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 be a scheme.
The big étale site of is the over category of schemes over equipped with the coverage given by étale covers (after forgetting the maps to ).
The small étale site is the full subcategory of on the étale morphisms .
The abelian sheaf cohomology of the étale site is called étale cohomology.
For an affine scheme and an étale cover, then there exists a refinement to an étale cover such that each is an affine scheme.
The inverse image restriction functor on the categories of sheaves with values in Ab
is an exact functor
maps injective objects to injective objects.
For a scheme and an abelian sheaf on its big site, then the etale cohomology of with coefficients in may equivalently be computed on the small site:
This appears for instance in (deJong, prop. 3.4).
The derived geometry of the étale site is the étale (∞,1)-site. The precise statement is at derived étale geometry.
étale morphism, étale site, étale topos, étale cohomology, étale homotopy
The lisse-étale site of some consists of smooth morphisms .
fpqc-site fppf-site syntomic site étale site Nisnevich site Zariski site
The classical references are
Textbooks include
James Milne, Etale cohomology, Princeton Mathematical Series 33, 1980. xiii+323 pp.
A detailed survey is in chapter 34 of
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
Last revised on April 6, 2020 at 14:29:21. See the history of this page for a list of all contributions to it.