nLab
etale site

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Étale morphisms

Contents

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 XX be a scheme.

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

The small étale site i:X etSch/X,eti : X_{et} \hookrightarrow Sch/{X,et} is the full subcategory of Sch /XSch_{/X}on the étale morphisms UXU \to X.

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

Properties

Cofinal affine covers

Proposition

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

Cohomology

Proposition

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

Corollary

For XX a scheme and FSh(Sch /X,et,Ab)F \in Sh(Sch_{/{X,et}}, Ab) we have that the etale cohomology of XX with coefficients in FF may be computed on the small site:

H p(X et,F et)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.

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.

Textbooks include

A detailed survey is in chapter 34 of

Lecture notes include

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

Revised on November 25, 2013 06:21:45 by Urs Schreiber (89.204.153.236)