Contents

Idea

Traditional étale cohomology (e.g. Deligne 77) is the abelian sheaf cohomology for sheaves on the étale site of a scheme – which is an analog of the category of open subsets of a topological space $X$ , or rather the analog of the category of étale spaces over $X$, with finite fibers.

A certain inverse limit over étale cohomology groups for different coefficients yields ℓ-adic cohomology, which is a Weil cohomology theory.

More generally, there is étale generalized cohomology theory with coefficients in sheaves of spectra on the étale site (Jardine 97). Still more generally, there is étale generalized cohomology on the étale (∞,1)-site (Antieau-Gepner 12, Lurie).

Definition

Proposition

Given a scheme $X$ of finite type, the small étale site $X_{et}$ is the category whose objects are étale morphisms $Spec R \to X$ and whose morphisms $(f:Spec R\to X)\to (f':Spec R'\to X)$ are morphisms $\alpha: Spec(R)\to Spec(R')$ of schemes completing triangles: $f'\circ\alpha=f$ (notice that the morphisms between étale morphisms are automatically étale). This category naturally carries a Grothendieck topology that makes it a site, the étale site.

For $A \in Sh(X_{et}, Ab)$ an abelian sheaf on $X$, the étale cohomology $H_{et}^\bullet(X,A)$ of $X$ with coefficients in $A$ is the abelian sheaf cohomology with respect to this site.

Basic properties

The following are some basic properties of étale cohomology groups for various standard choices of coefficients.

Relation to Zariski cohomology

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 Zariski sheaf, but not necessarily conversely.

Remark

For $X$ a scheme, the inclusion

$\epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}$

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

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

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

With coefficients in coherent modules

Proposition

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

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

of the Leray spectral sequence of remark 2 is an isomorphism for all $p$, itentifying the abelian sheaf cohomology on the Zariski site with coefficients in $N$ with the étale cohomology with coefficients in $N_{et}$.

Moreover, for $X$ affine we have

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

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

$R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0 \,.$

By the discussion at direct image (also at abelian sheaf cohomology), $R^q \epsilon_\ast N$ is the sheaf on the Zariski topology which is the sheafification of the presheaf given by

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

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

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

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

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

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

is of the form

$0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots \,.$

known as the Amitsur complex.

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

With coefficients in a cyclic group

Proposition

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

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

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

\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 $F(-) = (-)^p$ is the Frobenius endomorphism. By prop. 2 the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish, and so from exactness we find an isomorphism

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

hence the claimed isomorphism

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

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 $\mathbb{G}_m$ in the first few degrees go by special names:

(…)

Main theorems

The following are the main theorems characterizing properties of étale cohomology. Together these theorems imply that étale cohomology, in its variant as l-adic cohomology, is a Weil cohomology theory.

Lefschetz fixed-point formula

Künneth formula + cycle map + Poincaré duality $\Rightarrow$ Lefschetz fixed-point formula

References

History, motivation and original accounts

Étale cohomology was conceived by Artin, Deligne, Grothendieck and Verdier in 1963. It was used by Deligne to prove the Weil conjectures. Some useful (and also funny) remarks on this are in the beginning of

The classical references include SGA, esp.

• Pierre Deligne et al., Cohomologie étale , Lecture Notes in Mathematics 569, Springer-Verlag, 1977.

• Barry Mazur, Notes on étale cohomology of number fields, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552 (Numdam, pdf)

Reviews and modern accounts

A modern textbook, though largely based on the material in SGA is

• Lei Fu, Étale cohomology theory, Nankai Tracts in Math. 13, World Sci. 2011; (toc pdf; Preface pdf; chap. 1 Descent theory pdf)

Lecture notes include

• Evan Jenkins, Étale cohomology seminar (web)

• Donu Arapura, An introduction to Étale cohomology (pdf)

• Antoine Ducros, Étale cohomology of schemes and analytic spaces, (pdf)