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. 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. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ...

The formal definition of étale cohomology is as the derived functor of the functor of sections, In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

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 mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, an abelian group is a commutative group, i. ... In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of general cohomology theories. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

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. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ...

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. In mathematics, in the field of calculus of several variables, the implicit function theorem says that for a suitable set of equations, some of the variables are defined as a function of the others. ... In mathematics, algebroid may mean algebroid branch, a formal power series branch of an algebraic curve algebroid multifunction Lie algebroid in the theory of Lie groupoids algebroid cohomology This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ... In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some...

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. For discussion of topoi in literary theory, see literary topos. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...

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 -adic cohomology. Here stands for any prime number with In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

≠ p

where p is the characteristic of F. One considers, for schemes V, the cohomology groups The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ...

Hⁱ(V, Z /^kZ)

and defines

Hⁱ(V, Z)

as their inverse limit. Here Z denotes the l-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/^kZ. In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... The title given to this article is incorrect due to technical limitations. ...

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-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 ≠ p is required: when = p the rank of the Tate module is at most g. In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...

To remove any torsion subgroup from the -adic groups (which can occur, and was applied by Mike Artin and David Mumford to geometric questions) the definition In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... Michael Artin is an American mathematician, known for his contributions to algebraic geometry. ... David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ...

Hⁱ(V, Q)

with the -adic numbers Q is typically used.

An application to curves

This is how the theory could be applied to the local zeta-function of an algebraic curve. In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta function comes via consideration of the logarithmic derivative . ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...

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 In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

,

where α_i are certain algebraic numbers satisfying . In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...

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 In mathematics, the projective line is a fundamental example of an algebraic curve. ...

According to the Lefschetz fixed point theorem, the number of fixed points of any morphism is equals to the sum In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some...

.

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 are those fixed by Fⁿ where F is the Frobenius automorphism in characteristic p. In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...

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 motives: formally [X] = [point]+[line]+[1-part], and [1-part] has something like points. In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ...