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Page "Grothendieck topology" ¶ 3
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cohomology and theory
His construction of new cohomology theories has left deep consequences for algebraic number theory, algebraic topology, and representation theory.
One of his most celebrated achievements is the discovery of the first arithmetic Weil cohomology theory: the ℓ-adic étale cohomology.
Collaborators on the SGA projects also included Mike Artin ( étale cohomology ) and Nick Katz ( monodromy theory and Lefschetz pencils ).
Jean Giraud worked out torsor theory extensions of non-abelian cohomology.
He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it.
Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology ( relevant also in categorical logic ).
From about 1955 he started to work on sheaf theory and homological algebra, producing the influential " Tôhoku paper " ( Sur quelques points d ' algèbre homologique, published in 1957 ) where he introduced Abelian categories and applied their theory to show that sheaf cohomology can be defined as certain derived functors in this context.
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.
While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.
His conjectures postulated that there should be a cohomology theory of algebraic varieties which gave number-theoretic information about their defining equations.
Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory which he suspected would be the Weil cohomology.
To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead.
Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups.
They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms ( e. g., a weak equivalence of spaces passes to an isomorphism of homology groups ), verified that all existing ( co ) homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.
The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck.

cohomology and was
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.
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.

cohomology and known
The five lemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology / cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object.
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.
Subsequently Tate introduced what are now known as Tate cohomology groups.
In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory.
A great deal is known about the cohomology of groups, including interpretations of low dimensional cohomology, functorality, and how to change groups.
Cartan is known for work in algebraic topology, in particular on cohomology operations, the method of " killing homotopy groups ", and group cohomology.
In mathematics, particularly algebraic topology and homology theory, the Mayer – Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups.
The set of all forms cohomologous to a given form ( and thus to each other ) is called a de Rham cohomology class ; the general study of such classes is known as cohomology.
Of course, the above equations are already known to be true in many senses, such as in the sense of CW-complex where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum.
The Steenrod cohomology operations form a ( non-commutative ) algebra under composition, known as the Steenrod algebra.
Yet another group of similar duality statements is encountered in arithmetics: étale cohomology of finite, local and global fields ( also known as Galois cohomology, since étale cohomology over a field is equivalent to group cohomology of the ( absolute ) Galois group of the field ) admit similar pairings.
To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence.
In de Rham cohomology, the cup product of differential forms is also known as the wedge product, and in this sense is a special case of Grassmann's exterior product.
In mathematics, more specifically in cohomology theory, a-cocycle in the cochain group is associated with a unique equivalence class known as the cocycle class or coclass of
His 1955 thesis at the University of Cambridge described a new theory termed " dihomology ", an algebraic structure associated to a topological space, containing both homology and cohomology, introducing what is now known as the Zeeman spectral sequence.
The earliest results identifiable as Galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves.

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