Inmathematics, theétale cohomology groups of analgebraic variety orscheme are algebraic analogues of the usualcohomology groups with finite coefficients of atopological space, introduced byGrothendieck in order to prove theWeil conjectures. Étale cohomology theory can be used to constructℓ-adic cohomology, which is an example of aWeil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction ofrepresentations of finite groups of Lie type.

Étale cohomology was introduced byAlexander Grothendieck (1960), using some suggestions byJean-Pierre Serre, and was motivated by the attempt to construct aWeil cohomology theory in order to prove theWeil conjectures. The foundations were soon after worked out by Grothendieck together withMichael Artin, and published as (Artin 1962) andSGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of the conjectures in 1960 usingp-adic methods), and the remaining conjecture, the analogue of theRiemann hypothesis was proved byPierre Deligne (1974) using ℓ-adic cohomology.

Further contact with classical theory was found in the shape of the Grothendieck version of theBrauer group; this was applied in short order todiophantine geometry, byYuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such asPoincaré duality and theLefschetz fixed-point theorem in this context.

Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such asGrothendieck toposes andGrothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, andDeligne (1977) gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved inZermelo–Fraenkel set theory) led to some speculation that étale cohomology and its applications (such as the proof ofFermat's Last Theorem) require axioms beyond ZFC. However, in practice étale cohomology is used mainly in the case ofconstructible 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 andGeorge Lusztig used it to constructrepresentations of finitegroups of Lie type; seeDeligne–Lusztig theory.

For complex algebraic varieties, invariants fromalgebraic topology such as thefundamental 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 thatWeil suggested that the Weil conjectures could be proved using such a cohomology theory.) In the case of cohomology ofcoherent sheaves, Serre showed that one could get a satisfactory theory just by using theZariski 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, forconstant 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 usualsingular 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 anycategory, 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 coefficientsZ/nZ whenn is coprime to thecharacteristic of the field one is working over.

Some basic intuitions of the theory are these:

• Theétale requirement is the condition that would allow one to apply theimplicit 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 anabelian variety, where the answers with constant sheaves of coefficients can be predicted (viaGalois cohomology andTate modules).

For anyschemeX the category Et(X) is the category of allétale morphisms from a scheme toX. 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" ofX. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps toX. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set.

Apresheaf on a topological spaceX is a contravariantfunctor from the category of open subsets to sets. By analogy we define anétale presheaf on a schemeX to be a contravariant functor from Et(X) to sets.

A presheafF on a topological space is called asheaf if it satisfies the sheaf condition: whenever an open subset is covered by open subsetsU_i, and we are given elements ofF(U_i) for alli whose restrictions toU_i ∩U_j agree for alli,j, then they are images of a unique element ofF(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 toU is said to coverU if the topological space underlyingU is the union of their images). More generally, one can define a sheaf for anyGrothendieck topology on a category in a similar way.

The category of sheaves of abelian groups over a scheme hasenough injective objects, so one can defineright derived functors ofleft exact functors. Theétale cohomology groupsHⁱ(F) of the sheafF of abelian groups are defined as the right derived functors of the functor of sections,

${\displaystyle F\to \mathrm{\Gamma }(F)}$

(where the space of sections Γ(F) ofF isF(X)). The sections of a sheaf can be thought of as Hom(Z,F) whereZ is the sheaf that returns the integers as anabelian group. The idea ofderived functor here is that the functor of sections doesn't respectexact sequences as it is not right exact; according to general principles ofhomological algebra there will be a sequence of functorsH 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). TheH ⁰ functor coincides with the section functor Γ.

More generally, a morphism of schemesf :X →Y induces a mapf_∗ from étale sheaves overX to étale sheaves overY, and its right derived functors are denoted byR^qf_∗, forq a non-negative integer. In the special case whenY is the spectrum of an algebraically closed field (a point),R^qf_∗(F ) is the same asH^q(F ).

Suppose thatX is aNoetherian scheme. An abelian étale sheafF overX is calledfinite locally constant if it is represented by an étale cover ofX. It is calledconstructible ifX can be covered by a finite family of subschemes on each of which the restriction ofF is finite locally constant. It is calledtorsion ifF(U) is a torsion group for all étale coversU ofX. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.

In applications to algebraic geometry over afinite fieldF_q with characteristicp, the main objective was to find a replacement for thesingular cohomology groups with integer (or rational) coefficients, which are not available in the same way as for geometry of analgebraic variety over thecomplex number field. Étale cohomology works fine for coefficientsZ/nZ forn co-prime top, but gives unsatisfactory results for non-torsion coefficients. To get cohomology groups without torsion from étale cohomology one has to take aninverse limit of étale cohomology groups with certain torsion coefficients; this is calledℓ-adic cohomology, where ℓ stands for any prime number different fromp. One considers, for schemesV, the cohomology groups

${\displaystyle H^i(V,\mathbf{Z}/{\ell }^k\mathbf{Z})}$

anddefines the ℓ-adic cohomology group

${\displaystyle H^i(V,{\mathbf{Z}}_{\ell })=\underleftarrow{\lim }{H}^i(V,\mathbf{Z}/{\ell }^k\mathbf{Z})}$

as their inverse limit. HereZ_ℓ denotes theℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficientsZ/ℓ^kZ. (There is a notorious trap here: cohomology doesnot commute with taking inverse limits, and the ℓ-adic cohomology group, defined as an inverse limit, isnot the cohomology with coefficients in the étale sheafZ_ℓ; the latter cohomology group exists but gives the "wrong" cohomology groups.)

More generally, ifF is an inverse system of étale sheavesF_i, then the cohomology ofF is defined to be the inverse limit of the cohomology of the sheavesF_i

${\displaystyle H^q(X,F)=\underleftarrow{\lim }{H}^q(X,F_i),}$

and though there is a natural map

${\displaystyle H^q(X,\underleftarrow{\lim }{F}_i)\to \underleftarrow{\lim }{H}^q(X,F_i),}$

this isnot usually an isomorphism. Anℓ-adic sheaf is a special sort of inverse system of étale sheavesF_i, wherei runs through positive integers, andF_i is a module overZ/ℓⁱ Z and the map fromF_i+1 toF_i is just reduction modZ/ℓⁱ Z.

WhenV is anon-singularalgebraic curve ofgenusg,H¹ is a freeZ_ℓ-module of rank 2g, dual to theTate module of theJacobian variety ofV. Since the firstBetti number of aRiemann surface of genusg is 2g, this is isomorphic to the usual singular cohomology withZ_ℓ 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 mostg.

Torsion subgroups can occur, and were applied byMichael Artin andDavid Mumford to geometric questions^{[citation needed]}. 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

${\displaystyle H^i(V,{\mathbf{Q}}_{\ell })=H^i(V,{\mathbf{Z}}_{\ell })\otimes {\mathbf{Q}}_{\ell }.}$

This notation is misleading: the symbolQ_ℓ on the left represents neither an étale sheaf nor an ℓ-adic sheaf. The etale cohomology with coefficients in the constant etale sheafQ_ℓ does also exist but is quite different from${\displaystyle H^i(V,{\mathbf{Z}}_{\ell })\otimes {\mathbf{Q}}_{\ell }}$. Confusing these two groups is a common mistake.

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 ofPoincaré 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. AKünneth 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 itsTate module.

There is one way in which ℓ-adic cohomology groups are better than singular cohomology groups: they tend to be acted on byGalois groups. For example, if a complex variety is defined over the rational numbers, its ℓ-adic cohomology groups are acted on by theabsolute Galois group of the rational numbers: they affordGalois representations.

Elements of the Galois group of the rationals, other than the identity andcomplex conjugation, do not usually actcontinuously 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 thefundamental 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 alsoGrothendieck's Galois theory.)

The main initial step in calculating étale cohomology groups of a variety is to calculate them forcomplete connectedsmooth algebraic curvesX overalgebraically closed fieldsk. The étale cohomology groups of arbitrary varieties can then be controlled using analogues of the usual machinery of algebraic topology, such as thespectral sequence of afibration. For curves the calculation takes several steps, as follows (Artin 1962). LetG_m denote the sheaf of non-vanishing functions.

The exact sequence of étale sheaves

${\displaystyle 1\to {\mathbf{G}}_m\to j_{\ast }{\mathbf{G}}_{m,K}\to \underset{x\in |X|}{⨁}{i}_{x\ast }\mathbf{Z}\to 1}$

gives a long exact sequence of cohomology groups

Herej is the injection of the generic point,i_x is the injection of a closed pointx,G_m,K is the sheafG_m onSpecK (the generic point ofX), andZ_x is a copy ofZ for each closed point ofX. The groupsH ⁱ(i_x* Z) vanish ifi > 0 (becausei_x* Z is askyscraper sheaf) and fori = 0 they areZ so their sum is just the divisor group ofX. Moreover, the first cohomology groupH ¹(X,j_∗G_m,K) is isomorphic to the Galois cohomology groupH ¹(K,K*) which vanishes byHilbert's theorem 90. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence

${\displaystyle K\to \mathrm{Div}(X)\to H^1({\mathbf{G}}_m)\to 1}$

where Div(X) is the group of divisors ofX andK is its function field. In particularH ¹(X,G_m) is thePicard group Pic(X) (and the first cohomology groups ofG_m are the same for the étale and Zariski topologies). This step works for varietiesX of any dimension (with points replaced by codimension 1 subvarieties), not just curves.

The same long exact sequence above shows that ifi ≥ 2 then the cohomology groupH ⁱ(X,G_m) is isomorphic toH ⁱ(X,j_*G_m,K), which is isomorphic to the Galois cohomology groupH ⁱ(K,K*).Tsen's theorem implies that the Brauer group of a function fieldK in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groupsH ⁱ(K,K*) vanish fori ≥ 1, so all the cohomology groupsH ⁱ(X,G_m) vanish ifi ≥ 2.

Ifμ_n is the sheaf ofn-th roots of unity andn and the characteristic of the fieldk are coprime integers, then:

where Pic_n(X) is group ofn-torsion points of Pic(X). This follows from the previous results using the long exact sequence

of the Kummer exact sequence of étale sheaves

${\displaystyle 1\to {\mu }_n\to {\mathbf{G}}_m\stackrel{(\cdot {)}^n}{\to }{\mathbf{G}}_m\to 1.}$

and inserting the known values

In particular we get an exact sequence

${\displaystyle 1\to H^1(X,{\mu }_n)\to \mathrm{Pic}(X)\stackrel{\times n}{\to }\mathrm{Pic}(X)\to H^2(X,{\mu }_n)\to 1.}$

Ifn is divisible byp this argument breaks down becausep-th roots of unity behave strangely over fields of characteristicp. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have ann-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.

By fixing a primitiven-th root of unity we can identify the groupZ/nZ with the groupμ_n ofn-th roots of unity. The étale groupH ⁱ(X,Z/nZ) is then a free module over the ringZ/nZ and its rank is given by:

whereg is the genus of the curveX. This follows from the previous result, using the fact that the Picard group of a curve is the points of itsJacobian variety, anabelian variety of dimensiong, and ifn is coprime to the characteristic then the points of order dividingn in an abelian variety of dimensiong over an algebraically closed field form a group isomorphic to (Z/nZ)^2g. These values for the étale groupH ⁱ(X,Z/nZ) are the same as the corresponding singular cohomology groups whenX is a complex curve.

It is possible to calculate étale cohomology groups with constant coefficients of order divisible by the characteristic in a similar way, using theArtin–Schreier sequence

${\displaystyle 0\to \mathbf{Z}/p\mathbf{Z}\to K\stackrel{\genfrac{}{}{0ex}{}{}{x\mapsto x^p-x}}{\to }K\to 0}$

instead of the Kummer sequence. (For coefficients inZ/pⁿZ there is a similar sequence involvingWitt vectors.) The resulting cohomology groups usually have ranks less than that of the corresponding groups in characteristic 0.

• IfX is the spectrum of a fieldK with absolute Galois groupG, then étale sheaves overX correspond to continuous sets (or abelian groups) acted on by the (profinite) groupG, and étale cohomology of the sheaf is the same as thegroup cohomology ofG, i.e. theGalois cohomology ofK.
• IfX 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 anyconstructible sheaf is the same.
• IfF is acoherent sheaf (orG_m) then the étale cohomology ofF is the same as Serre's coherent sheaf cohomology calculated with the Zariski topology (and ifX 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 theTate 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.

The étale cohomology groups with compact support of a varietyX are defined to be

${\displaystyle H_c^q(X,F)=H^q(Y,j_{!}F)}$

wherej is an open immersion ofX into a proper varietyY andj_! is the extension by 0 of the étale sheafF toY. This is independent of the immersionj. IfX has dimension at mostn andF is a torsion sheaf then these cohomology groups${\displaystyle H_c^q(X,F)}$ with compact support vanish ifq > 2n, and if in additionX is affine of finite type over a separably closed field the cohomology groups${\displaystyle H^q(X,F)}$ vanish forq > n (for the last statement, see SGA 4, XIV, Cor.3.2).

More generally iff is a separated morphism of finite type fromX toS (withX andS Noetherian) then thehigher direct images with compact supportR^qf_! are defined by

${\displaystyle R^q{f}_{!}(F)=R^q{g}_{\ast }(j_{!}F)}$

for any torsion sheafF. Herej is any open immersion ofX into a schemeY with a proper morphismg toS (withf = gj), and as before the definition does not depend on the choice ofj andY. Cohomology with compact support is the special case of this withS a point. Iff is a separated morphism of finite type thenR^qf_! takes constructible sheaves onX to constructible sheaves onS. If in addition the fibers off have dimension at mostn thenR^qf_! vanishes on torsion sheaves forq >2n. IfX is a complex variety thenR^qf_! is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves.

IfX is a smooth algebraic variety of dimensionN andn is coprime to the characteristic then there is a trace map

${\displaystyle \mathrm{Tr}:H_c^{2N}(X,{\mu }_n^N)\to \mathbf{Z}/n\mathbf{Z}}$

and the bilinear form Tr(a ∪b) with values inZ/nZ identifies each of the groups

${\displaystyle H_c^i(X,{\mu }_n^N)}$

and

${\displaystyle H^{2N-i}(X,\mathbf{Z}/n\mathbf{Z})}$

with the dual of the other. This is the analogue of Poincaré duality for étale cohomology.

This is how the theory could be applied to thelocal zeta-function of analgebraic curve.

Theorem. LetX be a curve ofgenusg defined overF_p, thefinite field withp elements. Then forn ≥ 1

${\displaystyle \mathrm{\#}X\left({\mathbf{F}}_{p^n}\right)=p^n+1-\sum _{i=1}^{2g}{\alpha }_i^n,}$

whereα_i are certainalgebraic numbers satisfying|α_i| =√p.

This agrees withP¹(F_pⁿ) being a curve of genus0 withpⁿ + 1 points. It also shows that the number of points on any curve is rather close (within2gp^{n / 2}) to that of the projective line; in particular, it generalizesHasse's theorem on elliptic curves.

According to theLefschetz fixed-point theorem, the number of fixed points of any morphismf :X →X is equal to the sum

${\displaystyle \sum _{i=0}^{2\dim (X)}(-1{)}^i\mathrm{Tr}\left(f{|}_{H^i(X)}\right).}$

This formula is valid for ordinary topological varieties and ordinary topology, but it is wrong for mostalgebraic topologies. However, this formuladoes hold for étale cohomology (though this is not so simple to prove).

The points ofX that are defined overF_pⁿ are those fixed byFⁿ, whereF is theFrobenius automorphism incharacteristicp.

The étale cohomologyBetti numbers ofX in dimensions 0, 1, 2 are 1, 2g, and 1 respectively.

According to all of these,

${\displaystyle \mathrm{\#}X\left({\mathbf{F}}_{p^n}\right)=\mathrm{Tr}\left(F^n{|}_{H^0(X)}\right)-\mathrm{Tr}\left(F^n{|}_{H^1(X)}\right)+\mathrm{Tr}\left(F^n{|}_{H^2(X)}\right).}$

This gives the general form of the theorem.

The assertion on the absolute values of theα_i is the 1-dimensional Riemann Hypothesis of the Weil Conjectures.

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

• Locally acyclic morphism
• Theorem of absolute purity

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• Cohomology theories
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