nLab étale cohomology

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 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. 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:

• $H^0_{et}(-, \mathbb{G}_m)$: group of units;

• $H^1_{et}(-, \mathbb{G}_m)$: Picard group (Hilbert's theorem 90, Tamme, II 4.3.1);

• $H^2_{et}(-, \mathbb{G}_m)$: Brauer group;

With coefficients in groups of roots of unity

• Kummer sequence

(Tamme, II, 4.4)

(…)

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.

Proper base change theorem

• proper base change theorem

(Milne, section 17)

Comparison theorem: Relation to singular cohomology

• comparison theorem (étale cohomology)

(Milne, section 21)

Künneth formula

(Milne, section 22)

Cycle map

(Milne, section 23)

Poincaré duality

(Milne, section 24)

Lefschetz fixed-point formula

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

(Milne, section 25)

Relation to motivic cohomology

We have the following, which is Theorem 10.2 in MazzaVoevodskyWeibel2006.

Theorem

Let $X$ be a smooth, separated scheme over a field $k$. Let $n$ be an integer prime to the characteristic of $k$. Then for every integer $p$ and every integer $q \geq 0$, we have that $H^{p,q}_{L}(X, \mathbb{Z} / n)$, the étale (or Lichtenbaum) motivic cohomology of $X$, is isomorphic to $H^{p}_{et}\left(X, \mathbb{Z}/n(q) \right)$, the étale cohomology of $X$ with coefficients in the $q$-th Tate twist of $\mathbb{Z}/n$.

The inverting of smashing with the Tate sphere as opposed to just with $S^{1}$ in the construction of the stable motivic homotopy category allows the $q$ part of motivic cohomology to be represented in it (by the motivic Eilenberg-MacLane spectrum), as opposed to just the $(p,0)$ part. Combining this with the above theorem, we see that inverting the Tate sphere as opposed to just $S^{1}$ allows the Tate twists of étale cohomology to be represented.

Related concepts

• basics of étale cohomology

• étale morphism, étale site, étale cohomology

• étale (∞,1)-site, étale topos

• étale homotopy

• Weil conjecture

• continuous étale cohomology

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

• Spencer Bloch, Review of Milne‘s Étale Cohomology (pdf; publisher’s book page)

See also

• MathOverflow Étale cohomology — Why study it?

The classical references include SGA, esp.

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

• James Milne, Étale Cohomology, Princeton Mathematical Series 33, 1980. xiii+323 pp.

See also

• 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

• Günter Tamme, Introduction to Étale Cohomology, 1994

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

• James Milne, Lectures on Étale Cohomology (html, pdf)

• Aise Johan de Jong, Étale cohomology 2009, in The Stacks Project (pdf)

• Evan Jenkins, Étale cohomology seminar (web)

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

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

See also

• Edgar Costa, Étale cohomology (pdf)

• Thomas H. Geisser, Weil-etale motivic cohomology, K-th archive

Discussion of generalized cohomology theory on the étale site but with coefficients in sheaves of spectra is in

• Rick Jardine, Generalized Étale cohomology theories, 1997 Progress in mathematics volume 146

Discussion of generalized étale cohomology over the étale (∞,1)-site (hence in higher topos theory/higher algebra) is in

• Benjamin Antieau, David Gepner, Brauer groups and étale cohomology in derived algebraic geometry (arXiv:1210.0290)

• Jacob Lurie, Descent theorems (pdf)

Other

• Mazza, Voevodsky, Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs 2. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. xiv, 216 p. (2006). zentralblatt pdf