# Étale cohomology including l-adic cohomology.

* Étale cohomology

Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry.

Étale cohomology quickly found other applications, for example Deligne and Lusztig used it to construct representations of finite groups of Lie type ; see Deligne – Lusztig theory.

Étale cohomology works fine for coefficients Z / nZ for n coprime to the characteristic, but gives unsatisfactory results for non-torsion coefficients.

* Archibald and Savitt Étale cohomology

# REDIRECT Étale cohomology

Étale cohomology is another cohomology theory for sheaves over a scheme.

# REDIRECT Étale cohomology

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck.

This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme.

This cohomology theory was known as the " Weil cohomology ", but using the tools he had available, Weil was unable to construct it.

Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions which imitated the cohomology functor H < sup > 1 </ sup >.

The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

The first of the Weil conjectures was proved by Bernard Dwork, and a completely different proof of the first two conjectures via l-adic cohomology was given by Alexander Grothendieck.

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology.

The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.

The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.

As the result of local contributions by Bernard Dwork, John Tate, Michiel Hazewinkel and a local and global reinterpretation by Jürgen Neukirch and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology free presentation of class field theory was established in the nineties, see e. g. the book of Neukirch.

The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to De Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product.

However, Grothendieck's standard conjectures remain open ( except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures ), and the analogue of the Riemann hypothesis was proved by, using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.

Grothendieck proved an analogue of the Lefschetz fixed point formula for l-adic cohomology theory, and by applying it to the Frobenius automorphism F was able to prove the following formula for the zeta function.

Artin was also an important expositor of Galois theory, and of the group cohomology approach to class ring theory ( with John Tate ), to mention two theories where his formulations became standard.

These varieties have been called ' varieties in the sense of Serre ', since Serre's foundational paper FAC on sheaf cohomology was written for them.

To show that A ( Q )/ 2A ( Q ) is finite, which is certainly a necessary condition for the finite generation of the group A ( Q ) of rational points of A, one must do calculations in what later was recognised as Galois cohomology.

The natural domains of definition of functions, continued to the limit, are called Stein manifolds and their nature was to make sheaf cohomology groups vanish.

Weil suggested that the conjectures would follow from the existence of a suitable " Weil cohomology theory " for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties.

( One reason for this is that Weil suggested that the Weil conjectures could be proved using such a cohomology theory.

Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.

The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck.

The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups ( which are called cohomology groups in this context and denoted by H < sup > n </ sup >) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.

In 1936 Norman Steenrod published a paper constructing Čech cohomology by dualizing Čech homology.

Two major foundational papers by Serre were Faisceaux Algébriques Cohérents ( FAC ), on coherent cohomology, and Géometrie Algébrique et Géométrie Analytique ( GAGA ).

where H < sup > p, q </ sup >( X ) is the subgroup of cohomology classes which are represented by harmonic forms of type ( p, q ).

That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z < sub > 1 </ sub >, ..., z < sub > n </ sub >, can be written as a harmonic function times.

More abstractly, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by α.

By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call, and the cap product can be computed by taking the cup product of and α and capping with the fundamental class of X.

It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations.

and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Artin for attacking the Weil conjectures, as outlined in.