Cohomology and homotopy
In higher category theory
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.
The abelian sheaf cohomology of the étale site is called étale cohomology.
Cofinal affine covers
For an affine scheme and an étale cover, then there exists a refinement to an étale cover such that each is an affine scheme.
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.
fpqc-site fppf-site syntomic site étale site Nisnevich site Zariski site
The classical references are
- Pierre Deligne et al., Cohomologie étale , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.
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