nLab étale site

Idea

The étale site of a scheme is an analog of the category of open subsets of a topological space. The corresponding cohomology is étale cohomology.

The étale topology has similar cohomological properties as the complex analytic topology, and in particular it is much finer for cohomological purposes than the Zariski topology.

Definition

Definition

Let $X$ be a scheme.

The big étale site $Sch_{/X,et}$ of $X$ is the over category $Sch_{/X}$ of schemes over $X$ equipped with the coverage given by étale covers (after forgetting the maps to $X$).

The small étale site $i : X_{et} \hookrightarrow Sch/{X,et}$ is the full subcategory of $Sch_{/X}$on the étale morphisms $U \to X$.

The abelian sheaf cohomology of the étale site is called étale cohomology.

Properties

Cofinal affine covers

Proposition

For $X = Spec(A)$ an affine scheme and $\{Y_i \to X\}$ an étale cover, then there exists a refinement to an étale cover $\{U_i \to X\}$ such that each $U_i$ is an affine scheme.

Cohomology

Proposition

The inverse image restriction functor $i^* Sh(Sch_{/X,et}, Ab) \to Sh(X_{et}, Ab)$ on the categories of sheaves with values in Ab

• is an exact functor

• maps injective objects to injective objects.

Corollary

For $X$ a scheme and $F \in Sh(Sch_{/{X,et}}, Ab)$ an abelian sheaf on its big site, then the etale cohomology of $X$ with coefficients in $F$ may equivalently be computed on the small site:

$$H^p(X_{et}, F|_{et}) \simeq H^p(X,F) \,.$$

This appears for instance in (deJong, prop. 3.4).

Derived geometry

The derived geometry of the étale site is the étale (∞,1)-site. The precise statement is at derived étale geometry.

Related concepts

• étale morphism, étale site, étale topos, étale cohomology, étale homotopy

• The lisse-étale site of some $X$ consists of smooth morphisms $U \to X$.

fpqc-site$\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site

• pro-étale site

• Weil-étale topology for arithmetic schemes

• étale (∞,1)-site

• constructible sheaf

• étale homotopy

References

The classical references are

• Pierre Deligne et al., Cohomologie étale , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.

Textbooks include

• Günter Tamme, Introduction to Étale Cohomology

• James Milne, Etale cohomology, Princeton Mathematical Series 33, 1980. xiii+323 pp.

A detailed survey is in chapter 34 of

• Aise Johan de Jong, The Stacks Project

Lecture notes include

• James Milne, Lectures on etale cohomology (pdf)

• Aise Johan de Jong, Étale cohomology (pdf)

A variant, the pro-étale site (locally contractible in some sense) is discussed in

• Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes (arXiv:1309.1198)