Étale cohomology
Encyclopedia
In mathematics
, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, 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.
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
(Dwork
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
was proved by Pierre Deligne
(1974) using ℓ-adic cohomology.
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.
Grothendieck originally developed étale cohomology in an extremely general setting, working with
concepts such as Grothendieck toposes and Grothendieck universe
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 ZFC) led to some uninformed speculation that étale cohomology and its applications (such as the proof of Fermat's last theorem
) needed axioms beyond ZFC. In practice étale cohomology is used mainly for constructible sheaves
over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC (and even in much weaker theories).
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
.
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
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, 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/nZ when n is coprime to the characteristic
of the field one is working over.
Some basic intuitions of the theory are these:
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. However, it is equivalent to a small category because étale morphisms are locally of finite presentation, so it is harmless to pretend that it is a small category.
A presheaf on a topological space X is a contravariant functor
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 Hi(F) of the sheaf F of abelian groups are defined as the derived functor
s of the functor of sections,
(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
. The idea of derived functor here is that the sheaf of sections doesn't respect exact sequence
s as it is not right exact; 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.
More generally, if f is a morphism of schemes from X to Y, it 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), Rqf*(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.
Fq, 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
over the complex number
field. Étale cohomology works fine for coefficients Z/nZ for n coprime to the characteristic, 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 ℓ-adic cohomology. Here "l" stands for any prime number different from p, where p is the characteristic of Fq. One considers, for schemes V, the cohomology groups
and defines the ℓ-adic cohomology group
as their inverse limit
. Here Zl denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/lkZ. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the ℓ-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the étale sheaf Zl; 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
and though there is a natural map
this is not usually an isomorphism.
An ℓ-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/liZ and the map from Fi+1 to Fi
is just reduction mod Z/liZ.
In the case that V is a non-singular algebraic curve
and i = 1, H1 is a free Zl-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, this is isomorphic to the usual singular cohomology with Zℓ coefficients for complex algebraic curves. It also shows one reason why the condition l ≠ p is required: when l = p the rank of the Tate module is at most g.
Torsion subgroup
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
(though this notation is misleading: Ql is neither an étale sheaf nor an ℓ-adic sheaf).
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. A Künneth formula 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 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 way in which ℓ-adic cohomology groups are better than singular cohomology groups: they tend to be acted on by Galois group
s. 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 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
.)
gives a long exact sequence of cohomology groups
Here j is the injection of the generic point, ix is the injection of a closed point x, Gm,K is the sheaf Gm on (the generic point of X), and Zx is a copy of Z for each closed point of X. The groups Hi(ix*Z) vanish if i>0 (because ix*Z is a "skyscraper sheaf") and if i=0 they are Z so their sum is just the divisor group of X. Moreover the first cohomology group H1(X, j*Gm,K) is isomorphic to the Galois cohomology group H1(K, K*) which vanishes by Hilbert's theorem 90
.
Therefore the long exact sequence of étale cohomology groups gives an exact sequence
where Div(X) is the group of divisors of X and K is its function field. In particular H1(X, Gm) is the Picard group Pic(X) (and the first cohomology groups of Gm 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.
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 Hi(K, K*) vanish for i ≥ 1, so all the cohomology groups Hi(X, Gm) vanish if i ≥ 2.
of the Kummer exact sequence of étale sheaves
and inserting the known values H0(X,Gm) = k*, H1(X,Gm) = Pic(X), and Hi(X,Gm)=0 for i ≥ 2. In particular we get an exact sequence
If n is divisible by p this argument breaks down because pth roots of 1 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 nth 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.
, 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/nZ)2g. These values for the étale group Hi(X,Z/nZ) are the same as the corresponding singular cohomology groups when X is a complex curve.
the characteristic in a similar way, using the Artin-Schreier
sequence
instead of the Kummer sequence. (For coefficients in Z/pnZ there is a similar sequence involving Witt vector
s.)
The resulting cohomology groups usually have ranks less than that of the corresponding groups
in characteristic 0.
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
with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups 
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 Rqf! are defined by
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 Rqf! takes constructible sheaves on X to constructible sheaves on S. If in addition the fibers of f have dimension at most n then Rqf! vanishes on torsion sheaves for q > 2n. If X is a complex variety then Rqf! 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 the there is a trace map
and the bilinear form Tr(a∪b) with values in Z/nZ identifies each of the groups
and
with the dual of the other. This is the analogue of Poincaré duality for étale cohomology.
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
where
are certain algebraic number
s
satisfying
.
Notes
Idea of proof
According to the Lefschetz fixed point theorem, the number of fixed points of any morphism
is equal to the sum
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 Fpn are those fixed by Fn, where F is the Frobenius automorphism in characteristic
p.
The étale cohomology Betti number
s 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
s is the 1-dimensional Riemann Hypothesis of the Weil Conjectures.
The whole idea fits into the framework of motives: formally [X] = [point]+[line]+[1-part], and [1-part] has something like
points.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory
Weil cohomology theory
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil...
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 suggested by , using some suggestions by J.-P. Serre, and was motivated by the attempt to construct a Weil cohomology theoryWeil cohomology theory
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil...
in order to prove the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
. The foundations were soon after worked out by Grothendieck together with Michael Artin
Michael Artin
Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...
, and published as and SGA 4
Grothendieck's Séminaire de géométrie algébrique
In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...
. Grothendieck used étale cohomology to prove some of the Weil conjectures
(Dwork
Bernard Dwork
Bernard Morris Dwork was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for the first general results on the Weil conjectures. Together with Kenkichi Iwasawa he received the Cole Prize in 1962.Dwork received his Ph.D. at Columbia...
had already managed to prove the rationality part of the conjectures in 1960 using p-adic
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
methods), and the remaining conjecture, the analogue of the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
was proved by Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
(1974) using ℓ-adic cohomology.
Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group
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 of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
; this was applied in short order to diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
, 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
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
and the Lefschetz fixed point theorem in this context.
Grothendieck originally developed étale cohomology in an extremely general setting, working with
concepts such as Grothendieck toposes and Grothendieck universe
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...
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 ZFC) led to some uninformed speculation that étale cohomology and its applications (such as the proof of Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
) needed axioms beyond ZFC. In practice étale cohomology is used mainly for constructible sheaves
Constructible sheaf
In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a twisted constant sheaf...
over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC (and even in much weaker theories).
Étale cohomology quickly found other applications, for example Deligne and Lusztig used it to construct representations
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of finite groups of Lie type; see Deligne–Lusztig theory
Deligne–Lusztig theory
In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by ....
.
Motivation
For complex algebraic varieties, invariants from algebraic topology such as the fundamental groupFundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
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
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
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, 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/nZ when n is coprime to the characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
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 theoremImplicit function theoremIn multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...
if it were true in algebraic geometry (but it isn't — implicit algebraic functions are called algebroidAlgebroidIn 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...
in older literature). - There are certain basic cases, of dimension 0 and 1, and for an abelian varietyAbelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
, where the answers with constant sheaves of coefficients can be predicted (via Galois cohomologyGalois cohomologyIn 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...
and Tate moduleTate moduleIn mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...
s).
Definitions
For any scheme X the category Et(X) is the category of all étale morphismÉtale morphism
In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not...
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. However, it is equivalent to a small category because étale morphisms are locally of finite presentation, so it is harmless to pretend that it is a small category.
A presheaf on a topological space X is a contravariant functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
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
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
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
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
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 Hi(F) of the sheaf F of abelian groups are defined as the derived functor
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
s of the functor of sections,
(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
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
. The idea of derived functor here is that the sheaf of sections doesn't respect exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
s as it is not right exact; according to general principles of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
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.
More generally, if f is a morphism of schemes from X to Y, it 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), Rqf*(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.
ℓ-adic cohomology groups
In applications to algebraic geometry over a finite fieldFinite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
Fq, 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 variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
field. Étale cohomology works fine for coefficients Z/nZ for n coprime to the characteristic, 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 ℓ-adic cohomology. Here "l" stands for any prime number different from p, where p is the characteristic of Fq. One considers, for schemes V, the cohomology groups
and defines the ℓ-adic cohomology group
as their inverse limit
Inverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
. Here Zl denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/lkZ. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the ℓ-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the étale sheaf Zl; 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
and though there is a natural map
this is not usually an isomorphism.
An ℓ-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/liZ and the map from Fi+1 to Fi
is just reduction mod Z/liZ.
In the case that V is a non-singular algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
and i = 1, H1 is a free Zl-module of rank 2g, dual to the Tate module
Tate module
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...
of the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
of V, where g is the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
of V. Since the first Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
of a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
of genus g is 2g, this is isomorphic to the usual singular cohomology with Zℓ coefficients for complex algebraic curves. It also shows one reason why the condition l ≠ p is required: when l = p the rank of the Tate module is at most g.
Torsion subgroup
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...
s can occur, and were applied by Michael Artin
Michael Artin
Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...
and David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
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
(though this notation is misleading: Ql is neither an étale sheaf nor an ℓ-adic sheaf).
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 dualityPoincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
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. A Künneth formula 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 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
Tate module
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...
.
There is one way in which ℓ-adic cohomology groups are better than singular cohomology groups: they tend to be acted on by Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s. 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
Absolute Galois group
In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
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 is related to the fact that the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
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
Grothendieck's Galois theory
In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry...
.)
Calculation of étale cohomology groups
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 fields 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 of a fibration. For curves the calculation takes several steps, as follows . (The sheaf Gm is the sheaf of non-vanishing functions.)Calculation of H1(X, Gm)
The exact sequence of étale sheavesgives a long exact sequence of cohomology groups
Here j is the injection of the generic point, ix is the injection of a closed point x, Gm,K is the sheaf Gm on (the generic point of X), and Zx is a copy of Z for each closed point of X. The groups Hi(ix*Z) vanish if i>0 (because ix*Z is a "skyscraper sheaf") and if i=0 they are Z so their sum is just the divisor group of X. Moreover the first cohomology group H1(X, j*Gm,K) is isomorphic to the Galois cohomology group H1(K, K*) which vanishes by Hilbert's theorem 90
Hilbert's Theorem 90
In abstract algebra, Hilbert's Theorem 90 refers to an important result on cyclic extensions of fields that leads to Kummer theory...
.
Therefore the long exact sequence of étale cohomology groups gives an exact sequence
where Div(X) is the group of divisors of X and K is its function field. In particular H1(X, Gm) is the Picard group Pic(X) (and the first cohomology groups of Gm 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, Gm)
The same long exact sequence above shows that if i ≥ 2 then the cohomology group Hi(X, Gm) is isomorphic to Hi(X, j*Gm,K), which is isomorphic to the Galois cohomology group Hi(K, K*). Tsen's theoremTsen's theorem
In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed. This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups H i vanish for...
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 Hi(K, K*) vanish for i ≥ 1, so all the cohomology groups Hi(X, Gm) vanish if i ≥ 2.
Calculation of Hi(X, μn)
If μn is the sheaf of nth roots of 1 and n is prime to the characteristic of the field k, then Hi(X,μn) is μn(k) if i=0, the group of n-division points of Pic(X) if i=1, and Z/nZ if i=2, and 0 if i≥3. This follows from the previous results using the long exact sequenceof the Kummer exact sequence of étale sheaves
and inserting the known values H0(X,Gm) = k*, H1(X,Gm) = Pic(X), and Hi(X,Gm)=0 for i ≥ 2. In particular we get an exact sequence
If n is divisible by p this argument breaks down because pth roots of 1 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 nth 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 Hi(X, Z/nZ)
By fixing a primitive nth root of 1 we can identify the group Z/nZ with the group μn of nth roots of 1. The étale group Hi(X,Z/nZ) is then a free module over the ring Z/nZ of rank 1 if i=0, 2g if i=1, 1 if i=2, and 0 if i≥ 3 (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 varietyJacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
, an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
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/nZ)2g. These values for the étale group Hi(X,Z/nZ) are the same as the corresponding singular cohomology groups when X is a complex curve.
Calculation of Hi(X, Z/pZ)
It is possible to calculate étale cohomology groups with constant coefficients of order divisible bythe characteristic in a similar way, using the Artin-Schreier
Artin-Schreier theory
In mathematics, Artin–Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p...
sequence
instead of the Kummer sequence. (For coefficients in Z/pnZ there is a similar sequence involving Witt vector
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.-Motivation:Any p-adic...
s.)
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 cohomologyGroup cohomologyIn abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
of G, i.e. the Galois cohomologyGalois cohomologyIn 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...
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 (or Gm) 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 moduleTate moduleIn mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...
, 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 bewhere 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
with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups 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 Rqf! are defined by
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 Rqf! takes constructible sheaves on X to constructible sheaves on S. If in addition the fibers of f have dimension at most n then Rqf! vanishes on torsion sheaves for q > 2n. If X is a complex variety then Rqf! 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 the there is a trace map
and the bilinear form Tr(a∪b) with values in Z/nZ identifies each of the groups
and
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-functionLocal zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
of an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
.
Theorem. Let X be a curve of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g defined over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with p elements. Then for every
n greater or equal 1 one has
where
are certain algebraic numberAlgebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s
satisfying
.Notes
- This agrees with the projective lineProjective lineIn mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
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
is equal to the sumThis 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 Fpn are those fixed by Fn, where F is the Frobenius automorphism in characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
p.
The étale cohomology Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s 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
s is the 1-dimensional Riemann Hypothesis of the Weil Conjectures.The whole idea fits into the framework of motives: formally [X] = [point]+[line]+[1-part], and [1-part] has something like
points.External links
- Archibald and Savitt Étale cohomology