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GENERIC AND q-RATIONAL REPRESENTATION THEORY

Edward Cline �, Brian Parshall��, and Leonard Scott��

Abstract. Part I of this paper develops various general concepts in generic repre-sentation and cohomology theories. Roughly speaking, we provide a general theory oforders in non-semisimple algebras applicable to problems in the representation theoryof �nite and algebraic groups, and we formalize the notion of a \generic" property inrepresentation theory. Part II makes new contributions to the non-describing repre-sentation theory of �nite general linear groups. First, we present an explicit Moritaequivalence connecting GLn(q) with the theory of q-Schur algebras, extending aunipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equiv-alence to study the cohomology groups H�(GLn(q); L), when L is an irreduciblemodule in non-describing characteristic. The generic theory of Part I then yieldsstability results for various groups Hi(GLn(q); L), reminscent of our general theory[CPSK] with van der Kallen of generic cohomology in the describing characteristiccase. (In turn, the stable value of such a cohomology group can be expressed in

terms of the cohomology of the a�ne Lie algebra bgln(C ).) The arguments entail newapplications of the theory of tilting modules for q-Schur algebras. In particular, weobtain new complexes involving tilting modules associated to endomorphism algebrasobtained from general �nite Coxeter groups.

Contents

IntroductionI: Generic representation theory

1. Representation theory on an open set2. Ext calculations: the ungraded case3. Ext calculations: the graded case4. Integral quasi-hereditary algebras5. Morita theory6. Derived category constructions and equivalences; tilting

modules7. Generic quiver theory

II: q-rational representation theory8. The Geck{Gruber{Hiss very large prime result9. A Morita equivalence for GLn(q)10. H1-cohomology in non-describing characteristic11. Resolutions

� Department of Mathematics, University of Oklahoma, Norman, OK 73019.�� Department of Mathematics, University of Virginia, Charlottesville, VA 22903.

Research supported in part by the National Science Foundation

Communicated by T. Miwa, May 1, 1998.

1991 Mathematics Subject Classi�cations: 20C15, 20C33, 20J06

1

2 GENERIC AND q-RATIONAL REPRESENTATION THEORY

12. Higher cohomology in non-describing characteristic13. Appendix: the constructible topology

This paper falls naturally into two parts. Part I (xx1{7,13) introduces and devel-ops some general concepts in generic representation theory. These ideas arose out of(and are applied to) issues in Part II (xx8{12), which deals with the non-describingcharacteristic representation and cohomology theories of the �nite general lineargroups GLn(q). We open here with a discussion of this latter topic.

Let G be a reductive group de�ned and split over a prime �eld Fr . Assume thatG has simply connected derived group G0. Fix q = rd and consider the split Cheval-ley group G(q) = G(Fq ) of Fq -rational points. Now let k be an algebraically closed�eld of positive characteristic p. The modular representation theory of the �nitegroups G(q) over k naturally breaks into two cases|the describing characteristictheory in which p = r, and the non-describing characteristic theory in which p 6= r.In both cases, the central issues include the classi�cation of the irreducible mod-ules, the determination of their characters (or decomposition numbers), and otherrepresentation-theoretic properties (e. g., cohomology and submodule structure ofnatural modules):

� In the describing characteristic theory, Steinberg's pioneering work shows thatthe classi�cation and characters of the irreducible modules are obtained (for allq = pd) from the solution of the analogous problem for the ambient algebraicgroup. In that case, the characters are known generically, i. e., if p is su�cientlylarge (depending on the root type ofG), thanks to [AJS], [KL2] and [KT].1 Similarly,questions concerning the FqG(q)-cohomology and submodule structure for naturalmodules can often be studied, and even explicitly answered for su�ciently large qor p, by passing to analogous problems for G [CPS1], [CPSK], [FP1,2].

� In the non-describing characteristic theory, however, even a natural parameter-ization of the irreducible modules remains problematic for some types. But, whenG = GLn, there is a parameterization of the irreducible modules, due to Dipper-James [DJ2], [D1]. Further, their work establishes that the Brauer characters ofthe irreducible kG(q)-modules are determined by decomposition numbers of certainqa-Schur algebras along with characteristic 0 character formulas due to Green [Gr].In cohomology theory, the study of the groups H�(G(q); L) (L simple) has largelyfocused on the untwisted L = k case, e. g., in the famous work of Quillen and others(cf. [AM] for discussion and references).

Part II of this paper makes new contributions to the non-describing characteristictheory in the case G = GLn. In particular, we extend the work of Dipper and Jamesdescribed above to obtain a more intimate connection between the representationtheory of kGLn(q) and that of q-Schur algebras. Namely, Theorem 9.17 presents aMorita equivalence:

(1) kG(q)=J(q)k �Morita

Ms2Css;p0

m(s)Oi=1

Sqai(s)(ni(s); ni(s))k

1However, no su�cient lower bound is known for p in any interesting cases.

GENERIC AND q-RATIONAL REPRESENTATION THEORY 3

between an algebra which is a direct sum of tensor products of certain qa-Schuralgebras and a quotient algebra kG(q)=J(q)k of kG(q) by an ideal J(q)k C kG(q).(See x9 for further explanation of notation.) Furthermore, the algebras kG(q) andkG(q)=J(q)k have the same irreducible modules. A similar Morita equivalencehas been obtained by Takeuchi [T] in the special case of unipotent blocks (in whichcase the right-hand side of (1) is replaced by a single q-Schur algebra).2 Our Moritaequivalence is derived from a second Morita equivalence:

(2) OG(q)=J(q) �Morita

Ms2Css;p0

m(s)Oi=1

Sqai(s)(ni(s); ni(s))O;

over a discrete valuation ring O with residue �eld k. (Here J(q) is an ideal inOG(q) such that J(q)O k is an ideal|namely, the ideal J(q)k of (1)|in kG(q) =OG(q) O k.) Cast this way, the correspondence between irreducible kGLn(q)-modules and their decomposition numbers and the corresponding issues for certainqa-Schur algebras becomes very conceptual|see (9.17(d)). The further connectionbetween the representation theory of q-Schur algebras and the quantum generallinear groupGLn;q(k) explains at least part of the title of this paper.

3 We emphasizethat these results rely heavily on the work [DJ2] as well as that of Fong-Srinivasan[FS].

We wish to use the above Morita equivalence to study the cohomology groupsH�(GLn(q); L), when L is an irreducible module in a non-describing characteristicp. When p divides jG(q)j, the algebra kG(q)=J(q)k has �nite global dimension,while kG(q) has in�nite global dimension. Hence, the precise relation between thecohomology of GLn(q) and that of q-Schur algebras by means of the Morita equiv-alence (1) is quite subtle. Sections 10,11,12 are devoted to attacking this problem.To our knowledge, our results represent the �rst progress in understanding the coho-mology of �nite reductive groups with general irreducible non-trivial coe�cients Lin non-describing characteristic. (It would be interesting to pursue Dwyer's generalstability results [Dw] with respect to n with these coe�cients in mind.)

For applications to �nite groups, e. g., to the study of maximal subgroups, theexplicit calculation of 1-cohomology for quasi-simple groups plays an importantrole [AS]. In this case, our results are particularly strong and quite easy to obtaindirectly from the Morita theorem in x9. This work is presented in x10. For example,Theorem 10.1 establishes a direct connection between H1-calculations for GLn(q)and Ext 1-calculations for q-Schur algebras. In Theorem 10.2, we use the work inPart I discussed below to prove, for a �xed n, and p su�ciently large (dependingon n), that there are only �nitely many possible values for H1(GLn(q); L), whenL is irreducible (or has bounded composition length), independent of q. The valueis the same for all irreducible modules with the same parametrization, and all q

2An announcement of our results here appears in the proceedings of the 1997 Newton InstituteNATO conference [CPS8]. After our announcement was submitted, another generalization ofTakeuchi's result was given in a preprint of Dipper [D3]. His result involves a double centralizerproperty, but does not include our Morita equivalence.

3As we hope this introduction will make clear, the title also recalls our paper [CPSK], writtenwith Wilberd van der Kallen. In that paper, somewhat similar topics were discussed in thedescribing characteristic case.


of the same multiplicative order in �Fp . The calculation of this p-stable value canthen be translated, using the generic results of Part I, �rst to an Ext 1-calculationfor a q-Schur algebra over the �eld C of complex numbers (taking q to be anappropriate root of unity), and then to an Ext 1-calculation in the category of

integrable modules for the quantum enveloping algebra eUq(gln). Using [KL2], thatcalculation translates into an equivalent Ext 1-calculation involving cohomology of

the a�ne Lie algebra bgln(C ) (see (10.3)). We expect that a precise answer can beobtained in the context of a�ne algebras,4 but we do not consider that problem (foreither Ext 1 or the higher Ext i discussed below) in this paper. Finally, Theorem10.5 computes 1-cohomology for SLn(q) in terms of the cohomology of GLn(q) inmost cases.

In x11 we begin to attack the higher cohomology groups of GLn(q) with nontriv-ial coe�cients, especially irreducible modules. The idea, of course, is to exploit theabove Morita equivalence and study instead questions for the q-Schur algebra. Tothat end, we develop an important projective resolution of the q-determinant rep-resentation of the q-Schur algebra Sq(n; n). The result hinges ultimately on earlierwork of Deodhar [De] on Hecke algebra complexes (reminiscent of the Solomon-Titscomplex for the associated Tits building). But to bring these results to bear on ourproblem, we require the tilting theory work of [DPS3] for q-Schur algebras over thering Z[q; q�1] of Laurent polynomials. In the process, we obtain another interestingcomplex, valid for endomorphism algebras associated to arbitrary �nite Coxetersystems (W;S); see Theorem 11.10.

We apply the results of xx9,11 in x12. For example, Theorem 12.4 establishesthat the cohomology groups Hi(GLn(q); L) can be computed in a range, subjectto arithmetic conditions on q and i, in terms of the cohomology of the q-Schuralgebra Sq(n; n). In order to achieve this goal, it is necessary to reinterpret theprojective resolution of detq given in x11 as a complex of GLn(q)-modules, usingthe Morita equivalence of x9. This is done in the technical Lemma 12.1, and it leadsto the arithmetic conditions on the cohomology calculations. As an application,Corollary 12.6 also presents a stability result for higher cohomology, similar to, butsomewhat weaker than, the H1-stability result given in (10.2). As with the H1-result above, the stable value of a given Hi(GLn(q); L) can be expressed in termsof the cohomology of complex q-Schur algebras and quantum enveloping algebras,

and then in terms of the cohomology of the a�ne Lie algebra bgln(C ).Generic cohomology and representation theory study behavior which stabilizes

for large values of the parameters q, p, etc. For example, one can ask if, for suf-�ciently large p, there is some some explicit formula for the Brauer characters ofthe irreducible kGLn(q)-modules, given in terms of ordinary characters in the samespirit as in the describing characteristic case (as proved by [AJS]). The answer isa�rmative, and much easier in the non-describing case here|in fact, this obser-vation has been made essentially in [GH; (10.2)]. We include a proof in x8 for

4We have in mind formulas like those in [CPS4; x3]. For example, under the assumption ofthe Lusztig conjecture for the characters of the modular irreducible rational representations of areductive group G, the groups Ext �G(L1; L2), for L1; L2 irreducible modules with regular highestweights in the Jantzen region, can be explicitly given in terms of Kazhdan-Lusztig polynomials[CPS4; (3.9.1)]. Similar remarks apply to the BGG category O for a complex semisimple Liealgebra g and all irreducible modules; cf. [CPS4; (3.8.2)].


completeness. As in the describing characteristic case, explicit formulas in termsof Kazhdan-Lusztig polynomials (and ordinary characters of GLn(q)) can be given,using the work [KL2].

More generally, Part I of this paper undertakes an examination of generic rep-resentation and cohomology theory. Our development is new, though the methodsare all quite easy. Roughly speaking, this work provides a general theory of ordersin non-semisimple algebras, suitable for application to problems related to the rep-resentation theory of �nite and algebraic groups. For example, let P be a propertyof �nite dimensional algebras A over �elds. Given a domain O with quotient �eldK and an algebra A over O, suppose that P holds for the K-algebra AK . Thenwe call P generic with respect to O provided there is a nonempty open subset � Spec O such that P holds for the residue algebras Ak(p) for all p 2 . Section1 studies this and related notions quite generally.

Sections 2,3 study the genericity of various cohomological properties for algebrasover domains O. For example, Theorem 2.1 states that, given �nitely generatedA-modules M , N , and an integer m � 0, there exists a nonempty open subset � Spec O such that, for p 2 ,

dim ExtmAK (MK ; NK) = dim ExtmAk(p)(Mk(p); Nk(p)):

The stability results obtained in Part II (i. e., (8.6), (10.2) and (12.6)) are eventualapplications of the above formula.

The paper contains a number of further general applications of our Ext -results;for example, we establish (in x3) that the property of being Koszul is generic, atleast in the presence of a �nite global dimension assumption.

Section 4 studies the genericity in the various senses of x1 for quasi-hereditaryalgebras. Given an algebra A such that AK is quasi-hereditary, we observe that theKazhdan-Lusztig polynomials associated to AK also exhibit a generic property, sothat the various parity conditions studied in [CPS3,4] are also generic phenomena.Finally, xx5-6 return to general considerations involving Morita equivalence (in x5)and derived equivalence (in x6), while x7 takes up the theory of quivers and relations.The appendix x13 introduces and brie y discusses the \constructible" topology. Byusing it, we are able to prove a new constructibility result for subsets of schemesarising in generic representation theory. Along the way, we recapture a classicaltheorem of Chevalley.

Part I: Generic representation theory

In the �rst seven sections (Part I) of this paper, O will denote a commutative,Noetherian domain. (Unless we state otherwise, a \domain" will always be assumedto be commutative and Noetherian.) In the remaining sections, we shall placefurther restrictions on O as needed. (In particular, in Part II, O will be a suitablediscrete valuation ring.) Let X = Spec O. For p 2 X, let Op be the localization ofO at p and k(p) = Op=pOp the associated residue �eld. Often we write K = k(0)(the quotient �eld of O). For 0 6= f 2 O, let Xf be the basic open set consisting ofthose p 2 X satisfying f 62 p. If we write Of for the localization of O at powers off , then Xf = Spec Of .


Unless otherwise indicated, all O-modules will be assumed to be �nitely gen-erated over O (i. e., they are O-�nite). Similarly, O-algebras A will be assumedto be O-�nite as modules. In particular, A-modules are all �nitely generated asA-modules, unless we say otherwise. (We freely repeat these assumptions for thesake of clarity.) Let A{mod (resp., mod{A) be the category of �nitely generatedleft (resp., right) A-modules. If M is an O-module and O ! R is a morphismof commutative rings, then MR = M O R is the R-module obtained by basechange. When 0 6= f 2 O (resp., p 2 X), we contract the notation for the module

MOf(resp., MOp

) to Mf (resp., Mp). A morphism Mg�! N induces morphisms

Mfgf�! Nf and Mp

gp�! Np by base change.

x1. Representation theory on an open set

We shall use continually, and often without comment, the following well knownand elementary lemmas. We include proofs for the convenience of the reader.

Lemma 1.1. Let V be a vector space over K and M an O-submodule of Vsuch that KM = V . Then the natural map

� :M O K ! V : m t 7! tm

is an isomorphism of vector spaces. If W is any K-subspace of V , the restrictionof � to (W \M)K de�nes an isomorphism (W \M)K �=W .

Proof. Since M spans V , it contains a K-basis for V , hence a free O-submoduleN whose quotient M=N is an O-torsion module. The exactness of localizationdetermines a short exact sequence

0! N O K !M O K !M=N O K = 0! 0:

The �rst statement follows from this and the isomorphism N O K �= V .For the second statement, we may regard (W \M)K as a subspace of W . If

w 2 W , then w is a linear combination of elements of M , and for some a 2 O,aw 2M . Thus, w 2 (W T

M)K , so (WTM)K =W . �

Lemma 1.2. ( [Ma, p. 185; CE, Ch. VII]) If M is a (�nite) O-module, thenthere exists a non-zero element a 2 O such that Ma is a free Oa-module. If M isan arbitrary torsion-free O{module, then the canonical map M !MK : m 7! m1is an injection.

Proof. If T (M) denotes the torsion submodule of M , then there is a non-zeroelement b 2 O which annihilates T (M), hence Mb

�= (M=T (M))b is Ob{torsion-free. The canonical map Mb ! MK becomes an isomorphism after base changeby �Ob

K. Let B = f b1; : : : ; br g be a subset of Mb which is mapped bijectivelyonto a K{basis BK of MK . Then the Ob{submodule N = ObB of Mb is free. SinceMbK

�= MK , the quotient module Mb=N is a �nite Ob{torsion module, hence isannihilated by a non-zero element a 2 Ob. Thus, Na

�=Ma is a free Oa{module.


The �nal assertion of the lemma is clear from the de�nition of the localizationof a module. �

Corollary 1.3. If M is a �nite O{module and X � MK is a �nite subsetof MK, then there exists a non-zero a 2 O such that, under the canonical mapMa !MK, Ma identi�es with a free Oa{submodule of MK containing X.

Proof. Choose 0 6= b 2 O so that Mb is Ob{free. Identify Mb with its image inMK and consider the Ob{submodule M 0

b generated by Mb together with the �niteset X. Then M 0

b=Mb is a �nite Ob{torsion module. Thus, there exists 0 6= a 2 Obsuch that (M 0

b=Mb)a = 0. This implies X �Ma. �

Corollary 1.4. Let A be an O{algebra and M;N two A{modules which areO-�nite. If g :MK ! NK is a homomorphism of AK{modules, then there exists anon-zero a 2 O and an Aa{module morphism h :Ma ! Na such that:

(a) Ma (resp., Na) is an Oa{free Aa{submodule of MK (resp., NK);

(b) hK = g.

If, in addition, g = fK for an A{module morphism f :M ! N , then

(c) h = fa.

Proof. Choose 0 6= b 2 O so that Mb (resp., Nb) is an Ob{free Ab{submodule ofMK (resp., NK). If X is an Ob{basis of Mb, then there exists 0 6= a 2 Ob such thatNa contains g(X). Then Ma; Na satisfy (a) and the restriction of g to Mb de�nesthe required map h :Ma ! Na of part (b). If g = fK for a morphism f :M ! N ,then the functoriality of localization implies that the restriction of g to Ma agreeswith fa proving (c). �

Let P be a property of �nite dimensional algebras over �elds. Let A be a �niteO-algebra which is O{torsion-free. There are at least two ways to say that \P holdsgenerically" for A:

(a) P holds for AK ; or

(b) There exists a nonempty open subset � X such that P holds for Ak(p) foreach p 2 .

Of course, (b) implies (a), and often (a) implies (b). When this latter implicationholds for all O-algebras A as above, the property P is called generic with respect toO. If condition P is true for all O in a class C of domains, then call P generic withrespect to C. (Example: Any property P for algebras over �elds is generic w.r.t. theclass C of discrete valuation rings.) If C consists of all (commutative, Noetherian)domains, then P is generic.

If P is a property of algebras over domains (and not just �elds), there is yet athird way to say that P holds generically for A:

(c) There exists a nonempty open subset � X = Spec O such that P holdsfor the Of -algebra Af for each non-zero f 2 O satisfying Xf � . Equivalently,there is a non-zero ideal I � A such that P holds for Af for each non-zero f 2 I.


Property P is integrally generic for a domain O if, whenever (a) above holds,then (c) holds with respect to some nonempty a�ne open subset of X.

Similarly, we can de�ne the notions of \integrally generic with respect to a classof domains," and \integrally generic".

There are many other variations possible when P is de�ned for algebras overdomains (e. g., one could ask if P holds for the localization Ap, for all p 2 , : : : ).However, we shall be content with the notions given, which seem the most useful.

To give a simple example, consider the property Mn : \A is isomorphic to then � n matrix algebra Mn(O) over O." Then Mn makes sense for all domains O.If AK

�= Mn(K), then for some non-zero h 2 O, the matrix units eij of AK liein Ah. Thus, we can regard Mn(Oh) as a submodule of Ah with torsion quotient,so Af

�= Mn(Of ) for some non-zero f 2 Oh. It follows that Mn is an integrallygeneric property. If Af

�= Mn(Of ), then Ak(p)�= Mn(k(p)) for any prime ideal p

satisfying f 62 p. Hence, Mn is also a generic property. Similarly, the property thatan algebra be split semisimple, i. e., that it is a direct product of matrix algebras,is both generic and integrally generic. However, we will see below in (1.7) that theproperty of being semisimple is not generic.

There are appropriate variations on the above theme for a property P whichmay hold for pairs (or triples, : : : ) of algebras, or for modules (or pairs of modules,: : : ) for an algebra (over a �eld or a domain). For example, let MORITA bethe property on pairs (A;B) of algebras which asserts that A and B are Moritaequivalent. Similarly, P might be a property holding for a morphism betweenmodules, or it might be a property holding for a complex of modules, : : : . Ratherthan write down a plethora of formal de�nitions, the following remarks presentsome basic examples along this line.

Examples 1.5. (a) Given a morphism Mg�! N of modules (for an algebra),

the property that g is surjective (resp., an injection, an isomorphism) is denotedSUR(g) (resp., INJ(g), ISO(g)). Now assume g is a morphism of modules for analgebra A over O.

Suppose �rst that SUR(gK) holds and N is O-�nite. By (1.4), we may choose0 6= b 2 O so that Mb (resp., Nb) is a free Ob{submodule of MK (resp., NK). IfX �MK is a �nite set mapped bijectively onto an Ob{basis of Nb by gK , then, by(1.3), we may choose 0 6= a 2 Ob such that X �Ma. It follows that for 0 6= c 2 Oathe map gc :Mc ! Nc is surjective. In particular SUR(gc) is true for these elementsc. Also SUR(gk(p)) holds for any p 2 Xa. Thus, the property SUR of morphismsof A-modules with O{�nite codomains is both generic and integrally generic.

Second, suppose that INJ(gK) holds and both modules M;N are �nite over O.We choose 0 6= a 2 O so that the canonical map Ma ! MK is an injection. Thefunctoriality of localization implies ga :Ma ! Na is also injective. We now choose0 6= b 2 Oa so that (Na=ga(Ma))b is a free Ob{module. Then each map gc; 0 6= c 2Ob, is a split injection of Oc{modules. Consequently, for p 2 Xb; gp : Mp ! Np isalso a split injection of Op{modules. This implies that gk(p) is an injective Ak(p){homomorphism for each such p. Therefore, INJ is both a generic and an integrallygeneric property for morphisms between A-modules which are O-�nite.

The statements for SUR and INJ, taken together, show that ISO is both genericand integrally generic for morphisms between A-modules which are O-�nite.


(b) The property SSUR (resp., SINJ) that a morphism of modules is a splitsurjection (resp., split injection) is both generic and integrally generic for A-modules

�nite over O. Suppose thatM g! N is a morphism such that gK is a split surjectionwith section map sK : NK ! MK . By (1.4) we may choose 0 6= a 2 O so thatMa (resp., Na) is an Oa{free Aa{submodule of MK (resp., NK). In addition,we may assume that the restriction h of sK to Na de�nes an Aa{homomorphismh : Na ! Ma. Evidently h is a section for the map ga. Further hc is a section forthe map gc for each 0 6= c 2 Oa. For p 2 Xa, the functoriality of base change byk(p) implies that hk(p) is a section for gk(p). This establishes our claim for SSUR;a similar proof handles the property SINJ.

(c) Suppose that A is an O-algebra and M;N are A-modules. If MK�= NK ,

then Mk(p) and Nk(p) have the same composition factors for all p belonging tosome nonempty open subset of X. In fact, by (1.4), there is an isomorphism

g : Ma�! Na of Oa{free Aa{modules for some non-zero a 2 O. Further, gk(p) :

Mk(p) ! Nk(p) is an isomorphism for all p 2 Xa; a fortiori the two modules havethe same composition factors.

(d) Let A be as in (c). Suppose that V is a �nite dimensional AK -module. Any�nite A-module M which is O-free and satis�es MK

�= V is called an A-lattice forV . Choose a �nite A-submodule M of V such that MK

�= V . By (1.2), there existsa non-zero a 2 O such thatMa is O-free. Thus, Ma is an Aa-lattice for V . Suppose0 6= b 2 O and we are given an Ab-lattice N for V . Then (c) implies there exists anonempty open set � Xa

TXb such that for p 2 the Ak(p)-modules Mk(p) and

Nk(p) are isomorphic (and hence have the same composition factors).

For the remainder of these examples, we assume that A is O-�nite and torsion-free.

(e) Let FREE (resp., PROJ, PGEN) denote the property that a (�nitely gener-ated) module for an algebra is free (resp., projective, a projective generator). Eachof these properties is generic and integrally generic.

If F is an A{module such that FK is a free AK{module, then (1.3) implies thereis a non-zero a 2 O such that Fa is an Aa{submodule of FK containing an AK{basisBK = f b1; : : : ; br g for FK . The set BK generates a free Aa{submodule F

0 of Fa.The quotient Fa=F

0 is a �nite Oa{torsion module. Thus there is 0 6= b 2 Oa suchthat Fb = F 0b is a free Ab{submodule of FK . Clearly, for 0 6= c 2 Ob, Fc remainsfree as an Ac{module. Further, for p 2 Xb, Fk(p) is free as well.

Now suppose P is an A{module and PK is a projective AK{module. Then there is

a free A{module F and a split surjection FKg�! PK . By (1.4) and (1.3) there exists

0 6= a 2 O such that Fa (resp., Pa) are Aa{submodules of FK (resp., PK), Fa isfree as an Aa{module, and there is a map h : Fa ! Pa such that hK = g. Applying(b) to the Oa{algebra Aa shows that h is generically and integrally generically asplit surjection. Hence PROJ is a generic and integrally generic property.

Finally, since PROJ is generic and integrally generic, (b) above implies that theproperty PGEN that a module is a projective generator for a �nite O-algebra isboth generic and integrally generic.

(f) Let PKn

dKn�1��! � � � dK0��! PK0 be a �nite complex of �nite dimensional AK-


modules. There exists a �nite complex Pndn�1��! � � � d0�! P0 of A-modules which

identi�es with the original complex PK� after applying the base change �OK. If

the PKi are AK -projective, then there exists a non-zero f 2 O such that each Pif

is a projective Af -module and free as a Of -module.To see this, use induction on the length n of the complex. If n = 0, let P0

be the A{submodule generated by a �nite set of AK{generators of PK0 . Applying

induction to the complex C� : PKn

dKn�1��! � � � dK1��! PK1 , we assume there exists an

A-module subcomplex Pndn�1��! � � � d1�! P1 of which identi�es with C� on base

change to K. Let P0 be the A-submodule of PK0 generated by the image of P1 in

PK0 , together with a K-basis for PK

0 . By (1.1), P0K identi�es with PK0 . Finally,

we de�ne d0 : P1 ! P0 to be the restriction of dK0 to P1.The remaining statement follows by applying example (e) to each term in the

complex P�.

(g) Let Mi; 0 � i � r, be O-torsion-free A-modules. Suppose there is given anexact complex

(1.5.1) MrK

dKr�1��!Mr�1K

dKr�2��! � � � dK0��!M0K :

Then there is a basic open subset Xf � X such that the restrictions dfi of dKi toMi+1f de�ne an exact complex

(1.5.2) Mrf

dfr�1��!Mr�1f

dfr�2��! � � � df0�!M0f

of Af -modules. Moreover, we may assume the sequence remains exact upon ten-soring with any residue �eld k(p), p 2 Xf .

To see this, we proceed as follows. By (1.2), we may assume, after identi�cation,that each Mi is an A{submodule of MiK . By (1.4), we may choose 0 6= a 2 O suchthat the restriction dai of d

Ki to Mi+1a has image in Mia. The exactness of (1.5.1),

together with the fact that the Mi are all torsion-free, implies that the sequence ofmodules and homomorphisms

Mra

dar�1��!Mr�1a

dar�2��! � � � da0�!M0a

is a complex.Assume r = 2 and let Ra denote the kernel of da0. Because localization is an

exact functor, we may identify RK with the kernel of dK0 . Since dK1 : M2K ! RK

is surjective, and SUR is an integrally generic property, there is a non-zero multiple

f of a such that df1 : M2 f ! Rf is surjective. As with dK0 , the exactness oflocalization implies Rf is the kernel of dK0 . Thus the �rst statement is true forr = 2. The general statement follows easily by induction. The second assertion isalso clear, since by (1.2), we can assume that all the kernels and cokernels in (1.5.2)are free Of -modules.

We now study the generic properties of being separable and split (semisimple).For simplicity, we will assume that the algebra A is O-torsion-free. Recall that A


is separable provided that A is a projective Ae = A Aop-module; equivalently, Ais separable if the multiplication map A A

�! A is a split surjection of (A;A)-bimodules. We say that A is split, it if is isomorphic to a direct sum of full matrixalgebras over O. If A is split, then it is also separable: if A = Mn(O), the mapA

�! AA given on matrix units by eij 7! ei1 e1j is a bimodule map splitting �above. We say that A is nil-separable (resp., nil-split) provided that there exists anilpotent ideal N of A such that A=N is a separable (resp., split) algebra over O.Let SPLIT (resp., SEP, NILSPLIT, and NILSEP) denote the property split (resp.,separable, nil-split, and nil-separable) for an algebra A over a domain O. The idealN for either NILSPLIT or NILSEP must necessarily be the Jacobson radical of A.

Lemma 1.6. The properties SPLIT, SEP, NILSPLIT, and NILSEP are genericand integrally generic properties.

Proof. We have already indicated earlier (above (1.5)) that SPLIT is both ageneric and an integrally generic property.

Suppose the algebra A is O-�nite and torsion-free. Assume that AK is separable,

so that the multiplication map AKAK�! AK is a split surjection as a (AK ; AK)-

bimodule morphism. By (1.5 (b)), the SSUR property of a morphism is both genericand integrally generic; whence, SEP is both generic and integrally generic.

Now assume that AK is nil-split, so there exists a nilpotent ideal NK of AK

such that AK=NK is a split semisimple algebra over K. The ideal N = NK

TA of

A satis�es N O K �= NK (so there is no ambiguity in notation) and (A=N)K �=AK=NK . Thus, there is a nonempty open set � X such that (A=N)f �= Af=Nf

is split if Xf � . It follows that NILSPLIT is an integrally generic property. Wecan also assume that (A=N)k(p) is a split k(p)-algebra for all p 2 . By genericfreeness, we can further assume, after possibly shrinking , that Np is an Op directsummand of Ap. Thus, for p 2 , Nk(p) is a nilpotent ideal in Ak(p) satisfyingAk(p)=Nk(p)

�= (A=N)k(p). This proves that NILSPLIT is generic.A similar argument establishes that NILSEP is generic and integrally generic. �

Example 1.7. The property that an algebra be semisimple (over a �eld) isnot a generic property, although the properties that an algebra be split semisimpleor separable are both generic. For example, let k be an algebraically closed �eldof positive characteristic p, and let O = k[t] be the polynomial algebra over k inan indeterminate t. Consider the O-algebra A = k[t1=p]. It is free of rank p overO. Then AK is a purely inseparable �eld extension of the function �eld K = k(t),obtained by adjoining a pth root of t. However, let p be any non-zero prime ideal inO. Then p = (t� �) for some � 2 k. The commutative k(p)-algebra Ak(p) is a freek(p)-module of rank p which is generated as a k(p)-algebra by the nilpotent elementt1=p � �1=p. Hence, Ak(p) is not semisimple, and \semisimple" is not generic.

We conclude this section by indicating some generic module-theoretic properties.We continue to assume that A is a �xed algebra which is �nite and torsion-free overO.

Lemma 1.8. Assume that AK is nil-separable. If M is an A-module such thatMK is a completely reducible AK-module, then there exists a nonempty open subset


� X such that Mk(p) is a completely reducible Ak(p)-module for all p 2 .Proof. The argument in (1.6) shows that there is an idealN ofA and a nonempty

open subset of X such that if p 2 then Nk(p) is the nilpotent radical of Ak(p)

and (A=N)k(p) �= Ak(p)=Nk(p) is a separable k(p)-algebra. SinceMK is a completelyreducible AK-module, it follows that NKMK = 0. By shrinking if necessary, wecan assume that Mp is Op{torsion-free, hence is an Ap{submodule of MK . Thus,NpMp = 0. Then Nk(p)Mk(p) = NpMk(p) = 0, so that Mk(p) is a completelyreducible Ak(p)-module. �

Lemma 1.9. Assume that AK is nil-split. Let LKi , i = 1; � � � ; n, be the distinctirreducible AK-modules. There exists a non-zero f 2 O such that each LKi has an

Af -lattice Lfi and such that for p 2 Xf , the set fLf1k(p); � � � ; Lfnk(p)g is a complete set

of representatives for the isomorphism classes of irreducible Ak(p)-modules. Each

Lfik(p) can be assumed to be absolutely irreducible.

Proof. Let fajg be a basis for AK formed by �rst taking a basis fa1; � � � ; amg fora Wedderburn complement S consisting of matrix units from each simple factor, andsecond taking a basis fam+1; � � � ; ang for rad(AK). For some non-zero f 2 O, Af

is a free Of -module with basis faig. Let S0 be the Of -subalgebra of Af generatedby fa1; � � � ; amg, so that S0 is a direct sum of matrix algebras of the form Mc(Of ),c 2 Z+. Also, fam+1; � � � ; ang is a basis for Af

Trad(AK). We have:8><>:

Af = S0L(Af

Trad(AK));

S0K�= S; and

(Af

Trad(AK))K �= rad(AK):

The lemma follows immediately from these observations. �

Lemma 1.10. Assume that AK is nil-split, and let 0 = MK0 � MK

1 � � � � �MK

n =MK be a composition series of an AK-module MK . If M 2 Ob(A{mod) is

any O-lattice for MK, there is a �ltration 0 = M0 � M1 � � � � � Mn = M and anonempty open subset � X such that:

(a) MiK�=MK

i for all i;

(b) For p 2 and any extension �eld E � k(p), 0 = M0E � M1E � � � � �MnE =ME is a composition series of ME.

(c) For 0 < i � n, set Li = Mi=Mi�1. If 0 < i; j � n, then LiE �= LjE if, andonly if, LiK �= LjK .

Proof. For each i, let Mi = MKi

TM , so that MiK

�= MKi . Choose a non-zero

f 2 O such that each fMf�=M

Mif=Mi�1f

is Of -free. If Li = Mi=Mi�1, the LiK are the composition factors of MK . IfLiK �= LjK , we can assume by (1.5a) that Li �= Lj. The lemma is now clear from(1.9).


The following result will be used below and again in x4.Lemma 1.11. LetM;N be O-torsion-free A-modules. There exists a nonempty

open set � X = Spec O such that if p 2 ,HomAk(p)(Mk(p); Nk(p)) �= HomA(M;N)k(p):

Also, if B is an O-algebra such that BK�= EndAK (MK), then we can assume that

Bk(p)�= EndAk(p)(Mk(p)) for all p 2 .

Proof. (Sketch) Let 0! L! Q! M ! 0 be a short exact sequence in whichQ is projective. Let X be the natural image of HomA(Q;N) in HomA(L;N). Forany non-zero f 2 O, we have HomA(M;N)f �= HomAf (Mf ; Nf ). So, by (1.2), afterreplacing O by a suitable localization (of the form Of ), we can assume that X is O-free. Certainly, HomA(Q;N)k = HomAk(Qk; Nk) for k = k(p), p 2 SpecO, becauseQ is projective. Because the exact sequence 0! HomA(M;N)! HomA(Q;N)!X ! 0 is O-split, HomA(M;N)k � HomAk(Mk; Nk) for any such k. Similarly,localizing O even further we can assume that HomA(L;N)k � HomAk(Lk; Nk).(Let L play the role of M in the previous discussion.) Now an elementary diagramchase (using the snake lemma) proves the �rst assertion of the lemma. We leavethe second easy assertion to the reader. �

Lemma 1.12. Let M 2 Ob(A�mod) be O-torsion-free with MK =Ln

i=1MKi

a decomposition of MK into a direct sum of AK{submodules. Put Mi =MTMK

i .Then MiK

�= MKi , and there exists a nonempty open subset of X such that

Mf =Ln

i=1Mif whenever Xf � and Mk(p) =Ln

i=1Mk(p) whenever p 2 .

Moreover, if each MKi is absolutely indecomposable, we may assume that each

Mik(p) is absolutely indecomposable and that for each i; j, Mik(p)�= Mjk(p) if and

only if MiK�= MjK.

Proof. This result is clear provided we observe that if M is an O-torsion-freeA-module such that MK is absolutely indecomposable, then there is a nonemptyopen subset of X such that if p 2 , Mk(p) is an absolutely indecomposableAk(p)-module. But let E = EndA(M). Then EK

�= EndAK (MK) is a nil-splitalgebra with radical quotient K. Thus, there is a nonempty open subset suchthat if p 2 then Ek(p) is nil-split with radical quotient k(p) by (1.9). By (1.11),we can assume that Ek(p)

�= EndAk(p)(Mk(p)). �

x2. Ext calculations: the ungraded case

In this section, we continue to assume that A is an algebra which is O-�niteand torsion-free for some domain O. The main result below shows that for two�xed objects M;N in A{mod, the dimensions of the spaces Ext nAk(p)(Mk(p); Nk(p))

are generically constant for n in a �nite range. As an immediate consequence, theproperty FGLDIM that an algebra B over a �eld k has �nite global dimension is ageneric property.

Theorem 2.1. Let M;N 2 Ob(A{mod). If m is a non-negative integer, thenthere is a nonempty open subset m � X, depending on m, such that for p 2 m,


any extension �eld E of the residue �eld k(p), and 0 � n � m

dim Ext nAE (ME; NE) = dim Ext nAK (MK ; NK):

Proof. Let F � !M denote a resolution of M by free A-modules of �nite rankand let di : F i+1 ! F i denote the ith di�erential of this resolution.

Let E be any commutative O-algebra. Since the terms of F � are �nite andA-free, we have natural isomorphisms of complexes

(2.1.1) HomA(F�; N)E �= HomA(F

�; NE) �= HomAE (F�E ; NE):

Let C�E denote the truncated complex

0! HomAE (F0E ; NE)

d0�E��! HomAE (F1E ; NE)

d1�E��! � � �(2.1.2)

dm�1�E��! HomAE (F

mE ; NE)

dm�E��! Im(dm�E )! 0:

When E = O, we delete the subscripts. Thus, C� denotes the truncation of theoriginal complex HomA(F

�; N) constructed in parallel with (2.1.2).By (1.2) we can choose 0 6= f 2 O so that the terms of the resolution F �f !Mf ,

each term of the complex C� f , the kernels and images of the di�erentials of C� f ,and the cohomology modules H�(C� f ) are all free Of -modules.

If p 2 Xf = and E is an extension �eld of k(p), it follows that F �E ! ME isa free resolution of the AE-module ME . Using the isomorphisms (2.1.1), it followsthat the truncated complex C�E de�ned in (2.1.2) is obtained from C� f by basechange from Of to E. Hence, for 0 � n � m,

dimExt nAE (ME; NE) = dimHn(C�E) = rankOfHn(C�f ): �

The above result will be applied in Part II to q-Schur algebras and to the problemof calculating the cohomology of GLn(q) in non-describing characteristic. See (8.6),(10.2) and (12.6).

Corollary 2.2. Assume that AK is nil-split. The following statements hold:

(a) The property FGLDIM is generic.

(b) If AK has �nite global dimension, then there exists a nonempty open set � X such that for each non-negative integer m and any twoM;N 2 Ob(A{mod),the function Dm

M;N : ! N de�ned by setting

DmM;N (p) = dimExtmAk(p)(Mk(p); Nk(p))

for p 2 is constant on .

Proof. For (a), suppose AK has �nite global dimension d. If LK1 ; : : : ; LKn are

(up to isomorphism) the distinct irreducible AK -modules, then, using (1.9), thereis a non-zero element g 2 O and, for each 1 � i � n, an Og-lattice Li in LKi , such


that for p 2 Xg, the set fLi k(p)g1�i�n is a complete set of representatives for theisomorphism classes of irreducible Ak(p)-modules.

For 1 � i � n, let P �Ki ! LKi be a minimal projective resolution of LKi . Ourassumption on the global dimension of AK implies that this resolution has �nitelength at most d. By (1.5f), there exists a a �nite complex P � ! Li of A-moduleswhich identi�es with the original complex PK

� ! LKi after applying the base change� O K. Since the PK

i are AK -projective, there exists a non-zero multiple f ofg such that each term Pif in the localization P �if of P �i is a projective Af -module

(see (1.5e)). We may also assume, at the same time, that Lif , each P jif and each

kernel and image in the sequence

(2.2.1) P �if ! Lif

is a free Of -module of �nite rank. Finally, by (1.5(g)), we may assume that P �if !Lif is a projective resolution of Lif as an Af -module. Since P �if ! Lif can beviewed as a sequence of short exact sequences, split as Of -modules, it follows forp 2 Xf that P �ik(p) ! Lik(p) is a projective resolution of the Ak(p)-module Lik(p),

of length at most d. Hence Ak(p) has global dimension at most d, proving (a).If M 2 Ob(A{mod), we use induction on the length of MK together with the

Cartan-Eilenberg construction to obtain, for the element f above, a projectiveresolution Q�

f !Mf (in the category of Af -modules) which has length at most d.

We now apply the argument of the theorem to this complex to establish (b). Notethat the choice of f is independent of M . This proves (b). �

x3. Ext calculations: the graded case

In this section A =L

i2ZAi is a Z-graded O{algebra, �nite and torsion-free asan O{module. We assume the structure map O ! A has image in the subalgebraA0 of A consisting of terms of grade 0. Let A-grmod denote the category of �nitelygenerated graded left A-modules. If M =

Li2ZMi 2 Ob(A{grmod), each Mi is

naturally a �nite O-module. Given a non-zero f 2 O, Mf =L

iMif is a gradedmodule for Af =

LiAif . Given M and j 2 Z, then M(i) denotes the graded A-

module whose nth-grade is given by setting M(i)n =Mn�i. The category A-grmodhas enough projectives. For M;N 2 Ob(A{grmod), we have for any integer n,

(3.1) Ext nA(M;N) �=Mi2Z

Ext nA{grmod(M;N(i)):

The main result below is to show that the Koszul property (for �nite O-algebras)is generic. To do this, we extend the results of x2 to the category A-grmod. Forexample, if MK 2 Ob(AK{grmod) and M 0 � M is an A-submodule such thatM 0K = MK , de�ne M 2 Ob(A{grmod) by setting Mj = M 0 \MK

j , j 2 Z. We

have MjK�= MK

j for all j. Since M has only �nitely many nonzero grades, thereexists a non-zero f 2 O such that the graded Af -moduleMf has O-free gradesMif ,by (1.2). It is now straightforward to extend the examples in (1.5) to the gradedcase. Thus, we obtain the following result whose proof we leave to the reader.


Theorem 3.2. Let M;N 2 Ob(A{grmod) be O-torsion-free. For m � 0 andall n, 0 � n � m, there is a nonempty open subset � X depending on m suchthat for each p 2 , each extension �eld E of the residue �eld k(p), we have:

dim Ext nAE�grmod(ME; NE) = dim Ext nAK�grmod(MK ; NK):

Now letB be a �nite dimensional algebra over a �eld k. (Our insistence that B be�nite dimensional is largely one of convenience here.) Assume that B =

Ln�0Bn is

positively graded and that B0 is a semisimple algebra. Each irreducible B-moduleL de�nes an irreducible graded B-module, still denoted L, by setting L = L0.Recall that B is a Koszul algebra provided that for any two simple B-modules L;L0

we have Ext nB{grmod(L(i); L0(i0)) 6= 0 implies that i0 � i = n for all integers n; i; i0.

There is an equivalent formulation which we will use in the proof below: B isKoszul provided that each irreducible B-module L (regarded as a graded B moduleconcentrated in grade 0) has a projective resolution � � � ! P1 ! P0 ! L ! 0 inthe category B-grmod in which the head of Pi is the graded term of Pi of gradei. For more discussion of the theory of Koszul algebras, we refer to [BGS], [CPS5],and [PS2].

Corollary 3.3. Assume that AK is a nil-split algebra. If AK is a Koszulalgebra over K with �nite global dimension, then there is an open subset � Xsuch that for each p 2 , the k(p)-algebra Ak(p) is Koszul. Thus, the propertyFKOSZ that a �nite dimensional graded algebra over a �eld is a Koszul algebrawith �nite global dimension is generic.

Proof. Let LKi ; 1 � i � n denote the (up to isomorphism) distinct irreducibleAK -modules. Using (1.9), we can replace O by a suitable localization (of the formOf for some non-zero f 2 O) so that each LKi has an A-lattice Li and such that forp 2 X = Spec O, the Lik(p), i = 1; � � � ; n, are the distinct irreducible Ak(p) (whichare absolutely irreducible).

Since AK is Koszul and of �nite global dimension, for each i, LKi has a gradedprojective resolution of the form

� � � ! PKij ! PK

ij�1 ! � � � ! PKi0 ! LKi ! 0

which has length � d (for some positive integer d) such that each term PKij has the

form PKij�= QK

ij (j), where QKij is a graded projective AK -module generated by its

grade 0 elements.

Adapting the proof of (2.2) to the graded setting yields a complex Pi� ! Liin A{grmod and 0 6= f 2 O such that the complex Pi�f ! Lif is a projectiveresolution in Af -grmod whose terms Pijf have the form Pijf �= Qijf (j), where Qijf

is a projective object in Af -grmod whose head is the term in grade 0. In addition, wecan choose f so that the freeness conditions of (2.2.1) hold. Then P �i k(p) ! Li k(p)is a graded projective resolution of Li k(p) of length at most d. The characterizationof Koszul algebras mentioned above implies that Ak(p) is Koszul. �


x4. Integral quasi-hereditary algebras

We begin this section by recalling the de�nition of a split quasi-hereditary algebraover O. That done, we will let QHA be the property that an algebra is a split quasi-hereditary algebra. After showing QHA is both generic and integrally generic, weapply (2.2) to obtain a result on generic constancy of dimensions of all Ext groups.Finally we relate these results to the variations on the notion of abstract Kazhdan-Lusztig theory introduced in [CPS4, CPS5]. Here are the details.

Let A be an O{algebra which is �nite and projective as an O-module. An idealJ in A is a split (resp., separable) heredity ideal the following properties hold:

(1) A=J is O-projective.(2) J is projective as a left A-module.

(3) J2 = J .

(4) The endomorphism algebra EndA(AJ) satis�es the property SPLIT (resp.,SEP).

By de�nition, A is a split (resp., separable) quasi-hereditary algebra over Oprovided there exists a chain 0 = J0 � J1 � � � � � Jt = A of ideals such that eachJi=Ji�1 is a split (resp., separable) heredity ideal in A=Ji�1 in the sense above.A detailed discussion of this notion is presented in [CPS3; x3]. When O = K isan algebraically closed �eld, the split quasi-hereditary property coincides with theusual notion of a quasi-hereditary algebra over K [CPS1]. In this case, each idealJi is an idempotent ideal; observe that when J = AeA for an idempotent e, thenEndA(AJ)

op �= eAe.

Let QHA denote the property of being a split quasi-hereditary algebra.

Theorem 4.1. QHA is both a generic and an integrally generic property.

Proof. Let A be an O{algebra which is �nite and torsion-free over O. Assumethat AK is a split quasi-hereditary algebra as de�ned above. Then we can choosea de�ning sequence 0 = JK0 � JK1 � � � �JKt = AK of idempotent ideals satisfyingthe above properties. Let Ji = A

TJKi , and consider the sequence 0 = J0 �

J1 � � � � � Jt = A of ideals in A. For each i, consider the short exact sequence0 ! J2i ! Ji ! Ji=J

2i ! 0. Then Ji=J

2i is a torsion O-module since JK2

i = JKi .Therefore, there exists a non-zero f 2 O such that each Jif is an idempotentideal in Af . By (1.2), we can further assume that Af=Ji�1f is a free Of -module.Since PROJ is an integrally generic property (1.5e), we may further assume thatJif=Ji�1f is a projective right Af -module. Finally, by (1.6), we can assume thatEndAf=Ji�1f

(Jif=Ji�1f ) satis�es SPLIT as does EndJik((p)(Ji�1k(p)=Jik(p)) for allp 2 Xf . Thus, Af is a split quasi-hereditary algebra over Of . Next, the localcharacterization of quasi-heredity [CPS3; (3.3)] implies that for p 2 Xf , the algebraAk(p) is a split quasi-hereditary algebra. �

Theorem 4.2. Let A be an O{algebra which is �nite and torsion-free overO. Assume that AK is a (split) quasi-hereditary algebra and let f 2 O be as inthe proof of theorem (4.1). Then there is a basic open set Xh � Xf such that for


�; � 2 �, p 2 Xh, we have:

(4.2.1) dim Ext nAK (L(�)K ; L(�)K) = dim Ext nAE (L(�)E; L(�)E);

(4.2.2) dim Ext nAK (�(�)K ; L(�)K) = dim Ext nAE (�(�)E; L(�)E);

(4.2.3) dim Ext nAK (L(�)K ;r(�)K) = dim Ext nAE (L(�)E;r(�)E)

for all integers n and all �eld extensions E of k(p).

Proof. It is enough to prove these statements taking E = k(p). First, if � is a�nite poset, and B is any quasi-hereditary algebra over a �eld k with poset �, thenB has �nite global dimension bounded above by 2c(�)� 2, where c(�) is the depthof �, i. e., the length of a maximal chain in �. See, for example, [DR]. It followsfrom (2.2a) that the global dimensions of the algebras Ak(p), p 2 Xf , are uniformlybounded above. Therefore, the three statements (4.2.x) all follow immediately from(1.5e,g) and (2.2a). �

Remark 4.3. In [CPS4,5,7] the authors studied various parity conditions for aquasi-hereditary algebra. Suppose that B{mod is a highest weight category with�nite poset �. Let ` : � ! Z be a function. The left and right Kazhdan-Lusztigpolynomials are de�ned, for �; � 2 �, as(

PL�;� = t`(�)�`(�)

Pn dim Homn

B(L(�);r(�))t�nPR�;� = t`(�)�`(�)

Pn dim Homn

B(�(�); L(�))t�n:

When these (Laurent) polynomials contain only even powers of t, then B{mod has aKazhdan-Lusztig theory (with respect to `). If B{mod has a Kazhdan-Lusztig andthe algebra B is Koszul, we say that B-mod has a graded Kazhdan-Lusztig theory.5

In view of (4.2) and x4, we conclude that the property of having a Kazhdan-Lusztigtheory (resp., a graded Kazhdan-Lusztig theory) is a generic property. Similarremarks apply to the strong Kazhdan-Lusztig theories (SKL) and (SKL0) studiedin [CPS7].

x5. Morita theory

Let A and B be algebras over the domainO (�nite and torsion-free asO-modules,as usual). Consider the property MORITA that A and B are Morita equivalent.

Theorem 5.1. MORITA is generic and integrally generic.

Proof. Suppose that the algebras AK and BK are Morita equivalent, so thatthere exists an (AK ; BK)�bimoduleMK which, as a left AK-module, is a projectivegenerator for AK-mod, and which satis�es EndAK (M

K) �= BK . Let M be the

5The original de�nition of a graded Kazhdan-Lusztig theory, discussed in [CPS4], was givenin terms of a parity condition involving graded Ext-groups. It was later proved to be equivalentto the one given above [CPS5].


AO Bop-submodule of MK generated by a K-basis for MK . Then MK�=MK as

(AK ; BK)-bimodules. By (1.5e), there is a nonempty open subset � X = SpecOsuch that if Xf � (resp., p 2 ), then Mf (resp., Mp) is a projective generatorfor Af (resp., Ap). By (1.11), we can also assume that Bp

�= EndAp(Mk(p)) for allp 2 . If Xf � , then the isomorphism Bf

�= EndAf (Mf ) is automatic. We haveshown that MORITA is both generic and integrally generic. �

x6. Derived category constructions and equivalences; tilting modules

Again, let O be a Noetherian integral domain with quotient �eld K. Let Aan O-algebra, which is torsion-free and �nite as an O-module. Let Kb(A) bethe triangulated category whose objects are �nite chain complexes X = X� : 0 !Xm ! � � � ! Xn ! 0,m � n, where each Xi 2 Ob(A{mod), and whose morphismsare homotopy classes of chain maps. The distinguished triangles X ! Y ! Z !in Kb(A) are those isomorphic to ones de�ned by taking Z to be the mapping coneof X ! Y . There are obvious localization functors Kb(A) ! Kb(AK), K

b(A) !Kb(Af ), 0 6= f 2 O, Kb(A) ! Kb(Ap), and specialization functors Kb(A) !Kb(Ak(p)), p 2 Spec O. If uK : XK ! Y K is a morphism in Kb(AK), there exists

X;Y 2 Ob(Kb(A)) such that XK�= XK , YK �= Y K . Also, there exists a morphism

u : Xf ! Yf in Kb(Af ) for some non-zero f 2 O such that uK = uK . Further,if uK is a quasi-isomorphism, then we can assume that u is a quasi-isomorphism.There is also a nonempty open � Spec Of such that uk(p) is a quasi-isomorphismfor all p 2 . All these facts follows easily from the methods in x1 and will be usedas needed below.

In this section, we will consider the derived category Db(A) = Db(A{mod). It isthe localization of Kb(A) by the multiplicative subset of quasi-isomorphisms. Thedistinguished triangles in Db(A) are those triangles isomorphic to the images inDb(A) of distinguished triangles in Kb(A). For more details, see [W; x10].

As with Kb(A), there are localization functors Db(A) ! Db(AK), Db(A) !

Db(Af ) and Db(A) ! Db(Ap), taking X to XK , Xf , and Xp (in the notationabove), respectively. (The categories Kb(A) and Db(A) have the same objects.)We have that X LO Of �= Xf and X LO Ap

�= Xp. The exactness of localizationalso implies that H�(XK) �= H�(X)K, H

�(Xf ) �= H�(X)f , and H�(X)) �= H�(X)p

for any non-zero f 2 O and any p 2 Spec O. Similarly, Hom�AK

(XK ; YK) �=Hom�

A(X;Y )K , etc. for X;Y 2 Db(A).For X 2 Ob(Kb(A)), Xk(p) is de�ned in Db(A) for any p 2 Spec O, though

X 7! Xk(p) is not generally a functor! (The correct derived category functor

is X 7! X L k(p).) However, there exists a non-zero f 2 O such that theterms Xi of X, the cohomology groups Hn(X), and the kernels and cokernelsof the di�erentials Xi ! Xi+1 become free upon localization at f . Then Xf

�=LnH

n(Xf )[�n]. For p 2 Spec Of ,Xk(p)�= XLk(p) andHn(Xk(p)

�= Hn(X)k(p).

Similarly, given X;Y 2 Db(A), there exists a nonempty, open � Spec A suchthat Hom�

Ak(p)(Xk(p); Yk(p)) �= Hom�

A(X;Y )k(p).

Recall that X 2 Ob(Db(A)) has �nite projective dimension provided there existsan integer N > 0 such that Homn

A(X;Y ) = 0 for all Y 2 Ob(A{mod) and alln > N . In this case, there exists a �nite complex P of �nitely generated projectiveA-modules and a chain map P ! X which is a quasi-isomorphism. (For example,


use the dual of [W; (10.7.2)].) Then for any Y 2 Ob(Db(A)), the derived complexRHom�

A(X;Y ) is represented by Hom�A(P; Y ), and so lies in Db(A) (rather than

in just D(A)). Of course, RHom�A(X;Y )f

�= RHom�Af(Xf ; Yf ) by exactness of

localization.The following elementary result summarizes some further basic features of this

localization/specialization process.

Proposition 6.1. (a) The property that two morphisms a; b : X ! Y inDb(A) be equal is both generic and integrally generic.

(b) Assume that X 2 Ob(Db(A)) has �nite projective dimension. For any Y 2Ob(Db(A)), there exists a nonempty open � Spec O so that

RHom�Ak(p)

(Xk(p); Yk(p)) �= RHom�A(X;Y )LO k(p) �= RHom�

A(X;Y )k(p)

for all p 2 .Proof. To prove (a), consider two morphisms a; b : X ! Y in Db(A) and assume

that aK = bK . A morphism a : X ! Y is represented by an equivalence class of

diagrams Xs Z

g! Y (in Kb(A)) in which s is a quasi-isomorphism; see [W;x10.3], for example. There are several ways to say that aK = bK in Db(AK). Oneway is to say that aK = bK means precisely that there is a commutative diagram

ZK

�� @

@@@@R

XK� WK

6

- YK

I@@@@@ �

��

Z 0K

?

in which XK ZK ! YK de�nes aK , XK Z 0K ! YK de�nes bK andWK ! XK

is a quasi-isomorphism. We can replace O by a localization Of to assume thatWK = WK for some W 2 Db(A), that all the maps in the above diagram arede�ned over O, and that the maps into X are quasi-isomorphisms (and remain soupon specialization to any residue �eld k(p)). Also, if two chain maps c; d : S ! Tare such that cK and dK are homotopic, then the chain homotopy can be de�nedover some Of . So, we can assume that a = b in Db(A), replacing O yet again bysome Of . Also, by our construction, ak(p) = bk(p) for any p 2 Spec O.

To prove (b), there is a chain map PK ! XK which is a quasi-isomorphism,where PK is a bounded complex of projective AK -modules. Replacing O by somelocalization Of , we can assume that PK �= PK for a bounded complex P of pro-jective A-modules. We can also assume that there is a quasi-isomorphism P ! Xwhich induces a quasi-isomorphism Pk(p) ! Xk(p) for all p 2 Spec O. Thus, after


replacing SpecO by a smaller open subset in order to assume that all kernels andcokernels in Hom�

A(P; Y ) are O-free, we have

RHom�Ak(p)

(Xk(p); Yk(p)) �= Hom�Ak(p)

(Pk(p); Yk(p))

�= Hom�A(P; Y )k(p)

�= RHom�A(X;Y )k(p)

�= RHom�A(X;Y )L k(p):

This completes the proof. �

Lemma 6.2. The property that a triangle T1 ! T2 ! T3 ! T1[1] in Db(A)be a distinguished triangle is both generic and integrally generic.

Proof. A diagram T1 ! T2 ! T3 ! T1[1] in Db(A) de�nes a distinguishedtriangle if and only if it is isomorphic to the image in Db(A) of a distinguishedtriangle in Kb(A). Thus, if T1K ! T2K ! T3K ! T1K [1] de�nes a distinguishedtriangle in Db(AK), there is a commutative diagram

(6.2.1)

T1K ! T2K ! T3K ! T1K [1]

XK1

?! XK

2

?! XK

3

?! XK

1 [1]

?

in Db(AK) in which the bottom row is the mapping cone sequence for a chainmap gK : XK

1 ! XK2 in Kb(AK) and the vertical maps are isomorphisms in

Db(AK). Now apply the remarks in the �rst paragraph of this section, togetherwith (6.1a). �

Let DERIVED be the property on pairs (A;B) of algebras (�nitely generatedand torsion-free over their common base ring) which holds if and only if the triangu-lated categories Db(A) and Db(B) are equivalent. Recall that Db(A) is equivalentto Db(B) if and only if there exists a bounded complex T of �nitely generatedprojective A-modules such that A lies in the triangulated subcategory of Db(A)generated by the direct summands of T and, in addition, Hom�

A(T; T )�= B, i. e.,

Homn(T; T ) �=�0 if n 6= 0

B if n = 0:See [R]. We will call such a T a projective triangu-

lated generator relating Db(A) to Db(B). Now we can prove:

Theorem 6.3. The property DERIVED is both generic and integrally generic.

Proof. Suppose DERIVED holds for the pair (AK ; BK), with A;B algebrasover O as above. Let TK be a projective triangulated generator relating Db(AK)to Db(BK). Using (1.5(e,f)), we replace O by some Of to assume that TK �= TK ,for a bounded complex T of �nitely generated projective AK -modules. Also, sinceAK arises from TK by a �nite number of steps of forming distinguished triangles


and direct summands, the same is true for A and T after a further appropriatelocalization of O, as follows using (6.2). (The fact that \M is a direct summandof N" is both a generic and an integrally generic property" is very easy to proveand is left to the reader.) Finally, using (6.1) and perhaps localizing O further,we obtain that B �= Hom�

A(T; T ) and Bk(p)�= Hom�

Ak(p)(Tk(p); Tk(p)). The theorem

now follows. �

x7. Generic quiver theoryAs in the previous secton, O is a Noetherian domain with quotient �eld K, and

A is an O-algebra which is torsion-free and �nite as an O-module.We will assume throughout this section that the following holds.

Hypothesis 7.1a. The algebra AK=rad(AK) is separable. (Equivalently, thealgebra AK is nil-separable in the terminology of x1.)

If (7.1a) holds, the Wedderburn principal theorem implies that there is a sub-algebra AK

0 complementary to the radical NK of AK . Clearly, NK = NK �= NK

where N = ATNK . Similarly, A

K0 = A0K �= A0K where A0 = A

TAK0 . We will

usually require, in addition to (7.1a) that

Hypothesis 7.1b. With the above notation, we have that A = A0

LN , and

N =ML

N2 for some (A0; A0)-bimodule M .

In the presence of (7.1a), we can always replace O by a localization Of in orderto assume that (7.1b) does hold. Indeed, we could even assume that A0 is separable,i. e., the algebra A is nil-separable. (In this case, standard homological argumentsalso show that A0 is uniquely determined up to an inner automorphism of A. Wedo not know any way to canonically choose M inside A, though as a bimodule it isisomorphic to N=N2.)

We will now take up the question of generators and relations for A and itslocalizations and specializations. Observe that in (7.1b), A is generated as a ring,by A0 and M . More precisely, since M is an (A0; A0)-bimodule, we can form thetensor ring

T = TA0(M) =

Mn

Tn;

where Tn = Mn = M A0� � � A0

M| {z }n times

and the multiplication is de�ned in the

obvious way. Then there is a unique surjection � : T ! A which respects theinclusions of A0 �M into both T and A. We view M as a direct analogy of the\quiver module" used in the theory of �nite dimensional algebras.

Now let R be any �nitely generated (A0; A0)-sub-bimodule of T . We say R isa module of relations for A with respect to A0 and M (or that the pair (M;R)gives generators and relations for A with respect to A0) if Ker(�) is the T -idealgenerated by R. (Notice that R necessarily is contained in only �nitely manygrades of T , as de�ned in terms of the number of tensors of M . We will usuallytake R to be contained in grades � 2, though the de�nition does not speci�callyrequire it.) Given a candidate �nitely generated A0-sub-bimodule R, we will say


the property QUIV(A0;M;R) holds if R is a module of relations for A. If, inaddition, R is a direct sum of its homogeneous projections of various grades, we sayHOMOG(A0;M;R) holds, and QUAD(A0;M;R) holds if R is entirely containedin grade 2. Also, if HOMOG(A0;M;R) (resp., QUAD(A0;M;R)) holds for someA0;M;R, we say that TIGHT (resp., QUADRATIC) holds for A.

Theorem 7.2. Assume that A satis�es (7.1a,b) and that R is a �nitelygenerated A0-submodule of T = TA0

(M). The properties

QUIV(A0;M;R); HOMOG(A0;M;R); QUAD(A0;M;R);

are all generic and integrally generic. Similarly, the properties TIGHT and QUA-DRATIC are generic and integrally generic for any algebra A satisfying (7.1a).

Proof. Suppose that QUIV(A0K ;MK; RK) holds. Choose n > 0 so that NnK =

0; thus Nn = 0, and TnK must be contained in the TK -ideal generated by RK . Thelatter ideal is the K-submodule generated by all products M iRM j with i+ j � 0.Upon suitable localization, Tn will be contained in the ideal generated by R. Inother words, we can assume that the T -ideal generated by R has as quotient anO-algebra A0, �nite as an O-module, and A0K

�= AK . Localizing further, we mayassume that A0f

�= Af for some non-zero f 2 O. Now QUIV(A0f ;Mf ; Rf) follows,

and certainly HOMOG(A0f ;Mf ; Rf), QUAD(A0f ;Mf ; Rf) follow from their coun-terparts over K just by requiring Rf to be torsion-free over O. It follows easilythat QUIV(A0;M;R), HOMOG(A0;M;R), and QUAD(A0;M;R) are generic andintegrally generic, as are TIGHT and QUADRATIC. (Notice that if we do not haveA0;M;R to begin with, it is easy to obtain them, assuming QUIV(AK

0 ;MK; RK)

for the algebra AK . We can write RK = RK , where R = TTRK once A0 and M

have been de�ned (passing to a suitable localization, if necessary). Observe thatthe intersection T

TRK involves only �nitely many terms.) �

As a corollary of the proof, we have

Corollary 7.3. Assume that A satis�es (7.1a,b). Then there is at least onemodule of relations R, in grades � 2, for A, with respect to some A0 and M asabove.

Proof. This follows from the proof of the theorem, since it is well-known thatQUIV(AK

0 ;MK ; RK) may be satis�ed using RK concentrated in grades � 2. This

property is inherited by R = TTRK and its further localizations in the above

proof. It is also easily to argue more directly: By adjoining �nitely many elementsto R, we may assume that the ideal it generates contains Tn (where n is as in theabove proof). Then, as above, the quotient by the ideal generated by R is a �nitelygenerated O algebra which clearly has A as a homomorphic image (from (7.1b)). Itis then an easy matter to add �nitely many elements (of grades � 2) to R so thatthis quotient is isomorphic to A. �

Part II: q-Rational representation theory

In this part, we let Z = Z[t; t�1] be the ring of Laurent polynomials in anindeterminate t. Let (W;S) be a �nite Coxeter system, and consider the associated


generic Hecke algebra H = H(W;Z) over Z. This algebra has basis f�wgw2Wsatisfying the relations

(1) �s�w =

��sw; if sw > w

t�sw + (t� 1)�w; otherwise(s 2 S;w 2W ):

For � � S, W� = hsis2� denotes the corresponding parabolic subgroup of W ,while H� = h�sis2� is the corresponding parabolic subalgebra of H. Thus, H� =H(W�;Z). Also, put

(2) x� =Xw2W�

�w and y� =Xw2W�

(�t)�`(w)�w:

The left ideals Hx� and Hy� have natural interpretations as induced modules.De�ne linear characters on H by

(3) IND : H ! Z; �w 7! t`(w); and SGN : H ! Z; �w 7! (�1)`(w)

for w 2W . Putting IND� = INDjH�and SGN� = SGNjH�

, we have

(4) �wx� = IND�(�w)x� and �wy� = SGN�(�w)y�; w 2W�;

so that Zx� (resp., Zy�) is a module realization of IND� (resp., SGN�). It followseasily that

(5) Hx� �= indHH�IND� and Hy� = indHH�

SGN�:

Similar comments hold for the right ideals x�H and y�H.

There is a Z-automorphism � : H ! H given on generators by �(�w) =(�t)`(w)��1

w�1 . If M is a (right or left) H-module, let M� denote the H-module

obtained by twisting the action of H on M by �. For example, (Hx�)� = Hy� for

all � � S [DPS3; (1.4c)]. Also, IND� = SGN.

Let R be a commutative ring and Z ! R be a ring homomorphism in whicht 7! q 2 R. The algebra H(W;Z)R = H(W;Z)Z R will be denoted by H(W;R; q)and sometimes simply by just by H(W; q)R or even H(W; q). Although it shouldalways be evident from context, we alert the reader that the \q" may have severaldi�erent meanings in what follows: for example, it is often a power q = rd of aprime r, while if k is a �eld of characteristic p 6= r, then in the Hecke algebraH(W; q)k, q = q � 1k is a root of unity in the �eld k.

For w 2 W , let �w denote the R-basis vector �w Z 1 2 H(W; q). The relations(1) above provide a presentation for the R-algebra H(W; q). The characters INDand SGN on H(W; q) de�ne linear characters IND : H(W; q) ! R and SGN :H(W; q)! R by base-change, and the isomorphisms (5) hold over R. Finally, theautomorphism � induces an automorphism on H(W; q).


x8. The Geck{Gruber{Hiss very large prime result

In this section, let (W;S) = (Sm; S), where Sm is the symmetric group of degreem and S = f(1; 2); (2; 3); � � � ; (m�1;m)g is the set of fundamental re ections. ThenH = H(Sm;Z), Z = Z[t; t�1].

Let V a free Z-module of rank n > 0. The Hecke algebra H acts on V m oncean ordered basis fv1; � � � ; vng for V has been �xed. If J = (j1; � � � ; jm) is a sequenceof integers satisfying 1 � ji � n, for all i, then write J� = (j��1(1); � � � ; j��1(m))for � 2 Sr, and put vJ = vj1 � � � vjm 2 V m. For s = (i; i+1) 2 S, the formula

(8.1) vJ�s =

�tvJs; if ji � ji+1

vJs + (t� 1)vJ ; otherwise

de�nes a right action of the generators �s, s 2 S, of H on V m. This actionextends to de�ne a right H-module structure on V m. (See [DD; (3.1.5)].) As aright H-module, V m decomposes into a direct sum of various copies of the inducedmodules x�H, � � S; see below. The endomorphism algebra

(8.2) St(n;m) = EndH(Vm)

is the t-Schur algebra of bidegree (n;m) over Z. For any commutative ring Rand homomorphism Z ! R, t 7! q, we often write Sq(n;m) or St(n;m)R for thealgebra St(n;m)R = St(n;m) Z R for the corresponding q-Schur algebra overR. The natural map St(n;m)! EndH(Sm;R;q)(V

mR ) de�nes (by base-change) an

isomorphism

(8.3) Sq(n;m)R �= EndH(Sm;R;q)(VmR )

of algebras over R. See [DJ1; (3.3)].The proof of (8.3) depends, in fact, on another description of the q-Schur algebras

that will be useful in the sequel. Let �(n;m) (resp., �+(n;m)) be the set ofcompositions (resp., partitions) of m with n (resp., at most n non-zero) parts.Thus, � 2 �(n;m) is a sequence � = (�1; � � � ; �n) of non-negative integers suchthat �1 + � � � + �n = m. If �1 � �2 � � � � , then � is a partition of m. Any� 2 �(n;m) determines a subset J(�) of S as follows. Let Y(�) be the Youngdiagram of shape �. Filling in the boxes in the �rst row of Y(�) consecutively withthe integers 1; � � � ; �1, the second row with the integers �1 + 1; � � � ; �1 + �2, etc.de�nes a standard tableau t� of shape �. Let J(�) be the subset of S consisting ofthose s which stablize the rows of t�. However, we usually write W�; x�; y�, etc. inplace of WJ(�); xJ(�); yJ(�), etc. Then

(8.4)

V m �=M

�2�(n;m)

x�H �=M

�2�+(n;m)

x�H�k�

�=M��S

x�H�r�

as right H-modules for some choice of non-negative integers k� = k�(n;m) (� 2�+(n;m)) and r� (� � S)[DD; (3.1.5)]. The second isomorphism above follows from


the fact that if W� and W� are conjugate in W , then Hx� �= Hx�. Sometimes itis useful to work with the algebra

(8.5) A = EndH(M

�2�+(n;m)

Hx�):

By (8.4), A is Morita equivalent to St(n;m).If � is a partition of m, write � ` m. The poset structure E on the set �+(m) =

f� j� ` mg of partitions of m is de�ned by � = (�1 � �2 � � � � ) E � = (�1 � �2 �� ) if and only if �1 � �1, �1 + �2 � �1 + �2, : : : . Thus, �

+(n;m) is a coideal in�+(m). By [DPS2; (2.5)], there exist (left) St(n;m)-modules �(�), � 2 �+(n;m),such that for every �eld k and algebra homomorphism Z ! k, t 7! q, Sq(n;m){modis a highest weight category with poset �+(n;m), and standard objects �(�)k =�q(�). In particular, the irreducible Sq(n;m)-modules Lk(�) = Lq(�) are indexedby the partitions � 2 �+(n;m).

We will need to know that the q-Schur algebras (or at least certain ones ofthem) are nil-split. Although this fact can be deduced using the theory of quantumenveloping algebras, it can also be established directly using the approach to q-Schur algebras given in [DPS2]. First, there exist (left) St(n;m)-modules r(�),� 2 �+(n;m), such that for any specialization Z ! k, t 7! q, into a �eld k,the r(�)k, � 2 �+(n;m), are the costandard objects in Sq(n;m)-mod. (Thesemodules can be de�ned �rst by taking Z-linear duals of the standard modulesfor the category mod-St(n;m) of right modules, using the right module version of[DPS2].) For k above, we have Ext iSq(n;m)(�(�)k;r(�)k) = 0, for i > 0, by a

standard highest weight category theory. By [DPS3;(4.4)],

HomSt(n;m)(�(�);r(�))k �= HomSq(n;m)(�(�)k;r(�)k):It is known that the Q (t)-algebra HQ(t) is split semisimple, so that St(n;m)Q(t)is a split semisimple algebra. In fact, it has irreducible modules �(�)Q(t) �=r(�)Q(t), � 2 �+(n;m). Thus, HomSt(n;m)Q(t)(�(�)Q(t);r(�)Q(t)) �= Q (t). Since

HomSt(n;m)(�(�);r(�)) is a �nitely generated projective module for Z (by anothercommutative algebra argument using [DPS2; (0.1)]), it follows that

HomSq(n;m)(Lk(�); Lk(�)) �= HomSq(n;m)(�(�)k;r(�)k) �= k;

so we conclude that Sq(n;m)=rad(Sq(n;m)) is a split semisimple algebra.

Now let ` be a positive integer, let q = � =p1 2 C be a primitive `th root of

unity, and set K = Q (q). Consider the Dedekind domain Z(`) of algebraic integersin the number �eld K = Q (q). There is a homomorphism Z ! Z(`) under whicht 7! q. Applying (1.9) and the previous paragraph, withK the quotient �eld of Z(`),there exists a non-zero f 2 Z(`) such that each LK(�) has an Sq(n;m)f -lattice L(�)such that for all p 2 = SpecZ(`)f , the Lf (�)k(p), � 2 �+(n;m), are the distinctirreducible Sq(n;m)k(p)-modules. In the previous sentence, Sq(n;m) denotes the

Z(`)-algebra St(n;m)Z(`). We can, in fact, assume that Lf (�)k(p) �= Lk(p)(�).We can now apply the results of xx1,2 to obtain the following result. As we

discuss in the remark following, an immediate consequence of the theorem is theGeck{Gruber{Hiss very large prime result.


Theorem 8.6. With the above notation, we can also assume � Spec Z(`)has the property that for p 2 and k = k(p), we have:

(8.6.1)

�[�(�)K : LK(�)] = [�(�)k : L

k(�)];

dim Ext nSt(n;m)K(LK(�); LK(�)) = dim Ext nSt(m;n)k

(Lk(�); Lk(�));

for all �; � 2 �, n 2 Z+.

Proof. The �rst assertion on composition factors follows from (1.10) since wehave already argued that St(n;m)K is nil-split. Finally, the second Ext -assertionfollows from (2.1), together with the fact that the algebras St(m;n)k have �niteglobal dimension uniformly bounded above by 2c(�+(n;m))�2, where c(�+(n;m))is the length of a maximal chain in �+(n;m) [DR]. �

Remark 8.7. The �rst assertion in (8.6) proves \generically" an old conjectureof James [Ja; x4]: Suppose that, modulo p, ` is equal to the smallest positiveinteger e with 1 + � + : : : + �e�1 = 0.6 (Thus, ` = p if � = 1 modulo p, and,otherwise, ` is the order of � modulo p. So we are requiring, simply, that ` bep whenever it happens to be divisible by p.) Then James conjectures that eachLK(�) remains irreducible upon reduction \modulo p", provided `p > r. Gruberand Hiss [GH; x10] remark that this follows for (` �xed and) p very large (thatis, su�ciently large, with no explicit bound) from an argument of Geck, madeoriginally in a Hecke algebra context. This conjecture is also a consequence of theabove theorem, particularly of the �rst part, which we found before we were awareof the Geck{Gruber{Hiss remark. The result implies there are generic characterformulas for modular charactrs of the �nite general linear groups in \non-describingcharacteristics," cf. [S].

There is an analogous large prime result in the \describing characteristic" casefor each type of root system, by the much more di�cult work of Andersen-Jantzen-Soergel [AJS]. It is an intriguing question as to whether or not an easier proof mightbe found of their result, better �tting the \generic" context of this paper.

x9. A Morita equivalence for GLn(q)

Recall that k is a �xed algebraically closed �eld of positive characteristic p. Let(O; K; k) be a p-modular system (i. e., O is a discrete valuation ring with quotient�eld K and residue �eld k). The following simple lemma is key to our approach.

Lemma 9.1. Let R be an O-algebra which is free of �nite rank over O and hasthe property that RK is a semisimple algebra over K. Consider an exact sequence0 ! N ! P ! M ! 0 of right R-modules which are free of �nite rank over O.Suppose that the RK-modules NK and MK have no composition factors in commonand that P is a projective R-module. Then M is a projective R=J-module, whereJ is the annihilator of M in R.

Proof. Because RK is semisimple, and NK and MK have no common compo-sition factors, we have that NKJK = NK . Thus, N=NJ is a torsion O-module.

6The formulation in [Ja] contains a misprint, omitting the \modulo p".


The surjective map P � M yields a surjective map � : P=PJ � M . The rightexactness of the functor �R R=J implies that Ker(�) is a homomorphic image ofN R R=J �= N=NJ , so Ker(�) is also a torsion O-module. But R=J is evidentlyO-torsion-free, so, because P is a projective R-module, P=PJ is also O-torsion-free. Since Ker(�) is a submodule of P=PJ , it is also O-torsion-free, so �nallyKer(�) = 0. Thus, M �= P=PJ is a projective R=J -module, as required. �

Now we can immediately obtain:

Theorem 9.2. Assume the hypotheses of (9.1). If, in addition, every irre-ducible R=J-module lies in the head of the right R-module M , then the functorF (�) = HomR=J (M;�) de�nes a Morita equivalence

(9.2.1) F : mod{R=J��!

Moritamod{EndR(M)

of right module categories.

Proof. We regardM as a left EndR(M)-module, so that in (9.2.1) HomR=J(M;X)is a right EndR(M)-module for any R=J -module X. The hypotheses imply thatevery irreducible R=J -module is a homomorphic image of M . Thus, by (9.1), M isa projective generator for the algebra R=J and the conclusion follows. �

For the rest of this section, we consider the group G(q) = GLn(q). It has ordergiven by

(9.3) jGLn(q)j = q

�n2

�� (q � 1)n �

nYj=2

qj � 1

q � 1:

Recall that q = rd is a power of a �xed prime r. In what follows, we assume thatthe quotient �eld K of O has been taken large enough so that it is a splitting �eldfor G(q). We will verify that the hypotheses of (9.2) above can be achieved forR = OG(q).

Fix a set C of representatives from the G(q)-conjugacy classes. If b is a non-negative integer, let Cb0 be the subset of x 2 C having order relatively prime tob. If Gss denotes the set of semisimple elements in the algebraic group G, putG(q)ss = Gss

TG(q) and Css = G(q)ss

T C. Let Css;p0 be the set of p0-elements inCss: Any s 2 G(q)ss has centralizer ZG(q)(s) in G(q) of the form:

(9.4) ZG(q)(s) �=m(s)Yi=1

GLni(s)(qai(s)); where

Xai(s)ni(s) = n:

Let n(s) = (n1(s); � � � ; nm(s)(s)). Put

�+(n(s)) = �+(ZG(q)(s))

for the set of multi-partitions � of n(s), i. e., � = (�(1); � � � ; �(m(s))), where �(1) `n1(s); � � � ; �(m(s)) ` nm(s)(s). For a �xed s 2 G(q)ss, we will make constant use of


the following con�guration of subgroups of G(q):

G(q) = GLn(q)

Ls(q)=

m(s)Yi=1

GLai(s)ni(s)(q)

�� @

@@@@

Hs(q)=

m(s)Yi=1

GLai(s)(q)(ni(s)) ZG(q)(s)=

m(s)Yi=1

GLni(s)(qai(s))

@@@@@ �

��

Ts(q) =

m(s)Yi=1

F�(ni (s))

qai (s)

In this set-up, N (m) = N � � � � � N (m times) for a subgroup N of G(q). Theinclusions are the natural ones; in particular, Ts(q) is a maximal torus of ZG(q)(s).

If L is a Levi subgroup of GLn de�ned over Fq , let RG(q)L(q) be the Deligne-Lusztig

induction functor from the (complex) character group X(L(q)) of L(q) to the char-acter group X(G(q)) of G(q). In case L(q) is the Levi subgroup of a parabolic

subgroup P (q) of G(q), then RG(q)L(q) agrees with Harish-Chandra induction from

mod{CL(q) to mod{CG(q), and so takes modules to modules. (More generally, for

any commutative ring Z, de�ne Harish-Chandra induction RG(q)L(q) : mod{ZL(q) !

mod{ZG(q) as follows: let P (q) = Ls(q)nV (q) be the standard parabolic subgroupassociated to Ls(q), and, for an RL(q)-module Q, let Q0 be the RP (q)-module ob-tained from Q by in ating the action of Ls(q) on Q through the natural homomor-

phism P (q)! Ls(q). Then RG(q)Ls(q)

Q = indG(q)P (q)Q

0.)

The irreducible ordinary characters of G(q) were �rst parameterized and deter-mined by Green [Gr], while a recasting of those results in terms of Deligne-Lusztigtheory7 is presented in [FS]: The distinct irreducible characters f�s;�g are indexedby pairs (s; �), where s 2 Css and � ranges over the irreducible unipotent charactersof the centralizer ZG(q)(s) described in (9.4). In turn the set �+(n(s)) indexes theirreducible unipotent characters of ZG(q)(s). For � 2 �+(n(s)), let �� denote thecorresponding unipotent character. For simplicity, write �s;� (or �s;�;G(q) in caseG(q) needs mentioning) in place of �s;�� . In the next paragraph, we elaborate onthe construction of �s;�.

As in [FS; (1.16)], s 2 Css can be associated with a linearK-valued character bs onZG(q)(s). For � 2 �+(n(s)), �R

Ls(q)ZG(q)(s)

bs�� = �s;�;Ls(q) is an irreducible character

7It would be possible to avoid Deligne-Lusztig theory, and just use Green's results, by takingthe approach in [DJ2; x7] to the results in [FS]. However, some aspects of the Deligne-Lusztigformalism are conceptually simplifying.


of Ls(q) for some choice � = �1. Further, RG(q)Ls(q)

�s;�;Ls(q) = �s;�. If s 2 Css;p0 andt 2 ZG(q)(s) is a p-element, then ZG(q)(st) � ZG(q)(s), while Lst(q) is a subgroupof Ls(q). Therefore, the transitivity of Harish-Chandra induction and the abovedescription of �s;� imply that

(9.5a) �st;� = RG(q)Ls(q)

�st;�;Ls(q); s 2 Css;p0 ; t 2 ZG(q)(s) a p-element:

Since the unipotent characters of ZG(q)(s) are just the constituents of the trivialcharacter induced from a Borel subgroup, the characters bs�� are precisely the irre-

ducible constituents of RZG(q)(s)

Ts(q)(bs), where we have written bs for bsjTs(q). Thus, by

transitivity of Deligne-Lusztig induction, the characters �s;� are just the characters

appearing with nonzero coe�cient (positive or negative) in RG(q)Ts(q)

(bs), i. e., in termsof the inner product on group characters, we have, for any irreducible character �on G(q),

(9.5b) (�; RG(q)Ts(q)

(bs)) 6= 0 () � = �s;�; some � ` n(s):

Let f(X) 2 Fq [X] be an irreducible monic polynomial of positive degree d anddistinct from X. For any root � 2 �Fq of f(X) = 0, [DJ2; p. 29] de�nes anirreducible cuspidal representation CK(�) of KGLd(q).

8 The representation CK(�)depends on a choice of an embedding F�

qd,! C � . They also de�ne an OGLd(q)-

lattice CO(�) of CK(�). By [DF; (4.4)], the character of the GLd(q)-module CK(�)

equals �RGLd(q)T 0(q) (b�) where b� is a (regular) linear (K-valued) character on a torus

T 0(q) �= F�qd

of GLd(q).

Now �x s 2 Css. We can assume that

s = diag(s1; � � � ; s1;| {z }n1 times

� � � ; sm; � � � ; sm| {z }nm times

);

where the semisimple matrices s1; � � � ; sm have distinct irreducible characteristicpolynomials f1(X); � � � ; fm(X) 2 Fq [X]. For each i, let �i 2 �Fq be a �xed root offi(X) = 0 and form the representation

(9.6) CK(s) =

m(s)Oi=1

CK(�i)ni(s)

of the Levi subgroup Hs(q) of Ls(q). It has an OHs(q)-lattice

(9.7) CO(s) =

m(s)Oi=1

CO(�i)ni(s):

Since Deligne-Lusztig induction behaves well with respect to group products, the

Hs(q)-module CK(s) has character �RHs(q)Ts(q)

(�), where � =Nm(s)

i=1b�ni(s)i .

8In the notation of [DJ2], the root � is denoted s.


As already mentioned above, [FS; (1.16)] associates a linear character bs of Ts(q)to s. Tracing through the construction used in [FS; (1.16)], we �nd that bs can be

de�ned as a product of characters � =Nm(s)

i=1 bsni(s)i . Each character bsi is linearand regular on F�qai obtained as follows: All �nite �elds in sight are identi�ed in a

�xed way with a sub�eld of �Fq , and a generator is chosen for the multiplicative groupof each such sub�eld. Thus, all �nite �elds under consideration may be regardedas having a �xed generator. The linear character bsi is obtained by sending thegenerator of F�qai to si 2 �F�q and following this map by a chosen �xed injection�F�q � �Q�p1

�= C � (for some prime p1, denoted ` in [FS]). We now rede�ne CK(s) to

be compatible with [FS]'s choice of bs. That is, b�i = bsi for each i, and thus we take� = bs above. None of the properties quoted from [DJ2] depend on their speci�cindexing of the cuspidal representations CK(�).

9

Recall again that Harish-Chandra induction RLs(q)Hs(q)

agrees with the Deligne-

Lusztig induction functor on characters.

Lemma 9.8. For s 2 Css, let Ms;Ls(q);R = RLs(q)Hs(q)

CR(s), where R = K or O.Then

(9.8.1) EndRLs(q)(Ms;Ls(q);R)�=

m(s)Oi=1

H(Sni(s); R; qai(s)):

Furthermore, Ms;Ls(q);K has character of the form

(9.8.2) chMs;Ls(q);K =X

�`n(s)

c��s;�;Ls(q)

with all multiplicities c� 6= 0.

Proof. Let m = m(s). In case m = 1, (9.8.1) is noted below [DJ2; (2.17)]. Thecase m > 1 follows formally from that fact. The second assertion (9.8.2) followsfrom (9.5b). �

For a partition � ` m and a commutative Z-algebra R (with t 7! q), let

y� =Xw2W�

(�q)�`(w)�w 2 H(Sm; R; q)

as in (2) in the introduction to Part II. Any � ` n(s) de�nes an element

y� = y�(1) � � � y�(m(s)) 2m(s)Oi=1

H(Sni(s); R; qai(s)):

9Also, the meaning of the original CK (�) is changed in this new notation only by a new

embedding F�

qd,! C

� , which is a matter of choice in any case.


If N is a submodule of an O-module M , letpN be the smallest O-submodule of

M containing N such that M=pN is O-torsion-free. With this notation, form the

OLs(q)-module

(9.9) cMs;Ls(q);O =M

�`n(s)

qy�Ms;Ls(q);O:

For R = K or k, put cMs;Ls(q);R = cMs;Ls(q);OOR. If � = ((1n1(s)); � � � ; (1nm(s)(s))),

then y� = 1, sopy�Ms;Ls(q);O = Ms;Ls(q);O. Hence,

(9.10) chcMs;Ls(q);K =X

�`n(s)

c0��s;�;Ls(q); where all c0� 6= 0

by (9.8.2).

Lemma 9.11. For s 2 Css and cMs;Ls(q);O as in (9.9), we have an isomor-phism:

EndOLs(q)(cMs;Ls(q);O)

�=m(s)Oi=1

Sqai(s)(ni(s); ni(s))O:

Proof. This is proved in [DJ2; (2.24vi)] for each factor. �

The main goal of [FS] is to classify the p-blocks of G(q). These are indexedby pairs (s; �), with s 2 Css;p0 = Css

T Cp0 and � an e-core of a multi-partition in�+(ZG(q)(s)). (Here e = (e1; � � � ; em(s)), where ei is de�ned in terms of the orderof si. We do not require the precise de�nition.) We have the following result, whichfollows immediately from remarks above and [FS; (7A)] (see also [DJ2; x7]):

Lemma 9.12. For distinct elements s; s0 2 Css;p0 , the characters �s;�; �s0;�0belong to distinct blocks of KG(q) for any choice of multi-partitions � ` n(s) and�0 ` n(s0). Thus, the characters in R

G(q)Ls(q)

cMs;Ls(q);O and RG(q)Ls0 (q)

cMs0;Ls0 (q)belong

to distinct blocks.

For s 2 Css;p0 , de�ne Bs;G(q) (resp., Bs;Ls(q)) to be the sum of the blocks of OG(q)(resp., OLs(q)) which contain a character of the form �st;� (resp., �st;�;Ls(q)), wheret 2 ZG(q)(s) is a p-element.

Lemma 9.13. Harish-Chandra induction de�nes a full embedding

(9.13.1) RG(q)Ls(q)

: mod{Btorsion-frees;Ls(q)

,! mod{Btorsion-frees;G(q)

from the category of O-torsion-free Bs;Ls(q)-modules to the category of O-torsion-free Bs;G(q)-modules. The functor induces a character isometry, takes projectiveindecomposable modules to projective indecomposable modules, preserving distinctisomorphism types.


Proof. The fact that Harish-Chandra induction RG(q)Ls(q)

from characters inBs;Ls(q)

to characters in Bs;G(q) is an isometry follows from (9.5a). Now let M;N be O-torsion-free Bs;Ls(q)-modules. The Mackey decomposition theorem implies that

HomOG(q)(RG(q)Ls(q)

M;RG(q)Ls(q)

N) �= HomOLs(q)(M;N)M

X;

where X is a torsion-free O-module. Since RG(q)Ls(q)

is an isometry, we have X = 0.

Thus,

HomOG(q)(RG(q)Ls(q)

M;RG(q)Ls(q)

N) �= HomOLs(q)(M;N);

so that the functor in (9.13.1) is a full embedding. Obviously, this functor takesindecomposable modules to indecomposable modules, and it preserves distinct iso-morphism types. It remains to check that if Q is a projective OLs(q)-module, thenRG(q)Ls(q)

Q is a projective OG(q)-module. Let P (q) = Ls(q) n V (q) be the standard

parabolic subgroup associated to Ls(q), and let Q0 be the OP (q)-module obtainedfrom Q by in ating the action of Ls(q) on Q through the natural homomorphismP (q)! Ls(q). Since V (q) has order prime to p, jV (q)j is a unit in O, and a standardargument shows that Q0 is a projective OP (q)-module. Thus, RG(q)

Ls(q)Q = ind

G(q)P (q)Q

0

is a projective OG(q)-module, as required. �

Corollary 9.14. For s 2 Css;p0, de�ne cMs;G(q);O = RG(q)Ls(q)

cMs;Ls(q);O. Then

EndOG(cMs;G(q);O) �=m(s)Oi

Sqai(s)(ni(s); ni(s))O:

Proof. This follows from (9.11) and (9.13). �

Lemma 9.15. For s 2 Css;p0, the OG(q)-module cMs;G(q);O satis�es the hy-pothesis of (9.1).

Proof. By [DJ2; (3.7)], the OLs(q)-module cMs;Ls(q);O satis�es the hypothesisof (9.1), so there is an exact sequence

(9.15.1) 0! bNs;Ls(q);O ! bPs;Ls(q);O ! cMs;Ls(q);O ! 0

in which the irreducible characters in bNs;Ls(q);O all have the form �st;�;Ls(q), where

1 6= t 2 ZG(q)(s) is a p-element. But RG(q)Ls(q)

�st;�;Ls(q) = �st;� and RG(q)Ls(q)

�s;�;Ls(q) =

�s;� by (9.5a). Thus, we can apply Harish-Chandra induction to (9.15.1) to obtainthe desired conclusion. �

For each s 2 Css;p0 , let Js(q) be the annihilator in Bs;G(q) of cMs;G(q);O. De�neJ(q) =

Ps2Css;p0

Js(q). Since OG(q) =L

s2Css;p0Bs;G(q), we have:

(9.16) OG(q)=J(q) �=M

s2Css;p0

Bs;G(q)=Js(q):


Now we can establish the following fundamental result.

Theorem 9.17. Let k be an algebraically closed �eld of characteristic p >0 and let G(q) = GLn(q), where p does not divide q. The algebras OG(q) andOG(q)=J(q) (de�ned in (9.16)) have the same irreducible modules over k. There isa Morita equivalence

(9.17.1) F : mod{OG(q)=J(q) �!�

Morita

mod{M

s2Css;p0

m(s)Oi=1

Sqai(s)(ni(s); ni(s))O;

de�ned by F (�) = HomOG(q)=J(q)(L

s2Css;p0(cMs;G(q);O;�). The functor F induces

a Morita equivalence

(9.17.2) �F : mod{kG(q)=J(q)k �!�

Morita

mod{M

s2Css;p0

m(s)Oi=1

Sqai(s)(ni(s); ni(s))k

by putting �F (�) = HomkG(q)=J(q)k(L

s2Css;p0cMs;G(q);k;�).

Proof. For s 2 Css;p0 , cMs;G(q);O is a projective Bs;G(q)=Js(q){module by (9.15)and (9.1). Hence, it is a projective OG(q)=J(1)-module. It follows that

cM =M

s2Css;p0

cMs;G(q);O

is a projective OG(q)=J(q)-module.By [DJ2; p. 356], the irreducible kLs(q)-modules appearing in the head ofcMs;Ls(q);O are indexed by the set of multi-partitions � ` n(s). Hence, using

(9.13), we see that the head of cMs;G(q);O = RG(q)Ls(q)

cMs;Ls(q);O has as many non-

isomorphic irreducible modules as there are multi-partitions � ` n(s). On the

other hand, (9.12) implies that for distinct s; s0 2 Css;p0 , the modules cMs;G(q);O

and cMs0;G(q);O belong to distinct blocks. Since the number of irreducible kG(q)-modules equals the number of p0-conjugacy classes in G(q), namely, the cardinality

of the set f(s; �) j s 2 Css;p0 ; � ` n(s)g, we see that cMs;G(q);O has all the irreduciblekG(q)-modules of Bs;G(q) in its head. Thus, OG(q)=J(q) and OG(q) have the sameirreducible modules over k.

We can apply (9.2) and (9.14) to conclude that the O-algebras Bs;G(q)=Js(q) and

EndBs;G(q)(cMs;G(q);O) �= EndOG(q)(cMs;G(q);O)

�=m(s)Oi

Sqai(s)(ni(s); ni(s))O

are Morita equivalent by the functor Fs(�) = HomOG(q)(cMs;G(q);O;�). Therefore,F (�) = HomOG(q)(cM;�) de�nes a Morita equivalence as required in (9.17.1).

Finally, base change de�nes the Morita equivalence �F in (9.17.2). �


As explained in x8, the irreducible modules for Sq(n;m) = St(n;m)k are natu-rally indexed by the poset �+(n;m) of partitions of m into at most n parts. For� 2 �+(n;m), we let Lq(�) or sometimes L

k(�) denote the corresponding irre-ducible Sq(n;m)-module. Thus, we can use (9.17) to obtain a natural indexingof the irreducible kG(q)-modules. Explicitly, given s 2 Css;p0 and � ` n(s), letD(s; �) 2 Irr(kG(q)) correspond, under the Morita equivalence �F to the irreducible

moduleNm(s)

i=1 Lqai(s)(�(i)) 2 Irr

�Nm(s)i=1 Sqai(s)(ni(s); ni(s))

�.

Remarks 9.18. (a) The category mod-Sq(n;m)k can be regarded as fully embed-ded in the module category for various versions of the quantum general linear groupor of a quantum enveloping algebra (see (e) below). For example, let GLn;q(k) bethe Manin quantum general linear group over k.10 As explained in [PW; x10],mod-Sq(n;m)k (or Sq(n;m)k-mod) identi�es with the full subcategory of rationalGLn;q(k)-modules which are homogeneous of degree m. We can assume that Lq(�),when regarded as a GLn;q(k)-module, is the irreducible module of highest weight� (in the usual Lie-theoretic sense).

(b) The algebra Sq(n; n)k has a unique one-dimensional representation Lq(�),obtained by taking � = (1n). This fact follows immediately from the connectiondescribed above with the representation theory of GLn;q(k) and the fact that theweights in any irreducible GLn;q(k)-module are stable under the Weyl group Sn

[PW; (8.8.2)]. In fact, Lq((1n)) corresponds precisely to the one-dimensional mod-

ule on GLn;q(k) de�ned by the quantum determinant detq. For this reason, wedenote the Sq(n; n)k-module Lq((1

n)) by simply detq. The quantum determinantis de�ned for the algebra St(n; n) over Z and is the unique group-like element dettin the coalgebra At(n; n) = HomZ(St(n; n);Z).

We give another description of dett. If ; 6= � � S, then HomH(x�H; SGN) = 0.Let V be a free Z-module of rank n (with �xed ordered basis). It follows from (8.4)with m = n that there is an isomorphism

HomH(Vn; SGN) �= HomH(x;H; SGN) �= Z

of Z-modules because in the decomposition (8.4) H �= x;H appears with multiplic-ity 1. Thus,

(9.18.1) dett �= HomH(Vm; SGN):

(c) Consider the trivial module k for kG(q). In the notation above, it has theform D(1; �) for some partition � ` n. We claim that � = (1n). To see this, itsu�ces from (b) to show, in the language of (9.17.2), that

dim �F (k) = dim HomkG(q)(cM1;G(q);k; k)

10Alternatively, one could use the quantum general linear group GL0n;q(k) studied in [DD].

Both quantum general linear groups lead to the same q-Schur algebra [DPW]. The action of H onV m used in [PW; (11.3c)] can be replaced by the equivalent action described in (8.1) above, sothat the quantum group GLn;q(k) can be used, rather than GL

n;q1=2(k).


is equal to 1. This is easily done by reducing to a corresponding problem over K.Here are the details. We have

cM1;G(q);k =

M�`n

qy�M1;G(q);O

!k

;

where M1;G(q);O = indG(q)B(q)O. Since (

py�M1;G(q);O)k = (ind

G(q)B(q)O)k �= ind

G(q)B(q)k

for � = (1n), the equality dim HomkG(q)(indG(q)B(q)k; k) = 1 shows that it su�ces to

check that

(9.18.2) HomkG(q)

��qy�ind

G(q)B(q)O

�k

; k

�= 0 for � 6= (1n):

Let M = indG(q)B(q)O. Because

py�M is a projective OG(q)=J(q)-module (as a

direct summand of the projective OG(q)=J(q)-module cM1;G(q);O) and because O isan OG(q)=J(q)-module, we have

HomOG(q)(py�M;O) �= HomOG(q)=J(q)(

py�M;O)k

�= HomkG(q)=J(q)k((py�M)k; k)

�= HomkG(q)((py�M)k; k):

Therefore, it su�ces to show that HomOG(q)(py�M;O) = 0 for � 6= (1n), or

equivalently, we must prove that

HomKG(q)((py�M)K ; K) �= HomKG(q)(y�ind

G(q)B(q)K;K) = 0

for � 6= (1n). If pn(q) =P

w2Snq`(w), then (

Pw2Sn

�w)2 = pn(q)

Pw2Sn

�w.Thus,

e = pn(q)�1

Xw2Sn

�w 2 EndKG(q)(indG(q)B(q)K) �= H(Sn; q)K ;

is the idempotent projection from indG(q)B(q)K onto its unique constituent isomorphic

to the trivial module. Obviously ey� = 0 for � 6= (1n). This proves our claim.It also follows that the trivial OG(q)-module O (which is annihilated by J(q))

corresponds under the Morita equivalence (9.17.1) to the quantum determinantrepresentation detq for Sq(n; n)O.

(d) The algebra on the right hand side of (9.17.2) has the following property:each of its irreducible modules can be written in the Grothendieck group as uniqueZ-linear combintation of the reductions \mod p" (i. e., tensoring with k) of (latticesof) irreducible modules over K of the algebra on the right hand side of (9.17.1).Therefore, the same result holds (with the same formulas) for the algebras on theleft hand side of (9.17.2) and (9.17.1). In this way, one obtains formulas in terms ofordinary characters for all the irreducible Brauer characters for kG(q). We regardthe problem of determining these Brauer character formulas as more important than


the dual problem of determining the decomposition numbers of ordinary characters.Theoretically, the latter problem can be solved from the former through charactervalues and linear algebra. In this way, the statement of Theorem 9.17 generalizesthe original decomposition number results of Dipper-James [DJ2] (though we usetheir results in deriving (9.17)). Finally, it follows from the results in [DJ2] and[D1], comparing decomposition numbers for kG(q) with those for qa-Schur algebras,that the indexing above for Irr(kG(q)) is compatible with that taken by Dipper andJames. However, we will not need to use this result in the sequel.

(e) Let eUq(gln) be the divided power Z[q; q�1]-form of the quantized envelopingalgebra of the Lie algebra gln(C ) of n � n matrices over C . For any �eld k andany specialization Z[q; q�1]! k, the corresponding q-Schur algebra Sq(n;m)k is a

homomorphic image of eUq(gln(C )) [Du]. Therefore, we could also index the irre-

ducible kG(q)-modules by using the quantum enveloping algebra eUq(gln(C )k . Thiswould lead to the same indexing by highest weights as above. More importantly,it provides, through the results of x2, together with the work [KL2] a connection

between the representation theory of G(q) and that of the a�ne Lie algebra bgln(C ).x10. H1-cohomology in non-describing characteristic

In this section, we show how the Morita equivalence (9.17) can be e�ectivelyapplied to study H1(GLn(q); V ), when V is an irreducible kGLn(q)-module and k isan algebraically closed �eld of characteristic p not dividing q. Such H1-cohomologyis important for the theory of maximal subgroups of �nite groups. As explained in[AS], the two basic ingredients needed to understand all maximal subgroups of agiven �nite group G, modulo easier or smaller problems, are:

(1) knowledge of the maximal subgroups of quasi-simple groups; and

(2) determination of the cohomology groups H1(G; V ) for quasi-simple groupsG and irreducible G-modules V .

We will show immediately below, that when p is prime to q(q � 1), the answerto (2) above for G = GLn(q) is given completely in terms of a corresponding Ext 1-calculation for q-Schur algebras Sq(n; n)k. As already remarked in the introduction,one can then expect the answers to be eventually known (via the theory of a�nealgebras) in terms of Kazhdan-Lusztig polynomials if p is large (allowing q to vary)using the generic results of xx2,8, together with [KL2].

We will use the description of the irreducible kG(q)-modules D(s; �) given in(9.17). Recall that s belongs to the set Css;p0 of representatives of the set ofsemisimple p0-conjugacy classes of GLn(q), for a prime p not dividing q, while� is a certain multipartition. We now prove the following result, which includes theH1-calculations alluded to above.

Theorem 10.1. Let G(q) = GLn(q). Assume that the characteristic p of the�eld k is relatively prime to q(q � 1). For any kG(q)=J(q)k-module V , we have

(10.1.1) H1(G(q); V ) �= Ext 1Sq(n;n)k(detq;�F (V ));


where �F (V ) is the image of V under the Morita equivalence �F de�ned in (9.17.2).Also, there is an injection

(10.1.2) Ext 2Sq(n;n)k(detq;�F (V )) ,! Ext 2G(q)(k; V )

�= H2(G(q); V ):

In particular, for any s 2 Css;p0 and any multi-partition � ` n(s), we have

(10.1.3) H1(G(q); D(s; �)) �=�Ext 1Sq(n;n)k(detq; L

k(�)); s = 1

0; s 6= 1:

Proof. Form the short exact sequence

(10.1.4) 0! U ! indG(q)B(q)k ! k! 0

of kG(q)-modules (where k denotes the trivial module). Since kjG(q)B(q) is a kG(q)-

direct summand of the module cM1;G(q);k de�ned in (9.9), the de�nition of J(q)kimplies that (10.1.4) is actually an exact sequence of kG(q)=J(q)k-modules in which

indG(q)B(q)k is a projective kG(q)=J(q)k-module. By hypothesis, p is relatively prime

to q(q � 1), so that k is a projective kB(q)-module, and hence indG(q)B(q)k is also a

projective kG(q)-module. Hence, writing M = kjG(q)B(q), the long exact sequence of

cohomology gives a commutative diagram

HomkG(q)=J(q)k(M;V )!HomkG(q)=J(q)k(U; V )!Ext 1kG(q)=J(q)k(k; V )! 0

HomG(q)(M;V )?

! HomG(q)(U; V )?

! Ext 1G(q)(k; V )

?! 0

in which the rows are exact and the two left vertical arrows are isomorphisms. Thisgives

H1(G(q); V ) �= Ext 1kG(q)(k; V )�= Ext 1kG(q)=J(q)k(k; V ):

Now (9.17) implies that

Ext 1kG(q)=J(q)k(k; V )�= Ext 1Sq(n;n)k(detq;

�F (V ));

using the fact proved in (9.18(c)) that �F (k) = �F (D(1; (1n)) �= detq. This proves(10.1.1).

Next, by general elementary principles, we have

Ext 1kG(q)=J(q)k(U; V ) � Ext 1kG(q)(U; V ):

By dimension shifting,

Ext 1kG(q)=J(q)k(U; V )�= Ext 2kG(q)=J(q)k(k; V ):


Another application of (9.17) yields (10.1.3).Finally, (10.1.3) is a special case of (10.1.1), taking V = D(s; �) and observing

that if s 6= 1, then D(s; �) and D(1; (1n)) belong to di�erent blocks. �

Now we prove the following stability result for H1 mentioned in the introduction.

Theorem 10.2. (H1-stability) Let n be a �xed positive integer and k an alge-braically closed �eld of characteristic p. Let q be a prime power not divisible by p,and consider the group GLn(q). For s 2 Css;p0 and � ` n(s), let D(s; �) denote theassociated irreducible kGLn(q)-module. Then:

(a) If 1 6= s 2 Css;p0 , then H1(GLn(q); D(s; �)) = 0.

(b) There exists an integer N(n), depending on n, such that if p > N(n), thendim H1(GLn(q); D(1; �)) depends only on � and the order ` of q modulo p.

Proof. First, (a) is clear, since the modules D(s; �) with s 6= 1 are not even inthe principal p-block.

To prove (b), we may �x the order ` of q modulo p to a �xed value � n, sincethese are the only cases for which p divides jGLn(q)j. Also, we can assume thatN(n) > n.

Case 1: ` > 1. In other words, p does not divide q � 1, so that (10.1) impliesthat

(10.2.1) dim H1(G(q); D(1; �)) = dim Ext 1Sq(n;n)k(detq; Lk(�)):

Consider the ring Z(`) of algebraic integers in K = Q(p1) as in (8.6). By (8.6),

there exists a non-empty open subset of Spec Z(`) such that if p 2 , then (1)each Lk(�), � 2 �+(n;m), is obtained by \reduction mod p" from a lattice forLK(�), and (2) there is an equality

(10.2.2) dim Ext 1Sq(n;n)k(p)(detq; Lk(�)) = dim Ext 1Sq(n;n)K (detq; L

K(�):

Thus,

(10.2.3)dim Ext 1Sq(n;n)k(detq; L

k(�)) = dim Ext 1Sq(n;n)k(p)(detq; Lk(�))

= dim Ext 1Sq(n;n)K (detq; LK(�);

provided that k(p) � k, which means precisely that p lies over (p), where now p isthe characteristic of k.

Since Z(`) is �nite over Z, for any prime integer p 2 Z, there exist only �nitelymany prime ideals p 2 Spec Z(`) satisfying pTZ = (p). Thus, we can assume thatN(n) is su�ciently large that if p > N(m) and p 2 Spec Z lies over (p), then p 2 .The theorem is proved in this case.

Case 2: ` = 1. For any subset � � S, let P�(q) the parabolic subgroup of G(q)whose Levi factor L�(q) has simple roots �. Then the dimension of H1(P�(q); k) �=Hom(P�(q)=P�(q)

0; k) equals the rank of the center of the Levi factor L� (sincep > n), and so depends only on �.


Now let O; K; k be as in x9. In particular, K is a splitting �eld for G(q), sothat KG(q)=J(q)K is a split semisimple algebra. Since kG(q)=J(q)k is also asplit semisimple algebra, we conclude that any D(1; �) lifts to an irreducible or-dinary unipotent representation. Also, [CR; (68.24)] implies that the ordinaryirreducible characters are Z-linear combinations of the characters of the transi-tive permutation modules C jG(q)P� (q)

, � � S. Hence, in the Grothendieck group

of kG(q)=J(q)k, any D(1; �) is a Z-linear combination of various kjG(q)P�(q). So,

by the Shapiro-Eckmann lemma, it su�ces to prove that the dimensions of the

H1(G(q); kjG(q)P�(q)) �= H1(P�(q); k) for � � S stabilize if p is a su�ciently large

prime (and p still divides q � 1). But for p > n, this is clear from the previousparagraph. �

Remarks 10.3. (a) The cohomology of the q-Schur algebras Sq(n;m)k can be

expressed in terms of the cohomology of the quantum enveloping algebra eUq(gln(k))because of the natural surjection eUq(gln(k)) ! Sq(n;m)k of algebras. Then wehave:

(10.3.1) Ext �Sq(n;m)k(V;W ) �= Ext �eUq(gln(k))(V;W );

for any Sq(n;m)k-modules V;W . In (10.3.1), on the right hand side, V;W are

regarded as eUq(gln(k))-modules by means of the above surjection, and the Ext �-group is computed in the category of integrable modules. For a proof of this fact, see[DS]. Variations on (10.3.1), involving the various quantum general linear groups,are also established in [Do2] and [PW].

(b) Using the generic result (8.6) as in Case 1 of the proof of (10.2), the dimen-sion of Ext nSq(n;n)k(detq; L

k(�)) can be equated, when p is su�ciently large, with

Ext nSq(n;n)K (detq; LK(�)) over a �eld K of characteristic 0 in which q is regarded as

a suitable root of unity. In turn, using the category equivalence given in [KL2], thislater dimension can be expressed as the dimension of an Ext n-group for the a�ne

algebra bgln(C ). It seems likely, as with the case of the category O for a complexsemisimple Lie algebra (see [CPS4; (3.8.1)]), that these dimensions can be given interms of values of certain Kazhdan-Lusztig polynomials. We hope to consider thisquestion in a later paper.

We next consider the relationship between the H1-cohomology of GLn(q) andthat of SLn(q). By Cli�ord theory, the restriction from GLn(q) to SLn(q) of anirreducible module is always completely irreducible, isomorphic to a sum of con-jugates of a single irreducible SLn(q)-module. In fact, all unipotent ordinary ormodular irreducible modules over an algebraically closed �eld restrict irreducibly.This may be argued as follows: The number of double cosets of a Borel subgroupdoes not change when passing from GLn(q) to SLn(q), so the ordinary unipotentcharacters, which are precisely the constituents of the associated permutation char-

acter of KjG(q)B(q) for G = GLn or SLn, must remain irreducible. Consequently,

EndOGLn(q)(cM1;G(q);O) �= EndOSLn(q)(

cM1;G(q);O):


So indecomposable GLn(q)-components of cM1;G(q);O remain indecomposable uponrestriction of SLn(q). In addition, they are projective for a quotient of the groupalgebra, just as in x9. Thus they have simple heads, and, since the restrictions oftheir heads are at least completely reducible, these restrictions must, in fact, beirreducible.

Clearly, non-isomorphic irreducible unipotent GLn(q)-modules restrict to non-isomorphic SLn(q)-modules. (It is possible for a non-unipotent character andunipotent character to have the same restriction, but this happens, by Frobeniusreciprocity, only for a character which is the product of a unipotent character witha linear character of GLn(q)=SLn(q). These non-unipotent irreducible modules,in the modular case, have the form D(s; �) where 1 6= s 2 Css;p0 is central.) Welet D(1; �) also denote the restriction of the irreducible GLn(q)-module D(1; �) toSLn(q).

De�nition 10.4. The modular representations of SLn(q) having compositionfactors of the form D(1; �) will be called the unipotent modular representations ofSLn(q). A similar terminology will be used for ordinary irreducible characters. Theassociated blocks of SLn(q) will be called unipotent blocks.

One corollary of the irreducibility is that, for unipotent blocks, there is a Moritaequivalence like that of (9.17) for SLn(q). In fact, the corresponding factor algebrasfor the group algebras over GLn(q) and SLn(q) are even isomorphic. Moreover, theprevious two theorems essentially hold in the SLn case. In more detail, we have

Theorem 10.5. Let L be an irreducible module for SLn(q) over the alge-braically closed �eld k of characteristic p which does not divide q. If L is notunipotent, then H�(SLn(q); L) = 0. If L = D(1; �) is unipotent, then:

H1(SLn(q); L) �= H1(GLn(q); D(1; �)):

Finally, if p divides q � 1, dim H1(SLn(q); L) depends only on � when p � n:

The proof is very similar to the GLn-proofs of the previous two theorems, withminor adjustments in the case where p divides q� 1. Further details are left to thereader. We note that, as a corollary, the stability Theorem 10.2 also holds for SLn.

Remark 10.6. Let G(q) = GLn(q), as above, but assume that q � 1 mod p, sothat the hypotheses of (10.1) fail. However, in case p > n (i. e., p does not divide theorder of the Weyl group of GLn), it is possible to use the methods of [CPS1; x6] toobtain some information concerning H�(G(q); V ) (and, in particular, H1(G(q); V ))for any kG(q)-module V . Namely, let T (resp., B) be the maximal torus (resp.,Borel subgroup) of GLn consisting of diagonal (resp., upper triangular) matrices.By (9.3), p does not divide [G(q) : B(q)]. As shown in [CPS1; x6], there is a rightaction of the Hecke algebra Hk = H(Sn; k; 1) on Hn(B(q); V ) given by

(10.6.1) ��w = �wjB(q)wTB(q)jB(q); w 2 Sn; � 2 Hn(B(q); V ):

Here jB(q) is corestriction (or transfer). But observe that Hk�= kSn, the group

algebra of Sn. Since T (q) contains a Sylow p-subgroup of G(q), the restriction


map H�(G(q); V ) ! H�(B(q); V ) is injective, and the main result in [CPS1; x6]establishes that the image of H�(G(q); V ) in H�(B(q); V ) identi�es with the spaceof �xed points, i. e.,

(10.6.2) H�(G(q); V ) �= H�(B(q); V )Sn :

The equality (10.6.2) provides a more conceptual reformulation of the classicalCartan-Eilenberg stability theorem for cohomology, cf. [CE; p.258]. Since B(q) =T (q)n U(q), where U is the unipotent radical of B, and p does not divide jU(q)j,an elementary spectral sequence argument shows that there is a natural identi�ca-tion H�(B(q); V ) �= H�(T (q); V U(q)). Thus, if we transfer the action of Sn fromH�(B(q); V ) to an action on H�(T (q); V U(q)), (10.6.2) can be rewritten in the form

(10.6.3) H�(G(q); V ) �= H�(T (q); V U(q))Sn :

Because the set S of simple re ections (i; i + 1) generates Sn, an element � 2H�(T (q); V U(q)) is Sn-�xed if it is �xed by every simple re ection s = (i; i + 1).This holds if the image of � in H�(T (q); V U(q)\U(q)s) is �xed by the usual actionof s. This condition (for all s) is necessary as well as su�cient for � to correspondto an element of H�(G(q); V ). See [CPS1; (6.1)] for further discussion.

x11. ResolutionsIn the previous section, we used the Morita equivalence proved in (9.17) to

relate the H1-cohomology of GLn(q) at modules in non-describing characteristicto Ext 1-calculations for q-Schur algebras. By using results from Part I, we wereled to the H1-stability Theorem 10.2. We wish to extend this work to include thehigher cohomology groups Hi(GLn(q); V ), i > 1. Section 12 below presents ourresults in that direction. The present section lays the foundation for that work.Our �rst main result, given in Theorem 11.10, develops an interesting complex of\tilting modules" for endomorphism algebras A associated to general �nite Coxetersystems (W;S).11 The complex itself comes about by \dualizing" the Deodharcomplex [De] for generic Hecke algebras. The proof of (11.10) makes essential useof certain aspects of the theory of Kazhdan-Lusztig cells; these facts are reviewedbelow. Next, we specialize to the special case in which W = Sm is a symmetricgroup. Our �nal result, Theorem 11.15, gives a very speci�c projective resolutionof the quantum determinant representation dett (see (9.18)) for the t-Schur algebraSt(m;m) over Z = Z[t; t�1]. In order to pass from the complex in (11.10) to thatin (11.15), we must use the theory of tilting modules for St(m;m) developed in[DPS3].

We will use the notation introduced in the preamble to Part II. Thus, let (W;S)be a �nite Coxeter system, and let H be the Hecke algebra over the ring Z =Z[t; t�1] of Laurent polynomials with basis f�wgw2W . Write m = jSj for the rankof W .

11A study of the homological properties of such endomorphism algebras A forms a centraltheme in previous papers [CPS6], [DPS1], [DPS2], [DPS3].


For an integer i � 0, form the left H-module

(11.1) Ni =M

��S;j�j=i

Hx�;

where x� is de�ned as in (2) (in the preamble to Part II). Observe that N0 = H,while Nm = Z with H acting through the index map IND : H ! Z, �w 7! t`(w).

If � � � � S, there is a unique H-module mapping Hx��;��! Hx� satisfying

��;�(x�) = x�.Fix a linear ordering < on the set S. If � � � and � n � = fs0g for some s0 2 S,

de�ne

(11.2) �(�; �) = (�1)jfs2� j s<s0gj:

Then de�ne the H-module homomorphism di : Ni �! Ni+1 by setting for any� � S satisfying j�j = i:

(11.3) dijHx� =X

��;j�j=i+1

�(�; �)��;�:

The following result summarizes the key property of the resulting complex.

Lemma 11.4. (Deodhar [De]) The above de�nitions de�ne a complex

(11.4.1) (N�; d�) : 0! N0d0�! N1

d1�! � � � dm�1�! Nm ! 0

of left H-modules which has cohomology groups satisfying

(11.4.2) Hi(N�) =

�SGN; i = 0

0; i > 0:

Actually, [De] proves this lemma for the complex N 0� = N�ZZ[t1=2; t�1=2] of left

modules for the Hecke algebra H0 = H Z Z[t; t�1]. However, since Z[t1=2; t�1=2]is a free Z-module, the validity of the lemma for N 0

� implies its truth for N�.12

There is a Z-automorphism � : H ! H given on generators by �(�w) =(�t)`(w)��1

w�1 . If M is a (right or left) H-module, let M� denote the H-module

obtained by twisting the action of H on M by �. For example, (Hx�)� = Hy�

for all � � S [DPS3; (1.4c)]. Since SGN� �= IND, the complex N�� has terms

N�i =

L��S;j�j=iHy� and cohomology which vanishes except in degree 0, where

it identi�es with IND.Recall the Z[t1=2; t�1=2]-bases

fCwgw2W and fC 0wgw2W12The paper [M1] by Mathas takes note of Deodhar's paper, but could be read as suggesting

that Deodhar obtains an exact complex only for in�nite Coxeter groups. That is not the case.We wish to make it clear that the exact complex is due to Deodhar for both the �nite and in�niteCoxeter groups.


for H0 = H Z Z[t1=2; t�1=2] de�ned in [KL1]. Explicitly,

Cw = (�t1=2)`(w)Xy�w

(�t)�`(y) �Py;w�y and C 0w = t�`(w)=2Xy�w

Py;w�y

where Py;w 2 Z is the Kazhdan-Lusztig polynomial corresponding to the pair(y; w) 2 W � W and �Py;w denotes the image of Py;w under the automorphismof Z satisfying t 7! t�1.

We put C+w = t`(w)=2C 0w and C�w = (�t1=2)�`(w)Cw for w 2 W .13 Then

fC�wgw2W , � = �, forms a Z-basis for H which satis�es the following multiplicative

rules (w 2W; s 2 S):

(11.4.3a) �sC�w =

( �C�w ; sw < w

tC�w � tC�sw � tP

z�w;sz<z �(z; w)t`(z)�`(w)+1

2 C�z ; sw > w:

and

(11.4.3b) �sC+w =

(tC+

w ; sw < w

�C+w + C+

sw +P

z�w;sz<z �(z; w)t`(w)�`(z)�1

2 C+z ; sw > w:

In both cases, z � w indicates that Pz;w has degree (`(w) � `(z) � 1)=2. When

z � w, �(z; w) denotes the coe�cient of t(`(w)�`(z)�1)=2 in Pz;w. There are com-pletely analogous formulas to both (11.4.3a,b) for computing C�w �s and C

+w �s. The

following lemma was essentially observed in [M2].

Lemma 11.5. For � � S, let w� be the long word in the parabolic subgroupW�. Let

�W be the set of distinguished left coset representatives of W� in W (i. e.,w 2 �W if and only if ws > w for all s 2 �). Then:

(a) x� = C+w�

and y� = C�w� .

(b) For � � S, write C�� = C�w� and C+� = C+

w�. The set fC�x C+

� gx2�W forms

a Z-basis for Hx� and the set fC+x C

�� gx2�W forms a Z-basis for Hy�.

(c) If x 62 �W , then C�x C+� = 0 = C+

x C�� .

Proof. (a) follows from the de�nition of Cw� and C 0w� given earlier, togetherwith the fact that Py;w� = 1 for any y � w� [KL1; (2.6vi)]. Next, we prove (c).Suppose that x 62 �W , so that there exists s 2 � with xs < x. By (11.4.3a,b) (andtheir right-hand versions), we obtain

�C�x C+� = C�x �sC

+� = tC�x C

+� ;

so C�x C+� = 0; as required. This proves the �rst equality in (c), while the second is

similar. Now since fC�x gx2W is a Z-basis for H and Hx� is Z-free of rank j�W j,the �rst assertion in (b) follows. The second assertion in (b) is similar. �

13Here we follow [DPS1], except that C�w may di�er by a sign from the de�nition given in[DPS1].


Given a Z-module M , write M� = HomZ(M;Z) for the corresponding Z-lineardual. Then Z-duality interchanges left and right H-modules, e. g., if M is a leftH-module, M� is a right H-module. It is well-known that Hx� �= (x�H)� andHy� �= (y�H)� for any � � S. Also, SGN� �= SGN and IND� �= IND, if welet SGN and IND denote the left/right H-modules de�ned by the sign and indexhomomorphisms. Since the Z-modules Ni and N�

i are Z-free, we can dualize thecomplexes (N�; d�) and (N

�� ; d�) to obtain complexes (N

�� ; d

��) and (N

�� ; d��) which

have sole non-vanishing homology groups SGN and IND, respectively, in degree 0.For each � � S, let r� be some �xed non-negative integer, and de�ne

(11.6) T =M��S

x�H�r� ; and A = EndH(T ):

We will assume that, given any � � S, the parabolic subgroup W� is W -conjugateto some W� in which r� > 0.14 We view T as a (A;H)-bimodule. Let (�)� denoteeither of the two contravariant functors

(11.7)

�(�)� = HomH(�; T ) : mod{H �! A{mod

(�)� = HomA(�; T ) : A-mod �! mod{H:

We will consider the complex

(11.8) (X�; @�) = (N�� ; d�� ):

Thus, its term in degree i is given as

(11.9) Xi = HomH(M

��S;j�j=i

y�H;T );

while the di�erential Xi@i�! Xi+1 is the map f 7! f � d�i , f 2 HomH(N

��i ; T ).

Then (X�; @�) is a complex of left A-modules.The next theorem determines the cohomology of the complex (11.8) of left A-

modules.

Theorem 11.10. Let (W;S) be an arbitrary �nite Coxeter system, and con-sider the corresponding complex (X�; @�) de�ned in (11.8){(11.9) above. It hascohomology groups as follows:

(11.10.1) Hi(X�) �=�IND� = HomH(IND; T ); i = 0

0; i > 0:

Proof. If M;N are right H-modules which are Z-free, we have HomH(M;N) �=HomH(N

�;M�). Also, M�� =M . Thus, we can rewrite (11.9) as:

Xi = HomH(N��i ; T ) = HomH(T

��; Ni)

=M��S

HomH(Hy�; Ni)�r� :

14Recall that, given subsets �; � � S, the right H-modules x�H and x�H are isomorphicprovided that the parabolic subgroups W� and W� are W -conjugate. See [DJ1; (4.3)]. Thus, ourassumption guarantees that the algebra A de�ned in (11.6) is Morita equivalent to the algebraEndH(

L��S x�H).


Thus, it is enough to show that, for a �xed � � S, the cohomology of the complex

0! HomH(Hy�; N0)! HomH(Hy�; N1)! � � � ! HomH(Hy�; Nm)! 0

(obtained by applying the functor HomH(Hy�;�) to the complex N�) vanishes inpositive degree, and equals HomH(Hy�; SGN) in degree 0. The assertion about itscohomology in degree 0 follows from the left exactness of HomH(Hy�;�), togetherwith (11.4). So, �x i > 0 and let Ki = Ker di. By the long exact sequence ofcohomology, it su�ces to check that the natural map

(11.10.2) Ext 1H(Hy�; Ki) �! Ext 1H(Hy�;M

��S;j�j=i

Hx�) = Ext 1H(Hy�; Ni)

is an injection. The Z-module Ext 1H(Hy�; Ki) is torsion (since H Q(t) is asemisimple algebra), so let 0 6= � 2 Z be in the annihilator of Ext 1H(Hy�; Ki). IfMis a Z-module, we will denote M=�M by �M below. Multiplication by � determines

short exact sequences 0 ! Ki�! Ki ! �Ki ! 0 and 0 ! Ni

�! Ni ! �Ni ! 0.(Recall Ni =

L��S;j�j=iHx� � Ki.) Since the Ni are Z-torsion free, and Ni=Ki

�=Ki+1 by (11.4), multiplication by � de�nes an injection Ni=Ki ! Ni=Ki. Hence,by the snake lemma, �Ki is a submodule of �Ni. Also, we obtain a commutativediagram

(11.10.3)

HomH(Hy�; Ki)g- HomH(Hy�; �Ki) - Ext 1H(Hy�; Ki) - 0

HomH(Hy�; Ni)

c

? e- HomH(Hy�; �Ni)

b

?- Ext 1H(Hy�; Ni)

a

?

with exact rows. In particular, the top row of (11.10.3) is exact because the element� annihilates the torsion module Ext 1H(Hy�; Ki).

Now any H-module morphism � : Hy� ! Ni =L

�Hx� is completely deter-mined by the image �(y�). In turn, �(y�) can be any vector � 2 Ni =

L�Hx�

which satis�es the condition

(11.10.4) �s� = �� for all s 2 �:

By (11.5b), Ni has a basis consisting of the products C�x C

+� , j�j = i, x 2 �W .

We claim that (11.10.4) holds if and only if � is a Z-linear combination of termsC�x C

+� , for j�j = i, x 2 �W , and � contained in the left-set L(x) = fs 2 S j sx <

xg of x. In fact, such a linear combination � does satisfy (11.10.4) by (11.4.3a).Conversely, suppose � satis�es (11.10.4). We can assume that, when expressed asa linear combination � =

Pax;�C

�x C

+� (where j�j = i and x 2 �W ) that ax;� = 0

if � � L(x). Among the x; � with ax;� 6= 0, choose x0; � so that x0 has maximallength. For s 2 � and s 62 L(x), (11.4.3a) and (11.5) imply, using �s� = ��, thattax0;� = �ax0;�, a contradiction. This establishes our claim.


Let �� : Hy� ! �Ni be the image of � under the map e : HomH(Hy�; Ni) !HomH(Hy�; �Ni) above. The previous paragraph establishes that ��(y�) lies in the Z-span of the images under the maps Ni ! �Ni of the C

�x C

+� for j�j = i and � � L(x).

Assume that ��(y�) 2 �Ki. For a given x 2 �W , let Ki(x) be the Z-submodule ofKi consisting of all terms which are expressible as a Z-linear combination of termsC�x C

+� , j�j = i. Then Ki =

LxKi(x) by (11.5) and the de�nition of the complex

N�. Also, �Ki =L

xKi(x). Again using (11.5), together with the above discussionof �(y�), this implies that ��(y�) 2 �Ki lifts to an element ! 2 Ki satisfying �s! = �!for all s 2 �. Therefore, ! = f(y�) for an H-module morphism f : Hy� ! Ki, andwe note that c(f) and � have the same image in HomH(Hy�; �Ni). Next, observethat the map b above is an injection. Finally, an elementary application of thesnake lemma to (11.10.3), or a direct diagram chase, shows that the map a is aninjection, i. e., the map (11.10.2) is an injection, as required. �

For the remainder of this section, we assume that (W;S) = (Sm; S), whereS = f(1; 2); � � � ; (m � 1;m)g. In particular, this means, contrary to the notationabove, that W has rank m � 1. We choose the integers r� so that T �= V m in(8.4) with n = m. Thus, in the notation of (11.6), the endomorphism algebra Aidenti�es with the t-Schur algebra St(m;m) de�ned in (8.2).

De�ne

(11.11) Y = HomH(M��S

y�H�r� ; T ); and E = EndA(Y ):

View Y as a left E-module and consider the contravariant functor

(11.12) G = HomA(�; Y ) : A{mod �! E{mod

We want to apply the functor G to the complex (11.8) of left A-modules. In orderto establish the next result concerning the behavior of the complex G(X�), we needto make use of the theory of tilting modules for the quasi-hereditary algebras Ak

for �elds k. The original reference for tilting module theory for quasi-hereditaryalgebras is [Ri], and many further results have been obtained by Donkin in thecontext of algebraic groups (especially GLn) [Do1]. However, [DPS3; x4] containsa brief summary of everything that is required. In addition, our proofs use someof the main results in [DPS3] which deal with tilting module theory for q-Schuralgebras over Z.

Lemma 11.13. The functor G carries the exact sequence

(11.13.1) 0! IND� ! X0 ! � � � ! Xm�1 ! 0

of left A-modules (guaranteed by (11.10)) to an exact sequence

(11.13.2) 0! G(Xm�1)! � � � ! G(X0)! G(IND�)! 0

of left E-modules.


Proof. Let k be a �eld which is a Z-algebra. By [DPS3; (7.4a)], the Ak-module(y�H)�k = HomH(y�H;T )k is a tilting module in the highest weight category Ak{mod for any subset � � S.15 This means that (y�H)�k has two �ltrations, onehaving sections which are standard modules �k(�) and the other having sectionswhich are costandard modules rk(�). By [CPS2], we therefore have:

(11.13.3) Ext iAk((y�H)�k; Yk) = 0; 8i > 0; 8� � S:

Suppose B is an arbitrary Z-algebra which is �nitely generated and projective asa Z-module, and letM;N be B-modules which are �nitely generated and projectiveover Z. It is proved in [DPS3; (4.4)] that if Ext iB(M;N) 6= 0 for some positiveinteger i, then there exists a �eld k which is a Z-algebra such that Ext iAk(Mk; Nk) 6=0. (The proof comes down to an elementary commutative algebra argument.) By(11.13.3), taking B = A, we conclude that

(11.13.4) Ext iA((y�H)�; Y ) = 0 8i > 0:

(It is well-known that A is a �nitely generated projective Z-module; in fact, theprojectivity of endomorphism algebras EndB(Z) for algebras B over regular ringsof Krull dim. at most 2 holds quite generally under mild hypotheses|see [DPS2;x1].)

For any j = 0; � � � ;m�1, Xj is a direct sum of various HomH(y�H;T ) = (y�H)�,so (11.13.4) implies that Ext iA(Xj; Y ) = 0 for all i > 0. Thus, working from rightto left in (11.13.1), we obtain that Ext iA(K;Y ) = 0, i > 0, for any kernel K of adi�erential @j in (11.8). The exactness of (11.13.2) is immediate from this fact. �

Since each Xi is a direct sum of various (y�H)� which are, in turn, direct sum-mands of Y , it follows that G(Xi) is a projective left E-module. In the followingresult, we identify G(Xi) in another way.

Lemma 11.14. There is an isomorphism E��! Aop such that the projective

left E-module G(Xi), when viewed as a right A-module by means of this isomor-phism, becomes isomorphic to HomH(T;

Lj�j=i x�H

�r�). Also, the left E-module

G(IND�) identi�es with the q-determinant representation dett of A over Z.Proof. The isomorphism E �= Aop is proved in [DPS3; (2.5)]. In [DPS3; (2.6.1)],

it proved that, under this isomorphism, G(Xi) identi�es with the left Aop-module

HomH(T�;Mj�j=i

y�H�r�) = HomH(T;

Mj�j=i

x�H�r�):

The identi�cation of G(Xi) as the right A-module described in the statement of

the theorem then follows immediately. Next, we can apply [DPS3; (2.6.1)] (with eNthere taken to be IND in the present notation) to obtain that G(IND�) identi�es

15The argument in [DPS3] applies to subsets � determined by partitions of m, but [DPS3;(1.4d)] establishes that the same assertion holds when the subset � is determined by a compositionof m, as explained above.


with HomH(T�; IND) as a right A-module. (For the de�nition of �, see below

(11.4.2).) But using (9.18.1),

HomH(T�; IND) �= HomH(T; SGN) �= dett:

Therefore, G(IND�) �= dett as A-modules. �

Putting everything together, we obtain the following important result:

Theorem 11.15. For the �nite Coxeter system (Sm;W ), let A = EndH(T ),where T =

L��S x�H

�r� and the positive integers r� are chosen so that A �=St(m;m). There exists a �nite resolution

(11.15.1) 0! Q(m� 1)! � � � ! Q(0)! dett ! 0

of dett by projective right A-modules Q(i), where

(11.15.2) Q(i) = HomH(T;Mj�j=i

x�H�r�):

For any commutative Z-algebra O, the complex Q(�)O de�nes a projective AO-resolution of detq over O.

Proof. This is immediate from (11.12) and (11.13). Observe that, because allmodules in (11.15.1) are Z-projective, it remains exact after tensoring with O. �

Remarks 11.16. (a) We emphasize that while (11.10) is valid for arbitrary �niteCoxeter systems (W;S), the above (11.15) has been proved only when W �= Sm.It would be interesting to extend (11.15) to include all �nite Coxeter systems.However, at present, the proof of the essential (11.13) uses highest weight categorytheory.

(b) Let V be a vector space of dimension m over a �eld k. For an integerj, let �j(V ) (resp., Sj(V )) be the jth exterior (resp., symmetric) power of V .For W = Sm, and consider the Schur algebra S(m;m) = EndSr

(V m). One canobtain a Koszul resolution of the S(m;m)-module det with terms in degree n givenby �m�j�1(V ) Sj+1(V ). Applying the standard duality, we obtain a resolutionP� of det . Over Q (t), P0 identi�es with the adjoint representation of GLm, andso, in general, P� (or, more precisely, its q-version) is distinct from our resolution(11.15.1). We have not seriously investigated the connection of (11.15.1) with othercommonly used complexes for GLm, e. g., the so-called Akin-Buchbaum resolution[AB].

(c) In the next section, we will see the utility of using the resolution (11.15)in precisely the form given. Thus, in (12.1), we will apply the Morita equivalence(9.17) to (11.15) to obtain yet another resolution involving the �nite groups GLn(q)in non-describing characteristic. This resolution looks very much like the Solomon-Tits complex associated to the building of type Am�1.


x12. Higher cohomology in non-describing characteristic

In this section, we apply (9.17) and (11.15) to obtain results on the highercohomology of G(q) = GLn(q) in non-describing characteristic. To begin, we mustinterpret the projective Sq(n; n)k-modules given in (10.13.1) as GLn(q)-modules.Let O be a commutative Z-algebra which is a discrete valuation ring and suchthat its quotient �eld K is a splitting �eld for G(q). Assume the residue �eld k ofO has characteristic p which does not divide q. As in (11.15), for each � � S =f(1; 2); � � � ; (n � 1; n)g, we �x a positive integer r� so that, with H = H(Sn;Z),we have EndH(

L��S x�H

�r�) �= St(n; n). Now we can state:

Lemma 12.1. There is a resolution of OG(q)-modules

(12.1.1) 0!M(n� 1)! � � � !M(1)!M(0)! O ! 0

where

(12.1.2) M(i) =M

��S;jJj=i

py�M

�r�with M = OjG(q)B(q)

is a projective OG(q)=J(q)-module.Proof. We will use the complex of right St(m;m)-modules given in (11.15.1).

Tensoring withO, gives a complex of Sq(m;m)O-modules which provide a projectiveresolution of detq. It su�ces to show that this complex corresponds under theMorita equivalence (9.17) to a complex of the form indicated in (12.1.1).

First, by (9.18a), the q-determinant module detq for Sq(n; n)O corresponds, un-der the Morita equivalence (9.17.1), to the trivial module O. Next, we need to iden-tify Q(i) = HomH(T;

Lj�j=i x�H) with an OG(q)=J(q)-module under the Morita

equivalence (9.17.1). By (9.17), this Morita equivalence is obtained by applying

the functor HomOG(q)=J(q)(cM;�), where cM = cM1;G(q);O as de�ned in (9.14), and

using the identi�cation EndOG(q)(cM) �= Sq(n; n)O given in (9.11) (following [DJ2]).

Letting M = OjG(q)B(q), the identi�cation arises through a natural isomorphism

HomOG(q)(py�M;

py�M) �= HomH(x�HO; x�HO);

see [DJ2; (2.24)]. Thus, HomOG(q)(cM;py�M) identi�es with HomH(T; x�H), as

required. �

For � � S, de�ne f�(q) 2 O by the condition that y2� = f�(q)y�. Since �wy� =

SGN(�w)y� = (�1)`(w)y� for w 2 W�, it follows that f�(q) =P

w2W�q�`(w). Let

� =Sai=1 �i be the decomposition of � into disjoint and connected (when identi�ed

with a set of simple roots in the root system of type An�1) subsets �i. Then

(12.2) f�(q) =aYi=1

fj�ij(q); where fj�ij(q) =

j�ijYi=1

q�i � 1

q�1 � 1:

Now we have


Lemma 12.3. Fix a non-negative integer m � n and assume that q � 1 andf�(q) is invertible in O for all � � S satisfying j�j = m. Then the OG(q)-moduleM(i) in (12.1.2) is projective. In particular, if p does not divide

Qm+1j=1 (q

j � 1),

then M(j) is a projective OG(q)-module for all j � m.

Proof. Because p does not divide q�1 (and is distinct from q), O is a projective

OB(q)-module and so M = OjG(q)B(q) is a projective OG(q)-module. The condition

that f�(q) be invertible for all � satisfying j�j = m implies that e� =1

f�(q)y� is an

idempotent in HO, so that py�M = y�M = e�M

is also a projective OG(q)-module. By (12.1.2), this proves the �rst assertion ofthe lemma. The �nal assertion follows immediately from formula (12.2). �

Now we are ready to prove the following important result which generalizes(10.1).

Theorem 12.4. Consider the group G(q) = GLn(q). Assume that p does not

divide qQm+1j=1 (q

j � 1) for some integer m � 0. Then for any kG(q)=J(q)k-module

V (e. g., any V = D(s; �), s 2 Css;p0 and � ` n(s)), we have an isomorphism

(12.4.1) Hi(G(q); V ) �= Ext iSq(n;n)k(detq;�F (V )); 0 � i � m+ 1

and an injection

(12.4.2) Extm+2Sq(n;n)k

(detq; �F (V )) ,! Hm+2(G(q); V ):

Proof. By (12.3), the OG(q)-modules M(0); � � � ;M(m) are projective. (Ofcourse, they are automatically projective as OG(q)=J(q)k-modules.) Since theacyclic complex (12.1.1) consists of projective O-modules, it remains exact afterapplying the functor �O k. Thus, we obtain an acyclic complex

(12.4.3) 0! �M(m+ 1)! � � � ! �M(0)! k! 0

in which, for j � m, the modules �M(j) = M(j)k are projective for both kG(q)and kG(q)=J(q)k, while �M(m + 1) = Im(M(m + 1)k ! M(m)k). The theoremnow follows by a dimension shifting argument extending that given in the proofof (10.1). More precisely, consider the double complex C�� obtained by applyingthe functor HomR(�; V ) to a Cartan-Eilenberg resolution of the complex M� in(12.4.3), where R = kG(q) or kG(q)=J(q)k. Filtering C�� by columns Cs� leads tothe well-known spectral sequence

(12.4.4) Es;t1R = Ext tR( �M(s); V )) Ext s+tR (k; V ):

For s � m and t > 0, we have Es;t1R = 0, so (12.4.1) follows from the equality

HomkG(q)(�;�) = HomkG(q)=J(q)k(�;�) of bifunctors on mod{kG(q)=J(q)k, to-gether with (9.17).


Also, we have

Extm+2kG(q)=J(q)k

(k; V ) �= Ext 1kG(q)=J(q)k(�M(m+ 1); V )

,! Ext 1kG(q)(k; V )�= Extm+2

kG(q)(k; V ):

In the above diagram, the isomorphisms are provided by the shapes of the spectralsequences Es;t

1R. Now (12.4.2) follows from this, together with (9.17). �

Remark 12.5. Although we have stated the above result for kG(q), the argumentalso establishes a similar result comparing the cohomology of OG(q) with thatof Sq(n; n)O. More precisely, let V of a OG(q)=J(q)-module. Then, under thearithmetic hypothesis of (12.4), we have

(12.5.1) Hi(G(q); V ) �= Ext iSq(n;n)O (detq; F (V )); 0 � i � m+ 1

and an injection

(12.5.2) Extm+2Sq(n;n)O

(detq; F (V )) ,! Hm+2(G(q); V );

where F is the Morita equivalence in (9.17.1).Combining (12.4) and (8.6), we obtain the following stability result, somewhat

analogous to H1-stability proved in (10.2).

Corollary 12.6. Let `; n be �xed positive integers. There exists an integerN = N(n; `) such that if k is an algebraically closed �eld of characteristic p > N ,� ` n, and 0 � i � ` and q has order ` modulo p, then Hi(GLn(q); D(1; �)) dependsonly on � and i.

As explained in (10.3b), using (8.6) (which is based on the results in Part I), thestable value of Hi(GLn(q); D(1; �)) in (12.6) above can be expressed in terms of

the dimension of a certain Ext i-group for the a�ne Lie algebra bgln(C ).We conclude this section with the following result concerning SLn-cohomology,

extending (10.7). We leave the proof to the interested reader.

Theorem 12.7. Let L be an irreducible unipotent module for SLn(q) overk, and let D(1; �) be the unique irreducible module of GLn(q) which restricts to it.Suppose p also does not divide

Qmi=1(q

i�1) for some non-negative integer m. Then

Hm(SLn(q); L) �= Hm(GLn(q); D(1; �)):

x13. Appendix: The constructible topologyConsider a property P for algebras A (or morphisms g) over domains O. For

example, in Part I, we de�ned P to be a generic property for algebras (which areassumed to be O-�nite and O-torsion free) provided that: given an algebra A overO such that P holds for AK (K = k(0)), then P holds for Ak(p) for p belonging to anonempty open subset of X = SpecO. In certain cases, it is possible to say moreabout the set of those points p 2 X at which P holds for Ak(p). This discussion


makes use of the constructible topology on X. We introduce this concept and givea brief exposition, though the results are not required elsewhere in this paper.

To consider a speci�c example, let A be an O-algebra, which is �nitely generatedand torsion-free as an O-module. Consider a morphism M

g! N of A-modules asin (1.5a). The set

Y = fp 2 X j Sur(gk(p)) holdsgis not necessarily open in X, but instead satis�es|at least|the following property(see also (13.6) below). If Y is a subset of a topological space, we let Y denote itsclosure.

Property GZ. If p 2 Y , there is a Zariski open neighborhood W of p in X =SpecO such that fpg \W � fpg \ Y .

This property follows easily from Nakayama's lemma, together with an elemen-tary localization argument. If p 62 Y , then gk(p) is not surjective, and in order totrack further information, it is useful to recast the surjectivity property in termsof the dimension of the image of gk(p). Thus, it makes sense to consider, for eachnon-negative integer n, the set Yn of points p 2 X such that the image of gk(p) hask(p){dimension n. The sets Yn satisfy (GZ) as well.

Since similar considerations apply to other properties of A{modules and maps,we de�ne a new topology based on the property (GZ). In fact, we do this for anarbitrary nonempty topological space.

Let T be a topology on a nonempty set X. De�ne T g to be the collection ofsubsets V � X with the following property:

Property C. If x 2 V , then there is aW 2 T containing x such that fxg\W �fxg \ V . In other words, for each x 2 V , fxg \ V is a neighborhood of x in fxg(given its subspace topology).

We verify directly that T � T g and that T g is a topology on X. We call T g theT {constructible topology onX. As we note in (13.2) below, T g has a basis consistingof the T {locally closed sets. By analogy with the Zariski topology, we say that asubset of X is T -constructible if it is the union of a �nite number of T -locally closedsubsets. Thus, T g may be regarded as a kind of \topological completion" of thecollection of T -constructible subsets of X.

The proofs of the following elementary statements about the T {constructibletopology will be left to the reader.

Lemma 13.1. Let T be a topology on a nonempty set X. The T -constructibletopology T g contains both the T -open and the T -closed subsets of T as members.In particular, if Y � X is either T -open or closed, then Y is both T g-open andT g-closed. For example, if x 2 X, then fxg is both open and closed in the T -constructible topology.

Lemma 13.2. Each of the two sets

B =ffxg \W jW 2 T gB0 =fC \W jC is T {closed and W is T {openg


is a basis for the T {constructible topology on X.

We describe the e�ect of passing to the constructible topology in several commonsituations. The proof is immediate from (13.2).

Lemma 13.3. The following statements hold:

(a) If T = fX; ;g is the minimal topology on X, then T = T g.(b) If (X; T ) is a T0{space, then (X; T g) is Hausdor�.(c) If (X; T ) is a T1{space, then T g is the discrete topology on X.

The Zariski topology on a scheme satis�es the T0-separation axiom, so that thefollowing result holds:

Corollary 13.4. Let T is the Zariski topology on a scheme X. Then theT {constructible topology is Hausdor�, and (T g)g is the discrete topology on X.

Example 13.5. For the a�ne scheme X = SpecZ, the constructible topologycoincides with the one-point compacti�cation of the discrete topological space whoseelements are the positive prime integers.

In the context of Noetherian schemes, we next wish to indicate a technique forparlaying the Property GZ, which is a kind of in�nite constructibility, into the �niteversion of constructibility.

Let X be a Noetherian scheme (or, more generally, any Zariski space in the senseof [H; Ex. (3.17)]). Then the following property holds for its Zariski topology:

Property Z. If Y is any nonempty closed subset of X, there is a point y 2 Ywhose closure fyg contains a nonempty open subset of Y .

Property Z, together with the usual descending chain condition on closed sets,allows for iteration and the construction of many �ltrations X = X0 � X1 � � � � �Xm = ; ofX by closed subsets Xi, with points xi 2 Xi (i = 0; � � � ;m�1), such thatXi nXi+1 � fxig. For example, consider our original example above involving thedimensions of images of maps gk(p) for p 2 SpecO and g :M ! N . By Property Z,

we can choose x 2 X whose closure fxg contains a nonempty Zariski open subset.Now applying Property GZ to this point x, we �nd that there is a closed subsetX1 with x 62 X1 and with dim Imgk(p) constant on XnX1. Continuing in this wayyields a �ltration X = X0 � X1 � � � � � Xm = ; by closed subsets such that thefunction p 7! dim Im gk(p) is constant on each stratum Xi nXi+1: Since these strataare all locally closed, we have proved the following (prototypic) result:

Corollary 13.6. Let A be an O-algebra, which is �nitely generated and

torsion-free as an O-module. Consider a morphism Mg! N of A-modules as in

(1.5a), and �x any positive integer n. Then the set

Yn = fp 2 X = SpecO j dim Im gk(p) = n g

is constructible (i. e., it is a �nite union of locally closed subsets).


To conclude, it is worthwhile to review the classical constructibility result ofChevalley. Although our proof is based on that given in [H; Ex. (3.19)], ourpresent point of view makes it more conceptual.

Proposition 13.7. (Chevalley) Let f : X ! Y be a morphism of �nite typeof Noetherian schemes. Then the image of f is constructible.

Proof. Induct on dim X + dim Y . Because f has �nite type, we can reduce, assuggested in [H; Ex. (3.19(a))], to the case f : X =SpecB ! Y = SpecA in whichcase f is induced by an inclusion A! B of domains. In addition, B can be takento be a �nitely generated A-algebra, so we can assume B = A[�].

Now we claim that Im f is open in the constructible topology on Y . Let y =q 2 Im f . If � is transcendental over A, then X �= Y � A 1 and f is projectiononto Y , so Im f = Y . Otherwise, there exists a nonzero a 2 A such that thelocalization B0 = Ba is integral over A0 = Aa. If a 62 q, then y belongs to the(Zariski) open subscheme SpecA0 � Im f of Y . Otherwise, f induces a morphismg : SpecB=(a) !SpecA=(a) in which y 2 Im g. By induction on dim A, Im g isopen in the constructible topology on SpecA=(a). Since SpecA=(a) is Zariski closedin SpecA, our claim is established.

Finally, Property Z yields an x 2 X so that fxg contains a nonempty Zariskiopen subset of X. By (13.2) and the claim, f(x) 2 V � Im f for a Zariski locallyclosed subset V of Y . Now x belongs to the Zariski locally closed subset f�1(V ), sof�1(V ) contains a nonempty Zariski open subset W of X. By induction, f(X nW )is constructible in Y . Since Im f = V

Sf(X nW ), Im f is also constructible. �

References

AB. K. Akin and D. Buchsbaum, Characteristic-free representation theory of the general

linear group II. Homological considerations, Adv. in Math. 72 (1988), 171|210.

AM. A. Adem and R. J. Milgram, Cohomology of �nite groups, Springer, 1994.

AJS. H. Andersen, J. Jantzen, and W. Soergel, Representations of quantum groups at a pth

root of unity and of semisimple groups in characteristic p: independence of p, Ast�erisque

220 (1994).

AS. M. Aschbacher and L. Scott, Maximal subgroups of �nite groups, J. Algebra 92 (1985),44{80.

BGS. A. Beilinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation

theory, J. Amer. Math. Soc. 9 (1996), 473{527.

B. N. Bourbaki, Commutative Algebra, Addison-Wesley, 1972.

BMM. M. Brou�e, G. Malle, and J. Michel, Repr�esentations unipotents g�en�eriques et blocs desgroupes r�eductifs �nis, Ast�erisque 212 (1993).

CE. H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, 1956.

CPS1. E. Cline, B. Parshall and L. Scott, Cohomology of �nite groups of Lie type, I, Publ.Math. I.H.E.S. 45 (1975), 160{191.

CPS2. E. Cline, B. Parshall and L. Scott, Finite-dimensional algebras and highest weight cate-

gories, J. reine angew. Math. 391 (1988), 85{99.

CPS3. E. Cline, B. Parshall, and L. Scott, Integral and graded quasi-hereditary algebras, J.Algebra 131 (1990), 126{160.

CPS4. E. Cline, B. Parshall, and L. Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math.J. 45 (1993), 511{534.

CPS5. E. Cline, B. Parshall, and L. Scott, The homological dual of a highest weight category,Proc. London Math. Soc. 68 (1994), 294{316.


CPS6. E. Cline, B. Parshall and L. Scott, Stratifying endomorphism algebras, Memoirs Amer.Math. Soc. 591 (1996).

CPS7. E. Cline, B. Parshall and L. Scott, Graded and non-graded Kazhdan-Lusztig theories, in\Algebraic groups and Lie groups," ed. G. Lehrer, Cambridge University Press (1997),105{125.

CPS8. E. Cline, B. Parshall and L. Scott, Endomorphism algebras and representation theory,in \Algebraic Groups and their Representations," ed. R. W. Carter and J. Saxl, KluwerAcademic Publishers (1998),131{149.

CPSK. E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology,Invent. math. 39 (1977), 143{163.

CR. C. Curtis and I. Reiner, Methods of Representation Theory, vol. 1,2, Wiley, 1981,1987.

De. V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue

of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), 483{506.

D1. R. Dipper, Polynomial representations of �nite general linear groups in non-describing

characteristic, Prog. Math. 95 (1991), 343{370.

D2. R. Dipper, On the decomposition numbers of the �nite general linear groups, Trans.Amer. Math. Soc. 290 (1985), 315{344.

D3. R. Dipper, On quotients of hom-functors and representations of �nite general linear

groups,II, preprint.

DD. R. Dipper and S. Donkin, Quantum GLn, Proc. London Math. Soc. 63 (1991), 165{211.

DF. R. Dipper and P. Fleischmann, Modular Harish-Chandra theory, I, Math. Z. 211 (1992),48{71.

DJ1. R. Dipper and G. James, Representations of Hecke algebras of general linear groups,Proc. London Math. Soc. 52 (1986), 20{52.

DJ2. R. Dipper and G. James, The q-Schur algebra, Proc. London Math. Soc. 59 (1989),23{50.

DJ3. R. Dipper and G. James, q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc.327 (1991), 251{282.

DR. V. Dlab and C. M. Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), 280{291.

Do1. S. Donkin, On tilting modules for algebraic groups, Math. Zeit. 212 (1993), 39{60.

Do2. S. Donkin, Standard homological properties for quantum GLn, J. Algebra 181 (1996),235{266.

Du. J. Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995),363{372.

DPS1. J. Du, B. Parshall, and L. Scott, Stratifying endomorphism algebras associated to Hecke

algebras, J. Algebra 203 (1998), 169{210.

DPS2. J. Du, B. Parshall, and L. Scott, Cells and q-Schur algebras, J. Transformation Groups3 (1998), 33{49.

DPS3. J. Du, B. Parshall, and L. Scott, Quantum Weyl reciprocity and tilting modules, Comm.Math. Physics 195 (1998), 321{352.

DPW. J. Du, B. Parshall, and J.-p. Wang, Two-parameter quantum linear groups and the

hyperbolic invariance of q-Schur algebras, J. London Math. Soc. 44 (1991), 420{436.

DS. J. Du and L. Scott, Lusztig conjectures, old and new, I, J. reine angew. Math. 455(1994), 141{182.

Dw. W. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. 111(1980), 239{251.

FS. P. Fong and B. Srinivasan, The blocks of �nite general linear and unitary groups, Invent.math. 69 (1982), 109{153.

FP1. E. Friedlander and B. Parshall, On the cohomology of algebraic and related �nite groups,Invent. math. 74 (1983), 85{117.

FP2. E. Friedlander and B. Parshall, Cohomology of in�nitesimal and discrete groups, Math.Ann. 273 (1986), 353{374.

GH. J. Gruber and G. Hiss, Decomposition numbers of �nite classical groups for linear

primes, J. reine angew. Math. 485 (1997), 55{91.


GHM. M. Geck, G. Hiss, G. Malle, Towards a classi�cation of the irreducible representations in

non-describing characteristic of a �nite group of Lie type, Math. Z. 221 (1996), 353{386.Gr. J.A. Green, The characters of the �nite general linear groups, Trans. Amer. Math. Soc.

80 (1955), 402{447.H. R. Hartshorne, Algebraic geometry, vol. 52, Springer-Verlag Graduate Texts in Mathe-

matics, 1977.J2. G. D. James, The decomposition matrices of GLn(q) for n � 10, Proc. London Math.

Soc. 60 (1990), 225{265.Ji. M. Jimbo, A q-analogue of U(gl(N +1)), Hecke algebra, and the Yang-Baxter equation,

Letters in Math. Physics 11 (1986), 247{252.KT. M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for a�ne Lie algebras with

negative level, Duke Math. J. 77 (1995), 21{62.KL1. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,

Invent. math. 53 (1979), 165{184.KL2. D. Kazhdan and G. Lusztig, Tensor structures arising from a�ne Lie algebras I{II,

III{IV, J. Amer. Math. Soc. 6, 7 (1993, 1994), 905{1011; 335|453.M1. A. Mathas, A q-analogue of the Coxeter complex, J. Algebra 164 (1994), 831{848.M2. A. Mathas, Some generic representations, W -graphs, and duality, J. Algebra 170 (1994),

322{353.

P. B. Parshall, Finite dimensional algebras and algebraic groups, In: Classical Groups andRelated Topics. Contemp. Math. 82 (1989), 97{114.

PS1. B. Parshall and L. Scott, Derived Categories, Quasi-hereditary Algebras, and Algebraic

Groups, Proc. Ottawa-Moosonee Workshop Algebra, Carleton{Ottawa Math. LNS 3

(1988), 1{105.PS2. B. Parshall and L. Scott, Koszul algebras and the Frobenius automorphism, Quart. J.

Math. Oxford 46 (1995), 345{384.PW. B. Parshall and J.-p. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 439

(1991).R. J. Rickard,Morita theory for derived categories, J. Lond. Math. Soc. 39 (1989), 436{456.Ri. C. Ringel, The category of modules with good �ltrations over quasi-hereditary algebras

has almost split sequences, Math. Zeit. 208 (1991), 209{223.S. L. Scott, Linear and nonlinear group actions, in \Algebraic groups and their represen-

tations," ed. R.W. Carter and J. Saxl, Kluwer Academic Publishers (1998), 1{23.So. L. Solomon, The Steinberg character of a �nite group with a BN-pair, in \The theory of

�nite groups," ed. R. Brauer and H. Sah (1989), 213{221.T. M. Takeuchi, The group ring of GLn(q) and the q{Schur algebra, J. Math. Soc. Japan

48 (1996), 259{274.W. C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced

Mathematics, vol. 38, 1994.

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