Étale morphism

Étale morphism in algebraic geometry, a field of mathematics, is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. It satisfies the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.

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.