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Étale cohomology

In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. This proved successful as a strategy, about a dozen years after the idea was mooted in the early 1960s.

The formal definition of étale cohomology is as the derived functor of the functor of sections, F -> Γ(F), for a type of sheaf. The sections of a sheaf can be thought of as Hom(Z,F) where Z is the sheaf returning always the integers as abelian group; the sheaf F is understood in the sense of a Grothendieck topology. The idea of derived functor here is that the sheaf of sections doesn't respect exact sequences; according to general principles of homological algebra there will be a sequence of functors Hi for i = 0,1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H0 functor coincides with the section functor Γ.

In these very abstract terms, the existence of such a theory comes down to some properties of étale morphisms in scheme theory, allowing us to use étale coverings as a Grothendieck topology; and some further proofs in homological terms, showing for example that injective resolutions are to be found in the sheaf category. To a very great extent, this attitude masks what is going on.

Some basic intuitions of the theory are these:

An application to curves

This is how the theory could be applied to the local zeta-function of an algebraic curve.

Theorem. Let X be a curve of genus g defined over the finite field with p elements. Then for every n greater or equal 1 one has

where are certain algebraic numbers satisfying .


Idea of proof

According to the Lefschetz fixed point theorem, the number of fixed points of any morphism is equals to the sum

This formula is valid for ordinary topological varieties and ordinary topology; it's wrong for most algebraic topologies. But this formula does hold for étale cohomology (don't think it's simple to prove!)

The points of X that are defined over are those fixed by Fn where F is the Frobenius automorphism in characteristic p.

The étale cohomology Betti numbers of X in dimensions 0, 1, 2 are resp. 1, 2g, and 1.

According to all of these,


This gives the general form of the theorem.

The assertion on the absolute values of the αs requires some deeper argument.

The whole idea fits into the framework of motivess: formally [X] = [point]+[line]+[1-part], and [1-part] has something like points.