�tale cohomology

In mathematics, the �tale 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 �tale 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 �tale morphisms in scheme theory, allowing us to use �tale 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:

The �tale requirement is the condition that would allow one to apply the implicit function theorem if it were true in algebraic geometry (but it isn't - implicit algebraic functions are called algebroid in older literature).
There are certain basic cases, of dimension 0 and 1, and for an abelian variety, where the answers with constant sheaves of coefficients can be predicted (via Galois cohomology and Tate modules).
As it turned out, these base cases in effect determined the theory (perhaps unexpectedly); but the case of a general sheaf on a curve is already complex.
Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group.
The job of the general theory is certainly both to integrate all this information, and to prove general results such as Poincar� duality and the Lefschetz fixed point theorem in this context.
With retrospect, much of the machinery of topos theory proved unnecessary for a minimal treatment of the �tale theory (though applicable to the more subtle crystalline and flat cohomology).
On the other hand �tale cohomology quickly found other applications, for example in representation theory, going beyond the initially planned application.

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 .

Notes

This agrees with the projective line being a curve of genus 0 and having pⁿ+1 points.
We see that number of points on any curve is 'rather close' to that of the projective line.

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 �tale 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 �tale 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.