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Derived Equivalences and Dade's Invariant Conjecture
Journal of Algebra 221, 513᎐527 Ž1999. Article ID jabr.1999.7982, available online at http:rrwww.idealibrary.com on
Derived Equivalences and Dade’s Invariant Conjecture* Andrei Marcus ‘‘Babes ¸-Bolyai’’ Uni¨ ersity, Department of Mathematics and Computer Science, Str. Mihail Kogalniceanu nr. 1, RO-3400 Cluj-Napoca, Romania ˇ Communicated by Michel Broue´ Received October 15, 1998
1. INTRODUCTION Let O be a complete discrete valuation ring and assume that the residue field k s OrJ Ž O . is algebraically closed of characteristic p ) 0, and the quotient field K has characteristic 0 and is also a splitting field for all the K-algebras considered below. Let further K be a normal subgroup of a finite group H, and let G s HrK. Consider a block b of O K with defect group P F K and let c be the block of O NK Ž P . corresponding to b. Dade stated in wD3x several forms of a conjecture and one of them, the invariant conjecture, involves the number of irreducible K-characters belonging to b and having a given defect and a given stabilizer in HrK. The aim of this paper is to provide a structural look to this conjecture, i.e., to find equivalences of categories preserving these invariants. Assume for the moment that b is G-invariant, and let O Hb and S s O NH Ž P . c. It follows that R and S are strongly G-graded O-algebras with R1 s O Kb and S1 s O NK Ž P . c. Then the group G acts on the category of Ž R1 , S1 .-bimodules. If C is a tilting complex of Ž R1 , S1 .-bimodules, then this equivalence is called G-equivariant if R g mR 1 C mS1 S gy1 , C in the bounded derived category of R1 mO S1op -mod. We investigate equivariant derived equivalences in Section 2 in a slightly more general setting, * This work was partly carried out while the author was visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge. The author thanks the Institute for its hospitality and also the referee for his observations. 513 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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and we show that they preserve stabilizers of simple K R1. It is well-known that derived equivalences also preserve defects of simple modules. In Section 3 we restrict ourselves to the case of group algebras and we show that an equivariant splendid derived equivalence between the blocks b and c of two groups K and K ⬘ having the same Brauer category induces, by applying the Brauer functor to the tilting complex, another equivariant derived equivalence between suitable blocks kCK Ž Q . and kCK ⬘Ž Q ., for any subgroup Q of P. As a consequence of this discussion, we conclude in Section 4 that an ‘‘invariant’’ form of Broue’s ´ conjecture implies Dade’s invariant conjecture in the case of blocks with abelian cyclic defect groups. The invariant conjecture was checked in many cases, one of them being that of blocks with cyclic defect groups in wD4x. We shall verify that Rouquier’s construction of a splendid derived equivalence between O Kb and O NK Ž P . c actually yields an equivariant derived equivalence. Note that the ‘‘equivariance’’ of a derived equivalence is a weaker condition than the ‘‘gradedness’’ considered in wM1x and wM2x, and we expect that graded derived equivalences preserve the invariants involved in the stronger forms of Dade’s conjecture. Concerning the terminology, rings will always be associative with unit element, and modules are unitary, finitely generated, and left, unless otherwise specified. The main reference for modular representation theory is wTx. We also refer to wBrx, wL1x, and wL2x for general facts on various types of equivalences between blocks, and to wD1x and wD2x for Clifford theory and representations of strongly graded algebras.
2. EQUIVARIANT DERIVED EQUIVALENCES 2.1. We fix a complete discrete valuation ring O with algebraically closed residue field k s OrJ Ž O . and quotient field of characteristic 0. We shall assume that all O-modules are free of finite rank. Fix also a finite group G and let R s [g g G R g and S s [g g G S g be two strongly G-graded O-algebras. We assume that R and S are symmetric O-algebras, such that the symmetrizing forms of R and S are G-invariant symmetrizing forms for R1 and S1 , respectively. We denote K R s K mO R, kR s k mO R, and we assume that K R and KS Žor equivalently K R1 and KS1 . are semisimple K-algebras, and that K is a splitting field for all the K-subalgebras of K R and KS. By an Ž R, S .-bimodule we mean a module over R mO S op . Observe that R mO S op is a strongly G = G-graded O-algebra and it has a strongly
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DERIVED EQUIVALENCES
G-graded subalgebra ⌬ s ⌬Ž R, S . s
[R ggG
g
mO S gop ,
where S gop s S gy1 . Note that an Ž R1 , S1 .-bimodule is the same as a ⌬ 1module. Similarly, the enveloping algebras R en s R mO R op and S en s S mO S op are G = G-graded O-algebras. 2.2. Let D b Ž K R1 . be the bounded derived category of the category K R1-mod. Since for any g g G, the functor K R g mK R 1 y : K R1-mod ª K R1-mod is an equivalence with inverse K R gy1 mK R 1 y , we have that G acts on R1-mod, on D b Ž K R1 ., and also on the corresponding Grothendieck groups RŽ K R1 . and RŽ D b Ž K R1 ... By wH, III, Lemma 1.2x, the canonical embedding of K R1-mod into D b Ž K R1 . induces an isomorphism R Ž K R1 . , R Ž D b Ž K R1 . . , which is clearly a G-isomorphism. We identify these two groups and we denote by w X x g RŽ K R1 . the class of an object X g D b Ž K R1 .. By definition, the stabilizer of X is the subgroup GX s g g G N K R g mK R 1 X , X in D b Ž K R1 . 4 of G. The group RŽ K R1 . is endowed with a scalar product defined by ² w X x , w X ⬘x : s
Ý Ž y1. i dim K Hom D Ž K R . Ž X w i x , X ⬘. . b
1
i
This scalar product is clearly G-invariant, and the G-set IrrŽ K R1 . of isomorphism classes of simple K R1-modules is an orthonormal ⺪-basis of RŽ K R1 .. 2.3. Let further D b Ž kR1 . be the bounded derived category of kR1-mod b Ž and let Dperf kR1 . be the full subcategory of D b Ž kR1 . consisting of perfect complexes Žthat is, complexes isomorphic to complexes of projective kR1modules., and denote by kR1-proj the subcategory of finitely generated projective kR1-modules. Again, G acts on these categories, since for any g g G the functor kR g mk R 1 y : kR1-mod ª kR1-mod
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is an autoequivalence. Consequently, the canonical embedding induces a G-isomorphism between the Grothendieck groups R Ž kR1 . , R Ž D b Ž kR1 . . and b R pr Ž kR1 . , R Ž Dperf Ž kR1 . . .
The G-set IrrŽ kR1 . of isomorphism classes of simple kR1-modules is a ⺪-basis of RŽ kR1 ., while the G-set PimŽ kR1 . of isomorphism classes of projective indecomposable kR1-modules is a ⺪-basis of R pr Ž kR1 .. There is a G-invariant duality between RŽ kR1 . and R pr Ž kR1 . defined by ²w P x , w X x: s
Ý Ž y1. i dim k Hom D Ž k R . Ž P w i x , X . , b
1
i b Ž where P g Dperf kR1 . and X g D b Ž kR1 ..
2.4. The Cartan᎐decomposition triangle T Ž R1 . of R1 is the commutative diagram 6
dec
R Ž kR1 . 6
t
dec
6
R Ž K R1 .
Car
R pr Ž kR1 .
where the maps are defined as follows. If P is a perfect complex of kR1-modules, then the Cartan matrix Car: R pr Ž kR1 . ª RŽ kR1 . sends the class w P x of P to its class in RŽ kR1 .. The decomposition map dec: RŽ K R1 . ª RŽ kR1 . sends the class w X x of a K R1-module X to the class of k mO X 0 g kR1-mod, where X 0 is an R1-lattice such that X , K mO X 0 . Reduction modulo J Ž O . is an isomorphism between R pr Ž R1 . and pr Ž R kR1 .. Using this, the adjoint t
dec: R pr Ž R1 . ª R Ž K R1 .
sends w P x to w K mO P x. These maps are clearly compatible with the metric structure of T Ž R1 ., and since kR g mk R 1 y , R g mR 1 y , and K R g mK R 1 y are autoequivalences of kR1-mod, R1-mod, and K R1-mod, respectively, we have that Car, dec, and tdec are G-maps, that is, Car g w P x sg Carw P x, decŽg w X x., and tdec g w X x sg Žtdecw X x. for any g g G.
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DERIVED EQUIVALENCES
We shall always consider T Ž R1 . endowed with the metric structure and the G-structure. 2.5. Definition. A derived equivalence induced by the complex C of Ž R1 , S1 .-bimodules is called G-equivariant if C is G-invariant; that is, R g mR 1 C mR 1 S gy1 , C
in D b Ž ⌬ 1 . .
2.6. THEOREM. Assume that the complex C of Ž R1 , S1 .-bimodules induces an equi¨ ariant deri¨ ed equi¨ alence between R1 and S1. Then there is a G-isomorphism between the Cartan᎐decomposition triangles T Ž R1 . and T Ž S1 .. 6
6
R Ž kS1 .
66
6
R Ž KS1 .
R Ž kR1 . 6
6
6
R Ž K R1 .
6
R pr Ž kS1 .
R pr Ž kR1 . Proof. The inverse equivalence is induced by the complex Hom R 1Ž C, R1 . of Ž S1 , R1 .-bimodules, which is naturally isomorphic to the O-dual C* of C. We first show that C* is G-invariant too. We have that Hom R Ž R mR 1 C, R . is a complex of G-graded Ž S1 , R .bimodules with Hom R Ž R mR 1 C, R . g , Hom R 1Ž R h mR 1 C, R h g . as complexes of Ž S1 , R1 .-bimodules of any g, h g G. Since R is strongly graded, it follows that Hom R Ž R mR 1 C, R . , Hom R 1Ž C, R1 . mR 1 R as complexes of G-graded Ž S1 , R .-bimodules, and consequently, Hom R 1Ž R g mR 1 C, R1 . , Hom R 1Ž C, R1 . mR 1 R gy1 as complexes of Ž S1 , R1 .-bimodules. On the other hand, Hom R 1Ž C mS1 S, R1 . is a complex of G-graded Ž S, R1 .-bimodules with Hom R 1Ž C mS1 S, R1 . g s f < f Ž C mS1 Sh . s 0 for h / gy1 4 , Hom R 1Ž C mS1 S gy1 , R1 . .
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Since S is strongly graded, we have that Hom R 1Ž C mS1 S, R1 . , S mS1 Hom R 1Ž C, R1 . as complexes of G-graded Ž S, R1 .-bimodules, and for any g g G, Hom R 1Ž C mS1 S gy1 , R1 . , S g mS1 Hom R 1Ž C, R1 . as Ž S1 , R1 .-bimodules. Finally, we obtain that for any g g G, Hom R 1Ž R g mR 1 C mS1 S gy1 , R1 . , S g mS1 Hom R 1Ž C, R1 . mR 1 R gy1 , which proves the claim. If X g D b Ž R1 ., then for any g g G, L
L
C* mR 1 Ž R g mR 1 X . , Ž S g mS1 C* mR 1 R gy1 . mR 1 Ž R g mR 1 X . L
, S g mS1 C* mR 1 X .
ž
/
By wBr, Proposition 4.2x, we have that K mO C and K mO C* induce a derived equivalence between K R1 and KS1 , while k mO C and k mO C* induces a derived equivalence between kR1 and kS1 , and these equivalences induce an isomorphism between the triangles T Ž R1 . and T Ž S1 .. Since K mO C and k mO C are G-invariant complexes, we see that the above isomorphism between T Ž R1 . and T Ž S1 . is a G-isomorphism, and since K R1 and KS1 are semisimple, we have a G-isomorphism ‘‘with signs’’ between IrrŽ K R1 . and IrrŽ KS1 .. 2.7. The group G = G acts on the category of Ž R1 , R1 .-bimodules by X ¬g X s R g mR 1 X mR 1 R hy1 , and G acts on the set of ideals of R1 by I ¬g I s R g IR gy1 . If B is a block of R1 and GB s g g G < R g BR gy1 s B4 is the stabilizer of B, then TrGGBŽ B . s
Ý
gg w GrG B x
is a G-invariant bimodule summand of R1.
g
B
DERIVED EQUIVALENCES
519
Using wBr, Proposition 4.3x and the theorem above we obtain 2.8. PROPOSITION.
Under the assumptions of Theorem 2.6, we ha¨ e
Ža. C mO C* induces a G = G-equi¨ ariant deri¨ ed equi¨ alence between R1en and S1en . Žb. If B is a block of R1 , then B⬘ s C* mR B mR C is a block of S1 , 1 1 GB s GB ⬘ , and BCB⬘ induces a GB-equi¨ ariant deri¨ ed equi¨ alence between B and B⬘, while C induces a G-equi¨ ariant deri¨ ed equi¨ alence between TrGGBŽ B . and TrGGBŽ B⬘.. 2.9. Remark. As in Definition 2.5, one can say that, by definition, the bimodule R 1 MS1 induces an equivariant stable Morita equivalence between R1 and S1 if for any g g G, ⌬ g m⌬ 1 M , M in the stable category ⌬ 1- mod of ⌬ 1-mod. ŽRecall that our assumptions force the inverse equivalence to be induced by the O-dual M* of M.. By adapting the proof of wRi1, Corollary 5.5x, one can easily see that if C induces an equivariant derived equivalence between R1 and C1 , then there is a bimodule M, projective as a left R1-module and as a right S1-module, inducing an equivariant stable Morita equivalence between R1 and S1. Indeed, truncating a projective resolution Žover ⌬ 1 . of C, we obtain for some degree n a bounded complex C⬘ s Ž ⭈⭈⭈ ª 0 ª Qyn ª Pynq1 ª Pynq2 ª ⭈⭈⭈ . isomorphic to C in D b Ž ⌬ 1 . with P i projective for i ) yn, such that M s ⍀yn Ž Qyn . induces a stable Morita equivalence between R1 and S1. This construction is functorial, and applied to ⌬ g m⌬ 1 C, we obtain the ⌬ 1-module ⌬ g m⌬ 1 M. The following observation will be needed in Section 4. 2.10. LEMMA. Let H be a normal p⬘-subgroup of G and C a complex of Ž R1 , S1 .-bimodules inducing a G-equi¨ ariant deri¨ ed equi¨ alence between R1 and S1. Assume also that C extends to ⌬ H s [h g H ⌬ h , and that the isomorphism between ⌬ g m⌬ 1 C and C holds in D b Ž ⌬ H .. If D s Ž R H mO S Hop . m⌬ H C, then D induces a GrH-equi¨ ariant deri¨ ed equi¨ alence between R H and SH . Proof. Since H is a normal subgroup of G wM2, Lemma 2.9 and Remark 2.10 c.x implies that ⌬ g m⌬ 1 C is naturally a ⌬ H -module. It follows by wM2, Theorem 4.8x that D induces a derived equivalence between R H and S H .
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ANDREI MARCUS
For g g G, we denote g D s R g H mR H D mS H SH gy1 , and we have to show that g
D , D in D b Ž R H mod SH . .
Observe first that g D , R g mR 1 D mS1 S gy1 is an H-graded Ž R H , SH .-bimodule, where for x g H,
Ž g D . x s R g mR 1 Dgy1 x g mS1 S gy1 . By assumption, there is an isomorphism in D b Ž ⌬ 1 . between D 1 s C and Žg D .1 , R g mR C mS S gy1 , and using wM2, Lemma 2.6x we conclude that 1 1 g D , D.
3. LOCAL STRUCTURE We recall from wL1x and wL2x the definition of a splendid equivalence, and we shall adapt it to our situation. 3.1. Let K be a normal subgroup of a finite group H and K ⬘ a normal subgroup of H⬘ such that G s HrK , H⬘rK ⬘, and let ␣ : HrK ª H⬘rK ⬘ be an isomorphism. It follows that O H and O H⬘ are strongly G-graded O-algebras. Let also b be a block idempotent of O K and c a block idempotent of O K ⬘. Let Hb and HcX be the stabilizers of b and c, respectively. If G b s HbrK and Gc s HcXrK ⬘, then R [ b O Hb s O Hb b is a strongly G b-graded algebra, and S [ c O HcX s O HcX c is a strongly Gc-graded algebra. We assume that b and c have a common defect group P F K, and that for each subgroup Q of P, ␣ induces an isomorphism
␣ Ž Q . : CH Ž Q . rCK Ž Q . ª CH ⬘ Ž Q . rCK ⬘ Ž Q . . Denoting GQ s CH Ž Q .rCK Ž Q ., we have that kCH Ž Q . and kCH ⬘Ž Q . are strongly GQ-graded k-algebras. 3.2. Let i g Ž O Kb. P and j g Ž O K ⬘c . P be primitive idempotents such that BrPK Ž i . / 0 and BrPK ⬘Ž j . / 0, where BrP is the Brauer map. For every subgroup Q of P there are unique block idempotents eQ g kCK Ž Q . and f Q g kCK ⬘Ž Q . such that BrPK Ž i . eQ / 0 and BrPK ⬘Ž j . f Q / 0. Define as above the stabilizer GŽQ , e Q . s hCK Ž Q . ¬ h g CH Ž Q . , h eQ s eQ ,
½
5
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DERIVED EQUIVALENCES
so eQ kCH Ž Q . eQ s eQ kCH Ž Q . G Ž Q, e . is a strongly GŽQ, e Q .-graded k-algebra. Q Similarly, f Q kCH ⬘Ž Q . f Q s f Q kCH ⬘Ž Q . G Ž Q, f . is a strongly GŽQ, f Q .-graded kQ algebra. By definition, for two subgroups Q and R of P, let EH , G ŽŽ Q, eQ ., Ž R, e R .. be the set of equivalence classes modulo inner automorphisms of R of group homomorphisms Q ª R of the form u ¬x u s xuxy1 for some x g H, satisfying xQxy1 : R and xeQ xy1 s e x Q xy1 . As in wKP, 2.8x, these maps are considered together with the action of H on G by left translation; it follows that homomorphisms induced by x, y g H such that xK / yK are different. We assume further that O Kb and O K ⬘c have G-equi¨ alent Brauer categories. We understand by this that for any subgroups Q and R of P, there is an equality EH , G Ž Ž Q, eQ . , Ž R, e R . . s EH ⬘, G Ž Ž Q, f Q . , Ž R , f R . . which is compatible with the isomorphism ␣ : HrK ª H⬘rK ⬘. Compatibility with ␣ means that whenever x g H induces by conjugation a homomorphism Q ª R such that x eQ s e x Q , there is x⬘ g H⬘ inducing the same homomorphism, such that ␣ Ž xK . s x⬘K ⬘. 3.3. The indecomposable Ž R1 , S1 .-bimodule M is called splendid if it is O K ⬘. a direct summand of the bimodule O Ki mO P jO Let M Ž Q . be the Ž kCK Ž Q ., kCK ⬘Ž Q ..-bimodule M Ž Q . s M Qr J Ž O . M Q q
ž
TrRQ Ž M R . ,
Ý R-Q
/
and we shall consider the Ž kCK Ž Q . eQ , kCK ⬘Ž Q . f Q .-bimodule eQ M Ž Q . f Q . By definition, a tilting complex of Ž R1 , S1 .-bimodules is splendid if the indecomposable summands of its components X i are splendid. 3.4. PROPOSITION. With the abo¨ e notations, assume that the Brauer categories of b and c are G-equi¨ alent. Then Ža. G b s Gc and GŽQ, e . s GŽQ, f . for any subgroup Q of P. Q Q Žb. If X is a splendid complex of Ž R1 , S1 .-bimodules inducing a G bequi¨ ariant deri¨ ed equi¨ alence between R1 and S1 , then the complex eQ X Ž Q . f Q induces a GŽQ, e Q .-equi¨ ariant deri¨ ed equi¨ alence between kCK Ž Q . eQ and kCK ⬘Ž Q . f Q . Proof. Ža. By the Frattini argument Žsee also wD1, Ž0.3a.x, we have that G b s NH Ž P , e P . KrK
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ANDREI MARCUS
and Gc s NH ⬘ Ž P , f P . K ⬘rK ⬘. For any Q F P we have, by Ž3.2., that NH Ž Q, eQ . rQCK Ž Q . , EH , G Ž Q, eQ . s EH ⬘, G Ž Q, f Q . , NH ⬘ Ž Q, f Q . rQCK ⬘ Ž Q . . Taking Q s P we obtain that G b s Gc . Since ␣ : HrK ª H⬘rK ⬘ restricts to the isomorphism ␣ Ž Q ., we deduce that GŽQ, e Q . s GŽQ, f Q . for any Q. Žb. The fact that the complex eQ X Ž Q . f Q induces a derived equivalence between kCK Ž Q . eQ and kCK ⬘Ž Q . f Q follows from wL1, Theorem 1.1x. We have to show that eQ X Ž Q . f Q is GŽQ, e Q .-invariant. Indeed, let h g CH Ž Q . and h⬘ g CH ⬘Ž Q . such that h eQ s eQ , h⬘ f Q s f Q , and ␣ Ž hK . s h⬘K ⬘. Then we have hCK Ž Q . eQ mk C K ŽQ. e Q eQ X Ž Q . f Q mk C K ⬘ŽQ. f Q kCK ⬘ Ž Q . f Q h⬘y1 , eQ Ž h mk X Ž Q . mk h⬘y1 . f Q , eQ Ž h mk X mk h⬘y1 . Ž Q . f Q , eQ X Ž Q . fQ . 3.5. Rickard gave in wRi2x a different definition of a splendid tilting complex, and his approach was refined by Harris wHax. We recall the main result of wHax, since it will be needed in the example of the next section. Let H, K, and G be as in Ž3.1., and let H⬘ be a subgroup of H, K ⬘ s H⬘ l K, and G⬘ s H⬘rK ⬘. Let further b and c be as in Ž3.1., and let P be a common defect group of b and c. A tilting complex X of Ž O Kb, O K ⬘c .-bimodules is called splendid, if the terms of X are relatively ␦ Ž P .-projective p-permutation O Ž K = K ⬘.-modules. Let Q F P and fix a c-subpair Ž Q, f Q . of K ⬘. Then by wHa, Theorem 4x there is a unique b-subpair Ž Q, eQ . of K such that eQ X Ž Q . f Q is a tilting complex of Ž kCK Ž Q . eQ , kCK ⬘Ž Q . f Q .-bimodules whose terms are p-permutation k Ž C K Ž Q . = C K ⬘ Ž Q ..-modules; moreover eXQ X Ž Q . f Q , 0 in D b Ž k Ž CK Ž Q . = CK ⬘Ž Q ... for any b-subpair Ž Q, eXQ . / Ž Q, eQ .. Let h g CH ⬘Ž Q .. It follows by the uniqueness of Ž Q, eQ . that if hCK ⬘Ž Q . fixes f Q , then hCK Ž Q . also fixes Ž Q, eQ .. Moreover, as in Proposition X 3.4Žb., we have that the equivalence induced by eQ X Ž Q . f Q is GŽQ, f Q .equivariant.
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DERIVED EQUIVALENCES
4. CONJECTURES FOR ABELIAN AND CYCLIC DEFECT GROUPS 4.1. Let H, K, G, b, and P be as in the preceding section and let H⬘ s NH Ž P ., K ⬘ s NK Ž P ., G⬘ s H⬘rK ⬘, and c be the Brauer correspondent of b. Let also ␣ : G⬘ ª G be the map induced by the inclusion of H⬘ in H. As we already have seen, we have G b s NH Ž P , e P . KrK , NH ⬘ Ž P , e P . K ⬘rK ⬘ s Gc . Moreover, if P is abelian, then by slightly adapting the proof of wT, Proposition 4.9.6x, one obtains that the Brauer categories of O Kb and O Kb⬘ are G b-equivalent. Therefore, it makes sense to formulate the following equivariant version of Broue’s ´ conjecture. 4.1.1. If b is a block of O K with abelian defect group P, then there is a G b-equivariant splendid derived equivalence between O Kb and O NK Ž P . c, where c is the Brauer correspondent of b. 4.2. Denote by
Ž P0 - P1 - ⭈⭈⭈ - Pn .
C:
a p-chain of K and by < C < s n the length of C. The groups K and H act on the set of p-chains of K, and let NK Ž C . and NH Ž C . be the normalizers of C in K and H, respectively. Denote also NG Ž C . s NH Ž C .rNK Ž C .. We have that CK Ž Pn . F NK Ž C . F NK Ž Pn . , and if  is a block of O NK Ž C ., then by wKR, Lemma 3.2x, the induced block  K of O K is defined. The group NG Ž C . acts on the set IrrŽ K NK Ž C .. of simple K NK Ž C .-modules, so if g IrrŽ K NK Ž C .., then the stabilizer NG Ž C, . is defined. Denote further by defŽ . the defect of and by  Ž . the block of O NK Ž C . to which belongs. If d F 0 is an integer and F is a subgroup of NG Ž C ., then denote by kŽ C, b, d, F . the number of characters g IrrŽ K NK Ž C .. satisfying def Ž . s d,
K
 Ž . s b,
and
NG Ž C, . s F ,
and by kŽ  , d, F . the number of characters g IrrŽ K . satisfying def Ž . s d
and
NG Ž  , . s F.
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ANDREI MARCUS
We have that kŽ C, b, d, F . depends only on the K-conjugacy class of C, and Dade’s invariant conjecture wD3, 2.5x states that 4.2.1. If Op Ž K . s 1 and defŽ b . ) 0 then < < Ý Ž y1. G k Ž C, b, d, F . s 0,
Cg FrK
where F is one of the families P, U , N , or E of p-chains of K introduced in wKRx. The argument of wKR, Proposition 5.5x now gives 4.3. PROPOSITION.
Conjecture Ž4.1.1. implies Conjecture Ž4.2.1..
Proof. Let b be a block of O K with abelian defect group P ) 1, let s Ž P0 - ⭈⭈⭈ - Pn . be an element of N , and let  be a block of O NG Ž C . inducing b. If Pn is a defect group of  , then, denoting C⬘ s Ž P0 - ⭈⭈⭈ - Pny1 ., we have that  NK ŽC ⬘. is defined, it has defect group Pn , and Ž  NK ŽC ⬘. . K s b. Then by hypothesis, there is an NG Ž C .-equivariant derived equivalence between  and  NK ŽC ⬘.. By wBr, Proposition 4.5x, defects of irreducible characters are preserved. It follows that k Ž  , d, F . s k Ž  NK ŽC ⬘. , d, F . . If Pn is not a defect group of  , then the defect group Pnq1 of  satisfies Pn - Pnq1 , and let C⬘ s Ž1 - P1 - ⭈⭈⭈ - Pn - Pnq1 . g N . Then NK Ž C⬘. s NNK ŽC .Ž Pnq1 . and there is a unique block  ⬘ of NK Ž C⬘. inducing  . Again we have that k Ž  ⬘, d, F . s k Ž  , d, F . . Consequently, Ž4.2.1. holds. 4.4. Assume from now on that P is cyclic and b is G-invariant. Rouquier proved in wRoux that there is a splendid derived equivalence Žin the sense of Rickard and Harris; see Ž3.5.. between O Kb and O K ⬘c. This result was a consequence of a series of derived equivalences, and in the remaining part of this paper we show that all those equivalences are equivariant. This verification, together with the fact that the composition of equivariant derived equivalences is equivariant too, will imply that Rouquier’s inductive proof can be used to conclude that Ž4.1.1. holds for blocks with cyclic defect groups. We denote R s Op Ž K ., and Žunder the hypothesis that P is non-normal. let Q be the subgroup of P such that w Q : R x s p. Let also I s NH Ž P, e P .rCK Ž P ., L s NH Ž Q ., I1 s NK Ž P, e P .rCK Ž P ., L1 s NK Ž Q ., and let c1 be the block of O L1 corresponding to b.
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We have that Q is a weakly closed subgroup of K with respect to H, and by the results of wD4, Sect. 3x we have that c1 is L-invariant, hence O Lc1 is a strongly G-graded algebra. Recall also that IrI1 , G. 4.5. Consider the block algebra O K ⬘c. Then we are in the situation of wRou, Proposition 2.15x: c is a block of CK Ž P . and it has a unique simple module V. Starting with V, one constructs an Ž O CK Ž P . c, O P .-bimodule M1 inducing a Morita equivalence between O CK Ž P . c and O P. Further, the I-graded O H⬘c-module O H⬘c mO C K Ž P .c M1 induces a graded Morita equivalence between O H⬘c and an I-graded crossed product of O P and I. By the proof of the same wRou, Proposition 2.15x, O K X c mO C K Ž P .c M1 induces an I1-graded Morita equivalence between O K ⬘c and O P i I1 Žthis amounts to the extendability of M1 to a certain ‘‘diagonal’’ subalgebra ., which is then G-invariant Žsince G , IrI1 .. 4.6. By wRou, Lemma 4.2x, if R s Op Ž K . s 1, then the bimodule b O Kc1 induces a Morita stable equivalence between O Kb and O L1 c1. Since K e H and b, c1 are G-invariant, we have that b O Kc1 is a G-invariant bimodule. Moreover, its unique nonprojective bimodule summand M1 is G-invariant too. 4.7. We show that if M is a G-invariant Ž O Kb, O Ž P i I1 .-bimodule inducing a stable Morita equivalence between O Kb and O Ž P i I1 ., then the construction of wRou, Sect. 3x provides an equivariant derived equivalence between O Kb and O Ž P i I1 .. Indeed, let ⑀ s < I1 < be the inertial index of b, S s M mO Ž P i I1 . O , and denote by ⍀ the Heller operator. Let ¨ i be the vertex of the Brauer tree corresponding to the character of ⍀ iS and let l 1 be the edge connecting ¨ i and ¨ iq1. Then
[
0FiF ⑀ y1
Ž P⍀ 2 i S mO P⍀U2 i O .
is a projective cover of M Žas a bimodule.. We have that S, ⍀ 2 iS, and P⍀ 2 i S are G-invariant O Kb-modules. One can easily see that P⍀ 2 i S m P⍀U2 i O is a G-invariant bimodule, and it follows that the tilting complex C defined in wRou, 3.4x is G-invariant.4.8. It remains to examine the case when R s OP Ž K . is not trivial. Assume first that 1 / R F ZŽ K ., and denote by ‘‘ ’’ the canonical projection from K to K s KrR and from L1 to L1 s L1rR. By the inductive assumption, there is an equivariant derived equivalence between O Kb and O L1 c1. Then the tilting complex of Ž O Kb, O L1 c1 .-bimodules constructed in wRou, 4.2.1x is clearly G-invariant.
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Assume finally that 1 / R g ZŽ K ., and we again show that the construction of wRou, 4.2.2x gives a G-invariant tilting complex of Ž O Kb, O L1 c1 .-bimodules. We have that O Kb and O L1 c1 are I1-graded algebras with 1-components O CK Ž R . b and O CL 1Ž R . c1 , respectively, where O CL 1Ž R . c1 is the Brauer correspondent of the block O CK Ž R . b. Since b and c1 are G-invariant, we have that NH=LŽ ␦ Ž R .. acts by conjugation on b O CK Ž R . c1 Žwhere ␦ Ž R . s Ž u, ¨ . ¬ u g R4 . Regarded as an O Ž CK Ž R . = CL 1Ž R ..r␦ Ž R .-module, b O CK Ž R . c1 has an indecomposable direct summand M with vertex ␦ Ž PrR., which extends to an 1 r␦ Ž R. O NK=L 1Ž R .r␦ Ž R .-module. Then Ind K=L is an I1 -graded N K= L 1 Ž R .r ␦ Ž R . M Ž O Kb, O L1 c1 .-bimodule. Inspecting further Rouquier’s arguments and using Lemma 2.10, we see that it is enough to show that the O NK= L 1Ž R .r␦ Ž R .-module M is I-invariant. This holds, since I1 is a p⬘-group, so M is an O NK=L 1Ž R .r␦ Ž R .-summand of b O CK Ž R . c1 and it still has vertex ␦ Ž PrR., while the O Ž CK Ž R . = CL 1Ž R ..r␦ Ž R .-summands of Ž b O CK Ž R . c1 .rM have smaller vertices. REFERENCES wBrx wD1x wD2x wD3x wD4x wHx wHax wKPx wKRx wL1x wL2x wM1x wM2x
M. Broue, ´ Equivalences of blocks of group algebras, in ‘‘Proceedings of the NATO Advanced Research Workshop on Representation of Algebras and Related Topics,’’ Kluwer Academic, Dordrecht, 1991. E. C. Dade, Block extensions, Illinois, J. Math. 17 Ž1973., 198᎐273. E. C. Dade, Extending group-modules in a relatively prime case, Math. Z. 186 Ž1984., 81᎐98. E. C. Dade, Counting characters in blocks, 2.9, in ‘‘Proceedings of the 1995 Conference on Representation Theory at Ohio State University.’’ E. C. Dade, Counting characters in blocks with cyclic defect groups, J. Algebra 186 Ž1996., 934᎐969. D. Happel, ‘‘Triangulated Categories in the Representation Theory of Finite Dimensional Algebras,’’ Cambridge Univ. Press, Cambridge, UK, 1986. M. Harris, Splendid derived equivalences for blocks of finite groups, J. London Math. Soc., to appear. B. Kulshammer and L. Puig, Extensions of nilpotent blocks, In¨ ent. Math. 102 Ž1990., ¨ 17᎐71. B. Knorr ¨ and G. R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. 39 Ž1989., 48᎐60. M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, preprint, 1996. M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, preprint, 1997. A. Marcus, On equivalences between blocks of group algebras: Reduction to the simple components, J. Algebra 184 Ž1996., 372᎐396. A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 Ž1998., 713᎐731.
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wRi1x
527
J. Rickard, Derived categories and stable equivalences, J. Pure Appl. Algebra 61 Ž1989., 303᎐317. wRi2x J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 Ž1994., 331᎐358. wRoux R. Rouquier, The derived category of blocks with cyclic defect groups ŽS. Konig ¨ and A. Zimmermann, Eds.., in ‘‘Derived Equivalences for Group Rings,’’ Springer-Verlag, Berlin, 1998. wTx J. Thevenaz, ‘‘G-Algebras and Modular Representation Theory,’’ Clarendon Press, ´ Oxford, 1995.
Derived Equivalences and Dade’s Invariant Conjecture* Andrei Marcus ‘‘Babes ¸-Bolyai’’ Uni¨ ersity, Department of Mathematics and Computer Science, Str. Mihail Kogalniceanu nr. 1, RO-3400 Cluj-Napoca, Romania ˇ Communicated by Michel Broue´ Received October 15, 1998
1. INTRODUCTION Let O be a complete discrete valuation ring and assume that the residue field k s OrJ Ž O . is algebraically closed of characteristic p ) 0, and the quotient field K has characteristic 0 and is also a splitting field for all the K-algebras considered below. Let further K be a normal subgroup of a finite group H, and let G s HrK. Consider a block b of O K with defect group P F K and let c be the block of O NK Ž P . corresponding to b. Dade stated in wD3x several forms of a conjecture and one of them, the invariant conjecture, involves the number of irreducible K-characters belonging to b and having a given defect and a given stabilizer in HrK. The aim of this paper is to provide a structural look to this conjecture, i.e., to find equivalences of categories preserving these invariants. Assume for the moment that b is G-invariant, and let O Hb and S s O NH Ž P . c. It follows that R and S are strongly G-graded O-algebras with R1 s O Kb and S1 s O NK Ž P . c. Then the group G acts on the category of Ž R1 , S1 .-bimodules. If C is a tilting complex of Ž R1 , S1 .-bimodules, then this equivalence is called G-equivariant if R g mR 1 C mS1 S gy1 , C in the bounded derived category of R1 mO S1op -mod. We investigate equivariant derived equivalences in Section 2 in a slightly more general setting, * This work was partly carried out while the author was visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge. The author thanks the Institute for its hospitality and also the referee for his observations. 513 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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and we show that they preserve stabilizers of simple K R1. It is well-known that derived equivalences also preserve defects of simple modules. In Section 3 we restrict ourselves to the case of group algebras and we show that an equivariant splendid derived equivalence between the blocks b and c of two groups K and K ⬘ having the same Brauer category induces, by applying the Brauer functor to the tilting complex, another equivariant derived equivalence between suitable blocks kCK Ž Q . and kCK ⬘Ž Q ., for any subgroup Q of P. As a consequence of this discussion, we conclude in Section 4 that an ‘‘invariant’’ form of Broue’s ´ conjecture implies Dade’s invariant conjecture in the case of blocks with abelian cyclic defect groups. The invariant conjecture was checked in many cases, one of them being that of blocks with cyclic defect groups in wD4x. We shall verify that Rouquier’s construction of a splendid derived equivalence between O Kb and O NK Ž P . c actually yields an equivariant derived equivalence. Note that the ‘‘equivariance’’ of a derived equivalence is a weaker condition than the ‘‘gradedness’’ considered in wM1x and wM2x, and we expect that graded derived equivalences preserve the invariants involved in the stronger forms of Dade’s conjecture. Concerning the terminology, rings will always be associative with unit element, and modules are unitary, finitely generated, and left, unless otherwise specified. The main reference for modular representation theory is wTx. We also refer to wBrx, wL1x, and wL2x for general facts on various types of equivalences between blocks, and to wD1x and wD2x for Clifford theory and representations of strongly graded algebras.
2. EQUIVARIANT DERIVED EQUIVALENCES 2.1. We fix a complete discrete valuation ring O with algebraically closed residue field k s OrJ Ž O . and quotient field of characteristic 0. We shall assume that all O-modules are free of finite rank. Fix also a finite group G and let R s [g g G R g and S s [g g G S g be two strongly G-graded O-algebras. We assume that R and S are symmetric O-algebras, such that the symmetrizing forms of R and S are G-invariant symmetrizing forms for R1 and S1 , respectively. We denote K R s K mO R, kR s k mO R, and we assume that K R and KS Žor equivalently K R1 and KS1 . are semisimple K-algebras, and that K is a splitting field for all the K-subalgebras of K R and KS. By an Ž R, S .-bimodule we mean a module over R mO S op . Observe that R mO S op is a strongly G = G-graded O-algebra and it has a strongly
515
DERIVED EQUIVALENCES
G-graded subalgebra ⌬ s ⌬Ž R, S . s
[R ggG
g
mO S gop ,
where S gop s S gy1 . Note that an Ž R1 , S1 .-bimodule is the same as a ⌬ 1module. Similarly, the enveloping algebras R en s R mO R op and S en s S mO S op are G = G-graded O-algebras. 2.2. Let D b Ž K R1 . be the bounded derived category of the category K R1-mod. Since for any g g G, the functor K R g mK R 1 y : K R1-mod ª K R1-mod is an equivalence with inverse K R gy1 mK R 1 y , we have that G acts on R1-mod, on D b Ž K R1 ., and also on the corresponding Grothendieck groups RŽ K R1 . and RŽ D b Ž K R1 ... By wH, III, Lemma 1.2x, the canonical embedding of K R1-mod into D b Ž K R1 . induces an isomorphism R Ž K R1 . , R Ž D b Ž K R1 . . , which is clearly a G-isomorphism. We identify these two groups and we denote by w X x g RŽ K R1 . the class of an object X g D b Ž K R1 .. By definition, the stabilizer of X is the subgroup GX s g g G N K R g mK R 1 X , X in D b Ž K R1 . 4 of G. The group RŽ K R1 . is endowed with a scalar product defined by ² w X x , w X ⬘x : s
Ý Ž y1. i dim K Hom D Ž K R . Ž X w i x , X ⬘. . b
1
i
This scalar product is clearly G-invariant, and the G-set IrrŽ K R1 . of isomorphism classes of simple K R1-modules is an orthonormal ⺪-basis of RŽ K R1 .. 2.3. Let further D b Ž kR1 . be the bounded derived category of kR1-mod b Ž and let Dperf kR1 . be the full subcategory of D b Ž kR1 . consisting of perfect complexes Žthat is, complexes isomorphic to complexes of projective kR1modules., and denote by kR1-proj the subcategory of finitely generated projective kR1-modules. Again, G acts on these categories, since for any g g G the functor kR g mk R 1 y : kR1-mod ª kR1-mod
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is an autoequivalence. Consequently, the canonical embedding induces a G-isomorphism between the Grothendieck groups R Ž kR1 . , R Ž D b Ž kR1 . . and b R pr Ž kR1 . , R Ž Dperf Ž kR1 . . .
The G-set IrrŽ kR1 . of isomorphism classes of simple kR1-modules is a ⺪-basis of RŽ kR1 ., while the G-set PimŽ kR1 . of isomorphism classes of projective indecomposable kR1-modules is a ⺪-basis of R pr Ž kR1 .. There is a G-invariant duality between RŽ kR1 . and R pr Ž kR1 . defined by ²w P x , w X x: s
Ý Ž y1. i dim k Hom D Ž k R . Ž P w i x , X . , b
1
i b Ž where P g Dperf kR1 . and X g D b Ž kR1 ..
2.4. The Cartan᎐decomposition triangle T Ž R1 . of R1 is the commutative diagram 6
dec
R Ž kR1 . 6
t
dec
6
R Ž K R1 .
Car
R pr Ž kR1 .
where the maps are defined as follows. If P is a perfect complex of kR1-modules, then the Cartan matrix Car: R pr Ž kR1 . ª RŽ kR1 . sends the class w P x of P to its class in RŽ kR1 .. The decomposition map dec: RŽ K R1 . ª RŽ kR1 . sends the class w X x of a K R1-module X to the class of k mO X 0 g kR1-mod, where X 0 is an R1-lattice such that X , K mO X 0 . Reduction modulo J Ž O . is an isomorphism between R pr Ž R1 . and pr Ž R kR1 .. Using this, the adjoint t
dec: R pr Ž R1 . ª R Ž K R1 .
sends w P x to w K mO P x. These maps are clearly compatible with the metric structure of T Ž R1 ., and since kR g mk R 1 y , R g mR 1 y , and K R g mK R 1 y are autoequivalences of kR1-mod, R1-mod, and K R1-mod, respectively, we have that Car, dec, and tdec are G-maps, that is, Car g w P x sg Carw P x, decŽg w X x., and tdec g w X x sg Žtdecw X x. for any g g G.
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We shall always consider T Ž R1 . endowed with the metric structure and the G-structure. 2.5. Definition. A derived equivalence induced by the complex C of Ž R1 , S1 .-bimodules is called G-equivariant if C is G-invariant; that is, R g mR 1 C mR 1 S gy1 , C
in D b Ž ⌬ 1 . .
2.6. THEOREM. Assume that the complex C of Ž R1 , S1 .-bimodules induces an equi¨ ariant deri¨ ed equi¨ alence between R1 and S1. Then there is a G-isomorphism between the Cartan᎐decomposition triangles T Ž R1 . and T Ž S1 .. 6
6
R Ž kS1 .
66
6
R Ž KS1 .
R Ž kR1 . 6
6
6
R Ž K R1 .
6
R pr Ž kS1 .
R pr Ž kR1 . Proof. The inverse equivalence is induced by the complex Hom R 1Ž C, R1 . of Ž S1 , R1 .-bimodules, which is naturally isomorphic to the O-dual C* of C. We first show that C* is G-invariant too. We have that Hom R Ž R mR 1 C, R . is a complex of G-graded Ž S1 , R .bimodules with Hom R Ž R mR 1 C, R . g , Hom R 1Ž R h mR 1 C, R h g . as complexes of Ž S1 , R1 .-bimodules of any g, h g G. Since R is strongly graded, it follows that Hom R Ž R mR 1 C, R . , Hom R 1Ž C, R1 . mR 1 R as complexes of G-graded Ž S1 , R .-bimodules, and consequently, Hom R 1Ž R g mR 1 C, R1 . , Hom R 1Ž C, R1 . mR 1 R gy1 as complexes of Ž S1 , R1 .-bimodules. On the other hand, Hom R 1Ž C mS1 S, R1 . is a complex of G-graded Ž S, R1 .-bimodules with Hom R 1Ž C mS1 S, R1 . g s f < f Ž C mS1 Sh . s 0 for h / gy1 4 , Hom R 1Ž C mS1 S gy1 , R1 . .
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Since S is strongly graded, we have that Hom R 1Ž C mS1 S, R1 . , S mS1 Hom R 1Ž C, R1 . as complexes of G-graded Ž S, R1 .-bimodules, and for any g g G, Hom R 1Ž C mS1 S gy1 , R1 . , S g mS1 Hom R 1Ž C, R1 . as Ž S1 , R1 .-bimodules. Finally, we obtain that for any g g G, Hom R 1Ž R g mR 1 C mS1 S gy1 , R1 . , S g mS1 Hom R 1Ž C, R1 . mR 1 R gy1 , which proves the claim. If X g D b Ž R1 ., then for any g g G, L
L
C* mR 1 Ž R g mR 1 X . , Ž S g mS1 C* mR 1 R gy1 . mR 1 Ž R g mR 1 X . L
, S g mS1 C* mR 1 X .
ž
/
By wBr, Proposition 4.2x, we have that K mO C and K mO C* induce a derived equivalence between K R1 and KS1 , while k mO C and k mO C* induces a derived equivalence between kR1 and kS1 , and these equivalences induce an isomorphism between the triangles T Ž R1 . and T Ž S1 .. Since K mO C and k mO C are G-invariant complexes, we see that the above isomorphism between T Ž R1 . and T Ž S1 . is a G-isomorphism, and since K R1 and KS1 are semisimple, we have a G-isomorphism ‘‘with signs’’ between IrrŽ K R1 . and IrrŽ KS1 .. 2.7. The group G = G acts on the category of Ž R1 , R1 .-bimodules by X ¬g X s R g mR 1 X mR 1 R hy1 , and G acts on the set of ideals of R1 by I ¬g I s R g IR gy1 . If B is a block of R1 and GB s g g G < R g BR gy1 s B4 is the stabilizer of B, then TrGGBŽ B . s
Ý
gg w GrG B x
is a G-invariant bimodule summand of R1.
g
B
DERIVED EQUIVALENCES
519
Using wBr, Proposition 4.3x and the theorem above we obtain 2.8. PROPOSITION.
Under the assumptions of Theorem 2.6, we ha¨ e
Ža. C mO C* induces a G = G-equi¨ ariant deri¨ ed equi¨ alence between R1en and S1en . Žb. If B is a block of R1 , then B⬘ s C* mR B mR C is a block of S1 , 1 1 GB s GB ⬘ , and BCB⬘ induces a GB-equi¨ ariant deri¨ ed equi¨ alence between B and B⬘, while C induces a G-equi¨ ariant deri¨ ed equi¨ alence between TrGGBŽ B . and TrGGBŽ B⬘.. 2.9. Remark. As in Definition 2.5, one can say that, by definition, the bimodule R 1 MS1 induces an equivariant stable Morita equivalence between R1 and S1 if for any g g G, ⌬ g m⌬ 1 M , M in the stable category ⌬ 1- mod of ⌬ 1-mod. ŽRecall that our assumptions force the inverse equivalence to be induced by the O-dual M* of M.. By adapting the proof of wRi1, Corollary 5.5x, one can easily see that if C induces an equivariant derived equivalence between R1 and C1 , then there is a bimodule M, projective as a left R1-module and as a right S1-module, inducing an equivariant stable Morita equivalence between R1 and S1. Indeed, truncating a projective resolution Žover ⌬ 1 . of C, we obtain for some degree n a bounded complex C⬘ s Ž ⭈⭈⭈ ª 0 ª Qyn ª Pynq1 ª Pynq2 ª ⭈⭈⭈ . isomorphic to C in D b Ž ⌬ 1 . with P i projective for i ) yn, such that M s ⍀yn Ž Qyn . induces a stable Morita equivalence between R1 and S1. This construction is functorial, and applied to ⌬ g m⌬ 1 C, we obtain the ⌬ 1-module ⌬ g m⌬ 1 M. The following observation will be needed in Section 4. 2.10. LEMMA. Let H be a normal p⬘-subgroup of G and C a complex of Ž R1 , S1 .-bimodules inducing a G-equi¨ ariant deri¨ ed equi¨ alence between R1 and S1. Assume also that C extends to ⌬ H s [h g H ⌬ h , and that the isomorphism between ⌬ g m⌬ 1 C and C holds in D b Ž ⌬ H .. If D s Ž R H mO S Hop . m⌬ H C, then D induces a GrH-equi¨ ariant deri¨ ed equi¨ alence between R H and SH . Proof. Since H is a normal subgroup of G wM2, Lemma 2.9 and Remark 2.10 c.x implies that ⌬ g m⌬ 1 C is naturally a ⌬ H -module. It follows by wM2, Theorem 4.8x that D induces a derived equivalence between R H and S H .
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For g g G, we denote g D s R g H mR H D mS H SH gy1 , and we have to show that g
D , D in D b Ž R H mod SH . .
Observe first that g D , R g mR 1 D mS1 S gy1 is an H-graded Ž R H , SH .-bimodule, where for x g H,
Ž g D . x s R g mR 1 Dgy1 x g mS1 S gy1 . By assumption, there is an isomorphism in D b Ž ⌬ 1 . between D 1 s C and Žg D .1 , R g mR C mS S gy1 , and using wM2, Lemma 2.6x we conclude that 1 1 g D , D.
3. LOCAL STRUCTURE We recall from wL1x and wL2x the definition of a splendid equivalence, and we shall adapt it to our situation. 3.1. Let K be a normal subgroup of a finite group H and K ⬘ a normal subgroup of H⬘ such that G s HrK , H⬘rK ⬘, and let ␣ : HrK ª H⬘rK ⬘ be an isomorphism. It follows that O H and O H⬘ are strongly G-graded O-algebras. Let also b be a block idempotent of O K and c a block idempotent of O K ⬘. Let Hb and HcX be the stabilizers of b and c, respectively. If G b s HbrK and Gc s HcXrK ⬘, then R [ b O Hb s O Hb b is a strongly G b-graded algebra, and S [ c O HcX s O HcX c is a strongly Gc-graded algebra. We assume that b and c have a common defect group P F K, and that for each subgroup Q of P, ␣ induces an isomorphism
␣ Ž Q . : CH Ž Q . rCK Ž Q . ª CH ⬘ Ž Q . rCK ⬘ Ž Q . . Denoting GQ s CH Ž Q .rCK Ž Q ., we have that kCH Ž Q . and kCH ⬘Ž Q . are strongly GQ-graded k-algebras. 3.2. Let i g Ž O Kb. P and j g Ž O K ⬘c . P be primitive idempotents such that BrPK Ž i . / 0 and BrPK ⬘Ž j . / 0, where BrP is the Brauer map. For every subgroup Q of P there are unique block idempotents eQ g kCK Ž Q . and f Q g kCK ⬘Ž Q . such that BrPK Ž i . eQ / 0 and BrPK ⬘Ž j . f Q / 0. Define as above the stabilizer GŽQ , e Q . s hCK Ž Q . ¬ h g CH Ž Q . , h eQ s eQ ,
½
5
521
DERIVED EQUIVALENCES
so eQ kCH Ž Q . eQ s eQ kCH Ž Q . G Ž Q, e . is a strongly GŽQ, e Q .-graded k-algebra. Q Similarly, f Q kCH ⬘Ž Q . f Q s f Q kCH ⬘Ž Q . G Ž Q, f . is a strongly GŽQ, f Q .-graded kQ algebra. By definition, for two subgroups Q and R of P, let EH , G ŽŽ Q, eQ ., Ž R, e R .. be the set of equivalence classes modulo inner automorphisms of R of group homomorphisms Q ª R of the form u ¬x u s xuxy1 for some x g H, satisfying xQxy1 : R and xeQ xy1 s e x Q xy1 . As in wKP, 2.8x, these maps are considered together with the action of H on G by left translation; it follows that homomorphisms induced by x, y g H such that xK / yK are different. We assume further that O Kb and O K ⬘c have G-equi¨ alent Brauer categories. We understand by this that for any subgroups Q and R of P, there is an equality EH , G Ž Ž Q, eQ . , Ž R, e R . . s EH ⬘, G Ž Ž Q, f Q . , Ž R , f R . . which is compatible with the isomorphism ␣ : HrK ª H⬘rK ⬘. Compatibility with ␣ means that whenever x g H induces by conjugation a homomorphism Q ª R such that x eQ s e x Q , there is x⬘ g H⬘ inducing the same homomorphism, such that ␣ Ž xK . s x⬘K ⬘. 3.3. The indecomposable Ž R1 , S1 .-bimodule M is called splendid if it is O K ⬘. a direct summand of the bimodule O Ki mO P jO Let M Ž Q . be the Ž kCK Ž Q ., kCK ⬘Ž Q ..-bimodule M Ž Q . s M Qr J Ž O . M Q q
ž
TrRQ Ž M R . ,
Ý R-Q
/
and we shall consider the Ž kCK Ž Q . eQ , kCK ⬘Ž Q . f Q .-bimodule eQ M Ž Q . f Q . By definition, a tilting complex of Ž R1 , S1 .-bimodules is splendid if the indecomposable summands of its components X i are splendid. 3.4. PROPOSITION. With the abo¨ e notations, assume that the Brauer categories of b and c are G-equi¨ alent. Then Ža. G b s Gc and GŽQ, e . s GŽQ, f . for any subgroup Q of P. Q Q Žb. If X is a splendid complex of Ž R1 , S1 .-bimodules inducing a G bequi¨ ariant deri¨ ed equi¨ alence between R1 and S1 , then the complex eQ X Ž Q . f Q induces a GŽQ, e Q .-equi¨ ariant deri¨ ed equi¨ alence between kCK Ž Q . eQ and kCK ⬘Ž Q . f Q . Proof. Ža. By the Frattini argument Žsee also wD1, Ž0.3a.x, we have that G b s NH Ž P , e P . KrK
522
ANDREI MARCUS
and Gc s NH ⬘ Ž P , f P . K ⬘rK ⬘. For any Q F P we have, by Ž3.2., that NH Ž Q, eQ . rQCK Ž Q . , EH , G Ž Q, eQ . s EH ⬘, G Ž Q, f Q . , NH ⬘ Ž Q, f Q . rQCK ⬘ Ž Q . . Taking Q s P we obtain that G b s Gc . Since ␣ : HrK ª H⬘rK ⬘ restricts to the isomorphism ␣ Ž Q ., we deduce that GŽQ, e Q . s GŽQ, f Q . for any Q. Žb. The fact that the complex eQ X Ž Q . f Q induces a derived equivalence between kCK Ž Q . eQ and kCK ⬘Ž Q . f Q follows from wL1, Theorem 1.1x. We have to show that eQ X Ž Q . f Q is GŽQ, e Q .-invariant. Indeed, let h g CH Ž Q . and h⬘ g CH ⬘Ž Q . such that h eQ s eQ , h⬘ f Q s f Q , and ␣ Ž hK . s h⬘K ⬘. Then we have hCK Ž Q . eQ mk C K ŽQ. e Q eQ X Ž Q . f Q mk C K ⬘ŽQ. f Q kCK ⬘ Ž Q . f Q h⬘y1 , eQ Ž h mk X Ž Q . mk h⬘y1 . f Q , eQ Ž h mk X mk h⬘y1 . Ž Q . f Q , eQ X Ž Q . fQ . 3.5. Rickard gave in wRi2x a different definition of a splendid tilting complex, and his approach was refined by Harris wHax. We recall the main result of wHax, since it will be needed in the example of the next section. Let H, K, and G be as in Ž3.1., and let H⬘ be a subgroup of H, K ⬘ s H⬘ l K, and G⬘ s H⬘rK ⬘. Let further b and c be as in Ž3.1., and let P be a common defect group of b and c. A tilting complex X of Ž O Kb, O K ⬘c .-bimodules is called splendid, if the terms of X are relatively ␦ Ž P .-projective p-permutation O Ž K = K ⬘.-modules. Let Q F P and fix a c-subpair Ž Q, f Q . of K ⬘. Then by wHa, Theorem 4x there is a unique b-subpair Ž Q, eQ . of K such that eQ X Ž Q . f Q is a tilting complex of Ž kCK Ž Q . eQ , kCK ⬘Ž Q . f Q .-bimodules whose terms are p-permutation k Ž C K Ž Q . = C K ⬘ Ž Q ..-modules; moreover eXQ X Ž Q . f Q , 0 in D b Ž k Ž CK Ž Q . = CK ⬘Ž Q ... for any b-subpair Ž Q, eXQ . / Ž Q, eQ .. Let h g CH ⬘Ž Q .. It follows by the uniqueness of Ž Q, eQ . that if hCK ⬘Ž Q . fixes f Q , then hCK Ž Q . also fixes Ž Q, eQ .. Moreover, as in Proposition X 3.4Žb., we have that the equivalence induced by eQ X Ž Q . f Q is GŽQ, f Q .equivariant.
523
DERIVED EQUIVALENCES
4. CONJECTURES FOR ABELIAN AND CYCLIC DEFECT GROUPS 4.1. Let H, K, G, b, and P be as in the preceding section and let H⬘ s NH Ž P ., K ⬘ s NK Ž P ., G⬘ s H⬘rK ⬘, and c be the Brauer correspondent of b. Let also ␣ : G⬘ ª G be the map induced by the inclusion of H⬘ in H. As we already have seen, we have G b s NH Ž P , e P . KrK , NH ⬘ Ž P , e P . K ⬘rK ⬘ s Gc . Moreover, if P is abelian, then by slightly adapting the proof of wT, Proposition 4.9.6x, one obtains that the Brauer categories of O Kb and O Kb⬘ are G b-equivalent. Therefore, it makes sense to formulate the following equivariant version of Broue’s ´ conjecture. 4.1.1. If b is a block of O K with abelian defect group P, then there is a G b-equivariant splendid derived equivalence between O Kb and O NK Ž P . c, where c is the Brauer correspondent of b. 4.2. Denote by
Ž P0 - P1 - ⭈⭈⭈ - Pn .
C:
a p-chain of K and by < C < s n the length of C. The groups K and H act on the set of p-chains of K, and let NK Ž C . and NH Ž C . be the normalizers of C in K and H, respectively. Denote also NG Ž C . s NH Ž C .rNK Ž C .. We have that CK Ž Pn . F NK Ž C . F NK Ž Pn . , and if  is a block of O NK Ž C ., then by wKR, Lemma 3.2x, the induced block  K of O K is defined. The group NG Ž C . acts on the set IrrŽ K NK Ž C .. of simple K NK Ž C .-modules, so if g IrrŽ K NK Ž C .., then the stabilizer NG Ž C, . is defined. Denote further by defŽ . the defect of and by  Ž . the block of O NK Ž C . to which belongs. If d F 0 is an integer and F is a subgroup of NG Ž C ., then denote by kŽ C, b, d, F . the number of characters g IrrŽ K NK Ž C .. satisfying def Ž . s d,
K
 Ž . s b,
and
NG Ž C, . s F ,
and by kŽ  , d, F . the number of characters g IrrŽ K . satisfying def Ž . s d
and
NG Ž  , . s F.
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ANDREI MARCUS
We have that kŽ C, b, d, F . depends only on the K-conjugacy class of C, and Dade’s invariant conjecture wD3, 2.5x states that 4.2.1. If Op Ž K . s 1 and defŽ b . ) 0 then < < Ý Ž y1. G k Ž C, b, d, F . s 0,
Cg FrK
where F is one of the families P, U , N , or E of p-chains of K introduced in wKRx. The argument of wKR, Proposition 5.5x now gives 4.3. PROPOSITION.
Conjecture Ž4.1.1. implies Conjecture Ž4.2.1..
Proof. Let b be a block of O K with abelian defect group P ) 1, let s Ž P0 - ⭈⭈⭈ - Pn . be an element of N , and let  be a block of O NG Ž C . inducing b. If Pn is a defect group of  , then, denoting C⬘ s Ž P0 - ⭈⭈⭈ - Pny1 ., we have that  NK ŽC ⬘. is defined, it has defect group Pn , and Ž  NK ŽC ⬘. . K s b. Then by hypothesis, there is an NG Ž C .-equivariant derived equivalence between  and  NK ŽC ⬘.. By wBr, Proposition 4.5x, defects of irreducible characters are preserved. It follows that k Ž  , d, F . s k Ž  NK ŽC ⬘. , d, F . . If Pn is not a defect group of  , then the defect group Pnq1 of  satisfies Pn - Pnq1 , and let C⬘ s Ž1 - P1 - ⭈⭈⭈ - Pn - Pnq1 . g N . Then NK Ž C⬘. s NNK ŽC .Ž Pnq1 . and there is a unique block  ⬘ of NK Ž C⬘. inducing  . Again we have that k Ž  ⬘, d, F . s k Ž  , d, F . . Consequently, Ž4.2.1. holds. 4.4. Assume from now on that P is cyclic and b is G-invariant. Rouquier proved in wRoux that there is a splendid derived equivalence Žin the sense of Rickard and Harris; see Ž3.5.. between O Kb and O K ⬘c. This result was a consequence of a series of derived equivalences, and in the remaining part of this paper we show that all those equivalences are equivariant. This verification, together with the fact that the composition of equivariant derived equivalences is equivariant too, will imply that Rouquier’s inductive proof can be used to conclude that Ž4.1.1. holds for blocks with cyclic defect groups. We denote R s Op Ž K ., and Žunder the hypothesis that P is non-normal. let Q be the subgroup of P such that w Q : R x s p. Let also I s NH Ž P, e P .rCK Ž P ., L s NH Ž Q ., I1 s NK Ž P, e P .rCK Ž P ., L1 s NK Ž Q ., and let c1 be the block of O L1 corresponding to b.
DERIVED EQUIVALENCES
525
We have that Q is a weakly closed subgroup of K with respect to H, and by the results of wD4, Sect. 3x we have that c1 is L-invariant, hence O Lc1 is a strongly G-graded algebra. Recall also that IrI1 , G. 4.5. Consider the block algebra O K ⬘c. Then we are in the situation of wRou, Proposition 2.15x: c is a block of CK Ž P . and it has a unique simple module V. Starting with V, one constructs an Ž O CK Ž P . c, O P .-bimodule M1 inducing a Morita equivalence between O CK Ž P . c and O P. Further, the I-graded O H⬘c-module O H⬘c mO C K Ž P .c M1 induces a graded Morita equivalence between O H⬘c and an I-graded crossed product of O P and I. By the proof of the same wRou, Proposition 2.15x, O K X c mO C K Ž P .c M1 induces an I1-graded Morita equivalence between O K ⬘c and O P i I1 Žthis amounts to the extendability of M1 to a certain ‘‘diagonal’’ subalgebra ., which is then G-invariant Žsince G , IrI1 .. 4.6. By wRou, Lemma 4.2x, if R s Op Ž K . s 1, then the bimodule b O Kc1 induces a Morita stable equivalence between O Kb and O L1 c1. Since K e H and b, c1 are G-invariant, we have that b O Kc1 is a G-invariant bimodule. Moreover, its unique nonprojective bimodule summand M1 is G-invariant too. 4.7. We show that if M is a G-invariant Ž O Kb, O Ž P i I1 .-bimodule inducing a stable Morita equivalence between O Kb and O Ž P i I1 ., then the construction of wRou, Sect. 3x provides an equivariant derived equivalence between O Kb and O Ž P i I1 .. Indeed, let ⑀ s < I1 < be the inertial index of b, S s M mO Ž P i I1 . O , and denote by ⍀ the Heller operator. Let ¨ i be the vertex of the Brauer tree corresponding to the character of ⍀ iS and let l 1 be the edge connecting ¨ i and ¨ iq1. Then
[
0FiF ⑀ y1
Ž P⍀ 2 i S mO P⍀U2 i O .
is a projective cover of M Žas a bimodule.. We have that S, ⍀ 2 iS, and P⍀ 2 i S are G-invariant O Kb-modules. One can easily see that P⍀ 2 i S m P⍀U2 i O is a G-invariant bimodule, and it follows that the tilting complex C defined in wRou, 3.4x is G-invariant.4.8. It remains to examine the case when R s OP Ž K . is not trivial. Assume first that 1 / R F ZŽ K ., and denote by ‘‘ ’’ the canonical projection from K to K s KrR and from L1 to L1 s L1rR. By the inductive assumption, there is an equivariant derived equivalence between O Kb and O L1 c1. Then the tilting complex of Ž O Kb, O L1 c1 .-bimodules constructed in wRou, 4.2.1x is clearly G-invariant.
526
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Assume finally that 1 / R g ZŽ K ., and we again show that the construction of wRou, 4.2.2x gives a G-invariant tilting complex of Ž O Kb, O L1 c1 .-bimodules. We have that O Kb and O L1 c1 are I1-graded algebras with 1-components O CK Ž R . b and O CL 1Ž R . c1 , respectively, where O CL 1Ž R . c1 is the Brauer correspondent of the block O CK Ž R . b. Since b and c1 are G-invariant, we have that NH=LŽ ␦ Ž R .. acts by conjugation on b O CK Ž R . c1 Žwhere ␦ Ž R . s Ž u, ¨ . ¬ u g R4 . Regarded as an O Ž CK Ž R . = CL 1Ž R ..r␦ Ž R .-module, b O CK Ž R . c1 has an indecomposable direct summand M with vertex ␦ Ž PrR., which extends to an 1 r␦ Ž R. O NK=L 1Ž R .r␦ Ž R .-module. Then Ind K=L is an I1 -graded N K= L 1 Ž R .r ␦ Ž R . M Ž O Kb, O L1 c1 .-bimodule. Inspecting further Rouquier’s arguments and using Lemma 2.10, we see that it is enough to show that the O NK= L 1Ž R .r␦ Ž R .-module M is I-invariant. This holds, since I1 is a p⬘-group, so M is an O NK=L 1Ž R .r␦ Ž R .-summand of b O CK Ž R . c1 and it still has vertex ␦ Ž PrR., while the O Ž CK Ž R . = CL 1Ž R ..r␦ Ž R .-summands of Ž b O CK Ž R . c1 .rM have smaller vertices. REFERENCES wBrx wD1x wD2x wD3x wD4x wHx wHax wKPx wKRx wL1x wL2x wM1x wM2x
M. Broue, ´ Equivalences of blocks of group algebras, in ‘‘Proceedings of the NATO Advanced Research Workshop on Representation of Algebras and Related Topics,’’ Kluwer Academic, Dordrecht, 1991. E. C. Dade, Block extensions, Illinois, J. Math. 17 Ž1973., 198᎐273. E. C. Dade, Extending group-modules in a relatively prime case, Math. Z. 186 Ž1984., 81᎐98. E. C. Dade, Counting characters in blocks, 2.9, in ‘‘Proceedings of the 1995 Conference on Representation Theory at Ohio State University.’’ E. C. Dade, Counting characters in blocks with cyclic defect groups, J. Algebra 186 Ž1996., 934᎐969. D. Happel, ‘‘Triangulated Categories in the Representation Theory of Finite Dimensional Algebras,’’ Cambridge Univ. Press, Cambridge, UK, 1986. M. Harris, Splendid derived equivalences for blocks of finite groups, J. London Math. Soc., to appear. B. Kulshammer and L. Puig, Extensions of nilpotent blocks, In¨ ent. Math. 102 Ž1990., ¨ 17᎐71. B. Knorr ¨ and G. R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. 39 Ž1989., 48᎐60. M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, preprint, 1996. M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, preprint, 1997. A. Marcus, On equivalences between blocks of group algebras: Reduction to the simple components, J. Algebra 184 Ž1996., 372᎐396. A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 Ž1998., 713᎐731.
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wRi1x
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J. Rickard, Derived categories and stable equivalences, J. Pure Appl. Algebra 61 Ž1989., 303᎐317. wRi2x J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 Ž1994., 331᎐358. wRoux R. Rouquier, The derived category of blocks with cyclic defect groups ŽS. Konig ¨ and A. Zimmermann, Eds.., in ‘‘Derived Equivalences for Group Rings,’’ Springer-Verlag, Berlin, 1998. wTx J. Thevenaz, ‘‘G-Algebras and Modular Representation Theory,’’ Clarendon Press, ´ Oxford, 1995.