪 X\mathcal{O}_X-modules and N etN_{et} the induced étale sheaf (by the discussion at étale topos – Quasicohetent sheaves), then the edge morphism

H Zar p(X,N)H et p(X,N et) H^p_{Zar}(X, N) \longrightarrow H^p_{et}(X,N_{et})

of the Leray spectral sequence of remark is an isomorphism for all pp, identifying the abelian sheaf cohomology on the Zariski site with coefficients in NN with the étale cohomology with coefficients in N etN_{et}.

Moreover, for XX affine we have

H et p(X,N et)0. H^p_{et}(X, N_{et}) \simeq 0 \,.

This is due to (Grothendieck, FGA 1). See also for instance (Tamme, II (4.1.2)).

Proof

By the discussion at edge morphism it suffices to show that

R qϵ *(N)=0,forp>0. R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0 \,.

By prop , R qϵ *NR^q \epsilon_\ast N is the sheaf on the Zariski topology which is the sheafification of the presheaf given by

UH q(X et|U,N), U \mapsto H^q(X_{et}|U, N) \,,

hence it is sufficient that this vanishes, or rather, by locality (sheafification) it suffices to show this vanishes for X=U=Spec(A)X = U = Spec(A) an affine algebraic variety.

By the existence of cofinal affine étale covers the full subcategory X et aX atX_{et}^{a} \hookrightarrow X_{at} on the étale maps with affien domains, equipped with the induced coverage, is a dense subsite. Therefore it suffices to show the statement there. Moreover, by the finiteness condition on étale morphisms every cover of X et aX_{et}^{a} may be refined by a finite cover, hence by an affine covering map

Spec(B)Spec(A). Spec(B) \longrightarrow Spec(A) \,.

It follows (by a discussion such as e.g. at Sweedler coring) that the corresponding Cech cohomology complex

N et(Spec(A))C 0({Spec(B)Spec(A)},N et)C 1({Spec(B)Spec(A)},N et) N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots

is of the form

0NN ABN AB AB. 0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots \,.

known as the Amitsur complex of ABA \to B, tensored with NN.

Since ABA \to B is a faithfully flat morphism, it follows again by the descent theorem, prop. that this is exact, hence that the cohomology indeed vanishes.

With coefficients in cyclic groups

Let XX be a reduced scheme of characteristic the prime number pp, hence such that for all points xXx \in X

p𝒪 X,x=0. p \cdot \mathcal{O}_{X,x} = 0 \,.

Write

Fid() p():(𝔾 a) X(𝔾 a) X F - id \coloneqq (-)^p - (-) \;\colon\; (\mathbb{G}_a)_X \longrightarrow (\mathbb{G}_a)_X

for the endomorphism of the additive group over the étale site X etX_{et} of XX (the structure sheaf regarded as just a sheaf of abelian groups) which is the Frobenius endomorphism F()() pF(-) \coloneqq (-)^p minus the identity.

Proposition

There is a short exact sequence of abelian sheaves over the étale site

0(/p) X(𝔾 a) XFid(𝔾 a) X0. 0 \to (\mathbb{Z}/p\mathbb{Z})_X \to (\mathbb{G}_a)_X \stackrel{F-id}{\to} (\mathbb{G}_a)_X \to 0 \,.

This is called the Artin-Schreier sequence (e.g. Tamme, section II 4.2, Milne, example 7.9).

Proof

By the discussion at category of sheaves – Epi-/Mono-morphisms we need to show that the left morphism is an injection over any étale morphism U YXU_Y \to X, and that for every element s𝒪 Xs \in \mathcal{O}_X there exists an étale site covering {U iX}\{U_i \to X\} such that () p()(-)^p- (-) restricts on this to a morphism which hits the restriction of that element.

The first statement is clear, since s=s ps = s^p says that ss is a constant section, hence in the image of the constant sheaf /p\mathbb{Z}/p\mathbb{Z} and hence for each connected U YXU_Y \to X the left morphism is the inclusion

/p𝒪 X \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathcal{O}_{X'}

induced by including the unit section e Xe_{X'} and its multiples re Xr e_{X'} for 0r<p0 \leq r \lt p. (This uses the “freshman's dream”-fact that in characteristic pp we have (a+b) p=a p+b p(a + b)^p = a^p + b^p).

This is injective by assumption that XX is of characteristic pp.

To show that () p()(-)^p - (-) is an epimorphism of sheaves, it is sufficient to find for each element s𝒪 X=As \in \mathcal{O}_X = A an étale cover Spec(B)Spec(A)Spec(B) \to Spec(A) such that its restriction along this cover is in the image of () p():BB(-)^p - (-) \colon B \to B. The choice

BA[t]/(tt ps) B \coloneqq A[t]/(t- t^p - s)

by construction has the desired property concerning ss, the preimage of ss is the equivalence class of tt.

To see that with this choice Spec(B)Spec(A)Spec(B) \to Spec(A) is indeed an étale morphism of schemes it is sufficient to observe that it is a morphism of finite presentation and a formally étale morphism. The first is true by construction. For the second observe that for a ring homomorphism BTB \to T the generator tt cannot go to a nilpotent element since otherwise ss would have to be nilpotent. This implies formal étaleness analogous to the discussion at étale morphism of schemes – Open immersion is Etale.

Proposition

If X=Spec(A)X = Spec(A) is an affine reduced scheme of characteristic a prime number pp, then its étale cohomology with coefficients in /p\mathbb{Z}/p\mathbb{Z} is

H q(X,(/p) X){A/(Fid)A ifq=1 0 ifq>0. H^q(X, (\mathbb{Z}/p\mathbb{Z})_X) \simeq \left\{ \array{ A/(F - id)A & if\; q = 1 \\ 0 & if \; q \gt 0 } \right. \,.
Proof

Under the given assumptions, the Artin-Schreier sequence (see there) induces a long exact sequence in cohomology of the form

0 H 0(X et,/p)H 0(X et,𝒪 X)FidH 0(X et,𝒪 X) H 1(X et,/p)H 1(X et,𝒪 X)FidH 1(X et,𝒪 X) H 2(X et,/p)H 2(X et,𝒪 X)FidH 2(X et,𝒪 X), \begin{aligned} 0 & \to H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X) \\ & \to H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X) \\ & \to H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots \end{aligned} \,,

where F()=() pF(-) = (-)^p is the Frobenius endomorphism. By prop. the terms of the form H p1(X,𝒪 X)H^{p \geq 1}(X, \mathcal{O}_X) vanish, and so from exactness we find an isomorphism

H 0(X et,𝒪 X)/(Fid)(H 0(X et,𝒪 X))H 1(X et,/p), H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X)) \stackrel{\simeq}{\to} H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \,,

hence the claimed isomorphism

A/(Fid)(A)H 1(X et,/p). A/(F-id)(A) \stackrel{\simeq}{\to} H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \,.

By the same argument all the higher cohomology groups vanish, as claimed.

With coefficients in the multiplicative group

the étale cohomology groups with coefficients in the multiplicative group 𝔾 m\mathbb{G}_m in the first few degrees go by special names:

Outlook: The main theorems

What makes étale cohomology interesting in a broader context is that is verifies a collection of good structural theorems, which we just list now. In their totality these properties make étale cohomology (in its incarnation as ℓ-adic cohomology) qualify as a Weil cohomology theory. This in turn means that using étale cohomology one can give a proof of the Weil conjectures – a number of conjectures about properties of the numbers of points in algebraic varieties, hence of the numbers of solutions to certain polynomial equations over certain rings – , and this was historically a central motivation for introducing étale cohomology in the first place.

These theorems are

  1. proper base change theorem (Milne, section 17)

  2. comparison theorem (étale cohomology) (Milne, section 21)

  3. Künneth formula (Milne, section 22)

  4. cycle map theorem (Milne, section 23)

  5. Poincaré duality (Milne, section 24)

Together these imply the central ingredient for a proof of the Weil conjectures, a Lefschetz fixed-point formula

For more on this see… elsewhere.

References

Last revised on December 4, 2013 at 08:03:56. See the history of this page for a list of all contributions to it.