Étale cohomology

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.

History

Étale cohomology was discovered by Grothendieck (with the help of some suggestions by J.-P. Serre), motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were worked out by Grothendieck together with Michael Artin in 1963-1964, and published as 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 universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and Harvtxt|Deligne|1977 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.

Motivation

For complex algebraic varieties, invariants from algebraic topology such as the fundamental groupand 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 needed 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/"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 and Tate modules).

Definitions

For any scheme "X" the category Et("X") is the category of all étale morphisms 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 "U""i", and we are given elements of "F"("U""i") for all "i" whose restrictions to "U""i"∩"U""j" 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 derived functors of the functor of sections,

:F o 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. The idea of "derived functor" here is that the sheaf of sections doesn't respect exact sequences 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"R""q""f"*, 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 "H""q"("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 field F"q", 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/"n"Z 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 F"q". One considers, for schemes "V", the cohomology groups

:H^i(V, mathbb{Z}/ell^kmathbb{Z})

and "defines" the ℓ-adic cohomology group

:H^i(V,mathbb{Z}_ell) = lim_{leftarrow}H^i(V, mathbb{Z}/ell^kmathbb{Z})

as their inverse limit. Here Z denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/ℓ"k"Z. (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 Z; the latter cohomology group exists but gives the "wrong" cohomology groups.)

More generally, if "F" is an inverse system of étale sheaves "F""i", then the cohomology of "F" is defined to be the inverse limit of the cohomology of the sheaves "F""i":H^q(X,F) = lim_leftarrow H^q(X, F_i)and though there is a natural
H^q(X,lim_leftarrow F_i) o lim_leftarrow H^q(X, F_i)this is not usually an isomorphism. An l-adic sheaf is a special sort of inverse system of étale sheaves "F""i", where "i" runs through positive integers, and "F""i" is a module over Z/"l""i"Z and the map from "F""i"+1 to "F""i"is just reduction mod Z/"l""i"Z.

In the case that "V" is a non-singular algebraic curve and "i" = 1, "H""1" is a free Z-module of rank 2"g", 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 2"g", this is isomorphic to the usual singular cohomology with Z 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".

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,mathbb{Q}_ell)=H^i(V,mathbb{Z}_ell)otimesmathbb{Q}_ell(though this notation is misleading: Q is neither an étale sheaf nor an ℓ-adic sheaf).

Torsion can occur, and was applied by Michael Artin and David Mumford to geometric questions.

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 duality 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 groups. 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.)

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 be reduced to this case 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 "G""m" is the sheaf of non-vanishing functions.)

Calculation of H1("X", "G""m")

The exact sequence of étale sheaves :1 ightarrow G_m ightarrow j_*G_{m,K} ightarrowoplus_{xin X}i_{x*}Z ightarrow 1gives a long exact sequence of cohomology groups:0 ightarrow H^0(G_m) ightarrow H^0(j_*G_{m,K}) ightarrowoplus_{xin X}H^0(i_{x*}Z) ightarrow H^1(G_m) ightarrow H^1(j_*G_{m,K}) ightarrowcdotsHere "j" is the injection of the generic point, "i"x* is the injection of a closed point "x", and Z"x" 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:Klongrightarrow Div(X)longrightarrow H^1(G_m)longrightarrow 1 where Div("X") is the group of divisors of "X" and "K" is its function field. In particular H1("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 H"i"("X", "G""m")

The same long exact sequence above shows that if "i"≥2 then the cohomology group H"i"("X", "G""m") is isomorphic to H"i"("X", j*Gm,K), which is isomorphic to the Galois cohomology group H"i"("K", K*). Tsen's theorem implies that 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"≥1, so all the cohomology groups H"i"("X", "G""m") vanish if "i"≥2.

Calculation of H"i"("X", μ"n")

If μ"n" is the sheaf of "n"th roots of 1 and "n" is prime to the characteristic of the field "k", then H"i"("X",μ"n") is μ"n"("k") if "i"=0, the group of "n"-division points of Pic("X") if "n"=1, and Z/"n"Z if "n"=2, and 0 if "n"≥3. This follows from the previous results using the long exact sequence :0 ightarrow H^0(X, mu_n) ightarrow H^0(X, G_m) ightarrow H^0(X, G_m) ightarrow: ightarrow H^1(X, mu_n) ightarrow H^1(X, G_m) ightarrow H^1(X, G_m) ightarrow H^2(X, mu_n) ightarrow H^2(X, G_m)of the Kummer exact sequence of étale sheaves:1 ightarrowmu_n ightarrow G_mxrightarrow{n}G_m ightarrow 1.and inserting the known values H0("X","G""m") = "k"*, H1("X","G""m") = Pic("X"), and H"i"("X","G""m")=0 for "i"≥2. In particular we get an exact sequence:1 ightarrow H^1(X, mu_n) ightarrow Pic(X)xrightarrow{ imes n} Pic(X) ightarrow H^2(X, mu_n) ightarrow 1.If "n" is divisible by "p" this argument breaks down because "p"th 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 "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 1 we can identify the group Z/"n"Z with the group μ"n" of "n"th roots of 1. The étale group H"i"("X",Z/"n"Z) is then a free module over the ring Z/"n"Z of rank 1 if "i"=1, 2"g" 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 variety, 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 "n" 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 bythe characteristic in a similar way, using the Artin-Schreier sequence:0 ightarrow Z/pZ ightarrow K xrightarrow{xmapsto x^p-x} K ightarrow 0instead 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 groupsin 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 of "G".

*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 "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 the there is a trace
Tr: H_c^{2N}(X, mu_n^N) mapsto Z/nZand 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^{2N-i}(X,Z/nZ)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.

"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:#X(mathbb F_{p^n}) = 1 -sum_{i=1}^{2g} alpha_i^n+p^n,where alpha_i are certain algebraic numbers satisfying |alpha_i|=sqrt p.

"Notes"
* This agrees with the projective line being a curve of genus 0 and having "pn"+1 points.
* We see that number of points on any curve is 'rather close' to that of the projective line.

"Idea of proof"

According to the Lefschetz fixed point theorem, the number of fixed points of any morphism f:X o X is equal to the sum :sum_{i=0}^{dim X}(-1)^imathrm{Tr} f|_{H^i(X)}.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 F"p""n" are those fixed by "F""n", where "F" is the Frobenius automorphism in characteristic "p".

The étale cohomology Betti numbers of "X" in dimensions 0, 1, 2 are resp. 1, "2g", and 1.

According to all of these, :#X(mathbb F_{p^n}) = mathrm{Tr} F^n|_{H^0(X)} -mathrm{Tr} F^n|_{H^1(X)} +mathrm{Tr} F^n|_{H^2(X)}.

This gives the general form of the theorem.

The assertion on the absolute values of the αs requires some deeper argument.

The whole idea fits into the framework of motives: formally ["X"] = [point] + [line] + [1-part] , and [1-part] has something like sqrt p points.

References

*cite book
first = Michael
last = Artin
authorlink = Michael Artin
coauthors = Alexandre Grothendieck, Jean-Louis Verdier, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = xix+525
url=http://modular.fas.harvard.edu/sga/sga/4-1/index.html

*cite book
first = Michael
last = Artin
authorlink = Michael Artin
coauthors = Alexandre Grothendieck, Jean-Louis Verdier, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 2 (Lecture notes in mathematics 270)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = iv+418
url=http://modular.fas.harvard.edu/sga/sga/4-2/index.html

*cite book
first = Michael
last = Artin
authorlink = Michael Artin
coauthors = Alexandre Grothendieck, Jean-Louis Verdier, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3 (Lecture notes in mathematics 305)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = vi+640
url=http://modular.fas.harvard.edu/sga/sga/4-3/index.html

* springer
last=Danilov
first=V.I.
id=E/e036280
title=Étale cohomology

*citation|last=Deligne|first= Pierre |url=http://www.numdam.org/item?id=PMIHES_1974__43__273_0 |title=La conjecture de Weil. I|journal= Inst. Hautes Études Sci. Publ. Math. |volume= 43 |year=1974|pages=273--307|doi=10.1007/BF02684373
*citation|last=Deligne|first= Pierre|url=http://www.numdam.org/item?id=PMIHES_1980__52__137_0 |title=La conjecture de Weil : II|journal= Publications Mathématiques de l'IHÉS|volume= 52 |year=1980|pages= 137–252
* citation
editor-last=Deligne
editor-first=Pierre
editor-link=Pierre Deligne
title=Séminaire de Géométrie Algébrique du Bois Marie — Cohomologie étale (SGA 41⁄2)
series=Lecture notes in mathematics
number=569
publisher=Springer-Verlag
place=Berlin
year=1977
isbn=978-0-387-08066-6
language=French
url=http://modular.fas.harvard.edu/sga/sga/4.5/index.html
doi=10.1007/BFb0091516
volume=569
Chapter1: DOI|id= 10.1007/BFb0091518
*springer|id=l/l057020|title=l-adic cohomology|author=I.V. Dolgachev
*
* citation
last=Milne
first=James S.
author-link=
title=Étale Cohomology
year=1980
publisher=Princeton University Press
series=Princeton Mathematical Series 33
isbn=978-0-691-08238-7

*Günter Tamme "Introduction to Etale Cohomology" ISBN 0387571167

External links

*Archibald and Savitt [http://www.math.mcgill.ca/goren/SeminarOnCohomology/etale2.pdf "Étale cohomology"]
*citation
last=Milne
first=James S.
author-link=
title=Lectures on Étale Cohomology
date=1998
url=http://www.jmilne.org/math/CourseNotes/math732.html


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