nLab étale site



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


Étale morphisms



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.



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.


Cofinal affine covers


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.



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


For XX a scheme and FSh(Sch /X,et,Ab)F \in Sh(Sch_{/{X,et}}, Ab) an abelian sheaf on its big site, then the etale cohomology of XX with coefficients in FF may equivalently 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


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

Last revised on April 6, 2020 at 14:29:21. See the history of this page for a list of all contributions to it.