nLab
etale 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 to 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 ${\mathrm{Sch}}_{/X,\mathrm{et}}$ of $X$ is the over category ${\mathrm{Sch}}_{/X}$ of schemes over $X$ equipped with the coverage given by étale covers (after forgetting the maps to $X$).

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

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

Properties

Cohomology

Proposition

The inverse image restriction functor $i^{*}\mathrm{Sh}({\mathrm{Sch}}_{/X,\mathrm{et}},\mathrm{Ab})\to \mathrm{Sh}(X_{\mathrm{et}},\mathrm{Ab})$ 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 $F\in \mathrm{Sh}({\mathrm{Sch}}_{/X,\mathrm{et}},\mathrm{Ab})$ we have that the etale cohomology of $X$ with coefficients in $F$ may be computed on the small site:

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

• étale (∞,1)-site

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

References

The classical references are

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

• 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 are

• James Milne, Lectures on etale cohomology (pdf)

• Aise Johan de Jong, Étale cohomology (pdf)