SOURCE ALGEBRAS AND SOURCE MODULES J. L. Alperin, Markus Linckelmann, Raphael Rouquier May 1998 The aim of this note is to give short proofs in module

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1 SURCE ALGEBRAS AND SURCE MDULES J. L. Alperin, Markus Linckelmann, Raphael Rouquier May 1998 The aim of this note is to give short proofs in module theoretic terms of two fundamental results in block theory, due to L. Puig [4, 14.6]: rst, there is an embedding of the source algebra of the Brauer correspondent of a block of some nite group into a source algebra of that block, and second, the source algebras of the Brauer correspondent can be described explicitly. 1. Let p be a prime, a complete local principal ideal domain having a residue eld k = =J() of characteristic p; that is, either is a complete discrete valuation ring or = k. Throughout the paper, G is a nite group, b a block of G and P a defect group of b; that is, b is a primitive idempotent of Z(G) and P is a p?subgroup of G such that the subgroup P = f(u; u?1 )g u2p of G G 0 is a vertex of Gb as (G G 0 )?module, where G 0 is the group opposite to G and G is considered as (G G 0 )?module through (x; y):a = xay for any x; y 2 G and a 2 G. Note that Gb has the trivial P? module as source, since we have an isomorphism of (G G 0 )?modules G = Ind GG0 G () mapping x 2 G to (x; 1) 1. Denition 2. A source module of the block b of G is an indecomposable direct summand M of Res GG0 GP 0 (Gb) having P as vertex. Remark 3. Since G P 0 contains the vertex P of the (G G 0 )?module Gb, the block b has a source module M. The algebra End (G1) (M) 0 is then a source algebra of the block b (cf. [3]), which explains our terminology. Indeed, any direct summand of Gb as left G?module is of the form Gi for some idempotent i in Gb. Since M is a direct summand of Gb as (GP 0 )?module, we have in fact M = Gi for some P?stable idempotent i in Gb; moreover, as M is indecomposable, i is primitive in (Gb) P, and the condition that P be a vertex of M is then equivalent to requiring Br P (i) 6= 0, where Br P : (G) P?! kc G (P ) is the Brauer homomorphism (cf. [5, (37.5)(b)]). The map sending a 2 igi to right multiplication by a on M is an isomorphism of interior P?algebras igi = End (G1) (M) 0. Note that M is projective as left Gb?module and as right P?module. The block b can have more than one isomorphism class of source modules, but they are all conjugate under the action of N G (P ) in the sense, that if M 0 is another source module of b, there is an automorphism ' of P induced 1

2 2 J. L. ALPERIN, MARKUS LINCKELMANN, RAPHA EL RUQUIER by conjugation with an element of N G (P ) such that M 0 = ResId G'(M). This follows from the interpretation of source modules as actual sources of a module in Remark 6 below. 4. There is a block e of C G (P ) having Z(P ) as defect group such that C G (P )e is isomorphic to a direct summand of Res GG0 (Gb); any other CG(P )CG(P ) 0 block of C G (P ) with this property is then conjugate to e by some element of N G (P ). Equivalently, e is a block of C G (P ) satisfying Br P (be) 6= 0; that is, (P; e) is a maximal b?brauerpair. We set H = N G (P; e), the subgroup of all elements in N G (P ) which stabilize e, and E = H=P C G (P ), the inertial quotient of the block b (associated with e). Moreover, e is still a block of both P C G (P ) and H having P as defect group. If k is large enough for C G (P )e then E is a p 0?group (see e.g. [5, (37.12)]). bserve that N GP 0(P ) = P (C G (P )1) H P 0. Therefore, if we take a source module N of He, it follows from the Green correspondence [5, (20.8)], that Ind GP 0 HP (N) has, up to isomorphism, a unique indecomposable direct summmand 0 M with vertex P and trivial source, and then M is a source module for Gb. The next result makes this more precise: Theorem 5. (i) Up to isomorphism, there is a unique indecomposable direct summand X of Res GG0 (Gb) GH 0 with vertex P which is isomorphic to a direct summand of Ind GH0 HH 0 (He). (ii) If N is a source module for the block e of H, then M = X N is a H source module for the block b of G. Proof. The existence and uniqueness of X follows from the Green correspondence; similarly, since M is a direct summand of Ind GP 0 HP (N), the Green correspondence 0 implies that M = M 0 M 00, where M 0 is indecomposable with vertex P and all indecomposable direct summands of M 00 have vertices strictly smaller than P. But X is also isomorphic to a direct summand of Ind GH0 GP (Z) for some indecomposable (GP 0 )?module Z with vertex P. Thus, using Mackey's formula, 0 the (G P 0 )?module M = X N is isomorphic to a direct summand of H Res GH0 GP Ind GH0 0 GP 0 (Z) = (1;x) Z : x2[h 0 =P 0 ] Since H normalizes P, the module (1;x) Z has vertex (1;x) P for any x 2 H 0. Together with the fact that all indecomposable direct summands of M 00 have vertex strictly smaller than P follows that M = M 0, which concludes the proof. Remark 6. The Green correspondence does not require the ground ring to be commutative; one might therefore as well consider the group algebra AG 0 of G 0 over the ring A = G. We have an obvious isomorphism AG 0 = (G G0 ),

3 SURCE ALGEBRAS AND SURCE MDULES 3 which allows us to consider Gb as AG 0?module. With the notation of Theorem 5, as AG 0?module, Gb has vertex P 0, the source module M, considered as AP 0?module is a source of Gb and the AH 0?module X is the Green correspondent of Gb having M as source. Theorem 5 is a particular case of [1, 4.10], just translated to a module theoretic language: with the notation of the Theorem, let f be a primitive idempotent in (Gb) H such that Br P (ef) 6= 0 and let j be a primitive idempotent in (He) P such that Br P (j) 6= 0. Then X = Gf and N = Hj. Thus M = Gjf, and the assertion that M is indecomposable is equivalent to the assertion that the idempotent jf is primitive in (Gb) P. Note that, again by the Green correspondence, the unique indecomposable direct summand of Res GH0 (X) with vertex HH 0 P is isomorphic to He. From this and Theorem 5 we obtain the embedding of source algebras, due to L. Puig: Corollary 7. ([4, 14.6], [1, 4.10]) Set A = End (G1) (M) 0 and B = End (H1) (N) 0. The functor X? induces an injective homomorphism of H interior P?algebras B?! A which is split as homomorphism of B? B?bimodules; in particular, we have rk (A)=jP j rk (B)=jP j (mod p) : Proof. Since M = X N, tensoring by X induces a homomorphism of interior H P?algebras from B to A. This homomorphism is split injective as homomorphism of B? B?bimodules, because N is isomorphic to a direct summand of Res GP 0 (M) by the usual Green correspondence. Since moreover any other direct summand has a vertex of order strictly smaller than jp j, we get the statement HP 0 on the?ranks. If k is large enough, it is possible to be much more precise about source modules of He and the corresponding source algebras. For this, we recall some standard facts from block theory (see e.g. [5, 39]). 8. Assume that k is large enough for C G (P )e; that is, the semi-simple quotient of C G (P )e is isomorphic to a matrix algebra over k. Set C = C G (P )=Z(P ) and denote by e the image of e in C under the canonical surjection C G (P )?! C The idempotent e is a block of C having the trivial subgroup as defect group, and the block algebra Ce is isomorphic to a matrix algebra M n () over for some positive integer n; 8.2. as block of C G (P ), e has Z(P ) as defect group and the block algebra C G (P )e is isomorphic to the matrix algebra M n (Z(P )) over Z(P );

4 4 J. L. ALPERIN, MARKUS LINCKELMANN, RAPHA EL RUQUIER 8.3. as block of P C G (P ), e has P as defect group and the block algebra P C G (P )e is isomorphic to the matrix algebra M n (P ) over P. The next proposition is a translation of the statements in 8 to our terminology involving source modules, from which we obtain the source algebras in each of the cases in 8 (as originally determined by L. Puig in [4, 14.5, 14.6]): Proposition 9. Let V be a projective indecomposable Ce?module, and let V be the (P (C G (P ) 1))?module obtained by taking a projective cover of V as C G (P )e?module and restricting it through the group homomorphism P (C G (P ) 1)?! C G (P ) which maps (uz; u?1 ) to z for any u 2 P and any z 2 C G (P ). Set U = Ind P C G(P )P 0 (V ) : P (CG(P )1) (i) Up to isomorphism, V is the unique source module of Ce, and we have an algebra isomorphism End (C1) (V ) 0 =. (ii) Up to isomorphism, Res P (C G(P )1)) (V ) is the unique source module of CG(P )Z(P ) 0 C G (P )e, and the natural map is an algebra isomorphism. Z(P )?! End (C G(P )1)(V ) 0 (iii) Up to isomorphism, U is the unique source module of P C G (P )e, and the natural map P?! End (P C G(P )1)(U) 0 is an algebra isomorphism. Proof. Statement (i) is a trivial consequence of the fact that by 8.1 the block algebra Ce is a matrix algebra over and has defect group f1g. The restriction to C G (P ) of V is indecomposable, since it is a projective cover of the simple module k V. Thus V and its restriction to C G (P ) Z(P ) 0 are indecomposable. Moreover, P acts trivially on V. Thus P and Z(P ) are vertices of V and its restriction to C G (P )Z(P ) 0, respectively. This shows already, that Res P (C G(P )1 (V ) is a source module of C CG(P )Z(P ) 0 G (P )e. As the normalizer of the defect group Z(P ) acts transitively on the isomorphism classes of source modules (cf. Remark 3), this shows also the uniqueness, up to isomorphism in this situation. The natural algebra homomorphism in (ii) is injective, since V is projective as Z(P )?module. Since Z(P ) End (C G(P )1)(V ) = End (C1) (V ) =, it follows from Nakayama's lemma, that the algebra homomorphism in (ii) is also surjective. This shows (ii). The natural map P?! End (P C G(P )1)(U) 0 is?split injective, since U is projective as (1 P 0 )?module. In order to show the surjectivity, it suces to show that both sides have the same?rank. By Mackey's formula, we have Res P C G(P )P 0 (U) P CG(P )1 = Ind P C G(P )1 (V ). Thus Frobenius' reciprocity implies that CG(P )1

5 SURCE ALGEBRAS AND SURCE MDULES 5 End (P C G(P )1)(U) = Hom (C G(P )1)(V; (x;1) V ), whith x running over a right x transversal of C G (P ) in P C G (P ). Now (x;1) V = V as left C G (P )?modules for any x 2 P C G (P ), and whence the rank of End (P C G(P )1)(U) is [P C G (P ) : C G (P )]rk (End (C G(P )1)(V )) = [P : Z(P )]jz(p )j = jp j. This shows that U is indecomposable, even when restricted to P C G (P ) 1. Moreover, P is a vertex of V, and thus of U, too. Therefore U is a source module for P C G (P )e. Again by Remark 3, it is unique up to isomorphism. Before we go further, we recall some facts about central extensions of a - nite group by the group of invertible elements in. The reason for central extensions to occur in this context is the following: the module V, viewed as module for P C G (P ) via the canonical homomorphism P C G (P )?! P C G (P )=P = C G (P )=Z(P ), cannot, in general, be extended to H. It can, however, be extended to a twisted group algebra ^H for some central?extension ^H of H which arises naturally from the action of H on the matrix algebra Ce. 10. Assume that k is large enough for the block e of C G (P ). The group H acts on the matrix algebra S = Ce, since H normalizes P and stabilizes e. Let ^H be the subgroup of H S of all pairs (x; s) 2 H S satisfying x t = sts?1 for all t 2 S; that is, such that x and s induce the same automorphism of S. In other words, ^H is the pullback of the obvious maps from H and S to Aut(S). By the Skolem-Noether theorem, any automorphism of S is inner; in particular, the projection to the rst component ^H?! H mapping (x; s) to x is surjective. The kernel of this map is f(1 H ; 1 S )g 2 = ; thus ^H is a central extension of H by. We consider ^H as?group endowed with the map?! ^H sending 2 to ^ = (1 H ; 1 S ) and call it the?group dened by the action of H on S. Its opposite?group is the group H, which is equal to ^H as abstract group, endowed with the group homomorphism?! H sending 2 to = (1 H ;?1 1 S ). The twisted group algebra ^H is the quotient of the group algebra ^H by the ideal generated by the set of elements :^x? 1 :(^^x) with running over and ^x running over ^H. This is an?free algebra of?rank jhj; indeed, if for any x 2 H we denote by ^x an element of ^H lifting x, then the set f^xgx2h is an?basis of ^H, and multiplication is given by ^x^y = x;y cxy for any two x; y 2 H and uniquely determined coecients x;y 2 (this is a two-cocycle determining the central extension ^H of H). If the extension ^H of H splits; that is, ^H = H, any such isomorphism induces an isomorphism of?algebras ^H = H. We identify P C G (P ) with its canonical image in ^H (via the inclusion P CG (P ) H and the obvious homomorphism P C G (P )?! P C G (P )=P = C G (P )=Z(P )?! S ; see [4, 6.5]) and set ^E = ^H=P CG (P ). Similarly, we set E = H=P CG (P ). The?groups ^E and E are central?extensions of E. The projection to the second component ^H?! S mapping (x; s) to s induces a surjective algebra homomorphism ^H?! Ce which extends the canonical

6 6 J. L. ALPERIN, MARKUS LINCKELMANN, RAPHA EL RUQUIER homomorphism P C G (P )e?! Ce. Note that this homomorphism maps the elements of P to the unit element of Ce. In particular, any Ce?module can be viewed as ^H=P?module through this homomorphism. Note that k V is, up to isomorphism, the unique simple Ce?module, which necessarily remains simple when considered as ^H=P?module. Since E is a p 0?group, the inverse image of E in Aut(P ) has Int(P ) as normal Sylow-p?subgroup, thus has a complement isomorphic to E, unique up to conjugacy (cf. [5, (45.1)]), which we are going to denote again by E. We denote by P o E the corresponding semi-direct product. The?group E acts still on P via the canonical surjection E! E, and we denote again by P o E the corresponding semi-direct product, viewed as?group in the obvious way; that is, with the?group structure coming from E. We use the sign to denote the opposite product of a group. Proposition 11. Let : ^H?! H and : ^H?! ^E be the canonical surjections. (i) The product map (; ) : ^H?! H ^E is injective, and its image consists of all pairs (x; ^e) 2 H ^E with the property that the natural images of x and ^e in Aut(P )=Int(P ) coincide; equivalently, the triple ( ^H; ; ) is the pullback of the natural maps from H and ^E to Aut(P )=Int(P ). (ii) Set = NH(Po E) 0(P ). For any (x; u e) 2, where x 2 H, u 2 P, e 2 E, there is a unique ^x = (x; u e) 2 ^H such that (^x) = x and (^x)?1 = e, and this map is a surjective homomorphism of?groups :?! ^H which maps P to P and whose kernel is 1 Z(P ) 0. Proof. (i) Since ker() = and ker() = P C G (P ), the product map (; ) is injective, and for any ^x 2 ^H, the images of (^x) and (^x) in Aut(P )=Int(P ) coincide. Let x 2 H and ^e 2 ^E such that the images of x and ^e in Aut(P )=Int(P ) coincide. This means that the class ^e 2 ^E = ^H=P CG (P ) can be represented by an element of the form ^x = (x; s) of ^H for some s 2 S, which proves (i). (ii) An element (x; u e) 2 H (P o E)0 normalizes P if x and u?1 e?1, the latter now viewed as element of P o ^E, induce the same automorphism of P. In particular, the images of x and e?1 in Aut(P )=Int(P ) coincide, which by (i) implies that there is a unique ^x 2 ^H satisfying (^x) = x and (^x)?1 = e. >From this uniqueness follows also that the map thus dened is a group homomorphism. Conversely, given ^x 2 ^H, the elements x = (^x) 2 H and ^e = (^x) 2 ^E have same image in Aut(P )=Int(P ), which means that there is u 2 P such that (x; u ^e?1 ) normalizes P. This shows the surjectivity of. Moreover, if in this case both x = 1 and ^e = 1, then u 2 Z(P ), which implies that ker() = 1 Z(P ) 0. Finally, maps (u; u?1 ) 2 P to the unique element ^u of ^H satisfying (^u) = u and (^u) = 1 which forces ^u = u.

7 SURCE ALGEBRAS AND SURCE MDULES 7 The group homomorphism in the previous proposition allows us to consider any ^H=P?module as =P?module. In particular, the source module V for Ce from Proposition 9 becomes an =P?module in this way. The next Proposition shows, that the (C G (P ) 1))?module V from Proposition 9 extends to. Proposition 12. Let W be a projective cover of V viewed as =P?module, and consider then W as?module through the canonical map?! =P. Let V be the P (C G (P ) 1)?module as in Proposition 9. We have Res P (CG(P )1)(W ) = V : Proof. As left C G (P )?module, V is a projective cover of the simple module k V ; in particular, V has a unique maximal submodule rad(v ). Since k V is still simple as =P?module, W has a unique maximal submodule rad(w ), too, Res P (CG(P )1)(W ) has a direct summand isomorphic to V and we have Res P (CG(P )1)(W=rad(W )) = V=rad(V ) = k V : This isomorphism implies that rad(w ) is also a maximal submodule of Res P (CG(P )1)(W ). The group homomorphism :?! ^H from Proposition 11.(ii) induces an isomorphism =P (CG (P ) 1) = ^H=P C G (P ) = ^E. As E is a p 0?group, we have therefore J( ) = :J(P (CG (P ) 1)). This means that rad(w ) is also the radical and thus the unique maximal submodule of Res P (CG(P )1)(W ); in particular, the latter is indecomposable and whence isomorphic to V. We determine now the source modules of He and obtain from this description the structure of the source algebras of He as given by Puig in [4, 14.6] (extending earlier work of B. Kulshammer on blocks with normal defect groups [2]). We keep the notation introduced above. Theorem 13. Set N = Ind HP 0 P CG(P )P 0 (U), where U is a source module for P C G (P )e. (i) Up to isomorphism, N is the unique source module for He. (ii) We have N = Res H(Po E) 0 Ind H(Po E) 0 (W ). HP 0 (iii) As right (P o E)?module, Ind H(PoE) 0 (W ) is free of rank n. (iv) (Puig [14.6]) We have an isomorphism of interior P?algebras End (H1) (N) 0 = (P o E) ;

8 8 J. L. ALPERIN, MARKUS LINCKELMANN, RAPHA EL RUQUIER in particular, as an?algebra, He is isomorphic to the matrix algebra M n ( (P o E)). Proof. Since U is an indecomposable direct summand of P C G (P )e as (P C G (P ) P 0 )?module having P as vertex, the module N is isomorphic to a direct summand of He as (H P 0 )?module having any source module of He as direct summand. By Proposition 9 we have N = Ind HP 0 P (CG(P )1)(V ), and by Proposition 12 together with Mackey's formula, we get N = Ind HP 0 P (CG(P )1) ResP (CG(P )1)(W ) = Res H(Po E) 0 Ind H(Po E) 0 (W ) : HP 0 In particular, any such isomorphism shows that N has a right (P o E)? module structure, or equivalently, any such isomorphism induces a homomorphism of interior P?algebras (P o E)?! End(H1) (N) 0 mapping a 2 (P o E) to the endomorphism of N given by the action of a on N. Another application of Mackey's formula shows that Res H(Po E) 0 1(PoE) Ind H(Po E) 0 (W ) = Ind 1(Po E) 0 Res 0 1Z(P ) 0 1Z(P ) 0(W ) is free as right (P o E)?module by 12 and has therefore rank n by 8.2; in particular, the homomorphism 13.1 is?split injective. In order to show that 13.1 is an isomorphism it suces to show that both sides have the same?rank. By an appropriate version of Frobenius' reciprocity, we have an?linear isomorphism End (H1) (N) = Hom (P C G(P )1)(U; Res HP 0 (N)), and Mackey's P CG(P )1 formula implies that Res HP 0 (N) P CG(P )1 = (x;1) U, where x runs over a set of representatives in H of H=P C G (P ). x The direct summands (x;1) U in this sum are all isomorphic to U as left P C G (P )?modules since the elements of H stabilize e and since P C G (P )e has only one isomorphism class of projective indecomposable modules (cf. 8.3). This shows that rk (End (H1) (N)) = jejrk (End (P C G(P )1)(U)) = jejjp j, where the last equality comes from 9(iii). Therefore 13.1 is an isomorphism. Since we know already that every source module of He occurs as direct summand, up to isomorphism, of N, all we have to show is that N is indecomposable as (H P 0 )?module, or equivalently, that the unit element of (P E) is primitive in ( (P o E)) P. This is well-known, and there are various possibilities to show this: one way is to observe that the rank of any source algebra of He is at least jp jjej because for any x 2 H, any source algebra has a direct summand isomorphic to [P x] as (P P 0 )?module. A more direct proof goes as follows, involving the Brauer construction [5, 11]: rst, ( (P o E))(P ) = Z(P ) and so (P o E) has a unique block and a unique local point of P whose multiplicity is then necessarily 1; second, E is a p 0?group

9 SURCE ALGEBRAS AND SURCE MDULES 9 acting on the semi-simple quotient of (P o E) P permuting transitively the indecomposable direct algebra summands (because any E?orbit yields an E?stable idempotent which lifts to a block of (P o E), whence must be the unit element). Combining these observations shows that the unit element is primitive in (P o E)P. Yet another proof of this fact follows from decomposing (P o E) = e2e P e as (P P 0 )?module; the direct summands in this decomposition are indecomposable, mutually non isomorphic and transitively permuted by the action of E. Thus (P o E) is indecomposable as ((P o E) P 0 )?module, showing again that the unit element is primitive in ( (P o E))P. This concludes the proof of Theorem 13. References 1. Y. Fan, L. Puig, n blocks with nilpotent coecient extensions, preprint (1997). 2. B. Kulshammer, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985), 147{ L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265{ L. Puig, Pointed groups and construction of modules, J. Algebra 116 (1989), 7{ J. Thevenaz, G-Algebras and Modular Representation Theory, xford Science Publications, Clarendon Press, xford, J. L. Alperin University of Chicago Department of Mathematics 5734 University Avenue Chicago, Illinois U.S.A. Markus Linckelmann and Raphael Rouquier CNRS, Universite Paris 7 UFR de Mathematiques 2, place Jussieu Paris Cedex 05 FRANCE