Étale cohomology

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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 eventually proved successful as a strategy, although Dwork managed to prove the rationality part of the conjectures in 1960 using p-adic methods; the remaining conjectures awaited the development of étale cohomology. This theory is an example of a Weil cohomology theory in algebraic geometry, and as such it continues to play an important role in the more general theory of motives. Many further applications of the theory, for example to representation theory, have been found.

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 that returns the integers as an 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:

• The étale 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; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed point theorem in this context.

With hindsight, much of the general machinery of topos theory proved unnecessary for a minimal treatment of the étale theory (though applicable to the more subtle crystalline and flat cohomology) — this is Deligne's view as expressed for example in SGA4½. On the other hand, étale cohomology quickly found other applications, for example in representation theory, going beyond the initially planned application.

l-adic cohomology groups

In applications to algebraic geometry over a finite field F, the main objective was to find a replacement for the singular cohomology groups, which are not available in the same way as for geometry of an algebraic variety over the complex number field. The hope, which was generally upheld, was that a replacement would be found in the shape of [\ell]-adic cohomology. Here [\ell] stands for any prime number with

[\ell] ≠ p

where p is the characteristic of F. One considers, for schemes V, the cohomology groups

Hⁱ(V, Z /[\ell]^kZ)

and defines

Hⁱ(V, Z_[\ell])

as their inverse limit. Here Z_[\ell] denotes the l-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/[\ell]^kZ. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the [\ell]-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the sheaf Z_[\ell]; the latter cohomology group exists but gives the "wrong" cohomology groups.)

The reason that one might guess that this leads to the correct definition, is that in the case that V is a non-singular algebraic curve and i = 1, it can be shown that H¹ is a free Z_[\ell]-module of rank 2g, dual to the Tate module of the Jacobian variety of V, where g is the genus of V. Since the first Betti number of a Riemann surface of genus g is 2g, that value is reassuring. This becomes a kind of 'base case' for inductive study of the general case (that is, i > 1 or V of dimension > 1). It also shows why the condition [\ell] ≠ p is required: when [\ell] = p the rank of the Tate module is at most g.

To remove any torsion subgroup from the [\ell]-adic groups (which can occur, and was applied by Mike Artin and David Mumford to geometric questions) the definition

Hⁱ(V, Q_[\ell]) = Hⁱ(V, Z_[\ell]) ⊗ Q_[\ell]

with the [\ell]-adic numbers Q_[\ell] is typically used.

Properties

In general the l-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the l-adic integers (or numbers) rather than the integers (or rationals). They satisfy a form of Poincaré duality on non-singular projective varieties, and the l-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups.

For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first l-adic cohomology group of an elliptic curve over a finite field is a free module of rank 2 over the l-adic integers, provided l is not the characteristic of the field concerned, and is dual to its Tate module.

There is one important way in which l-adic cohomology groups are even better than singular cohomology groups: they tend to be acted on by Galois groups. For example, if a complex variety is defined over the rational numbers, its l-adic cohomology groups are acted on by the absolute Galois group of the rational numbers: they afford Galois representations.

Elements of the Galois group of the rationals, other than the identity and complex conjugation, do not usually act continuously on a complex variety defined over the rationals, so do not act on the singular cohomology groups. This phenomenon of Galois representations, foundationally speaking, is related to the fact that the fundamental group of a topological space acts on the singular cohomology groups, because Grothendieck showed that the Galois group can be regarded as a sort of fundamental group. (See also Grothendieck's Galois theory.)

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

[\#X(\mathbb F_) = 1 -\sum_^ \alpha_i^n+p^n],

where [\alpha_i] are certain algebraic numbers satisfying [|\alpha_i|=\sqrt p].

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 [f:X\to X] is equal to the sum

[\sum_^(-1)^i\mathrm f|_].

This formula is valid for ordinary topological varieties and ordinary topology, but it is wrong for most algebraic topologies. However, this formula does hold for étale cohomology (though this is not so simple to prove).

The points of X that are defined over [\mathbb F_] 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,

[\#X(\mathbb F_) = \mathrm F^n|_ -\mathrm F^n|_ +\mathrm F^n|_].

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 motives: formally [X] = [point]+[line]+[1-part], and [1-part] has something like [\sqrt p] points.

 

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