Cohomology of algebraic groups with coefficients in twisted representations
aa r X i v : . [ m a t h . K T ] O c t COHOMOLOGY OF ALGEBRAIC GROUPS WITHCOEFFICIENTS IN TWISTED REPRESENTATIONS
ANTOINE TOUZ´E
Abstract.
This article is a survey on the cohomology of a reductive algebraicgroup with coefficients in twisted representations. A large part of the paperis devoted to the advances obtained by the theory of strict polynomial func-tors initiated by Friedlander and Suslin in the late nineties. The last sectionexplains that the existence of certain ‘universal classes’ used to prove cohomo-logical finite generation is equivalent to some recent ‘untwisting theorems’ inthe theory of strict polynomial functors. We actually provide thereby a newproof of these theorems. Introduction
Let G be an affine algebraic group (scheme) over a field k of positive characteristic p . If V is a rational representation of G , we may twist it by base change along theFrobenius morphism k → k , x x p r . The representation obtained is denoted by V ( r ) . This article deals with the problem of computing rational cohomology of G with coefficients in such twisted representations. That is, we consider the followingnaive, basic question.How different is H ∗ ( G, V ( r ) ) from H ∗ ( G, V )?Cohomology with coefficients in twisted representations appears naturally in manyproblems regarding rational cohomology of algebraic groups and the connectionswith other topics. Unfortunately, the naive question raised above has no easyanswer. In recent years however, much effort has been made to find satisfactorypartial answers.The first four sections of this article are a survey of the problem of computingcohomology with twisted coefficients. We describe some motivations and someimportant partial answers to this problem. As we do not assume that the reader isan expert of algebraic groups, we begin section 2 by a short introduction to affinegroup schemes, their representations, and the associated cohomology theory. Thenwe present three motivations to the study of twisted representations, namely (1)the link with the cohomology of finite groups of Lie type, (2) the study of simplerepresentations of reductive algebraic groups, (3) cohomological finite generationtheorems. This list of motivations is not exhaustive. For example, we have leftaside the connections with algebraic topology (via strict polynomial functors andunstable modules over the Steenrod algebra [FFSS, FFPS]).In sections 3 and 4 we present classical and also more recent approaches andresults regarding the cohomology of reductive group schemes with coefficients in
Date : October 4, 2018.2010
Mathematics Subject Classification.
Primary 20G10, Secondary 18G15.This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). twisted representations. In particular, section 4 presents up-to-date results obtainedby the functorial approach initiated by Friedlander and Suslin in [FS]. Indeed, thetheory of strict polynomial functors has recently brought very clean answers [Cha2,Pha, Tou4] to the naive question formulated above. In order that non-experts ofstrict polynomial functors can easily use these answers, we have formulated thempurely in terms of (polynomial) representations of algebraic groups in section 4.4.Classical ‘group cohomologists’ may jump directly to this section to evaluate theinterest of strict polynomial functors.The last section is a bit more technical, but hopefully no less interesting. In thissection we prove that the existence of some ‘universal cohomology classes’ (whichare one of the key tools in cohomological finite generation theorems [TVdK]) isactually equivalent to the untwisting theorems of [Cha2, Pha, Tou4]. In particular,section 5.5 gives a new uniform proof of all these untwisting theorems, which doesnot rely on the adjoint to the precomposition by the Frobenius twist. The latteris a technical tool introduced by M. Cha lupnik [Cha2], which was essential in theearlier proofs. Our proof also separates general arguments from statements specificto strict polynomial functors. We hope that this makes it readable for non-experts,and helps to understand where the ‘strict polynomial magic’ really lies.
Acknowledgement.
The author thanks the anonymous referee for very carefullyreading a first version of the article and detecting several mistakes.2.
Problem and motivations
Group schemes and their representations.
We refer the reader to [Wat,Jan] for affine group schemes and their representations. A nice concise introductionto these topics is [Fri]. We recall here only a few basic definitions and conventions.Given a field k we let k − Alg be the category of commutative finitely generated k -algebras with unit. An affine algebraic group scheme over k is a representablefunctor G from k − Alg to the category of groups. All group schemes in this articlewill be affine algebraic without further mention. A morphism of group schemes isa natural transformation G → G ′ . Let k [ G ] denote the finitely generated k -algebrarepresenting a group scheme G , and let k be the algebraic closure of k . Then G is said to be smooth, resp. connected, if k [ G ] ⊗ k k is nilpotent-free, resp. has nonontrivial idempotent. For all k -vector spaces V , there is a functor GL V : k − Alg → Groupssending a k -algebra A to the group of invertible A -linear endomorphisms of V ⊗ k A .When V has dimension n this is a (smooth connected) group scheme sending A to GL n ( A ), and we denote it by GL n, k , or often by GL n when the ground field k isclear from the context. A representation of a group scheme G (or a G -module) isa k -vector space V equipped with a natural transformation ρ : G → GL V . Suchrepresentations are also called rational representations, but we most often drop theadjective ‘rational’.If k is algebraically closed, smooth group schemes (such as GL n , SL n , Sp n ,Spin n . . . ) identify with Zariski closed subgroups of some GL n ( k ). A finite di-mensional rational representation of a smooth group scheme G is then the sameas a representation of G such that the corresponding morphism G → GL ( V ) is amorphism of algebraic groups (i.e. a regular map). OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 3
All the problems and results described in this paper remain hard and interestingover an algebraically closed field, the hypothesis that k is algebraically closed doesnot bring any substantial simplification in the proofs.2.2. Rational cohomology with coefficients in twisted representations.
Wefix a field k of positive characteristic p >
0. The Frobenius morphism k → k , x x p and its iterates induce morphisms of group schemes (for r ≥ F r : GL n → GL n [ a ij ] [ a p r ij ] . If G is a subgroup scheme of GL n defined over the prime field F p , these morphismsrestrict to F r : G → G . If V is a vector space acted on by G via ρ : G → GL V , welet V ( r ) denote the same vector space, acted on by G via the composite ρ ◦ F r . Therepresentation V ( r ) is called the r th Frobenius twist of V , and we shall informallyrefer to such representations as twisted representations .Twisted representations play a prominent role in the representation theory ofaffine algebraic group schemes in positive characteristic. Given a group scheme G and a representation V , we denote by H ∗ ( G, V ) the extension groups Ext ∗ G ( k , V )between the trivial G -module k and V , computed in the category of rational rep-resentations of G (this category is abelian with enough injectives). The (rational)cohomology of G with coefficients in twisted representations appears naturally inmany situations, which motivates to study the following general problem. Main Problem 2.1.
Try to understand or to compute the cohomology of G withcoefficients in twisted representations. In particular, if we know H ∗ ( G, V ) , whatcan we infer about H ∗ ( G, V ( r ) ) , for r ≥ ? In sections 2.4-2.5, we give some concrete situations where cohomology withcoefficient in twisted representations naturally appear. Before this, we make somepreliminary remarks regarding problem 2.1.(1) By definition, the representation V (0) equals V . Thus, problem 2.1 is in-teresting for r >
0. However, it is sometimes convenient to allow r = 0 inorder to obtain uniform statements, see e.g. the results in section 4.4.(2) One could ask similar questions about extensions groups between twistedrepresentations, i.e. of the form Ext ∗ G ( V ( r ) , W ( r ) ). This is not a more gen-eral question, since the representations Hom k ( V ( r ) , W ( r ) ) and Hom k ( V, W ) ( r ) are isomorphic, so that there is a graded isomorphism:Ext ∗ G ( V ( r ) , W ( r ) ) ≃ H ∗ ( G, Hom k ( V, W ) ( r ) ) . (3) Finite groups are group schemes, but problem 2.1 is uninteresting for finitegroups. Indeed, the Frobenius morphism F r : G → G is then an isomor-phism, thus H ∗ ( G, V ( r ) ) is isomorphic to H ∗ ( G, V ).(4) Problem 2.1 may be asked for arbitrary affine algebraic group schemes, butin the sequel we concentrate mainly on reductive group schemes. We referthe reader to [Jan] for details about other kinds of group schemes.
Remark . Our twist is denoted by ‘ [ r ] ’ in [Jan], while ‘ ( r ) ’ refersthere to a different way of twisting representations, defined by postcomposing ρ bythe action of the Frobenius morphism. However, if V is defined over a subfield of F p r the two twists coincide: V ( r ) ≃ V [ r ] . As this is the case for most of the repre-sentations considered in this article, we have chosen to formulate all the motivations ANTOINE TOUZ´E and results by using only the construction of twists defined by precomposition bythe Frobenius morphism.2.3.
Cohomology of finite groups of Lie type.
We explain a first motivation tostudy problem 2.1, related to the cohomology of finite groups of Lie type. Recall firstsome basic definitions. Let k be a field with algebraic closure k . A smooth connectedgroup scheme G is called reductive if G ( k ) does not contain any nontrivial normalconnected unipotent subgroup (i.e. subgroup with unipotent elements only). It is split over k if all its maximal connected diagonalizable subgroups are isomorphic(over k ) to a product of n copies of the multiplicative group G m . The integer n is called the rank of G . Typical k -split reductive group schemes of rank n are theclassical matrix groups GL n , SL n +1 , Sp n or SO n,n and SO n,n +1 (which are splitforms of SO n and SO n +1 , see [Ste2] for further details).Let p be a prime and consider a F p -split reductive group G . The functor V V (1) is exact, so it induces for all i a morphism H i ( G, V ( r ) ) → H i ( G, V ( r +1) ) . It is known that this map is injective [CPS] [Jan, I.9.10], that is, twisting cre-ates cohomology . Moreover, Cline Parshall Scott and Van der Kallen discoveredin [CPSvdK] that this map is an isomorphism if r is big enough, and that thestable value colim r H i ( G, V ( r ) ), which they call generic cohomology and denote byH i gen ( G, V ), can be interpreted in terms of the cohomology of the finite groups ofLie type G ( F q ).To be more specific, for all finite extensions F q of F p , any rational representation V of G yields a representation V ⊗ F p F q of G ( F q ). The resulting functor { Rational representations of G } → { representations of G ( F q ) } V V ⊗ F p F q is exact, and sends V ( r ) and V to the same G ( F q )-modules provided q | p r (indeedthe r -th iterated Frobenius morphism is then the identity of F q ). Thus, if q | p r , the F q point functor induces a map H i ( G, V ( r ) ) ⊗ F q → H i ( G ( F q ) , V ⊗ F p F q ) . The main result in [CPSvdK] asserts that this map is an isomorphism if r and q are big enough (with respect to i and to explicit constants depending on G and V ). Example 2.3 (Quillen vanishing) . Assume that V = k is the trivial representation.Then k ( r ) = k for all r so the colimit of the H i ( G, k ( r ) ) equals H i ( G, k ). Kempfvanishing theorem implies that this is zero in positive degrees. Hence, Cline ParshallScott Van der Kallen’s theorem says in this case that H i ( G ( F q )) = 0 for q bigenough with respect to i . This vanishing was first discovered by Quillen [Qui] inthe case of the general linear group, and proofs for other classical groups can befound in [FP].The trivial representations are the only ones such that V = V ( r ) . Hence, toobtain further concrete computations in the style of example 2.3, one needs tostudy our main problem 2.1. Remark . The main theorem of [CPSvdK] can also be formulated with moreemphasis on finite groups, see section 3.2.
OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 5
Simple representations.
We now describe a second motivation for problem2.1, related to the representation theory of reductive group schemes. Let k be afield, and let G be a k -split reductive group scheme.Chevalley classified [Che] the simple representations of G , using the root systemassociated to G (the procedure used to classify simple representations is called highest weight theory ). However, our knowledge of simple representations is farfrom being satisfactory in positive characteristic. For example, for classical matrixgroups, we don’t know the dimensions of the simple representations in general, norcan we compute the Ext-quiver of G .Twisted representations naturally appear when studying simple representationsof G . For example, if V is a nontrivial simple representation of V , then V, V (1) , V (2) , . . . , V ( r ) , . . . is an infinite family of pairwise non isomorphic simple representations. More gen-erally, Steinberg tensor product theorem [ ? ] [Jan, II.3.17] says that all simple rep-resentations are of the form V ⊗ V (1)1 ⊗ · · · ⊗ V ( r ) r for some r ≥ V i belonging to a small set of simplerepresentations, namely those with p -restricted highest weight. Example 2.5.
Take G = SL n +1 . The simple representations of SL n +1 are inbijection with the set Λ + of partitions of integers in at most n parts (i.e. the set of n -tuples ( λ , . . . , λ n ) of nonnegative integers satisfying λ ≥ · · · ≥ λ n ). The simplerepresentations with p -restricted highest weight correspond through this bijectionwith the finite subset Λ ⊂ Λ + of partitions satisfying λ i − λ i +1 < p for 0 ≤ i < n and λ n < p .The structure of simple representations of G given by the Steinberg tensor prod-uct theorem implies that any general problem involving the cohomology of simplerepresentations of G requires to study our main problem 2.1.2.5. Cohomological finite generation.
Let G be a finite group, and k a fieldof positive characteristic p . The cohomology of G with trivial coefficients H ∗ ( G, k )is then a graded k -algebra, and Evens proved [Eve] that it is finitely generated.Finite generation is an important input to build a theory of support varieties, whichestablishes a bridge between geometry (subvarieties of Spec H ev ( G, k )) and algebra(representations of G ). The development of support varieties was a motivation tounderstand the cohomology of the wider class of finite group schemes . Finite groupschemes are the group schemes represented by finite dimensional k -algebras. Theyinclude finite groups, enveloping algebras of restricted Lie algebras, and kernels ofiterated Frobenius morphisms F r : G → G . Friedlander and Suslin proved thefollowing theorem. Theorem 2.6 ([FS, Thm 1.1]) . Let G be a finite group scheme over a field k and M a finite dimensional G -module. Then H ∗ ( G, k ) is a finitely generated algebra,and H ∗ ( G, M ) is a finite module over it. Cohomology with coefficients in twisted representations play an important role inFriedlander and Suslin’s proof. Let us briefly explain this in more detail. The prooffollows the same basic principle as Evens’ proof for finite groups. They consider aspectral sequence of graded algebras, with noetherian initial page and converging
ANTOINE TOUZ´E to the desired finite group scheme cohomology H ∗ ( G, k ). Then, each page of thespectral sequence is finitely generated, and to prove finite generation of H ∗ ( G, k ),one needs to prove that the spectral sequence stabilizes after a finite number ofpages. This is achieved by constructing sufficiently many permanent cocycles inthe spectral sequence. In Evens’ proof, the permanent cocycles are constructedwith Even’s norm map. For finite group schemes, there is no such map available.Instead, the permanent cocycles are constructed from cohomology classes livingin H ∗ ( GL n , gl ( r ) n ). The computation of this cohomology with twisted coefficients(see example 4.3 below) is a breakthrough of [FS], and solves an important case ofproblem 2.1.Actually, Evens (and later Friedlander and Suslin, although this is not explicitlystated in [FS]) proved a more general result. If A is a commutative k -algebraacted on by G by algebra automorphisms, then H ∗ ( G, A ) is a graded commutativealgebra. They prove that H ∗ ( G, A ) is finitely generated as soon as A is so. Themost general version of this result was proved in [TVdK]. Let k be the algebraicclosure of k and let G be a group scheme such that the identity component of thealgebraic group G ( k ) is reductive. Thus G may be reductive, finite, or an extensionof such groups schemes. It is known from invariant theory that these group schemesare exactly the ones with the finite generation (FG) property, that is the invariant k -algebra A G = H ( G, A ) is finitely generated as soon as A is. Theorem 2.7 ([TVdK, Thm 1.1]) . Let G be a group scheme over k , satisfying(FG) property. Let A be a finitely generated commutative algebra, acted on by G by automorphisms of algebras. Then H ∗ ( G, A ) is finitely generated. As in Friedlander and Suslin’s theorem, cohomology with coefficients in twistedrepresentations plays an important role in the proof. In particular the proof requiresthe construction of some cohomology classes in H ∗ ( GL n , Γ d ( gl n ) (1) ), d ≥
0, in orderto produce permanent cycles in spectral sequences. Here Γ d ( gl n ) denotes the d -thdivided power of gl n , i.e. Γ d ( gl n ) = ( gl ⊗ dn ) S d is the subrepresentation of elementsin gl ⊗ dn which are invariant under the action of the symmetric group (acting bypermuting the factors of the tensor product). The problem of constructing non-zero classes in H ∗ ( GL n , Γ d ( gl n ) (1) ) is clearly an instance of problem 2.1, but wewill explain in section 5 that the two problems are actually equivalent.3. Some results regarding the cohomology of twistedrepresentations
In this section k is a field of positive characteristic p and G is a reductive algebraicgroup over k , which is assumed to be split and defined over F p , for example oneof the classical group schemes GL n , SL n +1 , SO n,n , SO n,n +1 , Sp n . We describetwo approaches to the cohomology of G with coefficients in twisted representations.The first one is classical and described e.g. in the reference book [Jan], while thesecond one is more recent [PSS]. The content of this section is not needed in theremainder of the article. We have included this section in order to give an overviewof the techniques used to attack problem 2.1 without using strict polynomial functortechnology. The reader can compare these important results to the results basedon strict polynomial functor technology, described in section 4.3.1. Untwisting by using Frobenius kernels.
Let G r denote the r -th Frobeniuskernel of G , that is the scheme-theoretic kernel of the r -th iterated Frobenius F r : OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 7 G → G . Then G r is a normal subgroup scheme of G and for all G -module U , theLyndon-Hochschild-Serre spectral sequence of the extension G r → G ։ G/G r hasthe form E p,q ( U ) = H p ( G/G r , H q ( G r , U )) ⇒ H p + q ( G, U ) . If U = V ( r ) , this spectral sequence can be used to ‘untwist’ the action, and to ob-tain some information on H ∗ ( G, V ( r ) ) from cohomology with untwisted coefficients.Indeed, the quotient G/G r fits into a commutative diagram G F r / / π " " " " ❊❊❊❊❊❊❊❊ G ≃ (cid:15) (cid:15) G/G r . In particular, for all G -modules W the following three statements are equivalent:(i) G r acts trivially on W (ii) There is a G/G r -module V such that ρ W = ρ V ◦ π .(iii) There is a G module W ( − r ) such that W = ( W ( − r ) ) ( r ) .Thus the LHS spectral sequence can be rewritten as E p,q ( V ( r ) ) = H p ( G, H q ( G r , k ) ( − r ) ⊗ V ) ⇒ H p + q ( G, V ( r ) ) . ( ∗ )The cohomology of H ∗ ( G r , k ) is not known in general, and the differentials ofthe spectral sequence may very well be nonzero. However, one can overcome thesedifficulties when r = 1 and p is big enough (with some explicit bound, e.g. for G = SL n or GL n then p > n is big enough). Under these hypotheses Friedlanderand Parshall [FP1, FP2] and Andersen and Jantzen [AJ] determined H ∗ ( G , k ) ( − .As a very nice feature, this G -module has a ‘good filtration’, in particular it has nohigher G -cohomology. If in addition V also has a good filtration then the secondpage of the LHS spectral sequence satisfies E p,q ( V ( r ) ) = 0 for all positive p , hencethe spectral sequence collapses and one obtains the following result. Theorem 3.1.
Assume that G is a reductive group and that p > h ( h denotes theCoxeter number of G , if G = GL n or SL n , h = n ). If V has a good filtration, thereis an isomorphism: H ( G, H q ( G , k ) ( − ⊗ V ) ≃ H q ( G, V (1) ) . The approach to the computation of cohomology with twisted coefficients de-scribed above can be adapted to slightly more general G -modules as coefficients, asin [AJ, Prop 4.8]. Also the dimensions of the H q ( G, V (1) ) can be computed by ex-plicit combinatorial formulas. As it is done in [AJ], it is also possible to study somespecial cases when r = 2 or for p = h or p = h −
1. However, all this requires non-trivial technical work (analysis of weights or differentials in the spectral sequence)and cannot be pursued very far to obtain complete computations of H ∗ ( G, V ( r ) ). Remark . Although the technique of untwisting by using Frobenius kernelsreaches quickly its limits for complete computations, it can still be very usefulto obtain qualitative information on twisted representations. For example, it playsan important role in the proof of cohomological finite generation for reductive al-gebraic groups, see [vdK, TVdK]. As another example, by carefully analysing the
ANTOINE TOUZ´E weights of the G -modules appearing at the second page of the LHS spectral se-quence, it is sometimes possible to detect vanishing zones in E ∗ , ∗ , therefore leadingto vanishing results for H j ( G, V ( r ) ) as in [PSS, Thm 5.2]. See theorem 3.3 below.3.2. Untwisting by using finite groups of Lie type.
Let us come back toCline Parshall Scott van der Kallen’s comparison theorem [CPSvdK]. In section2.3, we presented this theorem as an interpretation of colim r H( G, V ( r ) ). The readerprimarily interested in finite groups might prefer the following alternative presen-tation. Take k = F p and let G = G ( k ) be a reductive algebraic group defined andsplit over F p (e.g. G = SL n +1 ). Let V be a rational representation of G ( k ). Allthe finite subgroups G ( F q ) act on V , and the generic cohomology of V isH i gen ( G, V ) = lim q H i ( G ( F q ) , V ) . The main theorem of [CPSvdK] asserts that H i gen ( G, V ) = H i ( G ( F q ) , V ) for q bigenough, and moreover this is equal to the rational cohomology H i ( G, V ( r ) ) for somebig enough r .Now assume that we are interested in finite group cohomology. We have a finitegroup G ( F q ) and a rational G -module V , and we wish to compute H ∗ ( G ( F q ) , V )by using CPSvdK comparison theorem and the rational cohomology of G . Thenwe face two practical problems:(1) We need to understand colim r H ∗ ( G, V ( r ) ) rather than H ∗ ( G, V ) (this is ourmain problem 2.1).(2) Maybe our field F q is not big enough in order that colim r H ∗ ( G, V ( r ) ) isisomorphic to H ∗ ( G ( F q ) , V ).A recent article of Parshall, Scott and Stewart [PSS] solves both problems at thesame time. Their result applies when V is a simple G -module or more generallywhen V = Hom k ( U, U ′ ) for simple G -modules U , U ′ . For the sake of simplicity, weexplain their method when V is simple. The results needs further mild restrictionson the reductive group G , namely that G is simply connected and semisimple (e.g. G = SL n +1 ), so we assume G satisfies these restrictions in this section.The approach of [PSS] can be decomposed in two steps. The first step usesthe untwisting technique of section 3.1 as explained in remark 3.2 to prove a co-homological vanishing result. To state this result, recall from section 2.4 that V decomposes as a tensor product V = V ⊗ V (1)1 ⊗ · · · ⊗ V ( k ) k . where the V i are simple modules with p -restricted highest weight. Since G is semi-simple the decomposition is unique. We call it the Steinberg decomposition of V inthe sequel. Theorem 3.3 ([PSS, Thm 5.2]) . For all i there exists an (explicit) integer d = d ( G, i ) such that for all V with more than d nontrivial simple factors V i in theirSteinberg decomposition, and all j ≤ i one has: H j ( G, V ( r ) ) = 0 = H j ( G, V ) . Using induction from G ( F q ) to G and relying on the filtration of Ind GG ( F q ) ( k )provided in [BNP], one can then deduce a similar vanishing result [PSS, Thm 5.4]for H j ( G ( F q ) , V ), which is valid even when q is not big enough to apply CPSvdKcomparison theorem. OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 9
The second step studies the remaining cases, i.e. when V is a tensor productof a small (i.e. ≤ d ) number of nontrivial twisted simple representations of G . Inthese remaining cases, comparison with G ( F q ) cohomology can be used to untwistthe coefficients in the following way. First, for all e ≥ F e : G → G induce isomorphisms of the finite groups G ( F q ) hence H ∗ ( G ( F q ) , V ( e ) )does not depend (up to isomorphism) on the twisting e . Second, all the V ( e ) aresimple G ( F q )-modules, but they need not be isomorphic in general. Of course if q = p u , F u is the identity map on G ( F q ), hence V ( u ) = V . Thus Frobenius twistinginduces a Z /u Z -action on the set of simple representations of G ( F q ). Example 3.4.
Take u = 5 and V = V ⊗ V (2)2 , with V and V non-isomorphic.Then the orbit of V under the action of Z / Z is: V ⊗ V (2)2 −→ V (1)0 ⊗ V (3)2 −→ V (2)0 ⊗ V (4)2 −→ V ⊗ V (3)0 −→ V (1)2 ⊗ V (4)0 . Moreover, the simple G ( F q )-representations in the orbit of V under the action of Z /u Z can be interpreted as restrictions of the simple G -modules V ( e ) , but one canalso interpret each of them in a unique way as the restriction of a simple G -modulewhose tensor product decomposition ( ∗∗ ) involves only k -th twisted G -modules for k < u . These simple G -modules are called the q -shifts of V in [PSS]. Example 3.5.
In example 3.4, the q -shifts of V are: V , V (1) , V (2) , W and W (1) ,where W = V ⊗ V (3)0 . Note that as a G -module , W is not isomorphic to a Frobeniustwist of V .Now since V has a small number d of nontrivial factors, it is possible to find a(computable) bound u = u ( G, i ) big enough with respect to d , such that for all u ≥ u and q = p u , and for all V with p u -restricted highest weight, there is a q -shift V ′ whose tensor product decomposition ( ∗∗ ) is such that: (i) it starts with a longenough chain of trivial simple representations and (ii) it ends with a long enoughchain of trivial representations. Condition (i) ensures that V ′ = W ( s ) is alreadytwisted enough to apply CPSvdK theorem without further twisting, while condition(ii) ensures that F q is big enough to apply CPSvdK theorem. This, together withtheorem 3.3 imply the following theorem. Theorem 3.6 ([PSS, Thm 5.8]) . Let G be a semi-simple group scheme, simply con-nected, split and defined over F p . Then there exists a nonnegative integer u ( G, i ) such that for all u ≥ u and q = p u , the following holds. For all simple G -modules V with p u -restricted highest weight (i.e. involving only k -th Frobenius twists for k < u in their Steinberg decomposition), one can find a q -shift V ′ such that for all j ≤ i , there is a chain of isomorphisms: H j ( G ( F q ) , V ) ≃ H j ( G ( F q ) , V ′ ) ≃ H j gen ( G, V ′ ) ≃ H j ( G, V ′ ) . This theorem removes the necessity of understanding Frobenius twists to com-pute finite group cohomology from rational cohomology of G , as well as problemswith the size of the field. If one is interested in rational cohomology of groupschemes rather than in the cohomology of finite groups, Then by taking r and q ′ big enough (both depending on V and with q ′ ≥ q with q as in theorem 3.6), oneobtains a chain of isomorphisms for all j ≤ i :H j ( G, V ( r ) ) ≃ H j gen ( G, V ) ≃ H j ( G ( F q ′ ) , V ) ≃ H j ( G ( F q ′ ) , V ′ ) ≃ H j gen ( G, V ′ ) ≃ H j ( G, V ′ ) . One may draw two opposite conclusions from such a chain of isomorphisms. Onthe one hand, one may think that it is possible to avoid studying further our mainproblem 2.1 when coefficients are simple representations. On the other hand, thiscan be seen as an additional motivation to study problem 2.1. Indeed, H ∗ ( G, V ′ ) ismysterious in general, and understanding some general properties of high Frobeniustwists of V might bring some interesting new information on V ′ .4. A functorial approach to the cohomology with twistedcoefficients
Polynomial representations of GL n and strict polynomial functors. We now give a quick introduction to polynomial representations and strict polyno-mial functors. For the sake of simplicity, we make the assumption (in this section4.1 only) that the ground field k is infinite of arbitrary characteristic. We refer thereader to [FS, Sections 2 and 3], [Tou2] or [Kra] for a presentation which is validover an arbitrary field (or even more generally over an arbitrary commutative ring).Since k is infinite, finite dimensional rational representations of GL n identifywith group morphisms ρ : GL n ( k ) → GL ( V ) ≃ GL m ( k ), [ a i,j ] [ ρ k,ℓ ( a i,j )] whosecoordinates functions may be written in the form ρ k,ℓ ( a i,j ) = P k,ℓ ( a i,j )det([ a i,j ]) α k,ℓ where P k,ℓ ( a i,j ) is a polynomial in the n -variables a i,j and α k,ℓ is a nonnegativeinteger. Thus the coordinate functions are rational functions of the matrix coordi-nates a i,j . The rational representation ( V, ρ ) is called polynomial (resp. of degree d ) if all the ρ k,ℓ are polynomial functions (resp. of degree d ). Finally, infinite di-mensional polynomial representations of degree d are the rational representationswhich can be written as a union of finite dimensional subrepresentations which arepolynomial of degree d . Example 4.1.
The defining representation V = k n of GL n is polynomial (ofdegree 1), its d -th tensor power V ⊗ d is polynomial (of degree d ) its d -th symmetricpower S d ( V ) = ( V ⊗ d ) S d is polynomial (of degree d ) and its d -th divided powerΓ d ( V ) = ( V ⊗ d ) S d is polynomial (of degree d ) as well.Strict polynomial functors can be thought of as a natural way to generalizeexample 4.1. To be more specific, let V k be the category of vector spaces over k and V f k its subcategory of finite dimensional vector spaces. A functor F : V f k → V k is called strict polynomial of degree d if for all vector spaces U , V , the coordinatefunctions of the map F U,V : Hom k ( U, V ) → Hom k ( F ( U ) , F ( V )) f F ( f )are polynomial functions of degree d . Remark . Strict polynomial functors are simply called ‘polynomial functors’in [McD], and it is not clear from the definition why the word ‘strict’ should beused. The reason is that the notion of a polynomial functor was already definedby Eilenberg and Mac Lane much before [EML]. Strict polynomial functors arepolynomial in the sense of Eilenberg-Mac Lane, but the converse is not true, whichjustifies the word ‘strict’.
OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 11
For V = k n , the map F V,V restricts to a polynomial action ρ : GL n ( k ) → GL k ( F ( V )) on F ( V ). The polynomial representations of example 4.1 are obtainedin this way from the degree d strict polynomial functors ⊗ d : V V ⊗ d , S d : V S d ( V ) and Γ d : V Γ d ( V ). Let us denote by P k the category of strict polynomialfunctors of finite degree and natural transformations. Evaluation on V = k n yieldsan exact functor ev k n : P k → Rational representations of GL n . The image of ev k n consists exactly of the polynomial representations. Friedlanderand Sulin proved [FS, Cor 3.13] that the graded map induced by evaluation:Ext ∗P k ( F, F ′ ) → Ext ∗ GL n ( F ( k n ) , F ′ ( k n ))is an isomorphism provided n is greater or equal to deg F and deg F ′ . As shown in[FS], it is often easier in practice to compute extensions between polynomial repre-sentations by computing them inside P k than by computing them in the categoryof rational GL n -modules. For example, the following fundamental computation isout of reach of the untwisting techniques described in sections 3.1 and 3.2. Example 4.3.
Let k be a field of positive characteristic p , and let E r denote thegraded truncated polynomial algebra: E r = S ∗ ( e , . . . , e r ) (cid:14) ( e p = · · · = e pr = 0)where the generators e i are homogeneous of degree 2 p i − . In particular, as a gradedvector space one has: E ir = ( k if i is even and 0 ≤ i < p r . By using strict polynomial functors, Friedlander and Suslin showed in [FS, Thm4.10] that for all r and all n ≥ p r , there is an isomorphism of graded algebras:H ∗ ( GL n , gl ( r ) n ) = Ext ∗ GL n ( k n ( r ) , k n ( r ) ) ≃ E r . Remark . Let P d, k be the full subcategory of P k whose objects are the strict polynomial functorswhich are homogeneous of degree d . Then P d, k is isomorphic to the category ofmodules over the Schur algebra S ( n, d ) = End k S d (( k n ) ⊗ d ) if n ≥ d . Modules overthe Schur algebra are a classical subject of representation theory, which goes back tothe work of Schur [Sch] in characteristic zero, and which is studied in Green’s book[Gre] in positive characteristic. Thus one may prefer to do Ext-computations inthe classical category of S ( n, d )-Modules rather than in the more unusual category P d, k of strict polynomial functors. This is not a good idea, for many computationsin P k ultimately rely on composing functors. Such a composition operation makessense with strict polynomial functors, but is harder to define and manipulate incategories of modules over Schur algebras.4.2. Frobenius twists.
Let k be a field of positive characteristic p . The r -thFrobenius twist functor I ( r ) is a strict polynomial functor of homogeneous degree p r , which may be described as the subfunctor of S p r such that for each finitedimensional vector space V , I ( r ) ( V ) is the subspace of S p r ( V ) spanned by all p r -th powers v p r , with v ∈ V . Precomposition by I ( r ) is the functorial version oftaking the r -th Frobenius twist of a representation of GL n . Indeed, if F is a strict polynomial functor of degree d , the composition F ◦ I ( r ) is a strict polynomial functorof degree dp r , and evaluating on k n yields an isomorphism of GL n -modules:ev k n ( F ◦ I ( r ) ) ≃ (ev k n F ) ( r ) . Composite functors of the form F ◦ I ( r ) are often denoted suggestively by F ( r ) and we shall informally refer to such functors as ‘twisted functors’. Problem 2.1translates as follows in the realm of strict polynomial functors. Problem 4.5.
Try to understand or to compute the
Ext between twisted functors.In particular, if we know
Ext ∗P k ( F, G ) , what can we infer about Ext ∗P k ( F ( r ) , G ( r ) ) ,for r > ? One of the recent important achievements of the theory of strict polynomialfunctors is a complete solution to problem 4.5. To state the solution we need anauxiliary operation on strict polynomial functors, which was introduced in [Tou3].Given a strict polynomial functor F and a finite dimensional vector space W , welet F W denote strict polynomial functor such that F W ( V ) = F ( W ⊗ V ). Now if W is graded, then F W canonically inherits a grading. We call this construction parametrization (indeed, the new functor F W is just the old one with a parameter W inserted). One may think of the grading on parametrized functors as follows.Let P ∗ k denote the category of graded strict polynomial functors, and morphismsof strict polynomial functors preserving the grading. Parametrization by W yieldsan exact functor P k → P ∗ k F F W . Thus, to understand the grading on parametrized functors, it suffices to understandthe grading on parametrized injectives. But injectives in P k are (direct summandsof products of) symmetric tensors of the form V S d ( V ) ⊗ · · · ⊗ S d k ( V ). Onsymmetric tensors, the grading induced by parametrization is simply the usualgrading that one obtains by considering W ⊗ V as a graded vector space (with V concentrated in degree zero) and applying the functor S d ⊗· · ·⊗ S d k . The followingresult was conjectured in [Tou3], where it was verified for several families of pairsof functors ( F, G ). The result generalizes many previously known computations[FS, FFSS, Cha1]. One may find two different proofs of it, namely in [Tou4] and in[Cha2]. Both proofs rely on an idea of M. Cha lupnik, namely using the (derived)adjoint of precomposition by the Frobenius twist. (But see section 5.5).
Theorem 4.6.
For all strict polynomial functors
F, G , there is a graded isomor-phism:
Ext ∗P k ( F ( r ) , G ( r ) ) ≃ Ext ∗P k ( F, G E r ) , where the degree on the right hand side is understood as the total degree obtainedby adding the Ext -degree with the degree of the graded functor G E r obtained byparametrizing G by the graded vector space E r = Ext ∗P k ( I ( r ) , I ( r ) ) explicitly de-scribed in example 4.3. Theorem 4.6 can be efficiently used in practical computations. For example,the computations of [Cha1] and most computations of [FFSS] are instances of thefollowing example.
Example 4.7.
Assume that for all vector spaces W , Ext > P k ( F, G W ) = 0. Sucha condition is satisfied in the following concrete cases: if F is projective, or if G OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 13 is injective, or if the GL d -module F ( k d ) has a standard filtration and the GL d -module G ( k d ) has a costandard filtration where d = max { deg F, deg G } . Then theExt computation between F ( r ) and G ( r ) can be recovered as a Hom computationvia the graded isomorphism:Ext ∗P k ( F ( r ) , G ( r ) ) ≃ Hom P k ( F, G E r ) . Moreover, in the concrete cases given above, the latter Hom computation is easyto perform.In general, it may not be easy to compute Ext ∗P k ( F, G E r ). However, even whenthese Ext-groups do not seem easy to compute, one can draw many interestingqualitative results from the isomorphism of theorem 4.6. For example, the gradedvector space Hom P k ( F, G E r ) is a direct summand of Ext ∗P k ( F ( r ) , G ( r ) ), which pro-vides lots of nontrivial cohomology classes. Here is another way to use theorem 4.6.Recall that E r = L p r − i =0 k [2 i ], where k [ s ] denotes a copy of k placed in degree s .In particular, the graded functor L p r − i =0 G k [2 i ] is a direct summand of G E r . If G is a homogeneous functor of degree d , then G k [2 i ] = G [2 di ] (a copy of G placed indegree 2 di ). Thus theorem 4.6 has the following consequence. Corollary 4.8. If G is homogeneous of degree d then Ext ∗P k ( F ( r ) , G ( r ) ) ≃ p r − M i =0 Ext ∗ +2 di P k ( F, G ) ⊕ Another graded summand . In other words, Ext ∗P k ( F ( r ) , G ( r ) ) contains p r shifted copies of Ext ∗P k ( F, G ) asdirect summands. This explains periodicity phenomena which were often observedempirically in computations. One can refine this idea at the price of using a bitof combinatorics. If G is a homogeneous functor of degree d , then the functor G ( V ⊕· · ·⊕ V N ) with N variables can be decomposed in a direct sum of homogeneousstrict polynomial functors of N variables. Applying this to E r ⊗ V = V [0] ⊕ V [2] ⊕· · · ⊕ V [2 p r − G E r . To state this decomposition,we denote by Λ( d, k ) the set of compositions of d into k parts (i.e. of tuples µ =( µ , . . . , µ k ) of nonnegative integers such that µ + · · · + µ k = d ) and by Λ + ( d, k ) thesubset of partitions into k parts (i.e. those compositions satisfying µ ≥ · · · ≥ µ k ).By reordering a composition µ , one obtains a partition which we denote by π ( µ ).Then there are strict polynomial functors G λ indexed by partitions of d and anisomorphism of graded functors (the integer in brackets indicates in which degreethe copy of G λ is placed) G E r = M µ ∈ Λ( d,p r ) G π ( µ ) " p r X i =1 i − µ i . For all partitions λ we let E λ = Ext ∗P k ( F, G λ ). This is a vector space of finite totaldimension provided F and G have finite dimensional values. We have the followinggeneralization of corollary 4.8. Indeed, that the GL d -module G ( k d ) has a costandard filtration is equivalent to the factthat the functor G has a Schur filtration, i.e. a filtration whose associated graded is a directsum of Schur functors as defined in [ABW]. It then follows from [ABW, Thm II.2.16] that theparametrized functor G W also has a Schur filtration. The Ext condition follows by a highestweight category argument. Corollary 4.9. If G is homogeneous of degree d then there is a finite number ofgraded vector spaces E λ indexed by partitions of d , such that for all r ≥ ∗P k ( F ( r ) , G ( r ) ) = M µ ∈ Λ( d,p r ) E π ( µ ) " p r X i =1 i − µ i . Note that there is only a finite number of E λ appearing in this decomposition,since there is only a finite number of partitions λ of d . The integer r plays a role onlyfor the number of factors E λ and the shift. One may draw interesting consequencesof this. For example there is a numerical polynomial f of degree ≤ d dependingof F , G but not on r , such that for r ≥ log p ( d ) the total dimension of the gradedvector space Ext ∗P k ( F ( r ) , G ( r ) ) is equal to f ( p r ).4.3. More groups and more general coefficients.
Although theorem 4.6 is asimple and complete solution to problem 4.5, the reader may feel unsatisfied becausetheorem 4.6 adresses only a small part of the original problem 2.1. Indeed:(a) it considers the group scheme G = GL n only,(b) it considers polynomial representations only,(c) it considers only stable representations (i.e. when n is big enough withrespect to the degrees of the representations).It seems hard to improve on problem (c), for strict polynomial technology reallyrelies on computational simplifications which are known to appear only in the stablerange, i.e. when n is big enough. But problems (a), (b) can be successfully adressed,and we wish to present solutions to these problems here.We first consider problem (b). Theorem 4.6 gives access to cohomology groupsof the form H ∗ ( GL n , Hom k ( V, W ) ( r ) ) ≃ Ext ∗ GL n ( V ( r ) , W ( r ) )for polynomial representations V and W . Many interesting and natural represen-tations of GL n are not covered by this theorem. For example symmetric powersof the adjoint representation S d ( gl n ) are not of the form Hom k ( V, W ). A solutionto cover more general coefficients by a functorial approach was found in [FF]. Theidea is to use strict polynomial bi functors. A strict polynomial bifunctor of bidegree( d, e ) is a functor B : ( V f k ) op × V f k → V k such that for all V the functor W B ( V, W ) is strict polynomial of degree e , andfor all W the functor V B ( V, W ) is strict polynomial of degree d . Example 4.10. If F and G are strict polynomial functors of respective degrees d and e , then Hom k ( F, G ) : (
V, W ) Hom k ( F ( V ) , G ( W )) is a strict polynomialbifunctor of bidegree ( d, e ). Let gl denote the strict polynomial bifunctor ( V, W ) Hom k ( V, W ) (of degree (1 , F of degree d , the composite F ◦ gl :( V, W ) F (Hom k ( V, W )) is a strict polynomial bifunctor of degree ( d, d ).We denote by P k (1 ,
1) the category of strict polynomial bifunctors of finite bide-gree (the notation (1 ,
1) suggests that the functors of this category have one con-travariant variable and one covariant variable). If B is a bifunctor, GL n acts on thevector space B ( k n , k n ) by letting a matrix g act as the endomorphism B ( g − , g ). OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 15
The resulting representation is a rational representation, but not a polynomialrepresentation in general. We get in this way an exact functorev ( k n , k n ) : P k (1 , → Rational representations of GL n . For example, ev ( k n , k n ) ( S d ◦ gl) is isomorphic to the symmetric power of the adjointrepresentation S d ( gl n ). More generally, all finite dimensional rational representa-tions of GL n lie in the image of this evaluation functor. Given a strict polynomialbifunctor B , we denote by its cohomology H ∗ gl ( B ) the extension groups:H ∗ gl ( B ) = M k ≥ Ext ∗P k (1 , (Γ k ◦ gl , B ) . Franjou and Friedlander constructed [FF, Thm 1.5] a graded map, which is anisomorphism as soon as n ≥ max { d, e } , where ( d, e ) is the bidegree of B :H ∗ gl ( B ) → H ∗ ( GL n , B ( k n , k n )) . Cohomology of strict polynomial bifunctors subsumes extensions of strict polyno-mial functors. Indeed, given strict polynomial functors F , G , there is an isomor-phism [FF, Thm 1.5]: H ∗ gl (Hom k ( F, G )) ≃ Ext ∗P k ( F, G ) . Remark . An alternative construction of the isomorphism between bifunctorcohomology and GL n -cohomology is given in [Tou2]. As shown in [Tou5], it alsoprovides a graded isomorphism for n ≥ max { d, e } :H ∗ gl ( B ) ≃ H ∗ ( SL n +1 , B ( k n +1 , k n +1 )) . Given a strict polynomial bifunctor B we denote by B ( r ) the bifunctor obtainedby precomposing each variable by the r -th Frobenius twist. That is, B ( r ) ( V, W ) = B ( V ( r ) , W ( r ) ), so that the rational GL n -module ev ( k n , k n ) ( B ( r ) ) is isomorphic to the r -th Frobenius twist of the rational GL n -module ev ( k n , k n ) ( B ). We also denote byΓ d,E r the graded strict polynomial functor such thatΓ d,E r ( V ) = Γ d Hom k ( E r , V ) = (Hom k ( E r , V ) ⊗ d ) S d ⊂ Hom k ( E r , V ) ⊗ d . (The grading on Γ d,E r is the one such that the inclusion above preserves gradings ifthe tensor product of graded vector spaces is defined as usual.) In [Tou4] we provethe following generalization of theorem 4.6. Theorem 4.12.
For all strict polynomial bifunctors B , there is a graded isomor-phism: H ∗ gl ( B ( r ) ) ≃ M k ≥ Ext ∗P k (1 , (Γ k,E r ◦ gl , B ) . In this isomorphism, the degree on the right hand side is understood as the totaldegree obtained by adding the
Ext -degree and the degree of the graded bifunctor Γ k,E r ◦ gl : ( V, W ) Γ k (Hom k ( E r , Hom k ( V, W ))) = Γ k (Hom k ( E r ⊗ V, W )) . As usual, E r is the graded vector space Ext ∗P k ( I ( r ) , I ( r ) ) described in example 4.3.Remark . The statement given here is slightly different from [Tou4, Thm 1.2]as the parameter E r is not at the same place in the formula. However, it is not hardto see that the two parametrizations used are adjoint, so that the two formulas areequivalent. See the end of section 5.6.4 for more details on this topic. Next we consider problem (a). Ext computations in P k can also be used in orderto compute G -cohomology when G is a classical group scheme Sp n , O n,n + ǫ , or SO n,n + ǫ , with ǫ ∈ { , } . We recall quickly the statements. Given strict polynomialfunctors X and F we denote by H ∗ X ( F ) the extension groups:H ∗ X ( F ) = M k ≥ Ext ∗P k (Γ k ◦ X, F ) . In [Tou2, Thm 3.17] and [Tou5, Thm 7.24] we proved the existence of a gradedmap, which is an isomorphism as soon as 2 n ≥ deg F :H ∗ Λ ( F ) → H ∗ ( Sp n , F ( k n )) . Here k n is the dual of the defining representation of Sp n . Similary, by [Tou2,Thm 3.24] and [Tou5, Thm 7.24, Cor 7.31] if p = 2, there are graded maps, whichare isomorphisms as soon as 2 n + ǫ ≥ deg F + 1:H ∗ S ( F ) → H ∗ ( O n,n + ǫ , F ( k n + ǫ )) res −−→ H ∗ ( SO n,n + ǫ , F ( k n + ǫ )) . The analogue of theorem 4.6 for extensions related to orthogonal and symplecticgroups was proved by Pham Van Tuan. To be more specific, on can deduce thefollowing statement from the results in [Pha].
Theorem 4.14. If X = S or Λ , and p > , there are graded isomorphisms(where the grading on the right hand side is obtained by summing the Ext degreewith the degree coming from the graded functor Γ k,E r ◦ X ): Ext ∗P k (Γ k ◦ X, F ( r ) ) ≃ Ext ∗P k (Γ k,E r ◦ X, F ) . Statements without functors.
Problem 2.1 was formulated in terms of thecohomology of group schemes, thus one might wish an answer in terms of thecohomology of group schemes. For the convenience of the reader, we now translatethe main results obtained with functorial techniques into ready-to-use functor-freestatements. We use the following conventions and notations. • A representation of a classical group scheme G ⊂ GL n is called polynomialof degree d if it may be written as the restriction to G of a polynomialrepresentation of degree d of GL n . • Let G n = GL n or SL n . A rational representation of G n × G m is said to be polynomial of bidegree ( d, e ) if its restriction to G n (resp. G m ) is polynomialof degree d (resp. e ). We let k n ⊠ k m be the polynomial representation ofbidegree (1 ,
1) acted on by G n × G m via the formula ( g, h ) · v ⊗ w = gv ⊗ hw . • Let G n = GL n or SL n . Using the embedding G n → G n × G n , g ( g − , g ),any polynomial representation V of G n × G n yields a (non polynomial)representation of G n , which we denote by V conj . For example, if G n = GL n then S d ( k n ⊠ k n ) conj equals S d ( gl n ), the symmetric power of the adjointrepresentation of GL n . • We define gradings on representations as follows. We denote by V [ i ] acopy of a representation, placed in degree i . Unadorned representationsare placed in degree zero, i.e. V = V [0]. Also, the k -linear dual of arepresention V (acted on by G via ( gf )( v ) = f ( g − v )) is denoted by V .Then theorems 4.12 and 4.14 can be reformulated as follows. Theorem 4.15 (Type A) . Let G n = SL n or GL n . Let V be a polynomialrepresentation of G n × G n of bidegree ( d, e ) . Let r ≥ and assume that n ≥ OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 17 max { dp r , ep r } + 1 . There is a graded isomorphism (we take the total degree on theright hand side): H ∗ ( G n , V ( r )conj ) ≃ H ∗ ( G n × G n , A ∗ r ⊗ V ) , where the graded representation A ∗ r denotes the symmetric algebra on the graded G n × G n -representation L p r − i =0 k n ⊠ k n [2 i ] . Theorem 4.16 (Type C) . Let V be a polynomial representation of Sp n of degree d . Let r ≥ and assume that n ≥ dp r , and p is odd. There is a a gradedisomorphism (we take the total degree on the right hand side): H ∗ ( Sp n , V ( r ) ) ≃ H ∗ ( GL n , C ∗ r ⊗ V ) , where the graded representation C ∗ r denotes the symmetric algebra on the graded GL n -representation L p r − i =0 Λ ( k n )[2 i ] . Theorem 4.17 (Types B and D) . Assume p = 2 , let G n,n + ǫ = O n,n + ǫ or SO n,n + ǫ with ǫ ∈ { , } , and let V be a polynomial representation of G n,n + ǫ of degree d . Let r ≥ and assume that n + ǫ ≥ dp r + 1 . There is a a graded isomorphism (we takethe total degree on the right hand side): H ∗ ( G n,n + ǫ , V ( r ) ) ≃ H ∗ ( GL n + ǫ , BD ∗ r ⊗ V ) , where the graded representation BD ∗ r denotes the symmetric algebra on the graded GL n -representation L p r − i =0 S ( k n + ǫ )[2 i ] .Remark . If r = 0 then V (0) = V and the results stated in this section can beinterpreted as polynomial induction formulas. These formulas follow directly fromthe results of [Tou2] (improved in [Tou5]). There is no such interpretation if r ≥ Cohomology of twisted representations versus cohomologyclasses
As recalled in section 2.5, one of the crucial ingredients of the proof [TVdK] ofcohomological finite generation for reductive group schemes is the construction ofsome cohomology classes living in H ∗ ( GL n , Γ d ( gl n ) (1) ), for d ≥ n ≫
0. Thepurpose of this section is to show that this construction is equivalent to the solutionof problem 2.1 given in theorem 4.12.It is not difficult to use theorem 4.12 to construct the desired classes. This isexplained in [Tou4], and we briefly reiterate the argument in section 5.4.Conversely, we explain how the cohomology classes can be used to prove the-orem 4.12 in section 5.5. The strategy given here differs from the arguments of[Tou4, Cha2] in that it does not use the adjoint functor to the precomposition ofthe Frobenius twist (using this adjoint was a key idea for the earlier proofs, dueto M. Cha lupnik). Moreover, this proof can be adapted to cover theorem 4.14 re-lated to the cohomology of orthogonal and symplectic types, without relying onthe formality statements established in [Pha]. In order to deal with all settingssimultaneously (i.e. symplectic, orthogonal and general linear cases), we first needto introduce a few notations.
Some recollections and notations.
In the sequel, F stands for P k – thecategory of strict polynomial functors of bounded degree over a field k of character-istic p recalled in section 4.1, or its bifunctor analogue P k (1 ,
1) recalled in section4.3. While the letter ‘ F ’ suggests that we are working in a functor category, ourfunctor categories are very similar to categories of representations of finite dimen-sional algebras or groups. They are k -linear abelian with enough injectives andprojectives, Homs between finite functors (i.e. functors with values in finite di-mensional k -vector spaces) are finite dimensional, all objects are the union of theirfinite subobjects, and so on. Actually much of what is explained below can beunderstood while thinking of F as a category of representations.In order to deal with degrees without refering to the number of variables of thefunctors, we say that a bifunctor of bidegree ( d, e ) has (total) degree d + e .We will often have to compose (bi)functors. To avoid cumbersome notations, thecomposition symbol ‘ ◦ ’ will systematically be omitted in the sequel. Moreover the r -th Frobenius twist functor I ( r ) will most often be denoted by ‘ ( r ) ’. For examplethe composition S d ◦ I ( r ) ◦ gl will be denoted by S d ( r ) gl. We will also use concisenotations for tensor products of symmetric or divided powers. If µ = ( µ , . . . , µ n ) isa composition of d , i.e. a tuple of nonnegative integers with sum P µ i = d , then welet S µ = S µ ⊗ · · ·⊗ S µ n , and similarly for divided powers. Sometimes compositionswill be written as a matrix of nonnegative integers ( ν ij ) with 1 ≤ i ≤ ℓ , 0 ≤ j ≤ m ,and we let S ν be the tensor product N ≤ i ≤ ℓ N ≤ j ≤ m S ν ij .We will constantly use the divided powers Γ d V = ( V ⊗ d ) S d , d ≥ V . If µ is a composition of d , the tensor product of the inclusions Γ µ i ⊂ ⊗ µ i yields an injective morphism Γ µ ֒ → ⊗ d which we will call the canonical inclusion.More generally, the canonical inclusion Γ µ ֒ → Γ ν refers to the unique (if is exist)morphism factoring the canonical inclusion Γ µ ֒ → ⊗ d through the canonical inclu-sion Γ ν ֒ → ⊗ d . Recall also that L d ≥ Γ d is a Hopf algebra (dual to the maybemore usual symmetric algebra). We will mainly use the coalgebra structure. Thecomponent ∆ d,e : Γ d + e → Γ d ⊗ Γ e of the comultiplication is the canonical inclusion.Finally, divided powers satisfy a decomposition formula similar to that of symmetricpowers, which we shall refer to as the exponential isomorphism for divided powers :Γ d ( V ⊕ W ) ≃ M i + j = d Γ i V ⊗ Γ j W .
Fix an object X of F . Then we letH ∗ X ( F ) = M d ≥ Ext ∗F (Γ d X, F ) . More generally, if we fix a graded vector space E concentrated in even degrees andwith finite total dimension, we can replace the functor Γ d by the graded functorΓ d,E defined by Γ d,E ( V ) = Γ d Hom k ( E, V ). Then for all F in F , we letH ∗ E,X ( F ) = M d ≥ Ext ∗F (Γ d,E X, F ) . Note that this is a bigraded object: the first grading is the Ext grading, and thesecond grading is induced by the grading on Γ d Hom k ( E, X ). When refering toH ∗ E,X ( F ) as a graded vector space, we take the total grading. In particular if k isconsidered as a graded vector space concentrated in degree zero, then H ∗ k ,X ( F ) is OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 19 graded isomorphic to H ∗ X ( F ). These cohomology groups are equipped with a cupproduct: Ext i F (Γ d,E X, F ) ⊗ Ext i F (Γ e,E X, G ) ∪ −→ Ext i + j F (Γ d + e,E X, F ⊗ G )defined by c ∪ c ′ = ∆ ∗ d,e ( c ⊔ c ′ ), where ‘ ⊔ ’ refers to the external cup product. Thelatter is the derived version of the tensor product, defined for all F ′ , F , G ′ and G :Ext i F ( F ′ , F ) ⊗ Ext j F ( G ′ , G ) ⊔ −→ Ext i + j F ( F ′ ⊗ G ′ , F ⊗ G ) . In the sequel, we will mainly use these definitions for the functors X = S , X = Λ and for the bifunctor X = gl. Note that in these cases X commutes with Frobeniustwists, i.e. we have isomorphisms: ( r ) X ≃ X ( r ) .5.2. The cohomology classes.
In [Tou1, TVdK], we constructed some cohomol-ogy classes living in the bifunctor cohomology groups H ∗ gl (Γ ∗ (1) gl), or equivalentlyin the cohomology groups H ∗ ( GL n , Γ ∗ ( gl n ) (1) ) for n ≫
0. As recalled in section 2.5,these classes play a role in the proof of cohomological finite generation for reductivegroups. It is possible to construct analogous classes related to the cohomology ofclassical groups in types B,C,D. The uniform statement valid for all classical typesis given by the next theorem.
Theorem 5.1.
Assume that ( F , X ) equals ( P k (1 , , gl) or ( P k , Λ ) , or ( P k , S ) .In the last two cases, assume moreover that p is odd. There are graded k -linearmaps, for ℓ > : ψ ℓ : Γ ℓ (cid:0) H ∗ X ( (1) X ) (cid:1) −→ H ∗ X (Γ ℓ (1) X ) , satisfying the following properties: (1) ψ : H ∗ X ( (1) X ) → H ∗ X ( (1) X ) is the identity map, (2) for all positive ℓ, m , the following diagrams of graded vector spaces andgraded maps commutes: H ∗ X ( Γ ℓ + m (1) X ) (∆ ℓ,m ) ∗ / / H ∗ X (cid:0) (Γ ℓ ⊗ Γ m ) (1) X (cid:1) Γ ℓ + m (cid:0) H ∗ X ( (1) X ) (cid:1) ψ ℓ + m O O ∆ ℓ,m / / Γ ℓ (cid:0) H ∗ X ( (1) X ) (cid:1) ⊗ Γ m (cid:0) H ∗ X ( (1) X ) (cid:1) ψ ℓ ∪ ψ m O O , H ∗ X (cid:0) (Γ ℓ ⊗ Γ m ) (1) X (cid:1) mult ∗ / / H ∗ X ( Γ ℓ + m (1) X )Γ ℓ (cid:0) H ∗ X ( (1) X ) (cid:1) ⊗ Γ m (cid:0) H ∗ X ( (1) X ) (cid:1) ψ ℓ ∪ ψ m O O mult / / Γ ℓ + m (cid:0) H ∗ X ( (1) X ) (cid:1) ψ ℓ + m O O , Remark . For ( F , X ) = ( P k (1 , , gl), the classes in the image of ψ ℓ are called‘universal classes’. However it is not clear what universal property these classessatisfy. In particular, we don’t know if the morphisms ψ ℓ are uniquely determined.The problem of uniqueness is discussed in [Tou4, remark 4]. The classes are ‘versal’in some sense though, since any affine algebraic group scheme G is a subgroup of GL n for some n , hence receives by restriction some classes in H ∗ ( G, Γ d ( g ) (1) ). As already mentioned, a direct proof of theorem 5.1 when ( F , X ) = P k (1 ,
1) isgiven in [Tou1, TVdK]. The construction is combinatorial, that is, we constructan explicit resolution of the representation Γ d ( gl n ) (1) . The whole resolution iscomplicated but there is a small part of the resolution which is easy to understand,and where we can easily construct nontrivial cocycles. The idea to construct thisexplicit resolution is as follows.(1) The strict polynomial bifunctor Γ d (1) gl is too complicated, so we simplifythe problem. Instead of constructing an injective resolution of Γ d (1) gl, werather construct an injective resolution J of the strict polynomial functorΓ d (1) .(2) Having constructed J , we can evaluate on gl to obtain a resolution J gl ofΓ d (1) gl. Now J gl is not an injective resolution, nonetheless it is a resolu-tion by H ∗ gl ( − )-acyclic objects. Thus it is perfectly qualified to computecohomology.The category P k (1 ,
1) does not play a fundamental role in the construction of theclasses. Indeed, the construction would work as well if we replaced it by the categoryof rational GL n -modules, with gl replaced by the adjoint representation gl n andH ∗ gl ( − ) replaced by H ∗ ( GL n , − ). More generally the construction of the classeswould work verbatim in any situation where one wants to compute some cohomologyof the form H ∗ (Γ d (1) ( X )) provided this cohomology is equipped with cup products,and the objects S µ ( X ) are H ∗ ( − )-acyclic. In particular it works for ( F , X ) =( P k , S ) or ( P k , Λ ) in odd characteristic (in the later cases H ∗ X ( − )-acyclicity followsfrom the Cauchy filtration of [ABW, Thm III.1.4], see the proof of [Tou1, Lm 3.1].).We now provide yet another approach to prove theorem 5.1 for ( F , X ) = ( P k , S )or ( P k , Λ ) in odd characteristic, following the idea that the classes in H ∗ gl (Γ ℓ (1) gl)are ‘versal’ as mentioned in remark 5.2. To this purpose, we use restriction mapsres F : H ∗ gl ( F gl) → H ∗ X ( F X )natural with respect to the strict polynomial functor F and compatible with cupproducts. To be more specific, let F be homogeneous of degree d , and consider anextension e represented by (here we make apparent the variables U , V of the strictpolynomials bifunctors):0 → F (Hom k ( U, V )) → · · · → Γ d Hom k ( U, V ) → . By replacing U by V ∗ one gets an extension e ′ of F ⊗ by Γ d ⊗ . If ι : X ⇆ ⊗ : π are the canonical inclusion and projections, the restriction map is then defined byres F ( e ) = (Γ d ι ) ∗ ( F π ) ∗ ( e ′ ) . Next lemma collects some basic computations which can be found in the literature.The composition of the last three isomorphisms of lemma 5.3 is exactly the restric-tion map defined above, so that res I (1) is an isomorphism. Thus we can constructsimply the maps ψ ℓ for ( P k , X ) by restricting the maps ψ ℓ for ( P k (1 , , gl). Thatis, if we add decorations ‘ X ’ and ‘ gl ’ to distinguish the two cases we let: ψ Xℓ := res Γ ℓ (1) ◦ ψ gl ℓ ◦ Γ ℓ (res − I (1) ) . Lemma 5.3.
Assume that p is odd and X = Λ or S . Let P k (2) denote the cat-egory of strict polynomial functors with two covariant variables, and let ⊠ denote OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 21 the bifunctor ⊠ ( U, V ) = U ⊗ V . We have a chain of isomorphisms of graded vectorspaces: E ≃ H ∗ gl (gl (1) ) ≃ Ext ∗P k (2) (Γ p ⊠ , ⊠ ) ≃ H ∗ X ( ⊗ ) ≃ H ∗ X ( X (1) ) . Proof.
The first isomorphism is Friedlander and Suslin’s computation [FS, Thm4.10] for r = 1 (see example 4.3 for more details) translated in terms of bifunctorswith [FF, Thm 1.5] (see more explanations in section 4.3). The second isomorphismis induced by the equivalence of categories between P k (1 ,
1) and P k (2) given bydualizing the first variable of the bifunctors, i.e. it sends a bifunctor B to thebifunctor B ′ defined by B ′ ( U, V ) = B ( U ♯ , V ), where ♯ denotes k -linear duality.The third isomorphism is induced by evaluating both variables of the functor onthe same variable V , and then pulling back by the map Γ p ( ι ) where ι : X → ⊗ is the canonical inclusion. The fact that is is an isomorphism is (a very particularcase) of the proof [Tou2, Thm 6.6]. Finally the last isomorphism is induced by thequotient map ⊗ → X (1) and it is an isomorphism by [Tou2, Thm 6.6]. (cid:3) The untwisting isomorphisms.
Recall from section 4.1 that E r denotes thegraded vector space which equals k in degrees 2 i for 0 ≤ i < p r and zero in theother degrees. The notation E r reminds that this graded vector space is isomorphicto Ext ∗P k ( I ( r ) , I ( r ) ), as initially computed by Friedlander and Suslin [FS]. Thefollowing statement is an abstract form of theorems 4.12 and 4.14, where we havemade explicit the implicit naturalities and compatibilities with cup products (notall these properties are established in [Tou4, Cha2, Pha]). Theorem 5.4.
Assume that ( F , X ) equals ( P k (1 , , gl) or ( P k , Λ ) , or ( P k , S ) .In the last two cases, assume moreover that p is odd. For all positive r there areisomorphisms of graded vector spaces (take the total degree on the left hand side): φ F : H ∗ E r ,X ( F ) ≃ H ∗ X ( F ( r ) ) . Moreover, φ F is natural with respect to F and commutes with cup products.Remark . Theorem 5.4 does not hold for ( P k , Λ ) or ( P k , S ) in characteristic 2.Indeed, in both cases for F = I and r = 1 the source of φ F would be zero for degreereasons (there are no Ext between homogeneous functors of different degrees) whilethe target is nonzero as there are non split extensions 0 → I (1) → S → Λ → → I (1) → S → ⊗ → S → From theorem 5.4 to theorem 5.1.
We now assume that theorem 5.4 holds.Following [Tou4], we are going to prove theorem 5.1.Cup products define graded maps ∪ : H E ,X ( X ) ⊗ ℓ → H E ,X ( X ⊗ ℓ ). Since thegraded functor H E ,X ( − ) is left exact, H E ,X (Γ ℓ X ) identifies with the graded sub-space of H E ,X ( X ⊗ ℓ ) of elements invariant under the action of the symmetric group S ℓ (induced by letting S ℓ permuting the factors of the tensor product X ⊗ ℓ ). Thusthere is a unique graded map α ℓ making the following diagram commutative.H E ,X (Γ ℓ X ) (cid:31) (cid:127) / H E ,X ( X ⊗ ℓ )Γ ℓ (cid:0) H E ,X ( X ) (cid:1) α ℓ i i ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ∪ O O By construction α is the identity map of H E ,X ( X ), and the maps α ℓ fit intocommutative diagrams:H E ,X ( Γ ℓ + m X ) (∆ ℓ,m ) ∗ / / H E ,X (cid:0) (Γ ℓ ⊗ Γ m ) X (cid:1) Γ ℓ + m (cid:0) H E ,X ( X ) (cid:1) α ℓ + m O O ∆ ℓ,m / / Γ ℓ (cid:0) H E ,X ( X ) (cid:1) ⊗ Γ m (cid:0) H E ,X ( X ) (cid:1) α ℓ ∪ α m O O , H E ,X (cid:0) (Γ ℓ ⊗ Γ m ) X (cid:1) mult ∗ / / H E ,X ( Γ ℓ + m X )Γ ℓ (cid:0) H E ,X ( X ) (cid:1) ⊗ Γ m (cid:0) H E ,X ( X ) (cid:1) ψ ℓ ∪ ψ m O O mult / / Γ ℓ + m (cid:0) H E ,X ( X ) (cid:1) ψ ℓ + m O O . Now we can use theorem 5.4 to convert this rather trivial H E ,X ( − ) constructioninto the sought after H ∗ X ( − )-construction. Indeed by theorem 5.4, we have gradedisomorphisms (the equality on the right comes from the fact that X is an injectiveobject of F ): H ∗ X ( X (1) ) φ − X −−→ ≃ H ∗ E ,X ( X ) = H E ,X ( X ) . Theorem 5.4 also yields graded monomorphisms, compatible with cup products: ι ℓ : H E ,X (Γ ℓ X ) → H ∗ X (Γ ℓ X (1) ) . Since for our X we have a canonical isomorphism Γ ℓ X (1) ≃ Γ ℓ (1) X we define gradedmaps ψ ℓ satisfying the required properties by the formula: ψ ℓ = ι ℓ ◦ α ℓ ◦ Γ d ( φ − X ) . From theorem 5.1 to theorem 5.4: an overview.
Now we assume thattheorem 5.1 holds. We are going to use it to prove theorem 5.4. Recall that E r denotes the graded vector space with E ir = k for 0 ≤ i < p r and which is zero inother degrees. Let E (1) r be the same vector space with grading multiplied by p , i.e.( E (1) r ) pi = k for 0 ≤ i < p r and E (1) r is zero in the other degrees. Then for allpositive r we have an isomorphism of graded vector spaces: E r ≃ E (1) r − ⊗ E . In particular, for all r ≥ φ F : H ∗ E r ,X ( F ) ≃ H ∗ X ( F ( r ) ) of theorem 5.4 as the composition of the chain of r isomorphismsH ∗ E r ,X ( F ) −→ ≃ H ∗ E r − ,X ( F (1) ) −→ ≃ · · · −→ ≃ H ∗ E ,X ( F ( r ) ) = H ∗ X ( F ( r ) )which are provided by the next result. Theorem 5.6.
Let ( F , X ) equal ( P k (1 , , gl) or ( P k , Λ ) or ( P k , S ) . In the lasttwo cases, assume moreover that k has odd characteristic p . Let E be a gradedvector space of finite total dimension and concentrated in even degrees, and let E (1) denote the same vector space, with homothetic grading defined by E i = ( E (1) ) pi .Then there is a graded isomorphism, natural with respect to F and compatible withcup products: φ EF : H ∗ E (1) ⊗ E ,X ( F ) ≃ H ∗ E,X ( F (1) ) . OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 23
Thus, to prove theorem 5.4 it suffices to prove theorem 5.6. In the remainder ofsection 5.5, we describe the strategy of the proof of theorem 5.6. This strategy isfairly general, i.e. it is not properties specific to strict polynomial functors. Thereader can safely imagine that F is the category of rational GL n -modules, andreplace H ∗ E,X ( − ) by L d ≥ Ext ∗ GL n (Γ d Hom k ( E, gl n ) , − ). Functorial techniques arerelegated to section 5.6, in which we implement concretely the strategy.Recall that any functor in F splits as a finite direct sum of homogeneous func-tors, so it suffices to prove theorem 5.6 for F homogeneous. Moreover there areno nontrivial Ext between homogeneous functors of different degrees. Since X ishomogeneous of degree 2, both H ∗ E,X ( F (1) ) and H ∗ E (1) ⊗ E ,X ( F ) are zero unless F has even degree, say 2 d . Thus, to prove theorem 5.4, we have to construct a gradedisomorphism φ EF , natural with respect to F homogeneous of degree 2 d :H ∗ E (1) ⊗ E ,X ( F ) | {z } = Ext ∗F (cid:16) Γ d Hom k ( E (1) ⊗ E , X ) , F (cid:17) φ EF −−−−−−−−−→ H ∗ E,X (cid:16) F (1) (cid:17)| {z } = Ext ∗F (cid:16) Γ dp Hom k ( E, X ) , F (1) (cid:17) . Fix a basis ( b i ) ≤ i ≤ ℓ of E with the b i homogeneous, and a basis ( e , . . . , e p − ) of E , with deg e j = 2 j . Then, using the exponential isomorphism for divided powers,one may rewrite the domain and codomain of the sought-after φ EF as direct sums:H ∗ E (1) ⊗ E ,X ( F ) = M ν ∈ Λ( d,ℓ,p ) Ext ∗F (Γ ν X , F ) [ ps ( ν ) + t ( ν )] , H ∗ E,X (cid:16) F (1) (cid:17) = M µ ∈ Λ( pd,ℓ, Ext ∗F (cid:16) Γ µ , F (1) (cid:17) [ s ( µ )] . In these decompositions, Λ( a, ℓ, m ) denotes the set of matrices of nonnegative inte-gers ( ν ij ) with 1 ≤ i ≤ ℓ and 0 ≤ j < m such that P i,j ν i,j = a , and the bracketsindicate a shift of cohomological grading, for example Ext k F (cid:0) Γ µ , F (1) (cid:1) [ s ( µ )] is con-centrated in degree k + s ( µ ). The shifts are weighted sums of the coefficients of thematrices defined by s ( ν ) = P i,j ν i,j deg( b i ) and t ( ν ) = P i,j ν i,j deg( e j ).By summing the coefficients in each row of a matrix ν ∈ Λ( d, ℓ, p ) an multiplyingthe result by p , one obtains a matrix ν ∈ Λ( pd, ℓ, φ F as follows. For each ν ∈ Λ( d, ℓ, p ), we choose a cohomology class c ν ∈ Ext t ( ν ) F (Γ ν X, Γ ν X (1) ) ⊂ H t ( ν ) E,X (Γ ν X (1) ) . We define the restriction of φ F to the summand of H ∗ E (1) ⊗ E ,X ( F ) indexed by ν asthe following composition:Ext ∗F (Γ ν X, F ) [ ps ( ν ) + t ( ν )] φ F . . ❄ ❇ ❊ ❍ ❏ ▼ ❖ ◗ ❙ ❯ ❲ ❨ ❬ − (1) / / Ext ∗F (Γ ν X (1) , F (1) ) [ ps ( ν ) + t ( ν )] −◦ c ν (cid:15) (cid:15) Ext ∗F (Γ ν X, F (1) ) [ s ( ν )] (cid:127) _ (cid:15) (cid:15) H ∗ E,X (cid:0) F (1) (cid:1) where ‘ ◦ ’ stands for the Yoneda splice of extensions. At this point, we do notexplain which classes c ν we choose. However, whatever the choice of c ν is, thefollowing lemma is clear from the properties of Yoneda splices. Lemma 5.7.
The map φ E : H ∗ E (1) ⊗ E ,X ( − ) → H ∗ E,X ( − (1) ) is a morphism of δ -functors. To prove that φ E is an isomorphism of δ -functors, it suffices to prove that φ EF isan isomorphism for all injectives F , by the following basic lemma. Lemma 5.8.
Let A be an abelian category with enough injectives, let B be anabelian category, and let S ∗ , T ∗ : A → B be two δ -functors satisfying T i = S i = 0 for negative i . Then a morphism of δ -functors φ : S ∗ → T ∗ is an isomorphism ifand only if φ A : S ∗ ( A ) → T ∗ ( A ) is an isomorphism for all injective objects A in A . Finally, it is claimed in theorem 5.4 that φ EF is compatible with cup products. Let E ′ denote the tensor product E (1) ⊗ E . Then the homogeneous elements b (1) i ⊗ e j ,1 ≤ i ≤ ℓ , ≤ j < p form a basis of E ′ . For all d , e we have canonical decompositions(the first one has already been used in order to decompose the source of φ F ):Γ d + e Hom k ( E ′ , X (1) ) ≃ M ν ∈ Λ( d + e, ℓ, p ) Γ ν X (1) , Γ d Hom k ( E ′ , X (1) ) ⊗ Γ e Hom k ( E ′ , X (1) ) ≃ M λ ∈ Λ( d, ℓ, p ) µ ∈ Λ( e, ℓ, p ) Γ λ X (1) ⊗ Γ µ X (1) We let ∆ νλ,µ : Γ ν X (1) → Γ λ X (1) ⊗ Γ µ X (1) be the components of the canonicalinclusion ∆ d,e in this decomposition. Note that ∆ νλ,µ = 0 if ν = λ + µ . The nextlemma is a formal consequence of naturality properties of cup products. Lemma 5.9.
The morphism φ EF is compatible with cup products if and only if forall matrices λ and µ , the following equality holds in H ∗ E,X (Γ λ X (1) ⊗ Γ µ X (1) ) : c λ ∪ c µ = ∆ λ + µλ,µ ∗ c λ + µ . From theorem 5.1 to theorem 5.4: detailed proof.
In this section, weimplement the strategy just described, namely we define the classes c ν based onthe classes provided by theorem 5.1 and then we prove that these classes have allthe required properties. Checking the isomorphism on injectives requires classicalcomputation techniques which are specific to ‘functor technology’ (i.e. they haveno full equivalent in the category of representations of groups).5.6.1. Definition of the classes c ν . Recall that E is a graded vector space withhomogeneous basis ( e , . . . , e p − ) with deg( e j ) = 2 j . Lemma 5.3 yields a gradedisomorphism between E and H ∗ X ( X (1) ), and we still denote by ( e , . . . , e p − ) thecorresponding homogeneous basis of H ∗ X ( X (1) ). As there is a canonical isomorphism (1) X = X (1) , the graded morphisms ψ k of theorem 5.1 can be interpreted as a partialdivided power structure on H ∗ X (Γ ∗ X (1) ). We define ‘divided power classes’ γ k ( e i )by letting γ ( e i ) = 1 ∈ H X (Γ X (1) ) = k , and for k > γ k ( e i ) = ψ k ( e ⊗ ki ) ∈ H kiX (Γ k X (1) ) = Ext ki F (Γ pk X, Γ k X (1) ) . OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 25
For all matrices ν = ( ν i,j ) ∈ Λ( d, ℓ, p ), there are canonical inclusions ∆ ν : Γ ν X → Γ pν X and we define the class c ν ∈ Ext t ( ν ) F (Γ ν X, Γ ν X (1) ) by the formula: c ν = ∆ ∗ ν G i,j γ ν i,j ( e j ) . It is clear from the properties of the maps ψ ℓ given in theorem 5.1 that these classessatisfy c λ ∪ c µ = ∆ λ + µλ,µ ∗ c λ + µ . Hence by lemma 5.9 the morphisms φ F based on theclasses c µ are compatible with cup products. It remains to prove that the morphisms φ F are isomorphisms. By lemma 5.8, it suffices to prove the isomorphism when F is injective. We will proceed in several steps, starting with easy injectives (tensorproduct functors of low degree) and moving gradually towards general injectives.5.6.2. The maps φ F are isomorphisms (Step 1). If F denotes the category of strictpolynomial functors we let Y = ⊗ and if F denotes the category of strict polyno-mial bifunctors, we let Y = gl. In both cases Y is injective. As a first step towardsthe proof of theorem 5.6, we prove the following result. Proposition 5.10.
The map φ EY is an isomorphism. It is not hard to see that the vector space Hom F ( X, Y ) has dimension one, andwe denote by f a basis of it (we may take for f the canonical inclusion X → Y in thefunctor case, and the identity map in the bifunctor case). Recall that we have fixeda homogeneous basis ( b , . . . , b ℓ ) of E . We let ( b (1)1 , . . . , b (1) ℓ ) be the correspondinghomogeneous basis of E (1) . By definition φ EY is a map: E (1) ⊗ E ⊗ Hom F ( X, Y ) → M µ ∈ Λ( p,ℓ, Ext ∗F (Γ µ X, Y (1) ) , which sends each basis element b (1) i ⊗ e j ⊗ f of the source to the class φ EY ( b (1) i ⊗ e j ⊗ f ) = f ∗ ( e j ) ∈ Ext ∗F (Γ (0 ,..., ,p, ,..., X, Y (1) )where p occupies the i -th position in the ℓ -tuple (0 , . . . , , p, , . . . , φ EY induces an isomorphism onto the summand of the target indexed by thecompositions µ of p with exactly one nonzero coefficient. Thus, to finish the proofof proposition 5.10 it remains to prove that the other summands of the target arezero. This follows from the next lemma. Lemma 5.11.
Let µ be a composition of p into ℓ parts. If µ has at least twononzero coefficients then Ext ∗F (Γ µ X, Y (1) ) = 0 .Proof. If k < p the canonical inclusion Γ k ( V ) → V ⊗ k has a retract. Thus, underour hypotheses, Γ µ X is a direct summand of X ⊗ p which is itself a direct summandof Y ⊗ p . But Y ⊗ p is projective so Ext ∗F ( Y ⊗ p , Y (1) ) is concentrated in degree zero.Finally, it is not hard to see (use e.g. [FS, Cor 2.12]) that there is no nonzeromorphism Y ⊗ p → Y (1) . (cid:3) The maps φ F are isomorphisms (Step 2). As a second step towards the proofof theorem 5.6, we consider the case of the injective functors Y ⊗ d , for d ≥ Proposition 5.12.
For all d ≥ , the map φ EY ⊗ d is an isomorphism. In order to prove proposition 5.12, we will rely on the following lemma.
Lemma 5.13.
Let F be an object of F . The following conditions are sufficient toprove that φ EF ⊗ d is an isomorphism: (1) φ EF is an isomorphism, (2) the source and the target of φ EF ⊗ d have the same finite total dimension, (3) the following graded map is surjective: H ∗ E,X ( F (1) ) ⊗ d ⊗ End F ( F (1) ⊗ d ) → H ∗ E,X ( F (1) ⊗ d )( c ⊗ · · · ⊗ c d ) ⊗ f f ∗ ( c ∪ · · · ∪ c d ) . Proof. As φ E is a natural transformation compatible with cup products, we havea commutative square (where the lower horizontal arrow is the map appearing incondition (3), and the upper horizontal arrow is defined similarly):H ∗ E (1) ⊗ E ,X ( F ) ⊗ d ⊗ End F ( F ⊗ d ) / / ( φ EF ) ⊗ d ⊗ Id ≃ (cid:15) (cid:15) H ∗ E (1) ⊗ E ,X ( F ⊗ d ) φ EF ⊗ d (cid:15) (cid:15) H ∗ E,X ( F (1) ) ⊗ d ⊗ End F ( F (1) ⊗ d ) / / / / H ∗ E,X ( F (1) ⊗ d ) . In particular φ EF ⊗ d is surjective. Condition (2) ensures it is an isomorphism. (cid:3) We have seen in section 5.6.2 that the first condition of lemma 5.13 is satisfiedfor F = Y . We are now going to check conditions (2) and (3). We only provethe case F = P k , the bifunctor case being similar . For this computation, wemomentarily (until the end of section 5.6.3) change our notations and indicateexplicitly by the letter V the variable of the strict polynomial functors, e.g. wewrite Γ d Hom k ( E, X ( V )) instead of Γ d Hom k ( E, X ). We also use strict polynomialmultifunctors of 2 d (covariant) variables which we denote explicitly by V , . . . , V d .The category of multifunctors with 2 d covariant variables is denoted by P k (2 d ).The (derived) sum-diagonal adjunction (see e.g. the proof of [FFSS, Thm 1.7] or[Tou2, Section 5.3]) yields an isomorphism:Ext ∗P k (2 d ) (cid:16) Γ pd Hom k ( E, X ( V ⊕ · · · ⊕ V d )) , V (1)1 ⊗ · · · ⊗ V (1)2 d (cid:17) ≃ −−→ α Ext ∗P k (cid:16) Γ pd Hom k ( E, X ( V )) , V (1) ⊗ d (cid:17) . To be more specific, α sends an extension e V ,...,V d to the extension obtained byfirst replacing all the variables V i by V , and then pulling back by the map Γ pd ( δ ),where δ : V → V ⊕ d is the diagonal map which sends v to ( v, . . . , v ).Next we analyze the first argument of the source of α . We consider the partitionsof the set { , . . . , d } into d subsets of 2 elements. Each partition of this kind canbe uniquely represented as a d -tuple of pairs (( i , j ) , . . . , ( i d , j d )) of elements of { , . . . , d } satisfying i n < j n for all n and i < · · · < i d . We denote by Ω the setof such d -tuples. If I ∈ Ω we let Γ p,I ( V , . . . , V d ) denote the multifunctor:Γ p,I ( V , . . . , V d ) = O ≤ n ≤ d Γ p ( V i n ⊗ V j n ) . Moreover conditions (2) and (3) for bifunctors can also be deduced from computations alreadypublished in the literature. Indeed, for E = k (concentrated in degree zero) the statement followsfrom [FF, Thm 1.8], [Tou3, Thm Prop 5.4] or the computations of [Cha1, p. 781]. For anarbitrary E the computations can be deduced from the case E = k by using the isomorphismH ∗ E, gl ( B ) ≃ H ∗ gl ( B E ) explained at the end of section 5.6.4. OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 27
Using the exponential isomorphism for X and for divided powers, we obtain thatthe graded multifunctor Γ pd Hom k ( E, X ( V ⊕ · · · ⊕ V d )) is graded isomorphic to M I ∈ Ω Hom k ( E (1) , k ) ⊗ d ⊗ Γ p,I ( V , . . . , V d ) ⊕ other terms.The ‘other terms’ mentioned in the decomposition don’t bring any contribution tothe Ext by lemma 5.14 below, so that α actually induces a graded isomorphism: M I ∈ Ω E (1) ⊗ d ⊗ Ext ∗P k (2 d ) (cid:0) Γ p,I ( V , . . . , V d ) , V (1)1 ⊗ · · · ⊗ V (1)2 d (cid:17) ≃ −−→ α Ext ∗P k (cid:16) Γ pd Hom k ( E, X ( V )) , V (1) ⊗ d (cid:17) . Lemma 5.14.
For ≤ k ≤ N we consider a multifunctor X k ( V , . . . , V d ) = V i k ⊗ V j k with ≤ i k , j k ≤ d , and a nonnegative integer d k . We let F ( V , . . . , V d ) := O ≤ k ≤ N Γ d k X k ( V , . . . , V d ) . If Ext ∗P k (2 d ) ( F ( V , . . . , V d ) , V (1)1 ⊗ · · · ⊗ V (1)2 d ) is nonzero, then there exists a d -tuple I ∈ Ω such that the multifunctor F ( V , . . . , V d ) is isomorphic to Γ p,I ( V , . . . , V d ) .Proof. Since there is no nonzero Ext between homogeneous multifunctors of differ-ent multidegrees, F ( V , . . . , V d ) must have multidegree ( p, . . . , p ) in order that theExt is nonzero. In particular, this implies that d k ≤ p for all k . We claim that theExt is zero unless all the d k are equal to p . To this purpose, we follow the samestrategy as in the proof of lemma 5.11. Assume that d n < p for some n . Up torenumbering the variables we may assume that X n is non constant with respect tothe variable V . Let K ⊂ { , . . . , N } the set of the indices k such that X k is nonconstant with respect to V . As F ( V , . . . , V d ) has degree p with respect to thevariable V we have d k < p for all k ∈ K and in particular the canonical inclusion O k ∈ K Γ d k X k ( V , . . . , V d ) ֒ → O k ∈ K X k ( V , . . . , V d ) ⊗ d k has a retract. Thus we can write F ( V , . . . , V d ) as a retract of some multifunctor ofthe form V ⊗ p ⊗ G ( V , . . . , V d ). In particular by the K¨unneth formula this impliesthat Ext ∗P k (2 d ) ( F ( V , . . . , V d ) , V (1)1 ⊗ · · · ⊗ V (1)2 d ) is isomorphic to the tensor productExt ∗P k ( V ⊗ p , V (1)1 ) ⊗ Ext ∗P k (2 d − ( G ( V , . . . , V d ) , V (1)2 ⊗ · · · ⊗ V (1)2 d ) . But the factor on the left is zero by lemma 5.11. To sum up, we have proved that F ( V , . . . , V d ) must have multidegree ( p, . . . , p ) and that all the d k are equal to p .This implies that F ( V , . . . , V d ) must be equal to some Γ p,I ( V , . . . , V d ). (cid:3) Finally each term in the direct sum appearing at the source of α can be explic-itly computed. Let I = (( i , j ) , . . . , ( i d , j d )) be an element of Ω. Let σ I be thepermutation of { , . . . , d } defined by σ I ( j n ) = 2 n and σ I ( i n ) = 2 n −
1. There is acorresponding isomorphism of multifunctors (still denoted by σ I ): σ I : O ≤ n ≤ d V (1) i n ⊗ V (1) j n ≃ −→ O ≤ n ≤ d V (1)2 n ⊗ V (1)2 n +1 . If T (1) ,I ( V , . . . , V d ) denotes the source of σ I , we thus have a composite isomor-phism (where the first isomorphism is provided by lemma 5.3 and where κ refersto the K¨unneth map): E ⊗ d ≃ Ext ∗P k (2) (Γ p ( V ⊗ V ) , V (1)1 ⊗ V (1)2 ) ⊗ dκ −→ ≃ Ext ∗P k (2 d ) (cid:16) Γ p,I ( V , . . . , V d ) , T (1) ,I ( V , . . . , V d ) (cid:17) ( σ I ) ∗ −−−→ ≃ Ext ∗P k (2 d ) (cid:16) Γ p,I ( V , . . . , V d ) , V (1)1 ⊗ · · · ⊗ V (1)2 d (cid:17) . To sum up, we have constructed a completely explicit isomorphism: M I ∈ Ω E (1) ⊗ d ⊗ E ⊗ d ≃ −→ Ext ∗P k (cid:16) Γ pd Hom k ( E, X ( V )) , V (1) ⊗ d (cid:17) . (5.1)By following the same reasoning, one gets a similar isomorphism (which couldalso be computed by more down-to-earth methods. Indeed V ⊗ d is injective, hencethis isomorphism is actually a mere Hom computation): M I ∈ Ω E (1) ⊗ d ⊗ E ⊗ d ≃ −→ Ext ∗P k (cid:16) Γ d Hom k ( E (1) ⊗ E , X ( V )) , V ⊗ d (cid:17) . (5.2)By comparing (5.1) and (5.2), we see that condition (2) of lemma 5.13 is satisfied.To check condition (3) of lemma 5.13, we write down explicitly the effect ofisomorphism (5.1) on an element x = b k ⊗ · · · ⊗ b k n ⊗ a t ⊗ · · · ⊗ a t n belonging tothe term E (1) ⊗ d ⊗ E ⊗ d indexed by I . To be more explicit, let a ′ t n be the imageof a t n by the isomorphism E ≃ −→ Ext ∗P (Γ p X ( V ) , V (1) ⊗ ) provided by lemma 5.3.If we define an ℓ -tuple µ = (0 , . . . , , p, , . . . ,
0) with ‘ p ’ in n -th position, we caninterpret b k n ⊗ a ′ t n as an element ofExt ∗P (Γ µ X ( V ) , V (1) ⊗ ) [deg( b k n )] ⊂ H ∗ E,X ( V ) ( V (1) ⊗ )and by following carefully the explicit definition of isomorphism (5.1), we computethat isomorphism (5.1) sends x to( σ I ) ∗ (cid:0) ( b k ⊗ a ′ t ) ∪ · · · ∪ ( b k d ⊗ a ′ t d ) (cid:1) ∈ H ∗ E,X ( V ) ( V (1) ⊗ d ) . As isomorphism (5.1) is surjective, we obtain that condition (3) of lemma 5.13 issatisfied.5.6.4.
The maps φ F are isomorphisms (Step 3). To conclude the proof of theorem5.6, we now prove that the maps φ F are isomorphisms when F is an arbitraryinjective object of F , homogeneous of degree 2 d . Since the source and the targetof φ F commute with arbitrary sums, we may restrict ourselves to proving thisfor an indecomposable injective J . If F = P k , any indecomposable injective J homogeneous of degree 2 d is a direct summand of a symmetric tensor S µ for somecomposition µ of 2 d . If F = P k (1 , J homogeneousof total degree 2 d is a direct summand of a symmetric tensor S λ,µ : ( V, W ) S λ ( V ♯ ) ⊗ S µ ( W ) for some tuples of nonnegative integers λ = ( λ , . . . , λ s ) and µ = ( µ , . . . , µ t ) satisfying P λ i + P µ i = 2 d (and V ♯ is the k -linear dual of V ).The next lemma records a consequence of this fact. Lemma 5.15.
The map φ J is an isomorphism for all injectives J if and only if itis an isomorphism for all symmetric tensors J . OHOMOLOGY WITH COEFFICIENTS IN TWISTED REPRESENTATIONS 29
Moreover, if F = P k (1 , λ and µ are both compositionsof d , otherwise we have H ∗ X ( S λ,µ ( r ) ) = 0 = H ∗ E r ,X ( S λ,µ ) for degree reasons (thereare no nonzero Ext between homogeneous bifunctors of different bidegrees), so that φ S λ,µ is trivially an isomorphism.Now symmetric tensors are quotients of Y ⊗ d . Indeed if F = P k , then Y ⊗ d ( V ) = V ⊗ d , the symmetric group S d acts on Y ⊗ d by permuting the variables, and forall compositions µ of 2 d we have an isomorphism ( Y ⊗ d ) S µ ≃ S µ . Similarly in thebifunctor case Y ⊗ d ( V, W ) = ( V ♯ ) ⊗ d ⊗ W ⊗ d , the group S d × S d acts on Y ⊗ d bypermuting the variables, and we have an isomorphism: ( Y ⊗ d ) S λ × S µ ≃ S λ,µ . Let( J, S ) denote ( S λ,µ , S λ × S µ ) (in the bifunctor setting) or ( S µ , S µ ) (in the functorsetting). By naturality of φ F , we have a commutative diagramH ∗ E,X ( Y ⊗ d (1) ) S / / H ∗ E,X ( J (1) )H ∗ E (1) ⊗ E ,X ( Y ⊗ d ) S φ Y ⊗ d ≃ O O / / H ∗ E (1) ⊗ E ,X ( J ) φ J O O . The vertical map on the left is an isomorphism as we established it in section 5.6.3.The horizontal maps of the diagrams are isomorphisms by proposition 5.16 below,hence φ J is an isomorphism for all symmetric tensors J , which finishes the proof oftheorem 5.4. Proposition 5.16.
Assume that ( F , X ) = ( P k (1 , , gl) . If λ and µ are composi-tions of d , then for all r ≥ and all finite dimensional graded vector spaces E , thequotient map Y ⊗ d → S λ,µ induces an isomorphism H ∗ E,X ( Y ⊗ d ( r ) ) S λ × S µ ≃ −→ H ∗ E,X ( S λ,µ ( r ) ) If ( F , X ) = ( P k , S ) or ( P k , Λ ) , p is odd and µ is a composition of d the quotientmap Y ⊗ d → S µ induces an isomorphism H ∗ E,X ( Y ⊗ d ( r ) ) S µ ≃ −→ H ∗ E,X ( S µ ( r ) ) . The remainder of the section is devoted to the proof of proposition 5.16. As-sume first that ( F , X ) = ( P k (1 , , gl). If E = k (concentrated in degree zero)then the statement follows from the results of [FFSS], or alternatively of [Tou3].To be more specific, let = (1 , . . . ,
1) with ‘1’ repeated d times, hence ⊗ d = S . The map H ∗ gl ( S µ, ( r ) ) → H ∗ gl ( S µ,λ ( r ) ) identifies [FF, Thm 1.5] with the mapExt ∗P k (Γ λ ( r ) , S ( r ) ) → Ext ∗P k (Γ λ ( r ) , S µ ( r ) ). The latter becomes an isomorphism af-ter taking coinvariants under the action of S µ at the source. This follows from[FFSS, Thm 4.5], as explained in [Cha1]. An alternative proof without spec-tral sequences is given in [Tou3, Cor 4.7]. There is a similar result for the mapH ∗ gl ( S , ( r ) ) → H ∗ gl ( S µ, ( r ) ). Thus we obtain the isomorphism in two steps:H ∗ gl ( S λ,µ ( r ) ) ≃ H ∗ gl ( S λ, ( r ) ) S µ ≃ (H ∗ gl ( S , ( r ) ) S µ ) S λ = H ∗ gl ( S , ( r ) ) S λ × S µ . Now we prove the case of an arbitrary E . For this purpose we recall parametriza-tions of bifunctors by graded vector spaces. If E is a graded vector space of finitetotal dimension, there is an exact lower parametrization functor [Tou4, Section 3.1] − E : P k (1 , → P k (1 , ∗ B B E where B E is the bifunctor ( V, W ) B ( V, ( L i E i ) ⊗ W ), and the grading may bedefined in the same fashion as in the functor case (explained just before theorem4.6). Similarly, there is an exact upper parametrization functor − E : P k (1 , → P k (1 , ∗ B B E where B E ( V, W ) = B ( V, Hom k ( L i E i , W )). Lower and upper parametrizationsare adjoint, that is there is an isomorphism of graded k -vector spaces, natural withrespect to F , G and compatible with tensor products:Hom P k (1 , ( F E , G ) ≃ Hom P k (1 , ( F, G E ) . This property is easily checked when F is a standard projective and G is a stan-dard injective by using the Yoneda lemma, and the general isomorphism follows bytaking resolutions. Moreover as parametrization functors are exact, the adjunctionisomorphism induces a similar adjunction on the Ext-level. In particular, we havea graded isomorphism, natural with respect to B :H ∗ E, gl ( B ) ≃ H ∗ gl ( B E ) . Since the bifunctor ( S λ,µ ( r ) ) E splits as a direct sum of bifunctors of the form S ν,µ ( r ) ,it is now easy to prove that proposition 5.16 holds.Finally we prove the cases ( F , X ) = ( P k , S ) or ( P k , Λ ) and p is odd. In thesecases, X is a direct summand of ⊗ , hence the graded functor H ∗ E,X ( − ) is a directsummand of H ∗ E, ⊗ ( − ). Hence it suffices to prove the isomorphism for X = ⊗ . Byusing sum-diagonal adjunction and k -linear duality as in the proof of [Tou2, Thm6.6] we obtain isomorphisms natural with respect to the functor F :H ∗ E, ⊗ ( F ) ≃ H ∗ E, gl ( F ⊞ ) , where F ⊞ denotes the bifunctor ( V, W ) F ( V ♯ ⊕ W ). But S µ ( r ) ⊞ decomposes asa direct sum of bifunctors of the form S λ,ν ( r ) . Thus the statement for ( F , X ) =( P k , ⊗ ) can be deduced from the one for ( F , X ) = ( P k (1 , , gl). References
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