Étale morphism

Definition

The following conditions are equivalent for a morphism of schemes $f:X\to Y$:

1. $f$ is flat and unramified.
2. $f$ is flat and the sheaf of Kähler differentials is zero; $\Omega _{X/Y}=0$.
3. $f$ is smooth of relative dimension 0.

and $f$ is said to be étale when this is the case.

The small étale site

The category of étale $S$-schemes becomes a Grothendieck topology, if one defines the sets of coverings to be jointly-surjective collections of $S$-morphisms $\{f_{i}:U_{i}\to U\}$; i.e., such that the union of images $\bigcup f_{i}(U_{i})$ covers $U$. That this forms a grothendieck essentially follows from the following three facts:

1. Open immersions are étale.
2. The étale property lifts by base change: that is, if $f:X\to Y$ is an étale morphism, and $g:Y'\to Y$ is any morphism, then the canonical fibered projection $f'X\times _{Y}Y'\to Y'$ is again étale.
3. If $f:X\to Y$ and $g:Y\to Z$ are such that $g\circ f$ is étale, then $f$ is étale as well.

Étale cohomology

One begins by defining a presheaf to be a contravariant functor from the underlying category of a small étale site $T$ into an abelian category $A$. Am étale sheaf (or just sheaf if the étale site is implicit) on $T$ is then a presheaf $F$ such that for all coverings $\{U_{i}\to U\}\in cov(T)$, the diagram $0\to F(U_{i})\to \prod _{i}F(U_{i})\to \prod _{i,j}F(U_{i}\times _{U}U_{j})$ is exact.

Applications

Deligne proved the Weil-Riemann hypothesis using étale cohomology.