Étale Cohomology
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the étale cohomology groups of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
or scheme are algebraic analogues of the usual
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
groups with finite coefficients of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.


History

Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (
Bernard Dwork Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of ''p''-adic analysis to local zeta functions, and in particular for a proof of the first part of the Weil conjectures: the rationality ...
had already managed to prove the rationality part of the conjectures in 1960 using p-adic methods), and the remaining conjecture, the analogue of the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
was proved by
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoor ...
(1974) using â„“-adic cohomology. Further contact with classical theory was found in the shape of the Grothendieck version of the
Brauer group In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
; 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 In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...
and the
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is name ...
in this context. Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and
Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' ...
s. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
) led to some speculation that étale cohomology and its applications (such as the proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
) require axioms beyond ZFC. However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories. Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.


Motivation

For complex algebraic varieties, invariants from
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
such as the fundamental group and cohomology groups are very useful, and one would like to have analogues of these for varieties over other fields, such as finite fields. (One reason for this is that Weil suggested that the Weil conjectures could be proved using such a cohomology theory.) In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups (for coherent sheaves) as the much finer complex topology. However, for constant sheaves such as the sheaf of integers this does not work: the cohomology groups defined using the Zariski topology are badly behaved. For example, Weil envisioned a cohomology theory for varieties over finite fields with similar power as the usual singular cohomology of topological spaces, but in fact, any constant sheaf on an irreducible variety has trivial cohomology (all higher cohomology groups vanish). The reason that the Zariski topology does not work well is that it is too coarse: it has too few open sets. There seems to be no good way to fix this by using a finer topology on a general algebraic variety. Grothendieck's key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
, not just the category of open subsets of a space. He defined étale cohomology by replacing the category of open subsets of a space by the category of étale mappings to a space: roughly speaking, these can be thought of as open subsets of finite unbranched covers of the space. These turn out (after a lot of work) to give just enough extra open sets that one can get reasonable cohomology groups for some constant coefficients, in particular for coefficients Z/''n''Z when ''n'' is coprime to the characteristic of the field one is working over. 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 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. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
and Tate modules).


Definitions

For any scheme ''X'' the category Et(''X'') is the category of all
étale morphism In algebraic geometry, an étale morphism () is a morphism of Scheme (mathematics), schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topol ...
s from a scheme to ''X''. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of ''X''. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to ''X''. There is a rather minor set-theoretical problem here, since Et(''X'') is a "large" category: its objects do not form a set. A presheaf on a topological space ''X'' is a contravariant
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from the category of open subsets to sets. By analogy we define an étale presheaf on a scheme ''X'' to be a contravariant functor from Et(''X'') to sets. A presheaf ''F'' on a topological space is called a sheaf if it satisfies the sheaf condition: whenever an open subset is covered by open subsets ''Ui'', and we are given elements of ''F''(''Ui'') for all ''i'' whose restrictions to ''Ui'' ∩ ''Uj'' agree for all ''i'', ''j'', then they are images of a unique element of ''F''(''U''). By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to ''U'' is said to cover ''U'' if the topological space underlying ''U'' is the union of their images). More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way. The category of sheaves of abelian groups over a scheme has enough injective objects, so one can define right derived functors of left exact functors. The étale cohomology groups ''H''''i''(''F'') of the sheaf ''F'' of abelian groups are defined as the right derived functors of the functor of sections, :F \to \Gamma(F) (where the space of sections Γ(''F'') of ''F'' is ''F''(''X'')). 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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
. The idea of ''derived functor'' here is that the functor of sections doesn't respect
exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definit ...
s as it is not right exact; according to general principles of
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
there will be a sequence of functors ''H'' 0, ''H'' 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 ''H'' 0 functor coincides with the section functor Γ. More generally, a morphism of schemes ''f'' : ''X'' → ''Y'' induces a map ''f''∗ from étale sheaves over ''X'' to étale sheaves over ''Y'', and its right derived functors are denoted by ''Rqf''∗, for ''q'' a non-negative integer. In the special case when ''Y'' is the spectrum of an algebraically closed field (a point), ''R''''q''''f''∗(''F'' ) is the same as ''Hq''(''F'' ). Suppose that ''X'' is a Noetherian scheme. An abelian étale sheaf ''F'' over ''X'' is called finite locally constant if it is represented by an étale cover of ''X''. It is called constructible if ''X'' can be covered by a finite family of subschemes on each of which the restriction of ''F'' is finite locally constant. It is called torsion if ''F''(''U'') is a torsion group for all étale covers ''U'' of ''X''. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.


â„“-adic cohomology groups

In applications to algebraic geometry over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
F''q'' with characteristic ''p'', the main objective was to find a replacement for the singular cohomology groups with integer (or rational) coefficients, which are not available in the same way as for geometry of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
field. Étale cohomology works fine for coefficients Z/''n''Z for ''n'' co-prime to ''p'', but gives unsatisfactory results for non-torsion coefficients. To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called â„“-adic cohomology, where â„“ stands for any prime number different from ''p''. One considers, for schemes ''V'', the cohomology groups :H^i(V, \mathbf/\ell^k\mathbf) and ''defines'' the â„“-adic cohomology group :H^i(V,\mathbf_\ell) = \varprojlim H^i(V, \mathbf/\ell^k\mathbf) as their inverse limit. Here Zâ„“ denotes the â„“-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/â„“''k''Z. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the â„“-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the étale sheaf Zâ„“; the latter cohomology group exists but gives the "wrong" cohomology groups.) More generally, if ''F'' is an inverse system of étale sheaves ''Fi'', then the cohomology of ''F'' is defined to be the inverse limit of the cohomology of the sheaves ''Fi'' :H^q(X, F) = \varprojlim H^q(X, F_i), and though there is a natural map :H^q(X,\varprojlim F_i) \to \varprojlim H^q(X, F_i), this is not usually an isomorphism. An â„“-adic sheaf is a special sort of inverse system of étale sheaves ''Fi'', where ''i'' runs through positive integers, and ''Fi'' is a module over Z/â„“''i'' Z and the map from ''F''''i''+1 to ''Fi'' is just reduction mod Z/â„“''i'' Z. When ''V'' is a non-singular
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
of
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
''g'', ''H''1 is a free Zâ„“-module of rank 2''g'', dual to the Tate module of the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...
of ''V''. Since the first
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
of a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of genus ''g'' is 2''g'', this is isomorphic to the usual singular cohomology with Zâ„“ coefficients for complex algebraic curves. It also shows one reason why the condition â„“ â‰  ''p'' is required: when â„“ = ''p'' the rank of the Tate module is at most ''g''.
Torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...
s can occur, and were applied by Michael Artin and David Mumford to geometric questions. To remove any torsion subgroup from the ℓ-adic cohomology groups and get cohomology groups that are vector spaces over fields of characteristic 0 one defines :H^i(V,\mathbf_\ell)=H^i(V,\mathbf_\ell)\otimes\mathbf_\ell. This notation is misleading: the symbol Qℓ on the left represents neither an étale sheaf nor an ℓ-adic sheaf. The etale cohomology with coefficients in the constant etale sheaf Qℓ does also exist but is quite different from H^i(V,\mathbf_\ell)\otimes\mathbf_\ell. Confusing these two groups is a common mistake.


Properties

In general the â„“-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 â„“-adic integers (or numbers) rather than the integers (or rationals). They satisfy a form of
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...
on non-singular projective varieties, and the â„“-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A
Künneth formula Künneth is a surname. Notable people with the surname include: * Hermann Künneth (1892–1975), German mathematician * Walter Künneth (1901–1997), German Protestant theologian {{DEFAULTSORT:Kunneth German-language surnames ...
also holds. For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first â„“-adic cohomology group of an elliptic curve over a finite field is a free module of rank 2 over the â„“-adic integers, provided â„“ is not the characteristic of the field concerned, and is dual to its Tate module. There is one way in which â„“-adic cohomology groups are better than singular cohomology groups: they tend to be acted on by
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
s. For example, if a complex variety is defined over the rational numbers, its â„“-adic cohomology groups are acted on by the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
of the rational numbers: they afford Galois representations. Elements of the Galois group of the rationals, other than the identity and
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
, 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 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.)


Calculation of étale cohomology groups for algebraic curves

The main initial step in calculating étale cohomology groups of a variety is to calculate them for complete connected smooth algebraic curves ''X'' over
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
s ''k''. The étale cohomology groups of arbitrary varieties can then be controlled using analogues of the usual machinery of algebraic topology, such as the
spectral sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they h ...
of a fibration. For curves the calculation takes several steps, as follows . Let G''m'' denote the sheaf of non-vanishing functions.


Calculation of ''H''1(''X'', G''m'')

The exact sequence of étale sheaves :1\to \mathbf_m\to j_*\mathbf_\to \bigoplus_i_\mathbf\to 1 gives a long exact sequence of cohomology groups :\begin 0 &\to H^0(\mathbf_m)\to H^0(j_*\mathbf_)\to \bigoplus\nolimits_H^0(i_\mathbf) \to \\ &\to H^1(\mathbf_m)\to H^1(j_*\mathbf_)\to \bigoplus\nolimits_H^1(i_\mathbf) \to \\ &\to \cdots \end Here ''j'' is the injection of the generic point, ''ix'' is the injection of a closed point ''x'', G''m'',''K'' is the sheaf G''m'' on (the generic point of ''X''), and Z''x'' is a copy of Z for each closed point of ''X''. The groups ''H i''(''ix*'' Z) vanish if ''i'' > 0 (because ''ix*'' Z is a skyscraper sheaf) and for ''i'' = 0 they are Z so their sum is just the divisor group of ''X''. Moreover, the first cohomology group ''H'' 1(''X'', ''j''∗G''m'',''K'') is isomorphic to the Galois cohomology group ''H'' 1(''K'', ''K''*) which vanishes by Hilbert's theorem 90. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence :K\to \operatorname(X)\to H^1(\mathbf_m)\to 1 where Div(''X'') is the group of divisors of ''X'' and ''K'' is its function field. In particular ''H'' 1(''X'', G''m'') is the Picard group Pic(''X'') (and the first cohomology groups of G''m'' are the same for the étale and Zariski topologies). This step works for varieties ''X'' of any dimension (with points replaced by codimension 1 subvarieties), not just curves.


Calculation of ''Hi''(''X'', G''m'')

The same long exact sequence above shows that if ''i'' â‰¥ 2 then the cohomology group ''H i''(''X'', G''m'') is isomorphic to ''H i''(''X'', ''j''*G''m'',''K''), which is isomorphic to the Galois cohomology group ''H i''(''K'', ''K''*). Tsen's theorem implies that the Brauer group of a function field ''K'' in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups ''H i''(''K'', ''K''*) vanish for ''i'' â‰¥ 1, so all the cohomology groups ''H i''(''X'', G''m'') vanish if ''i'' â‰¥ 2.


Calculation of ''Hi''(''X'', ''μn'')

If ''μn'' is the sheaf of ''n''-th roots of unity and ''n'' and the characteristic of the field ''k'' are coprime integers, then: :H^i (X, \mu_n) = \begin \mu_n(k) & i =0 \\ \operatorname_n(X) & i = 1 \\ \mathbf/n\mathbf & i =2 \\ 0 & i \geqslant 3 \end where Pic''n''(''X'') is group of ''n''-torsion points of Pic(''X''). This follows from the previous results using the long exact sequence :\begin 0 &\to H^0(X, \mu_n)\to H^0(X, \mathbf_m)\to H^0(X, \mathbf_m)\to \\ &\to H^1(X, \mu_n)\to H^1(X, \mathbf_m)\to H^1(X, \mathbf_m)\to \\ &\to H^2(X, \mu_n)\to H^2(X, \mathbf_m)\to H^2(X, \mathbf_m) \to \\ &\to \cdots \end of the Kummer exact sequence of étale sheaves :1 \to \mu_n \to \mathbf_m \xrightarrow \mathbf_m \to 1. and inserting the known values : H^i (X, \mathbf_m) = \begin k^* & i = 0 \\ \operatorname(X) & i =1 \\ 0 &i \geqslant 2 \end In particular we get an exact sequence :1\to H^1(X, \mu_n)\to \operatorname(X)\xrightarrow \operatorname(X)\to H^2(X, \mu_n)\to 1. If ''n'' is divisible by ''p'' this argument breaks down because ''p''-th roots of unity behave strangely over fields of characteristic ''p''. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an ''n''-th root locally for the Zariski topology, so this is one place where the use of the étale topology rather than the Zariski topology is essential.


Calculation of ''H i''(''X'', Z/''n''Z)

By fixing a primitive ''n''-th root of unity we can identify the group Z/''n''Z with the group ''μn'' of ''n''-th roots of unity. The étale group ''H i''(''X'', Z/''n''Z) is then a free module over the ring Z/''n''Z and its rank is given by: :\operatorname(H^i(X, \mathbf/n\mathbf)) = \begin 1 & i =0 \\ 2g & i=1 \\1 & i = 2\\0 & i \geqslant 3 \end where ''g'' is the genus of the curve ''X''. This follows from the previous result, using the fact that the Picard group of a curve is the points of its
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...
, an abelian variety of dimension ''g'', and if ''n'' is coprime to the characteristic then the points of order dividing ''n'' in an abelian variety of dimension ''g'' over an algebraically closed field form a group isomorphic to (Z/''n''Z)2''g''. These values for the étale group ''H i''(''X'', Z/''n''Z) are the same as the corresponding singular cohomology groups when ''X'' is a complex curve.


Calculation of ''H i''(''X'', Z/''p''Z)

It is possible to calculate étale cohomology groups with constant coefficients of order divisible by the characteristic in a similar way, using the Artin–Schreier sequence :0\to \mathbf/p\mathbf\to K\ \xrightarrow\ K\to 0 instead of the Kummer sequence. (For coefficients in Z/''p''''n''Z there is a similar sequence involving Witt vectors.) The resulting cohomology groups usually have ranks less than that of the corresponding groups in characteristic 0.


Examples of étale cohomology groups

*If ''X'' is the spectrum of a field ''K'' with absolute Galois group ''G'', then étale sheaves over ''X'' correspond to continuous sets (or abelian groups) acted on by the (profinite) group ''G'', and étale cohomology of the sheaf is the same as the
group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...
of ''G'', i.e. the
Galois cohomology 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. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
of ''K''. *If ''X'' is a complex variety, then étale cohomology with finite coefficients is isomorphic to singular cohomology with finite coefficients. (This does not hold for integer coefficients.) More generally the cohomology with coefficients in any constructible sheaf is the same. *If ''F'' is a
coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
(or G''m'') then the étale cohomology of ''F'' is the same as Serre's coherent sheaf cohomology calculated with the Zariski topology (and if ''X'' is a complex variety this is the same as the sheaf cohomology calculated with the usual complex topology). *For abelian varieties and curves there is an elementary description of ℓ-adic cohomology. For abelian varieties the first ℓ-adic cohomology group is the dual of the Tate module, and the higher cohomology groups are given by its exterior powers. For curves the first cohomology group is the first cohomology group of its Jacobian. This explains why Weil was able to give a more elementary proof of the Weil conjectures in these two cases: in general one expects to find an elementary proof whenever there is an elementary description of the ℓ-adic cohomology.


Poincaré duality and cohomology with compact support

The étale cohomology groups with compact support of a variety ''X'' are defined to be :H_c^q(X, F) = H^q(Y, j_!F) where ''j'' is an open immersion of ''X'' into a proper variety ''Y'' and ''j''! is the extension by 0 of the étale sheaf ''F'' to ''Y''. This is independent of the immersion ''j''. If ''X'' has dimension at most ''n'' and ''F'' is a torsion sheaf then these cohomology groups H_c^q(X, F) with compact support vanish if ''q'' > 2''n'', and if in addition ''X'' is affine of finite type over a separably closed field the cohomology groups H^q(X, F) vanish for ''q'' > ''n'' (for the last statement, see SGA 4, XIV, Cor.3.2). More generally if ''f'' is a separated morphism of finite type from ''X'' to ''S'' (with ''X'' and ''S'' Noetherian) then the higher direct images with compact support ''R''''q''''f''! are defined by :R^qf_!(F)=R^qg_*(j_!F) for any torsion sheaf ''F''. Here ''j'' is any open immersion of ''X'' into a scheme ''Y'' with a proper morphism ''g'' to ''S'' (with ''f'' = ''gj''), and as before the definition does not depend on the choice of ''j'' and ''Y''. Cohomology with compact support is the special case of this with ''S'' a point. If ''f'' is a separated morphism of finite type then ''R''''q''''f''! takes constructible sheaves on ''X'' to constructible sheaves on ''S''. If in addition the fibers of ''f'' have dimension at most ''n'' then ''R''''q''''f''! vanishes on torsion sheaves for ''q'' > ''2n''. If ''X'' is a complex variety then ''R''''q''''f''! is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves. If ''X'' is a smooth algebraic variety of dimension ''N'' and ''n'' is coprime to the characteristic then there is a trace map :\operatorname: H_c^(X, \mu_n^N) \rightarrow \mathbf/n\mathbf and the bilinear form Tr(''a'' ∪ ''b'') with values in Z/''n''Z identifies each of the groups :H^i_c(X,\mu_n^N) and :H^(X,\mathbf/n\mathbf) with the dual of the other. This is the analogue of Poincaré duality for étale cohomology.


An application to curves

This is how the theory could be applied to the local zeta-function of an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
. Theorem. Let be a curve of
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
defined over , the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
with elements. Then for :\#X \left (\mathbf F_ \right ) = p^n + 1 -\sum_^ \alpha_i^n, where are certain
algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s satisfying . This agrees with being a curve of genus with points. It also shows that the number of points on any curve is rather close (within ) to that of the projective line; in particular, it generalizes Hasse's theorem on elliptic curves.


Idea of proof

According to the
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is name ...
, the number of fixed points of any morphism is equal to the sum :\sum_^ (-1)^i \operatorname \left (f, _ \right ). 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 that are defined over are those fixed by , where is the Frobenius automorphism in characteristic . The étale cohomology
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of in dimensions 0, 1, 2 are 1, 2''g'', and 1 respectively. According to all of these, :\#X \left (\mathbf F_ \right ) = \operatorname \left (F^n, _ \right )- \operatorname \left (F^n, _ \right ) + \operatorname \left (F^n, _ \right ). This gives the general form of the theorem. The assertion on the absolute values of the is the 1-dimensional Riemann Hypothesis of the Weil Conjectures. The whole idea fits into the framework of motives: formally 'X''ointnbsp;+  inenbsp;+  -part and -parthas something like points.


See also

* Locally acyclic morphism * Theorem of absolute purity


References

* * * * * * * * Chapter1: * * * * * *


External links

*Archibald and Savit
''Étale cohomology''
*Goresk
''Langlands Program For Physicists''
* * {{DEFAULTSORT:Etale cohomology Cohomology theories Homological algebra Topological methods of algebraic geometry