nLab
etale cohomology

Idea

Étale cohomology is the abelian sheaf cohomology for sheaves on the etale site? of a scheme (which is an analog of the category of open subsets of a topological space).

History

Etale cohomology was conceived by Artin, Deligne, Grothendieck and Verdier in 1963. It was used by Deligne to prove the Weil conjecture?s.

Some useful remarks on this are in the begining of

• Spencer Bloch, Review of Milne’s Étale cohomology (pdf)

Milne’s lectures (basically a slightly watered-down version of the text mentioned above) is available at

• John Milne, Lectures on Étale cohomology (pdf)

Details

Given a scheme $X$ of finite type, the small etale site? $\mathrm{Et}(X)$ is the category whose objects are étale morphisms $\mathrm{Spec}R\to X$ and whose morphisms $(f:\mathrm{Spec}R\to X)\to (f\prime :\mathrm{Spec}R\prime \to X)$ are morphisms $\alpha :\mathrm{Spec}(R)\to \mathrm{Spec}(R\prime )$ of schemes completing triangles: $f\prime \circ \alpha =f$ (notice that the morphisms between étale morphisms are automatically étale). This category naturally carries a Grothendieck topology that makes it a site.

The étale cohomology $H_{\mathrm{et}}^{•}(X,A)$ for $A\in \mathrm{Sh}(\mathrm{Et}(X),\mathrm{Ab})$ of $X$ is the abelian sheaf cohomology with respect to this site.