Frobenius Categories versus Brauer Blocks: The Grothendieck Group of the Frobenius Category of a Brauer Block

Progress in Mathematics Volume 274 Series Editors H. Bass J. Oesterlé A. Weinstein Lluís Puig Frobenius Categories ...

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Progress in Mathematics Volume 274

Series Editors H. Bass J. Oesterlé A. Weinstein

Lluís Puig

Frobenius Categories versus Brauer Blocks The Grothendieck Group of the Frobenius Category of a Brauer Block

Birkhäuser Basel · Boston · Berlin

Author: Lluís Puig CNRS, Institut de Mathématiques de Jussieu Université Denis Diderot (Paris VII) 175, Rue du Chevaleret 75013 Paris France e-mail: [email protected]

2000 Mathematics Subject Classification 20C11 Library of Congress Control Number: 2009921943 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978 3 7643 9997 9 Birkhäuser Verlag AG, Basel ∙ Boston ∙ Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-9997-9

e-ISBN 978-3-7643-9998-6

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 General notation and quoted results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Frobenius P -categories: the ﬁrst deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . 27 3 The Frobenius P -category of a block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nilcentralized, selfcentralizing and intersected objects in Frobenius P -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Alperin fusions in Frobenius P -categories . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exterior quotient of a Frobenius P -category over the selfcentralizing objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

39 47 57 73

Nilcentralized and selfcentralizing Brauer pairs in blocks . . . . . . . . . . . 93

8 Decompositions for Dade P -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9 Polarizations for Dade P -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10 A gluing theorem for Dade P -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11 The nilcentralized chain k*-functor of a block . . . . . . . . . . . . . . . . . . . . . . 151 12 Quotients and normal subcategories in Frobenius P -categories . . . . . 179 13 The hyperfocal subcategory of a Frobenius P -category . . . . . . . . . . . . .195 14 The Grothendieck groups of a Frobenius P -category . . . . . . . . . . . . . . . 211 15 Reduction results for Grothendieck groups . . . . . . . . . . . . . . . . . . . . . . . . 241 16 The local-global question: reduction to the simple groups . . . . . . . . . . 287 17 Localities associated with a Frobenius P -category . . . . . . . . . . . . . . . . . 319 18 19

The localizers in a Frobenius P -category . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Solvability for Frobenius P -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 sc

A perfect F-locality from a perfect F -locality . . . . . . . . . . . . . . . . . . . . 369 21 Frobenius P -categories: the second deﬁnition . . . . . . . . . . . . . . . . . . . . . . 389

20

22

The basic F-locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

23

Narrowing the basic F -locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

24

Looking for a perfect F -locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

sc

sc

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Introduction

I 1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem aﬃrming that, for a prime p and a ﬁnite group G , if the quotient of the normalizer by the centralizer of any p-subgroup of G is a p-group then, up to a normal subgroup of order prime to p , G is a p-group. Of course, it would be an anachronism to pretend that Frobenius, when doing this theorem, was thinking the category — noted FG in the sequel — where the objects are the p-subgroups of G and the morphisms are the group homomorphisms between them which are induced by the G-conjugation. Yet Frobenius’ hypothesis is truly meaningful in this category. I 2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotency of the so-called Frobenius kernel of a Frobenius group G with arguments — at that time completely new — which might be rewritten in terms of FG ; indeed, some time later, following these kind of arguments, George Glauberman [27] proved that, under some — rather strong — hypothesis on G , the normalizer N of a suitable nontrivial p-subgroup of G controls fusion in G , which amounts to saying that the inclusion N ⊂ G induces an equivalence of categories FN ∼ = FG . I 3 Thus, when about forty years ago we start working on ﬁnite groups, these kind of results pushed us to introduce the Frobenius category FG in the language of Claude Chevalley’s seminar. At the beginning, it was essentially a language to guide our research, as for instance in our reﬁnement [35, Ch. II and III] of the Alperin Fusion Theorem [1]. Moreover, we quickly realized that, in order to have a better insight into the structure of G — for instance, to follow Thompson’s arguments in Chapter IV of the so-called Odd paper [25] — we had to consider suitable extensions determined by G of this category, namely what in [35, Ch. VI] we call “localit´es a ` ´epimorphismes”. I 4 The next step in the gestation of the ideas contained in this book came thirty years ago, when Michel Brou´e talked us about his will of extending to the Brauer blocks our Frobenius categories for groups. As a matter of fact, we found that Richard Brauer already had partially realized this project in [9]; indeed, in this paper, for what Brauer calls a block of characters of G , he considers a new order relation reﬁning the inclusion between a suitable set of subgroups of a defect group of the block, already proving a suitable generalization of the Alperin Fusion Theorem in the new context.

2

Frobenius categories versus Brauer blocks

I 5 Then, Michel Brou´e† reformulated the new inclusion introduced by Brauer in terms of pairs — which freed Brou´e from the choice of a defect group — formed by a p-subgroup P of G and a block of characters e (for short, a block in the sequel) of the centralizer CG (P ) of P , and extended the new inclusion to all these pairs — ever since called Brauer pairs. At this point, we already had a Frobenius category for a block b of G , namely the category F(b,G) where the objects are all the Brauer pairs (P, e) containing ({1}, b) — with respect to the new inclusion — and where, again, the morphisms are the group homomorphisms between the p-subgroups induced by the G-conjugation, up to Brauer-Brou´e’s inclusion. I 6 The miracle happened again: the Frobenius Hypothesis on F(b,G) — all the automorphism groups of the objects are p-groups — was meaningful too; in this situation — where b is called a nilpotent block — with Michel Brou´e we proved in [12] that there is a bijection between the set of ordinary irreducible characters of G in b and the set of ordinary irreducible characters of any defect group of b . This immediately implies that the block algebra over C is Morita equivalent to the group C-algebra of the defect group of b ; in other words, denoting by (P, e) a maximal Brauer pair containing ({1}, b) and identifying b with the corresponding central idempotent, it implies that the categories of CGb- and CP -modules are equivalent. I 7 However, Brou´e guessed something more precise, namely that even over a complete discrete valuation ring O , the categories of OGb- and OP -modules were equivalent. In our joint work [12], we already proved that the centers Z(OGb) and Z(OP ) were isomorphic which, as Everett Dade pointed out to us, proved Brou´e’s conjecture whenever P is Abelian. Soon after, following James Green’s approach [29], we came to the idea of the source algebra of the block b — which is more than an O-algebra: it is the O-algebra i(OGb)i endowed with the homomorphism mapping u ∈ P on ui , for a primitive idempotent i of (OGb)P such that BrP (i) = 0 (cf. 1.12) — and succeeded in determining the source algebra of a nilpotent block [36], which in particular proved Brou´e’s conjecture. I 8 As for the Frobenius Theorem mentioned above, it could be claimed that, strictly speaking, the category F(b,G) is unnecessary to deﬁne a nilpotent block since we are just assuming that the quotient of the normalizer by the centralizer of any Brauer pair (Q, f ) fulﬁlling f BrQ (b) = 0 (cf. 1.13) is a p-group. Yet, the existence of the hyperfocal subalgebra in the source algebra of a block b — proved in [49] ten years ago — involves F(b,G) more seriously. †

A point of history. Before his stay at Chicago University where his joint paper with Jon

Alperin [3] comes from, Michel Brou´ e already had given a complete account of his reformulation in Chevalley’s seminar.

Introduction

3

I 9 Indeed, it is well-known that the direct limit lim FG is the maximal −→

Abelian p-quotient of G [28, Ch. 7,Theorem 3.4]: pushing it further, from the Frobenius category FG it is possible to compute, inside a Sylow p-subgroup, its intersection — called a hyperfocal subgroup of G — with the kernel of the maximal p-quotient of G . Mimicking this computation in F(b,G) , we can deﬁne the hyperfocal subgroup H(b,G) of the block b of G ; the point is that there exists an essentially unique P -stable subalgebra of the source algebra i(kGb)i which intersects P i in H(b,G) i and, together with P i , generates the whole source algebra [51, Theorems 14.7 and 15.10]. I 10 However, the step which led us to seek for an abstract setting behind all these constructions was the discovery — twenty years ago — of the localizer of a selfcentralizing Brauer pair, together with the localizing functor over the category of chains of such Brauer pairs [44]. The selfcentralizing Brauer pairs (Q, f ) are exactly those considered by Brauer in [9] and one of their possible deﬁnitions — justifying their name — is that CP (Q) ⊂ Q for any Brauer pair (P, e) containing (Q, f ) ; in this case, it follows from [34] that there exists a suitable extension LG (Q, f ) — the localizer of (Q, f ) — of the quotient NG (Q, f )/CG (Q) by the center Z(Q) . In the Frobenius category of a ﬁnite group G , these extensions were just the automorphism group of the objects in some of the “localit´es a ` ´epimorphismes” of G mentioned above. I 11 At that time there appeared the paper by Reinhard Kn¨ orr and Geoﬀrey Robinson [33] where they reformulated Jon Alperin’s Conjecture [2] in terms of an alternating sum over a set of chains of Brauer pairs — we are more precise below. Thus, quite naturally we considered the localizers LG (q) of chains q — here chain stands for totally ordered set throughout the inclusion — of selfcentralizing Brauer pairs, and in [44] we prove that this correspondence can be extended to a functor from the suitable category of chains — where the morphisms are deﬁned by the set inclusion and the G-conjugation — to the category of ﬁnite groups up to conjugation, namely (cf. 1.3). to the exterior quotient Gr I 12 Throughout all this work, it became clear that many arguments were inner arguments, in the sense that the blocks around it played no signiﬁcant role, and we decided to look for a suitable abstract formulation. Actually, this was a reason to delay publication of [44] since we could hope to recover its contents in a more general setting, as we did. A key point of the endeavour to ﬁnd such an axiomatic approach was the possibility to come back to Brauer’s point of view in [9], namely to the subgroups of a defect group; indeed, as for FG , if (P, e) is a maximal Brauer pair containing ({1}, b) , then F(b,G) is equivalent to the full subcategory over the set B(P,e) of Brauer pairs (Q, f ) contained in (P, e) and the correspondence mapping (Q, f ) on Q is an order-preserving bijection between B(P,e) and the set of subgroups of P [1, Theorem 3.4].

4

Frobenius categories versus Brauer blocks

I 13 In autumn 1990, we began to work in this direction when doing a series of lectures [44] at the MSRI, and in spring 1991, invited by James Green at Warwick University, we already could give a ﬁrst deﬁnition of an abstract Frobenius category†. The starting point was obvious: from a ﬁnite p-group P , we had to consider a subcategory of the category of ﬁnite groups Gr deﬁned over the set of subgroups of P , the problem being to ﬁnd suitable conditions tightening the situation enough to remain near the Frobenius categories of blocks of ﬁnite groups. I 14 That is to say, it was not diﬃcult to imagine reasonable necessary conditions, yet they should allow us to go far toward mimicking the usual constructions in groups. . . till where? Although the deﬁnition did not change from spring 1991, we spent some time developing the machinery of normalizers, centralizers, quotients, the translation of our reﬁnement of Alperin’s Fusion Theorem. . . till we were able to state a reasonable criterion of simplicity — in the sense that, up to a normal subgroup of order prime to p , a ﬁnite group G is simple whenever the Frobenius category of G fulﬁlls this abstract criterion (cf. 12.20). We wrote all this in a manuscript [46] which, in its ﬁrst half, essentially covered chapters 2, 4, 5 and 12 below. I 15 In chapter 2, we state the conditions deﬁning a Frobenius P -category — conditions admitting diﬀerent equivalent forms — and show the existence of normalizers and centralizers of the subgroups of P , fulﬁlling the same conditions. The title of this chapter — Frobenius P -categories: the ﬁrst definition — suggests that we have, at least, another deﬁnition; indeed, related with our eﬀort for proving the existence of a perfect locality explained below, we found a quite diﬀerent equivalent deﬁnition of a Frobenius P -category, stated in chapter 21. Although simpler from the formal point of view, this deﬁnition is farther from our main intuition and, for the moment, it takes second place. I 16 As announced in its title, the purpose of this book is not only to develop our abstract setting but to apply it to a better understanding of Brauer blocks. In chapter 3 we prove that the category F(b,G) considered above (cf. I5) fulﬁlls the conditions of a Frobenius P -category and, more generally, we illustrate in this case all the concepts introduced in 2. I 17 In chapter 4 we come back to the abstract Frobenius P -categories F to introduce the selfcentralizing objects, mimicking the deﬁnition of selfcentralizing Brauer pairs mentioned above (cf. I 10); indeed, these objects play the most important role in the structure of F and, actually, the full subsc category F of F over the set of them determines F , as we prove in this chapter. But, in many arguments, the selfcentralizing Brauer pairs (Q, f ) play a †

At present, there is some confusion with the terminology employed around this concept.

Although we hope that this book will contribute to ﬁx the original one, we mention some alternative names in footnotes and they can be found in italics in the Index.

Introduction

5

role simply because f is a nilpotent block of CG (Q) (cf. I 6), and this condition is preserved in a quotient of G by a central p-subgroup Z , whereas the selfcentralizing condition need not. Analogously, we introduce the nilcentralized nc objects — we note F the full subcategory over the set of them; this is possible since we already have centralizers of the objects in F and the nilpotency of a Frobenius P -category makes sense via the Frobenius Hypothesis above. I 18 Chapters 5 and 6 further illustrate the importance of selfcentralizing objects. In chapter 5 we develop in F our reﬁnement of the Alperin Fusion Theorem mentioned above, introducing the essential objects; actually, these objects already can be introduced in simpler structures called divisible P -categories, and then a suitable formulation of the Alperin Fusion Theorem becomes a necessary and “almost” suﬃcient condition to get a Frobenius P -category. In this case, the essential objects are selfcentralizing, proving sc again that F is completely determined by the subcategory F . Moreover, in our approach we show that the Alperin Fusion Theorem concerns the additive category ZF — where the morphisms between a pair of subgroups Q and R of P are the free Z-modules over F(Q, R) and the composition is distributive — rather than F itself, and it is somehow related to the projective resolution of the trivial contravariant functor over F ; we believe that this relationship deserves more consideration. sc I 19 Chapter 6 exploits a remarkable feature of the exterior quotient F˜ sc

sc

of F — the quotient of F by the inner automorphisms of the corresponding subgroups of P (cf. 1.3) — namely the fact that in this category any morphism is an epimorphism. This leads to a canonical partition of the set of morphisms between two objects relative to a morphism with the same origin (cf. Proposition 6.7); then, the existence of these partitions implies the sc sc existence of a direct product in the additive cover ac(F˜ ) of F˜ (cf. 6.2) — a construction introduced by Stefan Jackowski and James McClure in [32]. As a consequence, we get a vanishing result for positive cohomology, which is the key for the determination of the rank of the Grothendieck groups associated with F in chapter 14. I 20 As in chapter 3, in chapter 7 we analyze the nilcentralized — and the selfcentralizing — objects mentioned above (cf. I 17), inside the category F(b,G) associated with a block b of G . As a matter of fact, any nilcentralized Brauer pair (Q, f ) fulﬁlling f BrQ (b) = 0 (cf. 1.13) appears associated with two meaningful invariants, namely with a central extension Fˆ(b,G) (Q) of the automorphism group F(b,G) (Q) ∼ = NG (Q, f )/CG (Q)

I 20.1

by k ∗ — where k is a ﬁxed algebraically closed ﬁeld of characteristic p — ¯P (Q)-algebra SQ — a simple k-algebra endowed with an and with a Dade N ¯P (Q) = NP (Q)/Q which stabilizes a basis containing the unity action of N element (cf. 1.20).

6

Frobenius categories versus Brauer blocks

I 21 As we explain below, from the point of view of Alperin’s Conjecture [2] we were interested in the central extension Fˆ(b,G) (Q) rather than in the group F(b,G) (Q) itself, and this raised a huge problem of coherence. Explicitly, for any chain q (cf. I 11) of nilcentralized Brauer pairs fulﬁlling f BrQ (b) = 0 (cf. 1.12), we can deﬁne F(b,G) (q) as the stabilizer of q in NG (Q, f )/CG (Q) for the maximal element (Q, f ) in this chain, and then Fˆ(b,G) (q) is the corresponding converse image. It was clear that the map sending q to F(b,G) (q) might be extended to a functor from the corresponding category of chains to Gr ; yet, from the point of view of Alperin’s Conjecture, this functor was useless unless we were able to get a lifting of it sending q to Fˆ(b,G) (q) . I 22 Since the eighties, we knew how to prove the existence of a lifting Fˆ(b,G) (q) −→ Fˆ(b,G) (r)

I 22.1

for any morphism r → q between chains (cf. Theorem 7.16), and this was already announced in [44]; it is a consequence of a Splitting Theorem that Everett Dade announced in 1979 at the Santa Cruz Conference [20] and never published (cf. Theorem 9.21 below or [40]). But the huge problem is to prove that it is possible to do coherent choices in order to get a functor, a question that we only have solved when preparing this book. I 23 In the meanwhile, we tentatively have followed two strategies. In one of them, we tried to improve our proof in [40] of Dade’s Splitting Theorem, showing that it was possible to make a choice once for ever — a polarization — in the set of equivalence classes of Dade P -algebras for any ﬁnite p-group P , which then we could apply to our problem. Although what we obtained in this direction did not solve the problem, we explain our result in chapter 9; in particular, it includes a proof of Dade’s Splitting Theorem somewhat diﬀerent and more detailed than in [40]. Previously, in chapter 8 we recall the main facts on Dade P -algebras we need in the sequel. I 24 Still since the eighties, we already knew that the existence of a ¯P (Q)-algeF(b,G) (q)-stable Dade P -algebra S gluing together all the Dade N P bras SQ — namely, a Dade P -algebra S such that ResQ (S) and Res ϕ (S) are equivalent for any nilcentralized Brauer pair (Q, f ) contained in (P, e) and any F(b,G) -morphism ϕ from (Q, f ) to (P, e) , and that SQ is equivalent to the Brauer quotient S(Q) (see 11.6) — easily would solve our problem. Indeed, replacing kGb by the P -algebra S ◦ ⊗k ResG P (kGb) , we could ﬁnd inside the same extensions Fˆ(b,G) (q) and Fˆ(b,G) (r) — up to noncanonical isomorphisms — and then this P -algebra provided coherent extension group homomorphisms Fˆ(b,G) (q) −→ Fˆ(b,G) (r) I 24.1 (see Proposition 11.8 and Theorem 11.10 for the precise arguments).

Introduction

7

I 25 Thus, the other strategy was to look for such a Dade P -algebra. One way to ﬁnd S was from a conjectural gluing result on Dade P -algebras, which is indeed true when P is Abelian [45]; but presently, after the complete classiﬁcation of the Dade P -algebras in [17] and [6], such a general result seems impossible. In chapter 10 we state a result in this direction which should be combined with a suitable polarization. . . that does not exist. Another way to ﬁnd S could be from the P -algebra ResG P (kGb) itself and, although we have no construction to propose, we cannot close this possibility. I 26 Actually, the existence of such a Dade P -algebra S also would prove nc nc the existence of a suitable k ∗ -extension Fˆ of F — a question raised by Markus Linckelmann in the 2002 Durham Symposium — which naturally appears inside S ◦ ⊗k ResG P (kGb) . It is this fact that gives a solution to our ¯P (Q)-algebras SQ themselves, we are able to conproblem: from the Dade N nc struct k ∗ -extensions of suitable subcategories of F and the point is that these k ∗ -extensions are quite independent of our choice of Dade algebras; nc although we do not get a complete k ∗ -extension Fˆ , we get a complete coherent choice for the extension group homomorphisms I 24.1. All this is explained in chapter 11. I 27 In chapters 12 and 13, we pursue the development of the abstract Frobenius P -categories. In chapter 12 we discuss the existence of quotients of a Frobenius P -category F by suitable subgroups of P . If F = FG is the Frobenius category of a ﬁnite group G , any Sylow p-subgroup of a normal subgroup of G naturally determines one of those quotients of F ; although the converse is not true, using the Classiﬁcation of Finite Simple Groups, it is not diﬃcult to check that, provided P is not Abelian, there are not so many exceptions. But the main purpose of this chapter is to determine the p -quotients of F — the quotients reduced to a ﬁnite group of order prime to p — since we need them to state the simplicity criterion (cf. 12.20) and to talk about solvability in Frobenius P -categories. As a matter of fact, all the p -quotients of F have a kernel which is also a Frobenius P -category, and there is a smallest such Frobenius P -category (cf. Corollary 12.17). I 28 In chapter 13 we discuss the p-quotients of F ; as implicitly mentioned in I9 above, it is still possible to deﬁne the hyperfocal subgroup HF of P in F ; once again, P/HF is indeed the maximal p-quotient of F and there exists a suitable Frobenius HF -category F h — called the hyperfocal subcategory of F . Then, the contents of these two chapters allow us to state a deﬁnition of solvability for Frobenius P -categories, and one main point is that a solvable Frobenius P -category is necessarily the Frobenius category of a p-solvable ﬁnite group, which responds positively to our requirement of remaining near the Frobenius categories of ﬁnite groups (cf. I 13). This result does not appear till chapter 19 since its proof needs the existence of the localizer proved in chapter 18.

8

Frobenius categories versus Brauer blocks

I 29 Coming back to the Frobenius category F(b,G) of a block (b, G) , we already know from chapter 11 that there is a functor from the proper nc category of (F(b,G) ) -chains (cf. A2.8) to the category k ∗ -Gr of the central nc k ∗ -extensions of ﬁnite groups sending a (F(b,G) ) -chain q to the k ∗ -extension Fˆ(b,G) (q) above (cf. I 21). On the other hand, the modular Grothendieck group

evidently deﬁnes a contravariant functor from k ∗ -Gr to the category of free Z-modules. A main fact stated in this book is that the inverse limit of the composition of both functors — called the Grothendieck group of F(b,G) (cf. 14.3) — is a free Z-module tightly related with Alperin’s Conjecture.

I 30 Let us be more explicit. Recall that, following Alperin [2], a weight of the block b is a pair (Q, χ) formed by a p-subgroup Q of G and by an irreducible character χ of NG (Q) , associated with BrQ (b) (cf. 1.13), which comes ¯G (Q) = NG (Q)/Q of from a modular irreducible character χ ¯ in a block of N defect zero (cf. 1.17). Then, Alperin’s Conjecture aﬃrms that the number of G-conjugacy classes of weights of the block b coincides with the number |Irrk (G, b)| of modular irreducible characters in b . ¯G (Q)-module M aﬀording χ I 31 But, for a weight (Q, χ) , an N ¯ is simple ¯ and projective (cf. 1.17), and therefore its restriction to CG (Q) is semisimple and projective too, determining a set of blocksf¯ of C¯G (Q) of defect zero (cf. 1.17). That is to say, lifting f¯ to a block f in Z kCG (Q) , (Q, f ) is a selfcentralizing Brauer pair (cf I 10 and Corollary 7.3) and, up to G-conjugation, may assume it is contained in (P, e) ; moreover, it is not diﬃcult to prove that the set of weights determining the same selfcentralizing Brauer pair (Q, f ) bijectively correspond with the set of isomorphism classes of k∗ Fˆ(b,G) (Q)-modules, both simple and projective [53, Theorem 3.7]. Consequently, denoting by IrPrk Fˆ(b,G) (Q) the corresponding set of modular characters, Alperin’s Conjecture aﬃrms IrPrk Fˆ(b,G) (Q) |Irrk (G, b)| = I 31.1 Q

where Q runs over a set of representatives for the F(b,G) -isomorphism classes of F(b,G) -selfcentralizing objects. I 32 At this point, the arguments of Reinhard Kn¨ orr and Geoﬀrey Robinson in [33] prove, in the language above, that Alperin’s Conjecture is equivalent to the equality |Irrk (G, b)| = (−1)|q|−1 Irrk Fˆ(b,G) (q) I 32.1 q

where q runs over a set of representatives for the F(b,G) -isomorphism classes of sc (F(b,G) ) -chains (cf. I 11). But, in chapter 14 we prove that the Z-rank of the Grothendieck group of F(b,G) coincides with the right-hand member of this

Introduction

9

equality — actually, the statement makes sense for any Frobenius P -category F endowed with an analogous functor from the category of nilcentralized F-chains to k ∗ -Gr , and we prove that equality in this general context. Thus, Alperin’s Conjecture is equivalent to the assertion that the Z-ranks of the Grothendieck groups of the block (b, G) and the category F(b,G) coincide. I 33 Everyone understands that the sentence “an alternating sum of ranks which coincides with the rank of an inverse limit” necessarily suggests the possible existence of a diﬀerential complex with a unique nonzero cohomology group at degree zero. But, it has to be noticed that our consc travariant functor mapping an (F(b,G) ) -chain q on the Grothendieck group of Fˆ(b,G) (q) does not come from a contravariant functor deﬁned over the catesc gory (Fˆ(b,G) ) ; in other words, we are not dealing with the usual cohomology sc groups of (Fˆ(b,G) ) . Moreover, our sum runs over a set of representatives for

a set of suitable F(b,G) -isomorphism classes and this fact has to be integrated in our hypothetical diﬀerential complex.

I 34 All these remarks forced us to enlarge the usual construction of the cohomology groups of a category C in order to include our situation; the new cohomology groups we consider need not fulﬁll the long exact sequence condition (cf. A3.11.4), but they are useful for our purposes. We explain our point of view — which possibly has been already employed in other situations — in the Appendix. We have adopted the language of the 2-categories since, when constructing the cohomology groups of C , we dislike expressions such as “consider a sequence of n C-morphisms which can be composed”, that we replace by “consider a functor from ∆n to C”; but, if the simplex ∆n becomes a category then the simplicial category ∆ becomes a 2-category. . . I 35 Let us come back to our discussion on Alperin’s Conjecture. Of course, two free Z-modules with the same Z-rank are isomorphic, and therefore Alperin’s Conjecture is also equivalent to the assertion that the Grothendieck groups of the block (b, G) and the category F(b,G) are isomorphic. But, the stabilizer Out(G)b of b in the group of outer automorphisms of G has a natural action over both Grothendieck groups and then an obvious question arises: is there an Out(G)b -stable isomorphism between the Grothendieck groups of the block (b, G) and the category F(b,G) ? Actually, even a positive answer for a suitable scalar extension of the Grothendieck groups would be welcome. I 36 A good indication towards a positive answer to this question is that, up to a suitable scalar extension, both Grothendieck groups have the same behaviour throughout the restriction to the normal subgroups, provided we nc can “follow” the subcategory (F(b,G) ) in the normal subgroups; we expose our reduction results in chapter 15. Here there appears a signiﬁcant diﬀerence between our method and the method which consists of restricting any irreducible character in the block b individually; indeed, in the second one,

10

Frobenius categories versus Brauer blocks

we are forced to apply the so-called Cliﬀord Theory [31, Ch. V] which involves the unknown Cliﬀord extensions of the stabilizers, which makes any tentative induction enormously diﬃcult. I 37 In chapter 16 we develop a strategy toward reducing a possible positive answer to the question above, to a positive veriﬁcation of the same question “around” the noncommutative simple groups; by “around” we mean that, for any noncommutative simple group S , we have to consider the central k ∗ -extensions of suitable subgroups of Aut(S) containing S . Our strategy itself already needs the Classiﬁcation of Simple Groups since it quotes some facts which are only known from this classiﬁcation, as for instance the solvability of Out(S) . The precise result is stated in Theorem 16.45. I 38 The last part of this book deals again with an abstract Frobenius P -category F . In the second half of our manuscript [46], we investigated to what extent we still got localizers together with the localizing functor (cf. I 10) in our abstract setting — the main purpose and the crucial test in building it. The answer had been “almost” positive — as we explain in 18.5, it remained to prove that some 1-cocycle was a 1-coboundary, which now is done — and, since it was not reasonable to foresee a ﬁnite group as a possible direct limit of the localizing functor , we considered the possibility of the existence of a topological space as, roughly speaking, a direct limit of the functor deﬁned by the classifying spaces of the localizers. I 39 In 1994, we proposed this idea to Dave Benson, who already had constructed a topological space [5] from a conﬁguration considered by Ron Solomon when discussing ﬁnite simple groups with the same Sylow 2-subgroups P as the third Conway’s group [56]; actually, as it could be expected, Solomon’s conﬁguration is nothing but a Frobenius P -category [13]. After a sc while, Benson raised the question of the existence of an extension L of the sc full subcategory F mimicking a suitable “localit´e ` a ´epimorphismes” (cf. I 3), namely having the localizer as the automorphism group of any selfcentralizing object. I 40 After Benson’s publication [5], Carles Broto, Ran Levi and Bob Oliver became interested in our manuscript [46]† in order to prove that the sc topological space coming from the category L guessed by Benson had good enough properties to be a “classifying space” of F ; in [13] they proved the sc “good properties” of L but did not succeed in proving its existence and its uniqueness, just giving some suﬃcient conditions. †

A point of history. In december 1999, coming back from Wuhan, we found an e-mail sent

by Bob Oliver asking us for a copy of our manuscript. We personally gave him a copy on the basis of a possible collaboration. Only in October 2000, did we learn that Carles Broto and Ran Levi were not only interested but already deeply engaged in our manuscript.

Introduction

11

I 41 It has to be understood that the existence of a suitable extension sc of F , or even of F , already supplies localizers and localizing functors — namely, the automorphism groups of the objects in such an extension and the automorphism group functor over the corresponding category of chains (cf. Proposition A2.10). Thus, in chapter 17 we systematically consider extensions of F — called F-localities since they generalize our point of view in [35] — which we would like to be determined by F and P ; for some precise meaning of the word determined , this condition imposes a biggest possibility that we call perfect F-locality and corresponds to the category expected by Benson. In particular, we prove that if F holds a perfect F-locality L , then ¯ any quotient F¯ of F as considered in chapter 12 holds a perfect F-locality L¯ deﬁned as a quotient of L ; of course, had we the existence and uniqueness of a perfect F-locality such a result would be redundant. I 42 In this case, chapter 18 on the localizers would be somewhat redundant too, since the localizers we announce would be nothing but the automorphism groups of the objects in this category, as we said above. But, the localizer of a selfcentralizing object admits a direct group-theoretical characterization, given in [46] and coming from [34], which deserves to be stated. Moreover, as we mention above, this result is useful to prove, in chapter 19, that a solvable Frobenius P -category is necessarily the Frobenius category of a p-solvable ﬁnite group. In chapter 18 we also prove the existence and the uniqueness of the localizing functor locF mentioned above (cf. I 10), together with some kind of “universality” of it, which is quite useful in chapter 23. sc

sc

I 43 In chapter 20 we prove that the existence of a perfect F -locality L forces the existence of a perfect F-locality L by a direct necessarily unique sc construction of L from L ; we obviously proceed by induction, but cannot avoid the distinction between the fully centralized subgroups of P (cf. 2.6) and the others, as we cannot avoid the distinction between normal and ordinary inclusions. All this generates a long proof even if it is nothing but routine. Does there exist a general result guaranteeing that some kind of properties sc of F can be extended to F ? I 44 In chapter 21 we expose the second deﬁnition of a Frobenius P -category F , which leads to the basic F-locality. This equivalent deﬁnition comes from an original contribution of Broto, Levi and Oliver† to the behaviour of a Frobenius P -category F , namely the existence of a suitable P × P -set Ω , where P acts freely on the left and on the right, which has some precise F-stable property and P × P -orbits determined by F [13, Proposition 5.5] — that we call F-basic. Roughly speaking, Ω keeps some properties — which can be stated in terms of the Frobenius P -category FG — of the action of a Sylow p-subgroup P on a ﬁnite group G , by left and right multiplication. †

They credit Markus Linckelmann and Peter Webb for the original idea.

12

Frobenius categories versus Brauer blocks

I 45 But, the point is that we can deﬁne a basic P × P -set Ω independently of any Frobenius P -category, as a P × P -set with free actions on the left and on the right, fulﬁlling suitable extreme equalities — actually, the conditions are so simple that to give further details amounts to stating our deﬁnition here! Then, any basic P × P -set Ω supplies a Frobenius Ω P -category F and, by the Broto-Levi-Oliver result mentioned above, any Frobenius P -category comes from a basic P × P -set; of course, a Frobenius P -category F may come from two diﬀerent basic P × P -sets Ω and Ω , but then we can construct a third basic P × P -set Ω containing both Ω and Ω , and still fulﬁlling F

Ω

=F.

I 46 In particular, for any Frobenius P -category F and any F-basic Ω P × P -set Ω — a basic P × P -set such that F = F — we consider the group G of permutations σ of Ω which centralize the action of P on the right; then, by the action on the left, P becomes a subgroup of G and, by the very deﬁnition of a basic P × P -set, for any subgroup Q of P we have NG (Q)/CG (Q) ∼ = F(Q)

I46.1.

The elementary but careful work we do in chapter 22 consists of determining all the centralizers CG (Q) and the inclusions between them. I 47 Naturally, these centralizers contain full symmetric groups coming from the possible mutually isomorphic P × P -orbits of Ω , but fortunately the minimal normal subgroups in the centralizers containing these symmetric groups form a “localit´e” in the old sense of [35] and therefore they determine b an F-locality. Although this F-locality — called the basic F-locality L — is far from being perfect, it is canonically associated with F in the sense that it does not depend on the choice of Ω provided it is “big enough”. I 48 The “universal” property of the localizing functor proved in chapter 18 guarantees that, if a perfect F-locality does exist, it should be related b to the basic F-locality L — at least over the set of F-selfcentralizing subsc groups of P — because of the rich structure of the additive cover ac(F˜ ) of sc sc the exterior quotient F˜ of F developed in chapter 6. The problem with sc

the relationship between the basic and the possible perfect F -localities is the thickness of the ﬁrst one — in the sense that the kernel of the canonib,sc sc cal functor L → F is too big. In chapter 23 we start by showing that b,sc b of L over the set of F-selfcentralizing subgroups the full subcategory L sc c,sc of P admits a quotient — the polycentral F -locality L — already narb,sc rowing L .

I 49 As a matter of fact, the kernel of the corresponding canonical funcc,sc sc tor L → F admits a quite general description — we discuss it in our Appendix from A2.14 to A2.17 — in terms of representations and semidirect products. In our situation in chapter 23, this formulation leads to a

Introduction

13

vanishing result for stable positive cohomology groups — a type of nonstandard cohomology groups introduced in our Appendix (cf. I 34 and A3.8) — sc c,sc narrowing twice the polycentral F -locality L till we reach the reduced sc r,sc F -locality L . Moreover, in chapter 23 we explain our diﬃculties with sc r,sc ﬁnding a perfect F -sublocality of L and give a (strongly!) suﬃcient condition to overcome them. sc

I 50 Finally, in order to prove in chapter 24 that any perfect F -locality sc L is contained in the reduced F -locality, we have to exhibit a suitable basic P × P -set Ω . Where to ﬁnd such a P × P -set? The answer comes sc from the fact that any morphism in L is an epimorphism — actually, it is sc a monomorphism too — and, in particular, all the arguments on ac(F˜ ) in sc sc chapter 6 can be repeated in ac(L ) ; namely, the category ac(L ) admits a direct product, allowing us to consider the direct product of P by P which, sc being an ac(L )-object, involves some ﬁnite set Ω (cf. 6.2): this is the set we are looking for. sc

I 51 A last remark. The reader may ask himself whether or not it is possible to deﬁne an ordinary Grothendieck group for the Frobenius category F(b,G) of a block (b, G) ; namely, to carry out an analogous construction with the Grothendieck groups obtained from the categories of representations over a ﬁeld of characteristic zero, opening the possibility of dealing with Dade’s Conjecture [21]. Firstly note that, without any extra eﬀort, the direct sum of suitable scalar extensions of the Grothendieck group of the Frobenius categories F(g,CG (u)) when (g, u) runs over a set of representatives for the set of G-conjugacy classes of Brauer (b, G)-elements [11] provides a satisfactory deﬁnition for the ordinary Grothendieck group of F(b,G) . I 52 Does it coincide with the inverse limit of the composition of the F k -localizing functor loc with the corresponding ordinary Grothendieck (b,G) group functor? — here locF denotes the pull-back of the localizing func∗

(b,G)

tor locF(b,G) (cf. I 42) and the functor Fˆ(b,G) (•) in I 29 above. But, even if the answer was in the aﬃrmative, in order to deal with Dade’s Conjecture some extra idea would be necessary to ﬁt the defect of ordinary irreducible characters considered in [21] inside the functorial framework. Paris, October 2007

Chapter 1

General notation and quoted results 1.1 Since the title of this book, the word category appears often; but, we employ this concept in a very restricted sense. Mostly we bound ourself to consider categories where the isomorphism classes of objects form a set, as for instance the categories of ﬁnite groups — noted Gr — or of ﬁnitely generated Abelian groups — noted Ab — and the category of ﬁnite sets — noted ℵ . Although in the Appendix we expose all the results on categories and on their cohomology groups we need in the book, in this chapter we list some common terminology and standard notation, often employed without reference in the sequel. 1.2 When we consider a new category C , we simply assume that the objects form a set — often called a small category — and, if X and Y are C-objects, we denote by C(Y, X) the set of C-morphisms from X to Y , an order somewhat helpful when composing morphisms; also, we write C(X) instead of C(X, X) ; we call opposite category of C , and denote by C◦ , the category with the same objects and with the morphisms C◦ (Y, X) = C(X, Y )

1.2.1.

As usual, we call contravariant functors from C to Ab the functors a from C◦ to Ab . For any n ∈ N , let us denote by an the composition of a with the functor Ab → Ab determined by the n-th power. 1.3 Very often we deal with categories where it makes sense to talk about the inner automorphisms of the objects — the category Gr for instance; more precisely, an interior structure in a category C is a correspondence sending any C-object X to a subgroup I(X) of C(X) in such a way that, for any C-object Y and any f ∈ C(Y, X) , we have f ◦ I(X) ⊂ I(Y ) ◦ f

1.3.1.

˜ , the quotient In this case, we call exterior quotient of C , and denote by C category deﬁned by ˜ C(Y, X) = I(Y )\C(Y, X) 1.3.2 ˜ for any pair of C-objects X and Y ; usually, we denote by f˜ the C-morphism determined by f . 1.4 A subcategory D of a category C is a category where the set of objects is contained in the set of C-objects and, for any pair of D-objects X and Y , the set of D-morphisms D(Y, X) from X to Y is contained in C(Y, X) , the composition in D being induced by the composition in C ; we say that D is a full subcategory of C whenever D(Y, X) = C(Y, X) for any pair of D-objects X and Y .

16

Frobenius categories versus Brauer blocks

1.5 In particular, we denote by C∗ and Co the subcategories of C with the same set of objects as C and only with the C-isomorphisms and with the C-identities respectively. If D and D are subcategories of C then we can deﬁne the intersection D ∩ D in the obvious way. Note that a category where the isomorphism classes of objects form a set is equivalent to a full subcategory where the objects form a set, and therefore it is easy to translate results from the one to the other. If A and B are two small categories, we denote by A × B their direct product where the objects are the pairs (A, B) formed by an A-object A and a B-object B , and where the set of morphisms from (A, B) to (A , B ) is the direct product A(A , A) × B(B , B)

1.5.1.

Moreover, we denote by Fct(A, B) the small category of functors from A to B and, if f, g ∈ Fct(A, B) , by Nat(f, g) and Nat(f, g)∗ the respective sets of natural maps and natural isomorphisms from f to g . 1.6 Any functor f : B → C holds the inverse image throughout f f∗ : Fct(C◦ , Ab) −→ Fct(B◦ , Ab)

1.6.1,

which is a functor mapping any contravariant functor b : C → Ab on b ◦ f . Recall that f∗ admits a right adjoin called the direct image throughout f f∗ : Fct(B◦ , Ab) −→ Fct(C◦ , Ab)

1.6.2,

namely a functor (cf. A2.11 below) such that the evident functors Ab ←− Fct(B◦ , Ab) × Fct(C◦ , Ab)◦ −→ Ab

1.6.3

respectively mapping a ∈ Fct(B◦ , Ab) and b ∈ Fct(C◦ , Ab) on Nat f∗ (b), a and Nat b, f∗ (a) are naturally isomorphic. 1.7 For any C-object X we denote by CX the new category where the objects are the C-morphisms f : Y → X from any C-object Y to X and where the morphisms from f : Y → X to another CX -object f : Y → X are the C-morphisms g : Y → Y fulﬁlling f = f ◦ g , the composition being induced by the composition in C . Note that we have an obvious forgetful functor from CX to C . Moreover, ◦ CX = (C◦ )X 1.7.1 is the analogous category where the objects are the C-morphisms f : X → Y from X to any C-object Y and the morphisms from f : X → Y to another CX -object f : X → Y are the C-morphisms g : Y → Y fulﬁlling f = g ◦ f . 1.8 Throughout this book p is a ﬁxed prime number. The Frobenius category associated with a ﬁnite group G — noted FG — is the category

1. General notation and quoted results

17

where the objects are the p-subgroups of G and the morphisms are the group homomorphisms between them which are induced by the inclusion and by the G-conjugation (cf. I 1); actually, if P is a Sylow p-subgroup of G , it follows from the well-known Sylow Theorem(s) that FG is equivalent to the full subcategory over the set of subgroups of P — that we still denote by FG . More explicitly, if Q and R are p-subgroups of G and x ∈ G fulﬁlls R ⊂ Qx , we denote by κQ,R (x) : R −→ Q 1.8.1 (or by κQ (x) if R = Q) the group homomorphism induced by the conjugation by x . 1.9 As usual, we call order of G and denote by |G| the cardinal of a ﬁnite group G ; we call p -group any ﬁnite group of order prime to p . If H is a subgroup of G , we denote by ιG H : H → G the inclusion map and we set |G : H| = |G|/|H| ; occasionally, we write 1 instead of {1} for the trivial subgroup of G . Besides the standard notation NG (H) and CG (H) for the normalizer and the centralizer of H in G , for any subgroup X of the group Aut(H) of automorphisms of H , we call X-normalizer of H in G , and X denote by NG (H) , the converse image of X in NG (H) ; note that {idH }

NG

(H) = CG (H)

1.9.1;

moreover, it is handy to employ the notation ¯ X (H) = N X (H) H ∩ N X (H) N G G G

1.9.2.

If CG (H) ⊂ H , we say that H is a selfcentralizing subgroup of G . 1.10 Throughout this book k is an algebraically closed ﬁeld of characteristic p . A Brauer block , or a block for short, is a pair (b, G) formed by a ﬁnite group G and a primitive idempotent b of the center Z(kG) of the group algebra kG ; sometimes we also say that b is a block of G, but note that the same central idempotent may be called a block with respect to several groups. Then, a Brauer G-pair is a pair (P, e) formed by a p-subgroup P of G and a primitive idempotent e of Z kCG (P ) or, equivalently, a block of CG (P ) ; it is well-known that the image e¯ of e in Z k C¯G (P ) remains a primitive idempotent [50, Corollary 2.13]. Note that, for any subgroup H of G containing P ·CG (P ) , (P, e) is also a Brauer H-pair. 1.11 In order to deﬁne the inclusion between Brauer G-pairs, it is better to adopt the more general point of view of the G-algebras, that we also need in this book. Following Green [29], a G-algebra is a k-algebra A of ﬁnite dimension endowed with a G-action; we say that A is primitive if the unity element is primitive in AG . A G-algebra homomorphism from A to another

18

Frobenius categories versus Brauer blocks

G-algebra A is a not necessarily unitary algebra homomorphism f : A → A compatible with the G-actions; we say that f is an embedding whenever Ker(f ) = {0} and

Im(f ) = f (1A )A f (1A )

1.11.1.

We say that f is a strict semicovering — we only need this deﬁnition in Lemma 15.49 below — if f is unitary, Ker(f in the radical ) is contained J(A) of A and, for any p-subgroup P of G , f J(AP ) is contained in J(AP ) and f (i) is primitive in AP for any primitive idempotent i in AP [34, §3]. For any group homomorphism ϕ : G → G , we denote by Resϕ (A) the same algebra A endowed with the obvious G -action. 1.12 Recall that for any subgroup H of G , a point α of H on A is an (AH )∗ -conjugacy class of primitive idempotents of AH and the pair Hα is a pointed group on A [37, 1.1]; we denote by PA (H) the set of points of H on A . For any i ∈ α , iAi has an evident structure of H-algebra and we denote by Aα one of these mutually (AH )∗ -conjugate H-algebras, and by A(Hα ) the simple quotient of AH determined by α . A second pointed group Kβ on A is contained in Hα if K ⊂ H and, for any i ∈ α , there is j ∈ β such that [37, 1.1] ij = j = ji

1.12.1;

then, it is quite clear that the (AK )∗ -conjugation induces K-algebra embeddings fβα : Aβ −→ ResH 1.12.2. K (Aα ) We set P(A) = PA ({1}) and call points of A the points of {1} on A ; recall that, for any two-sided ideal a of A , the image in A¯ = A/a of a point of A not contained in a is a point of A¯ and that this correspondence determines an injective map [51, Proposition 3.23] ¯ −→ P(A) P(A)

1.12.3.

1.13 Following Brou´e, we consider the Brauer quotient and the Brauer (algebra) homomorphism [11, 1.2] P P BrA P : A −→ A(P ) = A

AP Q

1.13.1,

Q

where Q runs over the set of proper subgroups of P , and call local any point γ A of P on A not contained in Ker(BrP ) [37, 1.1]; we denote by LPA (P ) the set of local points of P on A . Recall that a local pointed group Pγ contained N (P )

in Hα is maximal if and only if BrP (α) ⊂ A(Pγ )P H γ [37, Proposition 1.3] and then the P -algebra Aγ — called a source algebra of Aα — is Morita

1. General notation and quoted results

19

equivalent to Aα [51, 6.10]; moreover, the maximal local pointed groups Pγ contained in Hα — called the defect pointed groups of Hα — are mutually H-conjugate [37, Theorem 1.2]. 1.14 Let us say that A is a p-permutation G-algebra if a Sylow p-subgroup of G stabilizes a basis of A [11, 1.1]. In this case, choosing a point α of G on A , we call Brauer (α, G)-pair any pair (P, eA ) formed by a p-subgroup P A ofG such that BrP (α) = {0} and by a primitive idempotent eA of the center Z A(P ) of A(P ) such that eA BrA P (α) = {0}

1.14.1;

note that any local pointed group Qδ on A contained in Gα determines the Brauer (α, G)-pair (Q, fA ) fulﬁlling fA BrA Q (δ) = {0} . 1.15 Then, it follows from Theorem 1.8 in [11] that the inclusion between the local pointed groups on A induces an inclusion between the Brauer (α, G)-pairs; explicitly, if (P, eA ) and (Q, fA ) are two Brauer (α, G)-pairs then we have (Q, fA ) ⊂ (P, eA ) 1.15.1 whenever there are local pointed groups Pγ and Qδ on A fulﬁlling Qδ ⊂ Pγ ⊂ Gα

,

A fA BrA Q (δ) = {0} and eA BrP (γ) = {0}

1.15.2.

Actually, according to the same result, for any p-subgroup P of G , any primitive idempotent eA of Z A(P ) fulﬁlling eA BrA and any subP (α) = {0} group Q of P , there is a unique primitive idempotent fA of Z A(Q) fulﬁlling eA BrA P (α) = {0}

and

(Q, fA ) ⊂ (P, eA )

1.15.3.

Once again, the maximal Brauer (α, G)-pairs are pairwise G-conjugate [11, Theorem 1.14]. 1.16 Obviously, the group algebra kG is a p-permutation G-algebra and, for any primitive idempotent b of Z(kG) , α = {b} is a point of G on A ; for any p-subgroup P of G , the Brauer homomorphism BrP = BrkG P induces a k-algebra isomorphism [42, 2.8.4] kCG (P ) ∼ = (kG)(P )

1.16.1;

thus, in a Brauer ({b}, G)-pair (P, e) — simply called a Brauer (b, G)-pair — e is nothing but a block of CG (P ) such that eBrP (b) = 0 , and we have an inclusion relation between them; note that ({1}, b) is a Brauer (b, G)-pair and that the condition eBrP (b) = 0 is equivalent to ({1}, b) ⊂ (P, e)

1.16.2.

20

Frobenius categories versus Brauer blocks

Then, the Brauer First Main Theorem aﬃrms that (P, e) is maximal if and only if the algebra k C¯G (P )¯ e is simple and the quotient NG (P, e)/P ·CG (P ) is a p -group [51, Theorem 10.14]. We still call Brauer G-pair a Brauer (b, G)-pair for some block b of G . 1.17 Recall that if (P, e) is a maximal Brauer (b, G)-pair then, in Brauer terms, P is a defect group of the block b and if |P | = pd , d is the defect of b . In particular, the sentence b has defect zero means that P = {1} which, by the very deﬁnition of defect pointed group, is clearly equivalent to the condition b ∈ (kG)G 1 ; moreover, according to Higman’s Criterion [51, Theorem 5.12], this condition is equivalent to any kGb-module is projective, which amounts to saying that kGb is simple. 1.18 More generally, we say that the block b is nilpotent whenever the quotients NG (Q, f )/CG (Q) are p-groups for all the Brauer (b, G)-pairs (Q, f ) [12, Deﬁnition 1.1]; by the main result in [41], the block b is nilpotent if and only if, for a maximal local pointed group Pγ on kGb , P stabilizes a simple unitary subalgebra S of (kGb)γ fulﬁlling (kGb)γ = SP ∼ = S ⊗k kP

1.18.1

where we denote by SP the obvious k-algebra u∈P Su and, for the isomorphism in the right, we consider the lifting to a group homomorphism P → S ∗ of the P -action on S [41, statement (1.8.1)]. Note that, although in [41] we argue over a complete discrete valuation ring O of characteristic zero, by the classiﬁcation of the Dade P -algebras in [17] and [6], S can be lifted to O and the comments in [41, 7.7-7.9] apply, allowing us to argue either in characteristic 0 or in characteristic p indiﬀerently. 1.19 Always in the case of the group algebra, for any p-subgroup P of G and any subgroup H of NG (P ) containing P ·CG (P ) , we have BrP (kG)H = (kG)(P )H

1.19.1

and therefore any block e of CG (P ) determines a unique point β of H on kG (cf. 1.12.3) such that Hβ contains Pγ for a local point γ of P on kG fulﬁlling [41, Lemma 3.9] eBrP (γ) = {0} 1.19.3. Moreover, if Q is a subgroup of P such that CG (Q) ⊂ H , then the blocks of CG (Q) = CH (Q) determined by (P, e) from G and from H coincide [11, Theorem 1.8]. Note that if P is normal in G then the kernel of the obvious k-algebra homomorphism kG → k(G/P ) is contained in the radical J(kG) and contains Ker(BrP ) ; thus, in this case, isomorphism 1.16.1 implies that any point of P on kG is local and that any block of G is contained in Z kCG (P ) .

1. General notation and quoted results

21

1.20 Let P be a ﬁnite p-group; a Dade P -algebra is a p-permutation P -algebra S which is a simple k-algebra and fulﬁlls S(P ) = {0} [43, 1.3]; since k is algebraically closed and Autk (S) ∼ = S ∗ /k ∗ , we have S ∼ = Endk (M ) for a suitable kP -module M that in [19] Everett Dade calls capped endopermutation kP -module. Since

¯P (Q) ∼ S(Q) N = S NP (Q)

1.20.1

for any subgroup Q of P [11, Proposition 1.5], ResP Q (S) is a Dade Q-algebra; moreover, according to 1.8 in [43], the Brauer quotient S(Q) is a Dade ¯P (Q)-algebra. N 1.21 In particular Q has a unique local point on S or, equivalently, ResP Q (M ) has a unique isomorphism class of direct kQ-summands of vertex Q . We say that two Dade P -algebras S∼ = Endk (M ) and S ∼ = Endk (M )

1.21.1

are similar if M and M have the same isomorphism class of direct kP -summands of vertex P or, equivalently, if S can be embedded (cf. 1.11) in the tensor product End(N ) ⊗k S for a suitable kP -module N with a P -stable basis [43, 1.5 and 2.5.1]; we denote by Dk (P ) the set of similarity classes. On the other hand, the tensor product S ⊗k S is a Dade P -algebra since we have [41, Proposition 5.6] (S ⊗k S )(P ) ∼ = S(P ) ⊗k S (P )

1.21.2,

and it induces a group structure on Dk (P ) — called the Dade group of P — where the opposite P -algebra S ◦ determines the inverse of the class of S since S ◦ ⊗k S ∼ 1.21.3 = Endk (S) is clearly similar to the trivial P -algebra k . 1.22 Now, for any subgroup Q of P , it is quite clear that the restriction induces a group homomorphism resP Q : Dk (P ) −→ Dk (Q)

1.22.1.

If Q is normal in P then the Brauer quotient functor BrP Q from the category of P -algebras to the category of P/Q-algebras, mapping a P -algebra A on the P/Q-algebra A(Q) , preserves the similarity and determines a group homomorphism P : Dk (P ) −→ Dk (P/Q) Br 1.22.2. Q

22

Frobenius categories versus Brauer blocks

Moreover, for a second normal subgroup R of P contained in Q , we have the obvious transitivity [11, Proposition 1.5] P/R

P

P

Br Q/R ◦ BrR = BrQ

1.22.3.

1.23 As announced in the Introduction (cf. I 20), we have to deal with ˆ of ﬁnite groups G by k ∗ ; however, note that G ˆ always central extensions G contains a ﬁnite subgroup G covering G [42, Lemma 5.5]. Precisely, we call k ∗ -group a group X endowed with an injective group homomorphism θ : k ∗ → Z(X) [42, 5], and k ∗ -quotient of (X, θ) the group X/θ(k ∗ ) ; we denote by X ◦ the k ∗ -group formed by X and by the composition of θ with the automorphism k ∗ ∼ = k ∗ mapping λ ∈ k ∗ on λ−1 ; we say that a k ∗ -group is ˆ a k ∗ -group ﬁnite whenever its k ∗ -quotient is ﬁnite. Usually, we denote by G ∗ ˆ by the x for the product of x ˆ∈G and by G its k -quotient, and we write λ·ˆ ∗ ˆ. image of λ ∈ k in G ˆ is a second k ∗ -group, we denote by G ˆ× ˆ the quotient of the ˆG 1.24 If G ˆ ˆ direct product G × G by the image of the inverse diagonal in k ∗ × k ∗ , which has an obvious structure of k ∗ -group with k ∗ -quotient G × G ; moreover, ˆ ∗G ˆ the k ∗ -group obtained from the converse if G = G then we denote by G ˆ ˆ ˆ image of ∆(G) ⊂ G × G in G × G , which is nothing but the so-called sum of both central extensions of G by k ∗ ; in particular, we have a canonical k ∗ -group isomorphism ˆ∗G ˆ◦ ∼ G 1.24.1. = k∗ × G ∗ ˆ → G ˆ is a group homomorphism which A k -group homomorphism ϕ : G preserves the k ∗ -multiplication. 1.25 Note that, for any k-algebra A of ﬁnite dimension, the group A∗ of invertible elements has a canonical k ∗ -group structure. If S is a simple algebra then Autk (S) coincides with the k ∗ -quotient of S ∗ ; in particular, any ˆ of ﬁnite group G acting on S determines — by pull-back — a k ∗ -group G ∗ ∗ k -quotient G , together with a k -group homomorphism [42, 5.7] ˆ −→ S ∗ ρ:G ∗

1.25.1.

It is clear that the inclusion k ⊂ k determines a k-algebra homomorphism ˆ, from the group algebra kk ∗ of k ∗ to k and that, for any ﬁnite k ∗ -group G ∗ ∗ ˆ ˆ the group algebra k G of the group G is also a kk -algebra; we call k -group ˆ the algebra algebra of G ˆ = k ⊗kk∗ k G ˆ k∗ G 1.25.2 ˆ which has dimension |G| . We also call block of G any primitive idempotent b ˆ , and denote by Irrk (G, ˆ b) the canonical Z-basis — formed by the of Z(k∗ G) ˆ isomorphism classes of simple k∗ Gb-modules — of the Grothendieck group of ˆ the category of ﬁnitely generated k∗ Gb-modules.

1. General notation and quoted results

23

1.26 In particular, for any local pointed group Pγ on kG — we are avoiding unnecessary generality — BrP (γ) is a (kG)(P )∗ -conjugacy class of primitive idempotents in (kG)(P ) and therefore it determines a simple quo¯G (Pγ ) on (kG)(Pγ ) tient (kG)(Pγ ) of this k-algebra; thus, the action of N ¯ˆ (P ) [42, 6.2]; but, the Brauer homomorphism determines a k ∗ -group N G

γ

BrP induces a group homomorphism C¯G (P ) −→ (kG)(Pγ )∗

1.26.1

and therefore an NG (Pγ )-stable group homomorphism ¯ˆ G (Pγ ) C¯G (P ) −→ N

1.26.2;

consequently, setting ˜G (Pγ ) = N ¯G (Pγ )/C¯G (P ) EG (Pγ ) = NG (Pγ )/CG (P ) and E

1.26.3,

˜ˆ G (Pγ ) = N ¯ˆ G (Pγ )/C¯G (P )†. we obtain the k ∗ -group E ˆ be a ﬁnite k ∗ -group; recall that we call G-interior ˆ 1.27 Let G algebra [42, 5.10] any k-algebra A of ﬁnite dimension endowed with a k ∗ -group homomorphism ˆ −→ A∗ ρ:G 1.27.1 ˆ and, as usual, we write x ˆ·a and a·ˆ x instead of ρ(ˆ x)a and aρ(ˆ x) for any x ˆ∈G ˆ and any a ∈ A . As above, a G-interior algebra homomorphism from A to ˆ another G-interior algebra A is a not necessarily unitary algebra homomor phism f : A → A this time fulﬁlling f (ˆ x·a) = x ˆ·f (a) and f (a·ˆ x) = f (a)·ˆ x

1.27.2; ∗

we say that f is an embedding if it fulﬁlls condition 1.11.1. For a k -group ˆ → G ˆ , we denote by Resϕ (A) the G ˆ -interior algebra homomorphism ϕ : G deﬁned by ρ ◦ ϕ . ˆ 1.28 The conjugation in A induces an action of the k ∗ -quotient G of G on A , so that A becomes an ordinary G-algebra; thus, all the pointed group ˆ ˆ is a k ∗ -subgroup language developed above applies to G-interior algebras; if H ∗ ˆα , ˆ of G and α a point of H on A , we call pointed k -group on A the pair H ˆ ˆα and note that now Aα becomes an H-interior algebra. More generally, if H ∗ ∗ ˆ and Kβ are two pointed k -groups on A , we say that an injective k -group ˆ →H ˆ is an A-fusion from K ˆ β to H ˆ α whenever there is homomorphism ϕˆ : K ˆ a K-interior algebra embedding ˆ

fϕˆ : Aβ −→ ResH ˆ (Aα ) K †

Note that we slightly modify our usual notation introduced in [39].

1.28.1

24

Frobenius categories versus Brauer blocks

such that the inclusion Aβ ⊂ A and the composition of fϕˆ with the incluˆβ, H ˆ α ) the set of them sion Aα ⊂ A are A∗ -conjugate; we denote by FA (K ˜ ˆ ˆ [39, Deﬁnition 2.5] and by FA (Kβ , Hα ) its quotient by the action of H†. ˆ α ) instead of FA (H ˆ α, H ˆ α ) ; choosing i ∈ α 1.29 As usual, we write FA (H and setting Aα = iAi , it follows from Corollary 2.13 in [39] that we have a group homomorphism H ∗ ˆ α ) −→ NA∗ (H·i)/(A ˆ FA (H α) α

1.29.1

ˆ α ) deﬁned by the pull-back and then we consider the k ∗ -group FˆA (H H ∗ ˆ α ) −→ ˆ FA (H NA∗α (H·i)/(A α) ↑ ↑ ˆ α ) −→ NA∗ (H·i) ˆ FˆA (H i + J(AH α) α

1.29.2.

It is clear that the group algebra kG is a (k ∗ ×)G-interior algebra and recall that, for any local pointed group Pγ on kG , we have a canonical k ∗ -group isomorphism [42, Proposition 6.12] ˆG (Pγ )◦ ∼ E = FˆkG (Pγ )

1.29.3.

ˆ ˆ 1.30 As usual, we denote by ResG ˆ (A) the corresponding H-interior alH ˆ gebra. Conversely, for any H-interior algebra B , we consider the induced ˆ G-interior algebra ˆ ˆ ˆ IndG ˆ B ⊗k∗ H ˆ k∗ G ˆ (B) = k∗ G ⊗k∗ H H

where the distributive product is deﬁned by the formula  ˆ x ˆ ∈ H ˆ ⊗ b.ˆ yx ˆ .b ⊗ yˆ if yˆx (ˆ x ⊗ b ⊗ yˆ)(ˆ x ⊗ b ⊗ yˆ ) =  0 otherwise

1.30.1,

1.30.2

ˆ and any b, b ∈ B , and where the structural homomorfor any x ˆ, yˆ, x ˆ , yˆ ∈ G phism ˆ −→ IndGˆˆ (B) G 1.30.3 H ˆ on ˆ running over a set of representatives maps x ˆ∈G ˆyˆ ⊗1B ⊗ yˆ−1 , yˆ ∈ G yˆ x ˆ H ˆ . Recall that if I is a G-orbit ˆ for G/ of pairwise orthogonal idempotents ˆ is the stabilizer in G ˆ of i ∈ I , we have a of A such that 1A = i and H i∈I

ˆ G-interior algebra isomorphism [53, Proposition 2.3] A∼ = IndG ˆ (iAi) H ˆ

†

Note that we slightly modify our usual notation introduced in [39].

1.30.4.

1. General notation and quoted results

25

1.31 Our general notation mainly concerns group theory — our standard reference being [28] — and homological algebra — our standard reference being [18]. In particular, if G is a ﬁnite group, recall that Op (G) , Op (G) , Op (G) and Op (G) respectively denote the minimal or the maximal normal subgroups of G with their index or their order being a power of p or prime ˆ except to p ; note that this notation still makes sense for a ﬁnite k ∗ -group G ˆ that Op (G) remains a p-group. For any pair of subgroups H and K of G , we denote by TG (K, H) the set of x ∈ G fulﬁlling xKx−1 ⊂ H . If P is a ﬁnite p-group, Φ(P ) denotes the Frattini subgroup, namely the intersection of all the maximal subgroups of P , and Ω1 (P ) denotes the subgroup of P generated by all the elements of order p . 1.32 We denote by ab : Gr → Ab the functor mapping G on its maximal Abelian quotient G/[G, G] and by ab◦ the contravariant Ab-valued functor deﬁned by the transfer [28, Ch. 7, Theorem 3.2] from the subcategory of Gr formed by all injective group homomorphisms. For any ﬁnite set Ω , we denote by SΩ the group of permutations of Ω and we also set Sn = SΩ whenever n = |Ω| ; if G is a subgroup of SΩ and H a group, H G denotes the wreath product (cf. 15.1 in [31, Ch. I]), namely the obvious semidirect product H G= H G 1.32.1. ω∈Ω

Finally, we denote by µ the so-called M¨ obius function mapping any ordered ﬁnite set X on Y (−1)|Y | where Y runs over the set of totally ordered subsets of X , setting µX (x) for the value of µ over the set of y ∈ X strictly smaller than x ∈ X .

Chapter 2

Frobenius P-categories: the ﬁrst deﬁnition 2.1 Let P be a ﬁnite p-group. In this chapter we introduce the main objects of this book, namely the Frobenius P -categories; actually, in chapter 21 below, we give an equivalent — somewhat easier — deﬁnition, but this ﬁrst deﬁnition is more directly related to the blocks, as we show in the next chapter. 2.2 Let us call P -category any subcategory F (cf. 1.4) of the category of ﬁnite groups Gr (cf. 1.1) containing the Frobenius category FP of P (cf. 1.8), where the objects are the FP -objects — the subgroups of P — and where all the homomorphisms are injective. Note that the intersection F ∩ F (cf. 1.5) of two P -categories is a P -category too, and that there is a unique maximal P -category, namely the P -category containing all injective group homomorphisms between the subgroups of P . 2.3 We say that a P -category F is divisible † whenever it fulﬁlls: 2.3.1 If Q , R and T are subgroups of P , for any ϕ ∈ F(Q, R) and any group homomorphism ψ : T → R the composition ϕ ◦ ψ belongs to F(Q, T ) (if and) only if ψ ∈ F(R, T ) . Or, equivalently, whenever for any subgroup Q of P , the category (F)Q (cf. 1.7) is a full subcategory of (Gr)Q (cf 1.4). Note that the maximal P -category is divisible, and that the intersection of two divisible P -categories is divisible too. Actually, all the P -categories we will consider are divisible. 2.4 Note that F is divisible if and only if, for any ϕ ∈ F(Q, R) which is a group isomorphism, the inverse ϕ−1 is also an F-morphism; moreover, in this case, if Q and R are respective subgroups of Q and R , the restriction of any ϕ ∈ F(Q, R) fulﬁlling ϕ(R ) ⊂ Q belongs to F(Q , R ) . In particular, a divisible P -category F is determined by the sets F(P, Q) where Q runs over the set of all the subgroups of P . Conversely, if H(Q) ⊂ Hom(Q, P ) is a set of injective homomorphisms containing FP (P, Q) for any subgroup Q of P and this family fulﬁlls the following condition 2.4.1 for any pair Q and R of P and any θ ∈ Hom(Q, R) of subgroups fulﬁlling H(Q) ∩ H(R) ◦ θ = ∅ , we have H(R) ◦ θ ⊂ H(Q) , there exists an evident divisible P -category F fulﬁlling F (P, Q) = H(Q) for any Q ∈ X . †

A divisible P-category is called a fusion system over P in [13].

28

Frobenius categories versus Brauer blocks

2.5 In a ﬁnite group G with a Sylow p-subgroup S , it is obvious that the centralizer or the normalizer in S of a subgroup Q of S need not be a Sylow p-subgroup of the centralizer or the normalizer of Q in G , and in our abstract setting we will determine when they are so. For our purpose, it is handy to introduce the following notation: if Q , R and T are subgroups of P and Q ⊂ T , any injective group homomorphism ψ : T → R determines a group isomorphism Aut(Q) ∼ = Aut ψ(Q) and we simply denote by ψK and ψχ the images of K ⊂ Aut(Q) and χ ∈ Aut(Q) respectively. 2.6 Let F be a divisible P -category, Q a subgroup of P and K a subgroup of Aut(Q) ; it is quite clear that, for any F-morphism ψ : Q·NPK (Q) → P (cf. 1.9), we have ψ ψ NPK (Q) ⊂ NPK ψ(Q) 2.6.1; then, we say that Q is fully K-normalized in F whenever it fulﬁlls ψ 2.6.2 For any ψ ∈ F P, Q·NPK (Q) , we have ψ NPK (Q) = NPK ψ(Q) . If K = {idQ } or K = Aut(Q) , we respectively say that Q is fully centralized or fully normalized in F ; note that K , K·FQ (Q) and K ∩ F(Q) play the same role in this condition, so that we always may assume that FQ (Q) ⊂ K ⊂ F(Q)

2.6.3.

Proposition 2.7 With the notation above, let R be a subgroup of Q·NPK (Q) containing Q and assume that an F-morphism ψ : R → P fulﬁlls ψ ψ 2.7.1 For any ψ ∈ F(P, R) we have NP K ψ (Q) ≤ NPK ψ(Q) . Then ψ(Q) is fully ψK-normalized in F . In particular there is ϕ ∈ F(P, Q) such that ϕ(Q) is both fully centralized and fully ϕK-normalized in F . Proof: Set Q = ψ(Q) ; since ψ(R) is contained in Q ·NPK (Q ) and F is ϕ divisible, any F-morphism ξ : Q ·NPK (Q ) → P determines an F-morphism ψ : R → P mapping v ∈ R on ξ ψ(v) and therefore simultaneously we have ψ

ψK ψ ψ ψ N ψ (Q) ≤ |NPK (Q )| and ξ NPK (Q ) ⊂ NP K ψ (Q) P

2.7.2,

ψ ψ so that we get the equality ξ NPK (Q ) = NP K ξ (Q ) . In particular, it is clear that there is an F-morphism ϕ : Q → P such that Q = ϕ(Q) is fully centralized in F , and then it is still clear that we can ﬁnd an F-morphism ξ : Q ·CP (Q ) → P such that ξ (Q ) is fully ξ ϕ ( K)-normalized in F ; but ξ (Q ) is fully centralized too, as it is easily checked. 2.8 We are ready to state our main deﬁnition; our conditions below mimic suitable formulations of the Sylow Theorem(s), as we show in the next

2. Frobenius P -categories: the ﬁrst deﬁnition

29

chapter. We say that a P -category F is a Frobenius P -category or a Frobenius category over P † if it is divisible and fulﬁlls the following two conditions: 2.8.1 The group FP (P ) of inner automorphisms of P is a Sylow p-subgroup of F(P ) . 2.8.2 For any subgroup Q of P , any subgroup K of Aut(Q) and any F-morphism ϕ : Q → P such that ϕ(Q) is fully ϕK-normalized in F , there are an F-morphism ψ : Q·NPK (Q) → P and χ ∈ K such that ψ(u) = ϕ χ(u) for any u ∈ Q . 2.9 Actually, in condition 2.8.2 we may assume that FQ (Q) ⊂ K ⊂ F(Q) (cf. 2.6.3) and that Q = ϕ(Q) is fully centralized too. Indeed, by Proposi¯ = ϕ(Q) tion 2.7, there is ϕ¯ ∈ F(P, Q) such that Q ¯ is both fully centralϕ ¯ ϕ ized and fully K-normalized in F ; then, setting K = K and denoting by ∼ ¯ ϕ : Q = Q the F-isomorphism fulﬁlling ϕ ϕ(u) = ϕ(u) ¯ for any u ∈ Q , assume that there are F-morphisms ψ¯ : Q·NPK (Q) −→ P

and ψ : Q ·NPK (Q ) −→ P

2.9.1,

and elements χ ¯ ∈ K and χ ∈ K such that we have ¯ ψ(u) = ϕ¯ χ(u) ¯ and ψ (u ) = ϕ χ (u )

2.9.2

for any u ∈ Q and any u ∈ Q ; since F is divisible and we have ψ ¯ and ψK = ϕ¯K ψ NPK (Q ) = NP K (Q)

2.9.3,

we ﬁnally get an F-isomorphism ¯ Pϕ¯K (Q) ¯ ∼ ψ ∗ : Q·N 2.9.4 = Q ·NPK (Q ) such that w = ψ ∗ ψ (w ) for any w ∈ Q ·NPK (Q ) ; thus, the F-morphism ¯ ψ : Q·NPK (Q) → P mapping w ∈ Q·NPK (Q) on ψ ∗ ψ(w) maps u ∈ Q on ψ ∗ ϕ¯ χ(u) ¯ = ψ ∗ ϕ ϕ χ(u) ¯ = χ−1 ϕ χ(u) ¯ = ϕ χ(u) 2.9.5

for a suitable χ ∈ K . 2.10 Moreover, condition 2.8.2 implies that 2.10.1 For any subgroup Q of P , any F-morphism ϕ : Q → P such that ϕ(Q) is fully centralized in F , and any subgroup R of NP (Q) such that Q ⊂ R and ϕ FR (Q) ⊂ FP ϕ(Q) , there is an F-morphism ρ : R → P extending ϕ . Indeed, ﬁrst of all note that if Q is a subgroup of P fully centralized in F and F (Q) R is a subgroup of NP (Q) containing Q then we have NP R (Q) = R·CP (Q) and, for any F-morphism η : R·CP (Q) → P , we get η F (Q) η(Q) 2.10.2, η R·CP (Q) = η(R)·CP η(Q) = NP R †

Called a saturated fusion system over P in [13].

30

Frobenius categories versus Brauer blocks

so that Q is also fully FR (Q)-normalized in F ; hence, in condition 2.10.1 the subgroup ϕ(Q) is fully FR (Q)-normalized in F where R is the converse ϕ image of FR (Q) in NP ϕ(Q) ; then, condition 2.8.2 implies the existence of ψ ∈ F(P, R) and χ ∈ FR (Q) such that ψ(u) = ϕ χ(u) for any u ∈ Q , and it suﬃces to choose w ∈ R lifting χ and to deﬁne ρ ∈ F(P, R) by ρ(v) = ψ(v w ) for any v ∈ R . Conversely, condition 2.8.2 can be replaced by condition 2.11.1 below†. Proposition 2.11 Let F be a divisible P -category such that FP (P ) is a Sylow p-subgroup of F(P ) . Then, F is a Frobenius P -category if and only if it fulﬁlls the condition 2.11.1 For any subgroup Q of P , any F-morphism ϕ : Q → P such that ϕ(Q) is both fully centralized and fully normalized in F , and any subgroup R of NP (Q) such that Q ⊂ R and ϕ FR (Q) ⊂ FP ϕ(Q) , there is an F-morphism ρ : R → P extending ϕ . In this case, for any subgroups Q of P and K of Aut(Q) , the following statements are equivalent: 2.11.2 2.11.3

The subgroup Q is fully K-normalized in F . ϕ For any ϕ ∈ F(P, Q) , we have N K ϕ(Q) ≤ |N K (Q)| . P

P

2.11.4 The subgroup Q is fully centralized in F and K ∩ FP (Q) is a Sylow p-subgroup of K ∩ F(Q) . Proof: We already know that statement 2.11.3 implies statement 2.11.2. Assume that F fulﬁlls statement 2.11.1 and let us prove that condition 2.8.2 holds; actually, it is easily checked from Proposition 2.7 that F fulﬁlls statement 2.10.1 too. Let Q be a subgroup of P , K a subgroup of Aut(Q) and ϕ : Q → P an F-morphism such that Q = ϕ(Q) is fully centralized and fully ϕ K-normalized in F ; by Lemma 2.12 below, setting K = ϕK we know that K ∩ FP (Q ) is a Sylow p-subgroup of K ∩ F(Q ) and therefore, for a suitable χ ∈ K , we have ϕ◦χ 2.11.5. K ∩ FP (Q) ⊂ K ∩ FP (Q ) Moreover, choose ϕ ∈ F(P, Q ) such that Q = ϕ (Q ) is fully centralized and fully normalized in F (cf. Proposition 2.7); once again, up to a modiﬁcation of our choice, we may assume that ϕ K ∩ FP (Q ) is a Sylow p-subgroup of ϕ K ∩ F(Q ) . Then, it follows from statement 2.11.1 that ϕ can be extended to an F-morphism ρ : Q ·NPK (Q ) → P and that there is an F-morphism ρ : Q·NPK (Q) → P fulﬁlling ρ (u) = ϕ (ϕ ◦ χ)(u) for any †

In [13] Broto, Levi and Oliver show that condition 2.8.2 can be replaced by condition 2.10.1

and Radu Stancu has noticed that the same is true by replacing “fully centralized” by “fully normalized”.

2. Frobenius P -categories: the ﬁrst deﬁnition

31

u ∈ Q ; but, since Q is fully K -normalized in F , we have ϕ ρ NPK (Q ) = NP K (Q ) ⊃ ρ Q·NPK (Q)

2.11.6;

consequently, since F is divisible, the F-morphism ρ : Q·NPK (Q) → P fulﬁl ling ρ ρ(w) = ρ (w) for any element w ∈ Q·NPK (Q) extends ϕ ◦ χ . Now, assume that F is a Frobenius P -category; we ﬁrstly prove that statement 2.11.2 implies statement 2.11.4; indeed, since F is divisible, any F-morphism ψ : Q·CP (Q) → P determines ϕ ∈ F P, ψ(Q) such that ϕ ψ(u) = u for any u ∈ Q (cf. 2.3) and therefore, setting Q = ψ(Q) and K = ψ K , by condition 2.8.2 there are an F-morphism ξ : NPK (Q ) → P and an element χ ∈ K such that we have ξ (u ) = ϕ χ (u ) for any u ∈ Q ; in particular, we have ξ CP (Q ) ⊂ CP (Q) and therefore ψ CP (Q) = CP (Q ) ; that is to say, Q is fully centralized in F and then statement 2.11.4 follows from Lemma 2.12 below. Furthermore, if Q is a subgroup of P and ϕ : Q → P an F-morphism such that Q = ϕ(Q) is fully centralized and fully normalized in F then, for any subgroup R of NP (Q) fulﬁlling Q ⊂ R and ϕFR (Q) ⊂ FP (R ) , denoting by R the converse image of ϕFR (Q) in NP (Q ) so that ϕFR (Q) = FR (Q ) , it is easily checked that Q is also fully FR (Q )-normalized in F ; hence, condition 2.8.2 implies the existence of ψ ∈ F(P, R) and χ ∈ FR (Q) such that ψ(u) = ϕ χ(u) for any u ∈ Q ; thus, in order to prove statement 2.11.1, it suﬃces to choose w ∈ R lifting χ and to deﬁne ρ ∈ F(P, R) by ρ(v) = ψ(v w ) for any v ∈ R . Finally, assume that statement 2.11.4 holds; it follows from the divisibility of F and from condition 2.8.2 that, if ϕ ∈ F(P, Q) andwe set Q = ϕ(Q) , there is an F-morphism ψ : Q ·CP (Q ) → P such that ψ ϕ(u) = u for any u ∈ Q , and therefore we have ψ CP (Q ) ⊂ CP (Q) ; hence, we get ϕK NP (Q ) = |CP (Q )|·|ϕK ∩ FP (Q )| ≤ |CP (Q)|·|K ∩ F(Q)|p = |NPK (Q)|

2.11.7

which proves statement 2.11.3 (cf. Proposition 2.7). We are done. Lemma 2.12 Let F be a divisible P -category such that FP (P ) is a Sylow p-subgroup of F(P ) , and X a nonempty set of subgroups of P such that if Q ∈ X then any subgroup R of P fulﬁlling F(R, Q) = ∅ belongs to X . Assume that for any subgroup Q ∈ X , any F-morphism ϕ : Q → P such that ϕ(Q) is fully centralized in F , and any subgroup R of NP (Q) fulﬁlling Q ⊂ R and ϕ FR (Q) ⊂ FP ϕ(Q) , there is an F-morphism ρ : R → P extending ϕ . Then, for any Q ∈ X and any subgroup K of Aut(Q) such that Q is fully centralized and fully K-normalized in F , K ∩ FP (Q) is a Sylow p-subgroup of K ∩ F(Q) .

32

Frobenius categories versus Brauer blocks

Proof: We may assume that Q = P and argue by induction on |P : Q| ; set R = NPK (Q) . In the case where K = Aut(Q) , denoting by J the set of automorphisms of R stabilizing Q , it is clear that NPJ (R) = R and therefore, since Q is fully normalized in F , R is fully J-normalized in F so that, according to the induction hypothesis, J ∩ FP (R) = FR (R) is a Sylow p-subgroup of J ∩ F(R) . But, since Q is fully centralized in F , it follows from our hy pothesis that any element of NF (Q) FR (Q) can by lifted to J ∩ F(R) ; con sequently, FR (Q) is a Sylow p-subgroup of NF (Q) FR (Q) , so it is a Sylow p-subgroup of F(Q) . In the general case, choose an F-morphism ϕ : Q·CP (Q) → P such that Q = ϕ (Q) is fully normalized and fully centralized in F (cf. Proposition 2.7); thus, by the above argument, FP (Q ) is a Sylow p-subgroup of F(Q ) and therefore it contains a Sylow p-subgroup of τ ◦ϕ K for a suitable τ ∈ F(Q ) . That is to say, up to a modiﬁcation of our choice of ϕ , we may assume that ϕ K ∩ FP (Q ) is a Sylow p-subgroup of ϕ K ∩ F(Q ) containing ϕ FP (Q) ; in this case, according to our hypothesis, ϕ can be extended to an F-morphism ρ : Q·R → P and moreover, if Q is fully K-normalized in F , we actually

ϕ

have ρ (R) = NP K (Q ) , so that we get

ϕ

K ∩ FP (Q) = ϕ K ∩ FP (Q )

2.12.1;

hence, since we also have ϕ K ∩ F(Q) = ϕ K ∩ F(Q ) , K ∩ FP (Q) is a Sylow p-subgroup of K ∩ F(Q) . We are done. Corollary 2.13 Let F be a divisible P -category such that FP (P ) is a Sylow p-subgroup of F(P ) . Then, F is a Frobenius P -category if and only if it fulﬁlls the following two conditions: 2.13.1 For any pair of F-isomorphic subgroups Q and Q of P fully normalized and fully centralized in F , there is an F-isomorphism NP (Q) ∼ = NP (Q ) mapping Q onto Q . 2.13.2 For any subgroup Q of P fully normalized and fully centralized in F and any subgroup R of NP (Q) containing Q·CP (Q)†, denoting by F(R)Q the stabilizer of Q in F(R) , the group homomorphism F(R)Q → NF (Q) FR (Q) induced by the restriction is surjective. Moreover, in this case, for any pair of subgroups Q and Q of P , and any subgroups K of Aut(Q) and K of Aut(Q ) , if Q and Q are respectively fully K- and K -normalized and there is an F-isomorphism ϕ : Q ∼ = Q such that K = ϕK , then there is an F-isomorphism Q·NPK (Q) ∼ = Q ·NPK (Q ) mapping Q onto Q and K onto K . †

In [52, Corollary 2.14] the hypothesis that R contains the centralizer of Q in P has been

forgotten.

2. Frobenius P -categories: the ﬁrst deﬁnition

33

Proof: Assume that F is a Frobenius P -category; if Q is a subgroup of P , K a subgroup of Aut(Q) and ϕ : Q → P an F-morphism, setting Q = ϕ(Q) and K = ϕK , it follows from condition 2.8.2 above that, if Q is fully K -normalized in F , then there is an F-morphism ψ : Q·NPK (Q) → P extending ϕ ◦ χ for a suitable χ ∈ F(Q) ∩ K , and therefore, if Q is fully K-normalized in F , we actually have ψ Q·NPK (Q) = Q ·NPK (Q ) (cf. 2.6.2); this proves condition 2.13.1 and the last statement. Moreover, in condi- tion 2.13.2, the surjectivity of the homomorphism F(R)Q → NF (Q) FR (Q) follows from Proposition 2.11; indeed, if σ ∈ F(Q) normalizes FR (Q) , it follows from condition 2.11.1 that there is ρ ∈ F(P, R) extending σ , which forces ρ(R) = R since R contains Q·CP (Q) . Conversely, assume that F fulﬁlls the two conditions above and let Q be a subgroup of P fully normalized and fully centralized in F ; ﬁrst of all, we claim that FP (Q) is a Sylow p-subgroup of F(Q) . Indeed, let ξ : NP (Q) → P be an F-morphism such that N = ξ NP (Q) is fully normalized in F (cf. Proposition 2.7); arguing by induction on |P : Q| , we may assume that FP (N ) is already a Sylow p-subgroup of F(N ) and therefore, up to a modiﬁcation of our choice of ξ , we still may assume that FP (N )Q is a Sylow p-subgroup of F(N )Q . On the other hand, since Q is fully centralized in F , N contains CP (Q ) . Then, it follows from condition 2.13.2 that the image of FP (N )Q is a Sylow p-subgroup of NF (Q ) FN (Q ) ; but, this image is contained in FN (Q ) ; consequently, FN (Q ) is a Sylow p-subgroup of its normalizer in F(Q ) , so that it is a Sylow p-subgroup of F(Q ) and therefore FP (Q) is a Sylow p-subgroup of F(Q) . Now, let Q be a subgroup of P , ϕ : Q → P an F-morphism such that Q = ϕ(Q) is fully normalized and fully centralized in F , and R a subgroup of ϕ NP (Q) such that FR (Q) ⊂ FP (Q ) ; according to Proposition 2.7, Q ⊂ R and there is ψ ∈ F P, NP (Q) such that Q = ψ(Q) is fully normalized and fully centralized in F , and then, by condition 2.13.1, there is an F-isomorphism ζ : NP (Q ) ∼ since F is divi= NP (Q ) such that ζ(Q ) = Q; in particular, sible, there is σ ∈ F(Q ) fulﬁlling ϕ(u) = σ ζ ψ(u) for any u ∈ Q . That is to say, setting R = ζ ψ(R) , the p-groups FR (Q )

and σ ◦ FR (Q ) ◦ σ −1 = ϕFR (Q)

2.13.3

are contained in FP (Q ) which is a Sylow p-subgroup of F(Q ) . At this point, it suﬃces to prove that there is θ ∈ F(P, R ) fulﬁlling θ (Q ) = Q and θ (u ) = σ (u ) for any u ∈ Q; indeed, in this case the F-morphism R → P sending v ∈ R to θ ζ ψ(v) extends ϕ and the corollary follows from Proposition 2.11. We apply the Alperin Fusion Theorem (cf. Theorem 2.6 in [28, Ch. 7] or chapter 5 below) to the group F(Q ) and

34

Frobenius categories versus Brauer blocks

argue by induction on the length of the decomposition of σ in Alperin’s statement (see 5.14 below). That is to say, we may assume that σ = τ ◦ σ where σ ∈ F(Q ) already fulﬁlls σ (u ) = θ (u ) for some θ ∈ F(P, R ) and any u ∈ Q , and where τ ∈ F(Q ) normalizes FT (Q ) for some subgroup T of NP (Q ) containing θ (R )·CP (Q ) ; then, it follows from the induction hypothesis and condition 2.13.2 that τ can be lifted to some ρ ∈ F(T )Q , so that we have ρ θ (u ) = σ (u ) for any u ∈ Q ; since θ (R ) ⊂ T , we are done. 2.14 The ﬁrst successful test for Frobenius P -categories — in order to mimic ﬁnite groups and, more generally, blocks — is the existence of normalizers and centralizers of the objects. Let F be a divisible P -category, Q a subgroup of P and K a subgroup of Aut(Q) ; assume that Q is fully K-normalized in F ; the K-normalizer — or the centralizer if K = {1} and the normalizer if K = Aut(Q) — of Q in F is the NPK (Q)-subcategory NFK (Q) where, for any pair of subgroups R and T of NPK (Q) , the set of morphisms from T to R is the set of ϕ ∈ F(R, T ) fulﬁlling the condition 2.14.1 There are an F-morphism ψ : Q·T → Q·R and an element χ ∈ K such that χ(u) = ψ(u) for any u ∈ Q and that ψ(v) = ϕ(v) for any v ∈ T . 2.15 It is quite clear that NFK (Q) is an NPK (Q)-category. Note that, since F is divisible, it is easy to check that the isomorphism T ∼ = ϕ(T ) K determined by ϕ ∈ F(R, T ) belongs to NF (Q) (R, T ) and therefore NFK (Q) is divisible too; actually, NFK (Q) (R, T ) also coincides with the set of group homomorphisms ϕ : T → R fulﬁlling condition 2.14.1. Moreover, if Q is a subgroup of P , K is a subgroup of Aut(Q ) , Q is fully K -normalized in F and there is an F-isomorphism Q·NPK (Q) ∼ = Q ·NPK (Q ) mapping Q onto Q and K onto K , from the divisibility condition it is straightforward to prove that such an F-isomorphism induces an equivalence of categories between NFK (Q) and NFK (Q ) . Proposition 2.16 Let F be a Frobenius P -category, Q a subgroup of P and K a subgroup of Aut(Q) . If Q is fully K-normalized in F then NFK (Q) is a Frobenius NPK (Q)-category. Proof: Set F = NFK (Q) and P = NPK (Q) ; since P is obviously fully normalized in F , denoting by K the subgroup of automorphisms of Q·P which stabilize Q and P , and act on Q via elements of K , it follows from Lemma 2.17 below that Q·P is fully K -normalized in F and then, it follows from Proposition 2.11 that K ∩ FP (Q·P ) is a Sylow p-subgroup of K ∩ F(Q·P ) ; but, by the very deﬁnition of F (cf. 2.14.1), the restriction to P determines a surjective homomorphism K ∩ F(Q·P ) → F (P ) mapping K ∩ FP (Q·P ) onto FP (P ) , so that F fulﬁlls condition 2.8.1. Let R be a subgroup of P , J a subgroup of Aut(R) and ϕ : R → P an F -morphism such that ϕ(R) is fully ϕJ-normalized in F , and assume that

2. Frobenius P -categories: the ﬁrst deﬁnition

35

ψ : Q·R → Q·P is an F-morphism and χ an element of K fulﬁlling ψ(v) = ϕ(v) and ψ(u) = χ(u)

2.16.1

for any v ∈ R and any u ∈ Q (cf. 2.14.1); set T = Q·R and denote by J the subgroup of automorphisms of T which stabilize Q and R , and act on them via elements of K and J respectively. According to Lemma 2.17 below, ψ(T ) is fully ψJ -normalized in F and therefore, it follows from condition 2.8.2 that there are an F-morphism ζ : T ·NPM (T ) → P and an element µ ∈ J such that ζ(w) = ψ µ(w) for any w ∈ T ; in particular, for any u ∈ Q we get ζ(u) = χ µ(u) and there fore the action of ζ on Q determines an element of K ; thus, ζ R·NPJ (T ) also normalizes Q and acts on it via a subgroup of K . Consequently, since NPJ (T ) = NPJ (R) (cf. Lemma 2.17 below) and the action of µ on R determines an element λ of J , the restriction of ζ over R·NPJ (R) determines an F -morphism R·NPJ (R) → P and, for any v ∈ R , we have ζ(v) = ψ µ(v) = ϕ λ(v)

2.16.2.

This proves that F fulﬁlls condition 2.8.2 too. We are done. Lemma 2.17 Let F be a Frobenius P -category, Q a subgroup of P and K a subgroup of Aut(Q) . Assume that Q is fully K-normalized in F . Let R be a subgroup of NPK (Q) and J a subgroup of Aut(R) , and denote by I the subgroup of automorphisms of Q·R which stabilize Q and R , and act on them via elements of K and J respectively. Then, we have NPI (Q·R) = NPJ (R) ∩ NPK (Q)

2.17.1

and if R is fully J-normalized in NFK (Q) then Q·R is fully I-normalized in F . Proof: Set F = NFK (Q) , P = NPK (Q) and T = Q·R ; ﬁrstly, the equality NPI (T ) = NPJ (R) is easily checked and needs no hypothesis on F . Secondly, for any F-morphism ψ : T ·NPI (T ) → P , set Q = ψ(Q) and consider the F-morphism Q → P obtained from the composition of the inclusion map ∼ ιP Q : Q → P and the inverse of the isomorphism Q = Q determined by ψ ; since Q is fully K-normalized in F , it follows from condition 2.8.2 that there ψ are an F-morphism ζ : Q ·NPK (Q ) → P and an element χ ∈ K such that ψ ζ ψ(u) = χ(u) for any u ∈ Q ; in particular, we have ζ NPK (Q ) ⊂ P and, since ψ(Q·P ) is contained in Q ·NPK (Q ) and F is divisible, the homomorphism η : T ·NPI (T ) = Q· R·NPJ (R) −→ Q·P 2.17.2 ψ

36

Frobenius categories versus Brauer blocks

mapping w ∈ T ·NPI (T ) on ζ ψ(w) belongs to F Q·P , T ·NPI (T ) ; moreover, ψ since ψ(R) ⊂ NPK ψ(Q) , it determines an F -morphism R·NPJ (R) → P (cf. 2.14.1). if weassume that R is fully J-normalized in F , we get JConsequently, η J η NP (R) = NP η(R) ; but, we already have NPI (T ) = NPJ (R) and, according to the same equality applied to η(Q) = Q , ηK = K , η(R) and ηJ , η η we still have NP I η(T ) = NPJ η(R) , so that we get ψ η ζ ψ NPI (T ) = η NPI (T ) = NP I η(T ) ⊃ ζ NPI ψ(T )

2.17.3,

ψ which forces ψ NPI (T ) = NPI ψ(T ) . We are done. 2.18 It is clear that, in a divisible P -category F , we can iterate the normalizer construction until to deﬁne the normalizer of an F-chain — C-chains are introduced in A2.8 for any small category C — provided it fulﬁlls a suitable iterated fully normalized condition. Explicitly, recall that for any n ∈ N , an F-chain — more precisely called a (n, F)-chain — is a functor q : ∆n −→ F

2.18.1

from the category ∆n formed by the objects 0 ≤ i ≤ n and the morphisms 0 ≤ j ≤ i ≤ n , with the obvious composition (cf. A2.2); then, arguing by induction on n , we say that q is fully normalized in F if q(n) is fully normalized in F and moreover, setting P = NP q(n) and F = NF q(n) , in the case where n ≥ 1 the F -chain q : ∆n−1 → F mapping 0 ≤ i ≤ n − 1 on the image of q(i • n) , and the ∆n−1 -morphisms on the corresponding inclusion maps, is fully normalized in F . Note that, by Proposition 2.7, any F-chain admits a ch(F)-isomorphic F-chain fully normalized in F . 2.19 Assume that q is an (n, F)-chain fully normalized in F . In the case where n ≥ 1 , we inductively deﬁne NP (q) = NP (q ) and NF (q) = NF (q )

2.19.1,

and it follows from Proposition 2.16 that if F is a Frobenius P -category then NF (q) is a Frobenius NP (q)-category. Actually, if F is a Frobenius P -category, denoting by F(q) the group of ch(F)-automorphisms of q — identiﬁed with the stabilizer in F q(n) of all the subgroups Im q(i • n) when i runs over ∆n (cf. A2.8) — it follows from Lemma 2.17 that q(n) is fully F(q)-normalized in F and that we have F (q)

NP (q) = NP

q(n)

F (q)

and NF (q) = NF

q(n)

2.19.2.

These comments are useful in chapter 18 for the construction of the localizing functor .

2. Frobenius P -categories: the ﬁrst deﬁnition

37

2.20 But in chapter 5, in order to exhibit a suﬃcient condition to get a Frobenius P -category from a suitable formulation of our version of the Alperin Fusion Theorem, we are also interested on the so-called normal F-chains, which can be reduced from the bottom. Let us say that an F-chain q : ∆n → F is normal if the image of q(i• n) is normal in q(n) for any i ∈ ∆n ; then, arguing by induction on n , we say that q is fully conormalized in F if the image Q of q(0 • n) is fully normalized in F and, setting P = NP (Q) and F = NF (Q), in the case where n ≥ 1 the F -chain q : ∆n−1 → F mapping 0 ≤ i ≤ n − 1 on the image of q(i + 1 • n) , and the ∆n−1 -morphisms on the corresponding inclusions, is fully conormalized in F . Note that, by Proposition 2.7, any F-chain admits a ch(F)-isomorphic F-chain fully conormalized in F . Once again, if q is a normal (n, F)-chain fully conormalized in F , in the case where n ≥ 1 we inductively deﬁne NP (q) = NP (q ) and NF (q) = NF (q )

2.20.1;

if q is also fully normalized in F , it is not diﬃcult to check from condition 2.14.1 that both deﬁnitions coincide. Corollary 2.21 Let F be a Frobenius P -category. For any normal F-chain q : ∆n → F there is an F-morphism ϕ : q(n) → P such that the image of ϕ ◦ q(i• n) is fully centralized in F for any i ∈ ∆n . Moreover, any subgroup Q of P having a selfcentralizing subgroup R fully centralized in F is also fully centralized in F . Proof: We may assume that n ≥ 1 and that q is fully conormalized in F , and argue by induction on n ; with the notation in 2.20 above, we already know that the image T of q(0•n) is fully centralized in F since it is fully normalized (cf. Proposition 2.11); then, it follows from Proposition 2.16 that F = NF (T ) is a Frobenius P -category, and from the induction hypothesis applied to the normal F -chain q that there is an F -morphism ϕ : q (n − 1) → P such that the image of ϕ ◦ q (i • n −1) is fully centralized in F for any i ∈ ∆n−1 ; since T is fully normalized in F and normal in the image of ϕ ◦ q(i + 1• n) , it follows from Lemma 2.17 that this image is also fully centralized in F . In order to prove the last statement arguing by induction on |Q : R| , we clearly may assume that Q normalizes R ; in this case, we already know that there is an F-morphism ϕ : Q → P such that ϕ(Q) and ϕ(R) are both fully centralized in F . But, since ϕFQ·CP (R) (R) = Fϕ(Q) ϕ(R) , it follows from condition 2.10 that there is an F-morphism ρ : Q·CP (R) → P extending ϕ ; moreover, since CQ (R) ⊂ R (cf. 1.9), we necessarily have ρ(Q) = ϕ(Q) and therefore, since ϕ(R) is fully centralized in F , we get ρ CP (Q) = ρ CCP (R) (Q) = Cρ(CP (R)) ρ(Q) 2.21.1. = CCP (ϕ(R)) ϕ(Q) = CP ϕ(Q) Consequently, since ϕ(Q) is fully centralized in F , Q is fully centralized too (cf. Proposition 2.7). We are done.

Chapter 3

The Frobenius P-category of a block 3.1 Let G be a ﬁnite group, k an algebraically closed ﬁeld of characteristic p , b a primitive idempotent of the center Z(kG) of the group algebra kG — so that (b, G) is a block , or b is a block of G (cf. 1.10) — and (P, e) a maximal Brauer (b, G)-pair (cf. 1.15). In this chapter, we describe the Frobenius P -category F(b,G) associated with (b, G) ; although strictly speaking it depends on the choice of (P, e) , it will be immediately clear that the diﬀerent choices determine equivalent categories and that this dependence is ultimately unessential. 3.2 Let Q and R be a pair of subgroups of P ; recall that there are unique blocks f and g of the respective centralizers CG (Q) and CG (R) fulﬁlling (cf. 1.15) (Q, f ) ⊂ (P, e) and (R, g) ⊂ (P, e) 3.2.1; the uniqueness of f and g allows us to deﬁne F(b,G) (Q, R) as the set of group homomorphisms ϕ : R → Q such that there is x ∈ G fulﬁlling (R, g) ⊂ (Q, f )x

and ϕ(v) = xvx−1 for any v ∈ R

3.2.2;

the transitivity of the inclusion between Brauer (b, G)-pairs (cf. 1.15) guarantees that the sets F(b,G) (Q, R) deﬁne a subcategory F(b,G) of Gr over the set of subgroups of P and it is clear that all these group homomorphisms are injective. Moreover, for any u ∈ P , we still have (Q, f )u ⊂ (P, e) and therefore if R ⊂ Qu then we get (R, g) ⊂ (Q, f )u (cf. 1.15.3); hence, we have FP (Q, R) ⊂ F(b,G) (Q, R)

3.2.3,

which proves that F(b,G) is a P -category. Note that if b is the so-called principal block of G then F(b,G) coincides with FG (cf. Theorem 3.13 in [3]). Proposition 3.3 The P -category F(b,G) is divisible. Proof: Let Q , R and T be subgroups of P , ϕ an element of F(b,G) (Q, R) and ψ an element of F(b,G) (Q, T ) fulﬁlling ψ(T ) ⊂ ϕ(R) , so that there is a group homomorphism η : T → R such that ψ = ϕ ◦ η ; thus, denoting by f , g and h the respective blocks of the centralizers CG (Q) , CG (R) and CG (T ) such that (P, e) contains (Q, f ) , (R, g) and (T, h) , there are x, y ∈ G fulﬁlling −1

⊂ (Q, f )

and ϕ(v) = xvx−1 for any v ∈ R

−1

⊂ (Q, f )

and ψ(w) = ywy −1 for any w ∈ T

(R, g)x (T, h)y

3.3.1;

40

Frobenius categories versus Brauer blocks −1

−1

consequently, we get xη(w)x−1 = ywy −1 for any w ∈ T , so that T y ⊂ Rx −1 −1 and therefore, by the uniqueness of the blocks g x and hy , we still get −1 −1 (T, h)y ⊂ (R, g)x (cf. 1.15.3), so that η and x−1 y fulﬁll condition 3.2.2; thus, η belongs to F(b,G) (R, T ) . 3.4 Let (Q, f ) be a Brauer (b, G)-pair contained in (P, e) and K a subK group of Aut(Q) ; since the converse image NG (Q, f ) of K in NG (Q, f ) conK tains CG (Q) , it is clear that f is also a block of NG (Q, f ) (cf. 1.19); moreover, the uniqueness of f implies that NP (Q) — and a fortiori NPK (Q) — ﬁxes f . On the other hand, since CG Q·NPK (Q) is a normal subgroup of K CNGK (Q,f ) NPK (Q) = CG NPK (Q) ∩ NG (Q, f )

3.4.1,

the block g of CG Q·NPK (Q) such that (P, e) contains (Q·NPK (Q), g) deter mines a block g K of CNGK (Q,f ) NPK (Q) , so that (NPK (Q), g K ) is a Brauer K f, NG (Q, f ) -pair (cf. 1.16). Proposition 3.5 With the notation above, Q is fully K-normalized in F(b,G) K if and only if (NPK (Q), g K ) is a maximal Brauer f, NG (Q, f ) -pair. Proof: We know that for any F(b,G) -morphism ψ : Q·NPK (Q) → P there is y ∈ G fulﬁlling (Q·NPK (Q), g)y ⊂ (P, e)

and ψ(v) = v y for any v ∈ Q·NPK (Q)

3.5.1;

K hence, setting Ky = ψK and denoting by gy the block of CG Qy ·NP y (Qy ) K

such that (P, e) contains (Qy ·NP y (Qy ), gy ) , we have (cf. 1.15.3) K

(Q·NPK (Q), g)y ⊂ (Qy ·NP y (Qy ), gy )

3.5.2;

K

moreover, it is clear that f y is a block of NG y (Qy ) and that this inclusion of Brauer (b, G)-pairs implies the inclusion K

(NPK (Q), g K )y ⊂ (NP y (Qy ), (gy )K )

3.5.3

K of Brauer f y , NG y (Qy , f y ) -pairs (cf. 1.15) where, as above, (gy )K denotes K the block of CN Ky (Qy ,f y ) NP y (Qy ) determined by gy . But, if (NPK (Q), g K ) G K is a maximal Brauer f, NG (Q, f ) -pair then it is clear that (NPK (Q), g K )y K is a maximal Brauer f y , NG y (Qy , f y ) -pair too. In particular, in this case we get ψ K ψ NPK (Q) = NPK (Q)y = NP y (Qy ) = NPK ψ(Q) 3.5.4, which proves that Q is fully K-normalized in F(b,G) (cf. 2.6.2).

3. The Frobenius P -category of a block

41

Conversely, assume that Q is fully K-normalized in F(b,G) ; any maximal K (Q, f ) -pair (T, hK ) containing (NPK (Q), g K ) is determined Brauer f, NG by a Brauer (b, G)-pair (Q·T, h) which contains (Q·NPK (Q), g) (cf. 3.4) and therefore there is y ∈ G such that (Q·T, h)y ⊂ (P, e) . Thus, we have the inclusions (Q·NPK (Q), g)y ⊂ (Q·T, h)y ⊂ (P, e) 3.5.5 and, in particular, the group homomorphism ψ : Q·NPK (Q) → P mapping v ∈ Q·NPK (Q) on v y belongs to F(b,G) P, Q·NPK (Q) ; hence, since Q is fully K-normalized in F(b,G) , we get ψ K NPK (Q)y = ψ NPK (Q) = NPK ψ(Q) = NP y (Qy ) 3.5.6. K

But, it is clear that T y ⊂ P ∩ NG y (Qy , f y ) which, with equality 3.5.6, forces the equality (NPK (Q), g K ) = (T, hK ) . We are done. Corollary 3.6 With the notation above, assume that Q is fully K-normalized in F(b,G) . Then, the K-normalizer NFK(b,G) (Q) of Q in F(b,G) coincides with K the NPK (Q)-category F(f,NGK (Q,f )) associated with the block f of NG (Q, f ) . K Proof: We know that (NPK (Q), g K ) is a maximal Brauer (f, NG (Q, f ))-pair K and therefore that F(f,NGK (Q,f )) is a NP (Q)-category; moreover, if (T, hK ) is K a Brauer (f, NG (Q, f ))-pair contained in (NPK (Q), g K ) and ψ : T → NPK (Q) K is an F(f,NGK (Q,f )) -morphism, there is n ∈ NG (Q, f ) fulﬁlling

(T, hK ) ⊂ (NPK (Q), g K )n

and ψ(w) = nwn−1 for any w ∈ T

3.6.1.

On the other hand, we know that the block h of CG (Q·T ) such that the Brauer (b, G)-pair (Q·T, h) is contained in (Q·NPK (Q), g) , determines the z block hK (cf. 1.15). In particular, we have hK = z∈Z h for a suitable K subset Z ⊂ CG (T ) ∩ NG (Q, f ) and, similarly, it follows from the inclusion in 3.6.1 that there is z ∈ Z such that we have the inclusion of (b, G)-pairs (Q·T, hz ) ⊂ (Q·NPK (Q), g)n

3.6.2;

hence, the group homomorphism ξ : Q·T → mapping u ∈ Q·T −1 −1 on nz uzn is an F(b,G) -morphism; ﬁnally, since ξ extends ψ and the restriction of ξ to Q induces an element of K , ψ is an NFK(b,G) (Q)-morphism (cf. 2.14.1). Q·NPK (Q)

Conversely, if ψ : T → NPK (Q) is an NFK(b,G) (Q)-morphism, we know that ψ can be extended to an F(b,G) -morphism ξ : Q·T → Q·NPK (Q) such that its restriction to Q induces an element of K (cf. 2.14.1); thus, with the notation above, there is x ∈ G fulﬁlling (Q·T, h) ⊂ (Q·NPK (Q), g)x

and ξ(u) = xux−1 for any u ∈ Q·T

3.6.3;

42

Frobenius categories versus Brauer blocks

in particular, x normalizes Q and induces on it an element of K ; moreover, we have (Q, f ) ⊂ (P, e)x and therefore f = f x (cf. 1.15.3). Then, with K the notation above, (NPK (Q), g K )x is a maximal Brauer (f, NG (Q, f ))-pair, K K whereas (T, h ) is a Brauer (f, NG (Q, f ))-pair and we still have (cf. 1.15) (T, hK ) ⊂ (NPK (Q), g K )x

3.6.4;

now, inclusion 3.6.4 shows that ψ is a F(f,NGK (Q,f )) -morphism too. Consequently, for any subgroup T of NPK (Q) we have obtained

NFK(b,G) (Q) NPK (Q), T = F(f,NGK (Q,f )) NPK (Q), T

3.6.5

and, since the NPK (Q)-categories NFK(b,G) (Q) and F(f,NGK (Q,f )) are both divisible (cf. 2.14 and Proposition 3.3), they coincide (cf. 2.4). Theorem 3.7 F(b,G) is a Frobenius P -category. Proof: By Proposition 3.3, we already know that F(b,G) is divisible and, according to condition 3.2.2, we have F(b,G) (P ) ∼ = NG (P, e)/CG (P )

3.7.1;

thus, since p does not divide |NG (P, e)/P ·CG (P )| (cf. 1.16), F(b,G) fulﬁlls condition 2.8.1. Let Q be a subgroup of P , K a subgroup of Aut(Q) containing FQ (Q) and ϕ : Q → P an F(b,G) -morphism such that ϕ(Q) is fully ϕK-normalized in F(b,G) ; in particular, denoting by f the block of CG (Q) such that (P, e) contains (Q, f ) , there is x ∈ G fulﬁlling (cf. 3.2.2) (Q, f )x ⊂ (P, e)

and ϕ(u) = ux for any u ∈ Q

3.7.2.

As above, denote by g the block of CG NPK (Q) such that (P, e) contains the Brauer (b, G)-pair (NPK (Q), g) . Moreover, setting Q = ϕ(Q) , K = ϕK and f = f x in kG and denoting by g the block of CG NPK (Q ) such that (P, e) contains the Brauer (b, G)-pair (NPK (Q ), g ) (cf. 1.15), it follows from Proposition 3.5 K that (NPK (Q ), g ) is a maximal Brauer (f , NG (Q , f ))-pair; thus, since K K (NPK (Q), g)x is also a Brauer (f , NG (Q , f ))-pair, there is n ∈ NG (Q , f ) such that (cf. 1.15)

(NPK (Q), g)xn ⊂ (NPK (Q ), g )

3.7.3.

3. The Frobenius P -category of a block

43

But, this inclusion is an inclusion of Brauer (b, G)-pairs too (cf. 1.19). Consequently, (P, e) contains the Brauer (b, G)-pair (NPK (Q), g)xn and there fore the group homomorphism ψ : NPK (Q) → P mapping v ∈ NPK (Q) on v xn is an F(b,G) -morphism; moreover, denoting by χ : Q ∼ = Q the group automor

−1

phism mapping u∈ Q on uxn x , it is clear that χ belongs to K and that we have ψ(u) = ϕ χ(u) for any u ∈ Q . So, F(b,G) fulﬁlls condition 2.8.2 too. We are done. 3.8 As a matter of fact, a similar argument covers a more general situation, namely the case of the p-permutation G-algebras (cf. 1.14) introduced in [11]. Precisely, let A be a G-algebra over k such that a Sylow p-subgroup of G stabilizes a basis of A ; consider a point α of G on A (cf. 1.12) and the Brauer (α, G)-pairs (cf. 1.14). Recall that the inclusion between the local pointed groups on A (cf. 1.12) contained in Gα induces an inclusion between the Brauer (α, G)-pairs (cf. 1.15), and that all the maximal Brauer (α, G)-pairs are G-conjugate (cf. Theorem 1.14 in [11]); denote by (P, eA ) one of them and let us describe the P -category F(α,G) associated with A and α . 3.9 Let Q and R be a pair of subgroups of P ; once again, recall that there are unique primitive idempotents fA of Z A(Q) and gA of Z A(R) fulﬁlling (cf. 1.15) (Q, fA ) ⊂ (P, eA ) and

(R, gA ) ⊂ (P, eA )

3.9.1;

the uniqueness of fA and gA allows us to deﬁne F(α,G) (Q, R) as the set of group homomorphisms ϕ : R → Q such that there is x ∈ G fulﬁlling (R, gA ) ⊂ (Q, fA )x

and ϕ(v) = xvx−1 for any v ∈ R

3.9.2;

the transitivity of the inclusion between Brauer (α, G)-pairs guarantees that the sets F(α,G) (Q, R) deﬁne a subcategory F(α,G) of Gr over the set of subgroups of P and it is clear that all these group homomorphisms are injective. Moreover, for any u ∈ P , we still have (Q, fA )u ⊂ (P, eA ) and therefore if R ⊂ Qu then we get (R, gA ) ⊂ (Q, fA )u (cf. 1.15); hence, we still have FP (Q, R) ⊂ F(α,G) (Q, R)

3.9.3

which proves that F(α,G) is a P -category. As above, it is easily proved that 3.9.4 F(α,G) is a divisible P -category. Remark 3.10 As in the particular case of the Frobenius P -category F(b,G) , this divisible P -category F(α,G) depends on the choice of the maximal Brauer (α, G)-pair (P, eA ) ; although this dependence is unessential, it is sometimes e more handy to consider the equivalent category — noted F(α,G) — where the objects are all the Brauer (α, G)-pairs, avoiding the choices of (P, eA ) and of e a functorial section F(α,G) → F(α,G) . This point of view will be very useful in chapter 11.

44

Frobenius categories versus Brauer blocks

Proposition 3.11 Let Q be a subgroup of P , K a subgroup of Aut(Q) con- taining FQ (Q) and nA the primitive idempotent of the center of A NPK (Q) such that (NPK (Q), nA ) ⊂ (P, eA ) . If Q is fully K-normalized in F(α,G) then K K there is a point β of NG (Q, fA ) on A such that NG (Q, fA )β ⊂ Gα and K K (NP (Q), nA ) is a maximal Brauer (β, NG (Q, fA ))-pair. Proof: Let (T, mA ) be a Brauer (α, G)-pair which is maximal fulﬁlling K T ⊂ NG (Q, fA ) and

(NPK (Q), nA ) ⊂ (T, mA )

3.11.1;

K since mA BrT (α) = {0} , we can consider a point β of NG (Q, fA ) on A such that (cf. 1.12) K NG (Q, fA )β ⊂ Gα

and mA BrT (β) = {0}

3.11.2;

then, we still have nA BrNPK (Q) (β) = {0} and fA BrQ (β) = {0} , and therefore K (T, mA ) , (NPK (Q), nA ) and (Q, fA ) are also Brauer (β, NG (Q, fA ))-pairs. y Moreover, we already know that we have (T, mA ) ⊂ (P, eA ) for some y ∈ G (cf. 1.15) and therefore we still have the inclusion

(NPK (Q), nA )y ⊂ (T, mA )y ⊂ (P, eA )

3.11.3;

in particular, the group homomorphism ψ : NPK (Q) → P mapping v ∈ NPK (Q) on v y belongs to F(α,G) P, NPK (Q) ; hence, since we assume that Q is fully K-normalized in F(α,G) , setting Ky = ψK we get ψ K NPK (Q)y = ψ NPK (Q) = NPK ψ(Q) = NP y (Qy )

3.11.4.

K

But, it is clear that T y ⊂ NP y (Qy ) which, with equality 3.11.4, forces the equality (NPK (Q), nA ) = (T, mA ) ; in particular, (NPK (Q), nA ) is a maximal K Brauer (β, NG (Q, fA ))-pair. We are done. 3.12 A specially interesting case where we ﬁnd the above situation concerns the group algebra kG but considered as an H-algebra with respect to some subgroup H of G . That is to say, in 3.8 above, we replace G by a subgroup H and consider A = kGb as an H-algebra; then, for any point β of H on kGb , choosing a maximal Brauer (b, G)-pair (Q, f ) such that f BrQ (β) = {0} , we get the divisible Q-category F(β,H) and, from the point of view of Remark 3.10, it is quite clear that we simply have an inclusion of categories e e F(β,H) ⊂ F(b,G) 3.12.1. Nevertheless, any element x ∈ G such that (Q, f ) ⊂ (P, e)x determines an inclusion functor from F(β,H) to F(b,G) noted ibβ : F(β,H) −→ F(b,G)

3.12.2,

3. The Frobenius P -category of a block

45

and, for another element x ∈ G such that (Q, f ) ⊂ (P, e)x , x x−1 determines a natural isomorphism between both inclusion functors. All this framework will be widely employed in chapter 11. K 3.13 A last remark; whenever H = NG (Q, f ) where (Q, f ) is a Brauer (b, G)-pair and K a subgroup of Aut(Q) , assuming that Q is fully K-norK malized in F(b,G) and denoting by ν the point of NG (Q, f ) on kGb determined by f (cf. 1.19), we have

F(ν,NGK (Q,f )) = NFK(b,G) (Q) = F(f,NGK (Q,f ))

3.13.1.

Chapter 4

Nilcentralized, selfcentralizing and intersected objects in Frobenius P-categories 4.1 Let P be a ﬁnite p-group and F a divisible P -category such that FP (P ) is a Sylow p-subgroup of F(P ) . As a matter of fact, in order to check whether or not F is a Frobenius P -category, we only need to check condition 2.11.1 above over a restricted set of subgroups of P — the so-called F-intersected subgroups — that we introduce in 4.11 below. 4.2 They appear inside a larger family of subgroups of P — the F-selfcentralizing subgroups (cf. 4.8 below) — which has an interest on its own: sc denoting by F the full subcategory of F over this family, all the morphisms sc in its exterior quotient F˜ (cf. 1.3) are epimorphisms (see Proposition 4.6 below). In chapter 6 we exhibit more interesting features of this exterior quotient, and give a meaningful interpretation of the F-intersected subgroups (see Proposition 6.16 below). 4.3 Actually, in Proposition 4.6 the key property of the F-selfcentralizing subgroups holds for a larger set of subgroups, namely for the F-nilcentralized subgroups of P ; we say that a subgroup Q of P is F-nilcentralized if, for any ϕ ∈ F(P, Q) such that Q = ϕ(Q) is fully centralized in F , the CP (Q )-categories CF (Q ) (cf. 2.14) and FCP (Q ) (cf. 1.8) coincide — loosely speaking, this amounts to saying that Q has a nilpotent centralizer in F , which motivates the terminology. 4.4 Note that, by 2.15 above, if F is a Frobenius P -category and this condition is fulﬁlled for some ϕ ∈ F(P, Q) , it is fulﬁlled for any. As a matter of fact, this condition is preserved by the central quotients of F (cf. Proposition 12.3 below and Theorem 3.4 in [28, Ch. 5]), whereas it need not be the case for the condition deﬁning the F-selfcentralizing subgroups (cf. 4.8 below). Proposition 4.5 Assume that F is a Frobenius P -category. A subgroup Q of P which contains an F-nilcentralized subgroup R is F-nilcentralized too. If moreover R is fully centralized in F , then Q is fully centralized in F too. Proof: Arguing by induction on |Q : R| , we may assume that R is normal in Q ; then, by Proposition 2.7, there is an F-morphism ξ : Q·CP (Q) → P such that ξ(R) is both fully centralized and fully normalized in F ; hence, up to replacing R and Q by ξ(R) and ξ(Q) , we may assume that R is fully centralized in F .

48

Frobenius categories versus Brauer blocks

Choose ϕ ∈ F(P, Q) such that Q = ϕ(Q) is fully centralized in F (cf. Proposition 2.7) and set R = ϕ(R) ; then, since we assume that R is fully centralized in F and we clearly have FQ (R) = FQ ·CP (R ) (R )

ϕ

4.5.1,

according to statement 2.10.1 there is an F-morphism ψ : Q ·CP (R ) → P such that ψ ϕ(v) = v for any v ∈ R ; moreover, setting U = FQ (R) , we already know that R is also fully U -normalized in F (cf. Proposition 2.11) and we clearly have NPU (R) = Q·CP (R) , so that we can consider the Frobenius Q·CP (R)-category NFU (R) (cf. Proposition 2.16). In particular, the composition U of ϕ with the restriction of ψ to Q determines an element of NF (R) Q·CP (R), Q ; but, we are assuming that CF (R) = FCP (R) and therefore we also have NFU (R) = FQ·CP (R) , as it is easily checked; consequently, up to a modiﬁcation of our choice of ψ , we may assume that ψ(ϕ(u) = u for any u ∈ Q and then we have ψ CP (Q ) ⊂ CP (Q) 4.5.2, which forces the equality (cf. 2.6.2) and proves that Q is fully centralized too. Since it is now clear that CF (Q) is a subcategory of CF (R) = FCP (R) , it is then straightforward to prove that CF (Q) = FCP (Q) . We are done. Proposition 4.6 Assume that F is a Frobenius P -category. Let Q be a subgroup of P and R an F-nilcentralized subgroup of Q . If ϕ , ϕ ∈ F(P, Q) fulﬁll ϕ(v) = ϕ (v) for any v ∈ R and the subgroup R = ϕ(R) = ϕ (R) is fully centralized in F , then there is u ∈ CP (R ) fulﬁlling ϕ (v) = ϕ(v)u for any v ∈ Q . Proof: We argue by induction on |Q : R| and may assume that R = Q ; moreover, from the divisibility of F , up to replacing Q and R by ϕ (Q) and ϕ (R) we may assume that ϕ is the inclusion map and that R is fully centralized in F . Set N = NQ (R) and U = FP (R) so that NPU (R) = NP (R) ; then, according to Proposition 2.11, R is also fully U -normalized in F and, since R is F-nilcentralized, it is not diﬃcult to see that NFU (R) = FNP (R) (cf. Theorem 3.2 in [28, Ch. 5]). In this situation, since ϕ(v) = v for any v ∈ R , the restriction of ϕ to N determines an element of NFU (R) NP (R), N and therefore there is w ∈ NP (R) such that ϕ(v)w = v for any v ∈ N ; but, by Proposition 4.5, N is F-nilcentralized and fully centralized in F too; consequently, it follows from the induction hypothesis that there is an element v ∈ CP (N ) ⊂ CP (R) such that ϕ(u)wv = u for any u ∈ Q . We are done. Corollary 4.7 Assume that F is a Frobenius P -category. Let Q be a subgroup of P and R an F-nilcentralized subgroup of Q fully centralized in F ; denote by F(Q)R the stabilizer of R in F(Q) . Then, the kernel of the restriction homomorphism F(Q)R → F(R) coincides with FCP (R) (Q) .

4. Nilcentralized, selfcentralizing and intersected objects

49

Proof: If σ ∈ F(Q) fulﬁlls σ(v) = v for any v ∈ R , then Proposition 4.6 states that σ belongs to FCP (R) (Q) . We are done. 4.8 We say that a subgroup Q of P is F-selfcentralizing† if we have CP ϕ(Q) ⊂ ϕ(Q) 4.8.1 for any ϕ ∈ F(P, Q) ; then, Q is clearly fully centralized in F and F-nilcentralized; moreover, any subgroup R of P such that F(R, Q) = ∅ is F-selfcentralizing too. Recall that F˜ denotes the exterior quotient of F (cf. 1.3). Corollary 4.9 Assume that F is a Frobenius P -category. For any F-selfcentralizing subgroups Q , R and T of P and any F-morphism ϕ : Q → R , the ˜ ˜ map F(T, R) → F(T, Q) determined by the composition with ϕ˜ is injective. ˜ In particular, any F-morphism from an F-selfcentralizing subgroup of P is an epimorphism. ˜ ψ˜ ∈ F(T, ˜ Proof: If two elements ψ, R) fulﬁll ψ˜ ◦ ϕ˜ = ψ˜ ◦ ϕ˜ , we may choose representatives ψ of ψ˜ and ψ of ψ˜ such that ψ ◦ ϕ = ψ ◦ ϕ , and then it follows from Proposition 4.6 that there is z ∈ Z(Q) fulﬁlling ψ (v) = ψ(v z ) = ψ(v)ψ(z)

4.9.1

for any v ∈ R , so that ψ˜ = ψ˜ . We are done. 4.10 Note that, for any subgroup Q of P fully centralized in F , Q·CP (Q) is clearly F-selfcentralizing; in particular, a subgroup Q of P fully centralized in F is F-selfcentralizing if and only if we have CP (Q) = Z(Q) , namely if Q is selfcentralizing in P (cf. 1.9). If F is a Frobenius P -category, a subgroup Q of P fully normalized in F is also fully centralized (cf. Proposition 2.11) and therefore it is F-selfcentralizing if and only if it is selfcentralizing in P ; in this case, by statement 2.10.1 and Proposition 4.6, if R is a subgroup of NP (Q) containing Q then any ϕ ∈ F(P, Q) such that ϕ FR (Q) ⊂ FP ϕ(Q) can be extended to R in a unique way up to conjugation by Z(Q) . 4.11 Actually, there is a maximal subgroup R of NP (Q) fulﬁlling the inclusion ϕ FR (Q) ⊂ FP ϕ(Q) for any ϕ ∈ F(P, Q) ; namely, for any subgroup Q of P fully centralized in F , we denote by IF (Q) the converse image in NP (Q) of the intersection ϕ∗ I¯F (Q) = FP ϕ(Q) 4.11.1, ϕ∈F (P,Q)

where ϕ∗ : ϕ(Q) ∼ = Q denotes the inverse of the isomorphism induced by ϕ , and we say that Q is an F-intersected subgroup of P whenever IF (Q) = Q . †

Called an F-centric subgroup in [13]

50

Frobenius categories versus Brauer blocks

Note that IF (Q) is an F-intersected subgroup of P , and that an F-intersected subgroup of P is F-selfcentralizing. We have the following criterion†. Theorem 4.12 A divisible P -category F is a Frobenius P -category if and only if the following conditions hold: 4.12.1 FP (P ) is a Sylow p-subgroup of F(P ) . 4.12.2 If Q is an F-intersected subgroup of P , R is a subgroup of NP (Q) containing Q and ϕ : Q → P is an F-morphism fulﬁlling ϕ FR (Q) ⊂ FP ϕ(Q) then there is an F-morphism ψ : R → P extending ϕ . 4.12.3 Any divisible P -category F fulﬁlling F (P, Q) ⊃ F(P, Q) for every F-intersected subgroup Q of P contains F . Proof: Conditions 4.12.1 and 4.12.2 are clearly necessary. In a Frobenius P -category F , condition 4.12.3 is necessary too since any F -morphism to P from a subgroup Q of P fully centralized in F can be extended to IF (Q) (cf. statement 2.10.1) which is an F -intersected subgroup (cf. 4.11). For any pair Q and Q of F-isomorphic subgroups of P , consider the set F (Q , Q) of elements ϕ ∈ F(Q , Q) such that there are subgroups U and U of P , F-isomorphic to Q and Q , which are both fully centralized and fully normalized in F and admit F-morphisms λ

NP (Q) −→ NP (U ) , IF (U ) −→ IF (U ) , NP (U ) ←− NP (Q ) λ

σ

fulﬁlling

λ(Q) = U , σ(U ) = U and λ ϕ(u) = σ λ(u) for any u ∈ Q .

U = λ (Q )

,

4.12.4 4.12.5

Note that FP (Q , Q) ⊂ F (Q , Q) ; indeed, if ϕ is the conjugation by some u ∈ P then, choosing an F-morphism λ : NP (Q) → P such that U = λ(Q) is fully centralized and fully normalized in F (cf. Proposition 2.7), it is clear that U = uU u−1 = uλ(Q ) is also fully centralized and fully normalized in F , and it suﬃces to consider λ = uλ and the isomorphism IF (U ) ∼ = IF (U ) determined by the conjugation by u . On the other hand, if Q is an F-intersected subgroup of P then we have F (Q , Q) = F(Q , Q) ; indeed, it suﬃces to choose F-morphisms ζ : NP (Q) −→ P

and ζ : NP (Q ) −→ P

4.12.6

such that the subgroups U = ζ(Q) and U = ζ (Q ) are both fully normalized and fully centralized in F (cf. Proposition 2.7); in this case, we have the equalities IF (U ) = U and IF (U ) = U , and therefore the existence of an F-morphism σ : U → U fulﬁlling the conditions above is clear. †

The interested reader will see that the hypothesis (*) in [15, Theorem 2.2, 331-339] forces

the set of subgroups H considered there to include all the F-intersected subgroups. Our proof here already appeared in [46] — quoted with the number 15 in [15] — except that in our old notes we considered all the F-selfcentralizing subgroups, and U ·CP (U ) instead of IF (U ).

4. Nilcentralized, selfcentralizing and intersected objects

51

From now on, we assume that F fulﬁlls the conditions above. More generally, for any pair Q and R of subgroups of P we set F (Q, R) =

ιQ ϕ(R) ◦ F ϕ(R), R

4.12.7,

ϕ∈F (Q,R)

where ιQ ϕ(R) denotes the corresponding inclusion map (cf. 1.9), and we will prove that F is a Frobenius P -category, so that F = F by condition 4.12.3. Since F (P, P ) = F(P ) , it suﬃces to prove that F is a category and fulﬁlls conditions 2.3.1 and 2.8.2. First of all, with the notation above we claim that if ϕ is an element of F (Q , Q) then ϕ−1 belongs to F (Q, Q ) ; indeed, since U is fully centralized in F , we have σ U ·CP (U ) = U ·CP (U ) and therefore we still have σ IF (U ) = IF (U ) (cf. 4.11.1), so that it suﬃces to consider the triple (λ , σ −1 , λ) of F-morphisms. Let Q be a third subgroup of P , F-isomorphic to Q and Q , and consider a homomorphism ϕ ∈ F (Q , Q ) ; in order to prove that ϕ ◦ ϕ belongs to F (Q , Q) , it follows from our argument above that we may assume Q , Q and Q are not F-intersected groups. Mutatis mutandis we have subgroups V and V of P , F-isomorphic to Q and Q , which are both fully centralized and fully normalized in F , and admit F-morphisms µ

µ

NP (Q ) −→ NP (V ) , IF (V ) −→ IF (V ) , NP (V ) ←− NP (Q ) 4.12.8 τ

fulﬁlling

µ(Q ) = V , τ (V ) = V , V = µ (Q ) 4.12.9 and µ ϕ (u ) = τ µ(u ) for any u ∈ Q ; in particular, denoting by λ∗ the inverse of the group isomorphism NP (Q ) ∼ = λ NP (Q ) induced by λ , we have the F-morphism µ ◦ λ∗ : λ NP (Q ) −→ NP (V ) 4.12.10 which induces an F-isomorphism θ : U ∼ =V . But, arguing by induction on |P : Q|, we may assume that the set X of all the subgroups of P of smaller index fulﬁlls the hypothesis in Lemma 4.13 below; then, since λ NP (Q ) belongs to X , since µ ◦ λ∗ extends θ and since V is also fully I¯F (V )-normalized in F (cf. 2.10), it follows from this lemma applied to U , to I¯F (U ) and to ιP V ◦ θ that θ can be extended to some F-morphism ρ : IF (U ) → IF (V )†; then, the existence of U , V , λ, µ and the F-morphism IF (U ) → IF (V ) mapping u ∈ U on τ ρ σ(u) proves that ϕ ◦ ϕ belongs to F (Q , Q) . †

This argument has been scratched in the proof of [52, Theorem 3.8].

52

Frobenius categories versus Brauer blocks

Let R and R be subgroups of P respectively containing Q and Q , and assume that ψ ∈ F (R , R) fulﬁlls ψ(Q) = Q ; we claim that the F-isomorphism ϕ : Q ∼ = Q induced by ψ (cf. 2.4) belongs to F (Q , Q) . We may assume that Q and Q are not F-intersected groups, that we have |R : Q| = 1 and that R and R respectively normalize Q and Q . We argue by induction on |R : Q| and we already know that there are ζ ∈ F(P, R) and ζ ∈ F(P, R ) such that V = ζ(Q) and V = ζ (Q ) are both fully centralized and fully normalized in F (cf. Proposition 2.7). Once again, since R and R belong to X , it follows from Lemma 4.13 below, applied to Q , to Aut(Q) and to ιP Q ◦ ϕ , that there are F-morphisms ν : NP (Q) −→ NP (V )

and ν : NP (Q ) −→ NP (V )

4.12.11

fulﬁlling ν(Q) = V and ν (Q ) = V ; moreover, that it is clear the F-isomorphism ω : ν(R) ∼ = ν (R ) deﬁned by ω ν(v) = ν ψ(v) for any v ∈ R maps V onto V and, since V is fully centralized in F , it is also fully I¯F (V )-normalized (cf. 2.10); then, it follows from Lemma 4.13 below, applied to V , to I¯F (V ) and to the restriction of ιP ν (R ) ◦ ω to V , that there ∼ IF (V ) fulﬁlling η ν(u) = ν ψ(u) for is an F-isomorphism η : IF (V ) =

any u ∈ Q ; now, the existence of V , V , ν , ν and η proves that ϕ belongs to F (Q , Q) . In conclusion, if R and T are subgroups of P , ϕ is an element of F (R, Q) and ψ an element of F (T, R) , we claim that ψ ◦ ϕ belongs to F (T, Q) ; indeed, setting Q = ϕ(Q) and Q = ψ(Q ) , and denoting by ϕ∗ : Q ∼ = Q ∼ and ψ∗ : Q = Q the corresponding F-isomorphisms, it follows from our deﬁnition that ϕ∗ belongs to F (Q , Q) and, by the arguments above, we already know that ψ∗ and ψ∗ ◦ ϕ∗ respectively belong to F (Q , Q ) and to F (Q , Q) ; hence, ψ ◦ ϕ belongs to F (T, Q) (cf. deﬁnition 4.12.7). It remains to prove that F fulﬁlls condition 2.8.2; let K be a subgroup of Aut(Q) containing FQ (Q) and ϕ ∈ F (P, Q) such that Q = ϕ(Q) is fully K -normalized in F where we set K = ϕ K ; actually, since I¯F (Q ) is normal ˆ = I¯F (Q )· K ∩ F(Q ) it is easily checked that Q is in F(Q ) , setting K ˆ -normalized in F (cf. 2.10) and therefore we may assume that K also fully K ¯ also contains I(Q) . Firstly assume that Q is an F-intersected group; since FP (Q ) ∩ K is a Sylow p-subgroup of F(Q ) ∩ K (cf. Lemma 2.12 and the induction hypothesis), there is χ ∈ F(Q) ∩ K such that ϕ◦χ

FP (Q) ∩ K ⊂ FP (Q ) ∩ K

4.12.12;

then, according to condition 4.12.2, there is an F-morphism ζ : NPK (Q) → P extending ϕ ◦ χ ; in this case†, we are done. †

This case has been forgotten in the proof of [52, Theorem 3.8].

4. Nilcentralized, selfcentralizing and intersected objects

53

Otherwise, since the isomorphism ϕ∗ : Q ∼ = Q induced by ϕ belongs to F (Q , Q) , as in 4.12.4 we have subgroups U and U of P , F-isomorphic to Q and Q , which are both fully centralized and fully normalized in F , and admit F-morphisms λ

NP (Q) −→ NP (U ) , IF (U ) −→ IF (U ) , NP (U ) ←− NP (Q ) 4.12.13 fulﬁlling equalities 4.12.5 and λ ϕ(u) = σ λ(u) for any u ∈ Q ; set λ

σ

R = IF (U ) and R = IF (U )

4.12.14 the inverse of the F-isomorphism NP (Q ) ∼ = λ NP (Q )

and denote by λ∗ induced by λ . Since U is fully centralized in F and we have σ I¯F (U ) = I¯F (U ) (cf. deﬁnition 4.11.1), we still have σ(R) = R (cf. 2.6.2); moreover, since Q is fully K -normalized in F , we get (cf. 2.6.2) λ λ NPK (Q ) = NP K (U ) ⊃ CP (U )

4.12.15

and therefore we still get R = IF (U ) ⊂ λ NP λ∗ (U ) = λ NP (Q )

4.12.16;

hence, there is an F-morphism ψ : R → P fulﬁlling ψ(v) = λ∗ σ(v) for any v ∈ R . Finally, since R belongs to X , since U B R and FR (U ) ⊂ λ K , and since ψ(U ) = Q is fully ψ (λK)-normalized in F , it follows from Lemma 4.13 below applied to U , to λ K and to the restriction to U of ψ that there are λ an F-morphism ξ : NPK (U ) → P and an element χ ∈ K such that, for any u ∈ Q , we have ∗ ξ λ(u) = ψ λ χ(u) = λ σ λ χ(u) = ϕ χ(u) 4.12.17. We are done. Lemma 4.13. Let X be a nonempty set of subgroups Q of P such that any subgroup T of P fulﬁlling F(T, Q) = ∅ belongs to X , and that, for any subgroup K of Aut(Q) and any F-morphism ϕ : Q → P such that ϕ(Q) is fully ϕK-normalized in F , the following condition holds: 4.13.1. There are an F-morphism ζ : Q·NPK (Q) → P and an element χ ∈ K such that ζ extends ϕ ◦ χ . A subgroup R of P , a subgroup J of Aut(R) and an F-morphism ψ : R → P such that ψ(R) is fully ψJ-normalized in F fulﬁll condition 4.13.1 provided there are Q ∈ X having R as a normal subgroup and stabilizing J , and an F-morphism η : Q → P extending ψ .

54

Frobenius categories versus Brauer blocks

Proof: Let Q be an element of X such that R ⊂ Q ⊂ NP (R) , J a Q-stable subgroup of Aut(R) such that ψ(R) is fully ψJ-normalized in F and η : Q → P an F-morphism extending ψ ; we argue by induction on |NP (R) : Q| and may assume that NPJ (R) ⊂ Q or, equivalently, that Q = Q·NPJ (R) . Denote by K the converse image of J·FQ (R) in the stabilizer Aut(Q)R of R in Aut(Q) . Choose an F-morphism ζ : NPK (Q) → P in such a way that ζ(R) is fully J·FQ (R) -normalized and ζ(Q) is fully ζK-normalized in F . This is possible since, applying Proposition 2.7 to Q ⊂ NPK (Q) , there is an F-morphism ¯ = ζ (Q) is fully ζ K-normalized in F ; then, setζ : NPK (Q) → P such that Q ¯ = ζ K , and applying Proposition 2.7 to ting J¯ = ζ J·FQ (R) and K

ζ

¯ = ζ (R) ⊂ N K¯ (Q) ¯ ⊂ N J¯(R) ¯ R P P

4.13.2,

¯ ¯ ¯ is fully there is an F-morphism ζ : NPK (Q) → P such that ζ (R) malized in F ; since we have (cf. 2.6.2) ¯ ¯ = N ζ K¯ ζ (Q) ¯ ζ NPK (Q) P

¯ is also fully ζ (Q)

ζ ¯

J-nor-

4.13.3,

ζ

¯ K-normalized in F .

Set R = ψ(R) , Q = η(Q) and J = ψJ ; it follows from condition 4.13.1 applied to Q , to K = ηK and to the group homomorphism Q → P deter mined by η and ζ that there are an F-morphism ξ : NPK (Q ) → P and an element χ ∈ K fulﬁlling ξ η(u) = ζ χ(u) for any u ∈ Q ; in particular, we have ξ(R ) = ζ(R) since χ(R) = R , and we set R = ξ(R ) = ζ(R) and J = ζJ . On the other hand, we still have

Q = NP (Q) ∩ Q·NPJ (R) = NPK (Q) and Q = NPK (Q )

4.13.4

and therefore, since R is J ·ζFQ (R)-fully normalized in F , it follows from the induction hypothesis applied to R , to NPK (Q) and to the restriction of ζ , and further to R , to NPK (Q ) and to the restriction of ξ , that there are F-morphisms α : R·NPJ (R) → P and α : R ·NPJ (R ) → P , and elements θ and θ of J such that, for any v ∈ R , we have α(v) = ζ θ(v)

and α ψ(v) = ξ ψ θ (v) = ζ χ θ (v)

4.13.5.

Moreover, since we are assuming that R is fully J -normalized in F , we get (cf. 2.6.2) α R·NPJ (R) ⊂ R ·NPJ (R ) = α R ·NPJ (R )

4.13.6

4. Nilcentralized, selfcentralizing and intersected objects

55

and therefore, denoting by ω : R·NPJ (R) → R ·NPJ (R ) the F-morphism fulﬁlling α ◦ ω = α (cf. 2.3.1), for any v ∈ R we still get ∗ ζ θ(v) = α ω(v) = ζ χ (θ ◦ ψ ) ω(v)

4.13.7

∼ R determined where ψ ∗ ∈ F(R, R ) is the inverse of the isomorphism R = by ψ ; thus, since ζ is injective and the image of χ in Aut(R) is contained in J·FQ (R) , there are θ ∈ J and u ∈ Q such that from equality 4.13.7 we obtain (cf. 1.8) θ(v) = χ θ ◦ θ ◦ κR (u) ◦ ψ ∗ ω(v) 4.13.8. Finally, since η extends ψ , for any v ∈ R we still obtain (cf. 1.8) ψ (θ−1 ◦ χ θ−1 ◦ θ)(v) = κ

P,R ·N J (R ) P

η(u) ◦ ω (v)

and, since θ−1 ◦ χ θ−1 ◦ θ belongs to J and κ

P,R ·N J (R ) P

4.13.9

η(u) ◦ ω is an

F-morphism from R·NPJ (R) to P , equality 4.13.9 proves that R , J and ψ fulﬁll condition 4.13.1. We are done.

Chapter 5

Alperin fusions in Frobenius P-categories 5.1 Let P be a ﬁnite p-group and F a divisible P -category such that FP (P ) is a Sylow p-subgroup of F(P ) . In this chapter, we prove that the framework in [50, Appendix] can be translated to this abstract setting and that a suitable reformulation of the main statement in that Appendix is then equivalent to saying that F is a Frobenius P -category. As we explain in the Introduction (cf. I 3), the origin of the concepts and the results below goes back to [35], where we formulate the ﬁrst systematic treatment of the Alperin Fusion Theorem [1]. 5.2 It is well-known that the Alperin Fusion Theorem [1] can be applied to compute the cohomology groups of a ﬁnite group G from the cohomology groups of its p-subgroups (cf. §2 in [35, Ch. III]) — precisely, they can be computed from the corresponding contravariant functors from FG to the category of ﬁnitely generated Abelian groups Ab . But, as a matter of fact, when dealing with contravariant functors a from F to Ab (cf. 1.2), it is handy to consider the category ZF still deﬁned over the set of the subgroups of P where, for any pair of subgroups Q and R of P , the set of morphisms from R to Q is the free Z-module ZF(Q, R) over F(Q, R) , with the distributive composition extending the composition in F ; indeed, it is clear that a can be extended to a contravariant functor Za : ZF −→ Ab

5.2.1.

5.3 Then, for any two diﬀerent elements ϕ , ϕ ∈ F(Q, R) , we call F-dimorphism from R to Q the diﬀerence ϕ − ϕ in the Z-module ZF(Q, R) ; moreover, let us call Alperin F-fusions the F-dimorphisms with Q = P ; it is clear that the set of F-dimorphisms is stable by left and right composition with F-morphisms; note that, for any ϕ ∈ F(Q, R) , the family {ϕ − ϕ}ϕ ∈F (Q,R)−{ϕ} is a Z-basis of the kernel of the evident augmentation Z-linear map εQ,R : ZF(Q, R) −→ Z 5.3.1 sending any ϕ ∈ F(Q, R) to 1 . The next elementary lemma relates any decomposition of an F-dimorphism with the partially deﬁned linear combinations introduced in [35, Ch. III]. Moreover, note that equalities 5.4.2 below coincide with Alperin’s original formulation in the case where Q = P , ϕ is P the inclusion map ιP R : R → P , and for any i ∈ I , we have µi = ιQi , Qi = Ri and ϕi = idRi (cf. 2 in [28, Ch. 7]).

58

Frobenius categories versus Brauer blocks

Lemma 5.4 With the notation above, let {Qi }i∈I and {Ri }i∈I be ﬁnite families of subgroups of P and, for any i ∈ I , ϕi − ϕi an F-dimorphism from Ri to Qi and µi : Qi → Q and νi : R → Ri two F-morphisms. We have ϕ − ϕ = µi ◦ (ϕi − ϕi ) ◦ νi 5.4.1 i∈I

if and only if there are n ∈ N and an injective map σ : ∆n → I fulﬁlling ϕ = µσ(0) ◦ ϕσ(0) ◦ νσ(0) µσ(6−1) ◦ ϕσ(6−1) ◦ νσ(6−1) = µσ(6) ◦ ϕσ(6) ◦ νσ(6) for any 1 ≤ D ≤ n µσ(n) ◦

ϕσ(n)

5.4.2

◦ νσ(n) = ϕ .

Proof: It is clear that equalities 5.4.2 imply equality 5.4.1. Conversely, equality 5.4.1 is obviously equivalent to ϕ + µi ◦ ϕi ◦ νi = ϕ + µi ◦ ϕi ◦ νi 5.4.3 i∈I

i∈I

and therefore, since ϕ = ϕ , there are i , i ∈ I (the possibility i = i is not excluded!) and a bijection π : I −{i } ∼ = I −{i } such that µi ◦ ϕi ◦ νi = ϕ µi ◦ ϕi ◦ νi = µπ(i) ◦ ϕπ(i) ◦ νπ(i) for any i ∈ I − {i }

ϕ = µi ◦

ϕi

5.4.4;

◦ νi

then, we inductively deﬁne σ setting σ(0) = i and σ(D + 1) = π σ(D) for any D ∈ N such that σ(D) is already deﬁned and diﬀerent from i , and we denote by n ∈ N the maximal D where σ is deﬁned (so that σ(n) = i ). It remains to prove that σ is injective; arguing by contradiction, assume that there are 0 ≤ D < D ≤ n such that σ(D) = σ(D ) ; that is to say, denoting by π ˆ the permutation of I extending π , we have π ˆ 6 (i ) = π ˆ 6 (i ) and therefore we get i = π ˆ 6 −6 (i ) , a contradiction. 5.5 On the other hand, by the so-called Yoneda’s Lemma (cf. [30, §1]), the contravariant functor hF : F → Ab mapping any subgroup Q of P on ZF(P, Q) and any morphism α : R → Q in F on the group homomorphism hF (Q) → hF (R) deﬁned by the composition with α , is projective in the category of functors Fct(F ◦ , Ab) ; moreover, still denoting by Z : F → Ab the trivial contravariant functor mapping all the F-objects on Z and all the F-morphisms on idZ , the family of augmentation maps εP,Q , when Q runs over the set of subgroups of P , deﬁne a surjective natural map εF : hF −→ Z

5.5.1

5. Alperin fusions in Frobenius P -categories

59

and therefore the kernel wF = Ker(εF ) is the Heller translate of the trivial functor Z ; in particular, for any contravariant functor a : F → Ab , we have the exact sequence (cf. §4 and §8 of Ch. V and §1 of Ch. VI in [18]) 0 −→ Nat(Z, a) −→ Nat(hF , a) −→ Nat(wF , a) −→ H1 (F, a) −→ 0 5.5.2 since the projectivity of hF forces ExtiF (hF , a) = {0} for any i ≥ 1 and we have Ext1F (Z, a) = H1 (F, a) (cf. A3.6.3 and A3.9.2). 5.6 Now, if a : F → Ab is a contravariant functor, let us say that a family S = {SQ }Q of subsets SQ ⊂ a(Q) , where Q runs over the set of proper subgroups of P , is a generator family of a whenever, for any proper subgroup Q of P , we have a(Q) = Z· a(ϕ) (a) 5.6.1, R

ϕ∈F (R,Q) a∈SR

where R runs over the set of subgroups of P (such that |R| ≥ |Q| ). Thus, according to Lemma 5.4, in the Frobenius category FG associated with a ﬁnite group G , it is clear that the genuine purpose of the Alperin Fusion Theorem (cf. [1]) is to describe suitable generator families of the contravariant functor wFG : FG −→ Ab

5.6.2.

5.7 With the analogous purpose, in the divisible P -category F let us set rF (P ) = wF (P ) and, for any proper subgroup Q of P , deﬁne rF (Q) =

wF (R) ◦ ZF(R, Q)

5.7.1

R

where R runs over the set of subgroups of P such that |R| > |Q| . Note that, since there is ψ ∈ F(P, R) such that ψ(R) is fully normalized in F (cf. Proposition 2.7) and since of the isomorphism R ∼ = ψ(R) the inverse determined by ψ belongs to F R, ψ(R) (cf. 2.4), in deﬁnition 5.7.1 it suﬃces to restrict the sum to the subgroups R which are fully normalized in F ; moreover, if Q is a subgroup of P and θ ∈ F(Q, Q ) is an isomorphism then we clearly have rF (Q) ◦ θ = rF (Q ) 5.7.2. We say that Q is F-essential when rF (Q) = wF (Q) and call F-irreducible the elements in wF (Q) − rF (Q) . 5.8 Coherently, the elements of rF (Q) are called F-reducible; actually, any element of rF (Q) is a sum of a family of F-reducible Alperin F-fusions from Q . Considering the canonical map hF (Q) −→ hF (Q)/rF (Q) = hF (Q)

5.8.1,

60

Frobenius categories versus Brauer blocks

it is clear that F(Q) acts on the image F(P, Q) of F(P, Q) in hF (Q)/rF (Q) by composition on the left; we denote by F(Q)ϕ the stabilizer of the image ϕ of ϕ ∈ F(P, Q) and, according to equality 5.7.2, we clearly have F(Q )ϕ◦θ = F(Q)ϕ

θ

5.8.2

for any F-isomorphism θ : Q ∼ = Q . Note that the correspondence mapping Q on rF (Q) deﬁnes a subfunctor rF of wF . Moreover, if P is a subgroup of P and F a divisible P -subcategory of F such that FP (P ) is a Sylow p-subgroup of F (P ) then, for any proper subgroup Q of P , it is clear that ιP P ◦rF (Q ) is contained in rF (Q ) and therefore the inclusion P ⊂ P induces a Z-module homomorphism hF (Q ) −→ hF (Q )

5.8.3

sending F (P , Q ) to F(P, Q ) . Proposition 5.9 Let S = {SQ }Q be a generator family of wF , where Q runs over the set of proper subgroups of P . The family formed by the F-irreducible elements of SQ , where Q runs over the set of proper subgroups of P , is also a generator family of wF . Moreover, for any F-essential subgroup Q of P , there is ϕ ∈ F(P, Q) such that Sϕ(Q) contains an F-irreducible element of wF ϕ(Q) . Proof: Let ω ∈ wF (Q) be F-irreducible and, setting SP = wF (P ) , assume that we have ω= σ ◦ αR,σ 5.9.1 R σ∈SR

for suitable αR,σ ∈ ZF(R, Q) , where R runs over the set of subgroups of P ; then, necessarily there are a suitable subgroup R of P such that |R| = |Q| and an F-irreducible element σ ∈ SR such that 0 = αR,σ ; in particular, we have R = ϕ(Q) for some ϕ ∈ F(P, Q) . On the other hand, if τ ∈ SQ is an F-reducible element then either Q = P or we have τ= θ ◦ βR,θ 5.9.2 R θ∈wF (R)

for suitable βR,θ ∈ ZF(R, Q) where R runs over the set of subgroups of P such that |R| > |Q| ; in the second case, considering an S-decomposition of any θ ∈ wF (R) , we still have τ= σ ◦ γR,σ 5.9.3 R σ∈SR

for suitable γR,σ ∈ ZF(R, Q) where R runs over the set of subgroups of P such that |R| > |Q| ; so that the new family where we replace SQ by SQ − {τ } is a generator family of wF too. We are done.

5. Alperin fusions in Frobenius P -categories

61

5.10 Let Q be an F-essential subgroup of P ; it is clear that FQ (Q) acts trivially on F(P, Q) and, if F is a Frobenius P -category, we prove below ˜ that the action of F(Q) = F(Q)/FQ (Q) on F(P, Q) is actually transitive and that any nontrivial p-subgroup ﬁxes a unique element in F(P, Q) . Then, since we also prove that Q is F-selfcentralizing, it is actually an F-intersected subgroup (cf. 4.11). Theorem 5.11 Assume that F is a Frobenius P -category. A subgroup Q of P is F-essential if and only if it fulﬁlls the two conditions 5.11.1 Q is F-selfcentralizing. ˜ ˜ such that p divides |M ˜ | and does not 5.11.2 F(Q) has a proper subgroup M σ ˜ ˜ ˜ ˜ ˜ divide |M ∩ M | for any σ ˜ ∈ F(Q) − M . ˜ In this case, the groups F(Q) ϕ , when ϕ runs over F(P, Q) , coincide with the ˜ of F(Q) ˜ minimal proper subgroups M in condition 5.11.2, and they contain ˜ Sylow p-subgroups of F(Q) . Moreover, F(P, Q) is a Z-basis of hF (Q) and ˜ F(Q) acts transitively on this set. Proof: Let ϕ : Q → P be an F-morphism such that Q = ϕ(Q) is fully normalized in F (cf. Proposition 2.7); for another F-morphism ϕ : Q → P , set R = NP ϕ (Q) and consider the isomorphism ϕ (Q) ∼ = Q determined by ϕ and ϕ ; since Q is also fully centralized in F (cf. Proposition 2.11), it follows from statement 2.10.1 that there ρ : R → P are an F-morphism and an element σ ∈ F(Q) such that ρ ϕ (u) = ϕ σ(u) for any u ∈ Q ; consequently, denoting by ψ : Q → R the group homomorphism determined by ϕ , we get (cf. 1.9) ϕ − ϕ ◦ σ = (ιP R − ρ ) ◦ ψ

5.11.3.

In particular, assuming that Q is F-essential, ϕ and ϕ ◦ σ have the same image in hF (Q) , so that the Z-linear map ZF(Q) −→ hF (Q)

5.11.4

sending τ ∈ F(Q) to the class of ϕ◦τ is surjective; thus, F(Q) acts transitively on the image of F(P, Q) ; moreover, from the very deﬁnition of F(Q)ϕ , we get the factorization Z F(Q)/F(Q)ϕ −→ hF (Q)

5.11.5,

where the left term denotes the free Z-module over F(Q)/F(Q)ϕ . Furthermore, if we assume that an element σ ∈ F(Q) does not ﬁx ϕ then the Alperin F-fusion ϕ − ϕ ◦ σ is F-irreducible; but, setting U = Q ·CP (Q )

62

Frobenius categories versus Brauer blocks

and considering the element ϕσ of F(Q ) determined by σ , it follows from statement that there is ρ ∈ F(U ) such that, for any u ∈ Q , we have 2.10.1 ρ ϕ(u) = ϕ σ(u) ; hence, denoting by ψ : Q → U the group homomorphism determined by ϕ , we get ϕ − ϕ◦ σ = ιP 5.11.6 U ◦ (idU − ρ) ◦ ψ which forces the equality Q = ϕ(Q) = U ; since ρ CP ϕ (Q) is contained in CP (Q ) , it is clear that Q is F-selfcentralizing, fulﬁlling condition 5.11.1. Set R = NP (Q ) ; according to Proposition 2.11, FR (Q ) is a Sylow p-subgroup of F(Q ) , which does not coincide with FQ (Q ) since Q = P (cf. 5.7); moreover, if v ∈ R and ν is the image of v in F(Q) by the isomorphism determined by ϕ , it is easily checked that ϕ − ϕ ◦ ν = idP − κP (v) ◦ ϕ 5.11.7 where κP (v) is the image of v in F(P ) (cf. 1.8), so that ν belongs to F(Q)ϕ ; that is to say, ϕF(Q)ϕ contains FR (Q ) and, in particular, F(Q)ϕ contains ˜ FQ (Q) and p divides |F(Q) ϕ| . Now, consider the intersection F = F(Q)ϕ ∩ F(Q)ϕ

5.11.8

and, assuming that p divides |F/FQ (Q)| , choose a p-subgroup V of F strictly σ containing FQ (Q) ; thus, since F(Q)ϕ = F(Q)ϕ and FR (Q ) is a Sy low p-subgroup of ϕ F(Q)ϕ , there are τ ∈ F(Q)ϕ and τ ∈ F(Q)ϕ such that ϕ◦τ V ⊂ FR (Q ) ⊃ ϕ◦σ◦τ V 5.11.9 and therefore, since we already have CP (Q ) = Z(Q ) , it follows from stateϕ◦τ ment 2.10.1 that, denoting by T the converse image of −1 V in P , there is an F-morphism ζ : T → R fulﬁlling ζ ϕ(u) = ϕ (σ◦τ ◦τ )(u) for any u ∈ Q . In conclusion, denoting by ξ : T → P the inclusion map ιP T , by ξ : T → P the composition of ζ with the corresponding inclusion map, and by η the F-morphism from Q to T determined by ϕ◦τ and by the inclusion ϕ(Q) ⊂ T , we have ξ ◦ η = ϕ ◦ τ and ξ ◦ η = ϕ ◦ σ ◦ τ 5.11.10

and therefore the Alperin F-fusions ϕ−ξ◦η

,

(ξ − ξ ) ◦ η

and ξ ◦ η − ϕ ◦ σ

5.11.11

are F-reducible, so that we get F(Q)ϕ = F(Q)ϕ◦σ = F(Q)ϕ

5.11.12.

5. Alperin fusions in Frobenius P -categories

63

Conversely, assume that Q fulﬁlls conditions 5.11.1 and 5.11.2, and de˜ a proper subgroup of F(Q) ˜ note by M as in condition 5.11.2 and by M its converse image in F(Q) ; in particular, for any p-subgroup V of M strictly containing FQ (Q) , M contains NF (Q) (V ) ; hence, a Sylow p-subgroup of M is also a Sylow p-subgroup of F(Q) and we have NF (Q) (M ) = M . Con sequently, with the notation above, there is σ ∈ F(Q) such that ϕ ◦σ M contains the image of R in F ϕ (Q) ; but, since CP ϕ (Q) = ϕ Z(Q) , this image strictly contains Fϕ (Q) ϕ (Q) and therefore such elements σ determine a unique class in F(Q)/M . Consider the map F(P, Q) → F(Q)/M sending ϕ to the class of σ ; we claim that the corresponding Z-linear map hF (Q) −→ Z F(Q)/M

5.11.13

annihilates rF (Q) . According to deﬁnition 5.8.1, it suﬃces to prove that, for any subgroup T of P such that |T | > |Q| , any ξ , ξ ∈ F(P, T ) and any η ∈ F(T, Q) , this map annihilates (ξ − ξ) ◦ η ; moreover, it is clear that we may assume that η(Q) is normal in T . Since M contains a Sylow p-subgroup ξ◦η◦τ of F(Q) , for suitable τ , τ ∈ F(Q) M and ξ ◦η◦τ M contain the groups the respective images of ξ(T ) in F (ξ ◦ η)(Q) and of ξ (T ) in F (ξ ◦ η)(Q) ; consequently, the image of T in F η(Q) is contained in the intersection η◦τ M ∩ η◦τ M = η (τM ∩ τ M ) and contains Fη(Q) η(Q) strictly by condi tion 5.11.1, so that we have τM = τ M which forces τ and τ to be in the same class since NF (Q) (M ) = M . In conclusion, Q is F-essential and, from the Z-linear maps 5.11.5 and 5.11.13, we get the composed Z-linear map Z F(Q)/F(Q)ϕ −→ hF (Q) −→ Z F(Q)/M

5.11.14

sending σF(Q)ϕ to σM ; in particular, this proves that F(Q)ϕ is contained in M and, applying it to the choice M = F(Q)ϕ , that the Z-linear map 5.11.5 is injective too. We are done. 5.12 If F is a Frobenius P -category, the F-essential subgroups Q of P behave as the “sous-groupes C-essentiels” in [35, Ch. II and Ch. III], namely we have ˜ of F(Q) ˜ ˜ 5.12.1 A proper subgroup M contains F(Q) ϕ for some ϕ ∈ F(P, Q) ˜ ˜ ˜ σ˜ | for σ ˜ ˜. if and only if p divides |M | and does not divide |M ∩ M ˜ ∈ F(Q) −M In particular, if Q is fully normalized in F then we have ˜ ˜ |R ˜ is a nontrivial subgroup of F˜P (Q) . F(Q) = NF˜ (Q) (R) ιP Q

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Frobenius categories versus Brauer blocks

˜ contains F(Q) ˜ ˜ ˜ Indeed, if M ϕ then, since the index |M : F(Q)ϕ | is prime ˜ to p and any nontrivial p-subgroup of F(Q) ﬁxes a unique element in F(P, Q) , ˜ is not contained in M ˜ σ˜ for any it is clear that any nontrivial p-subgroup of M ˜ ˜ ; the converse follows from Theorem 5.11. Moreover, it is quite σ ˜ ∈ F(Q)− M ˜ ﬁxes the unique element of F(P, Q) ﬁxed by R ˜ for any clear that NF˜ (Q) (R) ˜ of F˜P (Q) ; conversely, if Q is fully normalized in F , nontrivial subgroup R FP (Q) is a Sylow p-subgroup of F(Q) (cf. Proposition 2.11) and then it is not diﬃcult to check that the right-hand member of the equality above fulﬁlls the above condition. Similarly, let us translate Proposition 6 in [35, Ch. II] to our present context. Corollary 5.13 Assume that F is a Frobenius P -category and let Q be an ˜ F-essential subgroup of P . Then, the set of normal subgroups X of F(Q) such that p divides |X| has a unique minimal element XF˜ (Q) . In particular, the quotient XF˜ (Q)/Op XF˜ (Q) is simple and, denoting by XF (Q) the converse image of XF˜ (Q) in F(Q) , we have F(Q) = Op XF (Q) ·F(Q)ιP 5.13.1. Q

Proof: Arguing by contradiction, we may assume that there are two normal ˜ subgroups X and X of F(Q) such that p divides |X| and |X | but does not ˜ is a Sylow p-subgroup of F(Q) ˜ divide |X ∩ X | ; in particular, if R , the ιP Q

˜ and T˜ = X ∩ R ˜ are nontrivial and we have intersections T˜ = X ∩ R ˜ = {1} [T˜, T˜ ] ⊂ X ∩ X ∩ R

5.13.2,

so that T˜·T˜ contains a noncyclic subgroup of order p2 ; in this case, it is well-known that (cf. Theorem 3.16 in [28, Ch. 5]) X ∩ X = $CX∩X (˜ τ ) | τ˜ ∈ T˜·T˜ − {1}%

5.13.3.

˜ ⊂ F(Q) ˜ ˜ But, since T˜·T˜ ⊂ R and F(Q) fulﬁlls condition 5.11.2, this ιP ιP Q

Q

˜ stabilizer contains CF˜ (Q) (˜ τ ) for any τ˜ ∈ T˜·T˜ − {1} . Hence, F(Q) ρQ (ϕ) con tains X ∩ X . ˜ Moreover, the images of X and X in the quotient F(Q)/(X∩X ) central˜ ize each other and, in particular, the image of T centralizes the image of X , so that X normalizes (X ∩ X )·T˜ ; hence, since T˜ is a Sylow p-subgroup of this product, the Frattini argument shows that X = (X ∩ X ).NX (T˜) ; but, ˜ ˜ as above, F(Q) contains NF˜ (Q) (T˜) ; consequently, F(Q) contains X , a ιP ιP Q

Q

contradiction. Finally, equality 5.13.1 still follows from the Frattini argument. We are done.

5. Alperin fusions in Frobenius P -categories

65

Corollary 5.14 Assume that F is a Frobenius P -category. Let E be an F(P )-stable set of F-essential subgroups of P containing at least a representative for each F-isomorphism class. For any subgroup Q of P and any ϕ ∈ F(P, Q) , there are σ ∈ F(P ) , a ﬁnite family {Qi }i∈I of elements of E and, for any i ∈ I , a p -element σi ∈ XF (Qi ) not ﬁxing ιP Qi and an F-morphism νi : Q → Qi fulﬁlling ιP 5.14.1. ϕ = σ ◦ ιP Q+ Qi ◦ (σi − idQi ) ◦ νi i∈I

Proof: Setting E = E ∪ {P } and XF (P ) = F(P ) , we ﬁrstly prove that, for any ψ , ψ ∈ F(P, Q) , there are a ﬁnite family {Qj }j∈J of elements of E and, for any j ∈ J , a p -element ηj ∈ XF (Qj ) not ﬁxing ιP Qj and an F-morphism µj : Q → Qj fulﬁlling ψ − ψ = ιP 5.14.2. Qj ◦ (ηj − idQj ) ◦ µj j∈J

It is clear that, arguing by induction on |P : Q| , we may assume that Q = P and that ψ − ψ is F-irreducible; but, in this case, Q is F-essential and therefore we have an F-isomorphism θ : Q ∼ = Q for some Q ∈ E ; then, it follows from Theorem 5.11 and equality 5.13.1 that there are elements τ , τ ∈ Op XF (Q ) such that the Alperin F-fusions ρ = ψ ◦ θ−1 − ιP Q ◦ τ

and

ρ = ψ ◦ θ−1 − ιP Q ◦ τ

5.14.3

are F-reducible; thus, setting δτ = ιP Q ◦ (τ − idQ ) ◦ θ

5.14.4,

−1 δτ = ιP − idQ ) ◦ (τ ◦ θ) Q ◦ (τ

we already get the following decomposition ψ − ψ = (ρ − ρ ) ◦ θ + δτ + δτ 5.14.5; moreover, for any decomposition η = η ◦ η in Op XF (Q ) , we still have η − idQ = (η − idQ ) ◦ η + (η − idQ )

5.14.6;

hence, since τ and τ can be decomposed as suitable products of p -elements of XF (Q ) , it suﬃces to apply again the induction hypothesis. Now, we set ψ = ϕ and ψ = ιP Q and argue by induction on |J| ; we may assume that J = ∅ and, according to Lemma 5.4, there is j ∈ J such that ϕ = ιP Qj ◦ ηj ◦ µj and then, according to the induction hypothesis, the

F-morphism ϕ = ιP Qj ◦µj admits the announced decomposition 5.14.1. Thus, if Qj belongs to E then the equality ϕ = ϕ + ιP Qj ◦ (ηj − idQj ) ◦ µj

5.14.7

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Frobenius categories versus Brauer blocks

gives the announced decomposition for ϕ . If Qj = P then ιP Qj = idP , ηj belongs to F(P ) and we have ϕ = ηj ◦ µj and ϕ = µj ; in this case, it is easy to check that the equality ϕ = ηj ◦ ϕ still gives the announced decomposition for ϕ . We are done. 5.15 With the hypothesis and the notation of the corollary above, it is handy to introduce the E-length of ϕ for inductive purposes: it is the smallest integer DE (ϕ) such that we have a decomposition 5.14.1 with |I| = DE (ϕ) . Note that if DE (ϕ) ≥ 1 then there are R ∈ E , η ∈ F(R, Q) and a p -element τ ∈ XF (R) not ﬁxing ιP R such that we have ϕ = ιP R ◦τ ◦η

and DE (ιP R ◦ η) = DE (ϕ) − 1

5.15.1.

When E is the set of all the F-essential subgroups of P fully normalized in F , we simply write D(ϕ) and call it the length of ϕ . 5.16 To what extent does the behaviour of the F-essential subgroups of P characterize the Frobenius P -categories? In order to give some answer to this question, let us consider the normal F-chains (cf. 2.20); recall that, for any n ∈ N , an F-chain — more precisely called (n, F)-chain — is a functor q : ∆n −→ F

5.16.1

from the category ∆n formed by the objects 0 ≤ i ≤ n and the morphisms 0 ≤ j ≤ i ≤ n , with the obvious composition (cf. A2.2), and we say that q is normal if the image of q(i•n) is normal in q(n) for any i ∈ ∆n . Let us denote by F(q) the group of ch(F)-automorphisms of q — identiﬁed with the stabilizer in F q(n) of the image of q(i • n) for any i ∈ ∆n (cf. 2.19) — and, for any subgroup Q of P containing q(n) , we set FQ (q) = F(q) ∩ FQ q(n)

˜ and F(q) = F(q)/Fq(n) (q)

5.16.2.

5.17 Let q : ∆n → F be a normal F-chain; let us say that a normal F-chain r : ∆m → F extends q if n < m , q(n) ∼ r(m) , and there is a = ch∗ (F)-morphism (cf. A2.8) (ν, δ) : (q, ∆n ) −→ (r, ∆m )

5.17.1

such that δ(i) = i for any i ∈ ∆n ; in this case, it is clear that the F-morphism r(n • m) induces a group homomorphism ρrq : F(r) −→ F(q)

5.17.2;

note that, if F is a Frobenius P -category, there exists a normal F-chain r ex˜ tending q if and only if either q(n) is not F-selfcentralizing or p divides |F(q)|

5. Alperin fusions in Frobenius P -categories

67

(cf. Propositions 2.7 and 2.11). In general, we say that q is F-maximal if it cannot be extended; thus, if n ≥ 1 and q is fully conormalized in F (cf. 2.20), q is F-maximal if and only if we have q(n) = NP (q ◦ δnn−1 ) where as usual δnn−1 : ∆n−1 → ∆n is the injective order-preserving map which does not cover n in ∆n (cf. A3.1). 5.18 Similarly, we say that a normal F-chain is F-essential if it is ch(F)-isomorphic to a normal F-chain q : ∆n → F fully conormalized in F (cf. 2.20) such that q(n) is NF (q ◦ δnn−1 )-essential where for n = 0 we set NF (q ◦ δ0−1 ) = F . In particular, since we have NF (q ◦ δnn−1 ) q(n) = F(q) 5.18.1, if F is a Frobenius P -category then it follows from Proposition 2.16 and Theorem 5.11 that q is F-essential if and only if we have CP q(n) ⊂ q(n) ˜ ˜ such that p divides |M ˜ | and does not and F(q) has a proper subgroup M σ ˜ ˜ ∩M ˜ | for any σ ˜ ˜ ; otherwise, as in 5.12 above, it is divide |M ˜ ∈ F(q) −M easily checked that we have F(q) = NF (q) FR (q) | R 5.18.2 where R runs over the set of subgroups of FP (q) strictly containing Fq(n) (q) . Proposition 5.19 Assume that F is a Frobenius P -category. With the notation above, for any normal F-chain q : ∆n → F neither F-maximal nor F-essential, we have F(q) = ρrq F(r) | r 5.19.1 where r runs over the set of normal F-chains which extend q and are either F-maximal or F-essential. Proof: We may assume that q is conormalized in F (cf. 2.20) and argue by induction on |P : q(n)| and on n ; thus, up to replacing F by NF q ◦ δnn−1 if n ≥ 1 , we may assume that n = 0 where Q = q(0) is neither equal to P nor F-essential; thus, since NF (Q) is a Frobenius NP (Q)-category (cf. Proposition 2.16), it follows from Theorem 5.11 that Q is also not NF (Q)-essential. Then, according to Corollary 5.14, for any σ ∈ F(Q) there are a ﬁnite family {Ri }i∈I of subgroups of NP (Q) which are fully normalized in NF (Q) and contain Q strictly, and two families of elements τi ∈ NF (Q) (Ri ) and νi ∈ NF (Q) (Ri , Q) , where i runs over I , such that N (Q) N (Q) N (Q) ιQP ◦ σ − ιQ P = ιRiP ◦ (τi − idRi ) ◦ νi 5.19.2. i∈I

But, for any i ∈ I , denoting by ri : ∆1 → F the normal F-chain obviously induced by the inclusion Q B Ri , it is clear that NF (Q) (Ri ) = F(ri ) ; moreover, assuming that our choice makes |I| minimal, it follows from Lemma 5.4

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Frobenius categories versus Brauer blocks

and equality 5.19.2 that, for a suitable total order in I , σ coincides with the composition of the family of group homomorphisms {ρrQi (τi )}i∈I , where we denote by Q the corresponding (0, F)-chain. Hence, we have obtained F(Q) = ρrQ F(r) | r

5.19.3

where r : ∆1 → F runs over the set of normal (1, F) -chains fulﬁlling r(0) = Q and r(1) ∼ Q . Finally, by our induction hypothesis, all these normal F-chains = which are neither F-maximal nor F-essential fulﬁll equality 5.19.1. We are done. 5.20 Let Q be a subgroup of P fully normalized and fully centralized in F ; for any normal NF (Q)-chain q : ∆n → NF (Q) such that q(0) contains Q·CP (Q) , denote by Fq (Q) the corresponding normal FF (Q) -chain mapping i ∈ ∆n on Fq(i) (Q) ; as a matter of fact, if Fq (Q) is neither FF (Q) -maximal nor FF (Q) -essential, Proposition 5.19 still may supply a generator set for the group NF (Q) Fq (Q) . 5.21 Indeed, considering the semidirect product L = Q F(Q) and denoting by Q Fq (Q) the corresponding normal FL -chain mapping i ∈ ∆n on Q Fq(i) (Q) , we have an evident surjective group homomorphism FL Q Fq (Q) −→ NF (Q) Fq (Q)

5.21.1.

But, since FL is a Frobenius Q FP (Q)-category (cf. 3.2 and Theorem 3.7), it follows from Proposition 5.19 that L FL Q Fq (Q) = ρrQFq (Q) FL (rL ) | rL

5.21.2

where rL runs over the set of normal FL -chains which extend Q Fq (Q) and are either FL -maximal or FL -essential; moreover, it is quite clear that rL = Q Fr (Q) where r is a normal NF (Q)-chain extending q and that we have (cf. 5.16.2) ¯F (Q) Fq (Q) F˜L Q Fq (Q) ∼ = NF (Q) Fq (Q) Fq(n) (Q) = N

5.21.3.

Consequently, the image in F(Q) of equality 5.21.2 yields NF (Q) Fq (Q) = NF (Q) Fr (Q) | r

5.21.4

where r runs over the set of normal NF (Q)-chains extending q such that ¯ such that p ¯F (Q) Fr (Q) either is a p -group or has a proper subgroup M N σ ¯ ¯ | and does not divide |M ¯ ∩M ¯ | for any σ ¯F (Q) Fq (Q) − M ¯. divides |M ¯∈N We are ready to state our answer to the question above.

5. Alperin fusions in Frobenius P -categories

69

Theorem 5.22 F is a Frobenius P -category if and only if it fulﬁlls the following two conditions 5.22.1 For any F-essential subgroup Q of P , F(Q) is transitive on F(P, Q) . 5.22.2 For any subgroup Q of P fully normalized and fully centralized in F , and any normalNF (Q)-chain q : ∆n → NF (Q) such that Q·CP (Q) ⊂ q(0) ¯ ¯ and that NF (Q) Fq (Q) either is a p -group or has a proper subgroup M σ ¯ ¯ ¯ ¯ such that p divides ¯ |M | and does not divide |M ∩ M | for any element σ ¯F (Q) Fq (Q) − M ¯ , the restriction induces a surjective group homomorin N phism

NF (Q) (q) −→ NF (Q) Fq (Q)

5.22.3.

Proof: If F is a Frobenius P -category then, it follows from Theorem 5.11 that F(Q) acts transitively on F(P, Q) , and from Corollary 2.13 and Proposition 2.16 that the restriction induces a surjective group homomorphism NF (Q) q(n) −→ NF (Q) Fq(n) (Q) 5.22.4; moreover, since Q·CP (Q) ⊂ q(0) , any element of the end term which normalizes Fq(i) (Q) comes from an element of the origin which stabilizes q(i) for any i ∈ ∆n . Conversely, we will prove that the two conditions above imply that F fulﬁlls both conditions in Corollary 2.13. Let Q and Q be F-isomorphic proper subgroups of P both fully normalized and fully centralized in F , and ϕ : Q → P an F-morphism such that ϕ(Q) = Q ; then, according to condition 5.22.1, we actually have ϕ = ιP Q ◦ σ for some σ ∈ F(Q) ; that is to say, the diﬀerence ϕ − ιP ◦ σ is F-reducible and therefore we have (cf. 5.8) Q ϕ − ιP Q◦σ =

θ ◦ αR,θ

5.22.5

R θ∈wF (R)

for suitable αR,θ ∈ ZF(R, Q) , where R runs over the set of subgroups of P such that |R| > |Q| . Consequently, it follows from 5.3 that we still have ϕ − ιP (ψj − ψj ) ◦ µj 5.22.6 Q◦σ = j∈J

where J is a ﬁnite set and, for any j ∈ J , we have ψj , ψj ∈ F(P, Rj ) and µj ∈ F(Rj , Q) for a suitable subgroup Rj of P such that |Rj | > |Q| ; more precisely, applying again condition 5.22.1 and arguing by induction on |P : Q| , we actually get ϕ − ιP ιP 5.22.7 Q◦σ = Ui ◦ (τi − idUi ) ◦ νi i∈I

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Frobenius categories versus Brauer blocks

where I is a ﬁnite set and, for any i ∈ I , τi is an element of F(Ui ) and νi an element of F(Ui , Q) for a suitable subgroup Ui of P such that |Ui | > |Q| . Then, it follows from Lemma 5.4 that, for a suitable D , we can identify ∆6 with a subset of I in such a way that, setting Q−1 = Q and Qi = τi (Qi−1 ) for any i ∈ ∆6 , we have Q6 = Q and, denoting by ϕi : Qi−1 ∼ = Qi the F-isomorphism induced by τi , the composition of all these isomorphisms coincides with the isomorphism Q ∼ = Q induced by ϕ ◦ σ −1 . Moreover, note that Ui contains Qi and Qi−1 for any i ∈ ∆6 . For any i ∈ ∆6 , choose ηi ∈ F P, NP (Qi ) such that Qi = ηi (Qi ) is fully normalized in F (cf. Proposition 2.7) and, setting Q−1 = Q and η−1 = ιP , denote by ϕi : Qi−1 ∼ = Qi the F-morphism mapping ηi−1 (u) NP (Q) on ηi ϕi (u) for any u ∈ Qi−1 ; we still may assume that Q6 = Q

and η6 = ιP NP (Q )

5.22.8.

Arguing by induction on |P : Q| , for any i ∈ ∆6 we claim that we can apply Lemma 4.13 to Qi−1 , to Aut(Qi−1 ) and to ϕi ; indeed, from the induction hypothesis we may assume that all the subgroups of P of smaller index fulﬁll condition 4.13.1; moreover, it is clear that Qi−1 is a proper normal subgroup of ηi−1 NUi (Qi−1 ) and this group clearly stabilizes Aut(Qi−1 ) ; ﬁnally, the F-morphism ηi−1 NUi (Qi−1 ) −→ ηi NUi (Qi ) 5.22.9 mapping ηi−1 (v) on ηi τi (v) for any v ∈ NUi (Qi−1 ) clearly extends ϕi . Hence, since Qi is fully normalized in F , it follows from this lemma that thereis an F-morphism ζi−1 : NP (Qi−1 ) −→ P

5.22.10

extending χi ◦ ϕi for some χi ∈ F(Qi ) ; moreover, since Qi−1 is fully normalized in F , we actually get ζi−1 5.22.11, NP (Qi−1 ) = NP (Qi ) so that ζi−1 induces an F-isomorphism ξi : NP (Qi−1 ) ∼ = NP (Qi ) . Finally, the composition of all these F-isomorphisms when i runs over ∆6 yields an F-isomorphism NP (Q) ∼ = NP (Q ) , proving condition 2.13.1. Let Q be a subgroup of P fully normalized and fully centralized in F and R a subgroup of NP (Q) containing Q·CP (Q) ; in order to prove condition 2.13.2, we may assume that Q = R ; then, it follows from equality 5.21.4 that NF (Q) FR (Q) = NF (Q) Fr (Q) | r 5.22.12

where r runs over the set of normal NF (Q)-chains such that r(0) = R and that ¯F (Q) Fr (Q) either is a p -group or has a proper subgroup M ¯ such that p N

5. Alperin fusions in Frobenius P -categories

71

¯ | and does not divide |M ¯ ∩M ¯ σ¯ | for any σ ¯F (Q) Fr (Q) − M ¯. divides |M ¯∈N But, according to condition 5.22.2, for such an r the restriction induces a surjective group homomorphism

NF (Q) (r) −→ NF (Q) Fr (Q)

5.22.13;

moreover, it is clear that the restriction still induces the following commutative diagram NF (Q) (r) −→ NF (Q) Fr (Q) 5.22.14. ↓ ∩ F(R)Q −→ NF (Q) FR (Q) Finally, equality 5.22.12 and the surjectivity of the top arrow in all these diagrams force the surjectivity of the bottom arrow. We are done.

Chapter 6

Exterior quotient of a Frobenius P-category over the selfcentralizing objects 6.1 Let P be a ﬁnite p-group and F a Frobenius P -category. Denote sc by F the full subcategory of F over the set of all the F-selfcentralizing subsc sc groups of P (cf. 4.8) and consider the exterior quotient F˜ † of F (cf. 1.3); that is to say, for any pair of F-selfcentralizing subgroups Q and R of P , ˜ F(Q, R) is the the set of Q-conjugacy classes in F(Q, R) . Although Corolsc lary 5.14 supplies a suitable decomposition for any morphism in F˜ , Proposition 6.7 below leads to a more precise description of the structure of this category inside its additive cover (cf. A2.7) sc ◦ sc ac(F˜ ) = pr (F˜ )◦ 6.1.1. ˜ sc 6.2 Recall that

the ac(F )-objects are the ﬁnite sequences {Qi }i∈I — denoted by Q = i∈I Qi — of F-selfcentralizing subgroups Qi of P , and

sc an ac(F˜ )-morphism from another object R = Rj to Q = Qi j∈J

i∈I

is a pair (˜ α, f ) formed by a map f : J → I and a family α ˜ = {˜ αj }j∈J of sc F˜ -morphisms α ˜ j : Rj → Qf (j) . The composition of (˜ α, f ) with another sc ac(F˜ )-morphism ˜ g) : T = (β, T6 −→ R 6.2.1 6∈L

˜ f ◦ g (cf. A2.6.3) where (˜ where β˜ = {β˜6 }6∈L , is the pair (˜ α ∗ g) ◦ β, α ∗ g) ◦ β˜ is the family {˜ αg(6) ◦ β˜6 }6∈L of composed morphisms α ˜ g(6) ◦ β˜6 : T6 −→ Rg(6) −→ Q(f ◦g)(6)

6.2.2.

6.3 Actually, with no extra eﬀort, we not only will describe the special structure of this category but, for any ﬁnite group K , the structure of sc the category of K-objects of ac(F˜ ) . Precisely, for any category C , let us call K-object of C any pair (ρ, C) formed by an object C in C and a group homomorphism ρ : K → C(C) ; then, the category of K-objects of C is the category — denoted by KC — where the objects are the K-objects of C and where the morphisms between two K-objects (ρ, C) and (ρ , C ) of C are the C-morphisms f : C → C such that ρ (x) ◦ f = f ◦ ρ(x) for any x ∈ K . In other words, 6.3.1 KC is the category of functors to C from the category with a unique object having K as the monoid of endomorphisms. †

˜ (cf. 1.3). Called the centric orbit category in [13], whereas their orbit category is our F

74

Frobenius categories versus Brauer blocks

Note that, for any z ∈ Z(K) and any K-object (ρ, C) of C , ρ(z) is a K C-automorphism of (ρ, C) and therefore the correspondence sending (ρ, C) to ρ Z(K) deﬁnes an inner automorphism structure in KC , so that KC adC (cf. 1.3). mits an exterior quotient — denoted by K 6.4 It follows from Corollary 4.9 that, for any triple of F-selfcentralizing ˜ subgroups Q , R and T of P , any F-morphism α ˜ : Q → R induces an injective ˜ ˜ ˜ map from F(T, R) to F(T, Q) and we will consider the elements of F(T, Q) which, even partially, cannot be extended via α ˜ ; precisely, we set ˜ ˜ ˜ F(T, Q)α˜ = F(T, Q) − F(T, 6.4.1, Q ) ◦ θ˜ θ˜

˜ where θ˜ runs over the set of F-nonisomorphisms θ˜ : Q → Q from Q — the set of nonﬁnal (F˜ ◦ )Q -objects (cf. 1.7) — fulﬁlling α ˜ ◦ θ˜ = α ˜ for some ˜ α ˜ ∈ F(R, Q ) which is then unique, and we simply say that θ˜ divides α ˜ ˜ ˜ ˜ setting α ˜ =α ˜ /θ ; thus, if α ˜ is an isomorphism, we have F(T, Q)α˜ = F(T, Q) . Note that the existence of α ˜ is equivalent to the existence of a subgroup of R which is F-isomorphic to Q and contains α(Q) for a representative α ∈ α ˜. ˜ 6.5 Actually, an element β˜ ∈ F(T, Q) which can be extended to Q via θ˜ , a fortiori it can be extended to NQ θ (Q) for a representative θ ∈ θ˜ ; ˜ hence, it follows from condition 2.10.1 that β˜ belongs to F(T, Q)α˜ if and only ˜ if, for some representative β ∈ β , we have ∗ α∗ FR α(Q) ∩ β FT β(Q) = FQ (Q) 6.5.1 where α∗ : α(Q) ∼ = Q and β ∗ : β(Q) ∼ = Q denote the inverse of the isomorphisms respectively induced by α and β (cf. 2.4) — which is a symmetric condition. That is to say, with the same notation we have ˜ ˜ 6.5.2 β˜ ∈ F(T, Q)α˜ is equivalent to α ˜ ∈ F(R, Q) ˜ . β

˜ ˜ 6.6 Note that, if F(P, Q)α˜ = F(P, Q) then, since ˜ιP ˜ obviously beR ◦α ˜ longs to F(P, Q) , α ˜ belongs to F(R, Q)˜ιPR ◦α˜ which forces α ˜ to be an isomorphism; conversely, by the very deﬁnition of the F-intersected subgroups of P in 4.11, it follows from equality 6.5.1 that 6.6.1 If F(P, Q)˜ιPQ = ∅ then Q is an F-intersected subgroup of P . Moreover, the quotient

α∗ ¯R α(Q) ∼ N = F˜R α(Q)

6.6.2

˜ clearly acts on F(T, Q)α˜ by composition on the right and whenever we have ∗ ˜ ˜ β◦α ˜ ◦κ ˜ α(Q) (v) ◦ α ˜ = β˜ (cf. 1.8) for some β˜ ∈ F(T, Q)α˜ and some v ∈ R , we still have α∗ ◦ κα(Q) (v) ◦ α = β ∗ ◦ κβ(Q) (w) ◦ β 6.6.3

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75

for a suitable w ∈ T , so that from equality 6.5.1 we get ¯R α(Q) acts freely on F(T, ˜ 6.6.4 N ˜ is not an Q)α˜ . In particular, if α ˜ ˜ F-isomorphism then p divides |F(T, Q)α˜ | . Proposition 6.7 For any triple of F-selfcentralizing subgroups Q , R and T ˜ of P and any α ˜ ∈ F(R, Q) , we have ˜ F(T, Q) =

˜ ˜ F(T, Q )α/ ˜ θ˜ ◦ θ

6.7.1

θ˜

where θ˜ : Q → Q runs over a set of representatives for the isomorphism classes of (F˜ ◦ )Q -objects dividing α ˜ . In particular, we have ˜ ˜ )| (mod p) |F(T, Q)| ≡ |F(T

6.7.2.

Proof: It is quite clear that arguing by induction on |T |/|Q| we get ˜ F(T, Q) =

˜ ˜ F(T, Q )α/ ˜ θ˜ ◦ θ

6.7.3

θ˜

˜ where θ˜ runs over the set of F-morphisms θ˜ : Q → Q from Q dividing α ˜; ˜ hence, it suﬃces to prove that, whenever for another such an F-morphism θ˜ : Q → Q we have

˜ ˜ = ∅ ˜ ˜ F(T, Q )α/ ˜ θ˜ ◦ θ ∩ F(T, Q )α/ ˜ θ˜ ◦ θ

6.7.4,

˜ there is an F-isomorphism η˜ : Q ∼ = Q fulﬁlling η˜ ◦ θ˜ = θ˜ and, in particular, we have ˜ ˜ ˜ ˜ F(T, Q )α/ 6.7.5. ˜ θ˜ ◦ θ = F(T, Q )α/ ˜ θ˜ ◦ θ We argue by induction on |R|/|Q| and may assume that |R| = |Q| , ˆ that Q and Q are subgroups of R containing Q = α(Q) for some α ∈ α ˜, and that the respective homomorphisms θ : Q → Q and θ : Q → Q determined by α are representatives of θ˜ and θ˜ , so that α ˜ /θ˜ = ˜ιR Q

and α/ ˜ θ˜ = ˜ιR Q

6.7.6;

˜ ˜ ˜ ˜ ˜ ˜ ˜ then, if β˜ ∈ F(T, Q )α/ ˜ θ˜ and β ∈ F(T, Q )α/ ˜ θ˜ fulﬁll β ◦ θ = β ◦ θ , choosing β ∈ β˜ and β ∈ β˜ such that β ◦ θ = β ◦ θ , and denoting by ˆ the element determined by the restriction of β ◦ θ = β ◦ θ , β ∈ F(T, Q) we have β ˆ ⊂ FT β(Q) ˆ ˆ ⊃ β FQ (Q) FQ (Q) 6.7.7.

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ˆ , N = NQ (Q) ˆ and N = $N , N % , we Hence, setting N = NQ (Q) still have β ˆ ⊂ FT β(Q) ˆ FN (Q) 6.7.8 and therefore, denoting by ν : Q → N the homomorphism induced by α , it ˜ follows from condition 2.10.1 that there is ξ˜ ∈ F(T, N ) such that ξ˜ ◦ ν˜ = β˜ ◦ θ˜ = β˜ ◦ θ˜

6.7.9;

˜ moreover, according to equality 6.7.3, there is an F-morphism θ˜ : N → Q R ˜ dividing ˜ιN such that ξ˜ = β˜ ◦ θ˜ for some β˜ ∈ F(T, Q )˜ιR /θ˜ . N

Consequently, respectively denoting by ν : Q → N and ν : Q → N the homomorphisms induced by α , we get

β˜ ◦ ˜ιQ ˜ = β˜ ◦ θ˜ = β˜ ◦ θ˜ ◦ ν˜ = β˜ ◦ θ˜ ◦ ˜ιN ˜ N ◦ ν N ◦ ν

β˜ ◦ ˜ιQ ˜ = β˜ ◦ θ˜ = β˜ ◦ θ˜ ◦ ν˜ = β˜ ◦ θ˜ ◦ ˜ιN ˜ N ◦ ν N ◦ ν

6.7.10

and therefore, according to Corollary 4.9, we still get

˜ ˜ ιN β˜ ◦ ˜ιQ N N = β ◦ θ ◦ ˜

˜ ˜ ιN and β˜ ◦ ˜ιQ N N = β ◦ θ ◦ ˜

6.7.11;

˜ now, it follows from the induction hypothesis that there are F-isomorphisms ∼ η˜ : Q ∼ Q Q and η ˜ : Q fulﬁlling = =

η˜ ◦ θ˜ ◦ ˜ιN ιQ N = ˜ N

and η˜ ◦ θ˜ ◦ ˜ιN ιQ N = ˜ N

6.7.12;

hence, setting η˜ = η˜ ◦ η˜−1 , we obtain

η˜ ◦ θ˜ = η˜ ◦ η˜−1 ◦ ˜ιQ ˜ = η˜ ◦ θ˜ ◦ ˜ιN ˜ N ◦ ν N ◦ ν

= η˜ ◦ θ˜ ◦ ˜ιN ˜ = ˜ιQ ˜ = θ˜ N ◦ ν N ◦ ν

6.7.13

and, in particular, we still obtain ˜ ˜ ˜ = F(T, ˜ ˜ ˜ ˜ F(T, Q )α/ Q )α/ ˜ θ˜ ◦ θ = F(T, Q )˜ ιR /θ˜ ◦ θ ◦ ν ˜ θ˜ ◦ θ N

6.7.14.

The last statement follows from statement 6.6.4 and equality 6.7.1. sc 6.8 The decomposition 6.7.1 allows us to consider in ac(F˜ ) (cf. 6.2) the exterior intersection of two F-selfcentralizing subgroups of P , which is just a direct product from the categorical point of view. Although, loosely speaking, the category of functors from any category to a category with a direct product inherits a direct product, as announced above we directly sc sc discuss the categories Kac(F˜ ) of K-objects of ac(F˜ ) (cf. 6.3), ﬁxing the notation. Actually, for ﬁnite p -groups K , we are mainly interested in the cohomological properties of some functors from these categories.

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6.9 Explicitly, ﬁx a ﬁnite group K and consider the category KF˜ of K-objects of F˜ — namely the pairs (ρ, Q) = Qρ formed by a subgroup Q ˜ of P and by a group homomorphism ρ : K → F(Q) (cf. 6.3) — and the full K ˜ sc subcategory (F ) over the K-objects where Q is F-selfcentralizing. First of all, if R and T are two F-selfcentralizing subgroups of P , we consider the ˜ where Q is an F-selfcentralizing subgroup of P set TR,T of triples (˜ α, Q, β) and we have ˜ ˜ α ˜ ∈ F(R, Q)β˜ and β˜ ∈ F(T, Q)α˜ 6.9.1; ˜ and (˜ we say that two triples (˜ α, Q, β) α , Q , β˜ ) are equivalent if there is an ˜ F-isomorphism θ˜ : Q ∼ = Q fulﬁlling α ˜ ◦ θ˜ = α ˜

and β˜ ◦ θ˜ = β˜

6.9.2;

R,T the set of equivalence classes of such triples. we denote by T ˜ and (˜ 6.10 Moreover, if two triples (˜ α, Q, β) α , Q , β˜ ) are equivalent, an ˜ F-isomorphism θ˜ : Q ∼ ˜ ◦ θ˜ = α ˜ and β˜ ◦ θ˜ = β˜ is unique. = Q fulﬁlling α Indeed, we may assume that the triples coincide with each other and, choos˜ ing α ∈ α ˜ , β ∈ β˜ and θ ∈ θ , it is easily checked that θ belongs to both α∗ β∗ FR α(Q) and FT β(Q) , and therefore it belongs to FQ (Q) (cf. equal˜ ity 6.5.1), so that θ˜ is the trivial element in F(Q) . sc 6.11 On the other hand, if Rσ and T τ are two K-objects of F˜ , it ˜ in TR,T and any x ∈ K , the triple is clear that, for any triple (˜ α, Q, β) ˜ σ(x) ◦ α ˜ , Q, τ (x) ◦ β still belongs to TR,T ; that is to say, σ and τ induce an action of K on TR,T preserving the equivalence relation, and therefore they ˜ R,T ; explicitly, if x ∈ K maps the equivalence induce an action of K on T ˜ on the equivalence class of x t = (x α ˜ , the triples class of t = (˜ α, Q, β) ˜ , x Q, x β) x x x˜ ˜ σ(x) ◦ α ˜ , Q, τ (x) ◦ β and ( α ˜ , Q, β) are equivalent and therefore there is ˜ x Q, Q) fulﬁlling a unique ρt,x ∈ F(

x

α ˜ ◦ ρt,x = σ(x) ◦ α ˜

and

x˜

β ◦ ρt,x = τ (x) ◦ β˜

6.11.1;

ˇ R,T of T R,T in TR,T , we have Consequently, for a set of representatives T obtained a group homomorphism sc ρ : K −→ ac(F ) Q 6.11.2 ˜ T ˇ R,T (α,Q, ˜ β)∈

ρ sc or, equivalently, a K-object of ac(F ) , and ﬁnally we ˜ T ˇ R,T Q (α,Q, ˜ β)∈ deﬁne the exterior intersection of Rσ and T τ by ρ Tτ = Rσ ∩ Q 6.11.3; ˜ T ˇ R,T (α,Q, ˜ β)∈

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Frobenius categories versus Brauer blocks

moreover, it follows from equalities 6.11.1 that the two families of F-mor˜ runs over T ˇ R,T determine phisms α ˜ : Q → R and β˜ : Q → T when (˜ α, Q, β) sc K two canonical ac(F )-morphisms T τ −→ T τ Rσ ←− Rσ ∩

6.11.4.

6.12 Note that, for another choice of the set of representatives, we get sc sc an isomorphic K-object of ac(F ) and a unique Kac(F )-isomorphism compatible with the canonical morphisms; actually, we may assume that any ˜ ∈T ˇ R,T fulﬁlls Q ⊂ R and α representative (˜ α, Q, β) ˜ = ˜ιR Q . Moreover, in the ˜ ˜ case where there are γ˜ ∈ F(P, R) and δ˜ ∈ F(P, T ) fulﬁlling γ˜ ◦ α ˜ = δ˜ ◦ β˜ , ˜ choosing respective representatives α , β , γ and δ of α ˜ , β , γ˜ and δ˜ fulﬁlling γ ◦ α = δ ◦ β , it follows from equality 6.5.1 that Nγ(R) (γ ◦ α)(Q) ∩ Nδ(T ) (γ ◦ α)(Q) = (γ ◦ α)(Q) 6.12.1 and therefore we get γ(R) ∩ δ(T ) = (γ ◦ α)(Q) , which motivates our terminology. 6.13 In our next result we prove the functorial nature of the exterior intersection deﬁned above. It is then easy by “distributivity” to extend this sc exterior intersection to all the category Kac(F˜ ) ; indeed, considering now sc two K-objects of the category ac(F˜ ) σ τ Rσ = Ri and T τ = Tj 6.13.1, i∈I

j∈J

the actions of K on I and on J determine an action of K on I × J and, moreover, if x ∈ K maps (i, j) ∈ I × J on (i , j ) then we have F-morphisms

σii (x) : Ri → Ri and τjj (x) : Tj → Tj ; thus, by the functoriality proved sc below, we get an ac(F˜ )-morphism

(i ,j ) Tj −→ Ri ∩ Tj ρ(i,j) (x) : Ri ∩

and therefore we still get an action ρ of K on the object sc in ac(F˜ ) ; then, we deﬁne Tτ = Rσ ∩

Tj Ri ∩

ρ

6.13.2

(i,j)∈I×J

Tj Ri ∩

6.13.3;

(i,j)∈I×J

sc Proposition 6.14 For any ﬁnite group K , the category Kac(F˜ ) admits a distributive direct product given by the exterior intersection. Moreover, for any homomorphism κ : K → K between ﬁnite groups, the restriction functor sc sc from K ac(F˜ ) to Kac(F˜ ) preserves the direct products.

6. Exterior quotient of a Frobenius P -category

79

Proof: With the notation above, in order to discuss the functorial nature of the exterior intersection, consider three F-selfcentralizing subgroups R , T ˜ ˜ and U of P and two morphisms ψ˜ ∈ F(R, U ) and η˜ ∈ F(T, U ) ; it follows from Proposition 6.7 that η˜ determines an isomorphism class of (F˜ ◦ )U -objects ˜ θ˜ , we have η˜ = η˜ ◦ θ˜ for θ˜ : U → U dividing ψ˜ such that, setting ψ˜ = ψ/ ˜ U )ψ˜ and, once again, η˜ is uniquely determined. a suitable η˜ ∈ F(T, ˜ η˜) determines That is to say, according to statement 6.5.2, the pair (ψ, R,T and, once we have chosen a set of an equivalence class of triples in T ˇ R,T , it determines a unique triple (ψ˜ , U , η˜ ) and a unique representatives T morphism θ˜ : U → U fulﬁlling ψ˜ = ψ˜ ◦ θ˜ and η˜ = η˜ ◦ θ˜ , so that the canonical map

˜ ˜ ˜ F(Q, U ) −→ F(R, U ) × F(T, U)

6.14.1

˜ T ˇ R,T (α,Q, ˜ β)∈

˜ sending ϕ˜ ∈ F(Q, U ) to (˜ α ◦ ϕ, ˜ β˜ ◦ ϕ) ˜ is bijective. ˜ In particular, considering two F-morphisms ζ˜ : R → R and ξ˜ : T → T ˜ ∈ TR,T , we have the morphisms and a triple (˜ α, Q, β) ζ˜ ◦ α ˜ : Q −→ R

and ξ˜ ◦ β˜ : Q −→ T

6.14.2

ˇ R ,T and an F-morphism ˜ and therefore we obtain a triple (˜ α , Q , β˜ ) in T ˜ θ : Q → Q fulﬁlling ζ˜ ◦ α ˜=α ˜ ◦ θ˜ and ξ˜ ◦ β˜ = β˜ ◦ θ˜

6.14.3.

Thus, we have obtained a map R,T −→ T R ,T t ˜ ˜ : T ζ,ξ

6.14.4

˜ and it is not diﬃcult to check that, for two other F-morphisms ζ˜ : R → R and ξ˜ : T → T , we get t ˜ ˜ ◦ t ˜ ˜ = t ˜ ˜ ˜ ˜ ζ ,ξ ζ,ξ ζ ◦ζ,ξ ◦ξ

6.14.5.

Moreover, if we consider respective K-actions σ , σ , τ and τ on R , ˜ R , T and T , and assume that ζ˜ and ξ˜ are KF-morphisms, it follows from equality 6.14.5 that the map tζ, ˜ ξ˜ is compatible with the actions of K on ˇ R,T the sets TR,T and TR ,T ; once we have chosen sets of representatives T ˇ R ,T for T R,T and T R ,T respectively, this map induces a new map and T

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Frobenius categories versus Brauer blocks

ˇ R,T → T ˇ R ,T and it is easily checked from equalities 6.11.1 that the ˇt ˜ ˜ : T ζ,ξ ˜ hence, from deﬁnimorphism θ˜ : Q → Q above becomes a KF-morphism; sc K ˜ tion 6.12.2, we obtain a ac(F )-morphism ξ˜ : Rσ ∩ T τ −→ Rσ ∩ T τ ζ˜ ∩

6.14.6.

Finally, always from equality 6.14.5, it is not diﬃcult to check that, for two ˜ other KF-morphisms ζ˜ : Rσ → Rσ and ξ˜ : T τ → T τ , we get ˜ = (ζ˜ ◦ ζ) ˜ ∩ ˜ ξ) (ξ˜ ◦ ξ) ξ˜ ) ◦ (ζ˜ ∩ (ζ˜ ∩

6.14.7. sc

More generally, if we now consider four objects in the category Kac(F˜ ) Rσ =

Ri

i∈I

Tτ =

Tj

σ

,

Rσ =

R i

σ

i ∈I

τ

and T τ =

Tj

6.14.8

τ

j ∈J

j∈J sc

and two Kac(F˜ )-morphisms ˜ f ) : Rσ −→ Rσ (ζ,

and

˜ g) : T τ −→ T τ (ξ,

6.14.9,

where f : I → I and g : J → J are K-compatible maps and moreover ζ˜ and ξ˜ ˜ , are K-compatible families of F-morphisms ζ˜i : Ri → Rf (i) and ξ˜j : Tj → Tg(j) i and j respectively running over I and J , then we clearly have a K-compasc tible family of ac(F˜ )-morphisms ξ˜j : Ri ∩ Tj −→ Rf (i) ∩ Tg(j) ζ˜i ∩

6.14.10

sc

which deﬁne a Kac(F˜ )-morphism

˜ f) ∩ ˜ g) : Rσ ∩ (ξ, T τ −→ Rσ ∩ T τ (ζ,

6.14.11.

Finally, it is not diﬃcult (but painful!) to check that bijections 6.14.1 imply the bijections K sc T τ , U ω) ac(F˜ ) (Rσ ∩ K sc sc ∼ = ac(F˜ ) (Rσ , U ω ) × Kac(F˜ ) (T τ , U ω )

6.14.12

sc for any Kac(F˜ )-object U ω , which proves that the exterior intersection is a direct product in this category. The proof of the last statement is straightforward.

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6.15 As any direct product, the exterior intersection is associative and commutative (from the categorical point of view!). Moreover, the exterior intersection i∈I Ri of any ﬁnite family R = {Ri }i∈I of F-selfcentralizing subgroups of P depends on the following set of indices; ﬁrst of all, for any F-selfcentralizing subgroup Q of P , let us consider the families α ˜ = {˜ αi }i∈I ˜ i , Q) fulﬁlling with α ˜ i ∈ F(R

FRi αi (Q) = FQ (Q)

(αi )∗

6.15.1

i∈I

where αi ∈ α ˜ i and (αi )∗ : αi (Q) ∼ = Q denotes the inverse of the isomorphism determined by αi — called the R-intersectional families at Q ; note that, for ˜ i∈I is an R-intersectional family at Q . ˜ any F-isomorphism θ˜ : Q ∼ αi ◦ θ} = Q , {˜ On the other hand, in order to state their relationship with the F-intersected |I| subgroups of P (cf. 4.11), we write P = Ri whenever Ri = P for i∈I

any i ∈ I .

Proposition 6.16 For any ﬁnite family R = {Ri }i∈I of F-selfcentralizing subgroups of P we have Ri ∼ Q 6.16.1 = i∈I (Q,α) ˜

˜ where (Q, α ˜ ) runs over a set of representatives for the F-isomorphism classes of pairs formed by an F-selfcentralizing subgroup Q of P and an R-intersec

tional family α ˜ at Q , and where the structural morphism (Q,α) ˜ Q → Ri is determined by α ˜ i : Q → Ri for any i ∈ I . In particular, Q is an F-intersected n subgroup of P if and only if it is a direct summand of P for some n ≥ 1 . sc

Proof: If |I| = 2 , the ac(F˜ )-isomorphism 6.16.1 follows from the very deﬁnition of the exterior intersection; otherwise, it suﬃces to argue by induction on |I| and to apply the distributivity and equality 6.5.1. In particular, from its very deﬁnition, an F-intersected subgroup Q of P |F˜ (P,Q)| is a direct summand of P ; conversely, it follows from equality 6.15.1 n that, for any n ≥ 1 , any direct summand Q of P is an F-intersected subgroup of P . sci sc Corollary 6.17 Denoting by F˜ the full subcategory of F˜ over the set of sci sc F-intersected subgroups of P , the subcategory ac(F˜ ) of ac(F˜ ) is closed with respect to the exterior intersection.

Proof: If Q and R are F-intersected subgroups of P then, by Proposition 6.16, n m they appear as direct summands in P and P for suitable n, m ∈ N−{0} n+m R appears in and therefore Q ∩ P by the distributivity of the exterior

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Frobenius categories versus Brauer blocks

sci R is an object in ac(F˜ ) intersection (cf. Proposition 6.14); consequently, Q ∩ by the same proposition. sc 6.18 As a matter of fact, for any K-object Qρ of ac(F˜ ) , the category K sc sc ac(F˜ ) Qρ (cf. 1.7) still admits a direct product or, equivalently, Kac(F˜ ) admits pull-backs; in order to show it, let us introduce the relative exterior intersection. For any triple of F-selfcentralizing subgroups R , R and T and

˜

˜

β β ˜ any pair of F-morphisms R → T ← R , denote by Tβ, ˜ β˜ the set of triples ˜ ˜ (˜ α, Q, α ˜ ) ∈ TR,R fulﬁlling β ◦ α ˜ = β ◦α ˜ (cf. 6.12); it is clear that Tβ, ˜ β˜ is a ˜ Rσ , union of equivalence classes in TR,R ; moreover, considering KF-objects σ τ K˜ R and T , and F-morphisms β˜

β˜

Rσ −→ T τ ←− Rσ

6.18.1,

˜ ˜ ˜ and therefore, in the set of represenit is quite clear that K stabilizes T β,β ˇ ˜ ˜ for T ˜ ˜ ˜ , the action ρ of K on R ∩ R determined by σ and σ tatives T β,β

β,β

ˇ ˜ ˜ . Then, we deﬁne induces an action ρβ, ˜ β ˜ on the subfamily indexed by Tβ, β ρ ˜ ˜ β T τ Rσ = Rσ β˜ ∩ β˜ Rσ = Rσ ∩ Q β, 6.18.2 ˇ ˜ ˜ (α,Q, ˜ α ˜ )∈T β,β sc T τ Rσ → Rσ determined endowed with the ac(F˜ )-morphisms Rσ ← Rσ ∩ ˇ ˜ ˜ . α, Q, α ˜ ) runs over T by the two families of F-morphisms α ˜ and α ˜ when (˜ β,β T R admits the following explicit description. As in 6.12 above, R ∩

Proposition 6.19 With the notation above, choosing representatives β of β˜ and β of β˜ , we have T R ∼ R∩ β(R)u ∩ β (R ) 6.19.1, = u

where u ∈ T runs over the set of elements such that β(R)u ∩ β (R ) is F-selfcentralizing in a set of representatives for β(R)\T /β (R ) , and we con˜ sider the F-morphisms from β(R)u ∩ β (R ) to R and to R determined by β , u and β . Proof: With the notation above, for any triple (˜ α, Q, α ˜ ) ∈ Tβ, ˜ β˜ it follows from 6.12 that β(R)u ∩ β (R ) = (β u ◦ α)(Q) 6.19.2, where α ∈ α ˜ , α ∈ α ˜ and u ∈ T fulﬁll β ◦ α = β u ◦ α . Conversely, for any w ∈ T such that β(R)w ∩ β (R ) is F-selfcentralizing, we already know ˜ that the respective F-morphisms γ˜ and γ˜ from β(R)w ∩ β (R ) to R and sc R , to R determined by β , w and β induce an ac(F˜ )-morphism to R ∩ and therefore they determine a triple which clearly belongs to Tβ, ˜ β˜ .

6. Exterior quotient of a Frobenius P -category

83

Consequently, by the argument above, there are a suitable element u ∈ T ˜ and an F-morphism (cf. 6.19.2) θ˜ : β(R)w ∩ β (R ) −→ β(R)u ∩ β (R )

6.19.3

˜ compatible with the F-morphisms determined by β , w , u and β ; actually, ˜ ˜ since γ˜ ◦ θ coincides with the F-morphism determined by the inclusion, we may assume that β(R)w ∩ β (R ) ⊂ β(R)u ∩ β (R )

6.19.4

and that θ˜ is determined by this inclusion; hence, there is v ∈ R fulﬁlling β(v)wtw−1 β(v)−1 = utu−1

6.19.5

for any t ∈ β(R)w ∩ β (R ) , and therefore the element u−1 β(v)w ∈ T centralizes β(R)w ∩ β (R ) ; since β(R)w ∩ β (R ) is F-selfcentralizing , we actually have u−1 β(v)w = β (v ) for some v ∈ R , so that u and w determine the same double class in β(R)\T /β (R ) and therefore we have β(R)w ∩ β (R ) = β(R)u ∩ β (R )

6.19.6.

We are done. 6.20 As above, we can extend the relative exterior intersection to the sc sc K-objects of ac(F˜ ) ; indeed, consider three Kac(F˜ )-objects Rσ =

Ri

σ

,

Rσ =

R i

σ

and T τ =

i ∈I

i∈I

Tj

τ

6.20.1

j∈J

sc

and two Kac(F˜ )-morphisms ˜ f ) : Rσ −→ T τ (β,

and

(β˜ , f ) : Rσ −→ T τ

6.20.2,

where f : I → J and f : I → J are K-compatible maps and β˜ and β˜ are ˜ K-compatible families of F-morphisms β˜i : Ri → Tf (i) and β˜i : Ri → Tf (i ) , i and i respectively running over I and I . Then, for any pair (i, i ) ∈ I × I β˜ Ri in the category such that f (i) = f (i ) , we already have deﬁned Ri β˜i ∩ i

sc

ac(F˜ ) and it is clear that K stabilizes the set of such pairs, namely the pull-back I ×J I ; moreover, it follows from Proposition 6.19 that, for any sc x ∈ K and any (i, i ) ∈ I ×J I , σ(x) and σ (x) induce an ac(F˜ )-morphism β˜ Ri −→ R(σ(x))(i) Ri β˜i ∩ i

β˜(σ(x))(i)∩β˜(σ (x))(i )

R(σ (x))(i )

6.20.3.

84

Frobenius categories versus Brauer blocks

sc That is to say, we get an action ρ of K over the ac(F˜ )-object (β˜ ,f ) R = β˜ Ri R (β,f Ri β˜i∩ ˜ )∩ i

(i,i )∈I×J I

6.20.4

sc

and then we deﬁne the relative exterior intersection in Kac(F˜ ) by T τ Rσ = Rσ (β,f (β˜ ,f ) Rσ = β˜ Ri ρ Rσ ∩ Ri β˜i∩ ˜ )∩ (i,i )∈I×J I

i

6.20.5

endowed with the evident structural Kac(F˜ )-morphisms to Rσ and Rσ . sc

sc Proposition 6.21 For any ﬁnite group K , the category Kac(F˜ ) admits distributive pull-backs given by the relative exterior intersection. Moreover, for any homomorphism κ : K → K between ﬁnite groups, the restriction sc sc functor from K ac(F˜ ) to Kac(F˜ ) preserves the pull-backs.

ω Proof: With the notation above, let U ω = be a fourth K-object 6∈L U6 sc of ac(F˜ ) and

(˜ γ , g) : U ω −→ Rσ

and

(˜ γ , g ) : U ω −→ Rσ

6.21.1

sc

two Kac(F˜ )-morphisms fulﬁlling ˜ f ) ◦ (˜ (β, γ , g) = (β˜ , f ) ◦ (˜ γ , g )

6.21.2;

in particular, the K-compatible maps g : L → I and g : L → I also fulﬁll f ◦ g = f ◦ g and thus they determine a K-compatible map h : L → I ×J I . ˜ Moreover, γ˜ and γ˜ are K-compatible families of F-morphisms γ˜6 : U6 −→ Rg(6)

and γ˜6 : U6 −→ Rg (6)

6.21.3,

where D runs over L , and equality 6.21.2 yields β˜g(6) ◦ γ˜6 = β˜g (6) ◦ γ˜6 ; then, it sc easily follows from Proposition 6.19 that there is a unique ac(F˜ )-morphism ε˜6 : U6 −→ Rg(6)

β˜g()∩β˜g ()

Rg (6)

6.21.4

sc compatible with γ˜ , γ˜ and the structural ac(F˜ )-morphisms, the uniqueness always being a consequence of Corollary 4.9; it is this uniqueness which sc guarantees that we still get a K-compatible family of ac(F˜ )-morphisms and sc therefore an Kac(F˜ )-morphism

T τ Rσ (˜ ε, h) : U ω −→ Rσ ∩

6.21.5.

All the remaining veriﬁcations and the last statement are straightforward.

6. Exterior quotient of a Frobenius P -category

85

6.22 From now on, we assume that K is a p -group. Let O be the unramiﬁed complete discrete valuation ring of characteristic zero admitting our ﬁeld k (cf. 1.10) as residue class ﬁeld, and respectively denote by O-mod and k-mod the categories of ﬁnitely generated O- and k-modules. We are interested in the vanishing cohomological properties of some contravariant sc functors to O-mod from full subcategories F of Kac(F˜ ) closed by direct sums, direct summands, direct products and pull-backs. sc

6.23 More explicitly,

it is clear that any K-object of ac(F˜ ) is the direct sum of K-objects Qρ = ( i∈I Qi )ρ such that K is transitive on I — called

σ the indecomposable K-objects; note that, if Rσ = j∈J Rj ) is another sc indecomposable K-object and (˜ α, f ) : Rσ → Qρ is an Kac(F˜ )-morphism, the map f : J → I is surjective and |I| divise |J| . We will consider the full sc subcategories F of Kac(F˜ ) over sets of indecomposable K-objects such that the subcategory ac(F) is closed by direct products and pull-backs. Recall that any contravariant functor m from F to O-mod can be uniquely extended to an additive contravariant functor ac(m) : ac(F) −→ O-mod

6.23.1

mapping the direct sums of K-objects on the direct sum of their images, and the ac(F)-morphisms on the corresponding O-module homomorphisms between the direct sums (cf. A4.10). 6.24 In [32], Stephan Jackowski and James McClure show a general result on vanishing cohomology, which applied to ac(F) and ac(m) would depend on the existence of a so-called Mackey complement for ac(m) — in order to get a Mackey functor from ac(F) , a concept introduced by Andreas Dress in [22]. A careful analysis of their proof shows that the point is just the existence of a natural section of some adjoinness natural map, which a priori could be easier to reach; this is the case in the proof our next result. This result is the key in chapter 14 for determining the O-rank of the so-called local Grothendieck group.

˜ sc 6.25 Let us consider the object x∈K P of the category ac(F ) , endowed with the K-action π deﬁned by the regular action of K on itself and

π by the identity on P between the corresponding terms, so that is x∈K P sc an indecomposable K-object of ac(F˜ ) . Note that, since |Z(K)| is invertible sc in O , if F is a full subcategory of Kac(F˜ ) and m : F → O-mod is a con˜ of F in the exterior quotravariant functor which factorizes via the image F sc ac(F˜ ) (cf. 6.3) throughout a contravariant functor m ˜ → O-mod , ˜ :F tient K it follows from Proposition A4.13 that, for any n ∈ N , we have ˜ m) ˜ = Hn (F, m) Hn (F,

6.25.1

86

Frobenius categories versus Brauer blocks

since, considering the subcategory Z of F formed by the same objects and ˜ m) ˜ clearly by the automorphisms of the F-objects induced by Z(K) , Hn (F, coincides with the Z-stable n-cohomology group of F over m (cf. A3.18). Theorem 6.26 Assume that K is a ﬁnite p -group. Let F be a full sub

π sc category of Kac(F˜ ) over indecomposable K-objects, including , x∈K P such that the subcategory ac(F) is closed by direct products and pull-backs, and m : F → O-mod a contravariant functor which maps any morphism on an isomorphism. Then, we have Hn (F, m) = {0} for any n ≥ 1 . Proof: First of all, we prove the theorem assuming that p·m = 0 or, equivalently, that m is a contravariant functor from F to k-mod . Moreover, it follows from Proposition A4.11 that for any n ≥ 1 we have Hn (F, m) ∼ 6.26.1, = Hn ac(F), ac(m) so that it suﬃces to prove that Hn ac(F), ac(m) = {0} for any n ≥ 1 .

Set S = x∈K P ; it follows from Proposition 6.14 that the exterior intersection with S π (cf. deﬁnition 6.13.3) deﬁnes a functor intS π : ac(F) −→ ac(F)

6.26.2;

S π → S π for any then, the existence of the structural ac(F)-morphism Qρ ∩ ρ ac(F)-object Q , shows that intS π factorizes throughout the evident forgetful functor (cf. 1.7) fgS π : ac(F)S π −→ ac(F) 6.26.3; explicitly, it suﬃces to consider the functor ac(F) → ac(F)S π mapping any ac(F)-object Qρ on the structural ac(F)-morphism above and any ac(F)-mor S π (cf. Proposition 6.14). But, since the category ˜ id phism α ˜ : Rσ → Qρ on α ˜∩ S π : S π → S π , it follows from Corollary A4.8 ac(F)S π has the ﬁnal object id that for any n ≥ 1 we have Hn ac(F)S π , ac(m) ◦ fgS π = {0} 6.26.4 and therefore, we still have (cf. A3.10.4). Hn ac(F), ac(m) ◦ intS π = {0}

6.26.5.

S π → Qρ Moreover, the existence of the structural morphism ω ˜ Qρ : Qρ ∩ ρ for any ac(F)-object Q , shows the existence of a natural map ω : intS π −→ idac(F)

6.26.6 sending Qρ to ω ˜ Qρ ; thus, in order to prove that Hn ac(F), ac(m) = {0} , it suﬃces to prove that the natural map ac(m) ∗ ω admits a natural section θ : ac(m) ◦ intS π −→ ac(m) so that ac(m) becomes a direct summand of ac(m) ◦ intS π .

6.26.7,

6. Exterior quotient of a Frobenius P -category

87

Explicitly, for any F-object Qρ = ( i∈I Qi )ρ , we have (cf. 6.13) ρˆ Sπ = Qρ ∩ Ti 6.26.8 ˇ Q ,P (i,x)∈I×K (˜ τi ,Ti ,˜ ιP )∈T T i i

ˇ Q ,P of T ˜ Q ,P in TQ ,P and a suitable action ρˆ for a set of representatives T i i i sc ˜ S ; in particular, K acts freely on the disjoint of K on the ac(F )-object Q ∩ union ˇ Q ,P Iˆ = 6.26.9 T i (i,x)∈I×K

ˆ ˆ , and let us denote by I/K the set of K-orbits on Iˆ and, for any O ∈ I/K ρˆO ρ π by (TO ) the corresponding indecomposable “direct summand” of Q ∩ S and by τ˜O the composition S π −→ Qρ τ˜O : (TO )ρˆO −→ Qρ ∩

6.26.10

of the structural ac(F)-morphism with ω ˜ Qρ . Moreover, we denote by Iˆ◦ /K sc ˆ the set of “special” orbits O ∈ I/K where the F˜ -morphisms determining τ˜O are isomorphisms; note that, according to Proposition 6.14, we have a canonical bijection ˜ Iˆ◦ /K ∼ 6.26.11. F(P, Qi ) = i∈I

Then, we consider the homomorphism S π ) −→ m(Qρ ) θQρ : ac(m) (Qρ ∩ + m (TO )ρˆO ˆ O∈I/K sending an element m = (mO )O∈I/K of this product to ˆ θQρ (m) = |Iˆ◦ /K|−1 · m(˜ τO )−1 (mO )

6.26.12

6.26.13,

O∈Iˆ◦ /K

so that we clearly have θQρ ◦ (ac(m) ∗ ω)Qρ = idm(Qρ )

6.26.14

By the distributivity of the exterior intersection (cf. 6.13), we easily can extend this correspondence to all the ac(F)-objects and then we claim that the extended correspondence is a natural map from ac(m) ◦ intS π to ac(m) ; once again, it suﬃces to consider an F-morphism α ˜ : Rσ → Qρ and to prove the commutativity of the following diagram θQ ρ S π ) −→ ac(m) (Qρ ∩ m(Qρ ) (ac(m))(α ˜ ∩ idS π )

↓ ↓ m(α) ˜ σ σ θ R S π ) −→ m(Rσ ) ac(m) (R ∩

6.26.15.

88

Frobenius categories versus Brauer blocks

Explicitly, if Rσ = ( j∈J Rj )σ then α ˜ is given by a K-compatible map ˜ f : J → I and by a K-compatible family of F-morphisms α ˜ j : Rj → Qf (j) where j runs over J , and as above we have Sπ = Rσ ∩

σˆ

Uj

6.26.16

ˇ R ,P (j,x)∈J×K (˜ υj ,Uj ,˜ ιP )∈T U j j

ˇ R ,P of T ˜ R ,P in TR ,P and a suitable action σ for a set of representatives T ˆ j j j sc S ; again, we set of K on the ac(F˜ )-object R ∩ ˇ R ,P Jˆ = T 6.26.17 j (j,x)∈J×K

ˆ and denote by J/K the set of K-orbits on Jˆ , by Jˆ◦ /K the set of “speˆ , by (U )σˆO the corresponding cial” K-orbits on Jˆ and, for any O ∈ J/K O S π and by υ˜O : (UO )σˆO → Rσ indecomposable “direct summand” of Rσ ∩ the analogous composition 6.26.10; moreover, it is clear that the map f and the family {˜ αj }j∈J determine a K-compatible map fˆ : Jˆ → Iˆ and, for any ˆ O ∈ J/K , an F-morphism ρˆ ˆ

α ˜ O : (UO )σˆO −→ (Tfˆ(O) )

f (O)

6.26.18.

It is easily checked from Propositions 6.14 and 6.21 that ω˜ Qρ (Qρ ∩ Sπ ) ∼ Sπ Rσ α˜ ∩ = Rσ ∩

6.26.19

and, by the distributivity, we may assume that the exterior intersection S π coincides with Rσ ∩ σˆ τ˜f (j) Tf (j) Rj α˜ j ∩ 6.26.20. (j,x)∈J×K (˜ τf (j) ,Tf (j) ,˜ ιP T

f (j)

ˇQ )∈T f (j) ,P

ˇ Then, for any (j, x) ∈ J × K and any tf (j) = (˜ τf (j) , Tf (j) , ˜ιP Tf (j) ) ∈ TQf (j) ,P , choosing τf (j) ∈ τ˜f (j) , αf (j) ∈ α ˜ f (j) and a set of representatives W(j,x,tf (j) ) in Qf (j) for the set of double classes τf (j) (Tf (j) ) Qf (j) αj (R) , and denoting sc by W(j,x,tf (j) ) the set of w ∈ W(j,x,tf (j) ) such that (cf. 1.8) −1 Uw = κQf (j) (w) ◦ αj τf (j) (Tf (j) )

6.26.21

sc

remains F-selfcentralizing, the ac(F˜ )-morphism idS π : Rσ ∩ S π −→ Qρ ∩ Sπ α ˜∩

6.26.22

6. Exterior quotient of a Frobenius P -category

89

sc is the “direct sum”, over the set of triples (j, x, tf (j) ) above, of the ac(F˜ )morphisms Uw −→ Tf (j) 6.26.23 sc

w∈W(j,x,t

f (j) )

sc deﬁned by the F˜ -morphisms β˜(j,x,tf (j) ,w) : Uw → Tf (j) determined by the compositions κQf (j) (w) ◦ αj (cf. 1.8).

Moreover, K acts on all the situation and we denote by O(j,x,tf (j) ,w) the K-orbit — which actually is regular — of (j, x, tf (j) , w) ∈ Jˆ , by O(f (j),x,t ) f (j)

the image in Iˆ via fˆ of O(j,x,tf (j) ,w) — which actually does not depend on w — and by β˜O(j,x,t ,w) and υ˜O(j,x,t ,w) the F-morphisms f (j)

f (j)

σ ˆO

(UO(j,x,t

f (j) ,w)

)

(j,x,tf (j) ,w)

f (j) ,w)

)

)

f (j) )

σ ˆO

(UO(j,x,t

ρˆO

−→ (TO(f (j),x,t

(j,x,tf (j) ,w)

−→ R

(f (j),x,tf (j) )

6.26.24

σ

R respectively determined by the K-orbits of β˜(j,x,tf (j) ,w) and ˜ιUwj ; then, the naturality of ac(m) ∗ ω forces

m(˜ υO(j,x,t

f (j) ,w)

) ◦ m(˜ α) = m(β˜O(j,x,t

f (j) ,w)

) ◦ m(˜ τO(f (j),x,t

f (j) )

)

6.26.25,

so that we still have m(˜ α) ◦ m(˜ τO(f (j),x,t

f (j) )

= m(˜ υO(j,x,t

)−1

f (j) ),w)

)−1 ◦ m(β˜O(j,x,t

f (j) ),w)

)

6.26.26

and therefore the right member of this equality does not depend on w . Now, we are ready to prove the commutativity of the diagram 6.29.15; according to our deﬁnition, the composition m(˜ α) ◦ θQρ sends the element m = (mO )O∈I/K = (m(i,x,ti ) )(i,x)∈I×K ,ti ∈Tˇ Q ˆ

i ,P

6.26.27,

τi , Ti , ˜ιP where mO ∈ m(TO ) and m(i,x,ti ) ∈ m(Ti ) if ti = (˜ Ti ) , to the sum |Iˆ◦ /K|−1 ·

O∈Iˆ◦ /K

(m(˜ α)) m(˜ τO )−1 (mO )

6.26.28,

90

Frobenius categories versus Brauer blocks

whereas we have idS π ) (m) (ac(m))(˜ α∩ = m(β˜O(j,1,t j

tf (j)

w

f (j) ,w)

) (mO(f (j),1,t

f (j) )

6.26.29,

)

ˇ Q ,P and w over W sc where j runs over J , tf (j) over T f (j) (j,1,tf (j) ) , and therefore, R

sc

j denoting by W(j,1,t the set of w ∈ W(j,1,tf (j) ) such that Uw = Rj , it follows f (j) ) from our deﬁnition of θRσ and from equality 6.26.25 that idS π ) (m) θRσ ◦ (ac(m))(˜ α∩

R

=

j | |W(j,1,t f (j) )

j

tf (j)

|Jˆ◦ /K|

·m(˜ α) m(˜ τO(f (j),1,t

f (j)

)−1 (mO(f (j),1,t )

f (j)

) )

6.26.30,

ˇ Q ,P . where j runs over J and tf (j) over T f (j) On the other hand, note that if O(f (j),1,tf (j) ) belongs to Iˆ◦ /K then we R

j have |W(j,1,t | = 1 ; moreover, we already know that α ˜ j induces an injective f (j) ) ˜ ˜ map from F(P, Qf (j) ) to F(P, Rj ) (cf. Corollary 4.9) and it is clear that

|f −1 (i)| = |J|/|I| for any i ∈ I . Hence, since according to bijection 6.26.11 we have (cf. 6.7.2) ˜ )| and |Jˆ◦ /K| ≡ |J||F(P ˜ )| (mod p) |Iˆ◦ /K| ≡ |I||F(P

6.26.31,

the sum of all these terms in the second member of equality 6.26.30 coincides with the sum 6.26.28 above. Consequently, in order to show the commutativity of diagram 6.26.14, it suﬃces to prove that, for any j ∈ J and any tf (j) = (˜ τf (j) , Tf (j) , ˜ιP Tf (j) ) ˇ in TQ ,P such that τ˜f (j) : Tf (j) → Qf (j) is not and isomorphism, p dif (j)

R

R

j j vides |W(j,1,t | ; but, it is clear that W(j,1,t is a set of representatives f (j) ) f (j) ) for the quotient set τf (j) (Tf (j) ) TQf (j) τf (j) (Tf (j) ), αj (Rj ) 6.26.32

¯Q τf (j) (Tf (j) ) acts freely on this set. and that the nontrivial p-group N f (j) This completes the proof of the naturality of θ and therefore the proof of the theorem in the case where p·m = 0 . In the general case, there is a subfunctor mtor : F → O-mod mapping any F-object Qρ on the torsion O-submodule of m(Qρ ) and then we have the quotient functor m/mtor : F −→ O-mod 6.26.33

6. Exterior quotient of a Frobenius P -category

91

which maps any object on a free O-module; consequently, since we have exact sequences (cf. A3.11.4) Hn (F, mtor ) −→ Hn (F, m) −→ Hn (F, m/mtor )

6.26.34,

we already may assume that either m = mtor or mtor = 0 . In the ﬁrst case we have p6 ·m = 0 for some D ∈ N and, considering the exact sequences (cf. A3.11.4) Hn (F, p·m) −→ Hn (F, m) −→ Hn (F, m/p·m) = {0}

6.26.35,

it suﬃces to argue by induction on D . In the second case, if c0 is an n-cocycle, we already have proved that c0 ≡ dn−1 (a0 ) (mod p)

6.26.36

for a suitable (n − 1)-cochain a0 , so that we have c0 − dn−1 (a0 ) = p·c1 for a suitable n-cocycle c1 since we are dealing with free O-modules; thus, we inductively can deﬁne n-cocycles ci and (n−1)-cochains ai fulﬁlling ci ≡ dn−1 (ai ) (mod p) and ci − dn−1 (ai ) = p·ci+1

6.26.37

and then, according to the completeness of O , it is quite clear that c0 = dn−1

i∈N

We are done.

pi ·ai

6.26.38.

Chapter 7

Nilcentralized and selfcentralizing Brauer pairs in blocks 7.1 As in chapter 3, let G be a ﬁnite group, k an algebraically closed ﬁeld of characteristic p , b a block of G and (P, e) a maximal Brauer (b, G)-pair, and denote by F(b,G) the corresponding Frobenius P -category deﬁned there. In this chapter, we give alternative descriptions of the F(b,G) -nilcentralized and the F(b,G) -selfcentralizing subgroups Q of P in terms of the blocks in their associated Brauer (b, G)-pairs. As a matter of fact, in this context the group F(b,G) (Q) can be canonically lifted to a k ∗ -group Fˆ(b,G) (Q) (cf. 1.23) ¯P (Q)-algebra SQ (cf. 1.20). and there exists a canonical Dade N Proposition 7.2 Let Q be a subgroup of P and f the block of CG (Q) such that (P, e) contains (Q, f ) . Then, Q is F(b,G) -nilcentralized if and only if f is a nilpotent block. Proof: By Proposition 2.7, there is ϕ ∈ F(b,G) (P, Q) such that Q = ϕ(Q) is fully centralized in F(b,G) ; in particular, there is x ∈ G such that Q = Qx and that, setting f = f x , (P, e) contains (Q , f ) (cf. 1.15) and therefore we may assume that Q is fully centralized in F(b,G) . In this case, by Proposition 3.5, CP (Q) is a defect subgroup of the block f of CG (Q) and, by Proposition 2.16, the centralizer CF(b,G) (Q) is a Frobenius CP (Q)-category; moreover, by Corollary 3.6, the Frobenius CP (Q)-categories CF(b,G) (Q) and F(f,CG (Q)) coincide. Thus, if Q is F(b,G) -nilcentralized then the CP (Q)-categories FCP (Q) and F(f,CG (Q)) coincide, so that f is a nilpotent block (cf. 1.18). Conversely, if f is a nilpotent block (cf. 1.18), it follows from statements (1.7.2) and (1.9.2) in [41] that the CP (Q)-categories F(f,CG (Q)) and FCP (Q) coincide, so that CF(b,G) (Q) = FCP (Q) and Q is F(b,G) -nilcentralized (cf. 4.3). Corollary 7.3 With the notation above, denote by f¯ the image of f in k C¯G (Q) . Then, Q is F(b,G) -selfcentralizing if and only if f¯ has defect zero. Proof: If Q is F(b,G) -selfcentralizing then it is F(b,G) -nilcentralized, so that f is a nilpotent block and CP (Q) = Z(Q) is a defect subgroup of f ; hence, f¯ has defect zero (cf. 1.17). Conversely, if f¯ has defect zero then Z(Q) is a defect subgroup of f (cf. 1.17 and 1.19); but, as above, we may assume that Q is fully centralized in F(b,G) and then, by Proposition 3.5, CP (Q) is a defect subgroup of the block f of CG (Q) ; hence, we have CP (Q) = Z(Q) and therefore Q is F(b,G) -selfcentralizing (cf. 4.10).

94

Frobenius categories versus Brauer blocks

7.4 Coherently, we say that a Brauer (b, G)-pair (Q, f ) is nilcentralized if f is a nilpotent block of CG (Q) (cf 1.17) and that it is selfcentralizing if f¯ is a block of defect zero of C¯G (Q) = CG (Q)/Z(Q) (cf. 1.17). Assume that (Q, f ) is nilcentralized; then, the quotient kCG (Q)f J kCG (Q)f is a simple k-algebra (cf. (1.9.1) in [41]) and therefore the action of the norˆG (Q, f ) (cf. 1.24) malizer NG (Q, f ) on this k-algebra determines a k ∗ -group N together with a k-algebra homomorphism (cf. 1.25) ˆG (Q, f ) −→ kCG (Q)f J kCG (Q)f k∗ N

7.4.1;

moreover, since we have an obvious NG (Q, f )-stable k-algebra homomorphism kCG (Q) −→ kCG (Q)f J kCG (Q)f 7.4.2, ˆG (Q, f ) we still have a NG (Q, f )-stable group homomorphism CG (Q) → N lifting the inclusion CG (Q) ⊂ NG (Q, f ) and therefore, assuming that (Q, f ) is contained in (P, e) , we get an exact sequence ˆG (Q, f ) −→ Fˆ(b,G) (Q) −→ 1 1 −→ CG (Q) −→ N

7.4.3

for a suitable k ∗ -group Fˆ(b,G) (Q) lifting F(b,G) (Q) . Proposition 7.5 Let Q and Q be F-isomorphic F(b,G) -nilcentralized subgroups of P . Any F-isomorphism ϕ : Q ∼ = Q determines a unique k ∗ -isomor ∼ ˆ ˆ phism F(b,G) (Q) = F(b,G) (Q ) induced by some x ∈ G such that Q = Qx , (Q, f )x ⊂ (P, e) and ϕ(u) = ux for any u ∈ Q . Proof: By condition 3.2.2, there is such an x ∈ G and, setting f = f x , it is clear that x induces isomorphisms compatible with the respective actions NG (Q, f ) ∼ = NG (Q , f ) kCG (Q)f J kCG (Q)f ∼ = kCG (Q )f J kCG (Q )f

7.5.1;

ˆG (Q, f ) ∼ ˆG (Q , f ) and a commutahence, x induces a k ∗ -isomorphism N =N tive diagram ˆG (Q, f ) −→ Fˆ(b,G) (Q) −→ 1 CG (Q) −→ N + + + ˆG (Q , f ) −→ Fˆ(b,G) (Q ) −→ 1 −→ CG (Q ) −→ N

1 −→ 1

7.5.2;

moreover, if x ∈ G is a second element fulﬁlling the same condition, the diﬀerence x x−1 centralizes Q and therefore it induces the identity map on Fˆ(b,G) (Q) .

7. Nilcentralized and selfcentralizing Brauer pairs

95

7.6 Recall that a maximal Brauer (f, Q·CG (Q))-pair is also a Brauer (b, G)-pair (cf. 1.10) and thus we may assume that (P, e) contains a maximal Brauer (f, Q·CG (Q))-pair (cf. 1.15). Denote by n the block of CG NP (Q) such that (P, e) contains the Brauer (b, G)-pair (NP (Q), n) (cf. 1.15) and always assume that (Q, f ) is nilcentralized. Then, since f is also a block of CG (Q)·NP (Q) (cf. 1.19), it is easily checked that (NP (Q), n) is also a maximal Brauer (f, CG (Q)·NP (Q))-pair (cf. 1.16) and, according to Proposition 6.5 in [34], the block (f, CG (Q)·NP (Q)) is nilpotent too. 7.7 Now, it follows from the Main Theorem in [41] that, denoting by ν the local point of NP (Q) determined by n on the NP (Q)-algebra kCG (Q)f (cf. 1.19), the quotient SQ = kCG (Q)f ν J kCG (Q)f ν

7.7.1

¯P (Q)-algebra (cf. 1.11 and 1.20) which, in particular, is a primitive Dade N ¯P (Q) (cf. 1.21); freely determines an element sQ in the Dade group Dk N using our results in chapter 8 below — namely, some statement depending on results of [17] — we can be more precise on the nature of sQ , but we will not need this precision in the sequel. Let us denote by C the ﬁeld of complex numbers, by κ the conjugation automorphism of C and by κG : CG ∼ = (CG)◦ −1 the κ-anti-isomorphism mapping any x ∈ G on x . Theorem 7.8 With the notation above, sQ is a torsion element in the Dade ¯P (Q) . group Dk N Proof: Let O be an unramiﬁed complete discrete valuation ring of characteristic zero lifting k ; without loss of generality, we may assume that k is the algebraic closure of the prime ﬁeld and, in particular, that O is contained in C ; let us denote by Oκ the conjugate of O , and set R = O ∩ Oκ so that R is a Dedekind ring. Set T = NP (Q) and H = CG (Q)·T , and as above denote by n the block of CG (T ) such that (P, e) contains (T, n) . Let fˇ be the unique idempotent lifting f to Z(OH) ; since the coeﬃcients of fˇ in the canonical O-basis of OH are contained in the extension of Z(p) by the p -roots of unity (cf. [8]), fˇ still belongs to Z(RH) ; then, it is easily checked that fˇκ remains a primitive idempotent in Z(OH) — O-block of H in the sequel — so that its image f κ in Z(kH) is again a block of H . Similarly, the Brauer (f κ , H)-pairs are the pairs (R, g κ ) where (R, g) runs over the set of Brauer (f, H)-pairs and therefore f κ is nilpotent too. In particular, (T, nκ ) is a maximal Brauer (f κ , H)-pair. It is well-known (cf. Lemma 3.13 in [12] or 7.11 in [41]) that the unique simple kHf -module comes from an ordinary absolutely irreducible character

96

Frobenius categories versus Brauer blocks

χ in the O-block fˇ and then, denoting by iχ the corresponding primitive idempotent in Z(CH) , the image of the canonical O-algebra homomorphism ρχ : OH fˇ −→ CHiχ

7.8.1

is isomorphic to the matrix algebra Mχ(1) (O) ; but, by the argument above, fˇ is also an Oκ -block of H and χ still belongs to this Oκ -block; thus, denoting by ρκχ : Oκ H fˇ −→ CHiχ the corresponding Oκ -algebra homomorphism and setting M = Ker(ρχ ) ∩ Ker(ρκχ ) 7.8.2, M is an ideal of RH fˇ such that RH fˇ = RH fˇ/M maps into CHiχ and we have O ⊗R RH fˇ ∼ = Mχ(1) (O)

and Oκ ⊗R RH fˇ ∼ = Mχ(1) (Oκ )

7.8.3.

From now on, we consider RH fˇ as an RH-module by the multiplication on the left; then, denoting by kκ the ﬁeld k endowed with the composed homomorphism κ R −→ R ⊂ O −→ k 7.8.4, it follows from the very deﬁnition of M that the kH-modules k ⊗R RH fˇ and kκ ⊗R RH fˇ are just the respective χ(1)-multiples of the simple kHf and kHf κ -modules. More precisely, considering the unique idempotent n ˇ lifting n to Z OCG (T ) , we restrict ourselves to the RT -submodule n ˇ RH fˇ ˇ which is a direct summand of ResH T (RH f ) . In particular, the endopermutation kT -module V involved in the Dade T -algebra SQ (cf. 1.20 and 7.7.1) is an indecomposable direct summand of the kT -module k ⊗R n ˇ RH fˇ and any indecomposable direct summand of vertex T of this kT -module is isomorphic to V (cf. 1.21). It is clear that κH (RH) = (RH)◦ and in particular κH induces an RH-module κ-isomorphism RH ∼ = (RH)◦ ; it is well-known that κH (fˇ) = fˇ , κH (iχ ) = iχ and κH (ˇ n) = n ˇ , and thus it is easily checked that the action of κH on CHiχ stabilizes the image of n ˇ RH fˇ ; in particular, n ˇ RH fˇ is a $κT %-stable RT -module. At the same time, it is clear that the R-algebras RH fˇκ and (RH fˇ )◦ are isomorphic, so that we have a kH-module isomorphism kκ ⊗R RH fˇ ∼ 7.8.5; = (k ⊗R RH fˇ )∗ in particular, the k-dual kT -module V ∗ is an indecomposable direct summand of the kT -module kκ ⊗R n ˇ RH fˇ and any indecomposable direct summand of vertex T of this kT -module is isomorphic to V ∗ .

7. Nilcentralized and selfcentralizing Brauer pairs

97

On the other hand, it follows from Propositions 8.11 and 8.18, and Corollary 8.22 below that, for a suitable n ∈ N , we have pn ·sQ = zU ·tenTU (r1U ) 7.8.6, U

where U runs over the set of nontrivial subgroups of T and, for such an U , zU is an integer. Moreover, for any nontrivial subgroup U of T , mimicking the deﬁnition of r1U (cf. 8.17.2) consider the RU -module µ(U/R) NU = Res πRU I R(U/R) 7.8.7, R

where R runs over the set of subgroups of U containing Φ(U ) (cf. 1.31)and, U for such an R , πR : U → U/R denotes the canonical map, I R(U/R) the augmentation ideal of R(U/R) , µ(U/R) the value of the M¨ obius function (cf. 1.32) over the set of proper subgroups of U strictly containing R , and µ(U/R) the µ(U/R)-th tensor power over R , with the conventional rule that the 0-th and the (−1)-th tensor powers of an R-torsion-free R(U/R)-module are R and the R-dual of this module respectively. Note that NU is R-torsionfree and $κU %-stable as RU -module. Consequently, the RT -module zU N= 7.8.8, TenTU NU U

where U runs over the set of nontrivial subgroups of T , is R-torsion-free and $κT %-stable as RT -module, and the scalar extensions k ⊗R N and kκ ⊗R N are mutually isomorphic endopermutation kT -modules. Moreover, according to equality 7.8.6, the tensor product pn pn (k ⊗R N )∗ ⊗k (k ⊗R n ˇ RH fˇ) ∼ n ˇ RH fˇ ) 7.8.9 = k ⊗R (N ∗ ⊗R admits the trivial kT -module k as a direct summand. Now, it follows from Lemma 7.10 below applied to the R-torsion-free pn $κT %-stable RT -module N ∗ ⊗R n ˇ RH fˇ that the tensor product pn pn (kκ ⊗R N )∗ ⊗k (kκ ⊗R n ˇ RH fˇ) ∼ n ˇ RH fˇ ) 7.8.10 = kκ ⊗R (N ∗ ⊗R also admits the trivial kT -module k as a direct summand and therefore the class of the Dade T -algebra Endk (V ∗ ) coincides with the class of the Dade T -algebra Endk (k ⊗R N ) ∼ 7.8.11, = Endk (kκ ⊗R N ) which by construction coincides with pn ·sQ ; that is to say, we get 2pn ·sQ = 0 . We are done. Remark 7.9 By 8.16 and Corollary 8.22 below, the torsion subgroup of ¯P (Q) is a 2-group and, if p = 2 , it is elementary Abelian. Dk N

98

Frobenius categories versus Brauer blocks

Lemma 7.10 With the notation above, if M is an R-torsion-free $κT %-stable RT -module then the trivial kT -module k is a direct summand of k ⊗R M if and only if it is a direct summand of kκ ⊗R M . ¯ = M/p·M and, following Green, consider the $κ%-stable R-moProof: Set M ¯ dule M [T ] deﬁned by the exact sequence ¯ ]T −→ M ¯ T −→ M ¯ [T ] −→ 0 0 −→ [T, M

7.10.1; ∼ since R/p·R = k × kκ , it is clear that we still have the exact sequences ¯ ]T −→ (k ⊗R M ¯ )T −→ k ⊗R M ¯ [T ] −→ 0 0 −→ [T, k ⊗R M 7.10.2, ¯ ]T −→ (kκ ⊗R M ¯ )T −→ kκ ⊗R M ¯ [T ] −→ 0 0 −→ [T, kκ ⊗R M so that, considering the analogous over k of deﬁnition 7.10.1, we get ¯ [T ] and (kκ ⊗R M )[T ] ∼ ¯ [T ] 7.10.3; (k ⊗R M )[T ] ∼ = k ⊗R M = kκ ⊗R M but, if k ⊗R M admits the trivial kT -module k as a direct summand, we have 7.10.4; (k ⊗R M )[T ] = {0} ¯ [T ] = {0} and, since M ¯ [T ] is a $κ%-stable R-module, consequently, we have M it follows from the right isomorphism in 7.10.3 that we still have (kκ ⊗R M )[T ] = {0} 7.10.5. T ¯ ) − [T, kκ ⊗R M ¯ ] , it is quite clear Then, for any element 1 ⊗ n in (kκ ⊗R M ¯ . We that k·(1 ⊗ n) is a trivial direct summand in the kT -module kκ ⊗R M are done. 7.11 With the notation in 7.6 and 7.7, it follows from Theorem 6.6 in [34] that we have an NP (Q)-algebra isomorphism (cf. 1.18.1) kCG (Q)f ∼ 7.11.1 = SQ CP (Q) ν

and, considering the unique group homomorphism CP (Q) → (SQ )∗ lifting the action of CP (Q) on SQ , as usual we get a new NP (Q)-algebra isomorphism (cf. (1.8.1) in [41]) kCG (Q)f ν ∼ 7.11.2. = SQ ⊗k kCP (Q) Recall that, if Q is a subgroup of P fully normalized in F(b,G) and F(b,G) -isomorphic to Q , there is an F(b,G) -morphism NP (Q) → NP (Q ) mapping Q onto Q (cf. condition 2.8.2 and Theorem 3.7). Proposition 7.12 Let Q and Q be F(b,G) -isomorphic F(b,G) -nilcentralized subgroups of P and assume that Q is fully normalized in F(b,G) . Then, for any F(b,G) -morphism ζ : NP (Q) → NP (Q ) mapping Q onto Q , we have an ¯P (Q)-algebra embedding SQ → Resζ¯(SQ ) and the equality sQ = resζ¯(sQ ) , N where ¯P (Q ) −→ Dk N ¯P (Q) resζ¯ : Dk N 7.12.1 ¯P (Q) → N ¯P (Q ) is the restriction map determined the homomorphism ζ¯ : N induced by ζ .

7. Nilcentralized and selfcentralizing Brauer pairs

99

Proof: Respectively denote by n and n the blocks of CG NP (Q) and of CG NP (Q ) such that (P, e) contains (NP (Q), n) and (NP (Q ), n ) (cf. 1.15); by condition 3.2.2, there is x ∈ G such that (NP (Q), n)x ⊂ (NP (Q ), n ) , Qx = Q and ux = ζ(u) for any u ∈ NP (Q) ; moreover, setting f = f x and denoting by ξ : NG (Q, f ) ∼ = NG (Q , f ) the isomorphism determined by x , it is clear that x induces an NG (Q, f )-algebra isomorphism kCG (Q)f ∼ 7.12.2; = Res ξ kCG (Q )f in particular, denoting by ν and ν the respective local points of NP (Q) and NP (Q ) determined by n and n on the NP (Q)- and NP (Q )-algebras ¯P (Q)-algebra embedding kCG (Q)f and kCG (Q )f (cf. 1.19), we get an N kCG (Q)f ν −→ Res ζ kCG (Q )f ν 7.12.3. ¯P (Q)-algebra embedding and equalConsequently, we get the announced N ity (cf. 1.21). 7.13 Let R be an F(b,G) -nilcentralized subgroup of Q ; in the Dade groups ¯P (Q) and Dk N ¯P (R) we have the respective elements sQ and sR ; Dk N then, setting NQ,R = NP (Q)∩NP (R) , we have evident homomorphisms from ¯Q,R = NQ,R /R to N ¯P (Q) and to N ¯P (R) , and we may ask ourthe quotient N ¯Q,R ) ; selves for the relationship between the restrictions of sQ and sR to DK (N if R is normal in Q , we can give an easy answer in terms of the homomorphism induced by the Brauer quotient functor (cf. 1.22) ¯ N

¯ Q,R : Dk (N ¯Q,R ) −→ Dk (N ¯Q,R /Q) ¯ ⊂ Dk (N ¯Q,R ) Br Q

7.13.1,

¯ the image of Q in N ¯Q,R and identify Dk (N ¯Q,R /Q) ¯ where we denote by Q ¯ with its image in Dk (NQ,R ) by the restriction homomorphism. Theorem 7.14 Let Q and R be F(b,G) -nilcentralized subgroups of P such that ¯ the respective images ¯Q,R and Q Q contains and normalizes R , denote by N ¯ ¯P (R) , and set N ¯ ¯ of NQ,R = NP (Q) ∩ NP (R) and Q in N Q,R = NQ,R /Q . Then, we have ¯ (Q) N

resN¯P

Q,R

¯Q,R N

¯ (sQ ) = Br Q

¯ (R) N resN¯PQ,R (sR )

7.14.1.

Proof: Denote by f and g the respective blocks of CG (Q) and CG (R) such that (P, e) contains (Q, f ) and (R, g) ; in particular, we have (R, g) B (Q, f ) kC (R) (cf. 1.15) and therefore we get f BrQ G (g) = f (cf. Theorem 1.8 in [11]). As above, respectively denote by n and m the blocks of CG NP (Q) and CG (NQ,R ) such that (P, e) contains (NP (Q), n) and (NQ,R , m) (cf. 1.15).

100

Frobenius categories versus Brauer blocks

Since we have CG (NQ,R ) ⊂ CG (Q) ⊂ CG (R) , (NQ,R , m) is also a Brauer (f, NQ,R ·CG (Q))-pair and a Brauer (g, NQ,R ·CG (R))-pair, and therefore m ¯ ¯ ¯ respectively determines local points ν of N Q,R and µ of NQ,R on the N Q,R ¯P (Q) ¯P (R) N N ¯Q,R -algebras Res ¯ and N kCG (Q)f and Res ¯ kCG (R)g (cf. 1.19), ¯ Q,R N

NQ,R

¯ ¯ so that we have N Q,R - and NQ,R -algebra isomorphisms (cf. 7.11.2) kCG (Q)f ν ∼ = (SQ )ν ⊗k kCP (Q) kCG (R)g µ ∼ = (SR )µ ⊗k kCP (R)

7.14.2,

¯ where we identify ν and µ with the respective unique local points of N Q,R ¯Q,R on SR (cf. Corollary 5.8 in [41]). on SQ and of N ¯Q,R -stable bases that we have Moreover, it follows from the existence of N ¯ an N Q,R -algebra isomorphism (cf. Proposition 5.6 in [41]) ¯ ∼ ¯ ⊗k kCP (Q) kCG (R)g µ (Q) 7.14.3 = (SR )µ (Q) ¯ and that BrSQ¯R (µ) is the local point of N Q,R on SR (Q) (cf. Corollary 5.8 in [41]), so that ¯ ∼ (SR )µ (Q) 7.14.4. = SR (Q)BrSR (µ) ¯ Q

¯ ¯ Since the Dade N Q,R - and NQ,R -algebras (SQ )ν and (SR )µ respectively de¯ ¯P (R) NP (Q) N ¯ ¯Q,R ) (cf. 1.22), termine res ¯ (sQ ) in Dk (N (sR ) in Dk (N Q,R ) and res ¯ ¯ Q,R N

NQ,R

equality 7.14.1 follows from isomorphism 7.14.4 above and Proposition 7.12. 7.15 A fundamental result on Dade P -algebras (cf. 9.2) allows us to answer the analogous of the question above on the k ∗ -groups Fˆ(b,G) (Q) introduced in 7.4. As above, let R be an F(b,G) -nilcentralized subgroup of Q and denote by F(b,G) (Q)R the stabilizer of R in F(b,G) (Q) ; it follows from the divisibility of F(b,G) (cf. Proposition 3.3) that the restriction induces a group homomorphism F(b,G) (Q)R −→ F(b,G) (R) 7.15.1; but the k ∗ -group Fˆ(b,G) (Q) above determines a k ∗ -subgroup Fˆ(b,G) (Q)R ; thus, we may ask for the relationship between Fˆ(b,G) (Q)R and Fˆ(b,G) (R) and, in the next result, the answer is that they agree. Actually, in chapter 11 this result will appear as a consequence of a more general framework. Theorem 7.16 If Q and R are F(b,G) -nilcentralized subgroups of P such that Q contains R , the restriction group homomorphism F(b,G) (Q)R → F(b,G) (R) can be lifted to a k ∗ -group homomorphism Fˆ(b,G) (Q)R −→ Fˆ(b,G) (R) and the kernel is a p-group.

7.16.1

7. Nilcentralized and selfcentralizing Brauer pairs

101

Proof: The last statement follows from Corollary 4.7. Arguing by induction on |Q : R| and setting N = NQ (R) , if R is not normal in Q then we may assume that we have k ∗ -group homomorphisms Fˆ(b,G) (Q)N −→ Fˆ(b,G) (N ) and Fˆ(b,G) (N )R −→ Fˆ(b,G) (R)

7.16.2

lifting the corresponding restriction homomorphisms; then, since F(b,G) (Q)R is contained in F(b,G) (Q)N and moreover its image in F(b,G) (N ) is still contained in F(b,G) (N )R , we get the announced lifting by composition. In the sequel, we assume that Q normalizes R ; then, by Proposition 2.7, we may assume that R is fully normalized in F(b,G) . Denote by f and g the respective blocks of CG (Q) and CG (R) such that (P, e) contains (Q, f ) and (R, g) (cf. 1.15), set A = kGb and consider the local points δ of Q and ε of R on A respectively determined by f and g (cf. 1.13 and 1.18)). The existence of the lifting 7.16.1 will follow from the coincidence of the k ∗ -group ˆG (Q, f ) structures on the intersection NG (Q, f ) ∩ NG (R, g) , coming from N ˆG (R, g) ; actually, it suﬃces to prove this coincidence over any and from N p -subgroup K of this intersection or, equivalently, to prove that the central extension of K by k ∗ which is the diﬀerence of the respective extensions ˆG (Q, f ) and N ˆG (R, g) is split. coming from N We borrow our notation from section 10 in [42]; more precisely, let K be a p -subgroup of F(b,G) (Q)R and denote by K δ and by K ε the respective extensions coming from Fˆ(b,G) (Q)R and from Fˆ(b,G) (R) via the restriction ◦

map F(b,G) (Q)R → F(b,G) (R) , and by K ε δ their diﬀerence; then, it follows from Proposition 10.13 in [42] that we have a k ∗ -group homomorphism ◦

ρ : Kε

δ

−→ Aδ (Rε )∗

7.16.3;

¯ = Q/R acts on the simple k-algebra moreover, since Q ﬁxes g (cf. 1.15), Q ¯ → Aδ (Rε )∗ Aδ (Rε ) (cf. 1.18); thus, we have a unique group homomorphism Q ε◦ δ lifting this action and then it is easily checked that ρ(K ) normalizes the ¯. image of Q On the other hand, we know that A(R) ∼ = kCG (R)BrR (b) (cf. 1.19) ¯ on A(R) ; then, according and that BrR (δ) still remains a local point of Q ¯ to 7.11.2, it is not diﬃcult to get the Q-algebra isomorphisms Aδ (R) ∼ = A(R)BrR (δ) ∼ = (SR )BrR (δ) ⊗k CP (R)

7.16.4,

¯P (R)-algebra introduced in 7.7.1 above and where SR denotes the Dade N ¯ on SR (cf. Corolwhere we identify BrR (δ) with the unique local point of Q ¯ lary 5.8 in [41]). Consequently, we ﬁnally get a Q-algebra isomorphism Aδ (Rε ) ∼ = (SR )BrR (δ)

7.16.5.

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Frobenius categories versus Brauer blocks

¯ But, it follows from the splitting theorem for Dade Q-algebras (cf. The∗ orem 9.21 below or [40]) that we have a k -group homomorphism ˆS (Q) ¯ −→ k ∗ ωQ,S ¯ R :F R

7.16.6

¯ (cf. 1.29), it is clear that and, by the very deﬁnition of the k ∗ -group FˆSR (Q) ◦ ¯ . We are done. ρ determines a k ∗ -group homomorphism K ε δ → FˆSR (Q)

Chapter 8

Decompositions for Dade P-algebras 8.1 Let P be a nontrivial ﬁnite p-group and k an algebraically closed ﬁeld of characteristic p . In the previous chapter we have seen that the Dade P -algebras over k (cf. 1.20) appear in a natural way when dealing with blocks. In order to handle them easily, in this chapter we show that any Dade P -algebra over k raised to some pn -th tensor power admits — up to similarity (cf. 1.21) — a canonical decomposition in terms of the tensor product of suitable tensor induced Dade Q-algebras over k from the subgroups Q of P . 8.2 More generally, if G is a ﬁnite group, H a subgroup of G and B an H-algebra over k (cf. 1.11), recall that, following Serre, the tensor induction of B from H to G is the G-algebra (cf. A2.2 in [48]) TenG H (B) =

(kC ⊗kH B)

8.2.1,

C∈G/H

where kC denotes the k-vector space over the class C ⊂ G , endowed with the (right) kH-module structure determined by the multiplication on the right, where we consider B as a kH-module via the action of H , and where the product and the action of G are deﬁned by the formulæ C∈G/H

(xC ⊗ aC ) (xC ⊗ aC ) = (xC ⊗ aC aC ) C∈G/H

C∈G/H

x (xC ⊗ aC ) = (x−1 xC ⊗ aC )

C∈G/H

8.2.2,

C∈G/H

for any xC ∈ C , any aC , aC ∈ B and any x ∈ G . 8.3 Note that we have an injective H-algebra homomorphism B G tH

G : B −→ ResG H TenH (B)

mapping a ∈ B on B G tH (a)

=

(xC ⊗ aC )

8.3.1

8.3.2

C∈G/H

where xH = 1 , aH = a and aC = 1 for any C ∈ G/H − {H} ; we write tG H (a) (a) for short, and note that, for any x ∈ G − H , the elements instead of B tG H G x tG H (a) and tH (a) centralize each other. Thus, for any subgroup K of G ,

104

Frobenius categories versus Brauer blocks

any a ∈ B K∩H and any set of representatives X ⊂ K for the set of classes (K ∩ H)\K , the product G x NrK tG H (a) K∩H tH (a) =

8.3.3

x∈X

does not depend neither on the choice of X nor on the choice of a total order K on X, and it is clearly an element of TenG H (B) . 8.4 All this construction is clearly functorial on B ; that is to say, any H-algebra homomorphism g : B → B determines a G-algebra homomorphism G G TenG 8.4.1, H (g) : TenH (B) −→ TenH (B ) this correspondence is compatible with the composition of homomorphisms and the H-algebra homomorphisms B tG H are natural on B . Moreover, we have a canonical P -algebra isomorphism G G ∼ TenG H (B ⊗k B ) = TenH (B) ⊗k TenH (B )

8.4.2

B G B G compatible with B⊗k B tG H and tH ⊗ tH . Furthermore, for any subgroup L of G containing H , we have a natural G-algebra isomorphism

L G ∼ TenG L TenH (B) = TenH (B)

8.4.3.

On the other hand, as in the ordinary induction, we have an evident Mackey decomposition; namely, for any subgroup K of G , there is a natural K-algebra isomorphism G ∼ ResG TenK K TenH (B) = K∩H x Resκx (B)

8.4.4,

x∈X

where X ⊂ G is a set of representatives for H\G/K and, for any x ∈ X , κx : K ∩ H x → H is the group homomorphism induced by the conjugation by x . 8.5 Let us come back to the the ﬁnite p-groups; for any P -algebra A over k and any σ ∈ Aut(k) , we set σA = Resσ (A) ; moreover, we denote by φ:k ∼ = k the Frobenius automorphism of k ; as a matter of fact, the Frobenius automorphism appears when computing the Brauer quotients (cf. 1.13) of a tensor induction†. †

When proving Proposition 8.6 in the eighties we were misled on the behaviour of the scalar

coeﬃcients via homomorphism 8.6.1 — we just forgot to check it — and it was Serge Bouc, ten years later, who mention to us that this behaviour involves the Frobenius automorphism. His independent version of Proposition 8.6 for Q-algebras B with a Q-stable basis appears in [6].

8. Decompositions for Dade P -algebras

105

Proposition 8.6 Let Q and R be subgroups of P and B a Q-algebra. Assume that R is normal in P , set P¯ = P/R , |R| = pvR and |Q∩R| = pvQ∩R , denote ¯ the image of Q in P¯ , and consider B(Q ∩ R) as a Q-algebra. ¯ by Q We have a unique P¯ -algebra homomorphism vR B(Q ∩ R) −→ φ TenP Q (B) (R)

¯ φvQ∩R

TenP ¯ Q

8.6.1

P ¯ R mapping tP for any a ∈ B Q∩R , which ¯ BrQ∩R (a) on BrR NrQ∩R tQ (a) Q is an isomorphism whenever Q stabilizes a basis of B . Proof: Firstly assume that R is contained in Q ; on the one hand, in that case the inclusion B R ⊂ B induces a P¯ -algebra homomorphism ¯ P P P R ∼ R R TenP ¯ (B ) = TenQ (B ) −→ TenQ (B) −→ TenQ (B) (R) Q

8.6.2

P ¯ R mapping tP ¯ (a) on BrR tQ (a) for any a ∈ B ; on the other hand, it is easily Q checked that the kernel of the canonical map ¯ P¯ P¯ B R TenP ¯ (BrR ) : TenQ ¯ (B ) −→ TenQ ¯ B(R) Q

8.6.3

¯ B u R is generated by {tP ¯ Ker(BrR ) }u∈P . But, if a = TrT (c) for a proper subQ R P group T of R and an element c ∈ B T , we have tP Q (a) = TrT tQ (c) since R ⊂ Q ; consequently, homomorphism 8.6.2 determines a P¯ -algebra homomorphism ¯ P TenP 8.6.4 ¯ B(R) −→ TenQ (B) (R) Q P ¯ R mapping tP ¯ BrR (a) on BrR tQ (a) for any a ∈ B . Moreover, starting Q from a Q-stable basis of B , it is not diﬃcult to obtain bases of both ends and to check that this homomorphism induces a bijection between them. By isomorphism 8.4.3, now it suﬃces to discuss the case where P = Q·R ; setting T = Q ∩ R , isomorphism 8.4.4 yields the R-algebra isomorphism Q P R ∼ ResP R TenQ (B) = TenT ResT (B)

8.6.5

and, up to the obvious identiﬁcation, we claim that the map φvT

vR

B T −→ φ

TenP Q (B) (R)

8.6.6

P ¯ sending a ∈ B T to BrR NrR is a Q-algebra homomorphism. T tQ (a) Indeed, for any a, a ∈ B T , we have P P tP Q (a + a ) = tQ (a) + tQ (a )

8.6.7

106

Frobenius categories versus Brauer blocks

and, choosing a set of representatives X ⊂ R for R/T , we get P NrR u ⊗ (a + a ) T tQ (a + a ) = u∈X

=

Y ⊂X

u⊗a ⊗

u ⊗ a

8.6.8;

u ∈X−Y

u∈Y

but, the tensor products aY = u ⊗ a and aX−Y =

u ⊗ a

8.6.9

u ∈X−Y

u∈Y

only depend on the image of Y in R/T and for any w ∈ R we have (cf. 8.2.2) (aY ⊗ aX−Y )w = aw−1 ·Y ⊗ aw−1 ·X−w−1 ·Y

8.6.10;

moreover, the stabilizer in R of this image by left multiplication is a proper subgroup of R unless we have Y = ∅ or Y = X ; hence, by the very deﬁnition of the Brauer quotient (cf. 1.13) we ﬁnally get P BrR NrR T tQ (a + a ) 8.6.11. P P = BrR NrR + BrR NrR T tQ (a) T tQ (a ) Furthermore, it is quite clear that P R P R P NrR T tQ (aa ) = NrT tQ (a) NrT tQ (a )

8.6.12

and that, for any λ ∈ k ∗ , we get P P pvR −vT NrR ·NrR T tQ (λ·a) = λ T tQ (a)

8.6.13.

Finally, for any w ∈ Q , we obtain (cf. 8.2.2) P w NrR u ⊗ aw T tQ (a ) = u∈X

=

w P w w·u·w−1 ⊗ a = NrR T tQ (a)

8.6.14,

u∈X

which completes the proof of the claim. On the other hand, if we have a = TrTU (c) for a proper subgroup U of T and an element c ∈ B U , considering the canonical map R/U → R/T , we get P −1 w NrR c = u ⊗ u s(u) c u⊗ T tQ (a) = u∈X

w

s

u∈X

8.6.15,

8. Decompositions for Dade P -algebras

107

where w runs over a set of representatives for T /U in T and s : R/T → R/U runs over the sections of the canonical map, once we have identiﬁed R/T with X and R/U with some set of representatives Y so that u−1 s(u) belongs to T for any u ∈ X . It is clear that R acts on R/T and R/U by the multiplication on the left, which is compatible with the canonical map; hence, R still acts on the set of sections of the canonical map and, for any v ∈ R , we have (cf. 8.2.2)

u⊗u

−1

s(u)

v −1 c = v −1 u ⊗ u s(u) c

u∈X

u∈X

=

u ⊗ u

−1 v

s (u )

8.6.16; c

u ∈X

consequently, choosing a set of representatives S for the set of R-orbits in the set of sections of the canonical map and, for any s ∈ S , denoting by Rs the stabilizer of s in R , equality 8.6.15 becomes P R −1 NrR TrRs u ⊗ u s(u) c T tQ (a) = s∈S

8.6.17

u∈X

P which proves that BrR NrR = 0 since R ﬁxes no section. T tQ (a) In conclusion, homomorphism 8.6.6 induces a P¯ -algebra homomorphism φv T

vR

B(T ) −→ φ

TenP Q (B) (R)

8.6.18.

As above, starting from a Q-stable basis of B , it is not diﬃcult to obtain bases of both ends and to check that this homomorphism induces a bijection between them. We are done. 8.7 In particular, if Q is a subgroup of P and S a Dade Q-algebra, it is P quite clear that P stabilizes a basis of TenP Q (S) and that TenQ (S) is a simple k-algebra, and it is easily checked from Proposition 8.6 that

TenP Q (S) (P ) = {0}

8.7.1;

that is to say, TenP Q (S) is a Dade P -algebra. Clearly, for any kQ-module N , we have an analogous deﬁnition for the tensor induction TenP Q (N ) , and a P -algebra isomorphism P ∼ TenP Q Endk (N ) = Endk TenQ (N )

8.7.2;

then, if N has a Q-stable basis, TenP Q (N ) has a P -stable basis too. Thus, according to isomorphism 8.4.2 above, for any Dade Q-algebra S similar

108

Frobenius categories versus Brauer blocks

P to S (cf. 1.21), TenP Q (S ) is also similar to TenQ (S) . That is to say, the tensor induction from Q to P determines a map between the Dade groups (cf. 1.21) tenP 8.7.3 Q : Dk (Q) −→ Dk (P )

which, according again to isomorphism 8.4.2, is a group homomorphism. 8.8 These group homomorphisms inherit the properties of the tensor induction; namely, for any pair of subgroups Q and R of P , we have the Mackey formula P resP tenR 8.8.1, R ◦ tenQ = R∩Qu ◦ resκu u∈X

where X ⊂ P is a set of representatives for Q\P/R and, for any u ∈ X , κu : R ∩ Qu → Q is the group homomorphism induced by u (cf. 1.8.1); also, whenever Q contains R we get (cf. 8.4.3) Q P tenP Q ◦ tenR = tenR

8.8.2.

Moreover, if R is normal in P , it follows from Proposition 8.6 that, respectively denoting by vR and vQ∩R the p-valuations of R and Q ∩ R , we have (cf. 1.22) Q P vQ∩R vR ¯ ◦ tenP tenP¯ ◦ φ = φ Br 8.8.3, Br Q

Q∩R

R

Q

where we also denote by φ the automorphism of Dk (P¯ ) determined by the Frobenius automorphism (cf. 8.5). 8.9 As we did in [42, §3] with the ordinary induction here, for the tensor induction, also we consider the residual Dade groups R◦ Dk (P ) =

P P Ker(resQ ) and RDk (P ) = Dk (P ) Im(tenQ ) 8.9.1

Q

Q

where Q runs over the set of proper subgroups of P . We denote by ◦ tdP P : R Dk (P ) −→ Dk (P ) and

rdP P : Dk (P ) −→ RDk (P )

8.9.2

the canonical maps and, more generally, for any subgroup Q of P , we set Q P tdP Q = tenQ ◦ tdQ

and

Q P rdP Q = rdQ ◦ resQ

8.9.3.

8.10 These two families of maps determine two group homomorphisms Q

P P P tdP td R◦ Dk (Q) −→ Dk (P ) and Dk (P ) −→ RDk (Q) Q

8.10.1,

8. Decompositions for Dade P -algebras

109

where Q runs over the set of subgroups of P , which actually have p-torsion kernels and cokernels; namely, denoting by Zp the ring Z localized over the set {pn }n∈N of powers of p , and setting Zp ⊗Z Dk (P ) = Dk (P )p

8.10.2,

the corresponding homomorphisms are isomorphisms, as we show below. Denote by su the obvious image of s ∈ Dk (Q)p in Dk (Qu )p for any u ∈ P . Proposition 8.11 The maps rdP and tdP induce Zp -module isomorphisms

R◦ Dk (Q)p

P

∼ = Dk (P )p ∼ =

Q

RDk (Q)p

P 8.11.1

Q

where Q runs over the set of subgroups of P . In particular, for any subgroup Q ◦ ∼ of P , rdQ Q induces an isomorphism R Dk (Q)p = RDk (Q)p and, denoting by P PQ : Dk (P )p −→ R◦ Dk (Q)p ⊂ Dk (Q)p

8.11.2

the composition of rdP Q with the inverse of this isomorphism, for any element s ∈ Dk (P )p we have |Q| P s= 8.11.3, ·tenP Q PQ (s) |P | Q

where Q runs over the set of all the subgroups of P . Proof: First of all, if Q and R are subgroups of P and s is an element of R◦ Dk (Q) , it follows from the Mackey formula (cf. 8.8.1) that P R u rdP rdR (s ) R tenQ (s) =

8.11.4,

u

where u ∈ P runs over a set of representatives for the set of classes {u ∈ P | Qu = R}/R ⊂ Q\P/R

8.11.5.

Now, we claim that it suﬃces to prove the isomorphism R◦ Dk (P )p ∼ = RDk (P )p

8.11.6.

Indeed, on the one hand it follows from equality 8.11.4 that for any P

◦ element Q sQ of , where sQ ∈ R◦ Dk (Q)p , we get Q R Dk (Q)p

(rdP )p ◦ (tdP )p

Q

sQ = |P/Q|·(rdQ Q )p (sQ ) Q

8.11.7

110

Frobenius categories versus Brauer blocks

where Q runs over the set of subgroups of P , so that (tdP )p is injective and (rdP )p is surjective. On the other hand, once isomorphism 8.11.6 is proved, by applying this isomorphism to any subgroup Q of P , we get Dk (Q)p = R◦ Dk (Q)p + tenQ 8.11.8, R Dk (R)p R

where R runs over the set of proper subgroups of Q , and therefore, arguing by induction on |Q| , we still get ◦ Dk (P )p = 8.11.9, tenP Q R Dk (Q)p Q

where Q runs over the set of subgroups of P ; moreover, if s = in Dk (P )p , where sQ ∈ R◦ Dk (Q)p , we still have s=

−1 1 tenP (sQu )u · Q |P | Q

Q

tenP Q (sQ )

8.11.10,

u∈P

so that s belongs to the image of (tdP )p , proving the claim. Let s be an element of the kernel of the map P ◦ (rdP 8.11.11; P )p ◦ (tdP )p : R Dk (P )p → RDk (P )p P in particular, s belongs to Q tenQ Dk (Q)p where Q runs over the set of proper subgroups of P ; arguing by induction on |Q| and applying again P equality 8.11.10, we get s = R tenR (sR ) where R runs over the set of proper subgroups of P and, for such an R , sR belongs to R◦ Dk (R)p and fulﬁlls (sR )u = sRu for any u ∈ P ; hence, by equality 8.11.4, we get Q ¯ 0 = (rdP Q )p (s) = |NP (Q)|·(rdQ )p (sQ )

8.11.12

and therefore, applying again the induction hypothesis, we obtain sQ = 0 . In order to prove the surjectivity of the map 8.11.11, it suﬃces to prove ◦ P that, for any s ∈ Dk (P )p , (rdP P )p (s) belongs to (rdP )p R Dk (P )p . Let Q be the set of subgroups Q of P such that resP Q (s) = 0 ; we argue by induction on |Q| and may assume that |Q| > 1 ; let Q be a minimal element of Q and set P 1 s = s − ¯ 8.11.13; ·tenP Q resQ (s) |NP (Q)| P since Q = P , we have (rdP P )p (s ) = (rdP )p (s) ; moreover, once again by the Mackey formula, for any subgroup R of P we get

P 1 P resP tenR · R (s ) = resR (s) − ¯ R∩Qu resR∩Qu (s) |NP (Q)| u

8.11.14,

8. Decompositions for Dade P -algebras

111

where u ∈ P runs over a set of representatives for Q\P/R , and therefore, by the minimality of Q , it is clear that resP R (s ) = 0 implies the existence of u u u ∈ P such that Q ⊂ R and Q = R ; hence, by our induction hypothesis, ◦ P (rdP P )p (s ) belongs to (rdP )p R Dk (P )p . We are done. Corollary 8.12 For any normal subgroup N of P , any subgroup Q of P ¯ = Q/N , and containing N and any s¯ ∈ Dk (P/N )p , setting P¯ = P/N and Q Q P ¯ ¯ denoting by πN : P → P and πN : Q → Q the canonical maps, we have P¯ |R| P P resπQ PQ P res ·tenQ s) = (¯ s ) ¯ (¯ πN R R N |Q|

8.12.1

R

where R runs over the set of subgroups of Q such that R·N = Q . Proof: It is clear that, for any subgroup R of Q such that R·N = Q , we have ◦ ¯ resQ = {0} 8.12.2 R resπ Q R Dk (Q) N

and therefore we still have (cf. Proposition 8.11) ◦ ¯ ⊂ resπQ R◦ Dk (Q) tenQ R R Dk (R) N

8.12.3,

R

where R runs over the set of subgroups of Q such that R·N = Q . Thus, P (¯ setting s = resπN s) , since (cf. 8.11.3) s¯ =

|Q| ¯ ¯ P¯ ·tenP s) ¯ PQ ¯ (¯ Q ¯ |P | ¯ Q

and s =

|R| R

|P |

P ·tenP R PR (s)

8.12.4,

¯ and R respectively run over the sets of subgroups of P¯ and P , where Q equality 8.12.1 follows from Proposition 8.11 and inclusion 8.12.3 above. Corollary 8.13 For any pair of subgroups Q and R of P such that R ⊂ Φ(Q) ¯ =N ¯P (R) and Q ¯ = Q/R we and Q normalizes R , setting N = NP (R) , N have N Q ¯ N N PQ 8.13.1. ¯ ◦ BrR = BrR ◦ PQ Proof: From equality 8.11.3 applied to Q , for any s ∈ Dk (Q) we get |T | vR N tenN PN (s) ·φ Br R T T |N | T v |T | N T ¯ φ T ∩R = res P Br ·tenN (s) ¯ θ T ∩R T T ,R T |N |

φvR

N

(s) = Br R

T

8.13.2,

112

Frobenius categories versus Brauer blocks

where T runs over the set of subgroups of N and, for such a T , we denote ¯ and by θT,R the canonical isomorphism T¯ ∼ by T¯ its image in N = T /(T ∩ R) ; ¯ forces T = Q , equality 8.13.1 follows from equality 8.11.3 now, since T¯ = Q ¯. applied to Q 8.14 Proposition 8.11 above states that, for any Dade P -algebra S , a suitable pn -th tensor power of S admits, up to similarity, a canonical decomposition as a tensor product of the tensor induction from the subgroups Q of P of a Dade Q-algebra in R◦ Dk (Q) ; thus, we are reduced to the description of RDk (P )p which, as a matter of fact, is very simple since it has at most Zp -rank one. Unfortunately, we have no direct proof of this fact and we need the strong classiﬁcation results in [19], [17] and [6]. 8.15 In the case where P is Abelian, for any subgroup Q of P and any Dade P -algebra S , S(Q) is a Dade P/Q-algebra and therefore, denoting P P S(Q) by πQ : P → P/Q the canonical map, ResπQ is a Dade P -algebra. Consequently, we get a group endomorphism of Dk (P ) P

P P ◦ BrQ : Dk (P ) −→ Dk (P/Q) −→ Dk (P ) = resπQ βQ

8.15.1

which is clearly an idempotent endomorphism; moreover, for another subgroup R of P , it is easily checked that (cf. equality 1.22.3) P P P P P βQ ◦ βR = βQ·R = βR ◦ βQ

8.15.2;

P hence, inductively deﬁning a family {αQ }Q of endomorphisms of Dk (P ) by P P βQ = αR 8.15.3, R

where R runs over the set of subgroups of P containing Q , it is quite easy to P check from equality 8.15.2 that {αQ }Q is a pairwise orthogonal idempotent decomposition of the identity; thus, we get P Dk (P ) = Dk (P ) αQ Q 8.15.4, P/Q P ∼ α Dk (P ) = α Dk (P/Q) Q

1

where Q runs over the set of subgroups of P . 8.16 Recall that, since we have a kP -module isomorphism Endk J(kP ) ∼ 8.16.1, = k ⊕ kP |P |−2 Endk J(kP ) is a Dade P -algebra (cf. 1.20) and, for any nontrivial subgroup Q of P , we have Endk J(kP ) (Q) ∼ 8.16.2; =k

8. Decompositions for Dade P -algebras

113

denote by j1P the element of Dk (P ) determined by this Dade P -algebra and, Q P for any subgroup Q of P , note that resP Q (j1 ) = j1 . When P is Abelian, it is quite clear that j1P belongs to α1P Dk (P ) and Dade’s Classiﬁcation in [19] states that this group is generated by this element; more precisely, he proves: 8.16.3 Dk (Z/2Z) ∼ = {0} . 8.16.4 If P is cyclic and |P | ≥ 3 then α1P Dk (P ) = Z·j1P ∼ = Z/2Z . P P ∼ 8.16.5 If P is Abelian noncyclic then α1 Dk (P ) = Z·j1 = Z . P/Q

P P (j 8.17 In this case, we set jQ = resπQ ) for any subgroup Q of P 1 and, by the equality and the isomorphisms in 8.15.4, we get

Dk (P ) =

P Z·jQ

8.17.1

Q

where Q runs over the set of proper subgroups of P . Moreover, denoting by µ(P ) the value of the M¨ obius function (cf. 1.32) on the ordered set of proper nontrivial subgroups of P , in Dk (P )p we still consider the following element r1P =

P µ(P/Q)·jQ

8.17.2,

Q

where Q runs over the set of proper subgroups of P . Proposition 8.18 If P is elementary Abelian we have R◦ Dk (P )p = Zp ·r1P

8.18.1.

In particular R◦ Dk (P )p ∼ = Zp unless |P | = p where either p = 2 and we have R◦ Dk (P )p ∼ = Z/2Z or p = 2 and we have R◦ Dk (P )p = {0} . Proof: For any proper subgroup R of P , it is quite clear that P resP R (r1 ) =

R µ(P/Q)·jQ∩R =

Q

T

µ(P/U ) ·jTR

8.18.2,

U

where Q runs over the set of (proper) subgroups of P , T runs over the set of (proper) subgroups of R and, for such a T , U runs over the set of subgroups of P containing T and fulﬁlling U ∩ R = T , which implies that the sum U µ(P/U ) is equal to zero. Moreover, since µ(P ) is invertible in Zp and we have Zp ·r1P ∼ = Zp ·j1P by statements 8.16.3, 8.16.4 and 8.16.5, it follows from equality 8.17.1 that Dk (P )p = Zp ·r1P ⊕

Q

P Zp ·jQ

8.18.3

114

Frobenius categories versus Brauer blocks

where Q runs over the set of nontrivial (proper) subgroups of P ; but, since for any nontrivial (proper) subgroup Q of P and any complement R of Q P R in P we have resP R (jQ ) = j1 , and since Q is clearly determined by the set of its complements in P , the restrictions to the proper subgroups R of P map

P injectively the direct summand Q Zp ·jQ above into R Dk (R)p , where R runs over the set of proper subgroups of P . We are done. Φ

Φ

P 8.19 In the general case, set P = P/Φ(P ) and denote by πΦ :P → P P the canonical map; since, for any proper subgroup Q of P , πΦ (Q) is also a Φ

proper subgroup of P , we have Φ resπΦP R◦ Dk (P ) ⊂ R◦ Dk (P )

8.19.1;

moreover, since the Brauer homomorphism is compatible with the restriction, it is quite clear that ◦ P Φ ◦ Br Φ(P ) R Dk (P ) ⊂ R Dk (P ) P

P = id and, since Br Φ(P ) ◦ resπΦ D

k (P

Φ

)

8.19.2

, we get the equality

◦ P Φ ◦ Br Φ(P ) R Dk (P ) = R Dk (P )

8.19.3.

P

8.20 Actually, we claim that Br Φ(P ) induces an isomorphism between Φ

R◦ Dk (P ) and R◦ Dk (P ) and now it suﬃces to prove the injectivity; that is to P (s) = 0 , say, it suﬃces to prove that, for any s ∈ R◦ Dk (P ) such that Br Φ(P )

we have s = 0 ; moreover, since for any normal subgroup Q of P , we have (cf. equality 1.22.3) P/Q

P

P

Φ

P

Br Φ(P/Q) ◦ BrQ = Brπ P (Q) ◦ BrΦ(P ) Φ

8.20.1,

arguing by induction on |P | we may assume that s belongs to the subgroup Q

P

) ∩ R◦ Dk (P ) = Ker(Br Q

NP (R)

Ker(Br R

) ∩ R◦ Dk (P )

8.20.2,

R

where Q runs over the set of nontrivial normal subgroups of P and R over the set of all the nontrivial ones. 8.21 As a matter of fact, if P is not elementary Abelian, this subgroup coincides with the kernel of homomorphism (2.2.1) in [43] where we prove that this kernel is ﬁnite, which already implies that Q ⊗Z R◦ Dk (P ) ∼ = Q ⊗Z R◦ Dk (P ) Φ

8.21.1;

8. Decompositions for Dade P -algebras

115

the main ingredient for proving this ﬁniteness was Carlson’s main theorem in [16] and, pushing further all the machinery Jon Carlson developed since, it is proved in [17] that 8.21.2 If P is not elementary Abelian, intersection 8.20.2 is trivial unless p = 2 and then it is a 2-group. Thus, the argument above and statements 8.16.5 and 8.21.2 yield: P

Corollary 8.22 With the notation above, Br Φ(P ) induces an isomorphism Φ R◦ Dk (P )p ∼ = R◦ Dk (P )p

8.22.1.

Thus, if P is not cyclic then R◦ Dk (P )p ∼ = Zp . Moreover, if p = 2 then Φ ◦ it also induces an isomorphism R Dk (P ) ∼ = R◦ Dk (P ) and Dk (P ) maps into Dk (P )p .

Chapter 9

Polarizations for Dade P-algebras 9.1 Let P be a ﬁnite p-group, k an algebraically closed ﬁeld of characteristic p , S a Dade P -algebra over k and s the class of S in the Dade group Dk (P ) (cf. 1.21); recall that S can be considered as a P -interior algebra (cf. 1.20 and 1.27), that P has a unique local point γ S on S (cf. 1.20) and that the group of S-fusions from Pγ S to Pγ S (cf. 1.28) — simply noted FS (P ) — coincides with the stabilizer Aut(P )s of s in Aut(P ) (cf. Proposition 2.18 in [39]). Choosing i ∈ γ S and denoting by Pi the canonical lifting to (iSi)∗ of the action of P on iSi , it follows from the very deﬁnition of the S-fusions that we have a group homomorphism (cf. homomorphism 1.29.1) FS (P ) −→ N(iSi)∗ (Pi )/(iS Pi)∗

9.1.1

and then, we deﬁne the k ∗ -group FˆS (P ) (cf. 1.29) by the pull-back FS (P ) −→ N(jSj)∗ (Pi )/(jS Pj)∗ ↑ ↑ ˆ FS (P ) −→ N(iSi)∗ (Pi ) i + J(iS Pi)

9.1.2.

9.2 Although we only have a poor reference [40], it is known that this k ∗ -group is split or, equivalently, that there is a k ∗ -group homomorphism ωP,S : FˆS (P ) −→ k ∗

9.2.1

and the main purpose of this chapter is to discuss the possibility of making coherent choices for ωP,S ; by the way we give a complete proof of the existence of such a k ∗ -group homomorphism, short cutting the reference [40]. 9.3 In order to formulate what we mean by coherent choices, let us consider the following category Dk . The Dk -objects are the pairs (P, S) formed by a ﬁnite p-group P and by a Dade P -algebra S over k ; for a second Dk -object (P , S ) , the Dk -morphisms from (P, S) to (P , S ) are the pairs (π, f ) formed by a surjective group homomorphism π : P → P such that Ker(π) is FS (P )-stable, and by a P -interior algebra embedding (cf. 1.11) f : Resπ (S ) −→ S

9.3.1.

9.4 Then, for a second Dk -morphism (π , f ) from (P , S ) to a third Dk -object (P , S ) , embedding 9.3.1 forces (cf. Proposition 2.14 in [39]) FS (P ) = FResπ (S ) (P )

9.4.1

118

Frobenius categories versus Brauer blocks

and, since FS (P ) stabilizes Ker(π) , it follows from the very deﬁnition of the S -fusions that the action of FS (P ) on P/Ker(π) ∼ = P is contained in FS (P ) , so that we get a group homomorphism Ff (π) : FS (P ) −→ FS (P )

9.4.2;

hence, FS (P ) stabilizes Ker(σ ) and thus, since Ker(σ ◦ σ) = σ −1 Ker(σ )

9.4.3,

FS (P ) stabilizes Ker(σ ◦ σ) too. Consequently, the pair π ◦ π, f ◦ Resπ (f ) is a Dk -morphism from (P, S) to (P , S ) and we deﬁne the composition in Dk by (π , f ) ◦ (π, f ) = π ◦ π, f ◦ Resπ (f ) 9.4.4, which clearly fulﬁlls the associative condition. 9.5 Now, we have a functor f : Dk → Gr from Dk to the category of ﬁnite groups Gr mapping any Dk -object (P, S) on the group FS (P ) and any Dk -morphism (π, f ) : (P, S) → (P , S ) on the group homomorphism Ff (π) above; actually, it follows from Proposition 6.8 in [42] that embedding 9.3.1 determines a k ∗ -group homomorphism Fˆf (π) : FˆS (P ) −→ FˆS (P )

9.5.1

lifting Ff (π) ; consequently, f can be lifted to a functor ˆf : Dk −→ k ∗ -Gr

9.5.2,

where k ∗ -Gr is the category of k ∗ -groups with ﬁnite k ∗ -quotients (cf. 1.23), mapping (P, S) on FˆS (P ) and (π, f ) on Fˆf (π) ; note that if S is a P -algebra with trivial P -action then we have FS (P ) = Aut(P ) and the pull-back 9.1.2 determines a k ∗ -group isomorphism FˆS (P ) ∼ = k ∗ × Aut(P )

9.5.3.

We are interested in the natural maps ω — called the polarizations — from this functor to the trivial one — namely, to the functor mapping (P, S) on k ∗ and (π, f ) on idk∗ — such that if S is a P -algebra with trivial P -action then ωP,S coincides with the ﬁrst projection in isomorphism 9.5.3. 9.6 While we shall prove the existence of a polarization, as we show in the following example for p = 2 it is not possible to get a polarization compatible with the tensor product, namely a polarization ω fulﬁlling

9. Polarizations for Dade P -algebras

119

9.6.1 If S and S are Dade P -algebras, the following diagram is commutative ˆ FˆS (P ) −→ FˆS (P ) ∩ ˆ ωP,S ωP,S ×

/

FˆS⊗k S (P ) 0 ωP,S⊗k S

k

∗

ˆ FˆS (P ) (cf. 1.24) of ˆ FˆS (P ) is the converse image in FˆS (P ) × where FˆS (P ) ∩ the diagonal image of FS (P ) ∩ FS (P ) in FS (P ) × FS (P ) , and where the top k ∗ -group homomorphism is given by Proposition 5.11 in [41] (see 9.15 below for more detail). Example 9.7 Assume that |P | = p = 2 and set S = Endk J(kP ) (cf. 8.16); then, it is quite clear that FS (P ) = Aut(P ) ∼ = Z/(p − 1)Z

9.7.1,

and it is well-known that the kP -module J(kP ) is selfdual or, equivalently, that there is an embedding (cf. 1.10) from k to S ⊗k S . Explicitly, choosing a nontrivial element u ∈ P , we have an obvious kP -module isomorphism J(kP ) ∼ = kP/k·(1 − u)p−1

9.7.2

mapping 1 − u on the class ¯ 1 of 1 and the scalar product in kP — deﬁned by the product and by the k-linear form π1 : kP → k mapping 1 ∈ P on 1 ∈ k and v ∈ P − {1} on 0 — induces a kP -module isomorphism ∗ kP/k·(1 − u)p−1 ∼ = J(kP )

9.7.3.

Since isomorphism 9.7.2 maps (1 − u)m on (1 − u)m−1 for any m ≥ 1 , then isomorphism 9.7.3 determines a scalar product in J(kP ) which, for any n ≥ 1 , sends (1 − u)n , (1 − u)m to

π1 (1 − u)

n+m−1

!

1 if n + m ≤ p

=

9.7.4; 0 if n + m > p

in particular, the family {(1 − u)m − (1 − u)m+1 }0<m

9.7.5

determined by isomorphisms 9.7.2 and 9.7.3 maps idJ(kP ) on r=

(1 − u)n ⊗ (1 − u)p−n − (1 − u)p−n+1

0
9.7.6.

120

Frobenius categories versus Brauer blocks

Thus, since we clearly have S = Ker(trJ(kP ) ) ⊕ k·idJ(kP )

9.7.7,

k·r is a trivial direct summand of the kP -module M = J(kP ) ⊗k J(kP ) . On the other hand, Aut(P ) acts naturally on kP , J(kP ) and M , and, for any σ ∈ Aut(P ), denoting by λσ ∈ k ∗ the element of the prime subﬁeld ˇ ˇ σ ∈ ∆p−1 of λσ , we have fulﬁlling σ(u) = uλσ for the lifting λ ˇ

σ(1 − u) = 1 − u

ˇσ λ

σ −1 λ

=

uh (1 − u)

9.7.8

h=0

and therefore we still have ˇ ˇ ˇ σ(r) = (1 − uλσ )n ⊗ (1 − uλσ )p−n − (1 − uλσ )p−n+1

9.7.9

0
which the scalar product maps on ˇ ˇ ˇ π1 (1 − uλσ )p−1 − (1 − uλσ )p = π1 (1 − uλσ )p−1 0
0
=

0
=

π1

ˇ σ −1 λ

uh

p−1

(1 − u)p−1

h=0

9.7.10

π1 (λσ )p−1 ·(1 − u)p−1

0
= −(λσ )−1 since u(1 − u)p−1 = (1 − u)p−1 and π1 (1 − u)p−1 = 1 . In conclusion, it is clear that P still ﬁxes the following element of M t= λσ ·σ(r) 9.7.11, σ∈Aut(P )

and it is easily checked that the scalar product maps t on 1 , so that k·t is also a trivial direct summand the kP -module M , and that σ(t) = (λσ )−1 ·t for any σ ∈ Aut(P ) . Now, the Dk -morphism (P, S ⊗k S) → ({1}, k) deﬁned by the embedding k −→ S ⊗k S = Endk (M )

9.7.12

induced by k·t determines a k ∗ -group homomorphism (cf. 9.5.1) FˆS⊗k S (P ) −→ k ∗

9.7.13

9. Polarizations for Dade P -algebras

121

which determines a k ∗ -group isomorphism FˆS⊗k S (P ) ∼ = k ∗ × Aut(P )

9.7.14;

moreover, since ωP,k = idk∗ , homomorphism 9.7.13 has to coincide with ωP,S⊗k S for any polarization ω . But, the analysis above shows that the action of Aut(P ) on J(kP ) induces a group homomorphism ˆ FˆS (P ) −→ FˆS⊗k S (P ) ∼ Aut(P ) −→ FˆS (P ) ∩ 9.7.15 = k ∗ × Aut(P ) mapping σ ∈ Aut(P ) on (λσ )−1 , σ ; thus, denoting by σ ˆ the image of σ in ˆ ˆ ˆ FS (P ) , in the diagram in 9.6.1 above, the images of σ FS (P ) ∩ ˆ in k ∗ coincide −1 2 with (λσ ) and with ωP,S (ˆ σ ) which cannot coincide over any generator of Aut(P ) . 9.8 Before going further, let us consider a second compatibility condition involving the Brauer quotient functor (cf. 1.22); this condition depends on a canonical k ∗ -group homomorphism deﬁned as follows. As above, let P be a ﬁnite p-group, S a Dade P -algebra and Q a normal subgroup of P ; set ¯ ˆ T = ResP Q (S) and P = P/Q , and denote by FS (P )Q the stabilizer of Q in FˆS (P ) ; if ϕ ∈ FS (P )Q and i ∈ S P is a primitive idempotent such that BrSP (i) = 0 , by the very deﬁnition of the S-fusions (cf. 1.28) there is an element a ∈ S ∗ fulﬁlling ia = i and ai·u = ϕ(u)·ai for any u ∈ P ; then, setting S ai = a i + J(iS P i) 9.8.1, S it follows from its very deﬁnition, that FˆS (P ) contains (ϕ, ai ) .

9.9 In particular, a acts on iSi and, since ϕ(Q) = Q , a still acts on (iSi)(Q) ; since this k-algebra is simple, this action can be lifted to an element of (iSi)(Q)∗ and therefore there is an element c ∈ (S Q )∗ such that ic = i and that ac−1 acts trivially on (iSi)(Q) ; then, choosing a primitive idempotent j in S Q such that ij = j = ji and BrSQ (j) = 0 , we have −1

BrSQ (j)ac

= BrSQ (j)

9.9.1 −1

and therefore we can modify our choice of c in such a way that j ac this case, since c ∈ (S Q )∗ , for any u ∈ Q we have ac−1 j·u = ϕ(u)·ac−1 j

= j ; in 9.9.2

which proves that the restriction ψ of ϕ to Q belongs to FT (Q) and, setting T ac−1 j = ac−1 j + J(jT Q j) T that (ψ, ac−1 j ) belongs to FˆT (Q) .

9.9.3,

122

Frobenius categories versus Brauer blocks

¯ 9.10 Finally, since BrQ S P ) = S(Q)P and S(Q) (P¯ ) ∼ = S(P ) (cf. iso¯ morphism 1.20.1), BrQ (i) is a primitive idempotent of S(Q)P and we have ∗ S(Q) (u·i) the canonical lifting of BrP¯ BrQ (i) = 0 ; moreover, denoting by Br Q u·i to (iSi)(Q)∗ for any u ∈ P , we get ∗ ∗ ∗ (u·i) = Br ϕ(u)·i a = Br ϕ(u)·i c Br Q Q Q

9.10.1,

which proves that the automorphism ϕ¯ of P¯ determined by ϕ belongs to FS(Q) (P¯ ) and, setting S(Q)

BrQ (ci)

= BrQ (c) BrQ (i) + J BrQ (i)S(Q)P BrQ (i)

S(Q)

that (ϕ, ¯ BrQ (ci)

9.10.2,

) belongs to FˆS(Q) (P¯ ) †.

Proposition 9.11 With the notation above, identify FˆT (Q) and FˆS(Q) (P¯ ) ˆ FˆS(Q) (P¯ ) . There is a unique k ∗ -group homowith their images in FˆT (Q) × morphism ˆ FˆS(Q) (P¯ ) ∆P,S,Q : FˆS (P )Q −→ FˆT (Q) ×

9.11.1

deﬁned by S(Q)

T

S

∆P,S,Q (ϕ, ai ) = (ψ, ac−1 j )·(ϕ, ¯ BrQ (ci)

)

9.11.2.

Proof: First of all, it is quite clear that this map does not depend on the choices of i and j ; moreover, note that we only can modify our choice of a by an element z ∈ (S P )∗ such that iz = i , so that BrP (zi) = λ·BrP (i) for some λ ∈ k ∗ ; then, we only can modify our choice of c by an element x ∈ (S Q )∗ such that ix = i , that BrQ (z −1 xi) = µ·BrQ (i) for some µ ∈ k ∗ and that jz

−1

x

S

= j ; these modiﬁcations simply multiply ai by λ and ac−1 j

B

by µ ,

S(Q)

and then it is easily checked that BrQ (ci) becomes multiplied by λµ−1 . Thus, the map ∆P,S,Q does not depend on our choices. For a second ϕ ∈ FS (P )Q , choosing a and c as above and denoting by ψ the restriction of ϕ to Q , it is clear that aa and cc are a suitable choice for ϕ ◦ ϕ , and that S

S

S

(ai )(a i ) = aa i S(Q)

BrQ (ci) †

S(Q)

BrQ (c i)

S(Q)

= BrQ (cc i)

9.11.3;

All these arguments can be suitably extended to any P-interior algebra as it is shown by

Proposition 10.3 in [42].

9. Polarizations for Dade P -algebras

123

moreover, we have ac−1 ja c−1 j = ac−1 a c−1 j = aa (cc )−1 [c−1 , a c−1 ] j

9.11.4

and, since a c−1 acts trivially on (iSi)(Q) , we still have BrQ [c−1 , a c−1 ]i = BrQ (i)

9.11.5,

so that we get T

T

(ac−1 j )(a c−1 j ) = aa (cc )−1 j

T

9.11.6.

Hence, the map ∆P,S,Q is a k ∗ -group homomorphism. We are done. Corollary 9.12 With the notation above, for any Dk -morphism (π, f˜) : (P, S) −→ (P , S )

9.12.1,

˜ ¯ setting Q = π(Q) , T = ResP ˜ = ResP Q (f ) and P = P /Q , and Q (S ) , g ¯ ¯ denoting by ρ : Q → Q and by π ¯ : P → P the group homomorphisms induced by π , we have the commutative diagram

FˆS (P )Q Fˆf˜(π)

∆P,S,Q

−−−−−−→

↓

ˆ FˆS(Q) (P¯ ) FˆT (Q) × ↓

FˆS (P )Q

∆P ,S ,Q

−−−−−−→

ˆ Fˆ ˜ π) Fˆg˜ (ρ) × f (Q) (¯

9.12.2.

ˆ FˆS (Q ) (P¯ ) FˆT (Q ) ×

Proof: With the notation above, for a representative f : Resπ (S ) → S of f˜ , we may assume that i = f (i ) for a suitable i ∈ S P , and that a and c centralize f (1S ) ; then, it is clear that the elements a , c , j ∈ S such that f (a ) = af (1S ) , f (c ) = cf (1S ) and f (j ) = j fulﬁll the corresponding conditions with respect to the images ϕ of ϕ , ψ of ψ and ϕ¯ of ϕ¯ ; hence, since S S Fˆf˜(π)(ϕ, ai ) = (ϕ , a i )

T T Fˆg˜ (ρ)(ψ, ac−1 j ) = (ψ , a c−1 j ) S(Q) S Fˆf˜(Q) (¯ π )(ϕ, ¯ BrQ (ci) ) = (ϕ¯ , BrQ (c i )

9.12.3,

(Q )

)

we are done. 9.13 When Q runs over the set of normal subgroups of P , the k ∗ -group homomorphisms ∆P,S,Q may be seen as a kind of coproduct for the functor fˆ above, and we need the coassociativity and the distributivity with respect to the tensor product. Firstly, we make the coassociativity explicit; let R be

124

Frobenius categories versus Brauer blocks

a second normal subgroup of P contained in Q , set U = ResP (S) , denote R by FˆS (P )Q,R the stabilizer of Q and R in FˆS (P ) , and identify S(R) (Q/R) with S(Q) (cf. isomorphism 1.20.1). Proposition 9.14 With the notation above, we have the following commutative diagram: ˆ FˆS(Q) (P/Q) FˆT (Q)R × ∆P,S,Q

1

/

FˆS (P )Q,R ∆P,S,R

ˆ id ˆ ∆Q,T ,R × F

S(Q) (P /Q)

ˆ FˆT (R) (Q/R) × ˆ FˆS(Q) (P/Q) FˆU (R) ×

/

1

idFˆ

U (R)

9.14.1.

ˆ ∆P /R,S(R),Q/R ×

ˆ FˆS(R) (P/R)Q/R FˆU (R) × Proof: With the notation above, assume that ϕ(R) = R and denote by η the T

S

restriction of ϕ to R ; then, (ϕ, ai ) and (ψ, ac−1 j ) respectively belong to FˆS (P )Q,R and to FˆT (Q)R , and, setting α = ∆P,S,Q , we already know that T

S

S(Q)

α(ϕ, ai ) = (ψ, ac−1 j )·(ϕQ , BrQ (ci)

)

9.14.2,

where ϕQ denotes the automorphism of P/Q induced by ϕ and where we idenˆ FˆS(Q) (P/Q) . tify FˆT (Q)R and FˆS(Q) (P/Q) with their images in FˆT (Q)R × Moreover, since ϕ(R) = R , it is clear that ac−1 i normalizes (iSi)R and acts on (iSi)(R) , so that there is e ∈ (S R )∗ fulﬁlling ie = i and such that ac−1 e−1 i acts trivially on (iSi)(R) ; in particular, choosing a primitive idempotent D in S R such that BrR (D) = 0 and Dj = D = jD , as above we can −1 −1 −1 modify our choice of e in such a way that j e = j and then that Dac e = D . In this situation, for any v ∈ R , we still have

so that, setting

ac−1 e−1 D·v = ϕ(v)·ac−1 e−1 D

9.14.3,

U ac−1 e−1 D = ac−1 e−1 D + J(DU R D)

9.14.4,

U the pair (η, ac−1 e−1 D ) belongs to FˆU (R) . ∗

(u·i) the canonical lifting to (iSi)(R)∗ of u·i As above, denoting by Br R for any u ∈ P , we get ∗ ∗ ∗ (u·i) = Br ϕ(u)·i ai = Br ϕ(u)·i eci Br R R R

9.14.5

9. Polarizations for Dade P -algebras

125

and therefore, setting S(R)

BrR (eci)

= BrR (ec) BrR (i) + J BrR (i)S(R)P BrR (i)

9.14.6

and denoting by ϕR the automorphism of P/R induced by ϕ , the pair S(R) ) belongs to FˆS(R) (P/R)Q/R . Thus, setting β = ∆P,S,R , (ϕR , BrR (eci) similarly we have S(R)

U

A

β(ϕ, ai ) = (η, ac−1 e−1 D )·(ϕR , BrR (eci)

)

9.14.7.

On the other hand, since ac−1 and e−1 centralize j , and ac−1 e−1 j acts ˆ idFˆ trivially on (jRj)(Q) , setting γ = ∆Q,T,R × we still have S(Q) (P/Q) A γ α(ϕ, ai ) U

T (R)

R

= (η, ac−1 e−1 D )·(ψ , BrR (ej)

S(Q)

)·(ϕQ , BrQ (ci)

9.14.8 )

R

where ψ denotes the automorphism of Q/R induced by ψ and, as above, we also identify FˆU (R) , FˆT (R) (Q/R) and FˆS(Q) (P/Q) with their images in ˆ FˆT (R) (Q/R) × ˆ FˆS(Q) (P/Q) FˆU (R) ×

9.14.9.

Finally, we know that BrR (i) is a primitive idempotent of S(R)P/R such that (cf. equality 1.22.3) S(R) BrP/R BrR (i) = 0

9.14.10,

that BrR (eci) normalizes (iSi)(R)Q/R (cf. equality 9.14.5), that BrR (e) be ∗ longs to S(R)Q/R and that, since ac−1 acts trivially on (iSi)(Q) (cf. 9.9) and ac−1 e−1 i acts trivially on (iSi)(R) , BrR (ei) still acts trivially on (cf. isomorphism 1.24.1) (iSi)(R) (Q/R) ∼ 9.14.11; = (iSi)(Q) ˆ ∆P/R,S(R),Q/R , we still get consequently, setting δ = idFˆU (R) × A δ β(ϕ, ai ) U

R

T (R)

= (η, ac−1 e−1 D )·(ψ , BrR (ej) We are done.

S(Q)

)·(ϕQ BrQ (ci)

9.14.12. )

126

Frobenius categories versus Brauer blocks

9.15 For the distributivity, let S be another Dade P -algebra and, as ¯ above, set T = ResP Q (S ) and P = P/Q ; moreover, identify (S⊗k S )(Q) with S(Q) ⊗k S (Q) (cf. 1.21.2). It follows from Propositions 5.9 and 5.11 in [41] that 9.15.1 FS (P ) ∩ FS (P ) ⊂ FS⊗k S (P ) ˆ FˆS (P ) ˆ FˆS (P ) the converse image in FˆS (P ) × and that, denoting by FˆS (P ) ∩ of the diagonal image of FS (P ) ∩ FS (P ) in FS (P ) × FS (P ) , this inclusion lifts to a natural k ∗ -group homomorphism ˆ FˆS (P ) −→ FˆS⊗k S (P ) νˆP,S,S : FˆS (P ) ∩

9.15.2

which, with the notation above and with analogous choices i and a for S , S S ˆ FˆS (P ) , maps the element (ϕ, ai )·(ϕ, a i ) , that clearly belongs to FˆS (P ) ∩ on S⊗k S S S νˆP,S,S (ϕ, ai )·(ϕ, a i ) = (ϕ, (a ⊗ a )a i ) 9.15.3

kS ) such that where i is a primitive idempotent in (S ⊗k S )P − Ker(BrS⊗ P

(i ⊗ i )i = i = i (i ⊗ i ) so that we have

9.15.4,

kS (i ) = BrSP (i) ⊗ BrSP (i ) BrS⊗ P

9.15.5,

kS ) fulﬁlling i(a⊗a )a and where a is an element of 1 + Ker(BrS⊗ P (cf. isomorphism 1.21.2).

= i

Proposition 9.16 With the notation above, let S be a Dade P -algebra and ˆ set T = ResP Q (S ) , α = ∆P,S⊗k S ,Q and β = ∆P,S,Q × ∆P,S ,Q . Then, we have the following commutative diagram: FˆS⊗k S (P )Q

α

−→

ˆ Fˆ(S⊗k S )(Q) (P¯ ) FˆT ⊗k T (Q) ×

ˆν ˆP ,S(Q),S (Q) ↑ νˆQ,T ,T × β ˆ FˆS(Q) (P¯ )∩ ˆ FˆT (Q) × ˆ FˆS (Q) (P¯ ) ˆ FˆS (P )Q −→ FˆT (Q)∩ FˆS (P )Q ∩ ν ˆP,S,S

↑

S

9.16.1.

T

Proof: With the notation above, we consider the pairs (ϕ, ai ) , (ψ, ac−1 j ) S(Q) ) , which respectively belong to FˆS (P )Q , to FˆT (Q) and and (ϕ, ¯ BrQ (ci) to FˆS(Q) (P¯ ) ; mutatis mutandis, assuming that ϕ ∈ FS (P )Q ∩ FS (P )Q , S

for S we choose i , a , j and c as above, and thus the pairs (ϕ, a i ) , S (Q) T (ψ, a c−1 j ) and (ϕ,Br ¯ Q (c i ) ) belong to FˆS (Q) , to FˆT (R) and to

9. Polarizations for Dade P -algebras

127

S S FˆS (Q) (P¯ ) respectively. Consequently, β maps (ϕ, ai )·(ϕ, a i ) on the element S(Q)

T

(ψ, ac−1 j )·(ϕ, ¯ BrQ (ci)

S (Q)

T

)·(ψ, a c−1 j )·(ϕ, ¯ BrQ (c i )

)

9.16.2.

kS On the other hand, choosing i ∈ (S ⊗k S )P and a ∈ 1 + Ker(BrS⊗ ) as P above, by equality 9.15.5 we have

S⊗k S S S νP,S,S (ϕ, ai )·(ϕ, a i ) = (ϕ, (a ⊗ a )a i ) = (ϕ, ai ⊗ a i

S⊗k S

9.16.3.

)

ac−1 ⊗ a c−1 acts trivially on (iSi ⊗k i S i )(Q) and a fortiori Moreover, on i (S ⊗k S )i (Q) ; then, choosing a primitive idempotent j in (S ⊗k S )Q such that BrQ (j ) = 0 and i j = j = j i 9.16.4,

kS we may ﬁnd c ∈ 1 + Ker(BrS⊗ ) fulﬁlling Q −1

i(c

⊗c−1 )c

thus, α maps (ϕ, ai ⊗ a i

= i

S⊗k S

⊗a c−1 )c

= j

9.16.5;

) on the element

T ⊗k T

(ψ, (ac−1 ⊗ a c−1 )c j

−1

and j (ac

(S⊗S )(Q)

)·(ϕ, ¯ BrQ (c−1 (c ⊗ c )i )

)

9.16.6.

As above, it is easily checked that (S⊗k S )(Q)

BrQ (c−1 (c ⊗ c )i )

(S⊗k S )(Q)

= BrQ (ci ⊗ c i )

9.16.7

and it is clear that S(Q) S (Q) νP,S(Q),S (Q) (ϕ, )·(ϕ, ¯ BrQ (c i ) ) ¯ BrQ (ci) (S⊗k S )(Q)

= (ϕ, ¯ BrQ (ci ⊗ c i )

9.16.8. )

On the other hand, since BrTQ (j) ⊗ BrTQ (j ) and BrTQ⊗T (j ) are both primitive in (T ⊗ T )(Q) , they are mutually conjugate and in FˆT ⊗k T (Q) up to suitable identiﬁcations we get (ψ, (ac−1 ⊗ a c−1 )c j

T ⊗k T

) = (ψ, ac−1 j ⊗ a c−1 j T

T ⊗k T

)

9.16.9; T

similarly, it is quite clear that νQ,T,T maps (ψ, ac−1 j )·(ψ, a c−1 j ) on ˆ νP¯ ,S(Q),S (Q) maps the element 9.16.2 this element. In conclusion, νQ,T,T × on the element 9.16.6. We are done.

128

Frobenius categories versus Brauer blocks

9.17 Now, we are ready to discuss our second condition. As in the case of the tensor product, if p = 2 there are no polarizations ω compatible with the kind of coproduct above and, more generally, we consider a relative condition with respect to a full subcategory Ek of Dk ; we say that a polarization ω is Ek -compatible with the Brauer quotient functor whenever it fulﬁlls the condition 9.17.1 If (P, S) is a Ek -object and Q a normal subgroup of P , the following diagram is commutative ∆P,S,Q ˆ FˆS(Q) (P¯ ) FˆS (P )Q −−−−−→ FˆT (Q) × ωP,S

/

ˆ ωP¯ ,S(Q) 0ωQ,T ×

k∗ ¯ where we set T = ResP Q (S) and P = P/Q . Similarly, we say that a polarization is Ek -compatible with the tensor product whenever it fulﬁlls the following condition 9.17.2 If (P, S) and (P, S ) are Ek -objects the following diagram is commutative ˆ FˆS (P ) −→ FˆS⊗k S (P ) FˆS (P ) ∩ ˆ ωP,S ωP,S ×

/

0 ωP,S⊗k S k∗

For any n ∈ N , let us denote by nDk the full subcategory of Dk over the objects (P, S) such that the class of S belongs to n·Dk (P ) ; we have to consider the full subcategories 2Dk and 2(p − 1)Dk . 9.18 In order to deﬁne a polarization ω , we start by discussing the elementary Abelian case. In this case, by 8.13.4 and 8.15 above, any element s ∈ Dk (P ) admits a unique decomposition s=

P/Q

P (j zQ ·resπQ 1

)

9.18.1,

Q

where Q runs over the set of subgroups of P of index at least 3 and, for any Q , zQ is an element of Z , which belongs to {0, 1} if |P/Q| = p ; consequently, any Dade P -algebra S in the class s can be embedded (cf. 1.11) in the tensor product (cf. 1.21) Endk (M ) ⊗k

zQ

Endk J(k(P/Q))

9.18.2,

Q

where Q runs over the set of subgroups of P of index at least 3 , where M is a kP -module admitting a P -stable basis with some ﬁxed element and where

9. Polarizations for Dade P -algebras

129

zQ

is the zQ -th tensor power over k , with the conventional rule that if zQ = 0 then we replace the corresponding tensor factor by k and if zQ < 0 ◦ −zQ by Endk J(k(P/Q)) . 9.19 If i is a primitive idempotent of S P such that BrSP (i) = 0 then we have FˆiSi (P ) ∼ = FˆS (P ) (cf. Proposition 2.14 in [39]) and the Dade P -algebra iSi can be embedded (cf. 1.11) in the above tensor factor (cf. 2.5.1 in [43]) T =

z

Q

Endk J(k(P/Q))

9.19.1,

Q

so that we may assume that M ∼ = k . Moreover, we know that (cf. Propositions 2.14 and 2.18 in [39]) FS (P ) = FT (P ) = Aut(P )s

9.19.2

and therefore, by the uniqueness of decomposition 9.18.1, FS (P ) coincides with the stabilizer of the family {zQ }Q ; but, for any subgroup Q of P , the sta bilizer Aut(P )Q of Q in Aut(P ) acts on J k(P/Q) and actually J k(P/Q) becomes a k P Aut(P )Q -module; consequently, the tensor product N=

z

Q

J(k(P/Q))

9.19.3,

Q

where Q runs over the set of subgroups of P of index at least 3 , becomes a k P FS (P ) -module. Since T ∼ = Endk (N ) , denoting by PN the image of P in T ∗ , we get a group homomorphism FS (P ) → NT ∗ (PN ) which determines a k ∗ -group homomorphism ωP,S : FˆS (P ) −→ k ∗

9.19.4.

mapping the corresponding image of FS (P ) on 1 ∈ k . Note that, if s = 0 , we have iSi ∼ = k and homomorphism 9.19.4 agrees with the ﬁrst projection in isomorphism 9.5.3. 9.20 Arguing by induction on |P | , we will extend this correspondence to all the pairs (P, S) , where P is a ﬁnite p-group and S a Dade P -algebra. We may assume that P is not elementary Abelian; then, setting Q = Φ(P ) and P¯ = P/Q , and denoting by π : P → P¯ the canonical map, it is clear that ∗ FS (P ) stabilizes Q and acts on P¯ ; thus, setting T = ResP Q (S) , the k -group homomorphisms ωQ,T and ωP¯ ,S(Q) are already deﬁned and we deﬁne ωP,S by the composition (cf. Proposition 9.11) ω

ˆ ×ω

¯ ,S(Q) Q,T ∆P,S,Q P ˆ FˆS(Q) (P¯ ) −−−−−−− FˆS (P ) −−−−−→ FˆT (Q) × −−−→ k ∗

9.20.1.

130

Frobenius categories versus Brauer blocks

Theorem 9.21 With the notation above, ω is a polarization 2Dk -compatible with the tensor product and 2(p − 1)Dk -compatible with the Brauer quotient functor. Proof: Let (π, f ) : (P, S) → (P , S ) be a Dk -morphism and ﬁrstly assume that P is elementary Abelian; thus, P is elementary Abelian too and, denoting by s ∈ Dk (P ) the class of S , we have a unique decomposition of s s =

P /Q

zQ ·resπP (j1 Q

Q

)

9.21.1

where Q runs over the set of subgroups of P of index at least 3 and, for such a Q , zQ is an element of Z , which belongs to {0, 1} if |P /Q | = p (cf. 9.18); then, we get s = resπ (s ) =

Q

zQ ·resπP−1 π

P/π −1 (Q )

(Q )

(j1

)

9.21.2,

where Q runs again over the set of subgroups of P of index at least 3 , and it is the unique decomposition of the class s ∈ Dk (P ) of S ; consequently, from the very deﬁnition of ω it is easily checked that ωP,S = ωP ,S ◦ Fˆf (π)

9.21.3.

Otherwise, the surjectivity of π forces π Φ(P ) = Φ(P ) and therefore, arguing by induction on |P | , setting Q = Φ(P ) , Q = Φ(P ) , P¯ = P/Q , P P P¯ = P /Q , T = ResP Q (S) , T = ResQ (S ) and g = resQ (f ) , and denoting ¯ : P¯ → P¯ the group homomorphisms induced by π , by ρ : Q → Q and by π we already have ωQ,T = ωQ ,T ◦ Fˆg (ρ)

and ωP¯ ,S(Q) = ωP¯ ,S (Q ) ◦ Fˆf (Q) (¯ π)

9.21.4.

On the other hand, from Corollary 9.12 we get the following commutative diagram ∆P,S,Q ˆ FˆS(Q) (P¯ ) FˆS (P ) −−−−−−→ FˆT (Q) ×   # Fˆg (ρ) × ˆ Fˆf (Q) (¯ Fˆf (π) # π) 9.21.5. ∆

P ,S ,Q FˆS (P ) −−−−−−→

ˆ FˆS (Q ) (P¯ ) FˆT (Q ) ×

Consequently, from the deﬁnition above, from equalities 9.21.4 and from the commutativity of this diagram, in this case we still get ωP,S = ωP ,S ◦ Fˆf (π) This already proves that ω is a polarization.

9.21.6.

9. Polarizations for Dade P -algebras

131

Now, let S and S be Dade P -algebras and denote by s and s their respective classes in Dk (P ) ; ﬁrst of all, assume that P is elementary Abelian, so that we have unique decompositions P/Q P/Q P (j P (j s= zQ ·resπQ ) and s = zQ ·resπQ ) 9.21.7, 1 1 Q

Q

where Q runs over the set of subgroups of P of index at least 3 and, for any Q , zQ and zQ are elements of Z , which belong to {0, 1} if |P/Q| = p P/Q

P (j (cf. 9.18). Hence, for such a Q , the coeﬃcient of resπQ 1

) in the unique

z Q + zQ

decomposition of s + s coincides with unless |P/Q| = p and we have zQ = 1 = zQ ; in this case this coeﬃcient is zero, whereas the factor J k(P/Q) ⊗k J k(P/Q)

9.21.8

appears in the tensor product of the corresponding decompositions 9.19.3; moreover, according to example 9.7, the natural action of P Aut(P )Q on this factor stabilizes a subspace JQ of dimension 1 where Aut(P )Q acts via the inverse of the natural monomorphism θP/Q : Aut(P/Q) → k ∗ . Consequently, denoting by FS (P ) ∩ FS (P ) Q the stabilizer of Q in FS (P ) ∩ FS (P ) , the composition of θP/Q with the canonical homomorphism

FS (P ) ∩ FS (P ) Q −→ Aut(P/Q)

9.21.9

determines, via the transfer homomorphism from FS (P ) ∩ FS (P ) Q to FS (P ) ∩ FS (P ) (cf. 1.32) a group homomorphism ζO : FS (P ) ∩ FS (P ) −→ k ∗ which describes the action of FS (P ) ∩ FS (P ) on Q ∈O JQ where FS (P ) ∩ FS (P )-orbit of Q . ˆ In conclusion, it is easy to check that, for any ϕˆ in FˆS (P ) ∩ we get ˆ ωP,S )(ϕ) (ωP,S⊗k S ◦ νˆP,S,S )(ϕ) ˆ = ζO (ϕ) (ωP,S × ˆ

9.21.10 O is the FˆS (P ) ,

9.21.11

O

where ϕ is the image of ϕˆ in FS (P ) ∩ FS (P ) and O runs over the set of orbits of FS (P ) ∩ FS (P ) on the set of subgroups Q of P such that |P/Q| = p and zQ = 1 = z Q . In particular, if s and s belong to 2Dk (P ) the corresponding diagram in 9.17.2 is commutative. Assume that P is not elementary Abelian and that (P, S) and (P, S ) are actually 2Dk -objects; setting Q = Φ(P ) and P¯ = P/Q , and denoting

132

Frobenius categories versus Brauer blocks

by π : P → P¯ the canonical map, it follows from Proposition 9.16 that the following diagram is commutative FˆS⊗k S (P )

α

ˆ Fˆ(S⊗k S )(Q) (P¯ ) FˆT ⊗k T (Q) ×

−→

ˆ νP ,S(Q),S (Q) ↑ νˆQ,T ,T ×ˆ β ˆ FˆS(Q) (P¯ )∩ ˆ FˆS (P ) −→ ˆ FˆT (Q) × ˆ FˆS (Q) (P¯ ) FˆT (Q)∩ FˆS (P )∩ ν ˆP,S,S

↑

9.21.12,

ˆ ∆P,S ,Q , T = ResP where we set α = ∆P,S⊗k S ,Q , β = ∆P,S,Q × Q (S) and P T = ResQ (S ) , so that (Q, T ) and (Q, T ) are 2Dk -objects too; thus, arguing by induction on |P | , we already have the commutative diagram ˆ FˆT (Q) FˆT (Q) ∩ ˆ ωQ,T ωQ,T ×

ν ˆQ,T ,T

−−−−−−→

/

FˆT ⊗k T (Q) 0 ωQ,T ⊗k T

9.21.13

k∗ Similarly, (P¯ , S(Q)) and (P¯ , S (Q)) are also 2Dk -objects and therefore, arguing by induction on |P | , we still have the commutative diagram ν ˆ

(Q) ¯ ˆ FˆS (Q) (P¯ ) −−P−,S(Q),S −−−−−−→ FˆS(Q)⊗k S (Q) (P¯ ) FˆS(Q) (P¯ ) ∩

ˆ ωP¯ ,S (Q) ωP ,S(Q) ×

/

0 ωP¯ ,S(Q)⊗k S (Q)

9.21.14.

k∗ Now, the k ∗ -product of diagrams 9.21.13 and 9.21.14, together with diagram 9.21.12, prove the commutativity of the diagram in 9.17.2. Finally, we prove that ω is 2(p − 1)Dk -compatible with the Brauer quotient functor . Consider S and s as above, assume that s belongs to 2(p − 1)·Dk (P ) , let R be a normal subgroup of P , set U = ResP R (S) and ¯ ¯ P = P/R , and denote by π : P → P the canonical map; ﬁrst of all, assume that P is elementary Abelian, so that we have a unique decomposition s=

P/Q

P (j 2(p − 1)zQ ·resπQ 1

)

9.21.15

Q

where Q runs over the set of subgroups of P of index at least p2 and, for such a Q , zQ is an element of Z (cf. 9.18); the group FS (P )R acts on this set of subgroups and obviously we still have s=

O

2(p − 1)zO ·sO

9.21.16

9. Polarizations for Dade P -algebras

133

where O runs over the set of FS (P )R -orbits and, for such an FS (P )R -orbit, P/Q P (j we set sO = Q∈O resπQ ) and zO = zQ for Q ∈ O . 1 2(p−1)zO Actually, we may assume that S ∼ SO where O runs = O over the set of FS (P )R -orbits and, for such an orbit, we set 9.21.17. SO = Endk J(k(P/Q)) Q∈O

But, it follows from Lemma 1.17 in [34] that we have FS (P )R = FSO (P )R

9.21.18,

O

where O runs over the set of FS (P )R -orbits, and then, iterating our deﬁnition ˆ , it follows from Proposition 5.11 in [41] that we still have in 9.15 of ∩ 2(p−1)zO ˆ FˆS (P )R = 9.21.19 FSO (P )R O

where O runs again over the set of FS (P )R -orbits and where, for such an orbit, if zO = 0 we set 0 ˆ FSO (P )R = k ∗ × Aut(P )R

9.21.20

whereas if zO < 0 we deﬁne (cf. 1.24) ◦ 2(p−1)zO ˆ −2(p−1)zO ˆ FSO (P )R FSO (P )R =

9.21.21.

Thus, since we already know that ω is 2Dk -compatible with the tensor 2 product, it suﬃces to discuss the case of SO ⊗ SO = SO for any orbit O ; clearly, either R ⊂ Q for any Q ∈ O or R ⊂ Q for any Q ∈ O . In the ﬁrst case, the tensor product 2 2 UO = ResP SO 9.21.22 R is a trivial R-algebra and therefore, by the very deﬁnition, ωR,⊗2 UO is the ﬁrst ¯ ∼ projection in 9.5.3; moreover, in this case we have J(k(P¯ /Q)) = J(k(P/Q)) for any Q ∈ O , so that we clearly have 2 2 SO = Resπ SO (R) 9.21.23 and therefore we get ωP,⊗2 SO = ωP¯ ,⊗2 SO (R) ◦ Fˆid

⊗ 2 SO

(π)

hence, in this case, for any ϕˆ ∈ FˆSO (P )R we simply get ˆ ωP¯ ,⊗2 SO (R) ) ∆P,⊗2 SO ,R (ϕ) ωP,⊗2 SO (ϕ) ˆ = (ωR,⊗2 UO × ˆ

9.21.24;

9.21.25.

134

Frobenius categories versus Brauer blocks

2 In the second case, we have SO (R) ∼ = k (cf. isomorphism 8.16.2) and 2 ¯ S (R) is a trivial P -algebra; moreover, for any Q ∈ O , we have therefore O a k R FSO (P )Q,R -module isomorphism J k(P/Q) ∼ = J k(R/(Q ∩ R)) ⊕ k (P − Q·R)/Q

9.21.26

2 SO and and therefore the splitting homomorphisms deﬁned in 9.19 for 2 UO agree provided |R/Q ∩ R| > p ; if |R/Q ∩ R| = p = 2 then we know from Example 9.7 that the image of FSO (P )Q,R in Aut R/(Q∩R) is mapped 2 UO but, since injectively into |Aut R/(Q ∩ R) | = p − 1

9.21.27,

2(p−1) UO . Consequently, for all the orbits it is ﬁnally mapped trivially in 2(p−1) SO and VO = ⊗2(p−1) UO we have the commutative O , setting TO = ⊗ diagram ∆P,TO ,R ˆ FˆT (R) (P¯ ) −→ FˆV (R) × FˆT (P )R −−−−− O

O

ωP,TO

/

O

0ωR,VO

ˆ ωP ,T (R) × O

9.21.28.

k∗ When P is not elementary Abelian, we argue by induction on |P | ; if R P contains Φ(P ) , setting Q = Φ(P ) , T = ResP Q (S) and U = ResR (S) , it follows from Proposition 9.14 that we have the commutative diagram ˆ FˆS(Q) (P/Q)R/Q FˆT (Q) × ∆P,S,Q

1

FˆS (P )R ∆P,S,R

/

/

idFˆ

T (Q)

ˆ ∆P /Q,S(Q),R/Q ×

ˆ FˆU (Q) (R/Q) × ˆ FˆS(R) (P/R) FˆT (Q) × 1

9.21.29

ˆ id ˆ ∆R,U,Q × F

S(R) (P /R)

ˆ FˆS(R) (P/R) FˆU (R)Q × and we already know that, for any ψˆ ∈ FˆU (R)Q and any ϕˆ¯ ∈ FˆS(Q) (P/Q)R/Q , we have ˆ = ωR,U (ψ) ˆ ˆ ωR/Q,U (Q) ) ∆R,U,Q (ψ) (ωQ,T × 9.21.30. ˆ P/R,S(R) ) ∆P/Q,S(Q),R/Q (ϕ) ˆ ˆ¯ ¯ = ωP/Q,S(Q)(ϕ) (ωR/Q,U (Q) ×ω Consequently, if ϕˆ ∈ FˆS (P )R and we have ˆ ϕˆ ¯ and ˆ = ψ· ∆P,S,Q (ϕ)

ˆ¯ ϕˆ¯ ∆P,S,R (ϕ) ˆ = ψ·

9.21.31,

9. Polarizations for Dade P -algebras

135

¯ ∈ FˆU (R)Q and ϕˆ¯ ∈ FˆS(R) (P/R) , where ψˆ ∈ FˆT (Q) , ϕˆ ¯ ∈ FˆS(Q) (P/Q)R/Q , ψˆ then we get (cf. 9.20.1, 9.21.29 and 9.21.30) ˆ¯ ˆ ωP/R,S(R) (ϕ) ˆ ωP/R,S(R) ) ∆P,S,R (ϕ) (ωR,U × ˆ = ωR,U (ψ) ˆ¯ ˆ ωP/R,S(R) (ϕ) ˆ R/Q,U (Q) ) ∆R,U,Q (ψ) = (ωQ,T ×ω ˆ ϕˆ¯ ˆ R/Q,U (Q) ×ω ˆ P/R,S(R) ) ∆R,U,Q (ψ)· = (ωQ,T ×ω ˆ ¯ P/Q,S(Q),R/Q (ϕ) ˆ R/Q,U (Q) ×ω ˆ P/R,S(R) ) ψ·∆ ˆ¯ = (ωQ,T ×ω ˆ ¯ R/Q,U (Q) ×ω ˆ P/R,S(R) ) ∆P/Q,S(Q),R/Q (ϕ) ˆ¯ = ωQ,T (ψ)(ω

9.21.32.

ˆ ¯ P/Q,S(Q) (ϕ) ˆ = ωQ,T (ψ)ω ¯ = ωP,S (ϕ) ˆ Finally, if R does not contain Φ(P ) , we set Q = Φ(P )·R and, as above, we have again by Proposition 9.14 the following commutative diagram ˆ FˆS(Q) (P/Q) FˆT (Q)R × ∆P,S,Q

1

FˆS (P )R ∆P,S,R

/

/

ˆ id ˆ ∆Q,T ,R × F

S(Q) (P /Q)

ˆ FˆT (R) (Q/R) × ˆ FˆS(Q) (P/Q) FˆU (R) × 1

idFˆ

U (R)

9.21.33;

ˆ ∆P /R,S(R),Q/R ×

ˆ FˆS(R) (P/R)Q/R FˆU (R) × similarly, for any ψˆ ∈ FˆT (Q)R and any ϕˆ ¯ ∈ FˆS(R) (P/R)Q/R , we already have ˆ = ωQ,T (ψ) ˆ ˆ Q/R,T (R) ) ∆Q,T,R (ψ) (ωR,U ×ω ˆ ωP/Q,S(Q) ) ∆P/R,S(R),Q/R (ϕ) ˆ ˆ¯ ¯ = ωP/R,S(R)(ϕ) (ωQ/R,T (R) ×

9.21.34;

now, as in equalities 9.21.32, it is easily checked that, for any ϕˆ ∈ FˆS (P )R , we still have ˆ ωP/R,S(R) ) ∆P,S,R (ϕ) (ωR,U × ˆ = ωP,S (ϕ) ˆ 9.21.35, We are done.

Chapter 10

A gluing theorem for Dade P-algebras 10.1 Let P be a ﬁnite p-group and k an algebraically closed ﬁeld of characteristic p . Mimicking the situation in Proposition 7.12 and Theorem 7.14, consider a P -stable nonempty set X of subgroups of P such that any subgroup Q of P containing some R ∈ X belongs to X , and that, for assume ¯P (Q) (cf. 1.20) any Q ∈ X , we have an element s(Q) in the Dade group Dk N ¯P (Qu ) in such a way that s(Qu ) coincides with the image of s(Q) in Dk N by the isomorphism induced by u ∈ P , and that, for any normal subgroup R of Q belonging to X , we have ¯ N ¯ (Q) N ¯ Q,R resN¯¯P (R) s(R) resN¯P s(Q) = Br 10.1.1, Q NQ,R Q,R

¯Q,R and Q ¯ the respective images of NP (Q) ∩ NP (R) where we denote by N ¯ ¯P (R) , and we set N Q,R = N ¯Q,R /Q ¯. and Q in N 10.2 It is clear that, for any s ∈ Dk (P ) , the correspondence mapping N (Q) P Q ∈ X on Br resP Q NP (Q) (s) fulﬁlls the conditions above (cf. equality 1.22.3), and we may ask for the correspondences above coming in this way from an element of Dk (P ) . If P is Abelian, simply all of them (cf. Proposition 3.6 in [45]) is the answer. In this chapter we prove that, whenever p = 2 , in the localized group Dk (P )p over the set {pn }n∈N of powers of p (cf. 8.10) the answer depends on the existence of the so-called balanced functions over the set of nontrivial elementary Abelian subgroups of P (cf. 10.10 below). Actually, for any n ∈ 2Z , prime to p if n = 0 , we may reformulate the analogous question in n Dk (P )p = Dk (P )p n·Dk (P )p 10.2.1 and we obtain the analogous answer with the particularity that when n = 2 the corresponding balanced functions always exist. Although only interested in the cases where n = 0 or n = 2 , we mainly formulate our statements n in Dk (P )p , keeping the same index for the corresponding maps. We need Carlson’s result (cf. statement 8.21.2) that we restate below. Let us call section of P any pair (R, Q) of subgroups of P such that R B Q , and we say that (R, Q) is proper if |Q/R| = |P | . Proposition 10.3 Assume that p = 2 and that P is not elementary Abelian. n Then, in Dk (P )p we have Q ◦ resP = {0} 10.3.1, Ker Br R Q (R,Q)

where (R, Q) runs over the set of proper sections of P .

138

Frobenius categories versus Brauer blocks n

Proof: If the class of s ∈ Dk (P )p in Dk (P )p belongs to this intersection, for P any proper subgroup Q of P we get PQ (s) = n·tQ for some tQ ∈ R◦ Dk (Q)p ; moreover, choosing Q = P and R = Φ(P ) , it follows from Corollary 8.22 that PPP (s) = n·tP for some tP ∈ R◦ Dk (P )p ; further, it is quite clear that we may choose a P -stable family {tQ }Q of such elements when Q runs over the set of subgroups of P ; then, it follows from Proposition 8.11 that there exists t ∈ Dk (P )p such that s = n·t . Remark 10.4 According to statements 8.15.4, 8.16.4 and 8.16.5, whenever P is elementary Abelian, equality 10.3.1 remains true provided |P | ≥ p3 . 10.5 Note that it suﬃces to formulate our question when X is the set ¯P (R) for of all the nontrivial subgroups of P , since otherwise we argue in N a subgroup R of P which is maximal outside X . Moreover, for any Q ∈ X , it is clear that we have an analogous correspondence, fulﬁlling the above conditions, for the set XQ of subgroups of Q in X . Thus, in the last step of a suitable induction argument, we may apply the following result. Theorem 10.6 Assume that p = 2 and that P is not elementary Abelian. n For any proper section (R, Q) of P , let s¯(R, Q) be an element of Dk (Q/R)p in such a way that the following conditions hold 10.6.1 For any u ∈ P , s¯(Ru , Qu ) is the image of s¯(R, Q) by the isomorphism n n ¯P (Q/R) ∼ ¯P (Qu /Ru ) Dk N = Dk N p p induced by the conjugation by u−1 . ¯ = Q/R 10.6.2 For any normal subgroup T of Q containing R , setting Q and T¯ = T /R , we have ¯ Q ¯ s¯(R, Q) = s¯(T, Q) Br T

and

¯ resQ s¯(R, Q) = s¯(R, T ) . T¯

n

Then there is a unique element s¯ ∈ Dk (P )p such that Q resP (¯ Br ¯(R, Q) R Q s) = s

10.6.3

for any proper section (R, Q) of P . Proof: For any proper section (R, Q) of P , note that either Q is contained in a maximal subgroup of P or R is normal in P , and therefore, in order to prove equality 10.6.3, it suﬃces to check that P

(¯ resP s) = s¯(1, M ) and Br ¯(N, P ) 10.6.4 N s) = s M (¯ for any maximal subgroup M of P and, setting Z = Ω1 Z(P ) , for any nontrivial subgroup N of Z .

10. A gluing theorem for Dade P -algebras

139

Thus, let us say that a section (R, Q) of P is dominant if R⊂Z ⊂QBP

10.6.5;

if (R, Q) and (R , Q ) are two dominant sections of P , then we say that (R , Q ) is smaller than (R, Q) and write (R , Q ) ≤ (R, Q)

10.6.6

whenever R ⊂ R ⊂ Q ⊂ Q , which deﬁnes an order relation in the set of dominant sections. Moreover, setting (R, Q) ∩ (R , Q ) = (R·R , Q ∩ Q )

10.6.7,

it is quite clear that (R, Q)∩(R , Q ) is smaller than both (R, Q) and (R , Q ) ; conversely, if (U, T ) is a dominant section of P such that (U, T ) ≤ (R, Q) and then we have

(U, T ) ≤ (R , Q )

10.6.8

(U, T ) ≤ (R, Q) ∩ (R , Q )

10.6.9.

Consider the M¨ obius function µD (cf. 1.32) over the set D of dominant proper sections of P endowed with the opposite order; that is to say, µD (1, P ) = 1 and, for any dominant proper section (U, T ) , we have

µD (R, Q) = 0

10.6.10,

(R,Q)

where (R, Q) runs over the set of dominant sections of P such that (U, T ) is smaller than (R, Q) . Then, for any pair of dominant proper sections (N, M ) and (U, T ) of P we claim that

(N,M )

µD (R, Q) = −δ(U,T )

10.6.11,

(R,Q)

when (R, Q) runs over the set of dominant proper sections such that (N, M ) ∩ (R, Q) = (U, T )

10.6.12.

Indeed, we may assume that (U, T ) ≤ (N, M ) and then we have

(U ,T ) (R ,Q )

µD (R , Q ) =

(R ,Q )

µD (R , Q ) = −1

10.6.13,

140

Frobenius categories versus Brauer blocks

where (U , T ) runs over the set of dominant sections of P such that (U, T ) ≤ (U , T ) ≤ (N, M )

10.6.14

and, for such a (U , T ) , (R , Q ) runs over the set of dominant proper sections of P such that (N, M ) ∩ (R , Q ) = (U , T ) 10.6.15, so that ﬁnally (R , Q ) runs over the set of all the dominant proper sections of P ; thus, arguing by induction on |T /U | , we may assume that

µD (R , Q ) = −δ(U ,T ) (N,M )

10.6.16

(R ,Q )

for any (U , T ) = (U, T ) fulﬁlling condition 10.6.14 and then it is clear that equality 10.6.13 forces equality 10.6.11. Now, we claim that it suﬃces to choose s¯ =

|Q| res µD (R, Q)·tenP s ¯ (R, Q) π Q (R,Q) |P |

10.6.17,

(R,Q)

where (R, Q) runs over the set of dominant proper sections of P and, for such a (R, Q) , π(R,Q) : Q → Q/R is the canonical map. Indeed, for any nontrivial subgroup N of Z we get P

(¯ Br N s) =

|Q| P ◦ tenP ) res π s ¯ (R, Q) µD (R, Q)·(Br N Q (R,Q) |P |

10.6.18,

(R,Q)

where (R, Q) runs over the set of dominant proper sections of P . But, according to equality 8.8.3 and to Corollary 8.22, we still get P ◦ tenP ) res π (Br s ¯ (R, Q) N Q (R,Q) Q P/N ) res π = (tenQ/N ◦ Br s ¯ (R, Q) N (R,Q) Q/R P/N = (tenQ/N ◦ res π((N ·R)/N,Q/N ) ) Br s ¯ (R, Q) (N ·R)/R

10.6.19;

moreover, by condition 10.6.2, we have Q/R Br ¯(R, Q) = s¯(N ·R, Q) (N ·R)/R s

10.6.20.

10. A gluing theorem for Dade P -algebras

141

¯ the quotient by N of any subgroup X of Consequently, denoting by X P containing N , from equality 10.6.11 we obtain P

(¯ Br N s) =

|Q| ¯ µD (R, Q)·(tenP ¯(R·N, Q) ¯ ◦ resπ(R·N ,Q) ) s Q |P |

(R,Q)

|Q | ¯ s ¯ (U, T ) res µD (R , Q ) ·tenP = ¯ π ¯ ,T ¯) (U T |P |

10.6.21,

(U,T ) (R ,Q )

= s¯(N, P ) where (R, Q) runs over the set of all the dominant proper sections of P , (U, T ) runs over the set of those which are smaller than (N, P ) and, for such a section (U, T ) , (R , Q ) runs over the set of dominant proper sections of P fulﬁlling 10.6.22 (N, P ) ∩ (R , Q ) = (U, T ) which implies that Q = T . It remains to prove that res P s) = s¯(1, M ) for any maximal subgroup M M (¯ of P . Firstly assume that M contains Z ; it follows from the Mackey formula 8.8.1 that |Q| P µD (R, Q)·(resP ¯(R, Q) M ◦ tenQ ) resπ(R,Q) s |P |

res P s) = M (¯

(R,Q)

10.6.23, |Q| u u = res s ¯ (R µD (R, Q)· tenM , M ∩ Q ) u π(Ru,M ∩Qu ) M ∩Q |P | u (R,Q)

where (R, Q) runs over the set of all the dominant proper sections of P and u over a set of representatives in M for P \M/Q . Hence, since Z ⊂ M , it follows from conditions 10.6.1 and 10.6.2, and from equality 10.6.11 that s) resP M (¯ |M ∩ Q| res µD (R, Q)·tenM s ¯ (R, M ∩ Q) = π(R,M ∩Q) M ∩Q |M | (R,Q)

|M ∩ Q | ¯(R, T ) µD (R, Q ) ·ten M = T resπ(R,T ) s |M | (R,T )

10.6.24,

Q

= s¯(1, M ) where (R, Q) runs over the set of all the dominant proper sections of P , (R, T ) runs over the set of those which are smaller than (1, M ) or, equivalently, fulﬁll

142

Frobenius categories versus Brauer blocks

T ⊂ M , and for such a (R, T ) , Q runs over the set of normal subgroups of P such that M ∩ Q = T or, equivalently, fulﬁll (1, M ) ∩ (R, Q ) = (R, T )

10.6.25.

Now, assume that M does not contain Z , so that we have P = Z·M . Let L be a subgroup of P such that P = Z·L , so that L is not elementary Abelian and we have Φ(L) = L ∩ Φ(P ) and

Ω1 Z(L) = L ∩ Z

10.6.26;

arguing by induction on |L ∩ Z| , we will prove that resP s) = s¯(1, L) . L (¯ Let (R, Q) be a proper section of L . If Q = L and K is a maximal subgroup of L containing Q , then either K contains L ∩ Z , so that K is contained in a maximal subgroup H of P containing Z , and therefore, by condition 10.6.2, we get P Q Q H resP (¯ Br s) R Q s) = BrR resQ resH (¯ Q resH s¯(1, H) = Br R Q Q Q s¯(1, Q) = Br resL s¯(1, L) = Br R R Q

10.6.27,

or we have L = (L∩Z)·K , so that P = Z·K , and therefore, by the induction hypothesis, we get res P s) = s¯(1, K) , so that we still get K (¯ P Q Q K resP (¯ Br res res s ) = Br (¯ s ) R R Q Q K Q resK s¯(1, K) = Br Q R Q Q s¯(1, Q) = Br resL s¯(1, L) = Br R R Q

10.6.28.

Finally, if Q = L then R is a nontrivial normal subgroup of P and, in ¯ = Q/N and R ¯ = R/N particular, we have N = R ∩ Z = 1 , so that setting Q we get (cf. 10.6.21 and equality 1.22.3) ¯ ¯ Q Q Q Q resP (¯ resP (¯ ¯ Br ¯ s¯(N, Q) Br s ) = Br s ) = Br R N Q Q R R

Q s¯(1, Q) . = s¯(R, Q) = Br R

10.6.29.

In conclusion, the diﬀerence resP s) − s¯(1, L) belongs to the intersection L (¯ in 10.3.1 for the p-group L ; hence, it follows from Proposition 10.3 applied

10. A gluing theorem for Dade P -algebras

143

to L that resP s) = s¯(1, L) . Thus, the element s¯ in 10.6.17 fulﬁlls stateL (¯ ment 10.6.3. n

Similarly, if s¯ ∈ Dk (P )p is another element fulﬁlling statement 10.6.3, then the diﬀerence s¯ − s¯ belongs to the left-hand member of equality 10.3.1 for P and therefore we have s¯ = s¯ . We are done. 10.7 Recall that when P is elementary Abelian, for any s ∈ Dk (P )p we have (cf. equality 8.17.1) P s= zQ ·jQ 10.7.1, Q

where Q runs over the set of proper subgroups of P and, for such a Q , zQ is an element of Zp , uniquely determined if |P/Q| ≥ p2 ; then, if |P | ≥ p2 , we denote by εP : Dk (P )p −→ Zp 10.7.2 the homomorphism mapping s on z1 , whereas if |P | = p we denote by 2

εP : Dk (P )p −→ Z = Zp /2Zp

10.7.3 2

the group homomorphism mapping s on the image z¯1 of z1 in Z , which is uniquely determined. Accordingly, we denote by εnP (s) the image of εP (s) n

2

n

in Zp = Zp /nZp or in Z , deﬁning a map εnP over Dk (P )p . Lemma 10.8 If p = 2 and P is elementary Abelian, for any s ∈ Dk (P )p we have P P (s) ·j s= εP /T Br 10.8.1 T T T

where T runs over the set of subgroups of P . Moreover, if Q and R are noncyclic subgroups of P , then we have P (s) εQ resP εP /T Br T Q (s) + T

= εR

resP R (s)

+

εP /U

P (s) Br U

10.8.2,

U

where T and U respectively run over the set of subgroups of P fulﬁlling T ∩ R = 1 = T ∩ Q

and

U ∩ Q = 1 = U ∩ R

10.8.3. 2

If they are nontrivial then the analogous equality with the images in Z of all the terms holds.

144

Frobenius categories versus Brauer blocks

Proof: It follows from equality 10.7.1 that, for any nontrivial proper subgroup T of P , we have P

(s) = Br T

P/T

zQ ·j Q/T

10.8.4,

Q

where Q runs over the set of proper subgroups of P containing T , and, P (s) coincides with z¯T or according to deﬁnitions 10.7.2 and 10.7.3, ε Br P /T

T

with zT according to P/T is cyclic or not; then, equality 10.8.1 follows from equality 10.7.1. Moreover, for any nontrivial subgroup Q of P , we have Q P resP Q (jT ) = jT ∩Q

10.8.5

and therefore, if Q is not cyclic, we get P (s) + ε (s) εQ resP zT = εP /T Br Q (s) = T P T

10.8.6,

T =1

whereas, in all the cases we still get P (s) + ε2 (s) ε2Q resP z¯T = ε2P /T Br T Q (s) = P T

10.8.7,

T =1

where T runs over the set of subgroups of P such that T ∩ Q = 1 . Now, if R is another nontrivial subgroup of P , the diﬀerence between the equalities 10.8.6 corresponding to Q and R yields equality 10.8.2 whenever they are not cyclic, whereas in all the cases the sum of the equalities 10.8.7 2 corresponding to Q and R yields the analogous equality in Z . We are done. 10.9 Let us come back to our starting setting and call (P, X, n)-gluing n ¯p (Q) fulﬁlling family any family ¯s = {¯ s(Q)}Q∈X of elements s¯(Q) ∈ Dk N p ¯ (Q) N

resN¯P

Q,R

¯ N ¯ Q,R resN¯¯P (R) s¯(R) s¯(Q) = Br Q NQ,R

10.9.1

¯Q,R and Q ¯ the for any R ∈ X such that R B Q , where we denote by N ¯P (R) , and where we set respective images of NP (Q) ∩ NP (R) and Q in N ¯ ¯ ¯ N Q,R = NQ,R /Q . As we show below, in order to determine a unique element n s¯ ∈ Dk (P )p from the family ¯s when X is the set of all the nontrivial subgroups of P , we also need the function mapping any nontrivial elementary Abelian P (¯ s) (cf. 10.7). We write Br subgroup E of P on εnE resP Q s) instead of E (¯ NP (Q) Br s) for short. resP Q NP (Q) (¯

10. A gluing theorem for Dade P -algebras

145

10.10 Let us be more explicit; denote by A(P ) the set of nontrivial elementary Abelian subgroups of P and set ¯ NP (F ) ε¯s (E, F ) = εnE/F resE/F s¯(F ) 10.10.1 for any E, F ∈ A(P ) such that F ⊂ E ; having regard to Lemma 10.8, we call (P, ¯s)-balanced function ε any P -stable map sending E ∈ A(P ) to an element 2

n

of Z or of Zp according to whether E is cyclic or not, in such a way that, for any pair of elements E and F of A(P ) centralizing each other, if they are n noncyclic then in Zp we have ε(E) + ε¯s (E·F, T ) = ε(F ) + ε¯s (E·F, U ) 10.10.2 T

U

where T and U respectively run over the set of subgroups of E·F fulﬁlling T ∩ F = 1 = T ∩ E

and U ∩ E = 1 = U ∩ F

10.10.3,

whereas in all the cases it holds the analogous equality with the images of all 2 the terms in Z . Proposition 10.11 Assume that p = 2 and that X is the set of all the nontrivial subgroups of P , and let ¯s = {¯ s(Q)}Q∈X be a (P, X, n)-gluing fan mily. For any (P, ¯s)-balanced function ε there is a unique element s¯ ∈ Dk (P )p fulﬁlling P

(¯ 10.11.1 For any nontrivial subgroup Q of P , we have Br ¯(Q) . Q s) = s 10.11.2 For any nontrivial elementary Abelian subgroup E of P , we have εnE resP s) = ε(E) E (¯ Proof: If P is elementary Abelian, it follows from equality 10.8.1 and from conditions 10.11.1 and 10.11.2 above that the good choice of s¯ should be P s¯ = εnP /T s¯(T ) ·¯ P 10.11.3, T + ε(P )·¯ 1 T

where T runs over the set of nontrivial subgroups of P . Indeed, for any nontrivial subgroup Q of P , we have (cf. 10.9.1) P

(¯ Br Q s) =

T

=

P/Q εnP /T s¯(T ) ·¯ T /Q P P/Q s¯(Q) ·¯ εnP /T Br T /Q = s¯(Q) T

T

where T runs over the set of subgroups of P containing Q .

10.11.4,

146

Frobenius categories versus Brauer blocks Moreover, for any nontrivial subgroup E of P , we have resP s) = E (¯

E εnP /T s¯(T ) ·¯ E T ∩E + ε(P )·¯ 1

10.11.5,

T

where T runs over the set of nontrivial subgroups of P ; in particular, if E is not cyclic, we get n εnE resP s) = εP /T s(T ) + ε(P ) E (¯

10.11.6,

T

where T runs over the set of nontrivial subgroups of P fulﬁlling T ∩ E = 1 , whereas in all cases it holds the analogous equality with the images of all the 2 terms in Z . Thus, we have εnE resP s) = ε(E) E (¯

10.11.7

since we are assuming that ε is (P, ¯s)-balanced (cf. 10.10.2). From now on, we assume that Φ(P ) = 1 and argue by induction on |P | . For any nontrivial proper subgroup Q of P , denoting by XQ the set of all the nontrivial subgroups of Q , it is easily checked that $ ¯P (T ) % ¯sQ = resN ¯(T ) T ∈X ¯Q (T ) s N

Q

10.11.8

is a (Q, XQ , n)-gluing family; hence, considering the (Q, ¯sQ )-balanced function εQ determined by ε , it follows from our induction hypothesis that there n is a unique element s¯Q ∈ Dk (Q)p fulﬁlling Q ¯P (T ) N (¯ Br ¯(T ) ¯Q (T ) s T sQ ) = res N

sQ ) = ε(F ) and εnF resQ F (¯

10.11.9

for any nontrivial subgroups T and F of Q , F being elementary Abelian. Now, if Q and R are nontrivial proper subgroups of P such that R ⊂ Q , we claim that resQ sQ ) = s¯R 10.11.10; R (¯ indeed, it is straightforward to prove that resQ sQ ) and s¯R determine the R (¯ same (R, XR , n)-gluing family; moreover, for any nontrivial elementary Abelian subgroup F of R , we have (cf. 10.11.9) εnF resQ sQ ) = ε(F ) = εnF resR sR ) F (¯ F (¯ hence, the claim follows from our induction hypothesis.

10.11.11;

10. A gluing theorem for Dade P -algebras

147

Then, for any proper section (R, Q) of P , we denote by s¯(R, Q) either Q ¯ (R) N BrR (¯ s¯(R) if R = 1 ; it follows from equalisQ ) if Q = P , or resQ¯ P ties 10.11.9 and 10.11.10 that the family {¯ s(R, Q)}(R,Q) , where (R, Q) runs over the set of all the proper sections of P , is well-deﬁned and fulﬁlls conditions 10.6.1 and 10.6.2. n

Consequently, by Theorem 10.6, there is a unique element s¯ ∈ Dk (P )p fulﬁlling Q resP (¯ Br ¯(R, Q) 10.11.12 R Q s) = s for any proper section (R, Q) of P ; in particular, this element fulﬁlls condition 10.11.1 and moreover, for any nontrivial elementary Abelian subgroup E of P , we have resP s) = s¯E , so that we still have E (¯ εnE resP s) = εnE (¯ sE ) = ε(E) Q (¯

10.11.13.

Finally, the uniqueness of s¯ follows from equality 10.11.3 whenever P is elementary Abelian, and from our induction argument together with Theorem 10.6 otherwise. We are done. Corollary 10.12 Assume that p = 2 . Consider a (P, X, 2)-gluing family 2 ¯s = {¯ s(Q)}Q∈X and an element z¯R ∈ Z for any subgroup R of P outside X . 2

Then, there exists a unique s¯ ∈ Dk (P )p such that 10.12.1 10.12.2 we have

P

(¯ For any Q ∈ X , we have Br ¯(Q) . Q s) = s

¯P (R)) For any subgroup R of P outside X , setting Z¯R = Ω1 Z(N

ε2Z

R

P ¯ (R) N (¯ Br R s)

resZ¯RP

= z¯R .

Proof: We may assume that X does not contain the trivial subgroup; let R be ¯ ¯ =N ¯P (R) , denote by Y a subgroup of P which is maximal outside X , set N ¯ ¯ ¯ the set of all the nontrivial subgroups of N and consider the (N , Y, 2)-gluing ¯ Q∈ family ¯t = {t¯(Q)} ¯ with ¯ Y ¯ = resN¯¯P (Q) t¯(Q) ¯(Q) ¯ s N ¯ (Q) N

10.12.3

¯ in NP (R) and where we identify N ¯N¯ (Q) ¯ where Q is the converse image of Q ¯ with its canonical image in NP (Q) . ¯ ) , it easily follows from condition 10.10.2 over Setting Z¯ = Ω1 Z(N ¯ , ¯t)-balanced function ε such that ε(Z) ¯ = z¯R ; Z/2Z that there is a unique (N ¯) thus, according to Proposition 10.11, there is a unique element s¯(R) ∈ Dk (N

148

Frobenius categories versus Brauer blocks

such that, setting X = X ∪ {Ru }u∈P , the extended family ¯s = {¯ s(Q)}Q∈X is a (P, X , 2)-gluing family too and that we have

ε2Z

R

¯P resN s) = z¯R ¯ BrR (¯ Z

10.12.4. 2

Hence, arguing by induction on |X| , we get the existence of s¯ ∈ Dk (P )p 2

fulﬁlling conditions 10.12.1 and 10.12.2; moreover, if s¯ ∈ Dk (P )p fulﬁlls the P (¯ same conditions, it follows from the argument above that Br ¯(R) ; R s ) = s then, when R = 1 we get s¯ = s¯ . We are done. Corollary 10.13 Assume that p = 2 , that P is not cyclic and that X is the set of all the nontrivial subgroups of P . Set Z = Ω1 Z(P ) and let D be a normal elementary Abelian subgroup of P of order p2 . Consider a (P, X, 0)-gluing family s = {s(Q)}Q∈X such that 2·s(Q) = 0 for any nontrivial subgroup Q of P , and a P -stable function mapping any elementary Abelian subgroup E of P of order p2 such that either E = D or Ω1 CP (E) = E on some element zE ∈ Zp . Then, there exists a unique s ∈ Dk (P )p such that P

(s) = s(Q) . For any nontrivial subgroup Q of P , we have Br Q P 10.13.2 We have εD resD (s) = zD and, for any elementary Abelian sub group E = D of P of order p2 such that Ω1 CP (E) = E , denoting by aE the element in {0, 1} fulﬁlling 10.13.1

a2E = ε2Z resP εs (E, F ) Z (s) + F

where F runs over the set of proper subgroups of E not contained in Z , we have εE resP (s) = aE + 2zE . E Proof: First of all, note that our hypothesis on s implies that εs (E, F ) = 0 for any E, F ∈ A(P ) such that F ⊂ E and |E/F | ≥ p2 (cf. 8.16.5, 10.7 and 10.10.1). If |Z| ≥ p2 then condition 10.10.2 and this remark forces any (P, s)-balanced function ε to fulﬁll the equality ε(E) = ε(Z) for any noncyclic elementary Abelian subgroup E of P ; thus, the value zD and the analogous condition 10.10.2 over Z/2Z determine a unique (P, s)-balanced function ε and, according to the remark above, its existence is clear. More2 over, if E is an elementary Abelian subgroup of P of order p such that Ω1 CP (E) = E , we have E = Z = D . Hence, in this case, the corollary follows from Proposition 10.11. Assume that |Z| = p , so that Z ⊂ D ; once again, condition 10.10.2 and our hypothesis on s forces any (P, s)-balanced function ε to fulﬁll the equality ε(E) = ε(D) for any noncyclic elementary Abelian subgroup E of CP (D) .

10. A gluing theorem for Dade P -algebras

149

2 Moreover, elementary Abelian subgroup E of P of order p such for any that Ω1 CP (E) = E , it is quite clear that the intersection CP (E) ∩ CP (D) contains an elementary Abelian subgroup F of order p2 and that the same argument forces ε(E) = ε(F ) = ε(D) 10.13.3. If Ω1 CP (E) = E then E is a maximal elementary Abelian subgroup 2

of P and the analogous condition 10.10.2 over Z forces 2

ε(E) = ε(Z) +

εs (E, F )

10.13.4

F

where F runs over the set of proper subgroups of E not contained in Z . Consequently, in all the cases denoting by aE the element in {0, 1} such that a2E coincides with right-hand member of equality 10.13.4, it is easily checked from the remark above that there exists indeed a unique (P, s)-balanced function ε fulﬁlling ε(D) = zD and ε(E) = aE + 2·zE for any such a subgroup E . Then, the corollary follows from Proposition 10.11. We are done. Remark 10.14 In [7] Serge Bouc and Jacques Th´evenaz prove a more precise result. Note that, even ﬁxing zD = 0 = zE for all the subgroups E as above, 2·s need not be zero since we have no control on the invariants aE .

Chapter 11

The nilcentralized chain k*-functor of a block 11.1 Let us come back to our setting in chapter 7. Let G be a ﬁnite group, b a block of G , (P, e) a maximal Brauer (b, G)-pair and F(b,G) the nc associated Frobenius P -category, and let us denote by (F(b,G) ) the full subcategory of F(b,G) over the set of nilcentralized objects. In chapter 7 we have seen that, for any F(b,G) -nilcentralized subgroup Q of P , the group F(b,G) (Q) can be canonically lifted to a k ∗ -group Fˆ(b,G) (Q) . In this chapter, we discuss the possibility of lifting whole subcategories of (F(b,G) ) to “k ∗ -categories”†. (F nc from the These results allow us to construct a suitable functor aut (b,G) ) nc

nc

proper category of chains of (F(b,G) ) (cf. A2.8) to the category of ﬁnite k ∗ -groups k ∗ -Gr , which lifts the automorphism functor aut(F(b,G) )nc introduced in Proposition A2.10 below. This is the main tool for the deﬁnition of (F nc ) in chapter 14. the Grothendieck group of the pair (F(b,G) , aut (b,G) ) 11.2 More explicitly, for any category C and any Abelian group Z let ˆ over the same objects us call regular central Z-extension of C any category C ˆ → C which is the identity over the objects, endowed with a full functor c : C and, for any pair of C-objects A and B , with a regular action of Z over the ﬁbers of the map ˆ C(A, B) −→ C(A, B) 11.2.1 induced by c , in such a way that these Z-actions are compatible with the comˆ position of C-morphisms. Note that, if C is a second category and f : C → C an equivalence of categories, we easily can obtain a regular central Z-extension ˆ of C and a Z-compatible equivalence of categories ˆf : C ˆ →C ˆ . For short, C ∗ ∗ we call k -category any regular central k -extension of a category. 11.3 Let H be a subgroup of G , β a point of H on kGb (cf. 1.12) and Qδ a defect pointed group of Hβ (cf. 1.13), and consider the divisible Q-category F(β,H) (cf. 3.12); recall that we have deﬁned a unique isomorphism class of faithful functors (cf. 3.12) ibβ : F(β,H) −→ F(b,G)

†

11.3.1.

For the full subcategory over the selfcentralizing objects, the question on the existence of

such an extension was raised by Markus Linckelmann at Durham in 2002.

152

Frobenius categories versus Brauer blocks

Actually, in the second part of this chapter, it will be more handy to follow the e e point of view of Remark 3.10, replacing F(β,H) and F(b,G) by F(β,H) and F(b,G) respectively, so that we simply have an inclusion e e F(β,H) ⊂ F(b,G)

11.3.2.

11.4 Assuming that Q is F(b,G) -nilcentralized, let X be a nonempty set of F(b,G) -nilcentralized subgroups of Q containing any subgroup R of Q such that F(β,H) (R, T ) = ∅ for some T ∈ X , and denote by F(β,H,X) the full subcategory of F(β,H) over X . In this chapter, we discuss some suﬃcient conditions to guarantee the existence of a regular central k ∗ -extension Fˆ(β,H,X) of F(β,H,X) such that, assuming that (P, e) contains the Brauer (b, G)-pair determined by Qδ (cf. 1.15), for any R ∈ X the inclusion F(β,H) (R) ⊂ F(b,G) (R) can be lifted to a (noncanonical!) k ∗ -group homomorphism (cf. 7.4.3) Fˆ(β,H,X) (R) −→ Fˆ(b,G) (R)

11.4.1.

Remark 11.5 Note that, since k ∗ is a p-divisible group and the category F(b,G) only involves a ﬁnite number of ﬁnite groups, there is n ∈ N such that, for any F(b,G) -nilcentralized subgroup R of P and any m ∈ N , we have a k ∗ -group isomorphism nm

∗p

Fˆ(b,G) (R) ∼ = Fˆ(b,G) (R)

11.5.1,

where ∗p Fˆ(b,G) (R) denotes the sum as k ∗ -extensions of pnm copies of the k ∗ -extension Fˆ(b,G) (R) (cf. 1.24); hence, it follows from Proposition 5.11 nm

in [41] that, in our discussion, we may replace the H-algebra kGb by its pnm pnm -th tensor power kGb . 11.6 In order to simplify our notation, we modify our choice of (P, e) in such a way that it contains the Brauer (b, G)-pair determined by Qδ (cf. 1.15). Our suﬃcient conditions are expressed in terms of a suitable Dade Q-algebra (cf. 1.20) or, equivalently, in terms of a suitable element in the Dade group Dk (Q) (cf. 1.21), and it is handy to name each condition. We say that an element s ∈ Dk (Q) is F(β,H,X) -stable if it fulﬁlls 11.6.1 For any R ∈ X and any ϕ ∈ F(β,H) (Q, R) we have resϕ (s) = resQ R (s) . Our second condition the family {sR }R we have introduced in 7.7 of involves ¯P (R) where R runs over the set of F(b,G) -nilcentralized elements sR ∈ Dk N subgroups of P . For any n ∈ N , we say that s ∈ Dk (Q) is a (X, n)-gluing element if it fulﬁlls N (R) ¯ (R) N Q 11.6.2 For any R ∈ X we have Br (s) = pn ·res ¯P (sR ) . resQ R

NQ (R)

NQ (R)

Of course, we are only interested in the integers n ∈ N fulﬁlling n ∗p Fˆ(b,G) (R) ∼ = Fˆ(b,G) (R)

called (b, G)-admissible integers.

11.6.3,

11. The nilcentralized chain k ∗ -functor

153

Remark 11.7 In condition 11.6.1, we could avoid the choice of Qδ by cone sidering the obvious contravariant functor dβk : F(β,H) → Ab mapping any local pointed group Rε on kGb contained in Hβ on the Dade group Dk (R) e and any F(β,H) -morphism on the corresponding restriction homomorphism. Then, moving from X to the set Xe of nilcentralized Brauer (β, H)-pairs e F(β,H) -isomorphic to some element of X , condition 11.6.1 simply says that s belongs to the inverse limit of the restriction of dβk to the full subcategory e e F(β,H,X e ) over X (cf. 11.15 below). Proposition 11.8 With the notation above, ﬁx n ∈ N , let S and S be Dade Q-algebras determining (X, n)-gluing elements in Dk (Q) and set n p ˆ = S ◦ ⊗k B (kGb)δ and S = S ◦ ⊗k S 11.8.1. ˆ ˆ and εS on S . Any R ∈ X has unique local points ε on (kGb)δ , εB on B ˆ ˆ and S , Then, denoting by δ B and δ S the respective local points of Q on B we have ˆ Bˆ (R Bˆ ) ∼ B 11.8.2. =k∼ = (S )δS (R) δ ε

Proof: Since any R ∈ X stabilizes bases on S and on kGb , it is well-known that (cf. Proposition 5.6 in [41]) pn ∼ ˆ B(R) (kGb)δ (R) 11.8.3. = S(R)◦ ⊗k Moreover, since R is F(b,G) -nilcentralized, the unique block g of CG (R) such that (P, e) contains (R, g) (cf. 1.15) is nilpotent (cf. Proposition 7.2) and therefore, up to isomorphism, there is a unique simple kCG (R)g-module (cf. 1.13 and isomorphism 1.18.1) or, equivalently, a unique local point ε of R on kGb fulﬁlling gBrR (ε) = {0} (cf. isomorphism 1.16.1). In particular, we have Rε ⊂ Qδ (cf. 1.15) and ε is the unique local point of ˆ ˆ (cf. Proposition 5.6 R on (kGb)δ , so that R has a unique local point εB on B and Corollary 5.8 in [41]) and we still have pn ◦ ∼ ˆ B(R (kGb)δ (Rε ) 11.8.4. ˆ ) = S(R) ⊗k εB Moreover, it follows from the deﬁnition of SR and sR in 7.7 that we have ¯Q (R)-algebra embedding SR → (kGb)δ (Rε ) (cf. embedding 1.12.2) and, an N ¯Q (R)-algebra S(R)◦ ⊗k pn SR determines by condition 11.6.2, the Dade N ¯Q (R) (cf. 1.21), so that N ¯Q (R) the trivial element in the Dade group Dk N ﬁxes a primitive idempotent in this simple k-algebra. On the other hand, since S is a Dade Q-algebra (cf. 1.20), R has a unique local point εS on S ¯Q (R) ﬁxes a primitive (cf. 1.20) and a similar argument holds, proving that N idempotent in S (RεS ) = S (R) .

154

Frobenius categories versus Brauer blocks

ˆ Bˆ or D = (S ) S ; in any case, in order to prove Consider either D = B δ δ isomorphisms 11.8.2, it suﬃces to prove that the unity element is primitive in the k-algebra D(R) for any R ∈ X ; we argue by induction on |Q : R| and the statement is clear when R = Q . Otherwise we may assume that, for any subgroup T of NQ (R) strictly containing R , the unity element is primitive in ¯Q (R) ﬁxes a primitive the k-algebra D(T ) . But, we have proved above that N idempotent D¯ in D(R) and then, setting T¯ = T /R , the primitivity of D¯ forces D(R) ¯ BrT¯ (D) = 0 (cf. Theorem 7.2 in [51]); hence, we get D(R)

BrT¯

¯ =0 (1 − D)

11.8.5.

That is to say, the idempotent 1 − D¯ of D(R) belongs to the intersection

D(R)

Ker(BrT¯

¯ (R) N

Q ) = D(R){1}

Q = BrD R (DR )

11.8.6

T¯

¯Q (R) (cf. Lemmas 1.11 where T¯ runs over the set of nontrivial subgroups of N and 1.12 in [11]); but, the unity is primitive in DQ and does not belong to Q Ker(BrD Q ) ⊃ DR since the involved point of Q is local, so that we have DQ ⊂ J(DQ ) ; hence, we still have 1 = D¯. We are done. R

11.9 In the situation of this proposition, if moreover the element of Dk (Q) determined by S is F(β,H,X) -stable, we claim that we can construct a regular central k ∗ -extension Fˆ(β,H,X) of F(β,H,X) ; before proving it, let us explicitly describe this construction. With the notation above, set pn B = S ◦ ⊗k kGb δBˆ

11.9.1

and, for any R ∈ X , denote by MR the kernel of the homomorphism B R → k determined by the unique local point of R on B (cf. isomorphism 11.8.2), and choose an idempotent iR in this point. For any T ∈ X and any F(β,H) -morphism ϕ : T → R , denote by TBϕ (R, T ) the set of elements aiT where a ∈ B ∗ fulﬁlls iT (iR )a = iT = (iR )a iT aiT v = ϕ(v)aiT and viT a−1 = iT a−1 ϕ(v) for any v ∈ T

11.9.2,

and set T = ϕ(T ) ; it is easily checked that the group iR + iR MT iR acts on the set TBϕ (R, T ) by the multiplication on the left; then, what shall ϕ (R, T ) of ϕ in our claimed regular central be the converse image Fˆ(β,H,X) ∗ ˆ k -extension F(β,H,X) is deﬁned by the quotient set ϕ Fˆ(β,H,X) (R, T ) = (iR + iR MT iR )\TBϕ (R, T )

11.9.3.

11. The nilcentralized chain k ∗ -functor

155

Theorem 11.10 With the notation above, assume that there exists a Dade Q-algebra S determining a F(β,H,X) -stable (X, n)-gluing element s in Dk (Q) for some admissible integer n . Then, there exists a unique regular central k ∗ -extension Fˆ(β,H,X) of F(β,H,X) such that, for any R, T ∈ X , the converse image of ϕ ∈ F(β,H,X) (R, T ) in Fˆ(β,H,X) (R, T ) coincides with the quotient set (iR + iR MT iR )\TBϕ (R, T )

11.10.1,

n that the composition is induced by the product in S ◦ ⊗k p kGb and that the inclusion F(β,H,X) (R) ⊂ F(b,G) (R) can be lifted to a k ∗ -group homomorphism Fˆ(β,H,X) (R) −→ Fˆ(b,G) (R)

11.10.2.

Moreover, the k ∗ -category Fˆ(β,H,X) does not depend on the choice of S . ˆ ˆ = S ◦ ⊗k pn (kGb)δ Proof: Denote by δ B the unique local point of Q on B (cf. Proposition 11.8) and set pn B = S ◦ ⊗k (kGb)δ δBˆ 11.10.3. Let R and T be elements of X and respectively denote by ε , εS , µ and µS the local points of R and T on (kGb)δ and S (cf. Proposition 11.8); we claim that the F(β,H,X) -stability of s forces F(β,H) (R, T ) ⊂ FS (RεS , TµS )

11.10.4;

indeed, for any ϕ ∈ F(β,H) (R, T ) , the F(β,H,X) -stability of s implies that the Dade T -algebras SµS and Resϕ (SεS ) are similar (cf. 1.21) or, equivalently, that there exists a T -algebra embedding fϕ : SµS → Resϕ (SεS ) (cf. 1.21) and, since S is a simple algebra, this amounts to saying that ϕ is an S-fusion from TµS to RεS (cf. 1.28 above and Corollary 2.4 in [37]). On the other hand, it follows from Theorem 3.1 in [39] that (cf. 1.28) F(b,G) (R, T ) = FkGb (Rε , Tµ )

11.10.5;

similarly, since (cf. Proposition 5.6 in [41]) pn pn kGb (R) ∼ (kGb)(R) = pn n p kGb (T ) ∼ (kGb)(T ) = n

11.10.6, n

ε and µ respectively determine local points ⊗p ε of R and ⊗p µ of T on pn kGb and then, the above result applied to the pn -th direct product pn × G and to the diagonal p-subgroups determined by R and T yields F(b,G) (R, T ) = F⊗pn kGb (R⊗pnε , T⊗pnµ )

11.10.7.

156

Frobenius categories versus Brauer blocks ˆ

ˆ

Moreover, denoting by εB , εB , µB and µB the respective local points of R ˆ (cf. Proposition 11.8), recall that (cf. Proposition 2.14 and T on B and B in [39]) FB (RεB , TµB ) = FBˆ (RεBˆ , TµBˆ ) 11.10.8 ˆ

ˆ

and that we actually identify εB with εB and µB with µB . Consequently, it follows from inclusion 11.10.4, equality 11.10.7 and Theorem 5.3 in [41] that F(β,H) (R, T ) ⊂ FB (RεB , TµB )

11.10.9.

In 11.9 above, we have chosen iR ∈ εB and iT ∈ µB , and then, for any homomorphism ϕ ∈ F(β,H) (R, T ) , we have considered the set TBϕ (R, T ) of elements aiT where a ∈ B ∗ fulﬁlls iT (iR )a = iT = (iR )a iT aiT v = ϕ(v)aiT and viT a−1 = iT a−1 ϕ(v) for any v ∈ T

11.10.10;

but, by the very deﬁnition of the B-fusions (cf. 1.28), inclusion 11.10.9 implies that there is a T -interior algebra embedding fϕ : BµB → Resϕ (BεB ) such that the inclusion BµB ⊂ B and the composition of fϕ with the inclusion BεB ⊂ B are B ∗ -conjugate, which amounts to saying that TBϕ (R, T ) is not empty. Moreover, setting T = ϕ(T ) , it is again easily checked that the group (iR B T iR )∗ acts on this set by the multiplication on the left. We claim that this action is transitive; indeed, if a ∈ B ∗ fulﬁlls the above condition, for any v ∈ T we have a iT a−1 = ϕ(v)a iT a−1 ϕ(v)−1

11.10.11,

so that a iT a−1 and the idempotents aiT a−1 and a iT a−1 belong to B T ; then, since we have (a iT a−1 )(a iT a−1 ) = a iT a−1 = (a iT a−1 )(aiT a−1 )

11.10.12

and the symmetric equalities on a and a still hold, the left multiplication by a iT a−1 determines a kT ⊗k B ◦ -module isomorphism h : (aiT a−1 )B ∼ = (a iT a−1 )B

11.10.13.

Then, the kT ⊗k B ◦ -modules (iR − aiT a−1 )B and (iR − a iT a−1 )B are isomorphic too; choosing such an isomorphism, we get a kT ⊗k B ◦ -module automorphism of iR B extending h ; but, such an automorphism is just the left multiplication by a suitable c ∈ (iR B T iR )∗ and, in particular, we get

11. The nilcentralized chain k ∗ -functor

157

caiT = a iT , which proves the claim. Consequently, k ∗ acts transitively on the quotient set ϕ Fˆ(β,H,X) (R, T ) = (iR + iR MT iR )\TBϕ (R, T )

11.10.14

and it is easily checked that this action is actually regular. If U is a third element of X and ψ : U → T an F(β,H) -morphism, we claim that TBϕ (R, T )·TBψ (T, U ) ⊂ TBϕ◦ψ (R, U ) 11.10.15. Indeed, if a, c ∈ B ∗ fulﬁll the corresponding equalities 11.10.10, so that aiT and ciU respectively belong to TBϕ (R, T ) and to TBψ (T, U ) , then we get aiT ciU = ac(iT )c iU = aciU

11.10.16

and from these equalities we still get iU (iR )ac = iU (iT )c (iR )ac = iU (iT )c = iU (iR )ac iU = (iR )ac (iT )c iU = iU aciU w = aiT ψ(w)ciU = (ϕ ◦ ψ)(w)aiT ciU = (ϕ ◦ ψ)(w)aciU

11.10.17.

wiU (ac)−1 = iU c−1 ψ(w)iT a−1 = iU (ca)−1 (ϕ ◦ ψ)(w) Finally, we claim that the product in B induces a map ϕ ψ ϕ◦ψ Fˆ(β,H,X) (R, T ) × Fˆ(β,H,X) (T, U ) −→ Fˆ(β,H,X) (R, U )

11.10.18

compatible with the product in k ∗ ; indeed, ﬁrst of all notice that, setting U = ψ(U ) and U = ϕ(U ) , and respectively denoting by µB and η B the unique local points of T and U on B , the inclusion U ⊂ T forces UηB ⊂ Tµ B , so that the image of µB in B(UηB ) ∼ = k is equal to {1} and therefore we have MT ⊂ MU . Moreover, with the the notation above, we get (iR + iR MT iR )aiT (iT + iT MU iT )ciU −1 −1 = iR + iR MT (iT )a + iR MT (iT MU iT )a aciU

11.10.19;

then, one the one hand we have −1

iR MT (iT )a

−1

⊂ iR MU (iT )a

⊂ iR MU iR

11.10.20;

on the other hand, since U ⊂ T , it is easily checked from condition 11.10.10 −1 −1 −1 that (iT B U iT )a = (iT )a B U (iT )a and therefore we get −1

(iT MU iT )a

−1

−1

= (iT )a MU (iT )a

11.10.21,

158

Frobenius categories versus Brauer blocks −1

so that we also obtain iR MT (iT MU iT )a ⊂ iR MU iR . We have proved the claim. Now, it suﬃces to consider the disjoint union Fˆ(β,H,X) (R, T ) =

ϕ (R, T ) Fˆ(β,H,X)

11.10.22,

ϕ∈F(β,H) (R,T )

endowed with the inverted action of k ∗ , to get the announced regular central k ∗ -extension. Indeed, the maps 11.10.18 deﬁne the composition in Fˆ(β,H,X) , which is associative and compatible with the action of k ∗ since it is induced by the product in B . Moreover, if T = R then it is clear that, for any ϕ ∈ F(β,H) (R) , we have TBϕ (R, R) ⊂ N(iR BiR )∗ (R·iR )

11.10.23,

so that, by its very deﬁnition, FˆB (RεB ) contains Fˆ(β,H,X) (R) (cf. 1.29.2), and therefore, since FˆS (RεS ) ∼ = k ∗ × FS (RεS ) (cf. Theorem 9.21) and n is admissible, we get (cf. 11.10.9 and 1.24 above, and Proposition 5.11 in [41]) Fˆ(β,H,X) (R) ⊂ FˆB (RεB ) ∼ = FˆS (RεS ) ∗ Fˆ⊗pn (kGb)δ (R⊗pnε )

11.10.24.

⊂ FˆkGb (Rε )

Note that, up to a unique equivalence, this category Fˆ(β,H,X) does not depend on the choice of iR ∈ εB for any R ∈ X . Let S be another Dade Q-algebra still determining a F(β,H,X) -stable ˆ

(X, n)-gluing element s in Dk (Q) ; mutatis mutandis denote by δ B the unique ˆ = S ◦ ⊗k pn (kGb)δ (cf. Proposition 11.8), set local point of Q on B pn B = S ◦ ⊗k (kGb)δ δBˆ

11.10.25

and, for any R ∈ X , choose an idempotent iR in the unique local point εB of R on B (cf. Proposition 11.8), denoting by MR the kernel of the corresponding homomorphism. Again, for any R, T ∈ X and any ϕ ∈ F(β,H) (R, T ) , consider the corresponding quotient set ϕ Fˆ(β,H,X) (R, T ) = (iR + iR MT iR )\TBϕ (R, T )

11.10.26. ϕ

In order to ﬁnd a relationship between the quotient sets Fˆ (R, T )(β,H,X) and Fˆ ϕ (R, T )(β,H,X) , set S = S ◦ ⊗k S and consider the Q-algebra B = S ◦ ⊗k S ⊗k S ◦ ⊗k

pn

ˆ (kGb)δ = S ⊗k B

11.10.27.

11. The nilcentralized chain k ∗ -functor

159

On the one hand, since we have a Q-algebra embedding k → S ⊗k S ◦ (cf. isomorphism 1.21.3), we still have a Q-algebra embedding from B to B and therefore, denoting by δ B the unique local point of Q on B (cf. Proposition 11.8), we can identify B with (B )δB and, in particular, the quotient ϕ set Fˆ(β,H,X) (R, T ) with the corresponding quotient set in B . δ

S

On the other hand, choosing an idempotent jQ in the unique local point of Q on S (cf. 1.21), we have an injective Q-algebra homomorphism ˆ = S ◦ ⊗k pn (kGb)δ −→ B t : B −→ B

11.10.28

ˆ Bˆ → B ˆ (cf. 1.12) and by the determined by the canonical embedding B = B δ ˆ injective homomorphism mapping a ∈ B on jQ ⊗a ; moreover, for any R ∈ X , t induces an injective k-algebra homomorphism t(R) : B(R) −→ B (R)

11.10.29

since, up to suitable identiﬁcations, we have the inclusion t(B) ⊂ (S )δS ⊗k B

11.10.30

and, according to Proposition 11.8, we get (S )δS (R) ∼ = k. Actually, since Q has a unique local point on (S )δS ⊗k B (cf. Proposiϕ tion 11.8 above and Proposition 5.6 in [41]) and the set Fˆ(β,H,X) (R, T ) does

not depend on the choice of iR in εB for any R ∈ X , we may assume that we also have the inclusion B ⊂ (S )δS ⊗k B = D

11.10.31

which allows us to replace B by D ; note that t(iQ ) = jQ ⊗ iQ is the unity element of D and that then we still may assume that we have iR t(iR ) = iR = t(iR )iR

11.10.32

for any R ∈ X . Furthermore, for any a ∈ B ∗ fulﬁlling condition 11.10.10 with respect R , T and ϕ , it follows from this condition that aiT a−1 is a primitive idempotent in B ϕ(T ) outside Mϕ(T ) , which is “contained” in iR — namely, it appears in a primitive pairwise orthogonal decomposition of iR in B ϕ(T ) ; hence, it follows from equality 11.10.31 that t(a)iT t(a−1 ) is a primitive idempotent in Dϕ(T ) “contained” in t(iR ) , which has a nontrivial image in D ϕ(T ) by the injectivity of homomorphism 11.10.29.

160

Frobenius categories versus Brauer blocks

But, denoting by εD the unique local point of R on D (cf. Proposition 11.8 above and Proposition 5.6 in [41]), it follows from Proposition 11.8 that D(RεD ) ∼ 11.10.33. = (S )δS (R) ⊗k B(RεB ) ∼ =k Thus, denoting by MR the maximal ideal in DR determined by εD , it follows from this isomorphism that there is an invertible element c ∈ jQ ⊗ iQ + MR such that the idempotent t(a)iT t(a−1 ) is “contained” in (iR )c . Moreover, since t(iQ ) = jQ ⊗ iQ , t(a) is invertible in D and therefore d = ct(a) is an invertible element of D which fulﬁlls iT (iR )d = iT = (iR )d iT

11.10.34;

furthermore, for any v ∈ T , the multiplication of t(aiT v) = t(ϕ(v)aiT ) by iT on the right and by c on the left yields diT v = ϕ(v)diT

11.10.35

and mutatis mutandis we get viT d−1 = iT d−1 ϕ(v) . On the other hand, denoting by iQ the unity element of B ⊂ D , so that B = iQ DiQ , the multiplication by d on the left determines a D◦ -module −1

isomorphism between iT D and (iT )d D which are both direct summands −1

of iQ D ; thus, the D◦ -modules (iQ − iT )D and (iQ − (iT )d )D are isomorphic too and, choosing such an isomorphism, we get a D◦ -module automorphism of iQ D which can be extended to a D◦ -module automorphism of D ; that is to say, we get an invertible element d ∈ D fulﬁlling d iQ d−1 = iQ

,

d iT = diT

and iT d−1 = iT d−1

11.10.36.

ϕ In conclusion, diT belongs to TBϕ (R, T ) and its image in Fˆ(β,H) (R, T ) does not depend on our choice of c since iR MR iR = iR MR iR ; thus, we ϕ get a map from T ϕ (R, T ) to Fˆ(β,H) (R, T ) sending ai to the class of di ; B

T

T

ﬁnally, it is easily checked that this map factorizes throughout the quotient ϕ set Fˆ(β,H,X) (R, T ) , determining a new map ϕ ϕ Fˆ(β,H,X) (R, T ) −→ Fˆ(β,H,X) (R, T )

11.10.37

which is compatible with the action of k ∗ and therefore it is bijective. Now, it is not diﬃcult to check that these maps, when R and T run over X , are compatible with the products in B and D and therefore with the maps 11.10.18; consequently, they deﬁne a canonical isomorphism between Fˆ(β,H,X) and the k ∗ -category constructed from S . We are done.

11. The nilcentralized chain k ∗ -functor

161

11.11 With the notation above, let L be a subgroup of H , ξ a point of L on kGb such that Lξ ⊂ Hβ and Rε a defect pointed group of Lξ ; as in 3.12, it is easily checked that an element y ∈ H such that Rε ⊂ (Qδ )y determines a unique isomorphism class of faithful functors iβξ : F(ξ,L) −→ F(β,H)

11.11.1.

As above, assume that R is F(b,G) -nilcentralized and let Y be a nonempty set of F(b,G) -nilcentralized subgroups of R , which contains any subgroup T of R such that F(ξ,L) (T, U ) = ∅ for some U ∈ Y and fulﬁlls iβξ (Y) ⊂ X ; again, in order to simplify our notation, we modify our choices of Qδ and (P, e) assuming that Rε ⊂ Qδ , so that we have Y ⊂ X . 11.12 Whenever S is a Dade Q-algebra determining a F(β,H,X) -stable (X, n)-gluing element in Dk (Q) for some admissible integer n , it is quite clear that ResQ R (S) still determines a F(ξ,L,Y) -stable (Y, n)-gluing element in Dk (R) , and the point is that we can exhibit a k ∗ -functor ˆiβ,X : Fˆ(ξ,L,Y) −→ Fˆ(β,H,X) ξ,Y

11.12.1.

lifting the corresponding restriction of iβξ ; before proving it, let us explicitly describe this functor. Borrowing the notation in 11.9, for any T, U ∈ Y and any F(ξ,L) -morphism ϕ : U → T , we have deﬁned a subset TBϕ (T, U ) of iT B ∗ iU where iT and iU respectively belong to the unique local points of T and U ˆ on B ; moreover, denoting by εD the unique local point of R on the Q-algebra n p ˆ = S ◦ ⊗k D kGb (cf. Proposition 11.8), we set pn D = S ◦ ⊗k kGb εDˆ

11.12.2,

we choose the idempotents iT and iU in D , and we identify D with iR BiR ; ϕ (T, U ) of iT D∗ iU and, with these analogously, we have deﬁned a subset TD ϕ choices we have iT DiU = iT BiU . Actually, we have TD (T, U ) = TBϕ (T, U ) as we prove below, and thus, in this situation we simply get ϕ ϕ Fˆ(ξ,L,Y) (T, U ) = Fˆ(β,H,X) (T, U )

11.12.3.

Proposition 11.13 With the notation and the hypothesis above, assume that there is a Dade Q-algebra S determining a F(β,H,X) -stable (X, n)-gluing element in Dk (Q) for some admissible integer n . Then, the Dade R-algebra ResQ R (S) determines a F(ξ,L,Y) -stable (Y, n)-gluing element in Dk (R) and the corresponding k ∗ -category Fˆ(ξ,L,Y) is a k ∗ -subcategory of Fˆ(β,H,X) .

162

Frobenius categories versus Brauer blocks

Proof: The ﬁrst statement is clear since conditions 11.6.1 and 11.6.2 for S and for the triple (β, H, X) imply that the same conditions for ResQ R (S) and for the triple (ξ, L, Y) hold. Moreover, with the notation above, if d ∈ D∗ fulﬁlls the corresponding ϕ condition 11.9.2 with respect T , U and ϕ , then diU belongs to TD (T, U ) ∗ and a = d + (iQ − iR ) belongs to B and fulﬁlls the same condition, so that the element diU = aiU still belongs to TBϕ (T, U ) ; consequently, we have ϕ TD (T, U ) ⊂ TBϕ (T, U ) and, since the group (iT DiT )∗ = (iT BiT )∗ acts regularly by left multiplication on both sets, we get the announced equality. In conclusion, we have the equality Fˆ ϕ (T, U ) = Fˆ ϕ (T, U ) for (ξ,L,Y)

(β,H,X)

any ϕ ∈ F(ξ,L) (T, U ) and therefore Fˆ(ξ,L,Y) (T, U ) is just the converse image of F(ξ,L) (T, U ) in Fˆ(β,H,X) (T, U ) . On the other hand, it is not diﬃcult to see that this fact does not depend on the choice of S . We are done. 11.14 Furthermore, let M be a subgroup of L , ζ a point of M on kGb such that Mζ ⊂ Lξ and Tµ a defect pointed group of Mζ (cf. 1.13); as above, we have unique isomorphism classes of faithful functors (cf. 3.12) iξζ : F(ζ,M ) −→ F(ξ,L)

and iβζ : F(ζ,M ) −→ F(β,H)

11.14.1

β ξ and it is clear that iβζ ∼ = iξ ◦ iζ . Once again, assume that T is F(b,G) -nilcentralized and let Z be a nonempty set of F(b,G) -nilcentralized subgroups of T which contains any subgroup U of T such that F(ξ,L) (U, V ) = ∅ for some

V ∈ Z and fulﬁlls iξζ (Z) ⊂ Y . In this situation, if there is a Dade Q-algebra S determining a F(β,H,X) -stable (X, n)-gluing element in Dk (Q) for some admissible integer n , then it follows from Proposition 11.13 applied three times that we have three faithful k ∗ -functors ˆiξ,Y : Fˆ(ζ,M,Z) −→ Fˆ(ξ,L,Y) ζ,Z

,

ˆiβ,X : Fˆ(ξ,L,Y) −→ Fˆ(β,H,X) ξ,Y

ˆiβ,X : Fˆ(ξ,L,Y) −→ Fˆ(β,H,X) ζ,Z

11.14.2

respectively lifting the corresponding restrictions of iβζ , iβξ and iξζ , and that we still have a natural isomorphism ˆiβ,X ∼ ˆβ,X ˆξ,Y ζ,Z = iξ,Y ◦ iζ,Z

11.14.3.

11.15 Actually, following the point of view of Remark 3.10, we simply get inclusions of categories e e e F(ζ,M ) ⊂ F(ξ,L) ⊂ F(β,H)

11.15.1;

moreover, it is clear that the sets X , Y and Z respectively determine sets Xe , e Ye and Ze of nilcentralized Brauer (b, G)-pairs, stable throughout F(β,H) -,

11. The nilcentralized chain k ∗ -functor

163

e e e e F(ξ,L) - and F(ζ,M ) -isomorphisms and then, denoting by F(β,H,Xe ) , F(ξ,L,Ye ) e e e and F(ζ,M,Ze ) the corresponding full subcategories over X , Y and Ze , it follows from 11.2 and Theorem 11.10 that we have k ∗ -categories Fˆ e e , (β,H,X )

e ∗ e e ˆe Fˆ(ξ,L,Y e ) and F(ζ,M,Ze ) which are regular k -extensions of F(β,H,Xe ) , F(ξ,L,Ye ) e and F(ζ,M,Ze ) respectively. Hence, it follows from Proposition 11.13 applied three times that we have a commutative diagram of faithful k ∗ -functors e Fˆ(ξ,L,Y e) 3

−→

e Fˆ(β,H,X e) 1

11.15.2

e Fˆ(ζ,M,Z e)

lifting inclusions 11.15.1. 11.16 The main situation we are interested in concerns the normalizers in G of the totally normal sets of nilcentralized Brauer (b, G)-pairs (cf. 7.4). We call totally normal set of nilcentralized Brauer (b, G)-pairs any nonempty set q of nilcentralized Brauer (b, G)-pairs, which is totally ordered by the inclusion and fulﬁlls q(i) B q(n) for any 1 ≤ i ≤ n where n = |q| and q(i) denote the i-th element of q — that is to say, totally normal sets of nilcentralized Brauer (b, G)-pairs is nothing but a particular kind of normal nc (F(b,G) ) -chains (cf. 2.20); then, the normalizer NG (q) of q is the stabilizer of q in G by G-conjugation. Moreover, if q(n) = (Q, f ) then, since CG (Q) ⊂ NG (q) ⊂ NG (Q, f )

11.16.1,

f is also a block of NG (q) (cf. 1.10) and determines a point ν q of NG (q) on kGb (cf. 1.19). 11.17 Let (P q )γ q be a defect pointed group of NG (q)ν q and denote by Xq the set of subgroups of P q containing Q ; recall that, since f is also a nilpotent block of CG (Q)·P q by Proposition 6.5 in [34], it follows from the main theorem in [41] that there exists a Dade P q -algebra S q such that we have a P q -interior algebra isomorphism (cf. isomorphism 1.18.1) k(CG (Q)·P q ) γ q ∼ 11.17.1. = SqP q Proposition 11.18 With the notation above, the Dade P q -algebra S q determines a F(ν q ,NG (q),Xq ) -stable (Xq , 0)-gluing element in Dk (P q ) . Proof: As in the notation above, we set n = |q| and q(n) = (Q, f ) ; since Q ⊂ P q , in particular we have CG (P q ) ⊂ CG (Q) and we can identify γ q with a local point of P q on kCG (Q)f (cf. 1.19); then, it follows from Theorem 1.12 in [34] that, with the notation there, we have ˆ kNG (q) γ q ∼ 11.18.1 = S q ⊗k k∗ L

164

Frobenius categories versus Brauer blocks

ˆ ; in particular, for any R ∈ Xq , it follows from for a suitable k ∗ -group L equality 3.13.1 above and from Theorem 3.1 in [39] that F(ν q ,NG (q)) (P q , R) = FkNG (q)f (P q )γ q , Rε 11.18.2 where we denote by ε the unique local point of R on kNG (q) γ q (cf. Proposition 11.8), and from Lemma 1.17 in [34] that FkNG (q)f (P q )γ q , Rε ⊂ FS q (P q )γ Sq , RεSq 11.18.3 q

q

where we respectively denote by εS and γ S the unique local points of R and P on S q (cf. 1.20). Thus, by the very deﬁnition of S q -fusions (cf. 1.28), for any homomorphism ϕ ∈ F(ν q ,NG (q)) (P q , R) there is an R-interior algebra embedding fϕ : (S q )εSq −→ Resϕ (S q )

11.18.4;

on the other hand, we have the structural embedding (1.12) q fεSq : (S q )εSq −→ ResP R (S )

11.18.5;

q consequently, Resϕ (S q ) and ResP R (S ) determine the same element in Dk (R) (cf. 1.21), proving condition 11.6.1. Note that, since Q is normal in CG (Q)·P q , Q acts trivially on S q and, setting P¯ q = P q /Q , S q becomes a Dade P¯ q -algebra; moreover, in order to prove condition 11.6.2, we may assume that (P, e) contains the Brauer (b, G)-pair determined by (P q )γ q ; in this case, it is quite clear that the ele¯ (Q) N ment of Dk (P¯ q ) determined by S q coincides with res ¯ qP (sQ ) (cf. 7.7); then, P

condition 11.6.2 is an easy consequence of Theorem 7.14. We are done. 11.19 Consequently, for any totally normal set q of nilcentralized Brauer (b, G)-pairs, it follows from Theorem 11.10 that we have the regular k ∗ -extension Fˆ(ν q ,NG (q),Xq ) of F(ν q ,NG (q),Xq ) ; similarly, denoting by Bq the set of Brauer (ν q , NG (q))-pairs containing q(n) , we have the corresponding regular k ∗ -extension Fˆ(νe q ,NG (q),Bq ) of F(νe q ,NG (q),Bq ) . Note that, if q has a unique element (Q, f ) , clearly we have e F(νe q ,NG (q),Bq ) (Q, f ) = F(b,G) (Q, f )

11.19.1

and then Theorem 11.10 allows us to ﬁx the choice of a k ∗ -group isomorphism e θˆ(Q,f ) : Fˆ(νe q ,NG (q),Bq ) (Q, f ) ∼ (Q, f ) = Fˆ(b,G)

11.19.2,

which we may assume compatible with the G-conjugation when (Q, f ) runs over the set of all nilcentralized Brauer (b, G)-pairs.

11. The nilcentralized chain k ∗ -functor

165

11.20 Moreover, for any totally normal set r of nilcentralized Brauer (b, G)-pairs such that q ⊂ r it is clear that we have NG (r) ⊂ NG (q) and it is easily checked that we still have NG (r)ν r ⊂ NG (q)ν q

and F(νe r ,NG (r)) ⊂ F(νe q ,NG (q))

11.20.1,

so that, since Br ⊂ Bq , we have the k ∗ -functor Fˆ(νe r ,NG (r),Br ) −→ Fˆ(νe q ,NG (q),Bq )

11.20.2

deﬁned above (cf. 11.15), which lifts the inclusion. Finally, for any totally normal set t of nilcentralized Brauer (b, G)-pairs such that r ⊂ t , in 11.15 above we already have exhibited a commutative diagram Fˆ(νe r ,NG (r),Br )

−→

Fˆ(νe q ,NG (q),Bq )

3

1

11.20.3.

Fˆ(νe t ,NG (t),Bt ) 11.21 All these results allow us to exhibit a more explicit lifting in Theorem 7.16 above, namely a lifting fulﬁlling a suitable transitive condition. For this, we consider the particular case where q , r and t respectively have one, two and three elements. Explicitly, let (Q, f ) and (R, g) be nilcentralized Brauer (b, G)-pairs such that (R, g) B (Q, f )

11.21.1

and set q = {(Q, f )}

,

r = {(R, g), (Q, f )}

and u = {(R, g)}

11.21.2;

for short, write (q) , (r) and (u) instead of (ν q , NG (q), Bq ) , (ν r , NG (r), Br ) and (ν u , NG (u), Bu ) , and consider the k ∗ -functors deﬁned in 11.20.2 above e e e Fˆ(u) ←− Fˆ(r) −→ Fˆ(q)

11.21.3;

then, since (Q, f ) belongs to Bq , to Br and to Bu , we get injective k ∗ -group homomorphisms e e χ ˆqr : Fˆ(r) (Q, f ) −→ Fˆ(q) (Q, f ) 11.21.4. e e χ ˆur : Fˆ(r) (Q, f ) −→ Fˆ(u) (Q, f ) 11.22 Moreover, since (R, g) and (Q, f ) belong to Bu , choosing a mor ) ˆe phism ˆι(Q,f lifting the inclusion of (R, g) in (Q, f ) , for (R,g) ∈ F(u) (Q, f ), (R, g)

166

Frobenius categories versus Brauer blocks

e e any element σ ˆ u in Fˆ(u) (Q, f ) there exists a unique τˆu ∈ Fˆ(u) (R, g) such that, ∗ e ˆ in the k -category F , the following diagram is commutative (u)

σ ˆu

(Q, f ) −→ (Q, f ) (Q,f ) ) ˆ ι(R,g) ↑ ↑ ˆι(Q,f (R,g) (R, g)

τˆu

−→

11.22.1;

(R, g)

thus, we get a k ∗ -group homomorphism e e ˆ(R,g) u : Fˆ(u) (Q, f ) −→ Fˆ(u) (R, g) (Q,f )

11.22.2

e such that, for any σ ˆ u ∈ Fˆ(u) (Q, f ) , we have ) ) ˆ(R,g) ˆι(Q,f ◦u (ˆ σu ) = σ ˆ u ◦ ˆι(Q,f (R,g) (Q,f ) (R,g)

11.22.3,

) which clearly does not depend on the choice of the lifting ˆι(Q,f (R,g) . Denote by e e Fˆ(b,G) (Q, f )R the stabilizer of R in Fˆ(b,G) (Q, f ) and note that, with our choice in 11.19.2, we have

e Im (θˆ(Q,f ) ◦ χ ˆqr ) = Fˆ(b,G) (Q, f )R

11.22.4.

Proposition 11.23 With the notation above, there is a unique k ∗ -group homomorphism e e (Q,f ) α ˆ (R,g) : Fˆ(b,G) (Q, f )R −→ Fˆ(b,G) (R, g)

11.23.1.

lifting the restriction map and fulﬁlling (Q,f ) ˆ(R,g) α ˆ (R,g) ◦ θˆ(Q,f ) ◦ χ ˆqr = θˆ(R,g) ◦ u ◦χ ˆur (Q,f )

11.23.2.

∗ ˆe ˆ(R,g) Proof: Clearly, θˆ(R,g) ◦ u (Q,f ) is a k -group homomorphism from F(u) (Q, f ) to Fˆ e (R, g) ; moreover, since (b,G)

F(νe r ,NG (r)) (Q, f ) = F(νe u ,NG (u)) (Q, f )

11.23.3,

actually χ ˆur is a k ∗ -group isomorphism; ﬁnally, according to equality 11.22.4 ˆqr induces a k ∗ -group isomorphism between above, the composition θˆ(Q,f ) ◦ χ Fˆ e (Q, f ) and Fˆ e (Q, f )R . We are done. (r)

(b,G)

Remark 11.24 The improvement with respect to Theorem 7.16 comes from the fact that here we only make independent choices over the objects.

11. The nilcentralized chain k ∗ -functor

167

11.25 Let (T, h) be another nilcentralized Brauer (b, G)-pair such that (R, g) S (T, h) B (Q, f )

11.25.1

and set t = {(T, h), (R, g), (Q, f )} p = {(T, h), (Q, f )}

,

v = {(T, h)} s = {(T, h), (R, g)}

and

11.25.2;

for short, we write again (t) , (v) , (p) and (s) instead of (ν t , NG (t), Bt ) , (ν v , NG (v), Bv ) , (ν p , NG (p), Bp ) and (ν s , NG (s), Bs ) , and consider the k ∗ functors deﬁned above together with the corresponding commutative diagrams (cf. 11.20.2 and 11.20.3) e e Fˆ(s) −→ Fˆ(u) 0 3 3 e e e Fˆ(v) Fˆ(t) −→ Fˆ(r) 3 0 0 Fˆ e −→ Fˆ e (p)

11.25.3.

(q)

11.26 Since (Q, f ) belongs to Bq , Br , Bt , Bu , Bv , Bs and Bp , the diagram 11.25.3 above determines the following commutative (k ∗ -Gr)-diagram u

χ ˆs e e Fˆ(s) (Q, f ) −→ Fˆ(u) (Q, f ) χ ˆv s

0

e Fˆ(v) (Q, f ) χ ˆv 3 p

3

χ ˆs t

3

χ ˆu r

r

χ ˆt e e Fˆ(t) (Q, f ) −→ Fˆ(r) (Q, f ) 0 χˆp 0 χˆq

e (Q, f ) Fˆ(p)

11.26.1;

r

t

χ ˆq p

−→

e Fˆ(q) (Q, f )

similarly, since (R, g) belongs to Bu , Bs and Bv , we get injective k ∗ -group homomorphisms e e ρˆus : Fˆ(s) (R, g) −→ Fˆ(u) (R, g) 11.26.2. v e e ρˆs : Fˆ(s) (R, g) −→ Fˆ(v) (R, g) 11.27 Moreover, since (T, h) , (R, g) and (Q, f ) belong to Bv , choosing e ˆι(R,g) ∈ Fˆ(v) (R, g), (T, h) (T ,h) e ) ∈ Fˆ(v) (Q, f ), (T, h) ˆι(Q,f (T ,h) e ) ˆι(Q,f ∈ Fˆ(v) (Q, f ), (R, g) (R,g)

11.27.1

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Frobenius categories versus Brauer blocks

lifting the inclusions of (T, h) and (R, g) in (R, g) and (Q, f ) respectively, we get k ∗ -group homomorphisms (cf. 11.22) e e ,h) ˆ(T v : Fˆ(v) (R, g) −→ Fˆ(v) (T, h) (R,g) e e ,h) ˆ(T v : Fˆ(v) (Q, f ) −→ Fˆ(v) (T, h) (Q,f )

11.27.2

e e ˆ(R,g) v : Fˆ(v) (Q, f ) −→ Fˆ(v) (R, g) (Q,f ) e e such that, for any τˆv ∈ Fˆ(v) (R, g) and any σ ˆ v ∈ Fˆ(v) (Q, f ), we have ,h) ˆ(T ˆι(R,g) ◦v (ˆ τ v ) = τˆv ◦ ˆι(R,g) (T ,h) (R,g) (T ,h) ) ,h) ) ˆ(T ˆι(Q,f ◦v (ˆ σv ) = σ ˆ v ◦ ˆι(Q,f (T ,h) (Q,f ) (T ,h)

11.27.3;

) ) ˆ(R,g) ˆι(Q,f ◦v (ˆ σ )=σ ˆ ◦ ˆι(Q,f (R,g) (Q,f ) (R,g)

v

v

,h) ˆ(T ˆ(T ,h) ˆ(R,g) moreover, it is easily checked that v (Q,f ) = v(R,g) ◦ v(Q,f ) .

11.28 Similarly, since (R, g) and (Q, f ) belong to Bs , choosing a mor ) ˆe phism ˆι(Q,f lifting the inclusion of (R, g) in (Q, f ) , we (R,g) in F(s) (Q, f ), (R, g) get a k ∗ -group homomorphism (cf. 11.22) e e ˆs(R,g) : Fˆ(s) (Q, f ) −→ Fˆ(s) (R, g) (Q,f )

11.28.1

such that, for any σ ˆ s ∈ Fˆ(νe s ,NG (s),Bs ) (Q, f ) , we have ) ) ˆι(Q,f ◦ ˆs(R,g) (ˆ σs ) = σ ˆ s ◦ ˆι(Q,f (R,g) (Q,f ) (R,g)

11.28.2.

e e e On the other hand, denote by Fˆ(b,G) (R, g)T , Fˆ(b,G) (Q, f )T and Fˆ(b,G) (Q, f )R,T e e e ˆ ˆ ˆ the respective stabilizers of T in F (R, g) , F (Q, f ) and F (Q, f )R . (b,G)

(b,G)

(b,G)

Note that, by Proposition 11.23, we have the k ∗ -groups homomorphisms e e (R,g) α ˆ (T : Fˆ(b,G) (R, g)T −→ Fˆ(b,G) (T, h) ,h) e e (Q,f ) α ˆ (T : Fˆ(b,G) (Q, f )T −→ Fˆ(b,G) (T, h) ,h)

11.28.3.

Theorem 11.29 With the notation above, we have e e (Q,f ) α ˆ (R,g) (R, g)T Fˆ(b,G) (Q, f )R,T ⊂ Fˆ(b,G) (Q,f ) (R,g) (Q,f ) e and α ˆ (T ˆ =α ˆ (T ˆ for any ϕˆ ∈ Fˆ(b,G) (Q, f )R,T . ˆ (R,g) (ϕ) ,h) α ,h) (ϕ)

11.29.1

11. The nilcentralized chain k ∗ -functor

169

e Proof: Since Im (θˆ(Q,f ) ◦ χ ˆqr ) = Fˆ(b,G) (Q, f )R (cf. 11.22.4), it is clear that

e e (θˆ(Q,f ) ◦ χ ˆqr ) Fˆ(r) (Q, f )T = Fˆ(b,G) (Q, f )R,T

11.29.2;

moreover, since we have F(νe r ,NG (r)) (Q, f )T = F(νe t ,NG (t)) (Q, f )

11.29.3,

the k ∗ -group homomorphism χ ˆrt induces a k ∗ -group isomorphism e e Fˆ(t) (Q, f ) ∼ (Q, f )T = Fˆ(r)

11.29.4.

ˆ for some ψˆ ∈ Fˆ e (Q, f ) ; Consequently, we have ϕˆ = (θˆ(Q,f ) ◦ χ ˆqr ◦ χ ˆrt )(ψ) (t) then, according to Proposition 11.23, we get (cf. diagram 11.26.1) (R,g) (Q,f ) (R,g) (Q,f ) ˆ α ˆ (T (ϕ) ˆ =α ˆ (T ◦ θˆ(Q,f ) ◦ χ ˆqr ◦ χ ˆrt )(ψ) α ˆ (R,g) (ˆ α(R,g) ,h) ,h) (R,g) ˆ ˆ(R,g) =α ˆ (T ◦χ ˆur ◦ χ ˆrt )(ψ) (θˆ(R,g) ◦ u ,h) (Q,f ) (R,g) ˆ ˆ(R,g) =α ˆ (T ◦χ ˆus ◦ χ ˆst )(ψ) (θˆ(R,g) ◦ u ,h) (Q,f )

11.29.5;

but, since (Q, f ) and (R, g) belong to both sets Bs and Bu , the functor e e Fˆ(s) → Fˆ(u) deﬁned above determines a map e e Fˆ(s) (Q, f ), (R, g) −→ Fˆ(u) (Q, f ), (R, g)

11.29.6

preserving the morphisms which lift the inclusion (R, g) ⊂ (Q, f ) ; at this point, choosing in the right-hand term the image of the choice in the lefthand term, it is easily checked that the following diagram is commutative ˆ(R,g) u

(Q,f ) e e Fˆ(u) (Q, f ) −−−−−−→ Fˆ(u) (R, g) & & u  χ ˆs ρˆu s

e Fˆ(s) (Q, f )

(R,g) ˆ s(Q,f )

−−−−−−→

11.29.7.

e Fˆ(s) (R, g)

Thus, from equalities 11.29.5, from the commutativity of this diagram and, once again, from Proposition 11.23 we get (R,g) (Q,f ) (R,g) ˆ α ˆ (T α ˆ (R,g) (θˆ(R,g) ◦ ρˆus ◦ ˆs(R,g) (ϕ) ˆ =α ˆ (T ◦χ ˆst )(ψ) ,h) ,h) (Q,f ) (R,g) ˆ ◦ θˆ(R,g) ◦ ρˆus ) (ˆs(R,g) ◦χ ˆst )(ψ) = (ˆ α(T ,h) (Q,f ) ,h) ˆ ˆ(T = (θˆ(T,h) ◦ v ◦ ρˆvs ) (ˆs(R,g) ◦χ ˆst )(ψ) (R,g) (Q,f ) ,h) ˆ ˆ(T = (θˆ(T,h) ◦ v ◦ ρˆvs ◦ ˆs(R,g) ◦χ ˆst )(ψ) (R,g) (Q,f )

11.29.8.

170

Frobenius categories versus Brauer blocks

Similarly, since (Q, f ) and (R, g) belong to both sets Bs and Bv , the functor Fˆ e → Fˆ e deﬁned above determines a map (s)

(v)

e e Fˆ(s) (Q, f ), (R, g) −→ Fˆ(v) (Q, f ), (R, g)

11.29.9,

and, once again, choosing our lifting of the inclusions from this map, it is easily checked that the following diagram is commutative ˆ(R,g) v

(Q,f ) e e Fˆ(v) (Q, f ) −−−−−−→ Fˆ(v) (R, g) & & v  χ ˆs ρˆv s

e (Q, f ) Fˆ(s)

(R,g) ˆ s(Q,f )

−−−−−−→

11.29.10.

e Fˆ(s) (R, g)

Finally, from equalities 11.29.8, from the commutativity of this diagram and from Proposition 11.23 again, we still get (cf. diagram 11.26.1) (R,g) (Q,f ) ,h) ˆ ˆ(T ˆ(R,g) α ˆ (T α ˆ (R,g) (ϕ) ˆ = (θˆ(T,h) ◦ v ◦v ◦χ ˆvs ◦ χ ˆst )(ψ) ,h) (R,g) (Q,f ) ,h) ˆ ˆ(T = (θˆ(T,h) ◦ v ◦χ ˆvp ◦ χ ˆpt )(ψ) (Q,f ) (Q,f ) ˆ = (ˆ α(T ◦ θˆ(Q,f ) ◦ χ ˆqp ◦ χ ˆpt )(ψ) ,h)

11.29.11.

(Q,f ) =α ˆ (T (ϕ) ˆ ,h)

We are done. 11.30 The transitive equality in Theorem 11.29 is the key for the construction of the announced functor (cf. 11.1) from the proper category of nc nc (F(b,G) ) -chains (cf. A2.8) to k ∗ -Gr . Recall that, for any n ∈ N , a (F(b,G) ) nc chain — more precisely called (n, (F(b,G) ) )-chain, or n-chain for short — is a functor nc q : ∆n −→ (F(b,G) ) 11.30.1 from the category ∆n formed by the objects 0 ≤ i ≤ n and the morphisms 0 ≤ j ≤ i ≤ n , with the obvious composition (cf. A2.2). Then, the proper nc nc category of (F(b,G) ) -chains (cf. A2.8) — denoted by ch∗ (F(b,G) ) — is nc the category formed by all the pairs (q, ∆n ) , where q is a (F(b,G) ) -chain, and by the morphisms (λ, δ) : (r, ∆m ) → (q, ∆n ) — sometimes noted (λ(r) , δ) to avoid confusion — where δ : ∆n → ∆m is a functor, or equivalently an order-preserving map, and λ : r ◦ δ ∼ = q a natural isomorphism, and where the composition with another morphism (µ, ε) : (t, ∆6 ) → (r, ∆m ) is deﬁned by (cf. A2.6.3) (λ, δ) ◦ (µ, ε) = λ ◦ (µ ∗ δ), δ ◦ ε 11.30.2.

11. The nilcentralized chain k ∗ -functor

171

11.31 Often, we write q instead of (q, ∆n ) and we denote by F(b,G) (q) nc or, equithe automorphism group of (q, ∆n ) in the category ch∗ (F(b,G) ) valently, the group of natural automorphisms of the functor q ; note that F(b,G) (q) can be identiﬁed to a subgroup of F(b,G) q(n) and we denote by Fˆ(b,G) (q) the corresponding k ∗ -subgroup of Fˆ(b,G) q(n) . On the other hand, recall that we have the automorphism functor (cf. Proposition A2.10) nc aut = aut(F(b,G) )nc : ch∗ (F(b,G) ) −→ Gr

11.31.1

mapping q on F(b,G) (q) . Theorem 11.32 With the notation above and our choice in 11.19.2, there is a unique functor nc ∗ = aut (F nc : ch aut (F(b,G) ) −→ k ∗ -Gr (b,G) )

11.32.1

nc which lifts the automorphism functor, maps any (F(b,G) ) -chain q on Fˆ(b,G) (q) and, for any nilcentralized Brauer (b, G)-pairs (Q, f ) and (R, g) contained (Q,f ) in (P, e) such that (R, g) B (Q, f ) , sends to α ˆ (R,g) the morphism determined by idR from the 1-chain determined by the inclusion R ⊂ Q to the 0-chain determined by R . nc Proof: Thus, for any n-chain q of (F(b,G) ) , we set aut(q) = Fˆ(b,G) (q) . Let nc ∗ (λ, δ) : (q, ∆n ) → (r, ∆m ) be a ch (F(b,G) ) -morphism, set Q = q(n) and R = r(m) , and denote by ϕ : R → Q the composition of the inverse of the isomorphism λm : q δ(m) ∼ = r(m) with the q-image of the ∆n -morphism from δ(m) to n — denoted by δ(m)•n (cf. A2.2). If R = ϕ(R) is normal in Q then, denoting by f and g the respective blocks of CG (Q) and CG (R ) such that (P, e) contains (Q, f ) and (R , g ) (cf. 1.15), Q ﬁxes g ; moreover, Fˆ(b,G) (q) is a k ∗ -subgroup of Fˆ(b,G) (Q)R and, denoting by ϕ∗ : R ∼ = R the isomorphism induced by ϕ , the restriction map sends F(b,G) (q) ⊂ F(b,G) (Q)R to

ϕ∗ ◦ F(b,G) (r) ◦ (ϕ∗ )−1 ⊂ F(b,G) (R )

11.32.2.

In this case, denoting by Fˆ(b,G) (ϕ∗ ) : Fˆ(b,G) (R) ∼ = Fˆ(b,G) (R )

11.32.3

the k ∗ -group isomorphism determined by an element of G inducing ϕ∗ , we deﬁne aut(λ, δ) : Fˆ(b,G) (q) −→ Fˆ(b,G) (r) 11.32.4 (Q,f ) as the corresponding restriction of Fˆ(b,G) (ϕ∗ )−1 ◦ α ˆ (R ,g ) (cf. 11.23.1).

172

Frobenius categories versus Brauer blocks

If R = Q , consider the maximal i ∈ ∆n such that q(i•n) maps q(i) on a proper subgroup Q of Q and then a F(b,G) (q)-stable normal proper subgroup U of Q containing Q ; since Q contains R , ϕ and q(i • n) respectively induce F(b,G) -morphisms R → U and q(i) → U ; these F(b,G) -morphisms and the inclusion U ⊂ Q determine chains nc

ˆq : ∆n+1 −→ (F(b,G) )

nc

and ˆr : ∆m+1 −→ (F(b,G) )

11.32.5

n m fulﬁlling ˆq ◦ δi+1 = q and ˆr ◦ δm+1 = r (cf. A3.1) and respectively mapping i +1 nc and m+1 on U . In this situation, we have evident ch∗ (F(b,G) ) -morphisms nc building a commutative ch∗ (F(b,G) ) -diagram m (idr ,δm+ 1)

(ˆr, ∆m+1 ) −−−−−−−→ & ˆ δ) ˆ (λ, (ˆq, ∆n+1 )

n (idq ,δi+ 1)

−−−−−−−→

(r, ∆m ) & (λ,δ)

11.32.6

(q, ∆n )

ˆ δ) ˆ is already λ, and, since U is normal in Q , the k ∗ -group homomorphism aut( deﬁned; moreover, arguing by induction on |Q|/|R| , we may assume that m r , δm+1 aut(id ) is deﬁned too; further, since we have ˆq(n + 1) = q(n) and U is F(b,G) (q)-stable, we get F(b,G) (ˆq) = F(b,G) (q) and therefore Fˆ(b,G) (ˆq) and Fˆ(b,G) (q) are the same k ∗ -subgroup of Fˆ(b,G) (Q) , so that we have (cf. 11.32.4) n q , δi+1 aut(id ) = idFˆ(b,G) (q)

11.32.7.

m ˆ δ) ˆ . r , δm+1 λ, Then, we deﬁne aut(λ, δ) = aut(id ) ◦ aut(

We claim that aut(λ, δ) does not depend on the choice of U ; indeed, for another choice U of an F(b,G) (q)-stable normal proper subgroup of Q containing Q , the intersection U = U ∩ U is also a possible choice and therefore we may assume that U ⊂ U ; once again, we have chains ˆq : ∆n+1 −→ (F(b,G) )

nc

and ˆr : ∆m+1 −→ (F(b,G) )

nc

11.32.8

n m fulﬁlling ˆq ◦ δi+1 = q and ˆr ◦ δm+1 = r (cf. A3.1) and respectively mapping nc i+1 and m+1 on U , and we have evident ch∗ (F(b,G) ) -morphisms building nc a commutative ch∗ (F(b,G) ) -diagram (ˆ r )

(idr

m ,δm+ 1)

(ˆr , ∆m+1 ) −−−−−−−−→ (r, ∆m ) & & (λ,δ) ˆ ,δ) ˆ (λ (ˆq , ∆n+1 )

(ˆ q ) n (idq ,δi+ 1)

−−−−−−−−→

(q, ∆n )

11.32.9.

11. The nilcentralized chain k ∗ -functor

173

But, the inclusion map U → U still determines another pair of chains nc ˆ ˆq : ∆n+2 −→ (F(b,G) )

nc ˆr : ∆m+2 −→ (F(b,G) ) and ˆ

11.32.10

n+1 n+1 m+1 ˆq ◦ δi+1 ˆq ◦ δi+2 fulﬁlling ˆ = ˆq and ˆ = ˆq , and analogously ˆˆr ◦ δm+1 = ˆr and nc m+1 ∗ ˆ ˆr ◦ δm+2 = ˆr ; as above we have evident ch (F(b,G) ) -morphisms building nc a commutative ch∗ (F(b,G) ) -diagram (id ,δ m+1 )

(id ,δ m+1 )

n+1 (idq ˆ ,δi+2 )

n+1 (idq ˆ ,δi+1 )

ˆ r m+1 ˆ r m+2 ˆr, ∆m+2 ) −−−−−−−→ (ˆr , ∆m+1 ) ←−−−−−−− (ˆ & & ˆ ˆ ˆ ,δ) ˆ ˆ δ) ˆ (λ (λ,

(ˆq , ∆n+1 )

ˆq, ∆n+2 ) (ˆ

←−−−−−−−

−−−−−−−→

(ˆr, ∆m+1 ) & (λ, ˆ δ) ˆ

11.32.11;

(ˆq, ∆n+1 )

moreover, as in the equality 11.32.7, we have n+1 n+1 qˆ , δi+2 qˆ , δi+1 aut(id ) = idFˆ(b,G) (q) = aut(id )

11.32.12.

On the other hand, since U and U are normal in Q , we already have deﬁned our functor to k ∗ -Gr over all the morphisms in diagram 11.32.11, and it is easily checked from Theorem 11.29 that the corresponding (k ∗ -Gr)-diagram is commutative; consequently, we have r) m (ˆ ˆ ˆ aut(id r , δm+1 ) ◦ aut(λ, δ) r) m n+1 (ˆ ˆ ˆ ˆ , δi+1 = aut(id ) r , δm+1 ) ◦ aut(λ, δ) ◦ aut(idq

ˆˆ ˆˆ r) m m+1 (ˆ r , δm+1 λ, = aut(id ) ◦ aut( δ) r , δm+1 ) ◦ aut(idˆ r ) m (ˆ ˆ ˆ aut(id r , δm+1 ) ◦ aut(λ , δ)

11.32.13.

r) m n+1 (ˆ ˆ ˆ ˆ , δi+2 = aut(id ) r , δm+1 ) ◦ aut(λ , δ) ◦ aut(idq

ˆˆ ˆˆ r ) m m+1 (ˆ r , δm+2 λ, = aut(id ) ◦ aut( δ) r , δm+1 ) ◦ aut(idˆ But, once again arguing by induction on |Q|/|R| , we may assume that nc the image in k ∗ -Gr of the commutative ch∗ (F(b,G) ) -diagram (ˆ r )

(idr

m ,δm+ 1)

(ˆr , ∆m+1 ) −−−−−−−−→ & (idˆr ,δ m+1 ) m+2

ˆr, ∆m+2 ) (ˆ

m+1 (idˆr ,δm+ ) 1

−−−−−−−−→

(r, ∆m ) & (ˆr) m (idr ,δm+ 1)

11.32.14

(ˆr, ∆m+1 )

is already deﬁned and commutative; that is to say, we may assume that r) m m+1 (ˆ r , δm+1 aut(id ) r , δm+1 ) ◦ aut(idˆ

r) m m+1 r , δm+2 (ˆ ) = aut(id r , δm+1 ) ◦ aut(idˆ

11.32.15.

174

Frobenius categories versus Brauer blocks

Consequently, it follows from equalities 11.32.13 that

r) m m ˆ δ) ˆ = aut(id r(ˆr) , δm+1 λ, (ˆ ˆ ˆ aut(id ) ◦ aut( r , δm+1 ) ◦ aut(λ , δ)

11.32.16

which proves that aut(λ, δ) does not depend on the choice of U . nc Finally, let (µ, ε) : (r, ∆m ) → (t, ∆6 ) be a second ch∗ (F(b,G) ) -morphism, set T = t(D) and denote of the inverse by ψ : T → R the composition of the isomorphism µ6 : r ε(D) ∼ = t(D) with r ε(D)•m ; we will prove that (µ, ε) ◦ (λ, δ) aut(µ, ε) ◦ aut(λ, δ) = aut 11.32.17 arguing by induction on |Q|/|T | ; set (cf. 11.30.2) (ν, η) = (µ, ε) ◦ (λ, δ) = µ ◦ (λ ∗ ε), ε ◦ δ

11.32.18.

If |Q|/|T | = 1 then aut(λ, δ) and aut(µ, ε) respectively coincide with the cor∗ responding restrictions of the k -group isomorphisms Fˆ(b,G) (ϕ) and Fˆ(b,G) (ψ) induced by elements x and y of G respectively determining ϕ and ψ , whereas aut(ν, η) clearly coincides with the corresponding restriction of the k ∗ -group isomorphism Fˆ(b,G) (ϕ ◦ ψ) induced by xy ; thus, in this case we are done. If R = Q , we have deﬁned m ˆ δ) ˆ r , δm+1 λ, aut(λ, δ) = aut(id ) ◦ aut(

11.32.19

from a suitable F(b,G) (q)-stable normal proper subgroup U of Q ; but, the same subgroup U of Q provides a good support in order to deﬁne aut(ν, η) ; that is to say, with the corresponding notation we still have 6 t , δ6+1 ν , ηˆ) aut(ν, η) = aut(id ) ◦ aut(ˆ

moreover, our induction hypothesis guarantees that m m r , δm+1 (µ, ε) ◦ (idr , δm+1 aut(µ, ε) ◦ aut(id ) = aut )

11.32.20;

11.32.21

and therefore we get m ˆ δ) ˆ (µ, ε) ◦ (idr , δm+1 λ, aut(µ, ε) ◦ aut(λ, δ) = aut ) ◦ aut(

11.32.22.

nc Similarly, consider the corresponding chain ˆt : ∆6 → (F(b,G) ) fulﬁlling 6 ˆt ◦ δ6+1 = t and mapping D + 1 on U ; as above, we have the corresponding nc F(b,G) -morphisms building a commutative ch∗ (F(b,G) ) -diagram

(ˆt, ∆6+1 ) & (ˆ µ,ˆ ε)

(idt ,δ+ 1)

−−−−−−−→ m (idr ,δm+ 1)

(ˆr, ∆m+1 ) −−−−−−−→

(t, ∆6 ) & (µ,ε) (r, ∆m )

11.32.23

11. The nilcentralized chain k ∗ -functor

175

and our induction hypothesis still guarantees that m 6 (µ, ε) ◦ (idr , δm+1 t , δ6+1 µ, εˆ) aut ) = aut(id ) ◦ aut(ˆ

11.32.24.

Moreover, from the commutative diagrams 11.32.6 and 11.32.23, it is ˆ δ) ˆ = (ˆ easily checked that (ˆ µ, εˆ) ◦ (λ, ν , ηˆ) ; but, since U is normal in Q , (Q,f ) ˆ ˆ aut(λ, δ) and aut(ˆ ν , ηˆ) are both deﬁned from α ˆ (U,i) (cf. 11.23.1), where (U, i) denotes the Brauer (b, G)-pair contained in (Q, f ) , and, since we assume that µ, εˆ) is deﬁned by the inclusion ˆr(m + 1) = U = ˆt(D + 1) , aut(ˆ Fˆ(b,G) (ˆr) ⊂ Fˆ(b,G) (ˆt) ⊂ Fˆ(b,G) (U )

11.32.25;

consequently, we get

ˆ δ) ˆ = aut ˆ δ) ˆ = aut(ˆ µ, εˆ) ◦ aut( λ, (ˆ ν , ηˆ) aut(ˆ µ, εˆ) ◦ (λ,

11.32.26.

In this case, equality 11.32.17 follows from equalities 11.32.20, 11.32.22, 11.32.24 and 11.32.26. If R = Q and T = ψ(T ) = R , in the argument above replacing (r, ∆m ) by (t, ∆6 ) and (λ, δ) by (ν, η) , we still may choose the same subgroup U of Q in the deﬁnition of aut(ν, η) ; again, we consider the chains nc

ˆq : ∆n+1 −→ (F(b,G) )

nc

and ˆt : ∆6+1 −→ (F(b,G) )

11.32.27

deﬁned above and therefore equality 11.32.20 still holds; moreover, since we have R = Q , aut(λ, δ) coincides with the restriction of the k ∗ -group isomorphism Fˆ(b,G) (ϕ) induced by a suitable element of G . On the other hand, let j ∈ ∆m be the maximal element such that the r(j • m) maps r(j) on a proper subgroup R of R , so that ϕ(R ) ⊂ Q ; thus, R is an F(b,G) (r)-stable subgroup of ϕ−1 (U ) B R and therefore we can choose an F(b,G) (r)-stable normal proper subgroup V of R containing R and contained in ϕ−1 (U ) ; as above, we have the corresponding chains nc

˜r : ∆m+1 −→ (F(b,G) )

nc

and ˜t : ∆6+1 −→ (F(b,G) )

11.32.28

m 6 = r and ˜t ◦ δ6+1 = t and respectively mapping j + 1 and fulﬁlling ˜r ◦ δj+1 nc D + 1 on V . Again, we have evident ch∗ (F(b,G) ) -morphisms building a nc commutative ch∗ (F(b,G) ) -diagram (˜ t)

(˜t, ∆6+1 ) & (˜ µ,˜ ε)

(idt ,δ+ 1)

−−−−−−−−→ (˜ r) m (idr ,δj+ 1)

(˜r, ∆m+1 ) −−−−−−−−→

(t, ∆6 ) & (µ,ε)

11.32.29

(r, ∆m )

r) m (˜ and, since aut(id r , δj+1 ) = idF(b,G) (r) , we have deﬁned (cf. 11.32.20) 6 (t˜t), δ6+1 µ, ε˜) aut(µ, ε) = aut(id ) ◦ aut(˜

11.32.30.

176

Frobenius categories versus Brauer blocks

This time, the inverse of the isomorphism ϕ−1 (U ) ∼ = U determined by ϕ induces an F(b,G) -morphism U → R and we replace the chain ˆr above by the corresponding chain nc ˆr◦ : ∆m+1 −→ (F(b,G) ) 11.32.31 m fulﬁlling ˆr◦ ◦ δj+1 = r and mapping j + 1 on U ; as above, we have evident nc nc ∗ ch (F(b,G) ) -morphisms building a commutative ch∗ (F(b,G) ) -diagram (ˆ t)

(idt ,δ+ 1)

−−−−−−−−→

(ˆt, ∆6+1 ) & (ˆ µ◦ ,ˆ ε◦ )

(ˆ r◦ )

(idr

(t, ∆6 ) & (µ,ε)

m ,δj+ 1)

(ˆr◦ , ∆m+1 ) −−−−−−−−→ (r, ∆m ) & & (λ,δ) ˆ ◦ ,δˆ◦ ) (λ

11.32.32.

(ˆ q)

n (idq ,δi+ 1)

−−−−−−−−→

(ˆq, ∆n+1 )

(q, ∆n )

Once again, it is clear that ˆ ◦ , δˆ◦ ) = (ˆ (ˆ µ◦ , εˆ◦ ) ◦ (λ ν , ηˆ)

11.32.33;

ϕ

ˆ ◦ , δˆ◦ ) coincides with λ moreover, since ˆr◦ (m + 1) = R ∼ = Q = ˆq(n + 1) , aut( the restriction of the k ∗ -group isomorphism Fˆ(b,G) (ϕ) and therefore we have ◦

r ) m ˆ ◦ , δˆ◦ ) (ˆ λ aut(λ, δ) = aut(id , δj+1 ) ◦ aut( r

11.32.34

n q(ˆq) , δi+1 ν , ηˆ) since aut(id ) = idFˆ(b,G) (q) ; similarly, since U is normal in Q , aut(ˆ (Q,f ) (Q,f ) µ◦ , εˆ◦ ) are respectively deﬁned from α and aut(ˆ ˆ (U,i) and α ˆ (U,i) ◦ Fˆ(b,G) (ϕ)−1 (cf. 11.23.1). Consequently, we get

ˆ ◦ , δˆ◦ ) ν , ηˆ) = aut(ˆ µ◦ , εˆ◦ ) ◦ aut( λ aut(ˆ

11.32.35

and therefore, by its very deﬁnition (cf. 11.32.20), we still get 6 ˆ ◦ , δˆ◦ ) (tˆt) , δ6+1 µ◦ , εˆ◦ ) ◦ aut( λ aut(ν, η) = aut(id ) ◦ aut(ˆ

11.32.36.

Finally, the Fˆ(b,G) -morphism V → U induced by ϕ still determines another pair of chains nc ˆ ˜r : ∆m+2 −→ (F(b,G) )

nc ˜t : ∆6+2 −→ (F(b,G) ) and ˆ

11.32.37

m+1 ˆr ◦ δ m+1 = ˆr◦ , and analogously ˆ˜t ◦ δ 6+1 = ˜t and ˜r ◦ δj+2 fulﬁlling ˆ = ˜r and ˜ j+1 6+2 ˆ 6+1 ˜t ◦ δ6+1 ˆ = t , and respectively mapping (j + 1) • (j + 2) and (D + 1) • (D + 2) nc on this homomorphism; as above, we have evident ch∗ (F(b,G) ) -morphisms

11. The nilcentralized chain k ∗ -functor

177

nc building the commutative ch∗ (F(b,G) ) -diagram (id

(˜ t)

,δ

)

+1 t (˜t, ∆6+1 ) −−−−−−−−→ (t, ∆6 ) (˜ µ,˜ ε) 1 (µ,ε) 1 & &   (ˆt) +1 (˜r, ∆m+1 ) −→ (id˜t ,δ+2 ) (r, ∆m ) (idt ,δ+1 ) +1 & (idˆ ,δ+1 ) m+1  ˜t, ∆6+2 ) −−−−t−− (ˆ ) −−→ (ˆt, ∆6+1 ) (id˜r ,δj+ 2  ˆ,εˆ (µ ˜ ˜) 1 ↑ 1 (ˆµ◦ ,ˆε◦ )

11.32.38.

m+1 (idˆr◦ ,δj+ ) 1

˜r, ∆m+2 ) −−−−−−−−→ (ˆr◦ , ∆m+1 ) (ˆ First of all, it follows from our induction hypothesis that the image in k ∗ -Gr of this diagram is already deﬁned and that it is also a commutative diagram; moreover, since F(b,G) (r) stabilizes V , we have m+1 ˆr◦ , δj+1 and aut(id ) = idF(b,G) (ˆr◦ )

˜r ) = Fˆ(b,G) (ˆr◦ ) Fˆ(b,G) ( ˆ

11.32.39;

consequently, we get (cf. deﬁnition 11.32.20 and diagram 11.32.38) 6 (tˆt), δ6+1 µ◦ , εˆ◦ ) aut(id ) ◦ aut(ˆ (ˆt)

6+1 6 ˆ˜, εˆ˜) t , δ6+1 ˆt , δ6+1 µ = aut(id ) ◦ aut(id ) ◦ aut( (˜t)

6+1 6 ˆ˜, εˆ˜) t , δ6+1 ˜t , δ6+2 µ = aut(id ) ◦ aut(id ) ◦ aut(

11.32.40

(˜t)

m+1 6 t , δ6+1 µ, ε˜) ◦ aut(id ˜r , δj+2 = aut(id ) ◦ aut(˜ ) m+1 ˜r , δj+2 = aut(µ, ε) ◦ aut(id )

and therefore, since r) m (˜ aut(id r , δj+1 ) = idF(b,G) (r)

m+1 ˆr , δj+1 and aut(id ) = idF(b,G) (ˆr )

11.32.41,

from equalities 11.32.34, 11.32.36 and 11.32.40 we ﬁnally obtain 6 ˆ ◦ , δˆ◦ ) (tˆt) , δ6+1 µ◦ , εˆ◦ ) ◦ aut( λ aut(ν, η) = aut(id ) ◦ aut(ˆ m+1 ˆ ◦ , δˆ◦ ) ˜r , δj+2 λ = aut(µ, ε) ◦ aut(id ) ◦ aut( r) m m+1 ˆ ◦ , δˆ◦ ) (˜ r , δj+2 λ = aut(µ, ε) ◦ aut(id ) ◦ aut( r , δj+1 ) ◦ aut(id˜

11.32.42.

◦

r ) m m+1 ˆ ◦ , δˆ◦ ) (ˆ ˆr◦ , δj+1 λ = aut(µ, ε) ◦ aut(id , δj+1 ) ◦ aut(id ) ◦ aut( r

= aut(µ, ε) ◦ aut(λ, δ) We are done. Remark 11.33 If we modify our choice of k ∗ -isomorphisms in 11.19.2, we clearly obtain an isomorphic functor.

Chapter 12

Quotients and normal subcategories in Frobenius P-categories 12.1 Let us come back to our abstract setting. Before going further into the blocks, we need to develop the analogy between our abstract setting and the groups by discussing homomorphisms, quotients and normal structures. Let P be a ﬁnite p-group and F a P -category; if P is a second ﬁnite p-group and F a P -category, we say that a group homomorphism α : P → P is (F, F )-functorial whenever, for any pair of subgroups Q and R of P and any ϕ ∈ F(Q, R) , we have ϕ R ∩ Ker(α) ⊂ Ker(α) 12.1.1 and the group homomorphism ϕ : α(R) → α(Q) determined by ϕ belongs to F α(Q), α(R) . In this case, α determines an evident functor

fα : F −→ F

12.1.2

that we call Frobenius functor ; clearly, the composition of Frobenius functors is a Frobenius functor. If P = P and F = F then idP is obviously (F, F)-functorial and, if F is divisible, any σ ∈ F(P ) is (F, F)-functorial and determines an evident natural isomorphism idF ∼ = fσ . 12.2 We say that a subgroup U of P is F-stable† if ϕ(Q ∩ U ) ⊂ U for any subgroup Q of P and any element ϕ of F(P, Q) ; note that, in this case, U is normal in P , and that the F-stability is a necessary condition to guarantee that U is the kernel of some (F, F )-functorial homomorphism α : P → P where P is a ﬁnite p-group and F a P -category; actually, the next result aﬃrms that, in the Frobenius P -categories, it is also a suﬃcient condition. In this chapter, from now on we assume that F is a Frobenius P -category. Proposition 12.3 Let U be an F-stable subgroup of P and set P¯ = P/U . There is a Frobenius P¯ -category F¯ = F/U — called the U -quotient of F ¯ = Q/U and R ¯ = R/U of P¯ , — such that, for any pair of subgroups Q ¯ ¯ ¯ ¯ ¯ F(Q, R) is the set of group homomorphisms ϕ¯ : R → Q induced by the homomorphism in F(Q, R) . Moreover, the canonical homomorphism P → P¯ ¯ is (F, F)-functorial. Proof: The above correspondence clearly deﬁnes a P¯ -category F¯ ; moreover, ¯ = Q/U , R ¯ = R/U and T¯ = T /U are subgroups of P¯ , ϕ¯ is an whenever Q †

We borrow this term from [18, Ch. XII, section 9]; the term “strongly closed” which has

been somewhat employed is inadequate: there is no “strong closure”!

180

Frobenius categories versus Brauer blocks

¯ Q, ¯ R) ¯ and θ¯ : T¯ → R ¯ is a group homomorphism such that ϕ¯ ◦ θ¯ element of F( ¯ ¯ ¯ ¯ belongs to F(Q, T ) , then ϕ¯ ◦ θ can be lifted to some ψ ∈ F(Q, T ) and in particular ψ(T ) is contained in ϕ(R) , so that there is a group homomorphism θ : R → T fulﬁlling ϕ ◦ θ = ψ , which implies that θ belongs to F(R, T ) since ¯ R, ¯ T¯) ; thus, F¯ is divisible too. F is divisible, and therefore θ¯ belongs to F( ¯ It is clear that F fulﬁlls condition 2.8.1. On the other hand, let Q be a ¯ = Q/U and subgroup of P containing U and ϕ an element of F(P, Q) , set Q ¯ → P¯ the group homomorphism determined by ϕ ; moreover, denote by ϕ¯ : Q ¯ be a subgroup of Aut(Q) ¯ and denote by K the converse image of K ¯ in let K ¯, the stabilizer Aut(Q)U of U in Aut(Q) ; although K need not map onto K ¯ ¯ K ϕ K it is clear that NP¯ (Q) is the image of NP (Q) . Set Q = ϕ(Q) , K = K , ¯ = ϕ( ¯ , and assume that Q ¯ is fully K ¯ and K ¯ = ϕ¯K ¯ -normalized in F¯ . Q ¯ Q) ¯ ¯ ¯ K¯ ¯ Since F P, Q ·N K ¯ (Q ) maps surjectively onto F P , Q ·N ¯ (Q ) , it P

P

is not diﬃcult to check that Q is fully K -normalized in F , and therefore, since F is a Frobenius P -category, there are an F-morphism ζ : Q·NPK (Q) −→ P

12.3.1

and an element χ ∈ K fulﬁlling ζ(u) = ϕ χ(u) for any u ∈ Q ; in particular, since χ(U ) = U , we get ζ(U ) = U , so that ζ determines a group homomorphism ¯ ¯ ¯ K ζ¯ : Q·N 12.3.2 P¯ (Q) −→ P ¯ ¯ ¯ u) = ϕ¯ χ(¯ ¯ K ¯. in F¯ P¯ , Q·N (Q) fulﬁlling ζ(¯ ¯ u) for any u ¯∈Q P¯ ¯ It remains to prove that the canonical map P → P¯ is (F, F)-functorial; ¯ since F is divisible, it suﬃces to prove that, for any subgroup Q of P and any ¯ = Q·U/U the homomorphism ϕ¯ : Q ¯ → P¯ F-morphism ϕ : Q → P , setting Q ¯ ¯ ¯ induced by ϕ belongs to F(P , Q) , namely to the image of F(P, Q·U ) in the ¯ to P¯ . We argue by induction on |P : Q| set of group homomorphisms from Q and on the length D = D(ϕ) of ϕ (cf. 5.15), and it is clear that we may assume P that U ⊂ Q ; if D = 0 then ϕ = σ ◦ ιP Q , where σ ∈ F(P ) and ιQ : Q → P is the inclusion map (cf. 1.9), and it suﬃces to consider the group homomorphism Q·U → P induced by σ . Assume that D ≥ 1 , so that we have ϕ = ιP R ◦ σ ◦ ν , where R is an F-essential subgroup of P fully normalized in F , σ is a p -element of F(R) and ν : Q → R is an F-morphism such that ιP R ◦ ν has length D − 1 (cf. 5.15); thus, it follows from the induction hypothesis that there is an F-morphism ¯ ¯ ψ : Q·U → P inducing the same homomorphism as ιP R ◦ ν from Q to P ; in ¯ ¯ ¯ particular, we have ψ(Q) ⊂ R and therefore we still have ψ(Q·U ) ⊂ R·U ; hence, since F is divisible, there is an F-morphism η : Q·U → R·U such that ¯ u) = ν¯(¯ ¯ . If U ⊂ R then σ ¯ R) ¯ and η¯(¯ u) = ψ(¯ u) for any u ¯∈Q ¯ belongs to F( ¯ P¯ , Q) ¯ . therefore ϕ¯ = ¯ιR ◦ σ ¯ ◦ η¯ belongs to F(

12. Quotients and normal subcategories

181

Otherwise, denote by H the subgroup of elements of F(R) acting trivially ¯ and set T = N H (R) ; since FP (R) is a Sylow p-subgroup of F(R) , on R P FT (R) is a Sylow p-subgroup of H and, by the Frattini argument, we get F(R) = H·NF (R) FT (R) 12.3.3; but, since R is fully centralized in F (cf. Proposition 2.11), it follows from statement 2.10.1 that any element in NF (R) FT (R) can be extended to T and actually determines an element of the stabilizer F(T )R of R in F(T ), which clearly stabilizes U ∩ T = NU (R) ; consequently, NF (R) FT (R) normalizes FU (R) and therefore, setting S = NR·U (R) = R·(U ∩ T )

12.3.4,

we still get

F(R) = H·NF (R) FS (R) 12.3.5; again by statement 2.10.1, any element in NF (R) FS (R) can be extended to S and actually determines an element of F(S)R . Thus, there are χ ∈ H S and τ ∈ F(P, S) such that ιP R ◦ σ = τ ◦ ιR ◦ χ ; but, since U ⊂ R , R is properly ¯ ¯ and therefore ϕ¯ = τ¯ ◦ ¯ιS ◦ η¯ contained in S ; hence, τ¯ belongs to F(P¯ , S) R ¯ ¯ ¯ belongs to F(P , Q) . We are done. Remark 12.4 Since U is clearly fully normalized in F , FP (U ) is a Sylow p-subgroup of F(U ) (cf. Proposition 2.11) and therefore P U ·CP (U ) is ˜ ) (cf. 1.3); thus, if U is F-selfcenisomorphic to a Sylow p-subgroup of F(U ˜ ) and it is tralizing (cf. 4.8), P¯ is isomorphic to a Sylow p-subgroup of F(U ¯ ˜ ) easily checked that F coincides with the Frobenius category of the group F(U ¯ (cf. 1.8). More generally, even when U is F-nilcentralized F is the Frobenius category of a ﬁnite group; indeed, it follows from Theorem 18.6 below that there are a group L and two group homomorphisms τU : P −→ L and πU : L −→ F(U )

12.4.1

fulﬁlling (cf. 1.8, 2.14 and 13.2 below) Ker(τU ) = HCF (U ) ,

Ker(πU ) = τU CP (U )

Im(πU ) = F(U ) and πU ◦ τU = κU

12.4.2;

thus, if U is F-nilcentralized (cf. 4.3), τU is injective (cf. statement 13.2.2) and it is not diﬃcult to check that F¯ coincides with the Frobenius category of the group L/τU (U ) . Corollary 12.5 Let U be an F-stable subgroup of P and set P¯ = P/U and ¯ F¯ = F/U . The converse image in P of an F-stable subgroup of P¯ is F-stable. In particular, if V is an F-stable subgroup of P then U ·V is F-stable too.

182

Frobenius categories versus Brauer blocks

¯ Proof: If V¯ = V /U is an F-stable subgroup of P¯ , setting P¯ = P¯ /V¯ it follows ¯ V¯ ; then, from Proposition 12.3 that we have a Frobenius P¯ -category F¯ = F/ it is clear that the composition of the canonical functors F −→ F¯ −→ F¯

12.5.1

¯ comes from the homomorphism P → P¯ ∼ = P/V which is (F, F)-functorial. On the other hand, if V is an F-stable subgroup of P , it is easily checked ¯ that V¯ = V ·U/U is F-stable. We are done. 12.6 Let us discuss normality; it seems reasonable to call F-normal any subgroup N of P fully normalized in F such that NF (N ) = F , and we consider them in chapter 19 when dealing with solvability. On the other hand, if P is an F-stable subgroup of P , we say that a divisible P -subcategory F of F is normal in F if F(P ) stabilizes F and, for any subgroup Q of P and any ϕ ∈ F(P, Q) , we have F ϕ(Q) = ϕ F (Q)

12.6.1;

in particular, F (Q) is then a normal subgroup of F(Q) . In this case, 12.6.2 A subgroup Q of P which is fully K-normalized in F for a subgroup K of Aut(Q) , is fully K-normalized in F too. Indeed, for any F-morphism ψ : Q·NPK (Q) → P , setting Q = ψ(Q) and K = ψK , the homomorphism ϕ : Q → P mapping ψ(u) on u , for any u ∈ Q , composed with a suitable element χ ∈ K can be extended to some F-morphism ζ : Q ·NPK (Q ) → P (cf. condition 2.8.2) and, since P is F-stable, we get ζ Q ·NPK (Q ) ⊂ Q·NPK (Q) ∩ P = Q·NPK (Q)

12.6.3

which forces the equality. 12.7 In this chapter, we prove the existence of a normal Frobenius P -subcategory F a of F — called the adjoin subcategory of F† — which is the analogous of Op (G) in a ﬁnite group G (but (FG )a need not coincide with FOp (G) !). We start by proving a general criterion on normality, which corresponds to the so-called Frattini argument. Proposition 12.8 Let P be an F-stable subgroup of P and F a divisible P -subcategory of F . Then, F is normal in F if and only if F(P ) stabilizes F and, for any subgroup Q of P we have F(P , Q) = F(P ) ◦ F (P , Q) †

The terminology comes from the Chevalley groups.

12.8.1.

12. Quotients and normal subcategories

183

Proof: First of all, assume that F is normal in F , so that F(P ) stabilizes F . We argue by induction on |P : Q| and may assume that Q = P ; let ϕ : Q → P be an F-morphism, set Q = ϕ(Q) and choose an F-morphism ψ : NP (Q ) → P such that Q = ψ(Q ) is fully F (Q )-normalized in F (cf. Proposition 2.7). Hence, since F is a Frobenius P -category, setting R = NP (Q) it is easily checked that thereexists an F-morphism ζ : R → P and χ ∈ F (Q) such that ψ ϕ(u) = ζ χ (u) for any u ∈ Q ; then, by our induction hypothesis, there are an F-morphism ζ : R → P and µ in F(P ) such that ζ = µ ◦ ζ , and therefore, denoting by ϕ∗ : Q ∼ = Q the F-isomorphism determined by ϕ , we get R ψ ◦ ϕ ∗ = ιP P ◦ µ ◦ (ζ ◦ ιQ ) ◦ χ

12.8.2.

Similarly, by the induction hypothesis we still get ψ = ιP P ◦ ν ◦ ξ for suitable ν ∈ F(P ) and ξ ∈ F (P , Q ) , and therefore we have ν ◦ ξ ◦ ϕ∗ = µ ◦ (ζ ◦ ιR Q) ◦ χ

12.8.3;

that is to say, setting λ = ν −1 ◦ µ and denoting by δ : µ−1 (Q ) ∼ = ν −1 (Q ) and ε : λ−1 (Q ) ∼ = Q

12.8.4,

the F-isomorphisms induced by λ , by ξ∗ : Q ∼ = ν −1 (Q )

12.8.5

the isomorphism determined by ξ , and by α : Q ∼ 12.8.6 = µ−1 (Q ) the isomorphism mapping u ∈ Q on ζ χ (u) , we ﬁnally obtain ξ∗ ◦ϕ∗ = δ◦α and therefore we get

P ϕ = ιP Q ◦ ϕ∗ = ιQ ◦ ξ∗

−1

◦ δ ◦ α

−1 = λ ◦ ιP ◦ ξ∗ λ−1 (Q ) ◦ (ε

where ε−1 ◦ ξ∗

−1

◦ δ) ◦ α

12.8.7,

−1

◦ δ belongs to −1 F λ−1 (Q ), µ−1 (Q ) = λ F Q , ν −1 (Q )

12.8.8

since F is normal in F . Conversely, if F(P ) stabilizes F and equality 12.8.1 holds, then, for any subgroup Q of P and any F-morphism ϕ : Q → P , we already know that ϕ = ιP P ◦ ψ for some F-morphism ψ : Q → P and therefore we have ψ = ν ◦ ψ for suitable ν ∈ F(P ) and ψ ∈ F (P , Q) ; thus, we get F ϕ(Q) = F ν ψ (Q) = ν F ψ (Q) = ϕ F (Q) 12.8.9. We are done.

184

Frobenius categories versus Brauer blocks

12.9 Actually, the Frattini argument can be pushed further till we reach a full functor . Explicitly, let P be an F-stable subgroup of P and F a normal Frobenius P -subcategory of F ; consider the full subcategory F|P of F over the set of subgroups of P — which is a divisible P -category — ∗ and denote by F|P the subcategory of F|P formed by the same objects and by all the isomorphisms, and by F ∗ the corresponding subcategory of F ∗ ∗ ∗ (cf. 1.5); ﬁnally, denote by F|P the quotient category of F|P /F where, for any pair of F-isomorphic subgroups Q and Q of P , we have ∗ ∗ (F|P )(Q , Q) = F(Q , Q)/F (Q) /F

12.9.1;

note that these quotient sets deﬁne indeed a category by the very deﬁnition of normality. 12.10 Always in order to iterate the Frattini argument, we have to consider the subgroups of P fully highnormalized in F . We say that a subgroup Q of P is fully highnormalized in F whenever N0 = Q and Ni = NP (Ni−1 ) for any i ≥ 1 are fully normalized in F ; from Proposition 2.7 it is easily checked that, for any subgroup Q of P , there is ϕ ∈ F(P, Q) such that ϕ(Q) is fully highnormalized in F . Then, let us denote by FF (P )|P the divisible P -category formed by all the homomorphisms induced by the action of the ∗ elements in F(P ) over the subgroups of P , and by HF (P )|P the subcate gory of FF (P )|P formed by the subgroups of P fully highnormalized in F and by all the FF (P )|P -isomorphisms between them. Similarly, we denote ∗ by HF (P )|P the corresponding subcategory and by ∗ ∗ HF (P )|P /HF (P )|P

12.10.1

the corresponding quotient. Note that, for any pair of subgroups Q and Q of P fully highnormalized in F , we have the evident inclusion ∗ ∗ HF (P )|P (Q , Q) ⊂ F|P (Q , Q)

12.10.2.

Proposition 12.11 Let P be an F-stable subgroup of P and F a normal ∗ ∗ Frobenius P -subcategory of F . The inclusion HF (P )|P ⊂ F|P induces a full functor ∗ ∗ ∗ ∗ HF 12.11.1 (P )|P /HF (P )|P −→ F|P /F and the restricted functor over the F-nilcentralized objects is faithful too. Proof: Let Q and Q be F-isomorphic subgroups of P fully highnormalized in F and θ : Q ∼ = Q an F-isomorphism; since Q is fully normalized in F and F (Q ) is a normal subgroup of F(Q ) , it follows from Proposition 2.11

12. Quotients and normal subcategories

185

that Q is also fully F (Q )-normalized in F and therefore, according to condition 2.8.2 and to equality 12.6.1, there are an F-morphism F (Q)

ζ : NP

(Q) −→ P

12.11.2

and an element χ ∈ F (Q) fulﬁlling θ(u) = ζ χ(u) for any u ∈ Q ; moreover, since Q is fully F (Q)-normalized in F too, we have F (Q) F (Q ) ζ NP (Q) = NP (Q )

12.11.3;

ﬁnally, since P is F-stable and we have F (Q)

P ∩ NP

F (Q )

(Q) = NP (Q) and P ∩ NP

(Q ) = NP (Q )

12.11.4,

it is clear that ζ induces an F-isomorphism ξ : NP (Q) ∼ = NP (Q )

12.11.5.

That is to say, denoting by F NP (Q ), NP (Q) Q ,Q the set of F-morphisms η : NP (Q) → NP (Q ) such that η(Q) = Q , we have proved that the map F NP (Q ), NP (Q) Q ,Q /F NP (Q) Q −→ F(Q , Q)/F (Q)

12.11.6

induced by the restriction is surjective; consequently, since F(P , P )Q ,Q maps NP (Q) onto NP (Q ) and, by the very deﬁnition of FF (P )|P , we have F(P , P )Q ,Q = FF (P )|P (Q , Q)

12.11.7,

it suﬃces to argue by induction on |P : Q| to get the surjectivity of the map FF (P )|P (Q , Q)/FF (P )|P (Q) −→ F(Q , Q)/F (Q)

12.11.8.

Moreover, if σ ∈ F(P ) induces an isomorphism θ : Q ∼ = Q belonging to F (Q , Q) , then, since Q and Q are also fully highnormalized in F by statement 12.6.2, and clearly we have θ

FN (Q) = FN (Q ) P (Q) P (Q )

12.11.9,

it follows from statement 2.10.1 that there exists an F -isomorphism ξ : NP (Q) ∼ = NP (Q ) extending θ . But, the diﬀerence between ξ and the isomorphism induced by σ acts trivially on Q and therefore, according to 2.14

186

Frobenius categories versus Brauer blocks

and to Proposition 2.16, it is an automorphism of NP (Q) in the Frobenius NP (Q)-category F

NP (Q)·CF (Q) = NFP

(Q)

(Q)

12.11.10

which coincides with FNP (Q) whenever Q is F-nilcentralized (cf. Remark 4.2); hence, this diﬀerence belongs to F NP (Q) , so that we may assume that ξ coincides with the isomorphism induced by σ . Consequently, once again arguing by induction on |P : Q| , we get σ ∈ F (P ) . We are done. Proposition 12.12 Let H be a subgroup of Aut(P ) containing F(P ) and ¯ = H/F(P ) is a p -group then there is a unique Frobenius stabilizing F . If H ¯ H P -category F such that, for any subgroup Q of P , we have ¯

F H (P, Q) = H ◦ F(P, Q)

12.12.1

¯

and then F is normal in F H . ¯

Proof: For any pair of subgroups Q and R of P , we deﬁne F H (Q, R) as the set of group homomorphisms ϕ : R → Q such that ιP Q ◦ ϕ ∈ H ◦F(P, R)

12.12.2;

¯

−1 thus, if T is a subgroup of P and ψ ∈ F H (R, T ) , we have ιP ◦η R ◦ψ = χ for suitable χ ∈ H and η ∈ F(P, T ) , so that we still have

ιP χ(R) ◦ (χR ◦ ψ) = η

12.12.3

where χR : R ∼ = χ(R) denote the group isomorphism determined by χ , and therefore χR ◦ ψ belongs to F χ(R), T ; hence, since P −1 χ ◦ ιP ◦ (χR ◦ ψ) Q ◦ (ϕ ◦ ψ) = χ ◦ (ιQ ◦ ϕ) ◦ (χR ) P −1 χ ◦ (ιQ ◦ ϕ) ◦ (χR ) ∈ H ◦ F P, χ(R)

12.12.4,

¯

H ιP is a P -category and Q ◦ (ϕ ◦ ψ) belongs to H ◦F(P, T ) ; that is to say, F we claim that it is a Frobenius P -category too. ¯

Indeed, if ϕ ◦ θ ∈ F H (Q, T ) for some group homomorphism θ : T → R then, for some ζ ∈ H and ν ∈ F(P, T ) , we have −1 ιP ◦ν Q◦ϕ◦θ =ζ

12.12.5;

moreover, since we have ιP Q ◦ ϕ = ξ ◦ µ for suitable ξ ∈ H and µ ∈ F(P, R) , denoting by ρ : R ∼ = (ζ ◦ ξ)(R) the group isomorphism determined by ζ ◦ ξ , we get (ζ ◦ ξ) ◦ µ ◦ ρ−1 ◦ (ρ ◦ θ) = ν 12.12.6

12. Quotients and normal subcategories

187

and therefore, since F is divisible, ρ ◦ θ belongs to F (ζ ◦ ξ)(R), T , so that the group homomorphism P ιP ρ(R) ◦ ρ ◦ θ = (ζ ◦ ξ) ◦ ιR ◦ θ

12.12.7

¯

belongs to F(P, T ) . Hence, F H is divisible too. ¯

¯

Since F H (P ) = H , F H fulﬁlls condition 2.8.1. On the other hand, let Q ¯ be a subgroup of P , K a subgroup of Aut(Q) and ϕ an element of F H (P, Q) ¯ such that ϕ(Q) is fully ϕK-normalized in F H ; thus, there are η ∈ H and ψ ∈ F(P, Q) such that ϕ = η ◦ ψ , and it is quite clear that ψ(Q) is fully ¯ ¯ ψ K-normalized in F H , and a fortiori in F (since F H contains F ); consequently, since F is a Frobenius P -category, there are an F-morphism ζ : Q·NPK (Q) −→ P

12.12.8

and χ ∈ K such that ζ(u) = ψ χ(u) for any u ∈ Q ; hence, setting ξ = η ◦ ζ , we get ξ(u) = ϕ χ(u) for any u ∈ Q . It is quite clear that F is normal ¯ in F H . We are done. 12.13 Recall that in chapter 5 we have introduced the F-essential subgroups of P (cf. 5.7) and, for such a subgroup R , the normal subgroup XF (R) of F(R) (cf. Corollary 5.13); in the analogy with the Chevalley groups we have evoked above, the corresponding quotients XF˜ (R) = XF (R)/FR (R)

12.13.1

play the role of “Levi subgroups of rank 1”. Proposition 12.14 If F is a divisible P -category contained in F and, for any F-essential subgroup R of P , F (R) contains XF (R) , then, for any subgroup Q of P , we have F(P, Q) = F(P ) ◦ F (P, Q)

12.14.1.

In particular, for any subgroup K of Aut(Q) , Q is fully K-normalized in F if and only if it is fully K-normalized in F . Moreover, Q is F -selfcentralizing if and only if it is F-selfcentralizing. Proof: Clearly we have F(P ) ◦ F (P, Q) ⊂ F(P, Q)

12.14.2;

conversely, we will prove that any ϕ ∈ F(P, Q) belongs to F(P ) ◦ F (P, Q) arguing by induction on the length D(ϕ) introduced in 5.15; since the inclusion ιP Q : Q → P belongs to F (P, Q) , we may assume that D(ϕ) ≥ 1 and therefore

188

Frobenius categories versus Brauer blocks

there are ψ ∈ F(P, Q) , an F-essential subgroup R of P fully normalized in F , an element η of F(R, Q) and a p -element τ of XF (R) such that (cf. 5.15) D(ψ) = D(ϕ) − 1

,

ϕ = ιP R ◦τ ◦η

and ψ = ιP R ◦η

12.14.3;

by our induction hypothesis, we get ψ = σ ◦ ψ for suitable σ ∈ F(P ) and ψ ∈ F (P, Q) and thus, setting R = σ −1 (R) and denoting by θ ∈ F(R , R) the isomorphism determined by σ −1 , we have ψ = ιP R ◦ (θ ◦ η) which implies that θ ◦ η belongs to F (R , Q) since F is divisible; but, denoting by τ the image of τ in XF (R ) throughout θ , we get σ −1 ◦ ϕ = ιP R ◦ τ ◦ (θ ◦ η)

12.14.4

and therefore σ −1 ◦ ϕ belongs to F (P, Q) . If Q is fully K-normalized in F and ψ : Q·NPK (Q) → P is an F -morphism, we have K ψ NPK (Q) = NP ψ ψ(Q) 12.14.5 since ψ is an F-morphism too. Conversely, if Q is fully K-normalized in F and ψ : Q·NPK (Q) → P is an F-morphism, by the above argument we get ψ = σ ◦ ψ for a suitable σ ∈ F(P ) and some F -morphism ψ : Q·NPK (Q) −→ P

12.14.6

and therefore we still get ψ ψ ψ NPK (Q) = σ NP K ψ (Q) = NPK ψ(Q)

12.14.7.

Moreover, if Q is fully centralized in both F and F , then Q is either F- or F -selfcentralizing if and only if CP (Q) ⊂ Q . We are done. Corollary 12.15 If F is a Frobenius P -category contained in F and we have XF (Q) ⊂ F (Q) for any F-essential subgroup Q of P , then Q is F -essential too and we have XF (Q) = XF (Q) 12.15.1. Moreover, an F -essential subgroup R of P such that F (R) is normal in F(R) is F-essential. Proof: If Q is F-essential, it is F-selfcentralizing (cf. Theorem 5.11), and we have (cf. equality 5.13.1) F (Q) = XF (Q)·F (Q)ιP

Q

12.15.2;

12. Quotients and normal subcategories

189

thus, the stabilizer F (Q)ιP of ιP Q is a proper subgroup of F (Q) fulﬁlling the Q

corresponding condition 5.11.2; moreover, Q is F -selfcentralizing by Proposition 12.14; hence, by Theorem 5.11 and Corollary 5.13, Q is F -essential too and we have XF (Q) = XF (Q) . Conversely, assume that Q is F -essential and that F (Q) B F(Q) ; then, Q is F -selfcentralizing (cf. 5.11.1), so that it is F-selfcentralizing by Proposition 12.14. Moreover, let M be a proper subgroup of F (Q) fulﬁlling condition 5.11.2; then, the Frattini argument proves that NF (Q) (M ) is a proper subgroup of F(Q) and it is easily checked that it fulﬁlls the corresponding condition 5.11.2. We are done. Theorem 12.16 Let P be an F-stable subgroup of P and F a divisible P -subcategory of F such that FP (P ) is a Sylow p-subgroup of F (P ) and that, for any F -selfcentralizing subgroup Q of P , we have F(P , Q) = F(P ) ◦ F (P , Q)

12.16.1.

Then, F is a Frobenius P -category if and only if it fulﬁlls the following two conditions 12.16.2 For any element σ ∈ F(P ) which, restricted to some F -selfcentralizing subgroup Q of P , determines an element of F (P , Q) , there is z in CP (Q) such that σ ◦ κP (z) belongs to F (P ) . 12.16.3 Any divisible P -category F fulﬁlling F (P , Q) ⊃ F (P , Q) for any F -selfcentralizing subgroup Q of P contains F . Proof: If F is a Frobenius P -category and, for some F -selfcentralizing subgroup Q of P , the restriction ρ : Q → P of σ ∈ F(P ) to Q belongs to F (P , Q) , then, since clearly we have ρ FNP (Q) (Q) ⊂ FP ρ(Q) 12.16.4, it follows from statement 2.10.1 that there is an F -morphism ζ : NP (Q) −→ P

12.16.5

extending ρ . Thus, the restriction ξ : NP (Q) → P of σ ∈ F(P ) to NP (Q) coincides with ζ over Q ; in particular, since Q is F -selfcentralizing, we get ζ NP (Q) = ξ NP (Q) 12.16.6 and then, denoting by ζ∗ : NP (Q) ∼ = ζ NP (Q)

and ξ∗ : NP (Q) ∼ = ξ NP (Q)

12.16.7,

the respective isomorphisms determined by ζ and ξ , (ξ∗ )−1 ◦ζ∗ is an F-automorphism of NP (Q) acting trivially on Q .

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But, since CP (Q) = Z(Q) , it follows from Theorem 3.2 in [28, Ch. 5] that, denoting by F NP (Q) Q the stabilizer of Q in F NP (Q) , the kernel of the restriction map F NP (Q) Q −→ F(Q)

12.16.8

is a p-group and thus it is contained in FP NP (Q) (cf. Proposition 2.11). Consequently, there is z ∈ CP (Q) fulﬁlling ζ = ξ ◦ κN (Q) (z) (cf. 1.8). P

That is to say, the restriction of the F-automorphism σ ◦ κP (z) of P to NP (Q) belongs to F P , NP (Q) ; at this point, arguing by induction on |P : Q| , we get an element z ∈ CP NP (Q) ⊂ CP (Q)

12.16.9

such that σ ◦ κP (zz ) belongs to F (P ) . On the other hand, if F is a Frobenius P -category, condition 12.15.3 follows from Corollary 5.14, arguing by induction on the length of ϕ ∈ F (P , Q) (cf. 5.15). From now on, we assume that F fulﬁlls these conditions; since FP (P ) is a Sylow p-subgroup of F (P ) and condition 12.16.3 implies condition 4.12.3, it suﬃces to prove that F fulﬁlls condition 4.12.2. Let Q be an F -selfcentralizing subgroup of P , R a subgroup of NP (Q) which contains Q and ϕ : Q → P an F -morphism such that ϕ FR (Q) is contained in FP ϕ (Q) . According to Proposition 2.7, there is an F-morphism ξ : R → P such that ¯ = ξ(Q) is fully normalized in F . Q ¯ = ξ(R) and denote by η : Q ¯ ∼ Set R = Q the inverse of the isomorphism ¯ induced by ξ ; since Q is also fully centralized in F (cf. Proposition 2.11) and ¯ → P extending ϕ ◦ η ϕ is an F-morphism too, there is an F-morphism ψ¯ : R ¯ (cf. statement 2.10.1) and therefore ψ = ψ ◦ξ extends ϕ . Then, since we have ψ(R) ⊂ P , it follows from equality 12.16.1 above that we get ψ = ιP P ◦σ ◦ψ for a suitable σ ∈ F(P ) and some F -morphism ψ : R → P ; hence, we still get σ ψ (u) = ϕ (u) for any u ∈ Q , so that σ induces an F -isomorphism Q = ψ (Q) ∼ = ϕ (Q)

12.16.10.

That is to say, the restriction of σ to Q belongs to F (P , Q ) and then, according to condition 12.16.2, there is z ∈ CP (Q ) such that σ ◦ κP (z) belongs to F (P ) ; ﬁnally, we obtain the F -morphism σ ◦ κP (z) ◦ ψ : R −→ P which still extends ϕ . We are done.

12.16.11

12. Quotients and normal subcategories

191

Corollary 12.17 The set of Frobenius P -categories F contained in F which fulﬁll XF (Q) ⊂ F (Q) for any F-essential subgroup Q of P has a smallest element F a . In particular, F a is normal in F , we have (F a )a = F a and Op F(R) ⊂ F a (R)

12.17.1

for any subgroup R of P . Moreover, any divisible P -category contained in F and containing F a is a Frobenius P -category. Proof: In order to prove the ﬁrst statement, it suﬃces to prove that if F and F are Frobenius P -categories which are contained in F and fulﬁll XF (Q) ⊂ F (Q) ∩ F (Q)

12.17.2

for any F-essential subgroup Q of P , then the smallest element of the set of divisible P -categories F which are contained in F and, for any F-selfcentralizing subgroup Q of P , fulﬁll F (P, Q) = F (P, Q) ∩ F (P, Q)

12.17.3

is a Frobenius P -category. Denote by F this smallest element; by Proposition 12.14, F fulﬁlls equality 12.16.1 for any F -selfcentralizing subgroup Q of P , and it is clear that F fulﬁlls condition 12.16.3. Assume that σ ∈ F(P ) restricted to Q belongs to F (P, Q) ; it follows again from Proposition 12.14 that F and F fulﬁll equality 12.16.1 and that Q is also F - and F -selfcentralizing. Then, it follows from Theorem 12.16 that σ belongs to the intersection F (P ) ∩ F (P ) = F (P )

12.17.4;

that is to say, F also fulﬁlls condition 12.16.2. Hence, the ﬁrst statement follows from Theorem 12.16. Now, since F a is unique, it is clear that F(P ) stabilizes F a ; moreover, it follows from Proposition 12.14 that F a fulﬁlls condition 12.8.1 and therefore, according to Proposition 12.8, F a is normal in F . In particular, for any subgroup Q of P , F a (Q) is a normal subgroup of F(Q) ; then, it follows from Corollary 12.15 that Q is F a -essential if and only if it is F-essential and that, in this case, we have XF a (Q) = XF (Q) ; consequently, we get (F a )a = F a . On the other hand, it follows from Propositions 2.11 and 12.14 that |F a (Q)|p = |F(Q)|p and therefore, since F a (Q) B F(Q) , we still get the inclusion Op F(Q) ⊂ F a (Q) . The last statement follows from Propositions 12.12 and 12.14. We are done. Corollary 12.18 If P is an F-stable subgroup of P and F is a normal Frobenius P -subcategory of F then F a is contained in F a and normal in F .

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Frobenius categories versus Brauer blocks

Proof: For any F -essential subgroup R of P , inclusion 12.17.1 implies that XF (R) ⊂ Op F (R) ⊂ Op F(R) ⊂ F a (R) 12.18.1; thus, in order to prove that F a ⊂ F a , it suﬃces to prove that the minimal divisible P -category F fulﬁlling F (P , Q) = F (P , Q) ∩ F a (P , Q)

12.18.2,

for any F -selfcentralizing subgroup Q of P , is a Frobenius P -category. Note that, according to Proposition 12.14 and to inclusion 12.18.1, F fulﬁlls equality 12.16.1 for any F -selfcentralizing subgroup of P , and that the minimality of F implies condition 12.16.3 with respect to F . Consequently, it suﬃces to prove that F fulﬁlls condition 12.16.2 too. Let σ be an element of F (P ) which restricted to Q belongs to F (P , Q) for some F -selfcentralizing subgroup Q of P ; setting Q = σ (Q) , consider an F a -morphism ψ : NP (Q ) → P such that Q = ψ(Q ) is fully normalized in F a (cf. Proposition 2.7); by Proposition 2.11, Q is fully centralized too and, since from the action of σ we get σ

FNP (Q) (Q) = FNP (Q ) (Q )

12.18.3,

a it follows from statement 2.10.1 that there is an F -morphism ζ : NP (Q) → P fulﬁlling ζ(u) = ψ σ (u) for any u ∈ Q .

Since F is normal in F , it follows from Proposition 12.8 that we still have CP (Q ) = Z(Q ) ; hence, according to equality 12.18.3, we get ζ NP (Q) = ψ NP (Q ) 12.18.4 and therefore there exists an F a -isomorphism ξ : NP (Q) ∼ = NP (Q )

12.18.5

fulﬁlling ξ(u) = σ (u) for any u ∈ Q ; thus, the diﬀerence between ξ and the isomorphism NP (Q) ∼ = NP (Q ) induced by σ acts trivially on Q ; but, since CP (Q) = Z(Q) , it follows from Theorem 3.4 in [28, Ch.5] that the kernel of the restriction map F NP (Q) Q −→ F(Q) 12.18.6 is a p-group and therefore it is contained in (cf. 12.17.1) Op F NP (Q) ⊂ F a NP (Q)

12.18.7;

hence, the restriction of σ to NP (Q) belongs to F P , NP (Q) . Consequently, arguing by induction on |P : Q| , we get σ ∈ F (P ) ; thus, F is a Frobenius P -category by Theorem 12.16.

12. Quotients and normal subcategories

193

On the other hand, since F is normal in F , F(P ) stabilizes F and therefore it stabilizes F a too; moreover, since F a is normal in F , for any subgroup Q of P we have (cf. Proposition 12.8) F(P , Q) = F(P ) ◦ F (P , Q) = F(P ) ◦ F a (P , Q)

12.18.8;

now, it follows from Proposition 12.8 that F a is also normal in F . We are done. Corollary 12.19 If U is an F-stable subgroup of P then, setting P¯ = P/U and F¯ = F/U , the image of F a in F¯ contains F¯ a . Proof: On one hand, it is clear that U is also F a -stable, and the image of F a in F¯ is just the U -quotient F a /U of F a , which is a Frobenius P¯ -category by Proposition 12.3; on the other hand, for any subgroup Q of P contain ¯ Q) ¯ , where Q ¯ = Q/U , and therefore, ing U , Op F(Q) maps onto Op F( a ¯ contains XF¯ (Q) ¯ whenever Q ¯ is according to inclusion 12.17.1, (F /U )(Q) ¯ F-essential. We are done. 12.20 We are ready to deﬁne the simple Frobenius P -categories; let us say that F is simple if F = F a and there is no nontrivial proper F-stable subgroup in P ; then, we have 12.20.1 A ﬁnite group G such that FG is simple and we have Op (G) = {1} is simple. Indeed, if P is a Sylow p-subgroup of G then, for any N B G , P ∩ N is an FG -stable subgroup of P and therefore, if FG is simple and P ∩ N = {1} , N contains P which implies that N contains Op (G) ; thus, since we are assuming that FG = (FG )a , N coincides with G .

Chapter 13

The hyperfocal subcategory of a Frobenius P-category 13.1 Let P be a ﬁnite p-group and F a Frobenius P -category. Denoting by iF the inclusion functor from F to the category Gr of groups, it is clear that we have a canonical surjective homomorphism from P to the direct limit of iF , and we call F-focal subgroup of P the kernel FF of this homomorphism P −→ lim iF −→

13.1.1;

actually, it easily follows from $ % Corollary 5.14 that FF is generated by the union of the sets u−1 σ(u) u∈Q where Q runs over the set of subgroups of P and σ over F(Q) . In the case we consider the Frobenius P -category FG associated with a ﬁnite group G having P as a Sylow p-subgroup (cf. 1.8), it is well-known (for instance, see Theorems 3.1 and 4.1 in [28, Ch.7]) that P/FFG is canonically isomorphic to the maximal Abelian p-quotient of G . 13.2 The iteration of the above construction in the ﬁnite groups leads us to introduce the F-hyperfocal subgroup as the subgroup HF of P generated by $ % the union of the sets u−1 σ(u) u∈Q where Q runs over the set of subgroups of P and σ over the set of p -elements of F(Q) . Indeed, it is clear that in a chain of normal subgroups of G starting on G and having successive Abelian p-quotients, all the terms contain all the p -elements of G , and then, it is not diﬃcult to prove that HFG = P ∩ Op (G) 13.2.1. In our abstract setting, it follows from Proposition 2.11 that 13.2.2 HF = {1} if and only if F = FP . Moreover, it is clear that F(P ) stabilizes HF and that Op F(P ) acts trivially on the quotient P/HF ; more precisely, we have the following result. Lemma 13.3 For any subgroup Q of P and any ϕ ∈ F(P, Q) , there is w ∈ P such that, for any u ∈ Q , we have ϕ(u) ≡ uw In particular, we have FF = HF ·[P, P ] .

mod HF

13.3.1.

Proof: We argue by induction on the length D of ϕ (cf. 5.15); since Op F(P ) acts trivially on the quotient P/HF , we may assume that D ≥ 1 and therefore that we have ϕ = ιP R ◦ σ ◦ ν , where R is an F-essential subgroup of P , σ is a p -element of F(R) and ν is an element of F(R, Q) such that ιP R ◦ ν has a length equal to D − 1 (cf. 5.15); then, according to our induction hypothesis, there is w ∈ P such that, for any u ∈ Q , we get ϕ(u) ≡ σ ν(u) ≡ ν(u) ≡ uw mod HF 13.3.2.

196

Frobenius categories versus Brauer blocks

13.4. Thus, any normal subgroup P of P containing the hyperfocal subgroup HF is F-stable and, setting P¯ = P/P , the corresponding P -quotient F¯ = F/P clearly is the Frobenius category associated with the group P¯ itself (cf. 1.8). In this chapter, we prove the existence of a Frobenius HF -subcategory F h of F — called the hyperfocal subcategory of F — which, whenever F is the Frobenius category associated with a ﬁnite group G , is the Frobenius category associated with the group Op (G) . Firstly, we need the following result. Lemma 13.5 For any subgroup Q of P fully normalized in F , FHF (Q) contains a Sylow p-subgroup of Op F(Q) . Proof: We already know that FP (Q) is a Sylow p-subgroup of F(Q) (cf. Proposition 2.11) and that the intersection FP (Q) ∩ Op F(Q) is generated ' ( by the union of the sets FR (Q), σ where R runs over the set of subgroups of NP (Q) and σ over the set of p -elements of NF (Q) FR (Q) (cf. 13.2.1); but, since Q is fully centralized too (cf. Proposition 2.11), it follows from statement 2.10.1 that such a σ can be lifted to a p -element τ of F(R)Q and ' ( $ % then FR (Q), σ is the image of the set v −1 τ −1 (v) v∈R which is contained in HF . We are done. Theorem 13.6 Let P be a normal subgroup of P containing HF . Then, we have a normal Frobenius P -subcategory F of F such that F (Q ) = FP (Q ) ◦ Op F(Q )

13.6.1

for any subgroup Q of P fully normalized in F . Proof:† For any subgroup Q of P , choose η ∈ F(P, Q) such that Q = η(Q) is fully normalized in F (cf. Proposition 2.7) and denote by F (Q) the subgroup of F(Q) deﬁned by F (Q) = FP (Q ) ◦ Op F(Q )

η

13.6.2;

note that, since FHF (Q ) contains a Sylow p-subgroup of Op F(Q ) by Lemma 13.5, we have 13.6.3 FP (Q ) is a Sylow p-subgroup of F (Q ) and then we claim that F (Q) does not depend on the choice of η . Indeed, for another choice η ∈ F(P, Q) , setting Q = η (Q) and denoting by σ : Q ∼ = Q the corresponding group isomorphism, we obviously have σ p O F(Q ) = Op F(Q ) 13.6.4 †

Our proof of this result in [52, Theorem 7.4] seems not correct.

13. The hyperfocal subcategory of a Frobenius P -category

197

and it follows from condition 2.8.2 that, for a suitable χ ∈ F(Q ) , there is an F-morphism ζ : NP (Q ) → P extending σ ◦ χ ; thus, since Q is fully normalized in F and P is F-stable, we get ζ NP (Q ) = NP (Q ) 13.6.5 and, since F (Q ) is normal in F(Q ) , we still get σ

FP (Q ) ◦ Op F(Q ) = FP (Q ) ◦ Op F(Q )

13.6.6.

Consequently, it is easily checked that, for any ϕ ∈ F(P, Q) , we have F (Q) = F ϕ(Q)

ϕ

13.6.7.

Moreover, if R is a subgroup of Q then we claim that: 13.6.8 The action over R of the stabilizer F (Q)R of R in F (Q) is contained in F (R). Indeed, according to Proposition 2.7, we may assume that Q is fully normalized in F and then, according to Proposition 2.11 and to statement 13.6.3, FP (Q) and FP (Q) are respective Sylow p-subgroups of F(Q) and F (Q) ; thus, since F(Q) = FP (Q) ◦ F (Q) , up to F (Q)-conjugation we may assume that FP (Q)R is a Sylow p-subgroup of F(Q)R and then, since FP (Q)R ∩ F (Q) ⊂ FP (Q)

13.6.9,

FP (Q)R is a Sylow p-subgroup of F (Q)R , so that we have F (Q)R = FP (Q)R ◦ Op F (Q)R

13.6.10;

but, the actions of FP (Q)R and of Op F (Q)R over R are respectively contained in FP (R) and in Op F(R) , so in F (R) . For any pair Q and Q of F-isomorphic subgroups of P , consider the set F (Q , Q) of ϕ ∈ F(Q , Q) fulﬁlling the following condition: 13.6.11 There are two sequences {Qi }0≤i≤n and {Ui }1≤i≤n of subgroups of P such that Q0 = Q

,

Qn = Q

and

Qi−1 , Qi ⊂ Ui for 1 ≤ i ≤ n ,

and a sequence {ϕi }1≤i≤n of F-isomorphisms ϕi : Qi−1 ∼ = Qi induced by F (Ui ) such that their composition coincides with ϕ . Note that ϕ−1 belongs to F (Q, Q ) for any ϕ ∈ F (Q , Q) , and that F (Q) and F (Q, Q) have been deﬁned independently; moreover, it is clear that F (Q) ⊂ F (Q, Q)

and FP (Q , Q) ⊂ F (Q , Q)

13.6.12.

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Frobenius categories versus Brauer blocks

Actually, by the tautological equality ϕ σ ◦ ϕ = ϕ ◦ σ for any σ ∈ F(Q) , it follows from equality 13.6.7 and statement 13.6.8 that we always may assume that Q0 = U1 = Q1 and that Qi = Qi+1 for any 1 ≤ i < n . More generally, for any pair of subgroups Q and R of P we set F (Q, R) =

ιQ ϕ(R) ◦ F ϕ(R), R

13.6.13

ϕ∈F (Q,R)

and we will prove that this correspondence F is a divisible P -category normal in F which fulﬁlls the two conditions in Corollary 2.13, and that, for any subgroup Q of P , we have (cf. 13.6.2) F (Q) = F (Q, Q)

13.6.14.

Actually, it is already clear that F (P ) = F (P , P ) and therefore, by statement 13.6.3, that F fulﬁlls condition 2.8.1. If Q , Q and Q are three F-isomorphic subgroups of P , ϕ an element of F (Q , Q) and ψ an element of F (Q , Q ) , it is quite clear from condition 13.6.11 that ψ ◦ ϕ belongs to F (Q , Q) . Moreover, if R and R are subgroups of P respectively containing Q and Q , and ψ ∈ F (R , R) fulﬁlls ψ(Q) = Q , then it easily follows from the same condition that the F-isomorphism ϕ : Q ∼ = Q induced by ψ belongs to F (Q , Q) . Consequently, if R and T are subgroups of P , ϕ an element of F (R, Q) and ψ an element of F (T, R) then ψ ◦ ϕ belongs to F (T, Q) ; indeed, setting Q = ϕ(Q) and Q = ψ(Q ) , and denoting by ϕ∗ : Q ∼ = Q and by the corresponding F-isomorphisms, it follows from our deﬁniQ ψ∗ : Q ∼ = tion that ϕ∗ belongs to F (Q , Q) and we already have proved that ψ∗ and ψ∗ ◦ ϕ∗ respectively belong to F (Q , Q ) and to F (Q , Q) . This already proves that F is a divisible P -category. Let Q and Q be F -isomorphic subgroups of P fully normalized and fully centralized in F ; in order to prove that they fulﬁll conditions 2.13.1 and 2.13.2, we argue by induction on |P : Q| and, in particular, we may assume that all the subgroups of P of smaller index fulﬁll condition 2.8.2 or, equivalently, condition 4.13.1 in Lemma 4.13. Let ϕ : Q ∼ = Q be an F -morphism and choose two sequences {Qi }0≤i≤n and {Ui }1≤i≤n of subgroups of P , and a sequence {ϕi }1≤i≤n of F-isomorphisms ϕi : Qi−1 ∼ = Qi , as in condition 13.6.11; note that, in order to prove condition 2.13.1, in the remark above about our choice we may assume that ϕ1 = {idQ0 } . According to Proposition 2.7, for any 1 ≤ i ≤ n we may choose an F -morphism ηi : NP (Qi ) −→ P 13.6.15 such that Qi = ηi (Qi ) is fully F (Qi )-normalized in F , and actually may P assume that Q1 = Q , Qn = Q , η1 = ιP N (Q) and ηn = ιN (Q ) . P

P

13. The hyperfocal subcategory of a Frobenius P -category

199

We claim that, for any 2 ≤ i ≤ n , we can apply Lemma 4.13 to the subgroup Qi−1 of P , to the subgroup F (Qi−1 ) of Aut(Qi−1 ) and, denoting ∗ by ηi−1 the inverse of the F -isomorphism induced by ηi−1 NP (Qi−1 ) ∼ = ηi−1 NP (Qi−1 )

13.6.16,

∗ to the F -isomorphism ϕi : Qi−1 ∼ (u)) . = Qi mapping u ∈ Qi−1 on ηi ϕi (ηi−1 Indeed, according to condition 13.6.11, ϕi is induced by an element of F (Ui ) and therefore it is still induced by a suitable F -isomorphism ψi : NUi (Qi−1 ) ∼ = NUi (Qi )

13.6.17;

thus, ϕi can be extended to an F -isomorphism ηi−1 NUi (Qi−1 ) ∼ = ηi NUi (Qi )

13.6.18

Note that, since we have chosen Qi−1 = Qi , Qi is properly contained in Ui and therefore in NUi (Qi ) . Hence, it follows from this lemma that, for any 2 ≤ i ≤ n , there are an F -morphism ζi : NP (Qi−1 ) −→ NP (Qi ) 13.6.19 and an element χi−1 ∈ F (Qi−1 ) such that ∗ ζi χi−1 (u) = ηi ϕi (ηi−1 (u))

13.6.20

for any u ∈ Qi−1 ; note that, since ζi determines a group isomorphism F (Qi−1 ) ∼ = F (Qi ) by equality 13.6.7, denoting by ζi χi−1 the image of χi−1 we still have ζi χi−1 (u) = ζi χi−1 ζi (u) 13.6.21. Consequently, the composition of the family {ζi }1≤i≤n deﬁnes an F -morphism ζ : NP (Q) −→ NP (Q ) 13.6.22 and denoting by χ the composition of the family of the corresponding images in F (Q) of all the elements χi−1 ∈ F (Qi−1 ) where 1 ≤ i ≤ n , it is not diﬃcult to check that ζ(u) = ϕ χ(u) 13.6.23 for any u ∈ Q ; in particular, we have ζ(Q) = Q proving statement 2.13.1. Moreover, allowing the possibility ϕ1 = {id}Q0 in the argument above, it still proves that, for any F -automorphism ψ : Q ∼ = Q there are an F -automorphism ξ : NP (Q) ∼ 13.6.24 = NP (Q)

200

Frobenius categories versus Brauer blocks

and θ ∈ F (Q) fulﬁlling ξ θ(u) = ψ(u) for any u ∈ Q ; that is to say, F (Q, Q) coincides with the set of compositions of all the elements in F (Q) with in F (Q, Q) of the stabilizer of Q in all the elements of the image F NP (Q), NP (Q) . But, since F (P ) = F (P , P ) , arguing by induction on |P : Q| we may assume that we already have F NP (Q) = F NP (Q), NP (Q) 13.6.25. Then, it follows from statement 13.6.8 that this image is contained in F (Q) , so that we get F (Q) = F (Q, Q) . In particular, it follows from equality 13.6.7 that F is normal in F . It remains to prove condition 2.13.2; let R be a subgroup of NP (Q) containing Q·CP (Q) ; since the restriction induces a surjective group homomorphism (cf. statement 2.10.1) F(R)Q −→ NF (Q) FR (Q) 13.6.26, it maps Op F(R)Q onto Op NF (Q) FR (Q) ; but, we have Op F(R)Q ⊂ Op F(R) ⊂ F (R) 13.6.27 and therefore we still have Op F(R)Q ⊂ F (R)Q ; similarly, we get Op NF (Q) FR (Q) ⊂ Op F(Q) ⊂ F (Q) 13.6.28 and therefore we still get Op NF (Q) FR (Q) ⊂ NF (Q) FR (Q) Hence, the image of the group homomorphism F (R)Q −→ NF (Q) FR (Q) contains Op NF (Q) FR (Q) .

13.6.29

13.6.30

Consequently, in order to prove that homomorphism 13.6.30 is surjective, it suﬃces to prove that its image still contains a Sylow p-subgroup of NF (Q) FR (Q) . But, according to statement 13.6.3, FP (Q) is a Sylow p-subgroup of F (Q) and therefore it contains a Sylow p-subgroup of an F (Q)-conjugate of NF (Q) FR (Q) ; thus, since F is a Frobenius P -category and F is normal in F , we can choose an F-morphism ρ : NP (Q) −→ P

13.6.31

such that, setting Q = ρ(Q) and R = ρ(R) , Q is fully normalized in F and in F (cf. Proposition 2.7), R contains ρ CP (Q) = CP (Q )

13. The hyperfocal subcategory of a Frobenius P -category

201

and NFP (Q ) FR (Q ) is a Sylow p-subgroup of NF (Q ) FR (Q ) (cf. Proposition 2.11). On the other hand, since CP (Q ) ⊂ R , FP (R )Q and NFP (Q ) FR (Q ) are both images of NP (R ) ∩ NP (Q ) and therefore the subgroup FP (R )Q of F (R )Q maps onto NFP (Q ) FR (Q ) . We are done. Remark 13.7† As a matter of fact, the argument above still proves the following more general fact. Theorem Let P be an F-stable subgroup of P and, for any subgroup Q of P , F (Q) a (normal) subgroup of F(Q) in such a way that the following three conditions hold 13.7.1 FP (P ) is a Sylow p-subgroup of F (P ) . 13.7.2 For any subgroup Q of P and any element ϕ ∈ F(P, Q) , we have ϕ F (Q) = F ϕ(Q) . 13.7.3 For any pair of subgroups Q and R of P such that CP (R) ⊂ Q and R B Q , the restriction induces a surjective group homomorphism F (Q)R −→ NF (R) FQ (R) where F (Q)R denotes the stabilizer of Q in F (Q) . Then, there exists a normal Frobenius P -subcategory F of F such that F (Q, Q) = F (Q) for any subgroup Q of P . In the corresponding argument, statement 13.6.8 is replaced by the existence of the group homomorphism in condition 13.7.3; moreover, the coincidence of condition 13.7.3 with condition 2.13.2 avoids the last part of the proof. 13.8 With the notation of Theorem 13.6, F is uniquely determined by condition 13.6.1 and it follows from Corollary 5.14 that if P is a normal subgroup of P such that HF ⊂ P ⊂ P then the Frobenius P -category fulﬁlling condition 13.6.1 is contained in F . In particular, whenever P = HF we set F = F h and call hyperfocal subcategory of F this subcategory F h ; note that HF h = HF and (F h )h = F h 13.8.1. Proposition 13.9 If P is a subgroup of P and F is a Frobenius P -subcategory of F then we have HF ⊂ HF

and

F h ⊂ F h

13.9.1.

Moreover, if P is F-stable and F is normal in F then HF is also F-stable and F h is still normal in F . †

This remark has been motivated by Michael Aschbacher’s preprint “Normal subsystems

of fusion systems” [4].

202

Frobenius categories versus Brauer blocks

Proof: The left inclusion is easily checked; moreover it follows from Proposition 2.7 that, for any subgroup Q of HF , we can choose ϕ ∈ F (P , Q) such that Q = ϕ(Q) is fully normalized in F ; then, since Q is also fully normalized in F h (cf. statement 12.6.2), we have (cf. equality 13.6.1) F h (Q ) = FHF (Q )Op F (Q ) ⊂ FHF (Q )·Op F(Q ) ⊂ F h (Q )

13.9.2

and therefore we get F h (Q) ⊂ F h (Q) ; now, the inclusion F h ⊂ F h follows from Corollary 5.14. On the other hand, if P is F-stable and F is normal in F , it is clear that F(P ) stabilizes HF and it also stabilizes F h by Theorem 13.6; moreover, it follows from Proposition 12.8 that for any subgroup Q of HF we have F(P , Q) = F(P ) ◦ F (P , Q)

13.9.3

and therefore, since F(P ) stabilizes HF , it is easily checked that HF is also F-stable and that we still have F(HF , Q) = F(HF ) ◦ F (HF , Q)

13.9.4;

ﬁnally, since F h is normal in F , the last statement follows from the corresponding equality obtained from equality 13.9.4 and Proposition 12.8. We are done. Proposition 13.10 If U is an F-stable subgroup of P , setting P¯ = P/U and F¯ = F/U , the natural homomorphism P → P¯ induces the group and the category isomorphisms HF¯ ∼ = HF /(HF ∩ U )

and

F¯ h ∼ = F h /(HF ∩ U )

13.10.1.

Proof: It easily follows from Proposition 12.3 that the natural homomorphism P → P¯ maps HF onto HF¯ , proving the ﬁrst isomorphism. On the other hand, set P = U ·HF and denote by F the Frobenius P -category fulﬁlling equality 13.6.1, and by F¯ the U -quotient of F ; ﬁrst of all, we will prove that F¯ = F¯ h . According to Proposition 2.7, for any subgroup Q of P containing U we can choose ϕ ∈ F(P, Q) such that Q = ϕ(Q) is fully normalized in F , so that we have (cf. equality 13.6.1) F (Q ) = FP (Q )·Op F(Q ) 13.10.2. ¯ ¯ ¯ But, setting F(hQ ) by the very deﬁnition Q = Q /U , since F(Q )p maps onto p ¯ ) and, since U ⊂ Q , ¯ Q ¯ ) ⊂ F¯ (Q of F¯ , O F(Q ) still maps onto O F(

13. The hyperfocal subcategory of a Frobenius P -category

203

¯ ) too. Thus, FP¯ (Q ¯ ) is a Sylow p-subgroup of F( ¯ Q ¯) FP (Q ) maps onto FP¯ (Q h ¯ ¯ ¯ and therefore FHF¯ (Q ) is a Sylow p-subgroup of F (Q ) ; hence, since FP (Q ) ¯ ) , we ﬁnally get the surjectivity of the canonical hostill maps onto FHF (Q ¯ , so that momorphism F (Q) → F¯ h (Q) ¯ = F¯ h (Q) ¯ F¯ (Q)

13.10.3,

and, since F¯ and F¯ h are both Frobenius HF¯ -categories, the equality F¯ = F¯ h follows from Corollary 5.14. Now, it suﬃces to prove that the inclusion F h ⊂ F induces an isomorphism F h = F h /(HF ∩ U ) ∼ 13.10.4; = F¯ but, if Q and R are subgroups of HF containing the intersection HF ∩ U it is clear that Q = (Q·U ) ∩ HF and R = (R·U ) ∩ HF 13.10.5 and therefore F(Q·U, R·U ) maps R into Q ; hence, setting Q = Q/(HF ∩ U )

and R = R/(HF ∩ U )

13.10.6,

¯ R) ¯ are deﬁned by the same in Proposition 12.3 the sets F h (Q, R) and F¯ (Q, quotient of F(Q·U, R·U ) . We are done. 13.11 A remarkable fact is that the hyperfocal subgroup HF of P is always F-nilcentralized; actually, we have the following slightly more general result. Proposition 13.12 For any subgroup Q of P fully centralized in F , the subgroup Q·HCF (Q) is F-nilcentralized and fully centralized in F . Moreover, for any subgroup R of NP (Q) containing Q·HCF (Q) and any F-morphism ψ : R → P such that ψ(Q) = Q , there is an F-morphism ζ : CP (Q)·R −→ P

13.12.1

extending ψ . In particular, the restriction induces an exact sequence 13.12.2. CP (Q·HCF (Q) ) −→ F Q·CP (Q) Q −→ F(Q·HCF (Q) )Q −→ 1 Proof: Actually, Q is also fully FQ (Q)-normalized in F (cf. 2.10) and thus, by Propositions 2.7 and 2.16, there is a morphism η : Q·HCF (Q) −→ Q·CP (Q)

13.12.3

in the Frobenius Q·CP (Q)-category F (Q)

Q·CF (Q) = NFQ

(Q)

13.12.4

204

Frobenius categories versus Brauer blocks

such that η(Q·HCF (Q) ) is fully centralized in this Q·CP (Q)-category; then, it follows from Lemma 2.17 that η(Q·HCF (Q) ) is also fully centralized in F and, F (Q)

since η is an NFQ

(Q)-morphism, it is clear that η(Q·HCF (Q) ) = Q·HCF (Q) .

Then, if U is a subgroup of CP (Q·HCF (Q) ) , any p -subgroup K of F(U ·Q·HCF (Q) ) which normalizes U and centralizes Q·HCF (Q) , induces a p -group of automorphisms of U ·HCF (Q) in the Frobenius CP (Q)-category CF (Q) and, in particular, from the very deﬁnition of HCF (Q) , we have ' ( K, U ·HCF (Q) ⊂ HCF (Q)

13.12.5;

hence, since K centralizes HCF (Q) , K centralizes U ·HCF (Q) (cf. Theorem 3.2 in [28, Ch.5)]), so that we have K = {idU ·Q·HCF (Q) } . Consequently, the CF (Q·HCF (Q) )-hyperfocal subgroup is trivial and the ﬁrst statement follows from statement 13.2.2 and from the very deﬁnition of F-nilcentralized subgroups in 4.3. Moreover, if R is a subgroup of NP (Q) containing Q·HCF (Q) and ψ an F-morphism from R to P such that ψ(Q) = Q , it follows from the divisibility of F and from statement 2.10.1 that there is an F-automorphism σ of Q·CP (Q) fulﬁlling σ(u) = ψ(u) for any u ∈ Q ; thus, σ stabilizes CP (Q) and then the Frobenius CP (Q)-category CF (Q) , so that it stabilizes HCF (Q) too. Hence, denoting by σ ¯ : Q·CR (P ) −→ Q·σ CR (Q)

13.12.6

the F-isomorphism determined by σ , and by ψ¯ the restriction to Q·CR (Q) of ψ (cf. 2.3), the composition ψ¯ ◦ σ ¯ −1 acts trivially on Q and therefore it deﬁnes a Q·CF (Q)-morphism (cf. 13.12.4) from Q·σ CR (Q) to Q·CP (Q) . But, since Q·HCF (Q) is F-nilcentralized, the group Q·σ CR (Q) is also F-nilcentralized (cf. Proposition 4.5) and therefore it is Q·CF (Q)-nilcentralized too. Consequently, ψ¯ ◦ σ ¯ −1 is determined by an element of Q·CP (Q) ; that is to say, we may choose σ in such a way that we have σ(u) = ψ(u) for any u ∈ Q·CR (Q) . On the other hand, the product Q·CP (Q) is F-selfcentralizing and normal in R·CP (Q) ; moreover, according to our choice, σ can be extended to a group homomorphism R·CP (Q) → P mapping v·z on ψ(v)·σ(z) for any v ∈ R and any z ∈ CP (Q) ; consequently, it follows from statement 2.10.1 that σ can be extended to an F-morphism ζ : R·CP (Q) → P and then, denoting by ζ¯ the restriction of ζ to R , the restrictions to Q·CR (Q) of ψ and ζ¯ coincide each other; at this point, it follows from Proposition 4.6 that we can modify our choice of ζ in such a way that we have ζ¯ = ψ . This proves the second statement above.

13. The hyperfocal subcategory of a Frobenius P -category

205

Furthermore, it is clear that any F-automorphism of Q·CP (Q) stabilizing Q still stabilizes CP (Q) and then the Frobenius CP (Q)-category CF (Q) , so that it stabilizes HCF (Q) too; that is to say, the restriction induces a group homomorphism F Q·CP (Q) Q −→ F(Q·HCF (Q) )Q 13.12.7 which is surjective by the argument above applied to R = Q·HCF (Q) . Finally, if σ is a p -element of the kernel of this homomorphism then, as above, it determines a p -automorphism of CP (Q) in the Frobenius CP (Q)-category CF (Q) and then it follows from Lemma 13.3 that σ acts trivially on the quotient CP (Q)/HCF (Q) ; since σ acts trivially on HCF (Q) , we get σ = idQ·CP (Q) (cf. Theorem 3.2 in [28, Ch. 5]) and therefore this kernel is a p-group. We are done. sc

13.13 If we restrict ourselves to the full subcategory F of F over the set of F-selfcentralizing subgroups of P , the correspondence mapping any sc object Q in F on its center Z(Q) is functorial since it is compatible with sc F-isomorphisms and, for another object R in F contained in Q , we have Z(Q) ⊂ Z(R) that is to say, we get a contravariant functor from F ﬁnitely generated Abelian groups Ab noted zF sc : F

sc

13.13.1; sc

to the category of

−→ Ab

13.13.2.

As a matter of fact, this contravariant functor can be extended to the whole Frobenius P -category F , and in chapter 20 below we discuss the existence of a suitable extension of F by this extended contravariant functor. Proposition 13.14 There is a contravariant functor chF from F to the ex of the category of ﬁnite groups, unique up to natural isoterior quotient Gr morphisms, which maps any subgroup Q of P fully centralized in F on the quotient CP (Q)/HCF (Q) , and maps any F-morphism ϕ : R → Q , between subgroups of P fully centralized in F , on the class of the group homomorphism ζ¯ : CP (Q)/HCF (Q) −→ CP (R)/HCF (R) 13.14.1 induced by an F-morphism ζ : ϕ(R)·CP (Q) → R·CP (R) 13.14.2 fulﬁlling ζ ϕ(v) = v for any v ∈ R . In particular, we have a contravariant functor cfF : F −→ Ab 13.14.3

mapping Q on CP (Q)/FCF (Q) and ϕ on the group homomorphism CP (Q)/FCF (Q) −→ CP (R)/FCF (R) determined by ζ¯ .

13.14.4

206

Frobenius categories versus Brauer blocks

Proof: Since F is equivalent to the full subcategory over the set of subgroups of P fully centralized in F (cf. Proposition 2.7), in order to deﬁne a con up to natural isomorphisms, it suﬃces to deﬁne travariant functor F → Gr it over this full subcategory. With the notation above, the existence of ζ follows from statement 2.10.1 applied to the inverseof the isomorphism R ∼ = ϕ(R) induced by ϕ ; then, it is clear that ζ CP (Q) ⊂ CP (R) and the restriction ξ : CP (Q) → CP (R) of ζ is CF (Q), CF (R) -functorial since, for any subgroup U of CP (Q) and any F-morphism θ : Q·U → Q·CP (Q) fulﬁlling θ(U ) ⊂ CP (Q) and θ(u) = u for any u ∈ Q , the group homomorphism η : R·ξ(U ) → R·ξ CP (Q) mapping vξ(u) on ζ ϕ(v)θ(u) , for any v ∈ R and any u ∈ U , clearly induces a CF (R)-morphism from ξ(U ) to ξ CP (Q) . Consequently, we have ζ(HCF (Q) ) ⊂ HCF (R)

13.14.5

and therefore ζ induces the announced homomorphism 13.14.1. Moreover, for choice ζ of ζ , it is quite clear that the isomor another phism ζ CP (Q) ∼ = ζ CP (Q) induced by ζ and ζ is a CF (R)-isomorphism and therefore, according to Lemma 13.3, there is z ∈ CP (R) such that, for any u ∈ CP (Q) , we have ζ (u) ≡ ζ(u)z

mod HCF (R)

13.14.6,

so that the group homomorphisms induced by ζ and by ζ determine the same Gr-morphism chF (ϕ) : CP (Q)/HCF (Q) −→ CP (R)/HCF (R)

13.14.7.

Finally, for a third subgroup T of P fully centralized in F and for any F-morphism ψ : T → R , it is easily checked that chF (ϕ ◦ ψ) = chF (ψ) ◦ chF (ϕ)

13.14.8.

The last statement follows from the last statement in Lemma 13.3. We are done. 13.15 A last remark. Denoting by p-iGr the subcategory of Gr formed by the ﬁnite p-groups and by the injective group homomorphisms between them, there is a functor z◦ : p-iGr −→ Ab 13.15.1 sc

mapping a ﬁnite p-group Q on Z(Q) , which restricted to F induces an “almost” Mackey complement (cf. 6.24) of the contravariant functor (cf. 6.2) sc

ac(˜zF sc ) : ac(F˜ ) −→ Ab determined by zF sc , as we explain below.

13.15.2

13. The hyperfocal subcategory of a Frobenius P -category

207

13.16 Explicitly, if Q is a ﬁnite p-group, N a normal subgroup of Q and U ⊂ Q a set of representatives for Q/N , then we have the group homomorphism (cf. 1.9) z◦ (ιQ 13.16.1 N ) : Z(N ) −→ Z(Q) mapping z ∈ Z(N ) on

TrQ N (z) =

uzu−1

13.16.2

u∈U

since all the terms of this product belong to Z(R) and the product does not depend on the choice of U , so that it belongs to Z(Q) . For any proper subgroup R of Q , we argue by induction on |Q : R| and, considering a proper normal subgroup N of Q containing R , we deﬁne ◦ Q ◦ N z◦ (ιQ R ) = z (ιN ) ◦ z (ιR )

13.16.3;

this composition does not depend on the choice of N since, for another proper normal subgroup N of Q containing R , we clearly have N Q Q N TrQ 13.16.4 N TrN ∩N (z) = TrN ∩N (z) = TrN TrN ∩N (z) for any z ∈ Z(N ∩ N ) and, by our induction hypothesis, we still have

◦ N ◦ N ∩N z◦ (ιN ) R ) = z (ιN ∩N ) ◦ z (ιR

◦ N ◦ N ∩N z◦ (ιN ) R ) = z (ιN ∩N ) ◦ z (ιR

13.16.5.

13.17 Finally, for any isomorphism θ : Q ∼ = Q , we deﬁne z◦ (θ) : Z(Q) ∼ = Z(Q )

13.17.1

as the isomorphism induced by θ . Since any injective group homomorphism ϕ : R → Q between two ﬁnite p-groups is the composition of an isomorphism ϕ∗ : R ∼ = ϕ(R) with the inclusion ϕ(R) ⊂ Q , it is easily checked that, setting ◦ z◦ (ϕ) = z◦ (ιQ ϕ(R) ) ◦ z (ϕ∗ )

13.17.2,

we get the announced functor which maps any ﬁnite p-group Q on Z(Q) and any injective group homomorphism ϕ : R → Q on z◦ (ϕ) . 13.18 Consider the restriction of z◦ to F z◦F sc : F

sc

sc

−→ Ab

13.18.1

and note that zF sc and z◦F sc obviously factorize throughout the exterior quosc sc tient F˜ of F (cf. 1.3). Let us show that the corresponding additive functor (cf. A4.10) sc ac(˜z◦F sc ) : ac(F˜ ) −→ Ab 13.18.2

208

Frobenius categories versus Brauer blocks

is an “almost” Mackey complement of ac(zF sc ) (cf. 6.24) in the sense speciﬁed sc by the following lemma; recall that the category ac(F˜ ) admits distributive pull-backs given by the relative exterior intersection (cf. Proposition 6.21). sc Lemma 13.19 With the notation above, for any F˜ -morphism ϕ˜ : R → Q we have ˜z◦F sc (ϕ) ˜ ◦ ˜zF sc (ϕ) ˜ = (idZ(Q) )|Q|/|R| 13.19.1. sc Moreover, for another F˜ -morphism ψ˜ : T → Q such that ϕ(R) ∩ ψ(T ) is ˜ we have F-selfcentralizing for any pair of representatives ϕ of ϕ˜ and ψ of ψ, the commutative diagram

Z(Q)

˜ z◦ sc (ϕ) ˜

˜

/ ˜zF sc (ψ) Z(T ) 1ac(˜z◦ sc )(δ) ˜

1

F

Z(R)

ac(˜ zF sc )(˜ γ )/ ˜Q T ) ac(˜zF sc ) (R ∩

13.19.2

F

˜ Q T is the relative exterior intersection of R and T over Q and where R ∩ δ˜

γ ˜

˜ Q T −→ T R ←− R ∩

13.19.3

sc

are the structural ac(F˜ )-morphisms. Proof: Equality 13.19.1 is easily checked when we have ϕ(R) B Q for a representative ϕ of ϕ˜ ; in general, it suﬃces to argue by induction on |Q|/|R| . For the second statement, we may assume that Q contains R and T , and that ϕ˜ and ψ˜ are induced by the inclusions. We argue by induction on |Q : R| and may assume that R = Q ; let N be a proper normal subgroup of Q containing R and, for any w ∈ Q , set Uw = N ∩ T and consider the group homomorphism κw : Uw → N determined by the conjugation by w ; then, since R ∩ (Uw )n = R ∩ T n is F-selfcentralizing for any n ∈ N , it follows from the induction hypothesis that we have the commutative diagram ˜ z◦ sc (˜ ιN R)

Z(N )

1

F

Z(R)

/ ˜zF sc (˜κw ) Z(Uw ) 1ac(˜z◦ sc )(˜εw )

ac(˜ zF sc )(˜ γw )/ ˜ N Uw ) ac(˜zF sc ) (R ∩

13.19.4

F

˜ N Uw is the relative exterior intersection of R and Uw over the where R ∩ morphisms ˜ιN ˜ w (cf. 6.18), and R and κ γ ˜w

ε˜

w ˜ N Uw −→ Uw R ←− R ∩ sc

are the structural ac(F˜ )-morphisms.

13.19.5

13. The hyperfocal subcategory of a Frobenius P -category

209

On the other hand, choosing a set of representatives W ⊂ Q for N \Q/T , it follows from Proposition 6.19 that ˜Q T ∼ N∩ Uw 13.19.6 = w∈W sc

and that the structural ac(F˜ )-morphisms κ ˜ N to N and ˜ιT to T are respecsc tively deﬁned by the families of F˜ -morphisms {˜ κw }w∈W and {˜ιTUw }w∈W ; but, for any z ∈ Z(N ) , we have TrQ u−1 zu = TrTUw (w−1 zw) 13.19.7 N (z) = u

w∈W

where u ∈ Q runs over a set of representatives for N \Q ; hence, we still have the commutative diagram Z(Q) ˜ z◦ sc (˜ ιQ ) N

Q

/ ˜zF sc (˜ιT ) Z(T ) 1ac(˜z◦ sc )(˜ιT )

1

F

Z(N ) ac(˜ zF

/

sc )(˜ κN )

˜Q T ) ac(˜zF sc ) (N ∩

13.19.8.

F

Finally, on the one hand, the commutative diagrams 13.19.4 when w runs over W yield to the following commutative diagram Z(N ) 1 / Z(R) w∈W Z(Uw ) ◦ / 1 ac(˜ z sc )(˜ εw ) w∈W F sc ˜ zF ) (R ∩N Uw ) w∈W ac(˜

˜ z◦ sc (˜ ιN R) F

13.19.9;

on the other hand, since the pull-backs fulﬁll an obvious transitive condition, we have ˜Q T ∼ ˜ N (N ∩ ˜Q T ) ∼ ˜ N Uw R∩ R∩ 13.19.10 = R∩ = w∈W sc

and the corresponding structural ac(F˜ )-morphisms agree; consequently, the commutative diagram 13.19.9 becomes (cf. 13.19.6) ˜ z◦ sc (˜ ιN R) F

Z(R)

1

Z(N ) N

/ ac(˜zF sc )(˜κ ) ˜Q T ) ac(˜zF sc ) (N ∩ 1

/ ac(˜ zF sc )(˜ γ ) ˜Q T ) ac(˜zF sc ) (R ∩

13.19.11.

Now, it is quite clear that the commutativity of diagram 13.19.2 follows from the commutativity of diagrams 13.19.9 and 13.19.11. We are done.

Chapter 14

The Grothendieck groups of a Frobenius P-category 14.1 Let P be a ﬁnite p-group and F a Frobenius P -category. As in nc chapter 11, let us denote by F the full subcategory of F over the F-nilcentralized subgroups of P (cf. 4.3) and let us consider the proper category of nc nc F -chains ch∗ (F ) (cf. A2.8); recall that we have the automorphism functor nc from ch∗ (F ) to the category of groups (cf. Proposition A2.10) autF nc : ch∗ (F ) −→ Gr nc

14.1.1

which maps any ch∗ (F )-object (q, ∆n ) on its ch∗ (F )-automorphismgroup — noted F(q) ; explicitly, we identify F(q) with the subgroup of F q(n) formed by the elements stabilizing the images q(i)n = q(i • n) q(i) in q(n) nc of all the groups q(i) determined by the functor q : ∆n → F for 0 ≤ i ≤ n . nc

nc

14.2 In Theorem 11.32 above, we have proved that when F comes from a block b of a ﬁnite group G , the functor aut(F(b,G) )nc can be lifted to a functor nc ∗ ∗ ∗ (F aut )nc from ch (F ) to the category k -Gr of ﬁnite k -groups (cf. 1.23); (b,G)

since our arguments mostly depend on the existence of such a lifting, in our abstract setting let us consider a functor F nc : ch∗ (F ) −→ k ∗ -Gr aut nc

14.2.1

lifting the functor autF nc to the category k ∗ -Gr and, for (q, ∆n ) as above, set ˆ F nc (q, ∆n ) F(q) = aut

14.2.2.

14.3 Let O be the unramiﬁed complete discrete valuation ring of characteristic zero admitting our ﬁeld k (cf. 1.10) as residue class ﬁeld, and denote by O-mod the category of ﬁnitely generated O-modules. For any ﬁnite ˆ , we denote by Gk (G) ˆ the scalar extension from Z to O of the k ∗ -group G so-called Grothendieck group (cf. Appendix in [54]) of the category of ﬁnite ˆ dimensional k∗ G-modules (cf. 1.25); it is well-known that we have a contravariant functor gk : k ∗ -Gr −→ O-mod 14.3.1 ˆ on Gk (G) ˆ and any k ∗ -group homomorphism ϕˆ : G ˆ →G ˆ on the mapping G ˆ ) → Gk (G) ˆ . Thus, we have the corresponding restriction map res ϕˆ : Gk (G composed functor a utF nc nc gk ch∗ (F ) −−−− −→ k ∗ -Gr −→ O-mod

14.3.2

212

Frobenius categories versus Brauer blocks

F nc ) as the inverse and we deﬁne the Grothendieck group of the pair (F, aut limit F nc ) = lim (gk ◦ aut F nc ) Gk (F, aut 14.3.3. ←−

14.4 It is a ﬁnitely generated free O-module and the main purpose of this chapter is to determine its O-rank (cf. Corollary 14.32 below); for this, we give alternative descriptions of this O-module. First of all, recall that if R ˆ then the restriction via the canonical homomoris a normal p-subgroup of G ∼ ˆ → G/R ˆ ˆ ˆ ; phism G determines an O-module isomorphism Gk (G/R) = Gk (G) nc ∗ in particular, for any ch (F )-object (q, ∆n ) , since the image q(0)n in q(n) of q(0) by q(0 • n) is contained, and therefore stabilizes, all the other images q(i)n for 1 ≤ i ≤ n . we get ˆ ˆ Gk F(q) Fq(0)n q(n) ∼ 14.4.1. = Gk F(q) nc

14.5 This allows us to replace the subcategory F by its exterior quonc nc tient F˜ (cf. 1.3); but, in chapter 6 we have seen that, rather than F˜ , it is sc the full subcategory F˜ of F˜ over the F-selfcentralizing subgroups which has remarkable properties; luckily, the next result shows that the inverse limit sc does not change when we restrict our functor to F . Recall that, for any category C and any contravariant functor m : ch∗ (C) → O-mod , the inverse limit lim m coincides with the 0-cohomology group of the diﬀerential complex ←−

given by the diﬀerential maps (cf. A3.11.2) dn : m(q, ∆n ) −→ m(r, ∆n+1 ) q

14.5.1,

r

where q and r respectively run over the sets of functors Fct(∆n, C) and Fct(∆n+1 , C) , sending any family a = (aq )q to the family dn (a) = dn (a)r r deﬁned by n+1 dn (a)r = (−1)i m(idr◦δin , δin ) (ar◦δin ) 14.5.2. i=0

Proposition 14.6 Let X be a nonempty set of F-nilcentralized subgroups X X nc of P , denote by F the full subcategory of F over X and by iX : F → F X = aut F nc ◦ ch∗ (iX ) and assume that the inclusion functor, set aut F 14.6.1 For any F-nilcentralized subgroup Q of P there are R in Xand ψ in F(R, Q) such that ψ(Q) is fully centralized in F , R contains CP ψ(Q) and we have ψ ◦ F(Q) ⊂ F(R) ◦ ψ . Then, the functor ch∗ (iX ) induces an O-module isomorphism F nc ) ∼ X lim (gk ◦ aut = lim gk ◦ aut ←−

←−

F

14.6.2.

14. The Grothendieck groups of a Frobenius P -category

213

F nc ) to Proof: First of all, we prove that the homomorphism from lim (gk ◦ aut ←− X induced by ch∗ (iX ) is injective; consider a family X = (XQ )Q , lim gk ◦aut F ←−

where Q runs over the set of all F-nilcentralized subgroups of P and XQ ˆ , and assume that d0 (X) = 0 where d0 denotes the belongs to Gk F(Q) corresponding diﬀerential map (cf. 14.5.2). For such a Q , choose R ∈ X and ψ ∈ F(R, Q) as in condition 14.6.1 nc above and consider the chain r : ∆1 → F mapping 0 on Q , 1 on R and 0•1 on ψ ; then, we get (cf. 14.5.2) 0 = resaut(id

r◦δ 0 0

,δ00 )

(XR ) − resaut(id

r◦δ 0 1

,δ10 )

(XQ )

14.6.3;

but, it follows from the inclusion ψ ◦ F(Q) ⊂ F(R) ◦ ψ in condition 14.6.1 that the homomorphism F(R)ψ(Q) → F(Q) induced by ψ is surjective and therefore the map resaut(id

r◦δ 0 1

,δ10 )

ˆ ˆ : Gk F(Q) −→ Gk F(r)

14.6.4

is injective; thus, if X belongs to the kernel of the homomorphism induced by ch∗ (iX ), we get XQ = 0 . In order to prove the surjectivity, assume that a family Y = (YR )R∈X , ˆ , belongs to the kernel of the corresponding difwhere YR ∈ Gk F(R) X

ferential map d0 ; we have to extend Y to a family X = (XQ )Q over the set of all F-nilcentralized subgroups of P , fulﬁlling d0 (X) = 0 . For any F-nilcentralized subgroup Q of P , choosing R ∈ X and ψ ∈ F(R, Q) asabove, the kernel of the homomorphism F(R)ψ(Q) → F(Q) coincides with CP ψ(Q) (cf. Corollary 4.7) and, in particular, homomorphism 14.6.4 is surjective too, so that with the same notation we can deﬁne XQ = resaut(id

−1

,δ 0 ) ˜ r◦sδ 0 1 1

resaut(id

,δ 0 ) ˜ r◦sδ 0 0 0

(YR )

14.6.5;

moreover, we assume that, for any subgroup of P which is F-isomorphic to Q and any isomorphism between them, we choose the same R ∈ X and the composed homomorphism; then, we claim that d0 (X) = 0 . Let q : ∆1 → F be a chain, set Q = q(0) , Q = q(1) and ϕ = q(0 • 1) ; arguing by induction on |P : Q| and on |Q |/|Q| , we will prove that we have d0 (X)q = 0 . Firstly note that, for another F-nilcentralized subgroup Q of P and an F-morphism ϕ : Q → Q , considering the chains sc

q : ∆1 −→ F

sc

,

q : ∆1 −→ F

sc

and qQ : ∆2 −→ F

sc

14.6.6

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Frobenius categories versus Brauer blocks

respectively mapping 0 on Q , Q and Q , 1 on Q , Q and Q , 0•1 on ϕ , ϕ ◦ ϕ and ϕ , 2 on Q , and 1•2 on ϕ , we have (cf. 14.5.2) resaut(id ,δ1 ) d0 (X)q qQ

= resaut(id

◦δ 1 1

1

,δ 1 ) qQ ◦δ 1 2

d0 (X)q + resaut(id

2

,δ 1 ) qQ ◦δ 1 0

d0 (X)q

14.6.7;

0

moreover, according to our choice above, we actually may assume that ϕ is not an isomorphism and then, according to our induction hypothesis, we already know that d0 (X)q = 0 . As above, let R ∈ X and ψ ∈ F(R, Q) be our choice for Q in condition 14.6.1 and, setting N = NR ψ(Q) , consider the chains n : ∆1 −→ F

sc

,

iR N : ∆1 −→ F

sc

and r : ∆1 −→ F

sc

14.6.8

where r is deﬁned as above, iR N by the inclusion from R to N and n by the restriction of ψ from Q to N ; then, we have d0 (X)r = 0 by equality 14.6.3, = 0. and it follows from the induction hypothesis that we still have d0 (X)iR N But, it follows from condition 14.6.1 above that the image in F(Q) of any element σ of F(N )ψ(Q) can be lifted to an element ρ of F(R) which stabilizes ψ(Q) and therefore it stabilizes N , so that the diﬀerence with σ of the restriction of ρ to N belongs to FCP (ψ(Q)) (N ) (cf. Proposition 4.6); consequently, since R contains CP ψ(Q) , any element of F(N )ψ(Q) can be lifted to an element of F(R) . Thus, mutatis mutandis, considering the chain sc nR : ∆2 −→ F extending n , and mapping 2 on R and 1•2 on the inclusion map N → R , we have a k ∗ -group isomorphism ˆ R) ∼ ˆ nR ◦δ1 , δ21 ) : F(n aut(id = F(n) 2

14.6.9;

hence, since nR ◦ δ11 = r , it follows from equality 14.6.7 that d0 (X)n = 0 . On the other hand, set N = NQ ϕ(Q) and consider the chains n : ∆1 −→ F

sc

,

iQ N : ∆1 −→ F

sc

and q : ∆1 −→ F

sc

14.6.10

where n is deﬁned by the restriction of ϕ from Q to N and iQ N by the inclusion from N to Q ; it follows from our induction hypothesis that we have d0 (X)iQ = 0 , and it is clear that any element of F(Q )ϕ(Q) stabiN

lizes N . As above, considering the chain nQ : ∆2 → Fsc extending n , and mapping 2 on Q and 1 • 2 on the inclusion map N → Q , we still have a k ∗ -group isomorphism ˆ Q ) ∼ ˆ nQ ◦δ1 , δ11 ) : F(n aut(id = F(q) 1

14.6.11;

14. The Grothendieck groups of a Frobenius P -category

215

consequently, according to equality 14.6.7, in order to prove that d0 (X)q = 0 , it suﬃces to prove that d0 (X)n = 0 . That is to say, we may assume that Q = N normalizes ϕ(Q) ; in this case, it follows from Proposition 2.7 that there is an F-morphism ζ : Q → P such that ζ ϕ(Q) is fully normalized in F and then, from condition 2.8.2, that there η : N → P and an element σ ∈ F(Q) fulﬁl are an F-morphism ling η ψ σ(u) = ζ ϕ(u) for any u ∈ Q ; actually, up to modifying our choice of ζ , we may assume that σ = idQ . Now, η(N ) and ζ (Q ) normalize ζ ϕ(Q) and we consider the group N = $η(N ), ζ (Q )% and the chains e : ∆1 −→ F

sc

n : ∆1 −→ F

,

sc

and nN : ∆2 −→ F

sc

14.6.12,

where e and n are respectively deﬁned by the homomorphisms N → N and Q → N determined by η and ζ ◦ ϕ , and nN extends n mapping 2 on N and 1• 2 on the homomorphism N → N determined by η . We already know that d0 (X)n = 0 and it follows from our induction hypothesis that d0 (X)e = 0 ; but, as above, it follows from condition 14.6.1 that any element of F(Q) can be lifted via ψ to an element of F(R) which stabilizes N . In particular, for any element σ in F(N )ζ (ϕ(Q)) = F(n ) , we have an element ρ in F(N ) such that η ◦ ρ coincides with σ ◦ η over ψ(Q) and therefore it follows from Proposition 4.6 that it suﬃces to modify our choice of ρ by composing it with the conjugation by a suitable element of CP ψ(Q) ⊂ N to get η ◦ ρ = σ ◦ η . In conclusion, the k ∗ -group homomorphism ˆ N ) −→ F(n ˆ ) nN ◦δ1 , δ11 ) : F(n aut(id 14.6.13 1

is an isomorphism; consequently, by equality 14.6.7, we get d0 (ξ)n = 0 . Similarly, consider the chains z : ∆1 −→ F

sc

n : ∆1 −→ F

,

sc

and qN : ∆2 −→ F

sc

14.6.14

where z is deﬁned by the homomorphism Q → N determined by ζ , and qN extends q = n mapping 2 on N and 1 • 2 on the homomorphism Q → N determined by ζ ; it follows from our induction hypothesis that d0 (X)z = 0 and we already know that d0 (X)n = 0 . Since N contains η CP ψ(Q) = CP ζ ϕ(Q)

14.6.15,

it follows from statement 2.10.1 that the automorphism of ζ ϕ(Q) determined by any σ ∈ F(Q )ϕ(Q) via ζ can be extended to an automorphism σ of N and then, arguing as above, it follows again from Proposition 4.6

216

Frobenius categories versus Brauer blocks

that we may choose σ fulﬁlling σ ζ (u ) = ζ σ (u ) for any u ∈ Q . In conclusion, the k ∗ -group homomorphism ˆ N ) −→ F(q) ˆ qN ◦δ1 , δ21 ) : F(q aut(id 2

14.6.16

is an isomorphism; consequently, again by equality 14.6.7, we get d0 (ξ)q = 0 . We are done. Corollary 14.7 The inclusion F

sc

⊂F

nc

induces an O-module isomorphism

F sc ) ∼ F nc ) lim (gk ◦ aut = Gk (F, aut ←−

14.7.1.

Proof: If Q is a F-nilcentralized subgroup of P and ψ an element of F(P, Q) such that Q = ψ(Q) is fully normalized in F (cf. Proposition 2.7), we already know that R = Q ·CP (Q ) is F-selfcentralizing (cf. 4.8) and it follows from statement 2.10.1 that any σ ∈ F(Q ) can be extended to some ρ ∈ F(P, R) ; since CP (Q ) ⊂ R , we have ρ(R) = R , so that ρ determines τ ∈ F(R) (cf. 2.3) fulﬁlling τ ◦ ψ = ψ ◦ σ . That is to say, the set of F-selfcentralizing subgroups of P fulﬁlls condition 14.6.1 and therefore the corollary follows from Proposition 14.6. nc sc 14.8 We are ready to move from F to the exterior quotient F˜ (cf. 1.3); as in 14.1.1, we have the corresponding automorphism functor (cf. Proposition A2.10) sc autF˜ sc : ch∗ (F˜ ) −→ Gr 14.8.1 sc sc mapping any ch∗ (F˜ )-object (˜q, ∆n ) on its ch∗ (F˜ )-automorphism group ˜ q) ; this time, it follows from Corollary 4.9 that F(˜ ˜ q) can — still noted F(˜ ˜ ˜ be identiﬁed with the subgroup of F ˜q(0) formed by the F-isomorphisms σ ˜0 : ˜q(0) ∼ = ˜q(0) which, for any i ∈ ∆n , fulﬁll

˜q(0•i) ◦ σ ˜0 ∈ F˜ ˜q(i) ◦ ˜q(0•i)

14.8.2.

˜ q) , since σ Indeed, for any element (˜ σ , id∆n ) of F(˜ ˜ : ˜q ∼ = ˜q is a natural isomor phism, for any 0 ≤ i ≤ n we have an element σ ˜i ∈ F˜ ˜q(i) in such a way that ˜q(0•i) ◦ σ ˜0 = σ ˜i ◦ ˜q(0•i) 14.8.3; but, according to Corollary 4.9, σ ˜i is then determined by σ ˜0 for any 1 ≤ i ≤ n . sc sc 14.9 Actually, denoting by esc : F → F˜ the canonical functor, it is sc quite clear that ˜q = esc ◦ q for some chain q : ∆n → F , and that a suitable representative σn ∈ F q(n) of σ ˜n determines a natural isomorphism σ : q ∼ =q

14. The Grothendieck groups of a Frobenius P -category

217

lifting σ ˜ ; that is to say, since Cq(0)n q(n) = Z q(n) (cf. 14.1), we get an evident exact sequence ˜ q) −→ 0 0 −→ Z q(n) −→ q(0) −→ F(q) −→ F(˜ 14.9.1. sc Moreover, if (˜r, ∆m ) is another ch∗ (F˜ )-object and (˜ ν , δ) : (˜r, ∆m ) → (˜q, ∆n ) sc sc a ch∗ (F˜ )-morphism, for any chain r : ∆m → F such that ˜r = esc ◦ r , sc it is not diﬃcult to check that (˜ ν , δ) can be lifted to a ch∗ (F )-morphism (ν, δ) : (r, ∆m ) → (q, ∆n ) and that we get a commutative diagram ˜ r) −→ 0 0 −→ Z r(n) −→ r(0) −→ F(r) −→ F(˜ ↓ ↓ ↓ α ↓α ¯ 14.9.2. ˜ q) −→ 0 0 −→ Z q(n) −→ q(0) −→ F(q) −→ F(˜

where α ¯ = autF˜ sc (˜ ν , δ) and α = autF sc (ν, δ) . 14.10 In other words, respectively denoting by vF sc : ch∗ (F ) −→ F sc

sc

⊂ Gr

and zch∗ (F sc ) : ch∗ (F ) −→ Gr sc

14.10.1

the evaluation functor mapping (q, ∆n ) on q(0) (cf. A3.7.3) and the evident functor mapping (q, ∆n ) on Z q(n) (cf. 13.13), we have obtained the exact sequence of functors 1 −→ zch∗ (F sc ) −→ vF sc −→ autF sc −→ autF˜ sc ◦ ch∗ (esc ) −→ 1

14.10.2.

In particular, considering the restricted functor (cf. 14.2.1) F sc : ch∗ (F ) −→ k ∗ -Gr aut sc

14.10.3,

F sc → autF sc is split the converse image over vF sc of the natural map aut sc ∗ sc since vF maps any ch (F )-object on a p-group, and therefore the natural F sc , deﬁning a functor map vF sc → autF sc can be lifted to aut F˜ sc : ch∗ (F˜ ) −→ k ∗ -Gr aut sc

14.10.4

sc which lifts the automorphism functor (cf. 14.8.1); as above, for any ch∗ (F˜ )object (˜q, ∆n ) , we set ˆ ˜ q) F˜ sc (˜q, ∆n ) = F(˜ aut 14.10.5.

F sc by aut F˜ sc ; indeed, for 14.11 As announced, we now can replace aut sc ∗ any ch (F )-object (q, ∆n ) , from the exact sequence 14.9.1 and this splitting ˆ˜ q) and therefore the ˆ we get a surjective k ∗ -group homomorphism F(q) → F(˜ restriction induces an injective O-module homomorphism ˆ ˜ q) −→ Gk F(q) ˆ Gk F(˜

14.11.1

218

Frobenius categories versus Brauer blocks

which is actually an isomorphism since the kernel of that k ∗ -group homomorphism is a p-group; that is to say, we get a natural isomorphism F˜ sc ◦ ch∗ (esc ) ∼ F sc gk ◦ aut = gk ◦ aut

14.11.2

and therefore, since the functor ch∗ (esc ) is full and essentially surjective, it follows from Corollary 14.7 that F nc ) ∼ F˜ sc ) Gk (F, aut = lim (gk ◦ aut ←−

14.11.3.

F nc ) , the other 14.12 In order to determine the O-rank of Gk (F, aut ingredient we need is the character decomposition of the functor gk (cf. 14.3 ). First of all, we introduce some notation; for any h ∈ N − pN , ﬁx a primitive ˆh = k ∗ × Uh ; note that h-th root of unity ξh in k and set Uh = $ ξh % and U the kernel of the canonical group homomorphism ˆh ) −→ Aut(Uh ) Autk∗ (U

14.12.1

can be identiﬁed with Hom(Uh , k ∗ ) and that ξh determines a group isomorphism Hom(Uh , k ∗ ) ∼ = Uh . Recall that we have a canonical group isomorphism (cf. Proposition 6 in [55, Ch. II]) O∗ ∼ 14.12.2 = k ∗ × 1 + J(O) and, as in 1.25, the inclusion O∗ ⊂ O and this isomorphism determine an evident O-algebra homomorphism Ok ∗ → O . ˆ with ﬁnite k ∗ -quotient G (cf 1.23), 14.13 As in 1.25, for any k ∗ -group G ˆ the group algebra of the group G ˆ , we set denoting by OG ˆ = O ⊗Ok∗ OG ˆ O∗ G

14.13.1

which is an O-algebra of O-rank |G| . Now, for any injective k ∗ -group hoˆh → G ˆ , the modular characters introduced by Brauer momorphism ηˆ : U ˆ → O map(cf. 18 in [54]) determine an O-module homomorphism Gk (G) ˆ ping the class of k ⊗O M in Gk (G) on the linear trace trM ηˆ(1, ξh ) for any ˆ ˆ O∗ G-module M ; moreover, if we have σ ˆ (1, ξh ) = (λ, ξh ) for some element σ in the kernel of homomorphism 14.12.1 then we still have trM (ˆ 14.13.2; η◦σ ˆ )(1, ξh ) = λ·trM ηˆ(1, ξh ) ˆh , G) ˆ the set of injective k ∗ -homomorin particular, denoting by Monk∗ (U ˆh to G ˆ , Uh acts on this set via the identiﬁcation above with phisms from U the kernel of homomorphism 14.12.1, and acts on O via isomorphism 14.12.2 together with the inclusion Uh ⊂ k ∗ .

14. The Grothendieck groups of a Frobenius P -category

219

ˆh , G), ˆ O the set of O-valued 14.14 Thus, denoting by FctUh Monk∗ (U functions which preserve the corresponding Uh -actions, we have obtained an O-module homomorphism ˆ −→ FctU Monk∗ (U ˆh , G), ˆ O Gk (G) 14.14.1; h ˆh , G) ˆ centralizing the action moreover, G acts by conjugation on Monk∗ (U ˆ ˆ ˆ = k ∗ ·G of Uh , so that it acts on FctUh Monk∗ (Uh , G), O ; actually, since G ˆ and then we have Gk (G) ˆ ∼ for a suitable ﬁnite subgroup G of G = Gk (kG e ) 1 ∗ where we set Z = k ∩ G and e = |Z | · z ∈Z z (cf Proposition 5.15 in [42]), it is not diﬃcult to prove that homomorphisms 14.14.1 when h runs over N − pN determine an O-module isomorphism ˆ ∼ ˆh , G), ˆ O G Gk (G) FctUh Monk∗ (U 14.14.2. = h∈N−pN

ˆh , G), ˆ O G vanishes on the k ∗ Note that any function χ ∈ FctUh Monk∗ (U ˆ mapping (1, ξh ) on an element x ˆ such ˆh → G ˆ∈G group monomorphisms ηˆ : U that the image of CGˆ (ˆ x) in G is a proper subgroup of CG (x) where x = k ∗ ·ˆ x. 14.15 Pushing it further, denote by Mon(Uh , G) the set of injective group homomorphisms from Uh to G and by ˆh , G) ˆ −→ Mon(Uh , G) Ph,Gˆ : Monk∗ (U

14.15.1

the canonical map, so that Uh acts regularly on the ﬁbers; thus, we still have the obvious decomposition ˆh , G), ˆ O ∼ FctUh Monk∗ (U FctUh (Ph,Gˆ )−1 (η), O 14.15.2 = η∈Mon(Uh ,G)

and therefore we get ˆ ∼ Gk (G) =

G FctUh (Ph,Gˆ )−1 (η), O

14.15.3.

h∈N−pN η∈Mon(Uh ,G)

Then, for any η ∈ Mon(Uh , G) , the term FctUh (Ph,Gˆ )−1 (η), O is a free O-module of rank 1, but note that the quotient of CG η(Uh ) by the image ˆh ) , where ηˆ ∈ (P ˆ )−1 (η) , acts faithfully on it. of CGˆ ηˆ(U h,G 14.16 As a matter of fact, isomorphisms 14.14.2 and 14.15.3 are natural . In order to prove this naturality, we have to develop a suitable functorial framework; respectively denote by k ∗ -iGr and by iGr the subcategories of k ∗ -Gr and Gr formed by the same objects and by the homomorphisms which are injective over the corresponding set of p -elements — called

220

Frobenius categories versus Brauer blocks

p -injective homomorphisms — and by qt : k ∗ -iGr → iGr the obvious k ∗ -quotient functor (cf. 1.23). As in 6.3 above, denote by Uh ℵ the category of ﬁnite sets endowed with a Uh -action and by Uh h resU ℵ −→ ℵ 1 :

14.16.1

the corresponding forgetful functor; then, we have two functors ˆh : k ∗ -iGr −→ Uh ℵ u

and uh : iGr −→ ℵ

14.16.2

ˆ (cf. 1.23) on the Uh -set Monk∗ (U ˆh , G) ˆ and which map any ﬁnite k ∗ -group G ˆ a ﬁnite group G on Mon(Uh , G) , and still have a natural map sending G to Ph,Gˆ (cf. 14.15.1) h ˆh −→ uh ◦ qt Ph : resU 1 ◦u

14.16.3.

14.17 Moreover, for any h dividing h , choosing the identiﬁcation be tween Hom(Uh , k ∗ ) and Uh determined by (ξh )h/h , the inclusion Uh ⊂ Uh induces a functor and two natural maps Uh h resU ℵ −→ Uh ℵ Uh : h ˆh −→ u ˆh ρˆh ,h : resU Uh ◦ u

and ρh ,h : uh −→ uh

14.17.1,

and it is easily checked that (cf. A1.5.1) h Ph ◦ (ˆ ρh ,h ∗ resU Uh ) = (ρh ,h ∗ qt) ◦ Ph

14.17.2.

Furthermore, identifying the category of sets ℵ with the full subcategory of the category of small categories CC (cf. A1.6) over the small categories which have no other morphisms than the corresponding identity morphisms, we can consider the functor uh ◦ qt : k ∗ -iGr −→ ℵ ⊂ CC

14.17.3

as a representation of k ∗ -iGr (cf. A2.2) and thus we can apply to it the construction of the corresponding semidirect product (cf. A2.7) uh

(k ∗ -iGr) = (uh ◦ qt) (k ∗ -iGr)

14.17.4

ˆ formed by a k ∗ -group G ˆ with ﬁnite where the objects are the pairs (η, G) k ∗ -quotient G , and by an injective group homomorphism η : Uh → G , and ˆ to another object (η , G ˆ ) are the p -injective hothe morphisms from (η, G) ˆ → G ˆ fulﬁlling η = qt(ϕ) momorphisms ϕˆ : G ˆ ◦ η . Moreover, for any h

14. The Grothendieck groups of a Frobenius P -category

221

dividing h , the natural map ρh ,h above determines a functor (cf. Proposition A2.17) (ρh ,h ∗ qt) idk∗ -iGr : uh (k ∗ -iGr) −→ uh (k ∗ -iGr)

14.17.5

ˆ on (η , G) ˆ where η : Uh → G ˆ is the restriction of η . mapping (η, G) Proposition 14.18 With the notation above, we have a functor ˆ h : uh (k ∗ -iGr) −→ Uh ℵ w

14.18.1

ˆ on the regular Uh -set (P ˆ )−1 (η) and (k ∗ -iGr)-object (η, G) h,G ˆ → (η , G ˆ ) on the bijective map any uh (k ∗ -iGr)-morphism ϕˆ : (η, G)

mapping any

uh

(Ph,Gˆ )−1 (η) ∼ = (Ph,Gˆ )−1 (η )

14.18.2

ˆ h via the strucˆh (ϕ) ˆh is the direct image of w determined by u ˆ . Moreover, u tural functor ph : uh (k ∗ -iGr) −→ k ∗ -iGr 14.18.3. Proof: As we mention above, the uh (k ∗ -iGr)-objects are the pairs formed ˆ and by an object of the “category” Mon(Uh , G) where by a k ∗ -iGr-object G ˆ G = qt(G) , namely an injective group homomorphism η : Uh → G ; then, in the map (cf. 14.15.1) ˆh , G) ˆ −→ Mon(Uh , G) Ph,Gˆ : Monk∗ (U

14.18.4,

Uh acts on the left end stabilizing and acting regularly on the ﬁbers, so that (Ph,Gˆ )−1 (η) is indeed a regular Uh -set. Analogously, the uh (k ∗ -iGr)-morˆ and (η , G ˆ ) are the pairs formed by a phisms between two objects (η, G) ˆ → G ˆ and by a “Mon(Uh , G )-morphism” from the k ∗ -iGr-morphism ϕˆ : G image of η by the functor (uh ◦ qt)(ϕ) ˆ to η , which actually forces the equality qt(ϕ) ˆ ◦ η = η . Then, it is clear that the map ˆh , G) ˆ −→ Monk∗ (U ˆh , G ˆ) ˆ(ϕ) u ˆ : Monk∗ (U

14.18.5

sends (Ph,Gˆ )−1 (η) bijectively onto (Ph,Gˆ )−1 (η ) , determining a Uh -set map. The proofs of the functoriality and of the last statement are straightforward. Remark 14.19 Note that, for any ξ ∈ Uh , the inner k ∗ -group automorphism ˆ determined by an element lifting η(ξ) to G ˆ acts trivially on (P ˆ )−1 (η) . of G h,G

222

Frobenius categories versus Brauer blocks

Proposition 14.20 With the notation above, for any h dividing h we have a natural map h ˆ ˆ τˆh ,h : resU Uh ◦ wh −→ wh ◦ (ρh ,h ∗ qt) idk∗ -iGr sending any

uh

14.20.1

ˆ to the Uh -morphism (k ∗ -iGr)-object (η, G)

−1 h h resU (η) −→ (Ph ,Gˆ )−1 ResU ˆ) Uh (Ph,G Uh (η)

14.20.2

ˆh , G) ˆ on its restriction to U ˆh . which maps any ηˆ ∈ (Ph,Gˆ )−1 (η) ⊂ Monk∗ (U ˆ→G ˆ , setting ϕ = qt(ϕ) Proof: For any k ∗ -group homomorphism ϕˆ : G ˆ the Uh u ∗ h ˆ ˆ ˆ h maps the functor resUh ◦ w (k -iGr)-morphism (η, G) → (ϕ ◦ η, G ) on the Uh -set map −1 −1 h h resU (η) −→ resU (ϕ ◦ η) ˆ) ˆ ) Uh (Ph,G Uh (Ph,G

14.20.3

h sending ηˆ ∈ (Ph,Gˆ )−1 (η) to ϕ◦ ˆ ηˆ whereas, setting η = ResU Uh (η) , the functor ˆ h ◦ (ρh ,h ∗ qt) idk∗ -iGr maps this morphism on the analogous Uh -set w map (Ph ,Gˆ )−1 (η ) −→ (Ph ,Gˆ )−1 (ϕ ◦ η ) 14.20.4;

thus, the corresponding diagram is indeed commutative since the restriction to Uh is compatible with the composition with ϕˆ on the left. 14.21 We are ready to discuss the naturality of isomorphisms 14.14.2 and 14.15.3. First of all, consider the evident contravariant functor FctUh : Uh ℵ −→ O-mod

14.21.1

mapping any ﬁnite Uh -set X on the O-module FctUh (X, O) of the O-valued functions over X preserving the Uh -actions; note that if ξ·x = x for some x ∈ X and some ξ ∈ Uh −{1} then we have f (x) = 0 for any f ∈ FctUh (X, O) . On the other hand, note that if we have a contravariant functor m : k ∗ -iGr −→ O-mod

14.21.2,

ˆ with ﬁnite k ∗ -quotient G (cf. 1.23) m(G) ˆ has an obvious for any k ∗ -group G OG-module structure, so that it makes sense to consider ˆ = m(G) ˆ G H0 G, m(G)

14.21.3;

ˆ →G ˆ is a p -injective k ∗ -group homomorphism (cf. 14.16), further, if ϕˆ : G ˆ )G on an O-submodule of m(G) ˆ G; it is easily checked that m(ϕ) ˆ maps m(G

14. The Grothendieck groups of a Frobenius P -category

223

that is to say, we get a new contravariant functor from k ∗ -iGr to O-mod — ˆ on m(G) ˆ G and ϕˆ on the map from m(G ˆ )G denoted by h0 (m) — mapping G ˆ G induced by m(ϕ) to m(G) ˆ . 14.22 With all this notation, it is now clear that isomorphism 14.14.2 actually deﬁnes a natural isomorphism

gk ∼ =

ˆh ) h0 (FctUh ◦ u

14.22.1.

h∈N−pN

Consequently, isomorphism 14.11.3 becomes F nc ) ∼ Gk (F, aut = lim ←−

F˜ sc ˆh ) ◦ aut h0 (FctUh ◦ u

14.22.2;

h∈N−pN

that is to say, we have

F nc ) ∼ Gk (F, aut =

F nc )h Gk (F, aut

14.22.3

F nc )h = lim h0 (FctUh ◦ u F˜ sc ˆh ) ◦ aut Gk (F, aut

14.22.4

h∈N−pN

where, for any h ∈ N − pN , we set ←−

which, as it is easily checked, yields F nc )h ∼ F˜ sc ) ˆh ◦ aut Gk (F, aut = lim (FctUh ◦ u ←−

14.22.5.

14.23 Recall that, according to Proposition 14.18, the functor ˆh : k ∗ -iGr −→ Uh ℵ u

14.23.1

is the direct image of the functor (cf. 1.6) ˆ h : uh (k ∗ -iGr) −→ Uh ℵ w

14.23.2

throughout the structural functor (cf. A2.7.1) ph : uh (k ∗ -iGr) −→ k ∗ -iGr

14.23.3.

F˜ sc = uh ◦ autF˜ sc , similarly considering the semiMoreover, since uh ◦ qt ◦ aut direct product sc uh ∗ ˜ sc ch (F ) = (uh ◦ autF˜ sc ) ch∗ (F˜ ) 14.23.4,

224

Frobenius categories versus Brauer blocks

we have the commutative diagram of functors (k ∗ -iGr) −→ k ∗ -iGr & & a iduh ◦aut ˜ sca utF˜ sc  utF˜ sc F sc uh ∗ ˜ ∗ ˜ sc ch (F ) −→ ch (F ) ph

uh

14.23.5.

F˜ sc ˆh ◦ aut 14.24 Then, it is not diﬃcult to check that the composition u is also the direct image of the composition (cf. 1.6) ˜ sc ) ˆ aut ˆ w h = wh ◦ (iduh ◦autF˜ sc autF

14.24.1

throughout the bottom structural functor in diagram 14.23.5; further, the direct image is clearly compatible with the functor FctUh : Uh ℵ −→ O-mod . Consequently, we ﬁnally get F nc )h ∼ F˜ sc ) ˆh ◦ aut Gk (F, aut = lim (FctUh ◦ u ←−

14.24.2.

∼ ˆ aut = lim (FctUh ◦ w h ) ←−

sc 14.25 The point is that the semidirect product uh ch∗ (F˜ ) admits anh sc sc other description in terms of the subcategory (F˜ ) ⊂ Uh (F˜ ) of faithful sc h sc Uh -objects of F˜ (cf. 6.3). Precisely, we denote by (F˜ ) the full subcatesc gory of Uh (F˜ ) over the objects Qη formed by an F-selfcentralizing sub˜ group Q of P and by an injective group homomorphism η : Uh → F(Q) ; we set for short h sc h sc vh = vh(F˜ sc ) : ch∗ (F˜ ) −→ (F˜ ) 14.25.1

for the evaluation functor of this subcategory. Note that we have a faithful h sc sc forgetful functor (F˜ ) → F˜ ; moreover, for any h dividing h , it is clear that the inclusion Uh ⊂ Uh induces a faithful functor h

sc

h

sc

rh ,h : (F˜ ) −→ (F˜ )

14.25.2. sc

14.26 On the other hand, since the category of chains ch(F˜ ) is already sc a semidirect product (cf. A2.8), the uh ch∗ (F˜ )-objects can be identiﬁed with sc the triples (η, ˜q, ∆n ) formed by a functor ˜q : ∆n → F˜ and by an injective ˜ q) . Moreover, for any i ∈ ∆n , denote group homomorphism η : Uh → F(˜ ˜ q ˜ q) → F˜ ˜q(i) the structural map (cf. 14.8), which actually coinby ιi : F(˜ sc cides with the image by the functor autF˜sc of the obvious ch∗ (F˜ )-morphism from (q, ∆n ) to (q(i), ∆0 ) .

14. The Grothendieck groups of a Frobenius P -category

225

Proposition 14.27 For any h ∈ N − pN , we have an equivalence of categories h sc sc jh : uh ch∗ (F˜ ) ∼ 14.27.1 = ch∗ (F˜ ) sc h sc ch∗ (F˜ )-object (η, ˜q, ∆n ) on the chain ˜qη : ∆n → (F˜ ) ˜ q h sc sending any i ∈ ∆n to the (F˜ )-object ˜q(i)ιi ◦η and any ∆n -morphism j •i to ˜q(j •i) . Moreover, for any h dividing h , we have the commutative diagram

which maps any

uh

ch∗ (F˜ ) &  ˜ sc sc

uh

(ρh ,h ∗autF˜ sc )idch∗ (F uh

jh

∼ =

ch∗

)

sc ch∗ (F˜ )

j

h ∼ =

h sc (F˜ ) & ch∗ (rh ,h )

ch∗

14.27.2.

h sc (F˜ )

Proof: It is easily checked that ˜q(j • i) is indeed a morphism between the ˜ q ˜ q sc sc (F˜ )-objects ˜q(j)ιj ◦η and ˜q(i)ιi ◦η . Moreover, a uh ch∗ (F˜ )-morphism to sc sc (η, ˜q, ∆n ) from a uh ch∗ (F˜ )-object (θ, ˜r, ∆m ) is deﬁned by a ch∗ (F˜ )-morphism (˜ µ, δ) : (˜r, ∆m ) −→ (˜q, ∆n ) 14.27.3

h

fulﬁlling autF˜ sc (˜ µ, δ) ◦ θ = η (cf. condition A2.6.2), and therefore (˜ µ, δ) is also a morphism from (˜rθ , ∆m ) to (˜qη , ∆n ) . Thus, we have a functor h sc sc jh : uh ch∗ (F˜ ) −→ ch∗ (F˜ )

14.27.4.

h sc h sc ˜q : ∆n → (F˜ ) clearly determines a On the other hand, any (F˜ )-chain ˆ sc h sc functor ˜q : ∆n → F˜ ; consequently, by the very deﬁnition of (F˜ ) , we have ˆ ˜q(i) = ˜q(i)ηi where ηi : Uh → F˜ ˜q(i) is an injective group homomorphism ˜q(j • i) is a morphism from ˜q(j)ηj to ˜q(i)ηi for for any i ∈ ∆n , and, since ˆ ˜ q) , so that there is a any 0 ≤ j ≤ i ≤ n , η0 (Uh ) is actually contained in F(˜ ˜ q ˜ q) fulﬁlling ι ◦ η = ηi for any i ∈ ∆n . unique η : Uh → F(˜ i h sc h sc ˜r : ∆m → (F˜ ) is an (F˜ )-chain and, for any j ∈ ∆m , we Similarly, if ˆ ˜ r ˜ r) , then ˜r(j) = ˜r(j)ιj ◦θ for an injective group homomorphism θ : Uh → F(˜ have ˆ h sc sc ˜r, ∆m ) → (ˆ˜q, ∆n ) induces a ch∗ (F˜ )-mor˜ˆ, δ) : (ˆ any ch∗ (F˜ ) -morphism (µ

phism (˜ µ, δ) : (˜r, ∆m ) → (˜q, ∆n ) such that ˜

autF˜ sc (˜ µi , id∆0 ) ◦ ι˜rδ(i) ◦ θ = ηi = ιqi ◦ η

14.27.5

for any i ∈ ∆n and therefore, since (cf. 14.25) ˜

autF˜ sc (˜ µi , id∆0 ) ◦ ι˜rδ(i) = ιqi ◦ autF˜ sc (˜ µ, δ)

14.27.6,

226

Frobenius categories versus Brauer blocks

we get autF˜ sc (˜ µ, δ) ◦ θ = η by the uniqueness of η . Consequently, the functor jh is an equivalence of categories. The commutativity of the diagram is easily checked. We are done. ˆ sc Proposition 14.28 For any h ∈ N − pN , we have a factorization of w h sc ch∗ (F˜ ) jh + h sc ch∗ (F˜ )

uh

ˆ aut w h

−→ vh

−→

&ℵ sh h sc (F˜ ) Uh

14.28.1

h sc h sc throughout a functor sh : (F˜ ) → Uh ℵ mapping any (F˜ )-object Qρ on the h sc Uh -set (Ph,F˜ˆ (Q) )−1 (ρ) and any (F˜ )-morphism ϕ˜ : Rσ → Qρ on a bijection

(Ph,F˜ˆ (R) )−1 (σ) ∼ = (Ph,F˜ˆ (Q) )−1 (ρ)

14.28.2.

In particular, we have F nc )h ∼ Gk (F, aut = H0

h sc (F˜ ), FctUh ◦ sh

14.28.3.

Moreover, for any h dividing h we have a natural map h θh ,h : resU Uh ◦ sh −→ sh ◦ rh ,h

14.28.4

sending Qρ to the Uh -morphism −1 h h resU (ρ) −→ (Ph ,F˜ˆ (Q) )−1 ResU ˆ (Q) ) Uh (Ph,F Uh (ρ) ˜

14.28.5.

induced by the restriction throughout the inclusion Uh ⊂ Uh . Proof: Let us denote by R and Q the obvious 0-chains and by ϕ˜ the 1-chain mapping 0 on R , 1 on Q and the ∆1 -morphism 0 • 1 on ϕ˜ ; thus, we have sc evident ch∗ (F˜ )-morphisms (R, ∆0 ) ←− (ϕ, ˜ ∆1 ) −→ (Q, ∆0 )

14.28.6

F˜ sc → autF˜ sc (cf. 14.10) sends them to the and then the natural map aut commutative diagram ˜ ˜ ϕ) ˜ F(R) ←− F( ˜ −→ F(Q) ↑ ↑ ↑ ˆ ˆ ˆ ˜ ˜ ˜ F(R) ←− F(ϕ) ˜ −→ F(Q)

14.28.7.

Since we have ρ(ξ) ◦ ϕ˜ = ϕ˜ ◦ σ(ξ) for any ξ ∈ Uh , and we have identi˜ ϕ) ˜ ˜ ϕ) ﬁed F( ˜ with its image in F(R) (cf. 14.8), actually we get σ(Uh ) ⊂ F( ˜

14. The Grothendieck groups of a Frobenius P -category

227

and the right-top group homomorphism maps σ(ξ) on ρ(ξ) ; hence, the rightbottom k ∗ -group homomorphism induces a Uh -set isomorphism sh (ϕ) ˜ : (Ph,F˜ˆ (R) )−1 (σ) ∼ = (Ph,F˜ˆ (Q) )−1 (ρ)

14.28.8.

h sc Now, we claim that the correspondence mapping the (F˜ )-morphism ˜ deﬁnes a functor; indeed, for a third ϕ˜ on the Uh -set isomorphism sh (ϕ) sc h sc Uh -object T τ of F˜ and an (F˜ )-morphism ψ˜ : T τ → Rσ , we have the sc evident commutative ch∗ (F˜ )-diagram

(Q, ∆0 ) = (Q, ∆0 ) ↑ (ϕ, ˜ ∆1 ) → ↑ ↑ (˜c, ∆2 ) → 0 (ϕ˜ ◦ ψ, ∆1 )

(R, ∆0 ) ↑ ˜ ∆1 ) → (ψ,

14.28.9 (T, ∆0 ) + (T, ∆0 )

→

sc sc where the F˜ -chain ˜c : ∆2 → F˜ maps 0 on T , 1 on R , 2 on Q , 0•1 on ψ˜ and 1•2 on ϕ˜ ; once again, the functor autF˜ sc maps it on the commutative diagram of groups sc sc F˜ (Q) = F˜ (Q) ↑ sc sc F˜ (ϕ) ˜ → F˜ (R) ↑ ↑ ↑ sc sc ˜ → F˜ sc (T ) ˜ ˜ F (˜c) → F (ψ) 0 + sc sc F˜ (ϕ˜ ◦ ψ) → F˜ (T )

14.28.10.

But, in this diagram it is quite clear that the middle square is a pullsc sc ˜ back ; thus, since σ(u) and τ (u) respectively belong to F˜ (ϕ) ˜ and to F˜ (ψ) for any u ∈ Uh , and then τ (u) is mapped on σ(u) , actually τ (u) belongs sc ˜ = sh (ϕ) ˜ . to F˜ (˜c) . At this point, it is easily checked that sh (ϕ˜ ◦ ψ) ˜ ◦ sh (ψ) aut ˆ h maps Moreover, it follows from Proposition 14.18 that the functor w −1 (η, ˜q, ∆n ) on the Uh -set (P ˜ˆ ) (η) ; but, the image by autF˜sc of the obh,F (˜ q)

vious ch∗ (F˜ )-morphism (˜q, ∆n ) → (˜q(0), ∆0 ) determines a lifting of ιq0 sc

˜

˜ ˆ ˜ q) −→ Fˆ˜ ˜q(0) ˆιq0 : F(˜

14.28.11;

˜

then, since ˆιq0 is injective (cf. 14.8), it is quite clear that ˆιq0 ◦ (Ph,F˜ˆ (˜q) )−1 (η) = (Ph,F˜ˆ (˜q(0)) )−1 (ιq0 ◦ η) ˜

˜

14.28.12,

228

Frobenius categories versus Brauer blocks ˜

where the left member denotes the set of compositions of ˆιq0 with all the elements of (Ph,F˜ˆ (˜q) )−1 (η) . On the other hand, by the very deﬁnition of sh , we actually have (Ph,F˜ˆ (˜q(0)) )−1 (ιq0 ◦ η) = (sh ◦ vh ◦ jh )(η, ˜q, ∆n ) ˜

14.28.13.

From these equalities it is easily checked that we have a natural isomorphism ∼ ˆ aut w h = sh ◦ vh ◦ jh

14.28.14.

Finally, the O-module isomorphism 14.28.3 follows from the O-module isomorphism 14.24.2, from Proposition 14.27 and from this natural isomorphism (cf. A3.9). The proof of the last statement is straightforward. Remark 14.29 By Remark 14.19, for any ξ ∈ Uh the functor sh maps the sc (F˜ )-automorphism ρ(ξ) of Qρ on the identity map of (Ph,F˜ˆ (Q) )−1 (ρ) ; in

h

sc h sc particular, sh factorizes via the exterior quotient h˜(F˜ ) of (F˜ ) introduced in 6.3. Moreover, since we have (cf. A3.9.1)

H0 we still have

h sc (F˜ ), FctUh ◦ sh = lim (FctUh ◦ sh )

14.29.1,

F nc )h ∼ Gk (F, aut = FctUh (lim sh , O)

14.29.2

←−

−→

for any h ∈ N − pN . In particular, we still get F nc ) ∼ F nc k ⊗O Gk (F, aut = lim (k ⊗O gk ) ◦ aut ←−

14.29.3.

Theorem 14.30 For any h ∈ N − pN and any n ≥ 1 we have Hn

h sc (F˜ ), FctUh ◦ sh = {0}

14.30.1.

Proof: In order to apply Theorem 6.26, let us consider the full subcatesc sc gory h F of Uh ac(F˜ ) — the category of Uh -objects of ac(F˜ ) (cf. 6.3) — over the set of faithful indecomposable Uh -objects, namely over the indecomposc sable Uh -objects Qρ of ac(F˜ ) (cf. 6.23) such that the group homomor˜ phism ρ : Uh → F(Q) is injective. Note that the indecomposable Uh -object

π P deﬁned in 6.25 from the regular action of Uh on itself, is faithful. u∈Uh

14. The Grothendieck groups of a Frobenius P -category

229

ρ σ σ Moreover, if Qρ = i∈I Qi ) and R = j∈J Rj ) are faithful indesc composable Uh ac(F˜ )-objects then, according to 6.11 and 6.13, the exterior

sc intersection of Q = i∈I Qi and R = j∈J Rj in ac(F˜ ) yields ˜R= Q∩

T

14.30.2

˜ T ˇ Q ,R (i,j)∈I×J (α,T, ˜ β)∈ i j

and, for any ξ ∈ Uh , ρ(ξ) and σ(ξ) induce an automorphism of this intersection. In particular, they induce a permutation of the disjoint union ) ˜ ∈T ˇ Q ,R is a ﬁxed element then ρ(ξ) and ˇ α, T, β) i j (i,j)∈I×J TQi ,Rj , and if (˜ ˜ σ(ξ) respectively ﬁx i and j , and induce F-automorphisms ρi (ξ) of Qi , σj (ξ) of Rj and τ (ξ) of T fulﬁlling ρi (ξ) ◦ α ˜=α ˜ ◦ τ (ξ) and σj (ξ) ◦ β˜ = β˜ ◦ τ (ξ)

14.30.3.

In particular, if τ (ξ) is trivial then it follows from Corollary 4.9 that ρi (ξ) and σj (ξ) are trivial too and therefore we get ξ = 1 ; thus, the extesc ˜ Rσ in the category Uh ac(F˜ ) is a direct sum of faithful rior intersection Qρ ∩ indecomposable Uh -objects. Consequently, since an indecomposable direct summand of a pull-back is also a direct summand of some exterior intersc section, the subcategory ac(h F) of Uh ac(F˜ ) is closed by direct products and pull-backs. h sc On the other hand, it is clear that (F˜ ) is a full subcategory of h F and we claim that the contravariant functor (cf. Proposition 14.28) h

sc

nh = FctUh ◦ sh : (F˜ ) −→ O-mod

14.30.4

can be extended to a contravariant functor mh from h F to O-mod . Indeed, ρ

sc for any faithful indecomposable Uh ac(F˜ )-object Qρ = choose i∈I Qi ˜ i) i ∈ I ; it is clear that ρ induces a group homomorphism ρi : Uhi → F(Q where Uhi denotes the stabilizer of i in Uh , and then we deﬁne mh (Qρ ) = FctUhi (ωh

ˆ ˜

i ,F (Qi )

)−1 (ρi ), O

14.30.5.

sc σ For any faithful indecomposable Uh ac(F˜ )-object Rσ = j∈J Rj ) h σ ρ and any F-morphism ϕ˜ : R → Q , ϕ˜ induces a necessarily surjective Uh -set sc map f : J → I and, for the chosen j ∈ J , an F˜ -morphism ϕ˜j : Rj → Qf (j) ; in particular, it is clear that Uhj ⊂ Uhf (j) or, equivalently, that hj divides hf (j) and therefore, denoting by ρj the restriction to Uhj of the group homomorphism ˜ f (j) ) ρf (j) : Uhf (j) −→ F(Q 14.30.6,

230

Frobenius categories versus Brauer blocks

sc ϕ˜j becomes an hj (F˜ )-morphism from (Rj )σj to (Qf (j) )ρj , so that from Proposition 14.28 we get a Uhj -set isomorphism

shj (ϕ˜j ) : (Ph

ˆ ˜

j ,F (Rj )

)−1 (σj ) ∼ = (Ph

ˆ ˜

j ,F (Qf (j) )

)−1 (ρj )

14.30.7.

On the other hand, it is clear that the inclusion Uhj ⊂ Uhf (j) determines the commutative diagram (cf. 14.17.2) M

ˆ ˆh , F(Q ˜ f (j) )) Monk∗ (U j & 

ˆ ˜ hj ,F (Qf (j) )

˜ f (j) )) Mon(Uhj , F(Q & 

−−−−−−−−−→

14.30.8;

M

ˆ ˜ hf (j) ,F (Qf (j) )

ˆ ˆh , F(Q ˜ f (j) )) Monk∗ (U f (j)

−−−−−−−−−→

˜ f (j) )) Mon(Uhf (j) , F(Q Uh

then, by applying FctUhj and the inclusion FctUhf (j) ⊂ FctUhj ◦ resUhf (j) to j

this diagram and to the bottom map respectively, we immediately get an O-module isomorphism ρ resQ f,j : FctUhj (Ph

ˆ ˜

j ,F (Qf (j) )

)−1 (ρj ), O

∼ = FctUhf (j) (Ph

)−1 (ρf (j) ), O ˆ ˜ f (j) ,F (Qf (j) )

14.30.9

since both members are free O-modules of rank 1 (cf. 14.15). Moreover, there is ξ ∈ Uh such that ρ(ξ) maps f (j) on the chosen element sc i ∈ I and therefore we have hf (j) = hi and ξ induces an hi (F˜ )-morphism f (j)

ρ(ξ)i

: (Qf (j) )ρf (j) −→ (Qi )ρi

14.30.10,

so that it follows again from Proposition 14.28 that we get a Uhi -set isomorphism f (j) shi ρ(ξ)i : (Ph

ˆ ˜

i ,F (Qf (j) )

)−1 (ρf (j) ) ∼ = (Ph

ˆ ˜

i ,F (Qi )

)−1 (ρi )

14.30.11

which clearly does not depend on the choice of ξ . Finally, we consider the composition ρ f (j) −1 mh (ϕ) ˜ = nhj (ϕ˜j ) ◦ (resQ ◦ nhi ρ(ξ)i f,j )

14.30.12

which deﬁnes an O-module isomorphism mh (Qρ ) ∼ = mh (Rσ ) ; we claim that this correspondence mh is the announced contravariant functor.

14. The Grothendieck groups of a Frobenius P -category

231

Indeed, it is clear that it extends nh and, for any faithful indecompos

sc τ h σ ˜ τ able Uh ac(F˜ )-object T τ = 6∈L T6 ) and any F-morphism ψ : T → R , mutatis mutandis we have a surjective Uh -set map g : L → J , a chosen elesc ment D in L and an h (F˜ )-morphism ψ˜6 : (T6 )τ → (Rg(6) )σ , together with a Uh -set isomorphism sh (ψ˜6 ) : (Ph

ˆ ˜

,F (T )

)−1 (τ6 ) ∼ = (Ph

ˆ ˜

,F (Rg() )

)−1 (σ6 )

14.30.13.

Analogously, we have an O-module isomorphism σ resR g,6 : FctUh (Ph

ˆ ˜

,F (Rg() )

)−1 (σ6 ), O

∼ = FctUhg() (Ph

ˆ ˜

g() ,F (Rg() )

)−1 (σg(6) ), O

14.30.14 .

Moreover, choosing ζ ∈ Uh such that σ(ζ) maps g(D) on j , as above we sc have hg(6) = hj and ζ induces an hj (F˜ )-morphism g(6)

σ(ζ)j

: (Rg(6) )σg() −→ (Rj )σj

14.30.15,

and therefore we get a Uhj -set isomorphism (cf. Proposition 14.28) g(6) shj σ(ζ)j : (Ph

ˆ ˜

g() ,F (Rg() )

)−1 (σg(6) ) ∼ = (Ph

ˆ ˜

j ,F (Rj )

)−1 (σj )

14.30.16.

Finally, we also get ˜ = nh (ψ˜6 ◦ (resRσ )−1 ◦ nh σ(ζ)g(6) mh (ψ) g,6 j j

14.30.17.

˜ ◦ mh (ϕ) In orderto compute mh (ψ) ˜ note that, since f is a Uh -set map, ρ(ζ) maps f g(D) on f (j) and we have hf (g(6)) = hf (j) = hi ; thus, ζ insc duces an hf (j) (F˜ )-morphism f (g(6))

ρ(ζ)f (j)

: (Qf (g(6)) )ρf (g()) −→ (Qf (j) )ρf (j)

14.30.18

and, as above, denoting by ρg(6) the restriction to Uhj of the group homomorphism ρf (g(6)) , the h F-morphism ϕ˜ : Rσ → Qρ forces the following commutative diagram (cf. 14.30.15 and 14.30.18) (Qf (g(6)) )ρg() & ϕ ˜g()  σg()

(Rg(6) )

−→ (Qf (j) )ρj & ϕ˜j −→

σj

(Rj )

14.30.19;

232

Frobenius categories versus Brauer blocks

hence, since nhj is a contravariant functor, we get g(6) g(6) nhj σ(ζ)j ◦ nhj (ϕ˜j ) = nhj ϕ˜j ◦ σ(ζ)j f (g(6)) = nhj ρ(ζ)f (j) ◦ ϕ˜g(6) f (g(6)) = nhj (ϕ˜g(6) ) ◦ nhj ρ(ζ)f (j)

14.30.20.

On the other hand, the natural map θh ,hj in Proposition 14.28 applied to sc the hj (F˜ )-morphism ϕ˜g(6) : (Rg(6) )σg() → (Qf (g(6)) )ρg() yields the following commutative diagram )−1 (σg(6) ) ˆ ˜ j ,F (Rg() )

(Ph

shj (ϕ ˜g() )

∼ =

(Ph

 #

sh (ϕ ˜g() )

∼ =

)−1 (σ6 ) ˆ ˜ ,F (Rg() )

(Ph

ˆ ˜

j ,F (Qf (g()) )

)−1 (ρg(6) )

 #

(Ph

ˆ ˜

,F (Qf (g()) )

14.30.21 )−1 (ρ6 )

where, as above, ρ6 is the restriction of the group homomorphism ρf (g(6)) to Uh ; then, as in 14.30.9 above, considering the O-module isomorphism determined by the inclusion Uh ⊂ Uhj ρ resQ g,6 : FctUh (Ph

)−1 (ρ6 ), O ˆ ˜ ,F (Qf (g()) )

∼ = FctUhj (Ph

)−1 (ρg(6) ), O ˆ ˜ j ,F (Qf (g()) )

14.30.22,

we still get ρ

σ

R nhj (ϕ˜g(6) ) ◦ resQ ˜g(6) ) g,6 = resg,6 ◦ nh (ϕ ρ

14.30.23.

ρ

Q Moreover, note that the composition resQ f,j ◦ resg,6 coincides with the O-module isomorphism determined by the inclusion Uh ⊂ Uhi

ρ resQ g◦f,6 : FctUh (Ph

ˆ ˜ ,F (Qf (g()) )

)−1 (ρ6 ), O

∼ = FctUhi (Ph

)−1 (ρf (g(6)) ), O ˆ ˜ i ,F (Qf (g()) )

14.30.24.

Uh

Finally, applying the natural inclusion FctUhi ⊂ FctUhj ◦resUhi to the Uhi -set j f (g(6)) (cf. 14.30.18), we easily get isomorphism shi ρ(ζ)f (j) ρ f (g(6)) f (g(6)) Qρ nhj ρ(ζ)f (j) ◦ resQ f,j = resf,j ◦ nhi ρ(ζ)f (j)

14.30.25.

14. The Grothendieck groups of a Frobenius P -category

233

In conclusion, from equalities 14.30.20, 14.30.23 and 14.30.25 we get ρ σ g(6) −1 −1 (resR ◦ nhj σ(ζ)j ◦ nhj (ϕ˜j ) ◦ (resQ g,6 ) f,j ) ρ σ f (g(6)) −1 −1 ◦ (resQ = (resR ◦ nhj (ϕ˜g(6) ) ◦ nhj ρ(ζ)f (j) g,6 ) f,j ) ρ ρ f (g(6)) −1 −1 = nh (ϕ˜g(6) ) ◦ (resQ ◦ (resQ ◦ nhj ρ(ζ)f (j) g,6 ) f,g(6) ) ρ f (g(6)) −1 = nh (ϕ˜g(6) ) ◦ (resQ ◦ nhj ρ(ζ)f (j) f ◦g,6 )

14.30.26

and therefore, from equalities 14.30.12 and 14.30.17 we obtain ˜ ◦ mh (ϕ) mh (ψ) ˜

ρ f (g(6)) −1 = nh (ϕ˜g(6) ◦ ψ˜6 ) ◦ (resQ ◦ nhi ρ(ζξ)6 f ◦g,6 )

14.30.27

˜ = mh (ϕ˜ ◦ ψ) which proves our claim. Moreover, it follows from the very deﬁnition of mh that it maps any h F-morphism on an O-module isomorphism; hence, it follows from Theorem 6.26 that, for any n ≥ 1 , we have Hn (h F, mh ) = {0}

14.30.28.

sc sc If h = 0 then 0 F = 0 (F˜ ) = F˜ and we are done. Otherwise, we consider the full subcategory h E of h F over the faithful indecomposable

ρ sc Uh ac(F˜ )-objects Qρ = such that |I| > 1 , namely over all the i∈I Qi h sc h ˜ F-objects which are not (F )-objects; note that there is no h F-morphism h sc from a (F˜ )-object to an h E-object. Denoting by lh : h E → O-mod the re-

striction of mh to h E , it is quite clear that Theorem 6.26 applies again to h E and lh , so that for any n ≥ 1 we get Hn (h E, lh ) = {0}

14.30.29.

Recall that, denoting by ∆ the simplicial 2-category (cf. A1.7), the cohoh sc mology groups Hn (h F, mh ) , Hn (h E, lh ) and Hn (F˜ ), nh are nothing but the homology groups of the respective obvious functors cmh , clh and cnh from ∆ to O-mod , mapping ∆n on (cf. A3.8) Cn (h F, mh ) = mh ˜q(0) ˜∈Fct(∆n ,h F) q

Cn (h E, lh ) = h sc Cn (F˜ ), nh =

mh ˜q(0)

˜∈Fct(∆n ,h E) q

˜ sc )) ˜∈Fct(∆n ,h(F q

mh ˜q(0)

14.30.30;

234

Frobenius categories versus Brauer blocks h

sc

then, the inclusion (F˜ ) ⊂ h F clearly determines a surjective natural map µh : cmh −→ cnh

14.30.31,

so that we obtain a fourth functor Ker(µh ) : ∆ −→ O-mod

14.30.32. h

sc

Moreover, since there is no h F-morphism from a (F˜ )-object to any h E-object, we have an O-module isomorphism Ker(µh ) (∆n ) ∼ mh ˜q(0) 14.30.33, = ˜ q

where ˜q runs over the set E h (∆n ) of functors from ∆n to h F such that ˜q(0) is a h E-object and therefore the inclusion h E ⊂ h F also determines a surjective natural map 14.30.34. λh : Ker(µh ) −→ clh But on the one hand, we already know that Hn (cmh ) = {0} = Hn (clh ) for any n ≥ 1 (cf. equalities 14.30.28 and 14.30.29) and on the other hand, setting H−n (cnh ) = H−n (cmh ) = H−n Ker(µh ) = {0} 14.30.35 for any n > 0 , there is a 1-graded connecting homomorphism (cf. A3.3.4) δ:

Hn (cnh ) −→

n∈Z

Hn Ker(µh )

14.30.36

n∈Z

such that we have the following exact triangle (cf. A3.3.5)

n∈Z

δ

Hn (cnh )

⊕n∈Z Hn (µh )

3

−→

n∈Z

Hn Ker(µh )

0

n∈Z

14.30.37.

Hn (cmh )

Consequently, in order to prove that Hn (cnh ) = {0} for any n ≥ 1 , it suﬃces to show that Hn (λh ) is injective. Actually, since the functor sh facsc h sc torizes throughout the exterior quotient h˜(F˜ ) of (F˜ ) (cf. Remark 14.29), it is easily checked that the contravariant functors nh , mh and lh respectively determine contravariant functors sc ˜h : h˜(F˜ ) −→ O-mod n

˜ h : h˜F −→ O-mod and ˜lh : h˜E −→ O-mod m

14.30.38,

14. The Grothendieck groups of a Frobenius P -category

235

where h˜F and h˜E denote the corresponding exterior quotients (cf. 6.3). Coherently, we get the corresponding functors cn˜h , cm ˜ h and c˜lh (cf. 14.30.30), and the corresponding natural maps (cf. 14.30.31 and 14.30.34) µ ˜ h : cm ˜ h −→ cn ˜h

˜ h : Ker(˜ and λ µh ) −→ c˜lh

14.30.39.

Then, it follows from equality 6.25.1 that, for any n ≥ 1 , it suﬃces to ˜ h ) is injective. First of all note that, up to isomorphisms, prove that Hn (λ

ρ ˜ h for a suitable any F-object has the canonical form Qρ = ¯ h /U Qξ¯ ξ∈U ˜ ˜ subgroup U of Uh , and, denoting by ˜jh : h E → h F the inclusion functor, we claim that we have a functor and a natural map ˜eh : h˜F −→ h˜E and ε˜h : ˜jh ◦ ˜eh −→ idh˜ F which respectively map Qρ on the

Uh

14.30.40

sc ˆ ρˆ formed by ac(F˜ )-object Q

ˆ=

Q ξ∈Uh Qξ¯

14.30.41

ˆ deﬁned by the regular acand by the k ∗ -group homomorphism ρˆ : Uh → F(Q) sc ˜ tion of Uh on itself, together with the F -isomorphisms from the ξ-summand to the ξ -summand induced by ρ(ξ ξ −1 ) , and on the h˜F-morphism ˆ ρˆ −→ Qρ (˜ εh )Qρ : Q

14.30.42

deﬁned by the canonical map Uh → Uh /U and by the identity automorphism of Qξ¯ for any ξ ∈ Uh .

σ Indeed, if Rσ = is another h˜F-object, an h˜F-morphism ˜ h /V Rξ˜ ξ∈U from Rσ to Qρ forces V ⊂ U and admits a canonical representative formed sc by the canonical map Uh /V → Uh /U and by an F˜ -morphism ϕ˜ξ˜ : Rξ˜ → Qξ¯ for any ξ˜ ∈ Uh /V ; then, the functor ˜eh above maps this h˜F-morphism on the ˜E-morphism h σˆ ρˆ ˆ σˆ =

ˆ ρˆ =

R −→ Q ξ∈Uh Rξ˜ ξ∈Uh Qξ¯

14.30.43

admitting a representative formed by the identity map of Uh and by the sc F˜ -morphism ϕ˜ξ˜ : Rξ˜ → Qξ¯ for any ξ ∈ Uh ; it is quite clear that this correspondence preserves the composition of h˜F-morphisms and is compatible with the h˜F-morphisms 14.30.42. ˜ h ◦ ˜jh ◦ ˜eh = ˜lh ◦ ˜eh Secondly, we consider the contravariant functor m which, as above, determines a functor (cf. 14.30.30) c˜lh ◦˜eh : ∆ −→ O-mod

14.30.44

236

Frobenius categories versus Brauer blocks

mapping ∆n on ˜ Cn (h F, ˜lh ◦ ˜eh ) =

˜ h ˆeh ˜q(0) m

˜∈Fct(∆n q

14.30.45,

˜ ,h F)

and then we get a natural map κ˜eh : cm ˜ h → c˜lh ◦˜ eh sending ∆n to the O-module homomorphism ˜ h ˜q(0) −→ ˜ h ˆeh ˜q(0) (κ˜eh )n : m m 14.30.46 ˜ F) ˜∈Fct(∆n ,h q

˜ F) ˜∈Fct(∆n ,h q

mapping m = (mq˜ )q˜∈Fct(∆n ,h˜ F) on (m˜jh ◦˜eh ◦˜q )q˜∈Fct(∆n ,h˜ F) . Note that if m ˜ h )∆ (cf. 14.30.39) then, for any ˜q ∈ Fct(∆n , h˜E) , we have belongs to Ker (λ n ˜ h maps any mq˜ = 0 and therefore we get (κ˜eh )n (m) = 0 . Moreover, since m ˜F-morphism on an O-module isomorphism (cf. 14.30.12), the natural map h (cf. 14.30.40) ˜ h ∗ ε˜h : m ˜ h −→ ˜lh ◦ ˜eh m

14.30.47

determines a natural isomorphism ∼ κε˜h : cm ˜ h = c˜lh ◦˜ eh

14.30.48

which sends ∆n to the O-module isomorphism (κε˜h )n :

˜∈Fct(∆n q

˜ F) ,h

˜ h ˆeh ˜q(0) m

˜ h ˜q(0) ∼ m =

˜∈Fct(∆n q

˜ h (˜ εh )q˜(0) m

mapping m = (mq˜ )q˜∈Fct(∆n ,h˜ F) on

14.30.49

˜ F) ,h

(mq˜ )

. ˜ F) ˜∈Fct(∆n ,h q

At this point, for any n ≥ 1 , following the notation introduced in Lemma A4.2 below we consider the O-module homomorphism hn−1 : cm ˜ h (∆n ) −→ c˜lh ◦˜ eh (∆n−1 )

14.30.50

mapping m = (mq˜ )q˜∈Fct(∆n ,h˜ F) ∈ cm ˜ h (∆n ) on hn−1 (m) =

n−1 i=0

(−1)i mhn−1 (˜εh ∗˜r) i

˜ F) ˜ r∈Fct(∆n ,h

14.30.51

Then, respectively denoting by dn and dˆn the diﬀerential maps for the functors cm ˜ h and c˜lh ◦˜ eh (cf. A3.2), we claim that (κε˜h )n (m) − (κ˜eh )n (m) = (dˆn−1 ◦ hn−1 + hn ◦ dn )(m)

14.30.52;

14. The Grothendieck groups of a Frobenius P -category indeed, setting θ = mh ˜eh ˜q(0•1) , for any ˜q ∈ Fct(∆n , h˜F) we have

237

dˆn−1 hn−1 (m) q˜ =

n−1

(−1)i dˆn−1 (mhn−1 (˜εh ∗˜r) )˜r∈Fct(∆n−1 ,h˜ F) q˜ i

i=0

=

n−1

(−1)i θ(mhn−1 (˜εh ∗(˜q◦δn−1 )) ) 0

i

i=0

+

n

(−1)j mhn−1 (˜εh ∗(˜q◦δn−1 ))

j=1

i

+

j

analogously, we still have hn dn (m) q˜ = hn θ(m˜r◦δ0n ) r∈Fct(∆ ,h˜ F) n+1 n+1

14.30.53;

˜ q

(−1)j hn (mr◦δjn )r∈Fct(∆n+1 ,h˜ F) q˜

j=1

=

n

14.30.54.

n+1 (−1)i θ(mhni (˜εh ∗˜q)◦δ0n ) + (−1)j mhni (˜εh ∗˜q)◦δjn

i=0

j=1

But from Lemma A4.2 we know that n n hni+1 (˜ εh ∗ ˜q) ◦ δi+1 = hni (˜ εh ∗ ˜q) ◦ δi+1 * n εh ∗ ˜q) ◦ δjn hi+1 (˜ hn−1 ε˜h ∗ (˜q ◦ δjn−1 ) = n n i hi (˜ εh ∗ ˜q) ◦ δj+1

if j ≤ i if i < j .

14.30.55.

Consequently, in equality 14.30.54 the terms where j = i and j = i + 1 cancel with each other for any 1 ≤ j ≤ n ; moreover, the term (i, j) in equality 14.30.53 cancels either with the term (i+1, j) if 1 ≤ j ≤ i ≤ n−1 , or with the term (i, j +1) if 0 ≤ i < j ≤ n in equality 14.30.54. Finally, the term (i, 0) in equality 14.30.53 cancels with the term (i + 1, 0) in equality 14.30.54 for any 0 ≤ i ≤ n − 1 , whereas the terms (0, 0) and (n, n + 1) respectively coincide with (κε˜h )n (m) and −(κ˜eh )n (m) , which proves the claim. ˜ In conclusion, if m ∈ Ker(d n ) lifts an element of Hn (λn ) then we may assume that m ∈ Ker (λh )∆n , so that we get (κ˜eh )n (m) = 0 , and therefore it follows from isomorphism 14.30.48 and from equality 14.30.52 that (κε˜h )n (m) = dˆn−1 hn−1 (m) −1 = dˆn−1 (κε˜h )n−1 ◦ (κε˜h )n−1 ◦ hn−1 (m) 14.30.56 −1 = (κε˜h )n ◦ dn−1 ◦ hn−1 (m) (κε˜h )n−1 which proves that m belongs to Im(dn−1 ) . We are done.

238

Frobenius categories versus Brauer blocks

14.31 Let us say that a ch∗ (F )-object (q, ∆n ) is regular if q(i − 1, i) is not an isomorphism for any 1 ≤ i ≤ n (cf. A5.2); note that there is a canonical bijection between a set of representatives for the isomorphism classes of sc regular ch∗ (F )-objects and a set of representatives for the F-isomorphism classes of nonempty sets of F-selfcentralizing subgroups of P , which are totally ordered by the inclusion. sc

Corollary 14.32 We have ˆ F nc ) = rankO Gk (F, aut (−1)n rankO Gk F(q)

14.32.1

(q,∆n )

where (q, ∆n ) runs over a set of representatives for the set of isomorphism sc classes of regular ch∗ (F )-objects. Proof: Denote by K the ﬁeld of quotients of O and by K nh the extension of nh = FctUh ◦ sh from O to K for any h ∈ N − pN . It is more or less well-known that for any n ∈ N we have (cf. Propositions A4.13 and A5.7) h sc h sc h sc Hn (F˜ ), K nh ∼ = Hn∗ (F˜ ), K nh ∼ = Hnr (F˜ ), K nh

14.32.2.

In particular, it follows from Theorem 14.30 that for any n ≥ 1 we get h sc Hnr (F˜ ), K nh = {0}

14.32.3

which amounts to saying that we have an inﬁnite exact sequence h sc 0 −→ H0r (F˜ ), K nh −→ . . . −→ C n → C n+1 −→ . . .

14.32.4

h sc where C n = Cnr (F˜ ), K nh is the set of elements (χq˜ )q˜∈Fctr (∆n ,h(F˜ sc )) ∈

K

nh ˜q(0)

14.32.5

h

˜ sc )) ˜∈Fctr (∆n , (F q

such that, for any natural isomorphism ν˜ : ˜q ∼ = ˜q between regular (F˜ )-vaν0 ) maps χq˜ on χq˜ . That is to say, we actually lued n-chains ˜q and ˜q , K nh (˜ have h sc F˜ (˜q) Cnr (F˜ ), K nh ∼ FctUh sk ˜q(0) , K 14.32.6 = h

sc

˜ q

where ˜q runs over a set of representatives for the set of isomorphism classes h sc in Fctr ∆n , (F˜ ) (cf. A5.3).

14. The Grothendieck groups of a Frobenius P -category

239

On the other hand, it is clear that for n big enough there are no reguh sc lar (F˜ )-valued n-chains and therefore, in the exact sequence above, only ﬁnitely many terms are not zero; thus, we still get h sc rankO H0 (F˜ ), nh =

n

(−1) dim K

F˜ (˜q) FctUh sk ˜q(0) , K

14.32.7

(˜ q,∆n )

where (˜q, ∆n ) runs over a set of representatives for the isomorphism classes h sc of regular ch∗ (F˜ ) -objects (cf. A5.3). Consequently, it follows from Propositions 14.27 and 14.28 that, deno˜ q)η the stabilizer of η in F(˜ ˜ q) for any uh ch∗ (F˜ sc )-object (η, ˜q, ∆n ) ting by F(˜ (cf. 14.23.4), since we have (Ph,F˜ˆ (˜q(0)) )−1 (ιq0 ◦ η) ∼ = (Ph,F˜ˆ (˜q) )−1 (η) ˜

14.32.8,

we actually have F nc ) rankO Gk (F, aut F˜ (˜q)η = (−1)n dim K FctUh (Ph,F˜ˆ (˜q) )−1 (η), K (η,˜ q,∆n )

14.32.9

h

where h runs over N − pN and (η, ˜q, ∆n ) runs over a set of representatives for sc the isomorphism classes of uh ch∗ (F˜ )-objects such that (˜q, ∆n ) is a regular sc ch∗ (F˜ )-object. sc On the other hand, for any regular ch∗ (F˜ )-object (˜q, ∆n ) , it follows from isomorphism 14.15.3 that h∈N−pN

∼ =

η

F˜ (˜q)η FctUh (Ph,F˜ˆ (˜q) )−1 (η), O

˜ (˜ h∈N−pN ρ∈Mon(Uh ,F q))

F˜ (˜q) FctUh (Ph,F˜ˆ (˜q) )−1 (ρ), O

14.32.10

ˆ ∼ ˜ q) ∼ ˆ = Gk F(q) = Gk F(˜

˜ q) on the set where η runs over a set of representatives for the orbits of F(˜ sc ˜ Mon Uh , F(˜q) and q : ∆n → F is a functor lifting ˜q . We are done.

Chapter 15

Reduction results for Grothendieck groups 15.1 Let k be an algebraically closed ﬁeld of characteristic p and O the unramiﬁed complete discrete valuation ring of characteristic zero with residue ˆ ﬁeld k ; denote by O-mod the category of ﬁnitely generated O-modules. Let G be a ﬁnite k ∗ -group — we already have seen the interest of replacing ﬁnite groups by k ∗ -groups with a ﬁnite k ∗ -quotient G (cf. 1.23) — and b be a ˆ — called a block of G ˆ (cf. 1.25). Recall that, primitive idempotent of Z(k∗ G) according to Proposition 5.15 in [42], b is also a block of a ﬁnite subgroup G ˆ fulﬁlling kG b = k∗ Gb ˆ and therefore all the terminology developed for G of G ˆ . Thus, let (P, e) be a maximal Brauer (b, G)-pair ˆ can be easily translated to G (cf. 1.16 and 1.28) and F = F(b,G) ˆ the corresponding Frobenius P -category nc

(cf. Theorem 3.7); recall that we denote by F the full subcategory of F over the set of F-nilcentralized subgroups of P (cf. 11.1). ˆ b) the scalar extension 15.2 As in chapter 14, let us denote by Gk (G, ˆ from Z to O of the Grothendieck group of the category of k∗ Gb-modules of ﬁnite dimension. Applying again the remark above, it follows from Theorem 11.32 that we have a functor F nc : ch∗ (F ) −→ k ∗ -Gr aut nc

15.2.1

lifting autF nc (cf. Proposition A2.10) and, in this situation, in chapter 14 we F nc ) of this pair (F, aut F nc ) . have deﬁned the Grothendieck group Gk (F, aut As we explain in the Introduction (cf. I 31), Corollary 14.32 shows that the F nc ) coincides with the O-rank that Jon Alperin conO-rank of Gk (F, aut ˆ jectures for Gk (G, b) and therefore Alperin’s Conjecture is equivalent to the ˆ b) ∼ F nc ) . existence of an O-module isomorphism Gk (G, = Gk (F, aut ˆ (cf. 1.10) is a normal sub-block 15.3 Let us say that a block (c, H) ∗ ˆ ˆ ˆ and we have cb = 0 — in 15.7 of (b, G) when H is a normal k -subgroup of G ˆ ﬁxes c . In this chapter, below we show that we actually may assume that G we prove that — independently of Alperin’s Conjecture — both Grothendieck ˆ b) and Gk (F, aut F nc ) have an analogous relationship with the groups Gk (G, ˆ . Grothendieck groups corresponding to suitable normal sub-blocks of (b, G) The point is that we prove enough such reduction results to show — in the next chapter — that a suitable reformulation strengthening the above conjecture can be veriﬁed just testing the blocks “around” the simple groups.

242

Frobenius categories versus Brauer blocks

15.4 As a matter of fact, some of the reduction steps really force us to employ ﬁnite k ∗ -groups instead of ordinary ﬁnite groups. First of all, F nc for the ﬁnite it is convenient to clarify the deﬁnition of the functor aut ∗ ˆ ˆ k -group G ; let (R, g) be a nilcentralized Brauer (b, G)-pair (cf. 7.4) contained in (P, e) (cf. 1.15); as in 7.4 above, the quotient k∗ CGˆ (R)g J k∗ CGˆ (R)g is a simple k-algebra — in particular, g determines a unique local point ε of R ˆ (cf. 1.19) — and therefore the action of the normalizer N ˆ (R, g) on k∗ Gb G ˆG (R, g) together with a k-algebra on this k-algebra determines a k ∗ -group N homomorphism (cf. 1.25) ˆG (R, g) −→ k∗ C ˆ (R)g J k∗ C ˆ (R)g k∗ N G G

15.4.1;

moreover, since we have an obvious NG (R, g)-stable k-algebra homomorphism k∗ CGˆ (R) −→ k∗ CGˆ (R)g J k∗ CGˆ (R)g 15.4.2, ˆG (R, g) we still have an NG (R, g)-stable k ∗ -group homomorphism CGˆ (R) → N lifting the inclusion CG (R) ⊂ NG (R, g) . 15.5 Consequently, we have an evident group homomorphism (cf. 1.23) ˆG (R, g)◦ ∆(R,g) : CG (R) −→ NGˆ (R, g) ∗ N

15.5.1

having a normal image and then, since (cf. 1.26) F(R) ∼ = EG (R, g) = NG (R, g)/CG (R)

15.5.2

ˆ where we identify the Brauer (b, G)-pair (R, g) with the unique local pointed group Rε , we set ∼ ˆ ˆG (R, g) = N ˆ (R, g) ∗ N ˆG (R, g)◦ ∆(R,g) CG (R) F(R) 15.5.3. =E G ˆ as a normal k ∗ -subgroup, denoting Similarly, if Fˆ is a k ∗ -group containing G by CFˆ (R, g) the stabilizer of g in CFˆ (R) , we deﬁne ◦ Fˆ ˆ CG ∆(R,g) CG (R) ˆ (R, g) = CFˆ (R, g) ∗ CF (R, g) nc

15.5.4.

nc

15.6 Then, for any F -chain q : ∆n → F (cf. 2.18), identifying F(q) to ˆ its image in F q(n) , as in 11.31 above we denote by F(q) the corresponding ∗ k -subgroup of Fˆ q(n) ; as mentioned above, it follows from Theorem 11.32 F nc lifting autF nc . that this correspondence can be extended to a functor aut Thus, as in 14.3, we have the composed functor a utF nc nc gk ch∗ (F ) −−−− −→ k ∗ -Gr −→ O-mod

15.6.1

15. Reduction results for Grothendieck groups

243

F nc ) as the and we have deﬁned the Grothendieck group of the pair (F, aut inverse limit F nc ) = lim (gk ◦ aut F nc ) Gk (F, aut 15.6.2. ←−

15.7 In this chapter, our standard setting is the pair formed by the ˆ and a normal sub-block (c, H) ˆ ; so, H ˆ is a normal k ∗ -subgroup block (b, G) ˆ and we assume that cb = 0 . In particular, we have bTrGˆˆ (c) = b where of G Gc

ˆ c denotes the stabilizer of c in G ˆ ; since we know that eBrP (b) = 0 (cf. 1.16), G we may assume that P stabilizes c ; then, considering the G-stable semisimple ˆ , where x ∈ G runs over a set of representatives k-subalgebra x k·bcx of k∗ G ˆ G ˆ c , it follows from Proposition 2.5 in [53] that (bc, Gc ) is a block for G/ (cf. 1.10) and, from Proposition 2.2 and Corollary 2.3 again in [53], that P remains a defect p-subgroup of this block and we have ˆc) ∼ ˆ and Gk (bc, G = Gk (b, G)

F(bc,Gˆ c ) = F(b,G) ˆ

15.7.1.

ˆ ﬁxes c Thus, for our purposes, in our standard setting we may assume that G and we have bc = b . ˆ 15.8 Furthermore, we choose a maximal Brauer (c, H)-pair (Q, f ) ; note ˆ ˆ that α = {c} is also a point of G on k∗ H and that, denoting by eH a block of CHˆ (P ) such that eH e = 0 , (P, eH ) and (Q, f ) are Brauer (α, G)-pairs (cf. 1.14). Thus, in our standard setting, we may assume that (P, eH ) and (Q, f ) are both contained in a maximal Brauer (α, G)-pair (O, o) (cf. 1.15); ˆ ∩ O , since there is a block f of C ˆ (Q ) such that moreover, setting Q = H H (cf. 1.15) (Q, f ) ⊂ (Q , f ) ⊂ (O, o) 15.8.1, we actually have ˆ ∩O Q=H

15.8.2.

Then, we consider the Frobenius P - and Q-categories F = F(b,G) ˆ

and H = F(c,H) ˆ

15.8.3;

although, in general, there is no easy relationship between F and H , in the nc sc situations we consider below we obtain some connection between F or F nc sc and H or H , which allows us to relate the corresponding Grothendieck F sc ) and Gk (H, aut Hsc ) . groups Gk (F, aut Proposition 15.9 In our standard setting, we have ˆ ∩P Q=H

and

BrP (f ) = 0

15.9.1.

244

Frobenius categories versus Brauer blocks

Proof: Setting R = Q ∩ P and denoting by g the block of CGˆ (R) such that (P, e) contains (R, g) (cf. 1.15), a maximal Brauer g, R·CGˆ (R) -pair (T, h) ˆ ˆ such that (cf. 1.15) is a Brauer (b, G)-pair too and therefore there is x ∈ G (R, g)x ⊂ (T, h)x ⊂ (P, e)

15.9.2 ;

ˆ ∩ P which forces Rx = R and then g x = g (cf. 1.15). thus, we have Rx ⊂ H That is to say, C¯P (R) is a defect group of the image g¯ of g in C¯Gˆ (R) ¯ (R) C (cf. 1.9.2), so that g¯ belongs to k∗ C¯ ˆ (R) ¯Gˆ (cf. Theorem 1.14 in [11]), G

CP (R)

and therefore, since we have C¯Hˆ (R) ∩ C¯P (R) = {1} , the restriction of any simple k∗ C¯Gˆ (R)¯ g -module to k∗ C¯Hˆ (R) is both semisimple (cf. Theorem 4.1 in [28, Ch. 3]) and projective. Consequently, since the block g H of CHˆ (R) such that (P, eH ) contains (R, g H ) (cf. 1.15) is P -stable and we have BrP (g H )e = 0 (cf. Theorem 1.8 in [11]), we still have g H g = 0 and therefore a simple k∗ C¯Hˆ (R)¯ g H -module is H ˆ projective, so that g¯ has defect zero (cf. 1.17) and the Brauer (c, H)-pair H (R, g ) is selfcentralizing (cf. 7.4); but, since (R, g H ) ⊂ (P, eH ) ⊂ (O, o) and

(Q, f ) ⊂ (O, o)

15.9.3,

we have (R, g H ) ⊂ (Q, f ) (cf. 1.15) and therefore R is H-selfcentralizing (cf. Corollary 7.3). Moreover, since we have (P, eH ) ⊂ (O, o) ⊃ (Q, f )

15.9.4,

¯Q (P ) = NQ (P )/R (cf. 1.9.2) NQ (P ) ﬁxes eH (cf. 1.15) and the quotient N H H ¯ acts on k∗ CGˆ (P )¯ e ; but, since the block e¯ has defect zero (cf. 1.17), it fol¯Q (P )-algebra isomorphism lows from Proposition 3.7 in [53] that we have a N (cf. deﬁnition 15.5.4) ˆ H G k∗ C¯Gˆ (P )¯ eH ∼ eH ⊗k k∗ CH = k∗ C¯Hˆ (P )¯ ˆ (P, e )

15.9.5

¯Q (P )-algebra isomorphism and, in particular, we still have a N ˆ G H Z k∗ C¯Gˆ (P )¯ eH ∼ = Z k∗ CH ˆ (P, e )

15.9.6;

¯Q (P ) acts trivially on G/H , it acts on the other hand, since the p-group N ˆ G H ¯ trivially on C (P, e ) . Hence, NQ (P ) acts trivially on Z k∗ C¯ ˆ (P )¯ eH and ˆ H

G

therefore it ﬁxes e ; consequently, since we have CQ (P ) ⊂ CQ (R) = Z(R) , this p-group maps into the p -group NGˆ (P, e)/P ·CGˆ (P ) (cf. 1.16) which forces ¯Q (P ) = {1} and therefore, since P acts on Q/R and contains R , this implies N that Q = R ; in particular, we get (Q, f ) ⊂ (P, eH ) . We are done.

15. Reduction results for Grothendieck groups

245

Proposition 15.10 In our standard setting, assume that CGˆ (Q, f )/CHˆ (Q) is a p-group. Then, we have b=c

and

CGˆ (Q, f ) = CHˆ (Q)·CP (Q)

15.10.1,

ˆ H ˆ and, for any selfcentralizing Brauer P/Q is a Sylow p-subgroup of G/ ˆ (c, H)-pair (R, g) , the quotient CGˆ (R, g)/CHˆ (R) is a p-group and, setting CG ˆ (R) G ˆ g = Tr (g) , (R, g G ) is a nilcentralized Brauer (b, G)-pair. In partiCG ˆ (R,g)

cular, H

sc

and H are respective subcategories of F

nc

and F .

Proof: We may assume that (R, g) ⊂ (Q, f ) ; since the correspondence sending (R, g) to (R, g G ) preserves the inclusion of Brauer pairs, the inclusion sc nc H ⊂ F will follow from the previous statement and then the inclusion H ⊂ F will follow from Corollary 5.14. Moreover, in order to prove ˆ that (R, g G ) is a nilcentralized Brauer (b, G)-pair, it suﬃces to prove that CGˆ (R, g)/CHˆ (R) is a p-group; indeed, since the block g, CHˆ (R) is nilpotent (cf. 1.10 and 7.4), it follows from Proposition 6.5 in [34] that g remains a nilpotent block of CGˆ (R, g) and then, by Proposition 3.5 in [53], g G is a nilpotent block of CGˆ (R) . ˆ determined by (R, g) Denote by Rε the local pointed group on k∗ Hc ˆ such that Rε ⊂ R·C ˆ (R)λ (cf. 15.4), by λ the point of R·CHˆ (R) on k∗ H H (cf. 1.19) and by Tν a defect pointed group of R·CHˆ (R)λ (cf. 1.15); we argue by induction on |Q : R| and, since CGˆ (Q, f )/CHˆ (Q) is a p-group, may assume that R = Q . By the Frattini argument, we get R·CGˆ (Rε ) = R·CHˆ (R)· NGˆ (Tν ) ∩ CGˆ (Rε )

15.10.2;

but, by Thompson’s Lemma (cf. Theorem 3.4 in [28, Ch. 5]), any k ∗ -subgroup ˆ of N ˆ (Tν ) ∩ C ˆ (Rε ) with k ∗ -quotient of order prime to p , centralizes T K G G ˆ. and therefore, by our induction hypothesis, it is contained in H In particular, since f remains a block of CGˆ (Q, f ) and any idempotent of Z k∗ NGˆ (Q, f ) belongs to Z k∗ CGˆ (Q, f ) (cf. 1.19), f remains a block of NGˆ (Q, f ) too and therefore it follows again from Proposition 3.5 in [53] N (Q)

that TrNGˆˆ (Q,f ) (f ) is a block of NGˆ (Q) , so that we get (cf. 1.15) G

N (Q)

N (Q)

BrQ (c) = TrNHˆˆ (Q,f ) (f ) = TrNGˆˆ (Q,f ) (f ) H

G

15.10.3

ˆ such that which forces BrQ (b) = BrQ (c) . Similarly, for any block b of G b c = b we still get BrQ (b ) = BrQ (c) which forces b = b , so that b = c .

246

Frobenius categories versus Brauer blocks

ˆ ˆ Then, since (k∗ Hc)(O) (cf. 15.8) maps into (k∗ Gb)(O) (cf. 1.19), the H ˆ Brauer (α, G)-pair (P, e ) in 15.8 above is maximal and therefore it follows from Proposition 5.3 and Corollary 6.3 in [34] that P/Q is a Sylow p-subgroup ˆ H ˆ ; moreover, P normalizes (Q, f ) (cf. 1.15) and therefore P/Q contains of G/ ˆ H ˆ , proving the right-hand equality the p-subgroup CGˆ (Q, f )/CHˆ (Q) of G/ in 15.10.1. 15.11 Thus, if CGˆ (Q, f )/CHˆ (Q) is a p-group and in our standard setting ˆ of G ˆ containing H ˆ , then b = c is still a block we consider a k ∗ -subgroup K ˆ F nc of K and, setting K = F(b,K) ˆ , it is possible to connect the functors aut Knc , suitably restricted to the F- and K-chains whose terms contain and aut H-selfcentralizing subgroups of Q ; actually, we only need this connection ˆ H ˆ is a p -group, which is easier to we discuss than the general case. when K/ ˆ H ˆ is a p -group then K is a Frobenius Q-category and a subgroup of If K/ Q is K-selfcentralizing if and only if it is H-selfcentralizing; more explicitly, ˆ if (R, g) is a selfcentralizing Brauer (c, H)-pair then, according to ProposiK ˆ tion 15.10, (R, g ) is a selfcentralizing Brauer (c, K)-pair and, by the Frattini argument, we have NKˆ (R, g K ) = CKˆ (R)·NKˆ (R, g)

15.11.1;

hence, we get K(R) ⊂ F(R) and then it follows from Corollary 5.14 and sc nc Proposition 15.10 that K is a subcategory of F ; in particular, denoting sc nc by j : K → F the inclusion functor, we have a functor and a natural map (cf. A2.8 and Proposition A2.10) ch∗ (j) : ch∗ (K ) −→ ch∗ (F ) sc

nc

ν : autKsc −→ autF nc ◦ ch∗ (j)

15.11.2.

Proposition 15.12 In our standard setting, assume that CGˆ (Q, f )/CHˆ (Q) ˆ be a k ∗ -subgroup of G ˆ containing H ˆ such that K/ ˆ H ˆ is a p-group and let K is a p -group. Then, with the notation above, ν can be lifted to a natural map Ksc −→ aut F nc ◦ ch∗ (j) νˆ : aut

15.12.1.

ˆ Proof: Let (R, g) be a selfcentralizing Brauer (c, H)-pair; it follows from Proposition 15.10 that the relative traces C (R)

C (R)

g K = TrCKˆˆ (R,g) (g) and g G = TrCGˆˆ (R,g) (g) K

G

15.12.2

are respective nilpotent blocks of CKˆ (R) and CGˆ (R) ; actually, since R is also K-selfcentralizing, g K is a block of CKˆ (R) of defect zero (cf. Corollary 7.3);

15. Reduction results for Grothendieck groups

247

hence, we get canonical embeddings (cf. 1.16.1 and 7.7.1 above and Proposition 3.5 in [53]) k∗ CKˆ (R)g K /J k∗ CKˆ (R)g K 1 k∗ C¯Hˆ (R)¯ g 15.12.3; / k∗ CGˆ (R)g G /J k∗ CGˆ (R)g G consequently, the actions of NKˆ (R, g) on these three simple k-algebras deˆ ˆ (R, g) (cf. 1.25) and therefore from the equaltermine the same k ∗ -group N K ity 15.11.1 successively applied to NKˆ (R, g K ) and to NGˆ (R, g G ) we get a canonical k ∗ -group homomorphism ˆ ˆ νˆR : K(R) −→ F(R)

15.12.4

which is clearly compatible with the K-isomorphisms. ˆ Moreover, for another selfcentralizing Brauer (c, H)-pair (T, h) contained in (R, g) , it is not diﬃcult to check that in 11.22.2 the k ∗ -group homomorK K ˆ and G ˆ , agree and ˆ(T,hK ) and u ˆ(T,hK ) , respectively deﬁned from K phisms u (R,g )

(R,g )

therefore, for coherent choices in 11.19.2, the uniqueness of the k ∗ -group ho(T,hK )

(T,hG )

momorphisms α ˆ (R,gK ) and α ˆ (R,gG ) obtained from Proposition 11.23 forces the commutative diagram (T ,hG ) (R,g G )

α ˆ

ˆ F(R) T ν ˆR

ˆ ) −−−−−−→ F(T

↑

↑ νˆT (T ,hK ) α ˆ (R,g K )

ˆ K(R) T

−−−−−−→

15.12.5.

ˆ ) K(T

sc

sc

nc

Pushing it further, for any K -chain q : ∆n → K , we have the F -chain nc j ◦ q : ∆n → F and a k ∗ -group homomorphism ˆ ˆ ◦ q) νˆq : K(q) −→ F(j determined by νˆq(n) (cf. 15.6); consequently, since from the composition F nc ◦ ch∗ (j) we can construct a functor aut ch∗ (K ) −→ k ∗ -Gr sc

15.12.6

lifting autKsc and fulﬁlling all the conditions in Theorem 11.32, suitably restated over the set of K-selfcentralizing subgroups of Q , it follows from the uniqueness part of this theorem that we get the announced natural map Ksc −→ aut F nc ◦ ch∗ (j) νˆ : aut

15.12.7.

248

Frobenius categories versus Brauer blocks

ˆ = H·C ˆ ˆ (Q, f ) and asProposition 15.13 In our standard setting, set K G ˆ K ˆ is a p -group and that C ˆ (Q, f )/C ˆ (Q) is a p-group. sume that C = G/ G H ˆ C the converse image in G ˆ of a complement in G/ ˆ H ˆ of the Sylow Denote by H p-subgroup, and by HC the Frobenius Q-category determined by the canonical isomorphism C∼ 15.13.1. = NGˆ (Q, f )/NKˆ (Q, f ) ∼ = F(Q)/H(Q) ˆ C , we have F ˆ C = HC , (HC )sc is a full Then, b = c is a block of H (b,H ) nc

sc

subcategory of F and, denoting by j : (HC ) we have a natural isomorphism

→F

nc

the inclusion functor,

(HC )sc ∼ F nc ◦ ch∗ (j) νˆ : aut = aut

15.13.2.

Moreover, the pair (j, νˆ) and the restriction induce O-module isomorphisms F nc ) ∼ (HC )nc ) Gk (F, aut = Gk (HC , aut

and

ˆ b) ∼ ˆ C , b) 15.13.3. Gk (G, = Gk (H

ˆ = H·C ˆ P (Q) Proof: It follows from Proposition 15.10 that b = c , that K sc nc and that H is a subcategory of F ; moreover, from the obvious isomorphism 15.13.1, we may identify F(Q)/H(Q) with the p -group C and then it follows from Proposition 12.12 that we have indeed a Frobenius Q-category HC fulﬁlling HC (Q) = F(Q) ; thus, since a subgroup of Q is HC -selfcentralizing if and only if it is H-selfcentralizing, it follows from Proposition 12.8 sc nc that (HC ) is also a subcategory of F . ˆ determined by f (cf. 1.17), Denoting by δ the local point of Q on k∗ H + ˆ recall that the group AutQ (k∗ H)δ of outer automorphisms of the Q-interior ˆ δ (cf. 1.13 and 1.27) of the block (b, H) ˆ is a p -group source algebra (k∗ H) (cf. Proposition 14.9 in [42]). On the other hand, for any element z of CGˆ (Qδ ) = CGˆ (Q, f ) = CHˆ (Q)·CP (Q)

15.13.4,

ˆ δ = j(k∗ H)j ˆ where j ∈ δ (cf. 1.12), we have j z = j a for a suitif (k∗ H) ˆ Q and then za−1 determines an element able invertible element a ∈ (k∗ H) + Q (k∗ H) ˆ δ ; it is clear that its image in Aut ˆ δ only depends of AutQ (k∗ H) on z and that this correspondence deﬁnes a group homomorphism from ˆ δ ) ; hence, this group homomorphism is the + Q (k∗ H) CGˆ (Qδ )/CHˆ (Q) to Aut ˆ δ. trivial one and we may assume that za−1 centralizes (k∗ H) ˆ = H·C ˆ P (Q) , K ˆ stabilizes all the isomorphism In particular, since K ˆ ˆ H ˆ is a p-group, any classes of simple k∗ Hb-modules (cf. 1.13) and, since K/ ˆ ˆ simple k∗ Hb-module can be extended to K in a unique way (cf. 1.25); moreˆ b) and Irrk (K, ˆ b) of over, this bijection between the respective sets Irrk (H,

15. Reduction results for Grothendieck groups

249

ˆ and k∗ K-modules ˆ isomorphism classes of simple k∗ H(cf. 1.25) is clearly compatible with the action of ˆ K ˆ ∼ ˆ C /H ˆ C = G/ =H

15.13.5.

Since C is a p -group, the right-hand isomorphism in 15.13.3 follows easily from the so-called Cliﬀord Theory (cf. §17 in [31, Ch. V]) applied to the ˆ of G ˆ and H ˆ of H ˆ C ; indeed, we get a canonical bijecnormal k ∗ -subgroups K ˆ C , b) tion, preserving the dimensions, between the set of elements of Irrk (H ˆ is a multiple of χ ∈ Irrk (H, ˆ b) , and the set such that their restriction to H ˆ ˆ of elements of Irrk (G, b) such that their restriction to H is a multiple of the ˆ b) extending χ . element χ ˆ of Irrk (K, On the other hand, it follows from Proposition 15.10, applied to the ˆ C , H) ˆ , that c = b is also a block of H ˆ C and that H is a normal pair (H C subcategory of F(b,Hˆ C ) ; then, since H (Q) = F(Q) , it follows from Proposisc

tion 12.8 that we have HC = F(b,Hˆ C ) ; moreover, since (HC ) is a subcategory of F

nc

, for any pair of HC -selfcentralizing subgroups R and T of Q , we have HC (R, T ) ⊂ F(R, T )

15.13.6;

ˆ = H ˆ C ·CP (Q) , respectively denoting by g and h the blocks but, since G of CHˆ (R) and CHˆ (T ) such that (Q, f ) contains (R, g) and (T, h) , for any ˆ fulﬁlling (T, h) ⊂ (R, g)x we have x = y·z for suitable y ∈ H ˆ C and x∈G y z ∈ CP (Q) , and therefore we still have (T, h) ⊂ (R, g) , both elements inducing the same group homomorphism since z centralizes (T, h) (cf. 1.15); hence, we have the equality HC (R, T ) = F(R, T )

15.13.7.

In particular, for any HC -selfcentralizing subgroup R of Q , we have H (R) = F(R) and therefore the natural map 15.12.1 here becomes a natural isomorphism (HC )sc ∼ F nc ◦ ch∗ (j) νˆ : aut 15.13.8. = aut C

Hence, since H fulﬁlls condition 15.14.2 in Lemma 15.14 below and j induces S nc an equivalence between HC and the full subcategory F of F over the set S of H-selfcentralizing subgroups of Q , from this lemma and from the inverse limit of the natural isomorphism gk ∗ νˆ we get the following O-module isomorphisms F nc ) ∼ F nc ◦ ch∗ (j) ∼ (HC )nc ) Gk (F, aut = lim gk ◦ aut = Gk (HC , aut ←−

We are done.

15.13.9.

250

Frobenius categories versus Brauer blocks

Lemma 15.14 Let P be a ﬁnite p-group, F a Frobenius P -category, S an F-stable subgroup of P and F nc : ch∗ (F ) −→ k ∗ -Gr aut nc

15.14.1

a functor lifting autF nc . Assume that the following condition holds 15.14.2 We have a normal Frobenius S-subcategory H of F such that any H-selfcentralizing subgroup of S is F-nilcentralized. nc

S

Then, denoting by F the full subcategory of F over the set S of H-self S : ch∗ (F S ) → k ∗ -Gr the correscentralizing subgroups of S , and by aut F ponding restricted functor, the restriction induces an O-module isomorphism F nc ) ∼ S Gk (F, aut = lim (gk ◦ aut F

←−

15.14.3.

Proof: Let X be the set of F-nilcentralized subgroups R of P such that R ∩ S is H-selfcentralizing (cf. 4.8); then, for any F-nilcentralized subgroup Q of P (cf. 4.3), choose ψ ∈ F(P, Q) such that T = ψ(Q ∩ S) is fully normalized in F (cf. Proposition 2.7); moreover, considering the Frobenius NP (T )-category N = NF (T ) (cf. Proposition 2.16), up to replacing U = ψ(Q) by an N -isomorphic subgroup, we may assume that U is fully normalized in N (cf. Proposition 2.7). Since T is also fully normalized in H , T ·CS (T ) is H-selfcentralizing (cf. Proposition 2.11); in particular, R = U ·CS (T ) belongs to X and, denoting by ϕ : Q → R the homomorphism determined by ψ , we claim that ϕ ◦ F(Q) ⊂ F(R) ◦ ϕ

15.14.4,

so that X fulﬁlls condition 14.6.1. Indeed, consider the respective Frobenius U NNP (T ) (U )- and R-categories NN (U ) and C = U ·CH (T ) = NH (T ) (cf. Proposition 2.16); it is easily checked that C is normal in NN (U ) (cf. 12.5); thus, it follows from Proposition 12.8 that we have

NN (U ) (R, U ) = NN (U ) (R) ◦ C(R, U )

15.14.5.

ϕ On the other hand, for any element σ ∈ F(Q) we have τ = σ ∈ F(U ) fulﬁlling τ ϕ(u) = ϕ σ(u) for any u ∈ Q and, since T = U ∩ S , the element of F(R, U ) determined by τ and the inclusion U ⊂ R belongs to the left-hand member of equality 15.14.5. Hence, there are ρ ∈ NN (U ) (R) and θ ∈ C(R, U ) fulﬁlling τ ϕ(u) = (ρ ◦ θ) ϕ(u) for any u ∈ Q and, since τ (U ) = U , we have θ(U ) = U ; but, since H is normal ' in F , (for any p -subgroup K of C(U ) we have [K, U ] ⊂ T and therefore K, [K, U ] = {1} which forces K = {idU } (cf. Theorem in [28, Ch. 5]); consequently, θ is

15. Reduction results for Grothendieck groups

251

induced by some v ∈ R and ﬁnally, denoting by κR (v) the conjugation by v on R , we get ϕ σ(u) = ρ ◦ κR (v) ϕ(u) 15.14.6 for any u ∈ Q , which proves inclusion 15.14.4. In conclusion, it follows from Proposition 14.6 that we have an O-module isomorphism F nc ) ∼ X Gk (F, aut 15.14.7, = lim (gk ◦ aut F ←−

X : ch (F ) → k -Gr denotes the corresponding restricted functor. where aut F ∗

∗

X

S

X

Moreover, according to our deﬁnition, the inclusion functor iS : F → F X S admits a section sS : F → F mapping any Q ∈ X on Q ∩ S and any X F -morphism on the corresponding restriction; thus, the O-module homomorphism induced by iS (cf. A3.9 and A3.11) X −→ lim gk ◦ aut S lim gk ◦ aut 15.14.8 F F ←−

←−

is surjective. Actually, we claim that it is also injective; indeed, if X = (XQ )Q∈X ˆ X , namely if XQ ∈ Gk F(Q) for any Q ∈ X and belongs to lim gk ◦ aut F ←−

we have (cf. A3.6 and A3.11) (po

d0

ch(F

X

)

)∗ (gk ◦a ut

F

X)

X

(X) = 0

15.14.9,

X

it suﬃces to consider the F -chain q : ∆1 → F mapping 0 on Q ∩ S , 1 on Q and 0 •1 on the inclusion map Q ∩ S → Q , to get 0 = resa ut

F

0 X (idq◦δ 0 ,δ0 ) 0

(XQ ) − resa ut

F

0 X (idq◦δ 0 ,δ1 ) 1

(XQ∩S )

15.14.10;

ˆ but, since S is F-stable, F(Q) stabilizes Q ∩ S and therefore the restriction map resa is an isomorphism; hence, if X also belongs to the 0 ut (id 0 ,δ ) F

X

q◦δ

0

0

kernel of the O-module homomorphism 15.14.8 then we obtain XQ = 0 . We are done. 15.15 In order to discuss statements analogous to Proposition 15.13 when CGˆ (Q, f )/CHˆ (Q) is a p -group, we need the following auxiliary result ˆ and on k∗ Hc ˆ on the relationship between the local pointed groups on k∗ Gb (cf. 1.28), which essentially depends on Fong’s reduction for interior algebras ˆ , the sta(cf. §3 in [53]); as in 15.5 above, if Rε is a local pointed group on k∗ H ˆ bilizer CG (Rε ) of ε in CG (R) acts on the simple algebra (k∗ H)(Rε ) (cf. 1.12), determining a k ∗ -group CˆG (Rε ) (cf. 1.25), and then we set (cf. 15.5.4) ˆ G ◦ ˆ CH 15.15.1 ∆Rε CH (R) ˆ (Rε ) ∗ CG (Rε ) ˆ (Rε ) = CG

252

Frobenius categories versus Brauer blocks

where, as above, ∆Rε is the group homomorphism mapping x ∈ CH (R) on ˆ CˆG (Rε )◦ where x the product of the two images of x ˆ and x ˆ−1 in CGˆ (Rε )× ˆ is ˆ a lifting to CHˆ (Rε ) = CH (Rε ) of x . ˆ H ˆ is a p -group. For Lemma 15.16 In our standard setting, assume that G/ ˆ , ε splits into a set {(ε, ϕ)} any local pointed group Rε on k∗ H ˆ ϕ∈P(k∗ C G (Rε )) ˆ H ˆ G ˆ of local points of R on k∗ G and, for any ϕ ∈ P k∗ CHˆ (Rε ) , we have a CGˆ (R)-interior k-algebra isomorphism CG ˆ (R) ∼ ˆ (k∗ G)(R (ε,ϕ) ) = IndC ˆ (Rε ) G

ˆ G ˆ k∗ CH ˆ (Rε ) (ϕ) ⊗k (k∗ H)(Rε )

15.16.1.

ˆ Moreover, the Brauer H-pair determined by Rε is selfcentralizing if and only ˆ if the Brauer G-pair determined by R(ε,ϕ) is selfcentralizing, and then we have Op EH (Rε ) = Op EG (R(ε,ϕ) ) 15.16.2. ˆ ⊂G ˆ induces an injective k-algebra homomorphism Proof: The inclusion H k∗ CHˆ (R) −→ k∗ CGˆ (R)

15.16.3

and, since CGˆ (R) acts on the semisimple k-algebra k∗ CHˆ (R)/J k∗ CHˆ (R) , setting J = J k∗ CHˆ (R) it is easily checked from Proposition 3.2 in [53] that C (R) ˆ k∗ CGˆ (R) k∗ CGˆ (R)·J ∼ IndCGˆˆ (Rε ) Sϕ ⊗k (k∗ H)(R = ε) G

15.16.4,

(ε,ϕ)

where ε runs over a set of representatives for the set of orbits of CGˆ (R) on ˆ (cf. 1.18.1) and for such an ε , since the set of local points of R on k∗ H ˆ G CGˆ (Rε )/CHˆ (R) is a p -group, ϕ runs over the set of points of k∗ CH ˆ (Rε ) ˆ

G (cf. 1.12) and Sϕ denotes the corresponding CH ˆ (Rε )-interior simple algebra (cf. 1.27 and 15.15.1). Moreover, since the right-hand member of isomorphism 15.16.4 is semisimple, we get the ﬁrst statement and isomorphism 15.16.1. ˆ On the other hand, if the Brauer H-pair determined by Rε is selfcentralizing then the corresponding simple k∗ C¯Hˆ (R)-module is also projective (cf. 1.17 and 4.7) and therefore, since CGˆ (R)/CHˆ (R) is a p -group, according to isomorphism 15.16.1, the simple k∗ C¯Gˆ (R)-module which corresponds to ˆ the local pointed group R(ε,ϕ) is projective too, so that the Brauer G-pair determined by R(ε,ϕ) is selfcentralizing too (cf. 1.17 and 4.7).

15. Reduction results for Grothendieck groups

253

Conversely, the projectivity of the simple k∗ C¯Gˆ (R)-module correspondˆ forces the projectivity of the siming to the local pointed group R(ε,ϕ) on k∗ G ˆ. ple k∗ C¯Hˆ (R)-module corresponding to the local pointed group Rε on k∗ H ˆ p G Finally, since O NH (Rε ) acts trivially on k∗ CHˆ (Rε ) , it ﬁxes any point ϕ and therefore Op EHˆ (Rε ) is a normal subgroup of EGˆ (R(ε,ϕ) ) of index prime to p ; hence, Op EHˆ (Rε ) contains Op EGˆ (R(ε,ϕ) ) and therefore we get Op EHˆ (Rε ) = Op EGˆ (R(ε,ϕ) ) 15.16.5. We are done. ˆ H ˆ is a p -group; 15.17 Now, in our standard setting, assume that G/ ˆ = H·C ˆ ˆ (Q, f ) and C = G/ ˆ K ˆ , and assume that moreover, set K G ∗ 15.17.1 The canonical k -group homomorphism ∗ CHˆ (Q) −→ k∗ C¯Hˆ (Q)f¯ can be extended to NKˆ (Q, f ) . In this situation, setting D = CG (Q, f )/CH (Q) it is clear that G ∼ ∗ CH ˆ (Q, f ) = k × D ˆ

15.17.2,

ˆH (Q, f ) . In particular, the group and that D acts trivially on E NKˆ (Q, f ) = NHˆ (Q, f )·CGˆ (Q, f )

15.17.3

ˆ G ˆ ˆ ˆ (Q, f ) by the acts trivially on P k∗ CH ˆ (Q, f ) (cf. 1.12) and, since G = H·NG Frattini argument (cf. 1.15), the group C ∼ = NGˆ (Q, f )/NKˆ (Q, f ) acts on the ˆ G set P k∗ CHˆ (Q, f ) . ˆ 15.18 On the other hand, denote by δ the local point of Q on k∗ Hc ˆ ˆ determined by f (cf. 1.17) and choose j ∈ δ such that j(k∗ H)j = (k∗ Hc)δ ˆ δ , recall that the exact sequence (cf. 1.12); setting B = (k∗ Hc) ˆH (Qδ )◦ −→ 1 1 −→ Q· j + J(B Q ) −→ NB ∗ (Q·j) −→ E

15.18.1

is split; more precisely, it follows from Theorem 12.8 in [51] that there is a unique (B Q )∗ -conjugacy class of sections† ˆH (Qδ )◦ −→ NB ∗ (Q·j) σ ˆδ : E †

15.18.2

Note that our notation in [51] and here do not agree; precisely, deﬁnition 12.6.2 in [51]

corresponds to deﬁnition 1.29.2 above.

254

Frobenius categories versus Brauer blocks

ˆH (Qδ )◦ (cf. 1.29). of the canonical k ∗ -group homomorphism NB ∗ (Q·j) → E Similarly, since we are assuming P = Q , it follows from Lemma 15.16 that ˆ G the block e of CGˆ (Q) (cf. 15.1) determines a point ϕ of P k∗ CH ˆ (Q, f ) such ˆ {b} (cf. 1.13) and then, choosing that Q(δ,ϕ) is a defect pointed group of G ˆ ˆ (δ,ϕ) and setting A = (k∗ Gb) ˆ (δ,ϕ) , there i ∈ (δ, ϕ) such that i(k∗ Gb)i = (k∗ Gb) is a unique (AQ )∗ -conjugacy class of sections ˆG (Q(δ,ϕ) )◦ −→ NA∗ (Q·i) σ ˆ(δ,ϕ) : E

15.18.3

ˆG (Q(δ,ϕ) )◦ (cf. 1.29 of the canonical k ∗ -group homomorphism NA∗ (Q·i) → E above and Theorem 12.8 in [51]). ˆ H ˆ is a p -group Proposition 15.19 In our standard setting, assume that G/ and that condition 15.17.1 holds. With the notation above, B is isomorphic ˆG (Q(δ,ϕ) )◦ -stable Q-interior subalgebra of A and, up to suitable idento an E ˆH (Qδ )◦ and we have tiﬁcations, σ ˆ(δ,ϕ) coincides with σ ˆδ over E A=

B·ˆ σ(δ,ϕ) (ˆ x)

15.19.1

x ˆ

ˆG (Q(δ,ϕ) )◦ runs over a set of representatives for the quotient where x ˆ ∈ E ˆG (Q(δ,ϕ) )◦ /E ˆH (Qδ )◦ . In particular, denoting by C(δ,ϕ) the stabilizer of E (δ, ϕ) in C , we have HC(δ,ϕ) = F . Proof: For any x ˆ ∈ NGˆ (Qδ ) there is an invertible element axˆ ∈ B Q such that −1 we have x ˆ·j·ˆ x = axˆ j(axˆ )−1 , so that (axˆ )−1 ·ˆ x centralizes j ; but, since we ˆ ˆ have G = H·NGˆ (Q, f ) by the Frattini argument, we get ˆ= k∗ G

ˆx ˆ k∗ H

15.19.2

x ˆ∈X

where X is a set of representatives for NGˆ (Qδ )/NHˆ (Qδ ) in NGˆ (Qδ ) ; consequently, we still get ˆ = j(k∗ G)j

−1 ˆ j(k∗ H)j(a ·ˆ x x ˆ)

15.19.3;

x ˆ∈X

ˆ , it follows actually, setting dxˆ = j(axˆ )−1 ·ˆ x which is an element of j(k∗ G)j from 15.18 that we can modify our choice of axˆ in such a way that dxˆ nor ˆ ˆ (Qδ )◦ . malizes σ ˆδ E H ˆ G ˆ ˆ (Qδ ) , it follows Moreover, since CH ˆ (Q, f ) (cf. 15.5.4) acts trivially on EH from Proposition 14.9 in [42] that dzˆ has an inner action on B for any zˆ

15. Reduction results for Grothendieck groups

255

in CGˆ (Qδ ) ; that is to say, we can modify our choice of azˆ in such a way that dzˆ centralizes B . In this situation, denoting by Z a set of representatives zˆ ∈ CGˆ (Qδ ) for D = CGˆ (Qδ )/CHˆ (Q) , it is easily checked that Cj(k∗ G)j Z(B)dzˆ 15.19.4; ˆ (B) = zˆ∈Z

ˆ of k ∗ -quotient D actually, since D is a p -group, for a suitable k ∗ -group D we get an isomorphism ∼ ˆ Cj(k∗ G)j ˆ (B) = Z(B) ⊗k k∗ D

15.19.5.

ˆ G Consequently, the point ϕ ∈ P k∗ CH ˆ (Q, f ) (cf. 1.12) determined by e ˆ (cf. 15.1) determines a direct factor Z(B) ⊗k (k∗ D)(ϕ) of Cj(k∗ G)j ˆ (B) (cf. 15.17.2) and, for any x ˆ ∈ NGˆ (Q(δ,ϕ) ) , we may assume that dxˆ normalizes this direct factor; then, since we may assume that dxˆ induces a p -action ˆ on (k∗ D)(ϕ) , it is not diﬃcult to prove that dxˆ centralizes a primitive idemˆ potent in Z(B)⊗k (k∗ D)(ϕ) ; since all the primitive idempotents in this direct factor are pairwise conjugate, we can modify our choice in such a way that, for any x ˆ ∈ NGˆ (Q(δ,ϕ) ) , dxˆ centralizes the same primitive idempotent i ˆ in Z(B) ⊗k (k∗ D)(ϕ) . Finally, it is clear that the multiplication by i induces a Q-interior algebra isomorphism B ∼ = Bi mapping j on i and, since we still have ˆ

Q·EH (Qδ ) iCj(k∗ G)j ˆ (B)i = Z(B)i and dx ˆ dx ˆ i ∈ dx ˆx ˆ iB

for any x ˆ, x ˆ ∈ NGˆ (Q(δ,ϕ) ) , it is easily checked that ˆ = A = i(k∗ G)i Bdxˆ i

15.19.6

15.19.7

x ˆ

where x ˆ ∈ NGˆ (Q(δ,ϕ) ) runs over a set of representatives for C(δ,ϕ) . Now, it is quite clear that in equality 15.19.7 we can replace dxˆ i by ˆG (Q(δ,ϕ) )◦ (cf. 15.18.3), that σ ˆ(δ,ϕ) (x ˆ˜) where x ˆ˜ denotes the image of x ˆ in E ˆG (Q(δ,ϕ) )◦ stabilizes Bi and contains the k ∗ -group E ˆH (Qδ )◦ (cf. ProposiE tion 6.21 in [42]), and that we have ˆG (Q(δ,ϕ) )◦ /E ˆH (Qδ )◦ ∼ E = C(δ,ϕ) ˆG (Q(δ,ϕ) )◦ = σ ˆH (Qδ )◦ ˆ(δ,ϕ) E Bi ∩ σ ˆ(δ,ϕ) E

15.19.8

which proves equality 15.19.1 and allows us to assume that σ ˆδ coincides with ˆH (Qδ ) . Since the Frobenius Q-category H is the restriction of σ ˆ(δ,ϕ) to E normal in F , the last statement follows from Propositions 12.8 and 12.12. We are done.

256

Frobenius categories versus Brauer blocks

ˆ H ˆ is a p -group Corollary 15.20 In our standard setting, assume that G/ ˆ be a k ∗ -suband that condition 15.17.1 holds. With the notation above, let G ˆ such that f BrQ (b ) = 0 , and f a ˆ containing H ˆ , b a block of G group of G ˆ = K ˆ ∩G ˆ block of CGˆ (Q) fulﬁlling f f = 0 and f BrQ (b ) = f . Setting K ˆ /K ˆ , and denoting by ϕ ∈ P k∗ C Gˆ (Q, f ) the point determined and C = G ˆ H

by f , assume that C(δ,ϕ ) = C(δ,ϕ) . Then Q(δ,ϕ ) is a defect pointed group ˆ of G , the Frobenius Q-category F = F ˆ coincides with F , we have {b }

(b ,G )

F nc ∼ F nc inducing an O-module isomorphism a natural isomorphism aut = aut F nc ) ∼ F nc ) Gk (F, aut = Gk (F , aut

15.20.1,

and the restrictions to the respective source algebras induce an O-module isomorphism ˆ b) ∼ ˆ , b ) Gk (G, 15.20.2. = Gk (G ˆ b Proof: It follows from Proposition 15.19 that the block algebras k∗ G ˆ have isomorphic source algebras and that F = HC(δ,ϕ) = F ; since and k∗ Gb ˆ b) F nc , Gk (F, aut F nc ) and Gk (F , aut F nc ) , and also Gk (G, autF nc and aut ˆ , b ) are completely determined from the respective source algebras and Gk (G (cf. Theorem 3.1 in [39] and Proposition 6.21 in [42]), we are done. ˆ C the converse 15.21 Always in our standard setting, let us denote by H C ˆ of a cyclic p -subgroup C of G/ ˆ H ˆ , and by H its k ∗ -quotient; image in G recall that Gk (C) and Gk (H C ) have ring structures and that the restriction determines a ring homomorphism Gk (C) −→ Gk (H C )

15.21.1,

ˆ C ) becomes a Gk (C)-module (cf. 5.13 in [42]); since clearly we so that Gk (H ∗ ˆ C ) and it is elehave C = Hom(C, k ∗ ) ⊂ Gk (C) , the group C ∗ acts on Gk (H mentary to check that the restriction determines an O-module isomorphism ˆ C )C ∗ ∼ ˆ C Gk (H = Gk (H)

15.21.2.

15.22 For our purposes, it is very useful to consider the intersection — ˆ C ) and called C-residual Grothendieck group of H ˆ C — of denoted by RHˆGk (H the kernels of all the O-module homomorphisms determined by the restriction ˆ C ) −→ Gk (H ˆ D) Gk (H

15.22.1

ˆ where D runs over the set of proper subgroups of C . Note that if CGˆ (Qδ ) ⊂ H D ˆ then it follows from Corollary 15.13 that (c, H ) is a block for any subgroup D of C (cf. 1.10) and therefore the C-residual Grothendieck group ˆ C , c) of (c, H ˆ C ) makes sense. RHˆGk (H

15. Reduction results for Grothendieck groups

257

15.23 Actually, the intersection — denoted by RGk (C) — of the kernels of all the O-module homomorphisms Gk (C) → Gk (D) determined by the restriction, where D runs over the set of proper subgroups of C , still makes sense and it is isomorphic to the following residue class ring of Gk (C) (cf. Proposition 3.4 in [42]) RGk (C) ∼ IndC 15.23.1, = Gk (C) D Gk (D) D

where D runs again over the set of proper subgroups of C . Now, it is quite ˆ C ) becomes a RGk (C)-module and then it is easily checked clear that RHˆGk (H that ˆ C) = ˆ C) R Gk (H RGk (C)·X ∼ 15.23.2, = RGk (C) ⊗G (C) Gk (H ˆ H

k

X

where X runs over a set of representatives for the set of C ∗ -orbits on the ˆ C -modules which remain simple set of isomorphism classes of the simple k∗ H ˆ . Moreover, it is clear that C acts on Gk (H ˆ D ) for any subrestricted to H group D of C , and it follows from isomorphism 15.21.2 that the restriction induces an O-module isomorphism ˆ C) ∼ ˆ D )C RGk (D) ⊗Gk (C) Gk (H 15.23.3; = RHˆGk (H

∼ hence, since Gk (C) = D RGk (D) (cf. Proposition 3.4 in [42]), we get ˆ C) ∼ ˆ D )C Gk (H RHˆGk (H 15.23.4 = D

where D runs over the set of subgroups of C . 15.24 More generally, let us denote by CG/ ˆ H ˆ the category where the ˆ ˆ objects are the cyclic p -subgroups C of G/H and the morphisms from C to ˆ H ˆ are the elements x ˆ H ˆ fulﬁlling another cyclic p -subgroup C of G/ ¯ ∈ G/ ¯ x C ⊂ C , with the composition determined by the product in this quotient. Clearly we have a contravariant functor ˆ • ) : C ˆ ˆ −→ O-mod Gk (H 15.24.1 G/H

ˆ C ) and the C ˆ ˆ -morphism x mapping C on Gk (H ¯ : C → C on the O-module G/H ˆ C ) → Gk (H ˆ C ) determined by the restriction via the homomorphism Gk (H ˆ lifting x conjugation by an element x ∈ G ¯ ; moreover, the restriction induces an O-module homomorphism ˆ −→ lim Gk (H ˆ •) Gk (G) 15.24.2. ←−

Note that the systematic consideration of isomorphisms 15.23.4 proves that G/ ˆ H ˆ ˆ •) ∼ ˆ C) lim Gk (H RHˆGk (H 15.24.3, = ←−

C

ˆ H ˆ. where C runs over the set of cyclic p -subgroups of G/

258

Frobenius categories versus Brauer blocks

Proposition 15.25 With the notation above, the restriction induces an O-module isomorphism ˆ ∼ ˆ •) Gk (G) 15.25.1. = lim Gk (H ←−

Proof: Borrowing our notation in 14.12, 14.13 and 14.14, we have a canonical O-module isomorphism (cf. isomorphism 14.14.2)

ˆ ∼ Gk (G) =

ˆh , G), ˆ O G FctUh Monk∗ (U

15.25.2;

h∈N−pN

in particular, for any h ∈ N − pN , denoting by ˆ• ˆh , H ˆ • ), O H : C ˆ ˆ −→ O-mod FctUh Monk∗ (U G/H

15.25.3

ˆ H ˆ on the the contravariant functor mapping any cyclic p -subgroup C of G/ ˆC H ˆh , H ˆ C ), O O-module FctUh Monk∗ (U and any CG/ ˆ H ˆ -morphism on the obvious O-module homomorphism, we have a natural isomorphism ˆ •) ∼ Gk (H =

ˆ• ˆh , H ˆ • ), O H FctUh Monk∗ (U

15.25.4.

h∈N−pN

On the other hand, for any h ∈ N − pN it is clear that (cf. 1.9) ˆh , G) ˆ = Monk∗ (U

ˆ ˆ ˆC ιG ˆ C ◦ Monk∗ (Uh , H ) H

15.25.5,

C

ˆ H ˆ , and therefore it is where C runs over the set of cyclic p -subgroups of G/ easily checked that ˆh , G), ˆ O G FctUh Monk∗ (U ˆ• ∼ ˆh , H ˆ • ), O H = lim FctUh Monk∗ (U

15.25.6.

←−

We are done. ˆ . Then, Corollary 15.26 In our standard setting, assume that CGˆ (Qδ ) ⊂ H we have b = c and the restriction induces an O-module isomorphism ˆ b) ∼ Gk (G, =

G/ ˆ H ˆ ˆ C , c) RHˆGk (H

C

ˆ H ˆ. where C runs over the set of cyclic p -subgroups of G/

15.26.1

15. Reduction results for Grothendieck groups

259

Proof: From isomorphisms 15.24.3 and 15.25.1 we get an O-module isomorphism G/ ˆ H ˆ ˆ ∼ ˆ C) Gk (G) RHˆGk (H 15.26.2 = C

and it is easily checked that this isomorphism is compatible with the “multiplication” by b = c (cf. Proposition 15.10). F nc ) , in 15.27 In order to obtain an analogous isomorphism for Gk (F, aut ˆ , so that b = c (cf. Proposiour standard setting assume that CGˆ (Q, f ) ⊂ H tion 15.10). As in 15.13.1 above, by the Frattini argument and this inclusion we get ˆ H ˆ ∼ G/ 15.27.1. = NGˆ (Q, f )/NHˆ (Q, f ) ∼ = F(Q)/H(Q) Let us identify both ends with each other; then, it follows from Proposiˆ H ˆ , we have a Frobenius tion 12.12 that, for any cyclic p -subgroup C of G/ C Q-category H containing H as a normal Q-subcategory and from Proposiˆ C and that then we have (cf. 3.2 and tion 15.10 that c is also a block of H Proposition 12.8) HC = F(c,Hˆ C ) 15.27.2. 15.28 Now, as in 15.11 above, we have an inclusion functor sc

jC : (HC ) −→ F

nc

15.28.1

and then a functor and a natural map sc nc ch∗ (jC ) : ch∗ (HC ) −→ ch∗ (F ) νC : aut(HC )sc −→ autF nc ◦ ch∗ (jC )

15.28.2;

moreover, it follows from Proposition 15.12 that this natural map can be lifted to a new natural map (HC )sc −→ aut F nc ◦ ch∗ (jC ) νˆC : aut sc

15.28.3;

sc

ˆ C (r) instead of as usual, for any (HC ) -chain r : ∆n → (HC ) , we write H (HC )sc (r, ∆n ) . Finally, it follows from Corollary 14.7 that the natural map aut gk ∗ νˆC determines an O-module homomorphism C nc nc resF HC : Gk (F, autF ) −→ Gk (H , aut(HC ) )

15.28.4.

ˆ H ˆ containing C then 15.29 Similarly, if C is a cyclic p -subgroup of G/ C C H is a subcategory of H and, as above, we successively get an inclusion functor and a natural map

sc

sc

C C jC C : (H ) −→ (H )

C (HC )sc −→ aut (HC )sc ◦ ch∗ (jC νˆC : aut C )

15.29.1;

260

Frobenius categories versus Brauer blocks

it is clear that jC ◦ jC C = jC and, once again, it follows from the uniqueness part of Theorem 11.32, suitably restated over the set of H-selfcentralizing subgroups of Q , that we get a natural isomorphism C ∼ νˆC ∗ ch∗ (jC ˆC 15.29.2. = νˆC C ) ◦ν C Then, it follows again from Corollary 14.7 that the natural map gk ∗ νˆC determines an O-module homomorphism C

C C nc nc resH HC : Gk (H , aut(HC ) ) −→ Gk (H , aut(HC ) )

15.29.3;

ﬁnally, it follows from isomorphism 15.29.2 that we have C

F F resH HC ◦ resHC = resHC

15.29.4.

15.30 Moreover, we claim that the quotient F(Q)/H(Q) acts, up to natural isomorphisms, on the family of categories {HC }C , where C runs over the set of cyclic p -subgroups of F(Q)/H(Q) ; explicitly, for any σ ∈ F(Q) , σ it is clear that the group automorphism σ : Q ∼ = Q is (HC , HC )-functorial (cf. 12.1), where σ ¯ denotes the image of σ in F(Q)/H(Q) , so that it determines a Frobenius functor (cf. 12.1.2) σ

fσ : HC −→ HC

15.30.1,

which is naturally isomorphic to the identity functor whenever σ belongs σ sc sc to H(Q) (cf. 12.1). As above, fσ induces a functor jσ from (HC ) to (HC ) and σ still determines a natural isomorphism (HC σ )sc ∼ (HC )sc ◦ ch∗ (jσ ) νˆσ : aut = aut

15.30.2,

so that we get an O-module isomorphism (HC )sc ) ∼ (HC )sc ) Gk (HC , aut = lim (gk ◦ aut ←− ∼ (HC )sc ◦ ch∗ (jσ ) = lim gk ◦ aut

15.30.3

←−

∼ (HC σ )sc ) ∼ (HC σ )sc ) = Gk (HC , aut = lim (gk ◦ aut σ

←−

which is the identity whenever σ ∈ H(Q) (cf. Proposition A4.5). 15.31 That is to say, as in 15.24 above, we get a contravariant functor (H• )sc ) : C ˆ ˆ −→ O-mod Gk (H• , aut G/H

15.31.1

ˆ H ˆ on Gk (HC , aut (HC )sc ) and any which maps any cyclic p -subgroup of G/ ¯ : C → C on the O-module homomorphism CG/ ˆ H ˆ -morphism x

(HC )sc ) −→ Gk (HC , aut (HC )sc ) res(C,x,C ) : Gk (HC , aut

15.31.2

15. Reduction results for Grothendieck groups

261

obtained as the composition of the O-module isomorphism above (cf. 15.30.3) x (HC )sc ) ∼ (HC x )sc ) Gk (HC , aut = Gk (HC , aut

15.31.3

C x

(cf. 15.29.3); the proof of the with the O-module homomorphism resH HC functoriality of this correspondence is straightforward. Moreover, since we clearly have a natural isomorphism jC ◦ jx¯ ∼ = jC x , it follows again from Proposition A4.5 that = resF res(C,x,C ) ◦ resF HC HC

15.31.4,

so that we ﬁnally obtain an O-module homomorphism F nc ) −→ lim Gk (H• , aut (H• )sc ) Gk (F, aut ←−

15.31.5.

ˆ . Then, Theorem 15.32 In our standard setting, assume that CGˆ (Q, f ) ⊂ H F the O-module homomorphisms resHC , where C runs over the set of cyclic ˆ H ˆ , induce an O-module isomorphism p -subgroups of G/ F nc ) ∼ (H• )nc ) Gk (F, aut = lim Gk (H• , aut ←−

15.32.1. S

Proof: Denoting by S the set of H-selfcentralizing subgroups of Q and by F the full subcategory of F over S , it follows from Lemma 15.14 that F nc ) ∼ S Gk (F, aut 15.32.2; = lim (gk ◦ aut F ←−

thus, denoting by j : F

S

→F

nc

the inclusion functor and setting

S = aut F nc ◦ ch∗ (j) aut F

15.32.3,

it suﬃces to prove that the O-module homomorphisms resF HC , where C runs ˆ ˆ over the set of cyclic p -subgroups of G/H , induce an injective O-module homomorphism S ) −→ (HC )sc ) lim (gk ◦ aut lim (gk ◦ aut 15.32.4 F ←−

C∈Obj(CG/ ˆ H ˆ)

←−

such that the image coincides with the corresponding inverse limit. Let Z = (ZR )R∈S be an element of the kernel of homomorphism 15.32.4; ˆ that is to say, for any R ∈ S , ZR is an element of Gk F(R) which has restric C ∼ F(Q)/H(Q) ; ˆ (R) , for any cyclic p -subgroup C of G/ ˆ H ˆ = tion zero in Gk H we claim that ZR = 0 . Indeed, since Z is stable by F-isomorphisms, we may assume that R is fully highnormalized in H (cf. 12.10) and then, since R is F-nilcentralized (cf. Proposition 15.10), it follows from Proposition 12.11

262

Frobenius categories versus Brauer blocks

that the restriction induces a group isomorphism F(Q)R /H(Q)R ∼ = F(R)/H(R)

15.32.5,

where F(Q)R and H(Q)R denote the corresponding stabilizers of R ; hence, ˆ of F(R) ˆ for any k ∗ -subgroup D with a cyclic k ∗ -quotient D of order prime ˆ is to p , we can ﬁnd a cyclic p -subgroup C of F(Q)/H(Q) such that D ˆ C (R) and therefore ZR still has restriction zero in Gk (D) ˆ . contained in H Now, the claim follows from isomorphism 14.14.2. From equalities 15.31.4, we already know that the image of homomorphism 15.32.4 is contained in the corresponding inverse limit. Conversely, let be an element of this limit; that is to say, for any cyclic Y = (YC )C∈Obj(CG/ ˆ H ˆ) ∼ G/ ˆ H ˆ , we have YC = (YC,R )R∈S where, for p -subgroup C of F(Q)/H(Q) = C ˆ (R) and, for another R ∈ S and any R ∈ S , YC,R is an element of Gk H C ˆ (r) we have any ψ ∈ HC (R , R) , in Gk H res a ut

sc (id ,δ 0 ) r◦δ 0 1 (HC ) 1

(YC,R ) = res a ut

sc (id ,δ 0 ) r◦δ 0 0 (HC ) 0

sc

(YC,R )

15.32.6,

sc

where r : ∆1 → (HC ) is the obvious (HC ) -chain deﬁned by ψ . Moreover, for any σ ∈ F(Q) , we have res(C σ ,σ,C) (YC ) = YC σ (cf. 15.31.2) or, equivalently, for any R ∈ S we still have res (ˆνσ )R (YC,R ) = YC σ ,σ−1 (R) where σ ¯ denotes the image of σ in F(Q)/H(Q) and ˆ C σ σ −1 (R) ∼ ˆ C (R) (ˆ νσ )R : H =H

15.32.7,

15.32.8

is the k∗ -group isomorphism determined by natural isomorphism νˆσ above; ﬁnally, for any subgroup D of C and any R ∈ S , we have ˆ C (Q) H

resHˆ D (Q) (YC,Q ) = YD,Q

15.32.9.

Thus, ﬁxing R ∈ S fully highnormalized in H , for any cyclic p -sub C ˆ (R) and, group C of F(Q)R /H(Q)R we have the element YC,R of Gk H moreover, for any σ ∈ F(R) and any subgroup D of C , we already know that res (ˆνσ )R (YC,R ) = YC σ ,R

and

ˆ C (R) H

resHˆ D (R) (YC,R ) = YD,R

15.32.10;

consequently, it follows from Proposition 15.25 and isomorphism 15.32.5 that ˆ fulﬁlling there exists XR ∈ Gk F(R) ˆ (R) F

resHˆ C (R) (XR ) = YC,R for any cyclic p -subgroup C of F(Q)R /H(Q)R .

15.32.11

15. Reduction results for Grothendieck groups

263

More generally, for any cyclic p -subgroup C of F(Q)/H(Q) and any ˆ of H ˆ C (R) ⊂ F(R) ˆ k -subgroup D with a cyclic k ∗ -quotient D of order prime ¯ the image of D ˆ in F(Q)R /H(Q)R via the composed to p , denoting by D group homomorphism ∗

ˆ C (R)/H(R) ˆ H ⊂ F(R)/H(R) ∼ = F(Q)R /H(Q)R

15.32.12,

ˆ C (R) and therefore we get (cf. 15.32.9) ˆ ⊂H ˆ D¯ (R) ⊂ H we have D ˆ (R) ˆ D (R) ˆ (R) F H F resHˆ C (R) (XR ) = resDˆ resHˆ D¯ (R) (XR )

ˆ C (R) H

resDˆ

ˆ D (R) H

= resDˆ

ˆ C (R) H

(YD,R ¯ ) = res ˆ D

15.32.13. (YC,R )

Thus, equality 15.32.11 holds for any cyclic p -subgroup C of F(Q)/H(Q) . Now, if R ∈ S is another element also fully highnormalized in H and θ:R ∼ = R is an F-isomorphism, we claim that res a ut

F

S (θ,id∆0 )

(XR ) = XR

15.32.14;

note that, according to Proposition 12.8, we may assume either that θ is induced by an element σ of F(Q) or that it is an H-isomorphism. Let C be a cyclic p -subgroup of F(R)/H(R) ; in the ﬁrst case, it follows from equality 15.32.7 that ˆ (R) F (XR ) resHˆ C (R) res a ut S (θ,id∆0 ) F Fˆ (R ) = (res(ˆνσ )R )−1 resHˆ C σ (R ) (XR )

15.32.15. ˆ (R) F

= (res(ˆνσ )R )−1 (YC σ ,R ) = YC,R = resHˆ C (R) (XR ) sc

sc

In the second case, denoting by r : ∆1 → (HC ) the obvious (HC ) -chain ˆ C (r) with H ˆ C (R) , it foldeﬁned by the H-isomorphism θ , and identifying H lows from equality 15.32.6 that ˆ (R) F (XR ) resHˆ C (R) res a ut S (θ,id∆0 ) F Fˆ (R ) resHˆ C (R ) (XR ) = res a ut(HC )sc (idq◦δ0 ,δ00 ) 0

= res a ut

(YC,R )

= res a ut

(YC,R ) = resHˆ C (R) (XR )

sc (id ,δ 0 ) q◦δ 0 0 (HC ) 0

sc (id ,δ 0 ) q◦δ 0 1 (HC ) 1

ˆ (R) F

By the transitivity of the equalities, the claim is proved.

15.32.16.

264

Frobenius categories versus Brauer blocks

Consequently, we can deﬁne a unique element X = (XR )R∈S , stable by F-isomorphisms, by setting XR = res a ut

F

S (θ,id∆0 )

(XR )

15.32.17

for a choice of R ∈ S fully highnormalized in H admitting an F-isomorphism θ:R ∼ = R ; actually, the arguments above also prove that, for any cyclic p -subgroup C of F(Q)/H(Q) , we still get ˆ (R) F

resHˆ C (R) (XR ) = YC,R

15.32.18.

S ) . Indeed, Finally, we claim that the element X belongs to lim(gk ◦ aut F ←−

S

S

for an F -chain r : ∆1 → F , set T = r(0) , R = r(1) and ψ = r(0 • 1) , and choose R ∈ S fully highnormalized in H admitting an H-isomorphism θ : R ∼ = R (cf. 12.10); according to Proposition 12.8, we have −1 ιQ ◦ ψ = τ ◦ ψ R ◦ θ

15.32.19

for suitable τ ∈ F(Q) and ψ ∈ H(Q, T ) ; set R = τ −1 (R ) and denote by η : R ∼ = R the isomorphism induced by τ , by ψ : T → R the homomorS phism determined by ψ , and by r : ∆1 → F the chain deﬁned by ψ . Then, it suﬃces to prove the equalities res a ut

F

0 S (idr ◦δ 0 ,δ1 ) 1

(XT ) = res a ut

F

XR = res a ut

F

0 S (idr ◦δ 0 ,δ0 ) 0 S (θ◦η,id∆0 )

(XR ) 15.32.20.

(XR )

The bottom equality follows from equality 15.32.14; moreover, since we have (cf. Proposition 12.11) and F(Q)R /H(Q)R ∼ = F(R )/H(R ) 15.32.21,

ˆ ) = F(R ˆ )ψ (T ) F(r

ˆ C (r ) for any it suﬃces to prove the restriction of the top equality to H cyclic p -subgroup C of F(Q)/H(Q) ; but, according to equalities 15.32.6 and 15.32.18, we get ˆ (r ) F resHˆ C (r ) res a (XT ) ut S (idr ◦δ0 ,δ10 ) F 1

= res a ut

sc (id ,δ 0 ) r ◦δ 0 1 (HC ) 1

ˆ (r ) F = resHˆ C (r ) res a ut

F

S

(YC,T ) = res a ut

(idr ◦δ0 ,δ00 )

(XR )

sc (id ,δ 0 ) r ◦δ 0 0 (HC ) 0

(YC,R )

15.32.22.

0

We have proved the claim. At this point, equality 15.32.18 shows that X maps on Y . We are done.

15. Reduction results for Grothendieck groups

265

ˆ. 15.33 As above, in our standard setting assume that CGˆ (Q, f ) ⊂ H sc ˆ H ˆ ∼ Let C be a cyclic p -subgroup of G/ = F(Q)/H(Q) ; for any (HC ) -chain C sc r : ∆n → (H ) , since r(n) is also H-selfcentralizing (cf. Lemma 15.16) and we have identiﬁed HC (r) with a subgroup of HC r(n) , it makes sense to con sider H(r) ⊂ H r(n) which is a normal subgroup of HC (r) , and then we set Cr = HC (r)/H(r) which can be identiﬁed with a subgroup of C . Thus, since C C ˆ (r) has a canonical Gk (Cr )-module structure (cf. 15.21), Gk H ˆ (r) Gk H becomes a Gk (C)-module via the restriction map Gk (C) → Gk (Cr ) ; furthermore, it is easily checked that these Gk (C)-module structures are compatible sc with the ch∗ (HC ) )-morphisms; ﬁnally, from its very deﬁnition (cf. 14.3.3), (HC )nc ) becomes a Gk (C)-module. Moreover, according to 15.29 Gk (HC , aut Hnc ) and we have a restriction and 15.30, the group C acts on Gk (H, aut O-module homomorphism (HC )nc ) −→ Gk (H, aut Hnc )C Gk (HC , aut

15.33.1

(HC )nc ) ; then, we claim we have an isosince C acts trivially on Gk (HC , aut morphism analogous to isomorphism 15.21.2. ˆ. Proposition 15.34 In our standard setting, assume that CGˆ (Q, f ) ⊂ H ˆ H ˆ , the restriction induces an O-module For any cyclic p -subgroup C of G/ isomorphism ∗ (HC )nc )C ∼ Hnc )C Gk (HC , aut 15.34.1. = Gk (H, aut sc

sc

Proof: It is clear that any (HC ) -chain r : ∆n → (HC ) is HC -isomorphic sc sc to a (HC ) -chain r : ∆n → (HC ) such that r (i−1) ⊂ r (i) and r (i−1•i) is the inclusion map for any 1 ≤ i ≤ n ; thus, since any subgroup of Q is HC -selfcentralizing if and only if it is H-selfcentralizing (cf. Lemma 15.16), sc the set C of these (HC ) -chains coincides with the corresponding set of sc H -chains and we have injective O-module homomorphisms (HC )nc ) −→ Gk (HC , aut

r∈C

Hnc ) −→ Gk (H, aut

C ˆ (r) Gk H ˆ Gk H(r)

15.34.2

r∈C

clearly compatible with the corresponding restriction homomorphisms. ˆ C and H ˆ deMoreover, it is clear that the actions of NHˆ C (Qδ ) on Q , H termine an action of this group on both O-module homomorphisms, which (HC )nc ) whereas it coincides with the action of C on is trivial on Gk (HC , aut

266

Frobenius categories versus Brauer blocks

Hnc ) ; hence, choosing a set of representatives R for the set of orbits Gk (H, aut of NHˆ C (Qδ ) over C , we have a commutative diagram C Cr ˆ (r) ˆ Gk H −→ r∈R Gk H(r) ↑ ↑ (HC )nc ) −→ Hnc )C Gk (HC , aut Gk (H, aut

r∈R

15.34.3

where the vertical arrows are still injective. On the other hand, the multiplicative subgroup C ∗ ⊂ Gk (C) (cf. 15.21) C ˆ (r) via its (HC )nc ) and, for any r ∈ R , it acts on Gk H acts on Gk (HC , aut ∗ image in (Cr ) (cf. 15.33); consequently, we still have a commutative diagram (cf. 15.21.2) C (Cr )∗ ˆ (r) Gk H ↑ (HC )nc )C ∗ Gk (HC , aut

r∈R

Cr ˆ Gk H(r) ↑ Hnc )C Gk (H, aut

∼ =

r∈R

−→

15.34.4

Finally, the obvious commutative diagrams

res

C (Cr )∗ ˆ (r) Gk H  #

∼ =

aut sc (ν,δ) (HC )

∗ ˆ C (t) (Ct ) Gk H

∼ =

Cr ˆ Gk H(r)  # res a ut sc (ν,δ) H Ct ˆ Gk H(t)

15.34.5,

where r and t run over R and (ν, δ) : (t, ∆m ) → (r, ∆n ) over the set of ch∗ (Hsc )-morphisms from (t, ∆m ) to (r, ∆n ) , proves the surjectivity of the bottom homomorphism in diagram 15.34.4. We are done. 15.35 At this point, we also consider the C-residual Grothendieck group (HC )nc ) — denoted by RHGk (HC , aut (HC )nc ) — as the of the pair (HC , aut intersection of kernels of all the restriction O-module homomorphisms (HC )nc ) −→ Gk (HD , aut (HD )nc ) Gk (HC , aut

15.35.1,

where D runs over the set of proper subgroups of C ; since they actually are Gk (C)-module homomorphisms via the restriction maps Gk (C) → Gk (D) , (HC )nc ) is a Gk (C)-submodule. the intersection RHGk (HC , aut (HC )nc ) , we also can 15.36 As the full Grothendieck group Gk (HC , aut C describe RHGk (H , aut(HC )nc ) as an inverse limit. In order to show this, sc

sc

for any (HC ) -chain r : ∆n → (HC ) such that Cr = C , consider the C ˆ (r) ; moreover, for any subgroup D of C and any O-module RH(r) Gk H ˆ

15. Reduction results for Grothendieck groups

267

sc ch∗ (HD ) -morphism (µ, δ) : (t, ∆m ) → (r, ∆n ) such that Ct = Cr = C , note that we still have the restriction homomorphism D D ˆ (r) −→ Gk H ˆ (t) res a : Gk H 15.36.1; ut(HD )sc (µ,δ) then, it is quite clear that C C ˆ ˆ (t) res a R G ⊂ RH(t) Gk H k H (r) ˆ ˆ H(r) sc (µ,δ) ut (HC )

15.36.2.

sc sc That is to say, denoting by ch∗C (HC ) the full subcategory of ch∗ (HC ) sc sc over the set of (HC ) -chains r : ∆n → (HC ) such that Cr = C , we get a contravariant functor C ˆ (•) : ch∗ (HC )sc −→ O-mod RH(•) Gk H 15.36.3. C ˆ On the other hand, for any subgroup D of C , from the very deﬁnition of (HD )nc ) (cf. 14.3.3) we get structural O-module homomorphisms Gk (HD , aut D ˆ (r) (HD )nc ) −→ Gk H Gk (HD , aut 15.36.4 compatible with homomorphisms 15.36.1 and ﬁnally, after routine veriﬁcations, we obtain an O-module homomorphism C ˆ (•) (HC )nc ) −→ lim R ˆ Gk H RHGk (HC , aut 15.36.5. H(•) ←−

ˆ. Proposition 15.37 In our standard setting, assume that CGˆ (Q, f ) ⊂ H ˆ H ˆ , the structural homomorphisms induce For any cyclic p -subgroup C of G/ O-module isomorphisms C ˆ (•) (HC )nc ) ∼ RHGk (HC , aut Gk H = lim RH(•) ˆ ←− 15.37.1. ∼ (HC )nc ) = RGk (C) ⊗Gk (C) Gk (HC , aut sc

sc

Proof: Actually, for any (HC ) -chain r : ∆n → (HC ) we have  C ˆ (r) if Cr = C Gk H ˆ C  RH(r) ˆ RGk (C) ⊗Gk (C) Gk H (r) = 15.37.2  {0} otherwise C C ˆ (r) → Gk H ˆ r (r) is an O-module since, by deﬁnition, the restriction Gk H isomorphism. Moreover, since RGk (C) is a direct factor of Gk (C) , the func C ˆ (•) is a direct summand of gk ◦ aut (HC )sc . Consetor RGk (C) ⊗Gk (C) Gk H quently, it is quite clear that we have C ˆ (•) lim RGk (C) ⊗Gk (C) Gk H ←− 15.37.3. ∼ (HC )nc ) = RGk (C) ⊗Gk (C) Gk (HC , aut

268

Frobenius categories versus Brauer blocks

But, on the other hand, we clearly have (HC )nc ) ⊂ RHGk (HC , aut (HC )nc ) RGk (C) ⊗Gk (C) Gk (HC , aut

15.37.4.

Then, the existence of homomorphism 15.36.5 and of equalities 15.37.2 completes the proof. ˆ. Proposition 15.38 In our standard setting, assume that CGˆ (Q, f ) ⊂ H D ˆ H ˆ acts on Gk (H , aut (HD )nc ) for any subgroup Any cyclic p -subgroup C of G/ D of C and, denoting by SC the set of them, the restriction induces an O-module isomorphism (HC )nc ) ∼ Gk (HC , aut =

(HD )nc )C RHGk (HD , aut

15.38.1.

D∈SC

Proof: By the Frattini argument, we have H C = H·NH C (Q, f ) and the conjugation by an element x ∈ NH C (Q, f ) deﬁnes an (HD , HD )-functorial automorphism of Q and therefore a Frobenius functor fx : HD → HD (cf. 12.1); as in 15.30 above, it follows from the uniqueness part of Theorem 11.32 that x determines a natural isomorphism (HD )sc ◦ ch∗ (fx ) (HD )sc ∼ νˆx : aut = aut

15.38.2.

Consequently, we get an O-module automorphism (HD )nc ) ∼ (HD )nc ) αx : Gk (HD , aut = Gk (HD , aut

15.38.3

which coincides with the identity map whenever x belongs to H since, in this case, fx is naturally isomorphic to the identity functor (cf. 12.1 and Proposition A4.5). Since αx ◦ αx = αx x for any x, x ∈ NH C (Qδ ) , we obtain an (HD )nc ) . action of C ∼ = NH C (Qδ )/NH (Qδ ) on Gk (HD , aut Moreover, as in 15.23 above, in order to prove isomorphism 15.38.1 it suﬃces to prove the O-module isomorphism (HC )nc ) ∼ (HD )nc )C RGk (D) ⊗Gk (C) Gk (HC , aut = RHGk (HD , aut

15.38.4

for any D in SC . But, it follows from Proposition 15.34 that ∗ (HC )nc )(C/D) ∼ (HD )nc )C/D Gk (HC , aut = Gk (HD , aut

15.38.5

and, since (C/D)∗ is a cyclic p -group and its group O-algebra is isomorphic to Gk (C/D) , it is quite clear that ∗ (HC )nc )(C/D) ∼ (HC )nc ) Gk (HC , aut = Gk (D) ⊗Gk (C) Gk (HC , aut

15.38.6.

15. Reduction results for Grothendieck groups

269

On the other hand, since C/D is a p -group, it follows from Proposition 15.37 that (HD )nc )C RHGk (HD , aut ∼ (HD )nc )C/D = RGk (D) ⊗Gk (D) Gk (HD , aut

15.38.7.

Now, isomorphism 15.38.4 follows from isomorphism 15.38.7 and from the isomorphisms obtained applying the functor RGk (D)⊗Gk (D) to isomorphisms 15.38.5 and 15.38.6. ˆ. Corollary 15.39 In our standard setting, assume that CGˆ (Q, f ) ⊂ H Then, the restriction induces an O-module isomorphism F nc ) ∼ Gk (F, aut =

F (Q)/H(Q) (HC )nc ) RHGk (HC , aut

15.39.1

C

where C runs over the set of cyclic p -subgroups of F(Q)/H(Q) . Proof: From Proposition 15.38, it is not diﬃcult to compute the inverse limit in the right-hand member of isomorphism 15.32 since all the homomorphisms we have to consider are direct sums of either zero homomorphisms or isomorphisms, and then we obtain the right-hand member of isomorphism 15.39.1 above. 15.40 Now, let us “double” our standard setting; that is to say, beˆ and (c, H) ˆ , (P, e) and (Q, f ) , and F and H , consider (b , G ˆ) sides (b, G) ˆ ) , (P , e ) and (Q , f ) , and F and H fulﬁlling the analogous and (c , H hypothesis. Let C be a cyclic p -group; we are only interested in the case where ˆ CGˆ (Q, f ) ⊂ H

,

ˆ CGˆ (Q , f ) ⊂ H

ˆ H ˆ ∼ ˆ /H ˆ 15.40.1; and G/ =C∼ =G

thus, we already know that b = c , P = Q , b = c and P = Q (cf. Proposition 15.10) 15.41 More precisely, we are interested in the pull-back ˆ = G ˆ× ˆ ˆC G G

15.41.1

ˆ = H ˆ× ˆ (cf. 1.24) as a ˆH which contains the direct sum of k ∗ -groups H ∗ normal k -subgroup; moreover, it is clear that c = c ⊗ c is a block of ˆ ∼ ˆ ⊗k k∗ H ˆ (cf. 1.24 and 1.25), that setting k∗ H = k∗ H Q = Q × Q

and f = f ⊗ f

15.41.2

270

Frobenius categories versus Brauer blocks

ˆ )-pair (cf. 1.16), and that we have (Q , f ) is a maximal Brauer (c , H ˆ CGˆ (Q , f ) ⊂ H

15.41.3.

ˆ ) is also a block (cf. Proposition 15.13), and we consider In particular, (c , G the Frobenius Q -categories F = F(c ,Gˆ )

and H = H(c ,Hˆ )

15.41.4.

ˆ× ˆ) ∼ ˆ ⊗k k∗ G ˆ (cf. 1.24 and 1.25), ˆG 15.42 First of all, since k∗ (G = k∗ G we get ˆ× ˆ) ∼ ˆ ⊗O Gk (G ˆ) ˆG Gk (G 15.42.1, = Gk (G) so that we still get ˆ× ˆ , c ) ∼ ˆ c) ⊗O Gk (G ˆ , c ) ˆG Gk (G = Gk (G,

15.42.2.

ˆ ⊂ G ˆ× ˆ , we have a restriction O-module homomorphism ˆG Thus, since G ˆ ˆ ˆ

×G ˆ c) ⊗O Gk (G ˆ , c ) −→ Gk (G ˆ , c ) resG : Gk (G, ˆ G

15.42.3.

15.43 On the other hand, it is easily checked that the projection maps Q ←− Q −→ Q

15.43.1

are (F , F)- and (F , F )-functorial respectively, so that we have Frobenius functors F ←− F −→ F 15.43.2, and therefore we get a faithful functor i : F → F × F (cf. 1.5) which clearly sc sc sc sends F to F × F . Then, consider the obvious functor F sc × F sc : ch∗ (F × F ) −→ k ∗ -Gr ˆ aut aut sc

sc

15.43.3

sc sc sc sc ˆ × ˆ Fˆ (r ) , where mapping any F × F -chain r × r : ∆n → F × F on F(r) sc sc sc r and r are respective F - and F -chains; since for any F -chain r which i maps on r × r we have an evident injective k ∗ -group homomorphism

ˆ × ˆ Fˆ (r ) Fˆ (r ) −→ F(r)

15.43.4,

it follows from Proposition 15.12 that we have an injective natural map

F ×F F sc ) ◦ ch∗ (i) F sc −→ (aut F sc × ˆ aut νˆF : aut

15.43.5.

15. Reduction results for Grothendieck groups

271

15.44 Moreover, it is clear that the structural functors from F × F sc sc to F and to F determine a functor sc

ch∗ (F × F ) −→ ch∗ (F ) × ch∗ (F ) sc

sc

sc

sc

sc

15.44.1

F sc × F sc above can be extended to the direct ˆ aut and that the functor aut sc ∗ ∗ sc product ch (F ) × ch (F ) (cf. 1.5) mapping an object (r, ∆n ) × (r , ∆n ) ˆ × ˆ Fˆ (r ) ; thus, since we already know that (cf. 15.42.1) on the k ∗ -group F(r) ˆ × ˆ ˆ Fˆ (r ) ∼ Gk F(r) ⊗O Gk Fˆ (r ) = Gk F(r) it is quite clear that we get an O-module homomorphism F sc ) F sc × ˆ aut lim gk ◦ (aut ←− &  F nc ) ⊗O Gk (F , aut F nc ) Gk (F, aut

15.44.2,

15.44.3.

15.45 But, we have the canonical O-module homomorphism (cf. A3.9) F sc × F sc ) ◦ ch∗ (i) ˆ aut lim gk ◦ (aut ←− &  15.45.1 F sc ) F sc × ˆ aut lim gk ◦ (aut ←−

F ×F and the natural map νˆF above determines an O-module homomorphism

F sc ) ◦ ch∗ (i) −→ Gk (F , aut F sc × F nc ) ˆ aut lim gk ◦ (aut ←−

15.45.2.

Consequently, by composition, we get the O-module homomorphism F nc ) ⊗O Gk (F , aut F nc ) −→ Gk (F , aut F nc ) Gk (F, aut

15.45.3

×F that we denote by resF . Moreover, denoting by SH and SH the reF spective sets of H- and H -selfcentralizing subgroups of Q and Q , it follows from Lemma 15.14 that the structural maps determine injective O-module homomorphisms

F nc ) −→ Gk (F, aut

ˆ Gk F(R)

R∈SH

F nc ) −→ Gk (F , aut

Gk Fˆ (R )

R ∈SH

F nc ) −→ Gk (F , aut

(R,R )∈SH ×SH

Gk Fˆ (R × R )

15.45.4.

272

Frobenius categories versus Brauer blocks

Theorem 15.46 With the notation and the hypothesis above, assume that ×F C is trivial. Then resF induces an O-module isomorphism F F nc ) ⊗O Gk (F , aut (F )nc ) ∼ (F )nc ) Gk (F, aut = Gk (F , aut

15.46.1.

Proof: If C = {1} then, for any R ∈ SH and any R ∈ SH , we have ˆ ˆ Fˆ (R ) Fˆ (R × R ) ∼ × = F(R)

15.46.2;

thus, according to isomorphism 15.42.1 and to the injectivity in 15.45.4, it is quite clear that we get the following commutative diagram with injective vertical arrows ∼ ˆ Gk (F(R)) ⊗O Gk (F (R )) Gk (Fˆ (R × R )) = (R,R )

(R,R )

& 

F nc ) ⊗O Gk (F , aut F nc ) −→ Gk (F, aut

& 

15.46.3,

F nc ) Gk (F , aut

where (R, R ) runs over (SH )C × (SH )C . Moreover, according to Remark 14.29, the three k-vector spaces F nc ) k ⊗O Gk (F, aut F nc ) and k ⊗O Gk (F , aut F nc ) k ⊗O Gk (F , aut

15.46.4

still coincide with the corresponding inverse limits and therefore, applying the functor k⊗O • = k⊗ • to diagram 15.46.3, the vertical arrows are injective too; hence, the bottom arrow yields an injective k-linear homomorphism k⊗

F nc )⊗k k⊗ Gk (F , aut F nc ) −→ k⊗ Gk (F , aut F nc ) Gk (F, aut

15.46.5.

Consequently, in order to prove isomorphism 15.46.1, it suﬃces to prove that the O-ranks of both sides coincide. But, according to Corollary 14.32, we have ˆ F nc ) = rankO Gk (F, aut (−1)n rankO Gk F(r) rankO

(r,∆n )

F nc ) = (−1)n rankO Gk Fˆ (r ) Gk (F , aut

15.46.6

(r ,∆n )

where (r, ∆n ) and (r , ∆n ) respectively run over sets of representatives for sc sc the isomorphism classes of regular ch∗ (F )- and ch∗ (F )-objects (cf. A5.2);

15. Reduction results for Grothendieck groups

273

actually, we may assume that all the morphisms r(j • i) and r (j • i ) are inclusion maps. Thus, we still have (cf. 15.42.1) F nc ) ⊗O Gk (F , aut F nc ) rankO Gk (F, aut ˆ = ⊗O Gk Fˆ (r ) (−1)n+n rankO Gk F(r) 15.46.7 (r,∆n ) (r ,∆n ) ˆ × ˆ Fˆ (r ) = (−1)n+n rankO Gk F(r) (r,∆n ) (r ,∆n )

where (r, ∆n ) and (r , ∆n ) run over the same sets of representatives. On the other hand, from any regular ch∗ (F )- and ch∗ (F )-objects sc sc (r, ∆n ) and (r , ∆n ) , we have the regular ch∗ (F )-chain r∗r : ∆n+n → F mapping any i ∈ ∆n on r(i) × r (0) , any n + i ∈ ∆n+n on r(n) × r (i ) , any ∆n -morphism j • i on r(j • i) × idr (0) and any ∆n+n -morphism (n+j )• (n+i ) on idr(n) × r (j • i ) ; then, it is easily checked that we have sc

sc

ˆ × ˆ Fˆ (r ) ∼ F(r) = Fˆ (r ∗ r )

15.46.8

and therefore, in order to prove isomorphism 15.46.1 it suﬃces to prove that we still have F nc ) rankO Gk (F , aut 15.46.9 = (−1)n+n rankO Gk Fˆ (r ∗ r ) (r,∆n ) (r ,∆n )

where (r, ∆n ) and (r , ∆n ) run over the same sets of representatives. But, it is clear that any regular ch∗ (F )-chain r : ∆n → F such that all the morphisms r (j • i ) are inclusion maps, composed with both sc sc functors 15.43.2 above determine regular ch∗ (F )- and ch∗ (F )-chains sc

r : ∆n −→ F

sc

and r : ∆n −→ F

sc

sc

15.46.10

where once again all the morphisms r(j • i) and r (j • i ) are inclusion maps. sc Consequently, it suﬃces to prove that, for any such regular ch∗ (F )- and sc ch∗ (F )-chains r and r , we have (−1)n rankO Gk Fˆ (r ) (r ,∆n ) n+n

= (−1)

rankO

Gk Fˆ (r ∗ r )

15.46.11

where (r , ∆n ) runs over a set of representatives R for the isomorphism sc sc classes of ch∗ (F )-objects such that r is a regular ch∗ (F )-chain, where all the morphisms r (j • i ) are inclusion maps, which maps on the pair (r, r ) .

274

Frobenius categories versus Brauer blocks First of all, let us prove that (−1)n rankO Gk Fˆ (r ) = 0

15.46.12

(r ,∆n )

when (r , ∆n ) ∈ R runs over the set of elements such that r (i ) is not a direct product for some i ∈ ∆n ; it suﬃces to exhibit an involutive permutation τ of this set such that if τ (r , ∆n ) = (t , ∆m ) then we have Fˆ (t ) ∼ = Fˆ (r ) and

(−1)m = −(−1)n

15.46.13.

Consider the biggest i ∈ ∆n such that r (i ) it is not a direct product and denote by R and R the respective images of r (i ) in Q and Q by the projection maps 15.43.1; if i = n and r (i + 1) = R × R then we set m = n − 1

−1 and t = r ◦ δin +1

15.46.14;

otherwise, if either i = n or i = n and r (i + 1) = R × R then we set sc sc m = n + 1 and denote by t : ∆n −→ F the regular ch∗ (F )-chain, with inclusion maps, fulﬁlling

t ◦ δin +1 = r

and t (i + 1) = R × R

15.46.15.

Since i ∈ ∆m is also the biggest element such that t (i ) it is not a direct product, it is easily checked that τ (t , ∆m ) = (r , ∆n ) ; moreover, it is quite clear that we have Fˆ (t ) ∼ = Fˆ (r ) . Now, in order to prove equality 15.46.11, it suﬃces to exhibit again an involutive permutation — still denoted by τ — fulﬁlling condition 15.46.13, over the set of elements (r , ∆n ) ∈ R such that r (i ) is a direct product for any i ∈ ∆n , whereas r does not coincide with r ∗ r . Consider the smallest i ∈ ∆n such that we have r (i ) = r(n) × r (i )

15.46.16

for some i ∈ ∆n , which actually forces i = n − n + i ; after, consider the smallest j ≤ i fulﬁlling r (j ) = r(i) × r (i ) for some i ∈ ∆n , which similarly forces i = n − i + j . Note that if j = 0 then i = 0 = i and r = r ∗ r , so that we get j = 0 ; moreover, r (j −1) is equal either to r(i) × r (i −1) or to r(i−1) × r (i −1) ; in the ﬁrst case, we set

m = n −1 and t = r ◦ δjn −1

15.46.17;

in the second case, we set m = n +1 and denote by t : ∆n −→ F sc regular ch∗ (F )-chain, with inclusion maps, fulﬁlling

t ◦ δjn = r

and t (j ) = r(i) × r (i −1)

sc

the

15.46.18.

15. Reduction results for Grothendieck groups

275

Then, it is easily checked that i+m−n ∈ ∆m is the smallest element such that we have t (i + m − n ) = r(n) × r (i ) 15.46.19 for some i ∈ ∆n which coincides with the corresponding element above, whereas the same j — which actually fulﬁlls j ≤ i + m − n — is the smallest element such that t (j ) = r(j) × r (i ) for some j ∈ ∆n and this time we get j = n − i − m + n + j = i − m + n 15.46.20. Thus, we still get t (j −1) =

  r(i)×r (i −1) = r(j −1)×r (i −1) if m = n −1 

r(i−1)×r (i −1) = r(j)×r (i −1)

if m = n +1

15.46.21

and then it is easily checked that τ (t , ∆m ) = (r , ∆n ) . We are done. Corollary 15.47 With the notation and the hypothesis above, the O-module ˆ× ˆ ˆG ×F homomorphisms resG and resF induce O-module isomorphisms F ˆ G ˆ c) ⊗RG (C) R Gk (G ˆ , c ) RHˆGk (G, ˆ k H ∼ ˆ , c ) = RHˆ Gk (G F nc ) ⊗RGk (C) R Gk (F , aut (F )nc ) RHGk (F, aut H

15.47.1.

∼ (F )nc ) = RHGk (F , aut Proof: Setting ∇(C) = (C × C)/∆(C) where ∆(C) is the diagonal subgroup, ˆ× ˆ )/G ˆ , ˆG it is clear that ∇(C) is canonically isomorphic to the quotient (G ∇(C) ˆ ˆ ˆ ˆG =G so that we have G × according to the notation above; hence, it follows from isomorphism 15.21.2 that ˆ× ˆ )∇(C)∗ ∼ ˆ )∇(C) ˆG Gk (G = Gk (G

15.47.2;

moreover, up to suitable identiﬁcations, ∇(C)∗ coincides with ∆(C ∗ ) and therefore, since they are p -groups and we have OC ∗ ∼ = Gk (C) , we get the O-module isomorphism ˆ ⊗G (C) Gk (G ˆ) ∼ ˆ× ˆ )∇(C) ˆG Gk (G) = Gk (G k

∗

15.47.3.

ˆ and k∗ G ˆ -modules M and M On the other hand, for any simple k∗ G ˆ ˆ which remain simple restricted to H and to H respectively, the composed ˆ G (C) Gk (G ˆ ) to Gk (G ˆ ) maps the tensor product homomorphism from Gk (G)⊗ k

276

Frobenius categories versus Brauer blocks

X ⊗X of their respective isomorphism classes X and X , on the isomorphism ˆ , which remains simple restricted class of the restriction of M ⊗k M to k∗ G ˆ . Moreover, in this form we obtain all the simple k∗ G ˆ -modules which to H ˆ remain simple restricted to H and this correspondence is compatible with the blocks. At this point, since isomorphisms 15.47.2 and 15.47.3 are actually Gk (C)-module isomorphisms, the top isomorphism in 15.47.1 follows from the equality in 15.23.2. In order to prove the bottom isomorphism, let us denote by F ⊗ F ˆ× ˆ ) ; it is clear ˆG the Frobenius Q -category associated with the block (c , G that the Frobenius Q -category F is a normal subcategory of F ⊗ F , and in fact we have F ⊗ F = F ∇(C) (cf. Proposition 12.12) since ∇(C) is canonically isomorphic to (F ⊗ F )(Q )/F (Q ) ; moreover, it follows from Theorem 15.46 that we have the canonical isomorphism F nc ) ⊗O Gk (F , aut (F )nc ) ∼ (F ⊗F )nc ) Gk (F, aut = Gk (F ⊗ F , aut

15.47.4.

Consequently, replacing isomorphism 15.21.2 by isomorphism 15.34.1, we get as in 15.47.2 and 15.47.3 above the O-module isomorphism F nc ) ⊗Gk (C) Gk (F , aut (F )nc ) ∼ F nc )∇(C) Gk (F, aut = Gk (F , aut

15.47.5.

On the other hand, it follows from Proposition 15.37 that, as in 15.45.4 above, we have an injective O-module homomorphism F nc ) −→ RHGk (F , aut

RHˆ (R×R ) Gk Fˆ (R × R )

15.47.6

(R,R ) C where R and R respectively run over the subsets SC H ⊂ SH and SH ⊂ SH of elements fulﬁlling CR = C = CR (cf. 15.36); but, according to isomorphisms 15.47.2 and 15.47.3 above, ∇(C) acts trivially on each right-hand factor RHˆ (R×R ) Gk Fˆ (R × R ) ; hence, ∇(C) still acts trivially on the left F nc ) and therefore it is quite clear that hand term R Gk (F , aut H

F nc ) ∼ F nc )∇(C) RHGk (F , aut = RGk (C) ⊗Gk (C) Gk (F , aut

15.47.7;

thus, according to Proposition 15.37, applying the functor RGk (C) ⊗Gk (C) • to isomorphism 15.47.5, we are done. 15.48 Besides our standard setting, we need reduction results on the quotients of k ∗ groups by central p-subgroups. More generally, let Z be a ˆ such that the centralizer H ˆ = C ˆ (Z) connormal Abelian p-subgroup of G G ˆ , so that the quotient G/ ˆ H ˆ is a p -group; it tains a Sylow p-subgroup of G ˆ ¯ = G/Z ˆ ¯ = G/Z and, is clear that G remains a k ∗ -group with k ∗ -quotient G

15. Reduction results for Grothendieck groups

277

ˆ¯ ; recall that we ˆ , we denote by R ¯ its image in G for any p-subgroup R of G ¯ˆ = H/Z ˆ set C¯Gˆ (Z) = CGˆ (Z)/Z (cf. 1.9.2) and coherently we also set H . ˆ ¯ ˆ Then, we claim that the canonical H-algebra homomorphism k∗ H → k∗ H ˆ , any is a strict semicovering (cf. 1.11); indeed, for any p-subgroup R of H ¯ can be lifted to a p -subgroup of C ˆ (R) (cf. Thep -subgroup of C ¯ˆ (R) H

H

orem 3.2 in [28, Ch. 5]) and therefore the kernel and the cokernel of the canonical k ∗ -group homomorphism (cf. 1.19.1) ¯ CHˆ (R) −→ CH¯ˆ (R)

15.48.1

are both p-groups. Hence, if follows from Example 3.9 in [34] that the corresponding k ∗ -group algebra homomorphism is a strict semicovering of k-alˆ , the canonical k-algebra gebras; that is to say, for any p-subgroup R of H homomorphism ¯ˆ R) ¯ ˆ 15.48.2 −→ (k∗ H)( (k∗ H)(R) is a strict semicovering; now, the claim follows from Theorem 3.16 in [34]. Lemma 15.49 In our standard setting, let Z be a normal Abelian p-subgroup ˆ and assume that H ˆ = C ˆ (Z) and that H ˆ contains a Sylow p-subgroup of G G ˆ . For any local pointed group Rε on k∗ G ˆ such that Z ⊂ R , ε ∩ k∗ H ˆ is a of G ˆ ¯ ˆ ˆ local point of R on k∗ H and the image of ε ∩ k∗ H in k∗ H is contained in a ¯ˆ which splits into a set {(¯ ¯ over k∗ H ε, ψ)}ψ∈P(k C Gˆ (R¯ )) of local point ε¯ of R ∗

ˆ H

ε

ˆ ˆ¯ ¯ on k∗ G ¯ . Moreover, the Brauer G-pair ¯ (¯ε,ψ) local points of R determined by R ˆ is selfcentralizing if and only if the Brauer G-pair determined by Rε is so and ∼ Op E ˜G (Rε ) = ˜H¯ (R ¯ ε¯) . In this case, we have fulﬁlls Op E ˆG (Rε ) ∼ Gk E =

ψ∈P

ˆ ¯ ε¯) k∗ C G (R ˆ H

NG (Rε ) ¯ (¯ε,ψ) ) ˆG¯ (R Gk E

15.49.1.

ˆ such that Z ⊂ R , we have Proof: If Rε is a local pointed group over k∗ G ˆ CGˆ (R) ⊂ H , so that we still have (cf. 15.5.4) ˆ G ∼ ∗ CH ˆ (Rε ) = k

15.49.2,

ˆ H ˆ is a p -group, it follows from 1.18 and from isomorand therefore, since G/ ˆ is a local point of R over k∗ H ˆ ; moreover, since the phism 15.16.1 that ε∩k∗ H ˆ ¯ is a strict semicovering (cf. 15.48), ˆ → k∗ H H-algebra homomorphism k∗ H ˆ ¯ is contained in a local point ε¯ of R ¯ over k∗ H ¯ˆ ˆ in k∗ H the image of ε ∩ k∗ H (cf. Proposition 3.15 in [34]); then, it follows from Lemma 15.16 again that ˆ¯ . ¯ (¯ε,ψ) } ¯ ε¯ splits into a set {R of local pointed groups over k∗ G R ˆ G ¯ ε )) ψ∈P(k∗ C ˆ (R H

278

Frobenius categories versus Brauer blocks

ˆ Moreover, since CGˆ (R) = CHˆ (R) , the Brauer G-pair determined by Rε ˆ is selfcentralizing if and only if the Brauer H-pair determined by Rε∩k∗ Hˆ is so (cf. 7.4) and then, according to Lemma 15.16, we have Op EG (Rε ) = Op EH (Rε∩k∗ Hˆ )

15.49.3.

On the other hand, it follows from Corollary 2.13 in [50] that the Brauer ˆ H-pair determined by Rε∩k∗ Hˆ is selfcentralizing and fulﬁlls (cf. 1.3) ∼ ˜ ¯ (R ¯ ε¯) ˜H (R E ˆ ) = EH ε∩k∗ H

15.49.4

¯ˆ ¯ ε¯ is selfcentralizing. But, if and only if the Brauer H-pair determined by R in any case, it follows from Theorem 2.9 in [50] and Theorem 3.1 in [39] that the group homomorphism ¯ ε¯) EH (Rε∩k∗ Hˆ ) −→ EH¯ (R

15.49.5

is surjective and moreover, since Z ⊂ Z(H) , its kernel is a p-group (cf. Theorem 3.2 in [28, Ch. 5]); hence, isomorphism 15.49.4 is equivalent to (cf. 1.3) ∼ ˜H (R ˜ ¯ (R ¯ ε¯) Op E ˆ ) = Op EH ε∩k∗ H

15.49.6.

¯ˆ Finally, it follows from Lemma 15.16 that the Brauer H-pair determined by ˆ ¯ ¯ ¯ (¯ε,ψ) Rε¯ is selfcentralizing if and only if the Brauer G-pair determined by R is so, and then we have ¯ ε¯) = Op EG¯ (R ¯ (¯ε,ψ) ) Op EH¯ (R

15.49.7.

ˆ ¯ ¯ (¯ε,ψ) is selfcentralizing Consequently, the Brauer G-pair determined by R ˆ if and only if the Brauer G-pair determined by Rε is so and fulﬁlls (cf. 1.3) ˜G (Rε ) ∼ ˜H¯ (R ¯ ε¯) Op E = Op E

15.49.8.

In this case, denoting by b(ε) the block of CGˆ (R) determined by ε , and by ¯b(ε) the image of b(ε) in k∗ C¯ ˆ (R) ¯ (cf. 1.10), it is quite clear that ¯ G ¯b(ε) =

¯b(¯ ε, ψ)

15.49.9,

ˆ ¯ ε )) ψ∈P(k∗ C G (R ˆ H

ˆ ¯ ¯ ¯ on k∗ G since any local point (¯ ε, ψ) of R involved in Rε¯ determines and is ¯ (cf. 1.17 and 7.4). determined by the block of defect zero ¯b(¯ ε, ψ), C¯G¯ˆ (R)

15. Reduction results for Grothendieck groups

279

Then, on the one hand, since the block ¯b(ε), C¯Gˆ (R) has defect zero (cf. 7.4) so that the k-algebra k∗ C¯Gˆ (R)¯b(ε) is simple (cf. 1.17), we have (cf. Theorem 3.7 in [53] and deﬁnition 15.5.3 above) ¯ ˆ (Rε )¯b(ε) ∼ ˆG (Rε ) k∗ N = k∗ C¯Gˆ (R)¯b(ε) ⊗k k∗ E G

15.49.10

and therefore we still have ˆG (Rε ) Gk NGˆ (Rε ), b(ε) ∼ = Gk E

15.49.11.

On the other hand, since NG (Rε ) stabilizes the decomposition 15.49.9, it follows from Proposition 3.2 and Theorem 3.7 in [53] that (cf. deﬁnition 15.5.3 above) ¯ ˆ (Rε )¯b(ε) k∗ N G ¯ (R ) N ∼ IndN¯Gˆ (R¯ ε = ˆ ¯ G

ψ

(ε,ψ) )

¯ ¯b(¯ ¯ (¯ε,ψ) ) ˆG¯ (R ε, ψ) ⊗k k∗ E k∗ C¯G¯ˆ (R)

15.49.12,

ˆ ¯ ¯ ε¯) runs over a set of representatives for the set of where ψ ∈ P k∗ C G¯ˆ (R H

orbits of NG (Rε ) , which this time implies that Gk NGˆ (Rε ), b(ε) ∼ =

ψ∈P

ˆ ¯ε ) k∗ C G (R ˆ H

NG (Rε ) ¯ (¯ε,ψ) ) ˆG¯ (R Gk E

15.49.13.

We are done. Proposition 15.50 In our standard setting, let Z be a normal Abelian ˆ contains a Syˆ and assume that H ˆ = C ˆ (Z) and that H p-subgroup of G G ˆ . With the notation above, denote by Pγ a defect pointed low p-subgroup of G ¯ˆ containing the group of b and by γ¯ the local point of P¯ = P/Z on k∗ H ˆ . Set Pγ¯ = P k∗ C G¯ˆ (P¯γ¯ ) and denote by P¯γ¯ the set of image of γ ∩ k∗ H ˆ ¯ H

NGˆ (Pγ )-orbits on Pγ¯ and by ϕ¯ the NGˆ (Pγ )-orbit of ϕ ∈ Pγ¯ . Then, there is a bijection from P¯γ¯ to the primitive decomposition of the image ¯b of b ¯ˆ such that, for any ϕ ∈ Pγ¯ , P¯(¯γ ,ϕ) is a defect pointed group of the in Z(k∗ G) ¯ˆ corresponding to ϕ¯ ∈ P¯γ¯ . Moreover, setting F¯ϕ¯ = F ¯ ¯ˆ for block ¯bϕ¯ of G (bϕ¯ ,G) any ϕ¯ ∈ P¯γ¯ , the restriction induces O-module isomorphisms ˆ b) ∼ Gk (G, =

ˆ ¯ ¯bϕ¯ ) Gk (G,

¯γ¯ ϕ∈ ¯ P

F nc ) ∼ Gk (F, aut =

¯γ¯ ϕ∈ ¯ P

(F¯ )nc ) Gk (F¯ϕ¯ , aut ϕ ¯

15.50.1.

280

Frobenius categories versus Brauer blocks

ˆ is local (cf. 1.19 Proof: First of all note that, since any point of Z on k∗ G above and Proposition 5.15 in [42]), P contains Z . It is clear that, for any ˆ , we have BrR¯ (¯b) = 0 only if BrR (b) = 0 , and therefore p-subgroup R of G ˆ ¯ has a local point on k∗ G ¯¯b only if R has a local point on k∗ Gb ˆ (cf. 1.13); R thus, it follows from Lemma 15.49 that any maximal local pointed group ˆ ¯¯b is G-conjugate to P¯(¯γ ,ϕ) for some ϕ ∈ P k∗ C G¯ˆ (P¯γ¯ ) ; then, the ﬁrst on k∗ G ˆ ¯ H

statement follows from the Brauer First Main Theorem (cf. 1.16) and the top isomorphism in 15.50.1 is an immediate consequence of this statement. ˆ contained in Pγ such Let Rε and Tη be local pointed groups over k∗ G that R and T are F-selfcentralizing, so that R and T contain Z ; we know ˆ fulﬁlling Tη ⊂ (Rε )x that any α ∈ F(R, T ) is induced by an element x ∈ G ¯ , T¯ and x (cf. 3.2) and therefore, denoting by R ¯ the respective images of R , T ˆ ¯ and x in G , we have ¯ x¯ ⊂ C ¯ˆ (T¯) and C ¯ˆ (R) ¯ x¯ ⊂ C ¯ˆ (T¯) CG¯ˆ (R) G H H

15.50.2,

so that x ¯−1 determines a group homomorphism ¯ ¯ −→ CG¯ (T¯)/CH¯ (T¯) cx : CG¯ (R)/C ¯ (R) H

15.50.3

ˆ . That is to say, we get a conwhich only depends on α since CGˆ (T ) ⊂ H travariant functor sc c : F −→ Gr 15.50.4 ¯ ¯ mapping R on the quotient CG¯ (R)/CH¯ (R) and α on cx . ¯ C¯ ˆ (R) is a Moreover, in 15.48 we have proved that the quotient C¯H¯ˆ (R)/ H p-group; hence, since the image ¯b(ε) in k∗ C¯Hˆ (R) of the block b(ε) determined by ε has defect zero (cf. 7.4), it follows from Proposition 6.5 in [34] that the ¯ of the block b(¯ ¯ determined by the local image ¯b(¯ ε) in k∗ C¯H¯ˆ (R) ε) of CH¯ˆ (R) ¯ˆ containing the image of ε ∩ k∗ H ¯ on k∗ H ˆ is nilpotent and then, point ε¯ of R it follows from Proposition 3.5 in [53] that we have an embedding (cf. 1.11) ¯ ¯b(¯ ¯ ¯b(¯ k∗ C¯Hˆ (R)¯b(ε) −→ k∗ C¯H¯ˆ (R) ε) J k∗ C¯H¯ˆ (R) ε) 15.50.5. ˆ ¯ ¯ ε¯) which lifts the quotient CG¯ (R)/C ¯ ¯ Consider the k ∗ -group C G¯ˆ (R ¯ (R) H H

(cf. 15.5.4); we claim that the functor c above can be lifted to a functor ˆc : F

sc

−→ k ∗ -Gr

15.50.6

ˆ ¯

¯ ε¯) . Indeed, since Z ⊂ R , we have C ˆ (R) = C ˆ (R) and mapping R on C G¯ˆ (R G H H

therefore we still have (cf. isomorphism 1.16.1) ∼ ¯ ˆ (R)¯b(ε) = k∗ C¯ ˆ (R)¯b(ε) ˆ (k∗ G)(R ε ) = k∗ CG H

15.50.7;

15. Reduction results for Grothendieck groups

281

consequently, from embedding 15.50.5 and this isomorphism, it follows that ¯ ε¯) obtained from the action of C ¯ˆ (R ¯ ε¯) on the simple the k ∗ -group CˆG¯ˆ (R G ¯ ¯ ¯ ¯ ¯ ¯ algebra k∗ CH¯ˆ (R)b(¯ ε) J k∗ CH¯ˆ (R)b(¯ ε) (cf. 15.5.4) is canonically isomorphic ∗ ˆ to a k -subgroup of N ˆ (Rε )/Z (cf. 1.26 above and Proposition 2.14 in [39]) G

ˆ ¯ ¯ ε¯) is (canonically isomorphic to) a k ∗ -subgroup of the and therefore C G¯ˆ (R H ˆ ¯ C¯H (R) (cf. 15.5). quotient of F(R) by C¯H¯ (R)/

ˆ contained in Pγ Now, if Rε and Tη are local pointed groups over k∗ G such that R and T are F-selfcentralizing, and α : T → R is an F-morphism, sc sc still denoting by α the F -chain ∆1 → F mapping 0 on T , 1 on R and 0•1 F nc supplied by Theorem 11.32, applied to the on α (cf. A2.2), the functor aut nc obvious ch(F )-morphisms from the 1-chain α to the 0-chains determined by R and T produces two k ∗ -group homomorphisms ˆ ) ←− F(α) ˆ ˆ F(T −→ F(R)

15.50.8;

ˆ ¯ ¯ ε¯) in F(R) ˆ but, it is quite clear that the converse image of C G¯ˆ (R is contained H ˆ in the image of F(α) ; then, it is not diﬃcult to prove that the left-hand k ∗ -group homomorphism in 15.50.8 induces a k ∗ -group homomorphism ˆ ¯ ¯ ε¯) −→ C Gˆ¯ˆ (T¯η¯) ˆc(α) : C G¯ˆ (R ¯ H

H

15.50.9

and that this correspondence deﬁnes the announced functor ˆc . ¯ˆ We claim that, if the Brauer (¯b, H)-pair determined by T¯η¯ is selfcentralizing too, ˆc(α) is actually an isomorphism; indeed, we may assume that T ⊂ R and that α is the inclusion map; moreover, arguing by induction on |R : T | , we also may assume that T is normal in R and then, that T is fully normalized in F (cf. Proposition 2.7); ﬁnally, arguing by induction on |P : T | , we still may assume that we have R = NP (T ) ; thus, denoting by ν the unique ˆ determined by η (cf. 1.19), Rε is a defect pointed point of NHˆ (Rε ) on k∗ G group of NHˆ (Tη )ν (cf. Proposition 3.5). Consequently, respectively denoting by CˆR and CˆT the converse images ¯ and C ¯ˆ (T¯) in G ˆ , and by λ the unique point of R·(H ˆ ∩ CˆT ) on k∗ G ˆ of C ¯ˆ (R) G

G

ˆ ∩ CˆT )λ determined by η (cf. 1.19), Rε is also a defect pointed group of R·(H and therefore, by the Frattini argument, we have ˆ ∩ CˆT )·N ˆ (Rε ) R·CˆT = R·(H CT

15.50.10.

¯ˆ But, since we assume that the Brauer (¯b, H)-pair determined by T¯η¯ is self¯ = Z(R) ¯ ⊂ T¯ (cf. Corollary 7.3); thus, it follows centralizing, we have CP¯ (R)

282

Frobenius categories versus Brauer blocks

from Theorem 3.4 in [28, Ch. 5] that any p -subgroup of NCˆR (Tτ ) centralˆ H ˆ is a p -group, we get izes T¯ , so that it is contained in CˆT ; hence, since G/ ˆ ∩ CˆR )·CˆT and therefore we still get CˆR = (H ¯ ¯ ∼ ˆ ∩ CˆR ) ∼ ˆ ∩ CˆT ) CG¯ (R)/C = CˆR /(H = CˆT /(H ¯ (R) H ∼ = CG¯ (T¯)/CH¯ (T¯)

15.50.11,

so that c(α) and ˆc(α) are isomorphisms, which proves our claim. ˆ which deterOn the other hand, for any local pointed group Rε on k∗ G ˆ ˜ mines a selfcentralizing Brauer (b, G)-pair and fulﬁlls Op EG (Rε ) = {1} , ˜H (R we have Op E ˆ ) = {1} (cf. Lemma 15.16) and always denoting by ε∩k∗ H

¯ on k∗ H ¯ˆ containing the image of ε ∩ k∗ H ˆ , we still have ε¯ the local point of R ¯ ε¯) = {1} since the kernel of the group homomorphism ˜H¯ (R Op E ˜H (R ˜ ¯ (R ¯ ε¯) E ˆ ) −→ EH ε∩(k∗ H)

15.50.12

is a p-group (cf. Theorem 3.2 in [28, Ch. 5]); then, it follows from Lemma 15.49 ¯ ε¯ determines a selfcentralizing Brauer (¯b, H)-pair. ¯ˆ that R Conversely, it fol¯ of P¯ lows from the same lemma that in this way we obtain all the subgroups R ˜ ¯ = {1} which, for some ϕ¯ ∈ P¯γ¯ , are F¯ϕ¯ -selfcentralizing and fulﬁll Op F¯ϕ¯ (R) ¯ and that, in this case, this holds for any ϕ¯ ∈ Pγ¯ . ¯ the set of such subgroups of P¯ , by X the set of Finally, denoting by X X their converse images in P , by F the full subcategory of F over X , by F¯ ¯ X X and F¯ the respective quotients of F and F by Z (cf. Proposition 12.3), ¯ X ¯ for any ϕ¯ ∈ P¯γ¯ , on the one and by (F¯ϕ¯ ) the full subcategory of F¯ϕ¯ over X hand it follows from Lemma 15.51 below and from Proposition 14.6 that we have F nc ) ∼ ¯ X¯ ) Gk (F, aut = lim(gk ◦ aut F ←−

(F¯ )nc ) ∼ ¯ Gk (F¯ϕ¯ , aut = lim(gk ◦ aut ϕ ¯ (F

¯ X ϕ ¯)

←−

) for any ϕ¯ ∈ P¯γ¯

On the other hand, it is clear that the restriction to F ¯ X above factorizes throughout F¯ giving a functor ¯ X

¯ ˆ ¯cX : F¯ −→ k ∗ -Gr ¯

X

15.50.13.

of the functor ˆc

15.50.14

X and we already know that it maps any F¯ -morphism on a k ∗ -group isomor¯ X phism; in particular, it induces a functor from F¯ to the category of ﬁnite sets (cf. 1.1) ¯ X ¯ ˆcX,P ¯ : F¯ −→ ℵ 15.50.15

15. Reduction results for Grothendieck groups

283

¯ on P k∗ C G¯ˆ (R ¯ ∈ X ¯ ε¯) where ε¯ is the unique local point of R ¯ mapping R ˆ ¯ H ¯ˆ such that R ¯ ε¯ ⊂ P¯γ¯ (cf. 1.17 and Corollary 7.3). on k∗ H As usual, we can identify ℵ with the 2-subcategory of CC (cf. A1.6) formed by ﬁnite categories with no other morphisms than the identity ones, ¯ X ¯ ¯cX,P becomes a representation of F¯ (cf. A2.2) which allows us to and then ˆ ¯ X ¯ ¯cX,P F¯ together with the structural functor consider the semidirect product ˆ (cf. A2.7.1) ¯ ¯ X X ¯ ˆcX,P p ¯ X¯ : ¯ F¯ −→ F¯ 15.50.16. F

¯ X

¯ ¯ formed by an element R ¯ of X Explicitly, a ˆcX,P F -object is a pair (ψ, R) ˆ ¯ ¯ ε¯) ; but, according to Lemma 15.49, the and an element ψ of P k∗ C G¯ˆ (R H ¯ determines and is determined by the local pointed group R ¯ (¯ε,ψ) pair (ψ, R) ¯ X

ˆ ¯ ¯ ; analogously, it is easily checked that any ˆcX,P over k∗ G F -morphism ˆ ¯ determines and is determined by the inclusion and the G-conjugation between ˆ¯ . the corresponding local pointed groups, so that it determines a block of G ¯ X

¯cX,P F¯ is the disjoint union of the cateConsequently, the category ˆ ¯ ¯ ¯ X X X ¯ ¯ ¯cX,P F¯ -objects and the ˆ¯cX,P F¯ -morphisms are the gories (F¯ϕ¯ ) — the ˆ ¯ ¯ X X disjoint union of the (F¯ϕ¯ ) -objects and the (F¯ϕ¯ ) -morphisms respectively — when ϕ¯ runs over P¯γ¯ , and therefore we still have ¯

¯ ¯ X,P X X ¯ ¯c F¯ = ch∗ ˆ ch∗ (F¯ϕ¯ )

15.50.17.

¯γ¯ ϕ∈ ¯ P

In particular, we have an evident union functor ˆX,P ¯ = aut ¯X ¯ c ¯ F

¯γ¯ ϕ∈ ¯ P ¯ X

¯ aut (F

¯ X ϕ ¯)

¯

X ¯ ¯cX,P F¯ ) −→ k ∗ -Gr : ch∗ (ˆ

¯cX,P F¯ -chain ¯qP : ∆n → ˆ ¯cX,P F¯ mapping any ˆ ˆ ¯ determined by ¯qP . bϕ¯ is the block of G ¯

¯

¯ X

15.50.18

on Fˆ¯ϕ¯ (pF¯ X¯ ◦ ¯qP ) , where

Moreover, it is easily checked that the direct image (cf. 1.6) of the con∗ ˆX,P ¯ via the structural functor ch (p X travariant functor g ◦ aut ¯ ¯ ) maps ¯X ¯ c ¯ F F ¯ X

any ch∗ (F¯ )-object (¯q, ∆n ) such that ¯q(i − 1) ⊂ ¯q(i) and ¯q(i − 1 • i) is the inclusion map for any 1 ≤ i ≤ n , on the direct sum ˆX,P ¯ ) (¯ ch∗ (pF¯ X¯ ) ∗ (gk ◦ aut q, ∆ n ) X ¯ ¯ ¯ c F 15.50.19 = Gk Fˆ¯ϕ¯ψ (¯q) ¯ ¯ ψ∈ˆ cX,P (¯ q(n))

284

Frobenius categories versus Brauer blocks

where, denoting by R the converse image of ¯q(n) in P — which is F-selfcenˆ such that Rε ⊂ Pγ tralizing — and by ε the unique local point of R on k∗ G ¯ ¯ (¯ε,ψ) ⊂ P¯(¯γ ,ϕ ) (cf. 1.17 and Corollary 7.3), ϕ¯ψ is the element in Pγ¯ fulﬁlling R ψ ˆ ¯ ¯ ¯ X,P G ˆ ¯q(n) = P k∗ C ¯ˆ (Rε¯) , and we identify for some ϕψ ∈ ϕ¯ψ and any ψ ∈ ¯c H ˆ ¯q with the corresponding F¯ -chain. ϕ ¯ψ

At this point, with our notation in 14.21, from Lemma 15.49 it is not diﬃcult to check that we have a natural isomorphism 0 ∗ ¯ X¯ ∼ ˆX,P ¯ ) ¯ ) gk ◦ aut ch (p (g ◦ aut 15.50.20; h = X X ¯ k ¯ ¯ ¯ ∗ F F F c then, as in 14.22 above, it is quite clear that ˆX,P ¯ ) lim h0 ch∗ (pF¯ X¯ ) ∗ (gk ◦ aut X ¯ ¯ ¯ c F ←− ∗ ∼ ˆX,P ¯ ) = lim ch (pF¯ X¯ ) ∗ (gk ◦ aut ¯X ¯ c ¯ F ←−

15.50.21.

∼ ˆX,P ¯ ) = lim (gk ◦ aut ¯X ¯ c ¯ F ←−

Now, since clearly we have ˆX,P ¯ ) ∼ lim (gk ◦ aut = ¯X ¯ c ¯ F ←−

¯γ¯ ϕ∈ ¯ P

¯ lim (gk ◦ aut (F ←−

¯ X ϕ ¯)

)

15.50.22,

the bottom isomorphism in 15.50.1 follows from the isomorphisms 15.50.13, 15.50.20 and 15.50.21. We are done. Lemma 15.51 Let P be a ﬁnite p-group and F a Frobenius P -category. For any F-nilcentralized subgroup Q of P there are an F-selfcentralizing sub ˜ group R fulﬁlling Op F(R) = {1} and an F-morphism ψ : Q → R such that ψ(Q) is fully centralized in F , R contains CP ψ(Q) and we have ψ ◦ F(Q) ⊂ F(R) ◦ ψ

15.51.1.

Proof: We argue by induction on |P : Q| ; let ψ : Q → P be an F-morphism such that Q = ψ(Q) is fully normalized in F (cf. Proposition 2.7); we already know that R = Q ·CP (Q ) is F-selfcentralizing (cf. Proposition 2.11) and it follows from statement 2.10.1 that any σ ∈ F(Q ) can be extended to some F--morphism ρ : R → P and then we have ρ(R) = R , so that ρ determines an element τ ∈ F(R) (cf. 2.3) fulﬁlling τ ◦ ψ = ψ ◦ σ . If Q is not an F-selfcentralizing subgroup of P , it follows from the induction hypothesis that there are an F-selfcentralizing subgroup T ful ˜ ) = {1} and an F-morphism η : R → T such that T contains ﬁlling Op F(T CP η(R) and we have η ◦ F(R) ⊂ F(T ) ◦ η 15.51.2;

15. Reduction results for Grothendieck groups

285

then, = η(Q ) is fully centralized in F , we have it is clear that (η ◦ ψ)(Q) CP η(Q ) ⊂ η(R) since Q is fully centralized too (cf. Proposition 2.11) and we get η ◦ ψ ◦ F(Q) ⊂ η ◦ F(R) ◦ ψ ∈ F(T ) ◦ η ◦ ψ 15.51.3. ˜ Thus, we may assume that Q is F-selfcentralizing but Op F(Q) = {1} ; then, FP (Q ) is a Sylow p-subgroup of F(Q ) (cf. Proposition 2.11) and, ˜ ) in P , it follows again from denoting by U the converse image of Op F(Q statement 2.10.1 that any σ ∈ F(Q ) can be extended to some F-morphism ζ : U → P which this time fulﬁlls ζ(U ) = U , so that ζ determines ξ ∈ F(U ) (cf. 2.3) fulﬁlling ξ ◦ ψ = ψ ◦ σ ; since Q = U , once again it suﬃces to apply the induction hypothesis.

Chapter 16

The local-global question: reduction to the simple groups 16.1 In this chapter we keep all the notation introduced in chapter 15. ˆ (cf. 1.23) and a primitive idempoIn particular, for a ﬁnite k ∗ -group G ˆ ˆ b) tent b of Z(k∗ G) (cf. 1.25), we consider the Grothendieck groups Gk (G, F nc ) — actually, they are O-modules — respectively coming and Gk (F, aut ˆ from the category of k∗ Gb-modules of ﬁnite dimension and, choosing a maxiˆ mal Brauer (b, G)-pair (P, e) and denoting by F = F(b,G) ˆ the corresponding Frobenius P -category, from the functor (cf. Theorem 11.32) F nc : ch∗ (F ) −→ k ∗ -Gr aut nc

16.1.1

lifting autF nc (cf. Proposition A2.10). ˆ of G ˆ 16.2 It is clear that the group of k ∗ -group automorphisms Autk∗ (G) ˆ (cf. 1.25); then, since for any element σ of the acts on the group algebra k∗ G ˆ ˆ ˆ M we have the k∗ Gb-module stabilizer Autk∗ (G)b of b and any k∗ Gb-module ˆ ˆ b acts on the category of k∗ Gb-modules Resσ (M ) , Autk∗ (G) of ﬁnite dimension and therefore, since the action of an inner automorphism determines a functor isomorphic to the identity, the corresponding group of exterior automorphisms (cf. 1.3) + k∗ (G) ˆ b = Outk∗ (G) ˆ b Aut

16.2.1

ˆ b) , so that Gk (G, ˆ b) becomes an still acts on the Grothendieck group Gk (G, ˆ OOutk∗ (G)b -module. ˆ (P,e) 16.3 On the other hand, it is clear that the stabilizer Autk∗ (G) of (P, e) acts on the Frobenius P -category F throughout Frobenius functors (cf. 12.1) and, once again, the Frobenius functors from F to F induced by the elements of NGˆ (P, e) are isomorphic to the identity (cf. 12.1). Moreover, it is nc clear that any Frobenius functor f : F → F stabilizes the full subcategory F and therefore it induces a functor ch∗ (f ) : ch∗ (F ) −→ ch∗ (F ) nc

nc

nc

16.3.1;

then, it follows from the uniqueness part of Theorems 11.10 and 11.32 above that we have a natural isomorphism nc F nc ∼ F nc ◦ ch∗ (f ) νˆf : aut = aut

16.3.2.

288

Frobenius categories versus Brauer blocks

ˆ (P,e) , we get the O-module isomor16.4 Hence, for any σ ∈ Autk∗ (G) phisms (cf. A3.9) nc F nc ) H0o (F , gk ◦ aut +

H0o (fσ ,gk ◦a utF nc ) nc

nc F nc ◦ ch∗ (fnc H0o F , gk ◦ aut σ ) +

H0o (F

16.4.1

nc

,νfσ )

F nc ) H0o (F , gk ◦ aut nc

and it is easily checked that this correspondence deﬁnes an action of the ˆ (P,e) on the O-module (cf. A3.9) stabilizer Autk∗ (G) nc F nc ) ∼ F nc ) = Gk (F, aut F nc ) H0o (F , gk ◦ aut = lim (gk ◦ aut

←−

16.4.2.

Moreover, if σ is induced by an element of NGˆ (P, e) then we have fσ ∼ = idF nc F nc ) (cf. 12.1) and thus the corresponding automorphism of H0o (F , gk ◦ aut ˆ acts transitively on the set of maximal is the identity. On the other hand, G ˆ Brauer (b, G)-pairs (cf. 1.15) and therefore we get an exact sequence ˆ (P,e) −→ Outk∗ (G) ˆ b −→ 1 NGˆ (P, e) −→ Autk∗ (G)

16.4.3.

ˆ b -module too. F nc ) becomes an OOutk∗ (G) Consequently, Gk (F, aut 16.5 Since Alperin’s Conjecture is equivalent to the existence of an ˆ b) ∼ F nc ) (cf I 32), in this new context O-module isomorphism Gk (G, = Gk (F, aut it seems reasonable to go farther and try to discuss the statement: ˆ b -module isomorphism (Q) There is an OOutk∗ (G) ˆ b) ∼ F nc ) Gk (G, = Gk (F, aut

16.5.1.

Although stronger than Alperin’s Conjecture, in this chapter we show that this statement has the advantage of admitting the possibility of an inductive proof ; that is to say, whether it is true, its proof can be reduced to the ˆ veriﬁcation of some conditions in the case where the k ∗ -quotient G of G (cf. 1.23) contains a normal noncommutative simple subgroup S such that CG (S) = {1} and p divides |S| without dividing |G : S| (cf. Theorem 16.45 below). Proposition 16.6 With the notation above, assume that there is a nontrivial ˆ of G ˆ and a non-G-stable ˆ ˆ such that characteristic k ∗ -subgroup N block c of N ˆ ˆ bc = 0 , and denote by H the stabilizer of c in G . The product bc is a block ˆ and if (Q) holds for (bc, H) ˆ then it holds for (b, G) ˆ . of H

16. The local-global question: reduction ˆ

289 ˆ

G Proof: It is clear that TrG ˆ (bc) = bTrH ˆ (c) = b and then it follows from H ˆ , that bc belongs to a point β of Proposition 3.5 in [53] that bc is a block of H ˆ on k Gb ˆ and that we have Gˆ and H-interior ˆ H algebra isomorphism (cf. 1.30) ˆ ˆ ∼ ˆ k∗ Gb = IndG ˆ (k∗ Hbc) and H

ˆ β∼ ˆ (k∗ Gb) = k∗ Hbc

16.6.1;

moreover, it follows from the same proposition that we may assume that ˆ β (cf. 1.15) determines the chosen maximal a defect pointed group Pγ of H ˆ Brauer (b, G)-pair (P, e) ; then, always from the same proposition, P is also a ˆ and the Frobenius P -category H = F ˆ defect group of the block bc of H (bc,H) associated with this block coincides with F . ˆ On the other hand, since we assume that a k ∗ -group automorphism of G ˆ , if it stabilizes b then it still stabilizes the G-orbit ˆ stabilizes N of c and it is ˆ ˆ ∗ clear that the stabilizer Autk (G)c of c acts on H ; hence, we get a canonical group homomorphism ˆ b −→ Outk∗ (H) ˆ bc Outk∗ (G)

16.6.2;

more generally, we get as above a commutative diagram of exact sequences ˆ γ NHˆ (Pγ ) −→ Autk∗ (H) + ↑ ˆ γ NGˆ (Pγ ) −→ Autk∗ (G)

ˆ bc −→ Outk∗ (H) ↑ ˆ b −→ Outk∗ (G)

−→ 1 16.6.3. −→ 1

Pushing it further, from the Frattini argument it is not diﬃcult to ˆ b and Outk∗ (H) ˆ bc to the groups get group homomorphisms from Outk∗ (G) of exterior automorphisms of the source P -interior algebras of the blocks ˆ and bc of H ˆ respectively. But, it follows from the isomorphism b of G ˆ β ∼ ˆ that the source P -interior algebras of the blocks b of G ˆ (k∗ Gb) = k∗ Hbc ˆ and bc of H are isomorphic throughout an isomorphism compatible with F nc and aut Hnc , Gk (F, aut F nc ) and homomorphism 16.6.2. Then, since aut ˆ ˆ nc Gk (H, autH ) , and Gk (G, b) and Gk (H, bc) are completely determined from the respective source algebras (cf. Theorem 3.1 in [39] and Proposition 6.21 in [42]), we are done. Proposition 16.7 With the notation above, assume that any block involved ˆ is G-stable, and that there is a in b of any characteristic k ∗ -subgroup of G ∗ ˆ ˆ with a block c of defect zero such nontrivial characteristic k -subgroup N of G ∗ ˆ that bc = b . Then there are a k -group G with k ∗ -quotient G and a k ∗ -group ˆ c)∗ extending the canonical map N ˆ c such ˆ → k∗ N homomorphism ˆG → (k∗ N ˆ ¯=G ˆ ∗ ˆG◦ /∆(N ) , we have a G-interior ˆ that, setting G algebra isomorphism ˆ¯¯b ˆ ∼ ˆ c ⊗k k∗ G k∗ Gb = k∗ N

16.7.1

ˆ ˆ¯ , it holds for (b, G) ¯ and if (Q) holds for (¯b, G) ˆ . for a suitable block ¯b of G

290

Frobenius categories versus Brauer blocks

ˆ c is a simple algebra (cf. 1.17), the action of G can be lifted Proof: Since k∗ N ∗ ˆ c)∗ extending the canonical map to a k -group homomorphism ρ : ˆG → (k∗ N ˆ , with the same ˆ → k∗ N ˆ c where ˆG is a suitable k ∗ -group containing N N ∗ ˆ ∗N ˆ◦ ∼ k -quotient G (cf. 1.25); then, the canonical decomposition N = k∗ × N ˆ ˆG◦ (cf. 1.24) isomorphic (cf. 1.24) determines a normal subgroup ∆(N ) of G∗ ˆ ˆ¯ has an ¯ = G∗ ˆ ˆG◦ /∆(N ) , the tensor product k∗ N ˆ c⊗k k∗ G to N and, setting G ˆ evident G-interior algebra structure (cf. 1.27) and it follows from Theorem 3.7 ˆ∼ ˆ ∗ ˆG◦ induces a in [53] that the obvious k ∗ -group isomorphism G = ˆG ∗ G ˆ G-interior algebra isomorphism ˆ¯¯b ˆ ∼ ˆ c ⊗k k∗ G k∗ Gb = k∗ N

16.7.2

ˆ ˆ¯¯b-modules ¯ , so that the categories of k∗ Gbˆ and k∗ G for a suitable block ¯b of G are equivalent. Moreover, it follows from the same theorem that P is isomorˆ ˆ¯¯b , ¯ , that γ determines a local point γ¯ of P¯ on k∗ G phic to its image P¯ in G that P¯γ¯ is a defect pointed group of ¯b and that this isomorphism α : P ∼ = P¯ induces a Frobenius equivalence fα : F ∼ = F(¯b,G) ˆ ¯

16.7.3.

As in the argument above, since we assume that a k ∗ -group automorˆ stabilizes N ˆ , if it stabilizes b then it ﬁxes c and also induces a phism of G ∗ k -group automorphism of ˆG , so that it induces a k ∗ -group automorphism ˆ ¯ stabilizing ¯b ; hence, we get a canonical group homomorphism of G ˆ¯ ¯ ˆ b −→ Outk∗ (G) ω : Outk∗ (G) b

16.7.4.

Pushing it further, from the Frattini argument it is not diﬃcult to get group ˆ¯ ¯ to the groups of exterior ˆ b and Outk∗ (G) homomorphisms from Outk∗ (G) b ˆ¯¯b) ˆ γ and (k∗ G automorphisms of the source P - and P¯ -interior algebras (k∗ Gb) γ ¯ ˆ ¯ ˆ ¯ of the blocks b of G and b of G respectively (cf. 1.13). But, it follows from isomorphism 16.7.2 that we have a canonical P -interior algebra embedding ˆ¯¯b) ˆ γ −→ k∗ N ˆ c ⊗k Resα (k∗ G (k∗ Gb) γ ¯

16.7.5,

and, since we have a P -interior algebra embedding (cf. 1.21.3) ˆ c ⊗k (k∗ N ˆ c)◦ k −→ k∗ N

16.7.6,

it is easily checked that we still have a canonical P -interior algebra embedding ˆ ¯¯b)γ¯ −→ (k∗ N ˆ c)◦ ⊗k (k∗ Gb) ˆ γ Resα (k∗ G

16.7.7.

16. The local-global question: reduction

291

ˆ γ Then, since the restriction induces Morita equivalences from (k∗ Gb) ˆ ˆ ¯¯b)γ¯ to k∗ Gb ¯¯b respectively (cf. 1.13), from the embeddings ˆ and k∗ G and (k∗ G ˆ b -module isomorphism above we get an OOutk∗ (G) ˆ ˆ b) ∼ ¯ ¯b) Gk (G, = Resω Gk (G,

16.7.8.

Moreover, it follows from the uniqueness part of Theorems 11.10 and 11.32, and from the embeddings above that we have a natural isomorphism (F F nc ∼ nc ◦ f aut = aut α ˆ ) (¯ b,G)

16.7.9

ˆ b -module isomorphism and therefore it is clear that we get an OOutk∗ (G) F nc ) ∼ (F nc ) Gk (F, aut 16.7.10 . = Resω Gk (F(¯b,G) ˆ , aut ¯ ¯ ˆ ) (b,G)

ˆ b -module isomorphisms 16.7.8 and 16.7.10 prove the last Now, the OOutk∗ (G) statement. We are done. Proposition 16.8 With the notation above, assume that any block involved ˆ is G-stable and has in b of any nontrivial characteristic k ∗ -subgroup of G ˆ ˆ positive defect. Set Z = Z Op (G) and H = CGˆ (Z) and, for any cyclic ˆ H ˆ , denote by H ˆ C the converse image of C in G ˆ . Then, p -subgroup C of G/ C C ˆ ˆ b is a block of H and if (Q) holds for (b, H ) for any cyclic p -subgroup C ˆ H ˆ , it holds for (b, G) ˆ . of G/ ˆ ; since b is contained in kC ˆ (U ) (cf. 1.19 above and Proof: Set U = Op (G) G Proposition 5.15 in [42]), it follows from our hypothesis that b is also a block ˆ ˆ and of C ˆ (U ) ; in particular, (U, b) is a Brauer (b, G)-pair contained of H G ˆ in (P, e) (cf. 1.15) and therefore, setting Q = H ∩ P , there is a block f of CGˆ (Q) = CHˆ (Q) such that (cf. 1.15) (U, b) ⊂ (Q, f ) ⊂ (P, e)

16.8.1;

ˆ actually, (U, b) and (Q, f ) are also Brauer (b, H)-pairs and (Q, f ) is a maximal one (cf. 1.15 and 1.19); moreover, according to Proposition 15.10, b is a block ˆ C for any cyclic p -subgroup C of G/ ˆ H ˆ. of H ˆ , Denote by H = F ˆ the Frobenius Q-category associated with (b, H) (b,H)

ˆ , it follows again which is clearly a subcategory of F . Since CGˆ (Q) ⊂ H from Proposition 15.10 that any H-selfcentralizing subgroup of Q is F-nilcensc nc tralized, so that H is also a subcategory of F , and by the Frattini argument we get (cf. 1.15) ˆ H ˆ F(Q)/H(Q) ∼ = G/ = NGˆ (Q, f )/NHˆ (Q, f ) ∼

16.8.2.

292

Frobenius categories versus Brauer blocks

Moreover, for any cyclic p -subgroup C of F(Q)/H(Q) , it follows from Proposition 12.12 that there is a unique Frobenius Q-category HC , containing and normalizing H , such that HC (Q) is the converse image of C in F(Q) ; ˆ H ˆ , it follows from Proposition 12.8 moreover, identifying F(Q)/H(Q) with G/ C that H coincides with the Frobenius Q-category associated with the block b ˆC . of H That is to say, we are in the situation discussed in 15.24 and thereˆ (P,e) on the set of fore, considering the action of the stabilizer Autk∗ (G) ˆ H ˆ , it follows from Corollaries 15.26 cyclic p -subgroups of F(Q)/H(Q) ∼ = G/ ˆ b -module isomorphisms and 15.39 that we get OOutk∗ (G) ˆ b) ∼ Gk (G, =

C

F nc ) ∼ Gk (F, aut =

G/ ˆ H ˆ ˆ C , b) RHˆGk (H F (Q)/H(Q) (HC )nc ) RHGk (H , aut

16.8.3

C

C

ˆ H ˆ. where C runs over the set of cyclic p -subgroups of F(Q)/H(Q) ∼ = G/ ˆ C )b -module isomorphism Now, let us assume that we have an OOutk∗ (H ˆ C , b) ∼ (HC )nc ) Gk (H = Gk (HC , aut

16.8.4

ˆ H ˆ ; since any group hofor any cyclic p -subgroup C of F(Q)/H(Q) ∼ = G/ momorphism from C to k ∗ determines a homomorphism to k ∗ from the ˆ C , and therefore it determines a k ∗ -group automorphism k ∗ -quotient H C of H C ˆ in the center of Autk∗ (H ˆ C ) , we have an O-algebra homomorphism of H ˆ C )b Gk (C) ∼ 16.8.5 = OHom(C, k ∗ ) −→ Z OOutk∗ (H compatible with the Gk (C)-module structures of both members in isomorphism 16.8.4 (cf. 15.21 and 15.33). ∼ G/ ˆ H ˆ , we Consequently, for any cyclic p -subgroup C of F(Q)/H(Q) = C ˆ still have an OOutk∗ (H )b -module isomorphism (cf. isomorphisms 15.23.2 and 15.37.1) ˆ C , b) ∼ (HC )nc ) RGk (C) ⊗Gk (C) Gk (H = RGk (C) ⊗Gk (C) Gk (HC , aut + ˆ C , b) RHˆGk (H

+

16.8.6.

(HC )nc ) RHGk (H , aut C

ˆ b -module isomorphism At this point, it is easy to construct an OOutk∗ (G) G/ F (Q)/H(Q) ˆ H ˆ ∼ ˆ C , b) (HC )nc ) RHˆGk (H RHGk (HC , aut 16.8.7 = C

C

ˆ H ˆ . We where C runs over the set of cyclic p -subgroups of F(Q)/H(Q) ∼ = G/ are done.

16. The local-global question: reduction

293

ˆ and Proposition 16.9 With the same notation as above, set Z = Z Op (G) ˆ¯ . Assume that any block ˆ ¯ = G/Z ˆ , and denote by ¯b the image of b in Z(k∗ G) G ˆ is G-stable involved in b of any nontrivial characteristic k ∗ -subgroup of G ˆ and has positive defect, and that the quotient G/CGˆ (Z) is a cyclic p -group. ˆ for any block c¯ of G ¯ ¯ˆ such that c¯¯b = c¯ , it holds Then, if (Q) holds for (¯ c, G) ˆ . for (b, G) ˆ = C ˆ (Z) and H ¯ˆ = H/Z ˆ Proof: Set H , and denote by Pγ the defect pointed G group of b determined by (P, e) , and by γ¯ the local point of P¯ = P/Z on ¯ˆ containing the image of γ ∩ k∗ H ˆ (cf. Lemma 15.49). It follows from k∗ H ˆ ¯ Proposition 15.50 that, setting Pγ¯ = P k∗ C G¯ˆ (P¯γ¯ ) and denoting by P¯γ¯ the H

set of NGˆ (Pγ )-orbits in Pγ¯ and by ϕ¯ the NGˆ (Pγ )-orbit of ϕ ∈ Pγ¯ , we have ¯b = ¯bϕ¯ 16.9.1 ¯γ¯ ϕ∈ ¯ P

ˆ ¯ ; moreover, according to the same where, for any ϕ ∈ Pγ¯ , ¯bϕ¯ is a block of G ¯ proposition, P(¯γ ,ϕ) is a defect pointed group of ¯bϕ¯ and, setting F¯ϕ¯ = F(¯b ,G) ˆ , ¯ we have the following O-module isomorphisms ˆ ˆ b) ∼ ¯ ¯bϕ¯ ) Gk (G, Gk (G, = ¯γ¯ ϕ∈ ¯ P

F nc ) ∼ Gk (F, aut =

(F¯ )nc ) Gk (F¯ϕ¯ , aut ϕ ¯

ϕ ¯

16.9.2.

¯γ¯ ϕ∈ ¯ P

ˆ¯ inducing ˆ acts on the k ∗ -group G More precisely, it is clear that Autk∗ (G) an O-algebra homomorphism ˆ¯ ˆ −→ OOutk∗ (G) OOutk∗ (G)

16.9.3

ˆ P of Pγ in Autk∗ (G) ˆ clearly acts on the and, since the stabilizer Autk∗ (G) γ ˆ ¯ k ∗ -group C G¯ˆ (P¯γ¯ ) (cf. deﬁnition 15.5.4), by the exact sequence 16.4.3 we get H ˆ b on P¯γ¯ ; then, for any ϕ¯ ∈ P¯ , denoting by Outk∗ (G) ˆ b,ϕ¯ an action of Outk∗ (G) ˆ the stabilizer of ϕ¯ in Outk∗ (G)b , on the one hand we get a new O-algebra homomorphism ˆ¯ ¯ ˆ b,ϕ¯ −→ OOutk∗ (G) OOutk∗ (G) 16.9.4; bϕ¯ ˆ b acts on the two families on the other hand, it is quite clear that Outk∗ (G) of O-modules ˆ ¯ ¯bϕ¯ )}ϕ∈ {Gk (G, ¯γ¯ ¯ P

(F¯ )nc )}ϕ∈ and {Gk (F¯ϕ¯ , aut ¯γ¯ ¯ P ϕ ¯

16.9.5

294

Frobenius categories versus Brauer blocks

and it is easily checked that these actions are compatible with the isomorˆ b -module isomorphisms phisms 16.9.2; hence, we obtain OOutk∗ (G) ˆ b) ∼ Gk (G, =

F nc ) ∼ Gk (F, aut =

ˆ ∗ (G)

Out

b IndOutk∗ (G) ˆ

ˆ¯ ¯b ) Gk (G, ϕ ¯

(F¯ )nc ) Gk (F¯ϕ¯ , aut ϕ ¯

b,ϕ ¯

k

ϕ ¯

b,ϕ ¯

k

ϕ ¯

ˆ ∗ (G)

Out

b IndOutk∗ (G) ˆ

16.9.6

ˆ b -orbits where ϕ¯ runs over a set of representatives for the set of Outk∗ (G) ¯ on Pγ¯ . ¯ˆ -module isoConsequently, if we assume that we have an OOut ∗ (G) k

bϕ¯

morphism ˆ ¯ ¯bϕ¯ ) ∼ (F¯ )nc ) Gk (G, = Gk (F¯ϕ¯ , aut ϕ ¯

16.9.7

for any ϕ¯ ∈ P¯γ¯ , it suﬃces to consider the restrictions throughout homomorˆ b -orbits on P¯γ¯ to phism 16.9.4 of a set of representatives for the Outk∗ (G) ˆ b -module isomorphism Gk (G, ˆ b) ∼ F nc ) . We obtain an OOutk∗ (G) = Gk (F, aut are done. 16.10 At this point, arguing by induction on |G| , in order to prove that ˆ fulﬁlls (Q) we may assume that a block b of a nontrivial ﬁnite k ∗ -group G ˆ is any block involved in b of any nontrivial characteristic k ∗ -subgroup of G ∗ ˆ G-stable and has positive defect, which forces Op (G) = k , and that we also have (cf. 1.31) ˆ = {1} Op (G) 16.10.1. Then, it is well-known that the product H of all the minimal nontrivial ˆ is a characteristic subgroup of G normal subgroups of the k ∗ -quotient G of G isomorphic to a direct product H∼ =

Hi

16.10.2

i∈I

of a ﬁnite family of noncommutative simple groups Hi of order divisible by p (cf. Theorem 1.5 in [28]). ˆ and H ˆ i the respective converse images of H and 16.11 Denoting by H ˆ Hi in G , it is quite clear that ˆ = H

i∈I

ˆi H

ˆ = k∗ and CGˆ (H)

16.11.1;

ˆ on H ˆ moreover, since this decomposition is unique, the action of Autk∗ (G) ˆ ˆ the kernel of the action induces an Autk∗ (G)-action on I and, denoting by W

16. The local-global question: reduction

295

ˆ on I , we have H ˆ ⊂W ˆ and get an injective group homomorphism of G ˆ /H ˆ −→ W

ˆ i) Outk∗ (H

16.11.2;

i∈I

thus, admitting the announced Classiﬁcation of the Finite Simple Groups, ˆ /H ˆ is solvable. the quotient W ˆ such that cb = b , recall that (P, e) is a 16.12 Let c be the block of H ˆ ˆ ; as in our standard setting maximal Brauer (b, G)-pair and set Q = P ∩ H in 15.7, it follows from Proposition 15.9 that Q is a defect group of c and that there is a block f of CHˆ (Q) such that we have eBrP (f ) = 0 and (Q, f ) ˆ is a maximal Brauer (c, H)-pair; then, we consider the Frobenius P - and Q-categories F = F(b,G) and H = F(c,H) 16.12.1. ˆ ˆ ˆ i , we have Q = Qi Since clearly c = ⊗i∈I ci where ci is a block of H i∈I where Qi is a defect group of ci , and f = ⊗i∈I fi where fi is a block of CHˆ i (Qi ) ˆ i )-pair. and (Qi , fi ) is a maximal Brauer (ci , H 16.13 Moreover, since we are assuming that any block involved in b ˆ has positive defect, for of any nontrivial characteristic k ∗ -subgroup of G particular, since any H-selfcenany i ∈ I the defect group Qi is nontrivial ; in tralizing subgroup T of Q contains Z(Q) = i∈I Z(Qi ) , CGˆ (T ) centralizes Z(Qi ) = {1} for any i ∈ I and therefore we get ˆ CGˆ (T ) ⊂ W

16.13.1.

ˆ = H·C ˆ ˆ (Q, f ) is a normal subgroup of G ˆ contained in W ˆ and Thus, N G ˆ /H ˆ is solvable (cf. 16.11). therefore the quotient N ˆ /H) ˆ by (cf. 1.31) 16.14 As usual, we deﬁne Op ,p (N ˆ ˆ ˆ /H)/O ˆ ˆ ˆ ˆ ˆ Op ,p (N p (N /H) = Op (N /H)/Op (N /H)

16.14.1.

ˆ of Op ,p (N ˆ /H) ˆ in G ˆ and the block d of K ˆ Consider the converse image K ˆ ∩ P ; it follows from Proposition 15.9 that R fulﬁlling db = b , and set R = K is a defect group of d and then, from Proposition 5.3 and Corollary 6.3 in [34], ˆ /H) ˆ . But, according to Proposithat R/Q is a Sylow p-subgroup of Op ,p (N ˆ tion 15.10, (Q, f ) remains a Brauer (d, K)-pair since f K = f , and, according to Proposition 15.9, we may assume that there exists a block g of CKˆ (R) such ˆ that (Q, f ) ⊂ (R, g) as Brauer (d, K)-pairs (cf. 1.15) and therefore we still G have (R, g ) ⊂ (P, e) (cf. Proposition 15.10). Then, it is clear that CGˆ (R, g)

296

Frobenius categories versus Brauer blocks

is contained in CGˆ (Q, f ) and therefore, since CGˆ (R) acts trivially on R/Q ˆ /H ˆ is solvable, we get (cf. Theorem 3.2 in [28, Ch. 6]) and the quotient N ˆ CGˆ (R, g) ⊂ K

16.14.2.

At this point, it follows from Proposition 15.10 that d = b and it is clear ˆ that (R, g) is also a maximal Brauer (b, K)-pair; denote by K = F(b,K) ˆ the ˆ Frobenius R-category associated with the block (b, K) , which is clearly a subcategory of F . From now on in this chapter, all this will be our standard setting. Proposition 16.15 In our standard setting, for any cyclic p -subgroup C ˆ . Then, b is a block ˆ K ˆ , denote by K ˆ C the converse image of C in G of G/ C C ˆ and if (Q) holds for (b, K ˆ ) for any cyclic p -subgroup C of G/ ˆ K ˆ , it of K ˆ holds for (b, G) . ˆ (cf. 16.14) and therefore, Proof: We already know that b is a block of K ˆ , it follows from Proposition 15.10 that b is also a block since CGˆ (R, g) ⊂ K C ˆ ˆ K ˆ and that any K-selfcentralizing of K for any cyclic p -subgroup C of G/ sc nc subgroup of R is F-nilcentralized, so that K is a subcategory of F , and together with the Frattini argument we get (cf. 1.15) ˆ K ˆ F(R)/K(R) ∼ = NGˆ (R, g)/NKˆ (R, g) ∼ = G/

16.15.1

ˆ since (R, g) is a maximal Brauer (b, K)-pair. Now, for any cyclic p -subgroup C of F(R)/K(R) , it follows from Proposition 12.12 that there is a unique Frobenius R-category KC , containing and normalizing K , such that KC (R) is the converse image of C in F(R) ; moreˆ K ˆ , it follows from Proposition 12.8 that over, identifying F(R)/K(R) with G/ ˆC . KC is the Frobenius R-category associated with the block b of K That is to say, we are in the situation discussed in 15.24 above and ˆ (P,e) on the set of therefore, considering the action of the stabilizer Autk∗ (G) ˆ K ˆ , it follows from Corollaries 15.26 cyclic p -subgroups of F(R)/K(R) ∼ = G/ ˆ and 15.39 that we get OOutk∗ (G)b -module isomorphisms ˆ b) ∼ Gk (G, =

C

F nc ) ∼ Gk (F, aut =

G/ ˆ K ˆ ˆ C , b) RKˆ Gk (K F (R)/K(R) (KC )nc ) RKGk (K , aut

16.15.2

C

C

ˆ K ˆ. where C runs over the set of cyclic p -subgroups of F(R)/K(R) ∼ = G/

16. The local-global question: reduction

297

ˆ C )b -module isomorphism Now, let us assume that we have an OOutk∗ (K ˆ C , b) ∼ (KC )nc ) Gk (K = Gk (KC , aut

16.15.3

ˆ K ˆ ; once again, since any for any cyclic p -subgroup C of F(R)/K(R) ∼ = G/ group homomorphism from C to k ∗ determines a homomorphism to k ∗ from ˆ C , and therefore it determines a k ∗ -group automorthe k ∗ -quotient K C of K C ˆ ˆ C ) , we get an O-algebra homomorphism of K in the center of Autk∗ (K phism ˆ C )b 16.15.4 Gk (C) ∼ = OHom(C, k ∗ ) −→ Z OOutk∗ (K compatible with the Gk (C)-module structures of both members in isomorphism 16.15.3 (cf. 15.21 and 15.33). ˆ K ˆ , we Consequently, for any cyclic p -subgroup C of F(R)/K(R) ∼ = G/ C ˆ still have an OOutk∗ (K )b -module isomorphism (cf. isomorphisms 15.23.2 and 15.37.1) ˆ C , b) ∼ (KC )nc ) RGk (C) ⊗Gk (C) Gk (K = RGk (C) ⊗Gk (C) Gk (KC , aut +

+

16.15.5.

(KC )nc ) RKGk (K , aut

ˆ C , b) RKˆ Gk (K

C

ˆ b -module isomorAt this point, it is not diﬃcult to construct an OOutk∗ (G) phism C

G/ F (R)/K(R) ˆ K ˆ ∼ ˆ C , b) (KC )nc ) RKˆ Gk (K RKGk (KC , aut =

16.15.6

C

ˆ K ˆ . We where C runs over the set of cyclic p -subgroups of F(R)/K(R) ∼ = G/ are done. 16.16 In our standard setting, it follows from this proposition that now ˆ K ˆ is a cyclic p -group, so that in parit suﬃces to discuss the case where G/ ˆ the converse image of Op (K/ ˆ H) ˆ ticular we have P = R ; let us denote by L ˆ ; since we have in K ˆ L ˆ∼ K/ 16.16.1 = P/Q ∗ and b = x λx ·ˆ x where x runs over the set of p -elements of the k -quotient ˆ ˆ (cf. 16.2 in [54]), b is also a block of L ˆ and K of K and x ˆ is a lifting of x to K we denote by L = F(b,L) ˆ the corresponding Frobenius Q-category. Moreover, ˆ ¯ ˆ acts trivially on the Frattini space V = P¯ /Φ(P¯ ) ; G acts on P = P/Q and K ˆV = V G ˆ , it is clear that b thus, considering the semidirect product G remains a block of the subgroups ˆV = V × K ˆ K

ˆV = V × L ˆ and L

16.16.2,

298

Frobenius categories versus Brauer blocks V

V

that the p-subgroups P = V × P and Q = V × Q are respective defect ˆ V , and that e remains a block of C ˆ V (P V ) . Then, ˆ V and L groups of b in K K since we still have (cf. 16.14.2) V

ˆ CGˆ V (P , e) ⊂ K

V

16.16.3,

ˆ V and it is clear it follows from Proposition 15.10 that b is also a block of G ˆV → G ˆ dethat the restriction throughout the canonical homomorphism G ˆ termines an OOut(G)b -module isomorphism ˆ b) ∼ ˆ , b) Gk (G, = Gk (G V

16.16.4.

16.17 Analogously, respectively denote by F

V

= F(b,Gˆ V )

V

K = F(b,Kˆ V )

,

V

and L = F(b,Lˆ V )

16.17.1

V V ˆ V ) , (b, K ˆ V ) and the Frobenius P - and Q -categories of the blocks (b, G V V V V V ˆ ) ; since G ˆ /K ˆ is a p -group, a subgroup of P is F -selfcentralizing (b, L V V if and only if it is K -selfcentralizing; moreover, it is clear that it is K -selfV centralizing if and only if it has the form T = V × T where T is a K-selfcentralizing or, equivalently, an F-selfcentralizing subgroup of P . In this case, denoting by h the block of CKˆ (T ) such that (T, h) ⊂ (P, e) (cf. 16.14), h is also a block of V CKˆ V (T ) = V × CKˆ (T ) 16.17.2

ˆ K ˆ is a p -group, it follows from Proposition 15.10 that we have and, since G/ V

CGˆ V (T , h) = V × CKˆ (T ) = V × CGˆ (T, h)

16.17.3

and that the relative traces V

V

C ˆ V (T )

hG = Tr

G

C ˆ V (T V ,h) G

C (T )

(h) and hG = TrCGˆˆ (T,h) (h) G

16.17.4

V

are blocks of CGˆ V (T ) and CGˆ (T ) respectively. 16.18 Consequently, we get V

V V V V F (T ) ∼ = NGˆ V (T , hG )/CGˆ V (T ) V V ∼ = NGˆ V (T , h)/CGˆ V (T , h) ∼ = N ˆ (T, h)/C ˆ (T, h) ∼ = F(T )

G

G

16.18.1;

16. The local-global question: reduction

299

moreover, since we have (cf. Proposition 3.5 in [53]) V V C ˆ V (T ) V ¯G ∼ ¯ k∗ C¯Gˆ V (T )h = IndCG (T V ,h) k∗ C¯Kˆ (T )h ˆV G

k∗ C¯Gˆ (T )h ∼ = ¯G

C (T ) ¯ IndCGˆˆ (T,h) k∗ C¯Kˆ (T )h G

16.18.2,

it is easily checked that we still get a canonical k ∗ -group isomorphism (cf. Proposition 6.18 in [42]) V V ˆ ) Fˆ (T ) ∼ 16.18.3; = F(T then, denoting by p : F V → F the obvious Frobenius functor , it is not diﬃcult to check that we get a natural isomorphism ∗ sc V sc ∼ sc aut (F ) = autF ◦ ch (p )

16.18.4.

ˆ b -moIn conclusion, according to Corollary 14.7, we have also an OOutk∗ (G) dule isomorphism V F nc ) ∼ V nc ) Gk (F, aut = Gk (F , aut (F )

16.18.5.

ˆ K ˆ is a Proposition 16.19 In our standard setting, assume that C = G/ ˆ the converse image of Op (K/ ˆ H) ˆ in K ˆ and cyclic p -group and denote by L C ˆ ˆ by L the converse image in G of a complement of the Sylow p-subgroup ˆ L ˆ . Then, b is a block of L ˆ C and if (Q) holds for (b, L ˆ C ) , it holds of G/ ˆ . for (b, G) ˆ , we have K ˆ = H·C ˆ ˆ (Q, f ) Proof: Since, from the very deﬁnition of K K V V V ˆ =L ˆ ·C ˆ V (Q , f ) (cf. 16.17.2) and the quotient (cf. 16.14), we still have K K V ˆV L ˆ ·C ˆ V (QV , f ) is also a cyclic p -group. Moreover, since C V (QV , f ) G G G V acts trivially on V = P¯ /Φ(P¯ ) , a p -subgroup of C V (Q , f ) acts trivially on G

(cf. Theorem 1.4 in [28, Ch. 5]) ˆ L ˆ∼ ˆ H) ˆ Op (G/ ˆ H) ˆ P¯ = P/Q ∼ = K/ = Op ,p (G/

16.19.1

V

and therefore it is contained in CLV (Q , f ) (cf. Theorem 3.2 in [28, Ch. 6]), V

V

so that the quotient CGV (Q , f )/CLV (Q , f ) is a p-group; thus, we have V

V

CGˆ V (Q , f ) = CKˆ V (Q , f ) and therefore we still have

ˆ V ·C ˆ V (QV , f ) L G V V V ∼ = NGˆ V (Q , f ) NLˆ V (Q , f )·CGˆ V (Q , f )

ˆV C∼ =G

V V V V ∼ = F (Q )/L (Q )

16.19.2.

300

Frobenius categories versus Brauer blocks V

V

Consequently, denoting by (L )C the Frobenius Q -category determined by this isomorphism (cf. Proposition 12.12), it follows from Proposition 15.13 that we have O-isomorphisms V V V nc ) ∼ V C nc Gk (F , aut = Gk (L )C , aut (F ) ((L ) ) V C ˆ ) ,b ˆ V , b) ∼ Gk (G = Gk (L

16.19.3.

ˆ acts transitively on the set of choices for L ˆC , On the other hand, since G it follows from the Frattini argument that we have canonical group homomorphisms ˆ b −→ Outk∗ (L ˆ C )b Outk∗ (G) 16.19.4. V C ˆ V )b −→ Outk∗ (L ˆ ) Outk∗ (G b Moreover, since it follows from Proposition 15.13 that the bottom isomorphism in 16.19.3 is induced by the restriction, this isomorphism is actually ˆ V )b -module isomorphism; but, since Autk∗ (G) ˆ stabilizes H ˆ, an OOutk∗ (G V ˆ b to Outk∗ (G ˆ )b . we have a canonical group homomorphism from Outk∗ (G) ˆ b -module isomorphisms (cf. 16.16.4) Thus, we get OOutk∗ (G) V C ˆ b) ∼ ˆ V , b) ∼ ˆ C , b) ˆ ) ,b ∼ Gk (G, = Gk (G = Gk (L = Gk (L

16.19.5.

ˆ b -module isomorphisms Similarly, we claim that we have OOutk∗ (G) V F nc ) ∼ V nc ) Gk (F, aut = Gk (F , aut (F ) V C ∼ V C nc ∼ (LC )nc = Gk (L ) , aut = Gk LC , aut ((L ) )

16.19.6;

indeed, it follows from 16.3 and 16.4 that the groups ˆ b Outk∗ (G) V C ˆ ) Outk∗ (L b

,

ˆ V )b Outk∗ (G

and

ˆ C )b Outk∗ (L

16.19.7

respectively act on the Grothendieck groups F nc ) , V nc ) Gk (F, aut Gk (F , aut (F ) V C C V C nc (LC )nc Gk (L ) , aut and Gk L , aut ((L ) ) V

16.19.8;

ˆ b acts on thus, according to all the group homomorphisms above, Outk∗ (G) all of these Grothendieck groups.

16. The local-global question: reduction

301

ˆ b -module isomorphism is just isoThen, in 16.19.6 the ﬁrst OOutk∗ (G) ˆ b -module isomorphism follows morphism 16.18.5. The second OOutk∗ (G) from the naturality of isomorphism 15.13.2 in Proposition 15.13. Finally, ˆ C and L ˆ in the note that the arguments in 16.17 and 16.18 still apply to L ˆ ˆ ˆ place of G and K , and therefore the last OOutk∗ (G)b -module isomorphism follows again from isomorphism 16.18.5. ˆ C ) guarantees the existence of a At this point, condition (Q) for (b, L ˆ C )b -module isomorphism OOutk∗ (L ˆ C , b) ∼ (LC )nc Gk (L = Gk LC , aut

16.19.9

ˆ b -module isomorphisms 16.19.5 and 16.19.6 imply and then the OOutk∗ (G) ˆ condition (Q) for (b, G) . We are done. 16.20 In our standard setting, it follows from this proposition that it ˆ H ˆ is a p -group and G/ ˆ K ˆ is cyclic; suﬃces to discuss the case where G/ ˆ ˆ ˆ ˆ in this case, we actually have K = H·CGˆ (Q, f ) and, since W contains K ˆ W ˆ is cyclic too. For any i ∈ I , we respectively denote by Gi (cf. 16.11), G/ i ˆ in Autk∗ (H ˆ i ) ; note that, since and K the respective images of W and K K i = Hi ·CK i (Qi , fi )

16.20.1

and Gi /K i is cyclic, Lemmas 16.21 and 16.27 below successively determine a ˆ i containing H ˆ i of k ∗ -quotient K i and a k ∗ -group G ˆ i containing K ˆ k ∗ -group K ∗ i of k -quotient G for any i ∈ I . Then, we set ˆN = K

i∈I

ˆi K

ˆN= and W

i∈I

ˆi G

16.20.2.

ˆ be a ﬁnite k ∗ -group such that Z(H) ˆ = k ∗ , c a block Lemma 16.21 Let H ˆ , (Q, f ) a maximal Brauer (c, H)-pair ˆ ˆ c of H and K a subgroup of Autk∗ (H) such that H ⊂ K and that K = H·CK (Q, f ) . Then there exists an essenˆ of K containing H ˆ , lifting the inclusion tially unique central k ∗ -extension K ∗ H ⊂ K and inducing a k -group isomorphism ∗ CKˆ (Q, f ) −→ k∗ C¯Hˆ (Q)f¯ 16.21.1 which extends the canonical k ∗ -group homomorphism from CHˆ (Q) and lifts the action of CKˆ (Q, f ) over the simple algebra k∗ C¯Hˆ (Q)f¯ . Proof: The action of CK (Q, f ) on the simple k-algebra k∗ C¯Hˆ (Q)f¯ (cf. 1.16) deﬁnes a k ∗ -group CˆK (Q, f ) containing CHˆ (Q, f ) as a normal k ∗ -subgroup (cf. 1.25) and then it suﬃces to consider ˆ = H ˆ CˆK (Q, f ) ∆∗ CˆH (Q, f ) K 16.21.2

302

Frobenius categories versus Brauer blocks

where ∆∗ CˆH (Q, f ) denotes the “inverse diagonal” subgroup (cf. deﬁnition 16.22.3 below); the uniqueness is clear. 16.22 At this point, similarly as in 15.17 above, assume that 16.22.1 For any i ∈ I , the canonical k ∗ -group homomorphism ∗ CHˆ i (Qi ) −→ k∗ CHˆ i (Qi )fi ˆ i )(Q ,f ) -stable k ∗ -group homomorphism from can be extended to an Autk∗ (K i i i ˆ )(Q ,f ) is the stabilizer of (Qi , fi ) in Autk∗ (K ˆ i) . N ˆ i (Qi , fi ) where Autk∗ (K K

i

i

ˆ N essentially contains K ˆ and that Then it follows from Lemma 16.21 that K ∗ the canonical k -group homomorphism ∗ ∗ CHˆ (Q) −→ k∗ CHˆ (Q)f ∼ k∗ CHˆ i (Qi )fi =

16.22.2

i∈I

can be extended to NKˆ (Q, f ) ; analogously, it follows from Lemma 16.27 ˆ N essentially contains W ˆ . Moreover, it is clear that we may below that W N ˆ ˆ ˆ N G ˆ , the assume that G acts on W ; then, in the semidirect product W inverse diagonal subgroup ˆ ) = {(ˆ ∆∗ (W x, x ˆ−1 )}xˆ∈W ˆ

16.22.3

is a normal subgroup; ﬁnally, we consider the k ∗ -group ∗ ˆ ˆ N = (W ˆ N G)/∆ ˆ G (W )

16.22.4

ˆ and W ˆ N. which, up to suitable identiﬁcations, contains G ˆ H ˆ is a p -group, Proposition 16.23 In our standard setting, assume that G/ ˆ ˆ that C = G/K is cyclic and that condition 16.22.1 holds. Denote by δ the ˆ determined by f , by ϕ the point of k∗ C Gˆ (Q, f ) such local point of Q on k∗ H ˆ H ˆ ˆϕ that (δ, ϕ) is the local point of Q = P on k∗ Gb determined by e , and by G ˆ the converse image in G of the stabilizer C(δ,ϕ) of (δ, ϕ) in C . Then, b is a ˆ ϕ and if (Q) holds for (b, G ˆ ϕ ) , it holds for (b, G) ˆ . block of G Proof: According to Lemma 15.16, the pair (δ, ϕ) has been identiﬁed indeed ˆ , so that e determines ϕ ; moreover, with a local point of Q = P on k∗ Gb ˆ G according to 15.17, C acts on the set of points of k∗ CH ˆ (Q, f ) and therefore ˆ ϕ of C(δ,ϕ) in G ˆ. it makes sense to consider the converse image G ˆ ⊂G ˆ ϕ , b is also a block of G ˆ ϕ (cf. Proposition 15.10); Since CGˆ (Q, f ) ⊂ K ˆ (P,e) of (P, e) moreover, since by the Frattini argument the stabilizer Aut(G)

16. The local-global question: reduction

303

ˆ b covers Outk∗ (G) ˆ b (cf. 1.15), we get a canonical group homomorin Aut(G) phism ˆ b −→ Outk∗ (G ˆ ϕ )b Outk∗ (G) 16.23.1. Now, setting F ϕ = F(b,Gˆ ϕ ) , it is easily checked from Corollary 15.20 that we ˆ b -module isomorphisms have OOut(G) ˆ b) ∼ ˆ ϕ , b) F nc ) ∼ (F ϕ )nc ) and Gk (G, Gk (F, aut = Gk (F ϕ , aut = Gk (G

16.23.2.

ˆ ϕ )b -module isomorphism Thus, if there is an OOutk∗ (G ˆ ϕ , b) (F ϕ )nc ) ∼ Gk (F ϕ , aut = Gk (G

16.23.3

ˆ b -module isomorphism Gk (F, aut ˆ b) . F nc ) ∼ then we get an OOutk∗ (G) = Gk (G, We are done. 16.24 With the notation and the hypothesis in the proposition above, it follows from this proposition that we are reduced to discuss the case where ˆ i )-stable k ∗ -group C ﬁxes (δ, ϕ) ; but, for any i ∈ I , choosing an Autk∗ (K homomorphism ∗ ρˆi : NKˆ i (Qi , fi ) −→ k∗ CHˆ i (Qi )fi 16.24.1 which extends the canonical k ∗ -group homomorphism ∗ CHˆ i (Qi ) −→ k∗ CHˆ i (Qi )fi

16.24.2,

we a get k ∗ -group isomorphism (cf. 15.17) K ∼ ∗ CH ˆ (Qi , fi ) = k × Di ˆi

16.24.3;

i

ˆ moreover, clearly we can choose a family {ρi }i∈I which is Autk∗ (G)-stable. In this situation, the family of trivial representations of Di for any i ∈ I ˆ N /K ˆ N -stable point ϕN of the k-algebra (cf. deﬁnition 16.22.4) determines a G ˆ; G ∼ k∗ CH ˆ (Q, f ) =

ˆi K ∼ k∗ CH ˆ (Qi , fi ) =

i

i∈I

kDi

16.24.4

i∈I

ˆ N (cf. Lemma 15.16) determines a and the local point (δ, ϕN ) of Q on k∗ G N N ˆ block b of G with defect pointed group Q(δ,ϕ; ) ; we set F N = F(b; ,Gˆ ; ) . ˆ H ˆ is a p -group, Proposition 16.25 In our standard setting, assume that G/ ˆ K ˆ is cyclic and that condition 16.22.1 holds. Moreover, with the that C = G/ ˆ N ) then it notation above, assume that C ﬁxes (δ, ϕ) . If (Q) holds for (bN , G ˆ . holds for (b, G)

304

Frobenius categories versus Brauer blocks

ˆ stabilizes both H ˆ and the family {H ˆ i }i∈I , Aut(G) ˆ b still Proof: Since Aut(G) i ˆ and therefore it acts on the family {K }i∈I ; hence, according to stabilizes K ˆ b still acts on the family {G ˆ i }i∈I Lemmas 16.21 and 16.27 below, Autk∗ (G) N ˆ (cf. deﬁnition 16.22.4); moreover, from (cf. 16.20), and therefore it acts on G ˆ b stabilizes bN . Consequently, our hypothesis it is easily checked that Autk∗ (G) we get a group homomorphism ˆ b −→ Outk∗ (G ˆ N )b; Outk∗ (G)

16.25.1.

ˆ b -moand, as above, it follows from Corollary 15.20 that we have OOutk∗ (G) dule isomorphisms ˆ b) ∼ ˆ N , bN ) F nc ) ∼ (F ; )nc ) and Gk (G, Gk (F, aut = Gk (F N , aut = Gk (G

16.25.2.

ˆ ∗ )b∗ -module isomorphism Hence, if there is an OOutk∗ (G ˆ ∗ , b) (F ∗ )nc ) ∼ Gk (F ∗ , aut = Gk (G

16.25.3

ˆ b -module isomorphism Gk (F, aut ˆ b) . F nc ) ∼ then we get an OOutk∗ (G) = Gk (G, We are done. ˆ H ˆ is 16.26 From now on, in our standard setting let us assume that G/ ˆ ˆ a p -group, that C = G/K is cyclic, that condition 16.22.1 holds, and that ˆ ∗ coincides with G ˆ (cf. 16.22.4), so that we have (cf. 16.20) G

W =

Gi

,

K=

i∈I

Ki

and b =

i∈I

bi

16.26.1,

i∈I

ˆ i and G ˆ i ; we set where, for any i ∈ I , bi is a block of K W = F(b,W ˆ)

,

F i = F(bi ,Gˆ i )

and Ki = F(bi ,K) ˆ

16.26.2.

It is clear that C¯ = G/W acts on I ; if I decomposes on a disjoint union of ¯ two nonempty C-stable subsets I and I then, setting ˆ= ˆ G ˆi W

ˆ = and W

i ∈I

ˆ ˆ i G

16.26.3,

i ∈I

ˆ the following lemma implies that there exist essentially unique k ∗ -groups G ˆ ˆ ˆ and G , respectively containing and normalizing W and W , such that ˆ /W ˆ∼ ˆ ˆ /W G = C¯ ∼ =G

ˆ × ˆ ˆ ∼ ˆ C¯ G and G =G

16.26.4.

16. The local-global question: reduction

305

Moreover, setting b = ⊗i ∈I bi and b = ⊗i ∈I bi , it follows from Propoˆ and G ˆ ; we set sition 15.10 that b and b are respective blocks of G F = F(b ,Gˆ )

and F = F(b ,Gˆ )

W = F(b ,W ˆ )

and W = F(b ,W ˆ )

16.26.5.

¯ of C¯ , we have an analogous situation with Note that, for any subgroup D ˆ D¯ , W ˆ D¯ and W ˆ D¯ of D ¯ in G ˆ, G ˆ and G ˆ . the respective converse images W ˆ a central (k ∗ )n -extension of a Lemma 16.27 Let n be a positive integer, H ∗ n ˆ ˆ ﬁnite group H such that Z(H) = (k ) , and K a subgroup of Aut(k∗ )n (H) such that, up to a suitable identiﬁcation, we have H ⊂ K and K/H is cyclic. ˆ of K containing There exists an essentially unique central (k ∗ )n -extension K ˆ H and lifting the inclusion map H → K . In particular, any automorphism ˆ stabilizing K can be lifted to an automorphism of K ˆ. σ ˆ of H Proof: We choose a cyclic subgroup C of K such that K = H·C and set ˆ of D in H ˆ is split, choosing a splitting, D = C∩H ; since the converse image D we can deﬁne an “inverse diagonal” subgroup ∆∗ (D) (cf. deﬁnition 16.22.3) ˆC; which is actually contained in the center of the semidirect product H then, it suﬃces to consider ˆ = (H ˆ C)/∆∗ (D) K

16.27.1.

ˆ ˆ is a central (k ∗ )n -extension of K containing H ˆ and lifting Moreover, if K ˆ ˆ ∗K ˆ ◦ contains H ˆ ∗H ˆ ◦ which is canonically the inclusion map H → K , then K ∗ n isomorphic to (k ) × H and therefore, up to suitable identiﬁcations, the ˆ ˆ ∗K ˆ ◦ )/H is a central (k ∗ )n -extension of the cyclic group K/H , quotient (K ˆ ˆ ∗K ˆ ◦ is also split and, more precisely, there is an which is split; hence, K ˆ ˆ ∼ ˆ inducing the identity on H ˆ. isomorphism K =K In particular, since σ ˆ induces an automorphism σ of K , it induces a ˆC ∼ ˆ σ(C) mapping ∆∗ (D) onto ∆∗ σ(D) and group isomorphism H =H therefore it determines an isomorphism ˆ ∼ ˆ σ(C) ∆∗ σ(D) K 16.27.2; = H but, the right member of this isomorphism is also a central (k ∗ )n -extension ˆ lifting the inclusion map H → K and therefore it admits an isomorphism of H ˆ inducing the identity on H ˆ . We are done. to K Proposition 16.28 With the notation and the hypothesis in 16.26 above, asˆ b stabilizes I and I . If (Q) holds for (b , W ˆ D¯ ) sume that the group Autk∗ (G) ˆ D¯ ) for any subgroup D ¯ of C, ¯ then it holds for (b, G) ˆ . and (b , W

306

Frobenius categories versus Brauer blocks

Proof: According to our hypothesis, we have canonical group homomorphisms ˆ )b ←− Outk∗ (G) ˆ b −→ Outk∗ (G ˆ )b Outk∗ (G

16.28.1;

then, since the homomorphisms from C¯ to k ∗ induce k ∗ -group automorphisms ˆ , which are contained in the centers of Autk∗ (G) ˆ , Autk∗ (G ˆ) ˆ,G ˆ and G of G ˆ ) respectively, it follows from Corollary 15.47 that we have and Autk∗ (G ˆ OOutk∗ (G)b -module isomorphisms ˆ , b ) ⊗RG (C) ˆ , b ) G (G RWˆ Gk (G ¯ RW ˆ k k ∼ ˆ b) = RWˆ Gk (G, F nc ) ⊗RG (C) RW Gk (F , aut ¯ RW Gk (F , aut F nc ) k

16.28.2.

∼ F nc ) = RWGk (F, aut ˆ )b - and OOutk∗ (G ˆ )b -module On the other hand, if we have OOutk∗ (G isomorphisms ˆ , b ) F nc ) ∼ Gk (F , aut = Gk (G

ˆ , b ) 16.28.3, F nc ) ∼ and Gk (F , aut = Gk (G

¯ since the restriction induces compatible Gk (C)-module structures on all the members of these isomorphisms (cf. 15.21 and 15.33), it follows from iso¯ ˆ morphisms 15.23.2 and 15.37.1 that we still have RGk (C)Out k∗ (G)b -module isomorphisms ˆ , b ) F nc ) ∼ RW Gk (F , aut = RWˆ Gk (G 16.28.4. ˆ , b ) F nc ) ∼ RW Gk (F , aut = RWˆ Gk (G ˆ b -module isomorThen, from isomorphisms 16.28.2 we get an OOutk∗ (G) phism ˆ b) F nc ) ∼ 16.28.5. RWGk (F, aut = RWˆ Gk (G, ¯ of C¯ we Consequently, according to our hypothesis, for any subgroup D ¯ D ˆ ∗ have an OOutk (W )b -module isomorphism ˆ D , b) (W D¯ )nc ) ∼ RWGk (W D , aut = RWˆ Gk (W ¯

¯

16.28.6;

ˆ stabilizes W ˆ , we have evident group homomorphisms but, since Autk∗ (G) ˆ D¯ )b ←− Autk∗ (G) ˆ b C¯ −→ Outk∗ (W

16.28.7

ˆ b contains and normalizes the image and it is clear that the image of Autk∗ (G) ˆ b -module isomorphism of C¯ ; hence, we still have an OOutk∗ (G) ¯ ˆ D¯ , b)C¯ (W D )nc )C¯ ∼ RWGk (W D , aut = RWˆ Gk (W

16.28.8.

16. The local-global question: reduction

307

Then, it follows from isomorphisms 15.23.4 and 15.38.1 that the direct sum of ¯ runs over the set of subgroups of C¯ supplies an isomorphisms 16.28.8 when D ˆ b -module isomorphism Gk (F, aut ˆ b) . We are done. F nc ) ∼ OOutk∗ (G) = Gk (G, 16.29 Now, with the notation and the hypothesis in 16.26 above, we may ˆ b acts transitively on I ; in particular, it acts assume that the group Autk∗ (G) ¯ ¯ transitively on the set of C-orbits of I and, for any C-orbit O we consider ∗ the k -groups and the block ˆO= ˆO = W Gi , K K i and bO = bi 16.29.1; i∈O

i∈O

i∈O

it follows from Lemma 16.27 that there exists an essentially unique k ∗ -group ˆ O containing W ˆ O and fulﬁlling C¯ ∼ ˆ O /W ˆ O ; then, it follows from PropoG =G O O ˆ and we set sition 15.10 that b is also a block of G F O = F(bO ,Gˆ O )

W O = F(bO ,W ˆ O)

,

and KO = F(bO ,Kˆ O )

16.29.2.

ˆ is isomorphic to the direct sum over C¯ of the family of k ∗ -groups Moreover, G O ˆ ¯ G when O runs over the set of C-orbits of I . Proposition 16.30 With the notation and the hypothesis in 16.26 above, ˆ b acts transitively on I and let O be a C-orbit ¯ assume that the group Autk∗ (G) O ¯ D O ˆ ) ¯ of C, ¯ then it holds of I . If (Q) holds for b , (W for any subgroup D ˆ for (b, G) . ˆ b on I induces an action of Proof: It is clear that the action of Autk∗ (G) ˆ ¯ ¯ ˆ b,O Outk∗ (G)b on the set I of C-orbits of I ; moreover, denoting by Outk∗ (G) ˆ b , it is quite clear that the restriction induces the stabilizer of O in Outk∗ (G) a group homomorphism ˆ b,O −→ Outk∗ (G ˆ O )bO Outk∗ (G)

16.30.1 .

ˆ b acts on the two families On the other hand, it is quite clear that Outk∗ (G) of O-modules ˆ O , bO )}O ∈I¯ and {R Gk (F O , aut (F O )nc )}O ∈I¯ {Gk (G WO

16.30.2

and then it is not diﬃcult to check from the canonical isomorphisms deﬁned ˆ b -module isomorphisms in Corollary 15.47 that we have OOutk∗ (G) ˆ b ¯ Out ∗ (G) RGk (C) ˆ O , bO ) Ten Outk∗ (G) RWˆ OGk (G ˆ k

¯ RGk (C)

ˆ ∗ (G)

Out

b Ten Outk∗ (G) ˆ k

b,O

b,O

∼ ˆ b) = RWˆ Gk (G, (F O )nc ) RW OGk (F O , aut ∼ F nc ) = RWGk (F, aut

16.30.3

308

Frobenius categories versus Brauer blocks

¯ ¯ where RGk (C)Ten denotes the tensor induction of RGk (C)-modules (cf. 8.2 above and A2.2 in [48]). ˆ O )bO -module isomorphism But, assume that we have an OOutk∗ (G

ˆ O , bO ) (F O )nc ) ∼ Gk (F O , aut = Gk (G

16.30.4;

¯ then, since the restriction induces compatible Gk (C)-module structures on both members of this isomorphism (cf. 15.21 and 15.33), it follows from iso¯ ˆ morphisms 15.23.2 and 15.37.1 that we still have an RGk (C)Out k∗ (G)b,O -module isomorphism ˆ O , bO ) (F O )nc ) ∼ RW OGk (F O , aut = RWˆ OGk (G

16.30.5.

ˆ b -module isomorphism Thus, from isomorphisms 16.30.3 we get an OOutk∗ (G) ˆ b) F nc ) ∼ RWGk (F, aut = RWˆ Gk (G,

16.30.6.

¯ of C¯ we Consequently, according to our hypothesis, for any subgroup D ¯ D ˆ have an OOutk∗ (W ) b -module isomorphism ¯ ˆ )D¯ , b ((W)D¯ )nc ∼ RWGk (W)D , aut = RWˆ Gk (W

16.30.7;

ˆ stabilizes W ˆ , we have evident group homomorphisms but, since Autk∗ (G) ˆ D¯ )b ←− Autk∗ (G) ˆ b C¯ −→ Outk∗ (W

16.30.8

ˆ b contains and normalizes the image and it is clear that the image of Autk∗ (G) ¯ ˆ b -module isomorphism of C ; hence, we still have an OOutk∗ (G) ¯ ¯ ¯ ˆ )D¯ , b C ((W)D¯ )nc C ∼ RWGk (W)D , aut = RWˆ Gk (W

16.30.9.

Then, it follows from isomorphisms 15.23.4 and 15.38.1 that the direct sum of ¯ runs over the set of subgroups of C¯ supplies an isomorphisms 16.30.9 when D ˆ b -module isomorphism Gk (F, aut ˆ b) . We are done. F nc ) ∼ OOutk∗ (G) = Gk (G, 16.31 With the notation and the hypothesis in 16.26 above, in the last ˆ W ˆ acts regularly step of our reduction, we may assume that the group C¯ = G/ i ¯ on I ; in this case, since W = i∈I G and C is a cyclic p -group, ﬁxing an element i ∈ I it is elementary to check that (cf. 1.32) G∼ = Gi C¯

ˆ∼ ˆ C¯ and G =W

16.31.1

16. The local-global question: reduction

309

where we may choose the right isomorphism in such a way that C¯ stabilizes (Q, f ) . Actually, many arguments we develop in this last situation could have been discussed in the more general setting of chapter 15, as the interested reader may check. 16.32 In this situation, we have to consider the direct product of groups ˆ ˆ = W

ˆi G

16.32.1;

i∈I

since G/W is cyclic, it follows from Lemma 16.27 that there exists an esˆ ˆˆ ˆ of G containing W sentially unique central (k ∗ )I -extension G and, ﬁxing an element i ∈ I as above, we have again (cf. 1.32) ˆ ˆ∼ ˆ i C¯ G =G

16.32.2.

Moreover, denoting by ∇k∗ : (k ∗ )I → k ∗ the group homomorphism induced ˆˆ by the product and considering the group algebras of (k ∗ )I and G over k and the k-algebra homomorphism k(k ∗ )I → k determined by ∇k∗ , it is quite clear that we have a k ∗ -group and a k-algebra isomorphisms ˆˆ ∼ ˆ G/Ker(∇ k∗ ) = G

ˆˆ ∼ ˆ and k ⊗k(k∗ )I k G = k∗ G

16.32.3.

ˆ ˆ acts transitively on the family {G ˆ i }i∈I and, 16.33 In particular, since G ˆ i (cf. 16.26.1), by the Frattini argument we for any i ∈ I , bi is a block of G get canonical group homomorphisms ˆ b −→ Outk∗ (G ˆ i )b Outk∗ (G) i

16.33.1.

Moreover, ﬁxing an element i ∈ I , it follows from equality 16.32.1 that ˆˆ ˆ i -module Mi can be considered as a k W any k∗ G -module, and the key point ˆ ˆ

is that, according to isomorphisms 16.32.3, the tensor induction TenGˆˆ (Mi ) W ˆ becomes a k∗ G-module (cf. 8.2). Proposition 16.34 With the notation and hypothesis above, let i be an eleˆ i -module Mi on the k∗ G-moˆ ment of I . The correspondence mapping any k∗ G ˆ ˆ G ˆ b -module isomorphism dule Ten ˆ (Mi ) induces an OOutk∗ (G) ˆ W

¯ ⊗O Gk (G ˆ i , bi ) ∼ ˆ b) RGk (C) = RWˆ Gk (G,

16.34.1.

310

Frobenius categories versus Brauer blocks

ˆˆ ˆˆ ˆˆ Proof: Recall that the tensor induction of Mi from W to G is the k G-module (cf. 8.2 above and A2.2 in [48]) ˆ ˆ

TenGˆˆ (Mi ) = W

(kX ⊗

ˆ ˆ kW

Mi )

16.34.2,

ˆ ˆ ˆ ˆ X∈G/ W

ˆˆ ˆˆ where kX denotes the k-vector space over the (right) W -class X ⊂ G , ˆ ˆ endowed with the (right) k W -module structure determined by the multiˆˆ plication on the right; actually, in our situation the image of C¯ in G by isomorphism 16.32.2 supplies a multiplicative closed set of representatives. ˆ ˆ

It is clear that in TenGˆˆ (Mi ) the multiplication by (λi )i∈I ∈ (k ∗ )I coinW cides with the multiplication by i∈I λi ∈ k ∗ , so that, according to isomorˆ

ˆ ˆ phisms 16.32.3, TenGˆˆ (Mi ) becomes a k∗ G-module. W

¯ = |I| and denoting by χi the modular characMoreover, setting c = |C| ter of Mi (cf. 18 in [54]), it is well-known and easily checked that, for any ˆ ˆ ˆ ˆ ˆ ˆ such that G ˆ=W ˆ ·$x p -element x ˆ∈G ˆ% , we have

i ˆ ˆ ˆ ˆˆ)c TenGˆˆ (χi ) (x ˆ) = χi (x W

16.34.3

c ci ˆˆ ˆ ˆ ˆ i ; in particular, this where (x ˆ) denotes the projection of (x ˆ) ∈ W in G ˆ ˆ

equality shows that the tensor induction TenGˆˆ induces an O-module homoW ˆ i ) to the O-module of modular characters of G ˆ restricted morphism from Gk (G ˆ ˆ ˆ to the set of p -elements x ˆ ∈ G such that G = W ·$ˆ x% ; but, it is easily checked that this restriction is equivalent to the projection (cf. 15.21.4) ˆ −→ R Gk (G) ˆ Gk (G) ˆ W

16.34.4;

that is to say, we ﬁnally get an O-module homomorphism ˆ i ) −→ R Gk (G) ˆ Gk (G ˆ W

16.34.5

ˆ has an RGk (C)-module ¯ and, since RWˆ Gk (G) structure (cf. 15.23), we still get ¯ an RGk (C)-module homomorphism ¯ ⊗O Gk (G ˆ i ) −→ R Gk (G) ˆ RGk (C) ˆ W which we claim is bijective.

16.34.6

16. The local-global question: reduction

311

ˆ = G ˆ i and therefore Indeed, if Mi is simple then, since we have W i∈I ˆ

ˆ ∼ ˆ i , it is clear that the restriction of TenGˆˆ (Mi ) to k∗ W ˆ is k∗ W = ⊗i∈I k∗ G ˆ W ¯ k ∗ ) determines k ∗ -group ausimple too; moreover, since the group Hom(C, ˆ tomorphisms of G , from 16.32 above it is not diﬃcult to check that this correspondence induces a bijection between the set of isomorphism classes of ˆ i -modules and the set of Hom(C, ¯ k ∗ )-orbits of isomorphism classes simple k∗ G ˆ ˆ . Then, the of the simple k∗ G-modules which remain simple restricted to W bijectivity of homomorphism 16.34.6 follows from the equality in 15.23.2; moreover, it is easily checked that Mi is associated with the block bi if and ˆ ˆ ˆ is associated with the block b ; only if the restriction of TenGˆ (Mi ) to k∗ W ˆ W

hence, isomorphism 16.34.6 induces the O-module isomorphism 16.34.1. ˆ stabilizes the On the other hand, since any k ∗ -automorphism σ ˆ of G ˆ ˆ j }j∈I family {Hj }j∈I , if σ ˆ stabilizes b then it also stabilizes both families {K ˆ j }j∈I , and therefore, according to Lemma 16.27, σ and {G ˆ can be lifted to ˆ ˆ ˆ ˆ an automorphism σ ˆ of G ; moreover, since G acts transitively on I , up to ˆ , we may assume that a modiﬁcation of σ ˆ by an inner automorphism of G ∗ ˆ ˆ i ; in this case, σ ˆ ﬁxes i and then σ ˆ determines a k -automorphism σ ˆi of G assuming that Mi is associated with the block bi , it is quite clear that ˆ ˆ ˆ ˆ TenGˆˆ Res σˆi (Mi ) ∼ = Res σˆˆ TenGˆˆ (Mi ) W

W

16.34.7

and therefore, since homomorphism 16.33.1 maps the class of σ ˆi on the class ˆ b -module homomorof σ ˆ , homomorphism 16.34.6 is actually an OOutk∗ (G) phism. We are done. 16.35 Now, always ﬁxing an element i in I , we need an analogous re F nc ) with Gk (F i , aut (F i )nc ) . Let us borrow our sult connecting RWGk (F, aut ˆ in the place of H ˆ . First of all recall that, since notation in 15.33 with W ¯ ¯ sc C F = W (cf. Proposition 12.12), denoting by ch∗C (W C ) the full subcate ¯ sc ¯ sc ¯ sc gory of ch∗ (W C ) over the set of (W C ) -chains r : ∆n → (W C ) such that C¯r = C¯ and considering the contravariant functor (cf. 15.36.3) C¯ ˆ (•) : ch∗¯ (W C¯ )sc −→ O-mod RW(•) Gk W ˆ C

16.35.1,

it follows from Proposition 15.37 that we have C¯ ˆ (•) F nc ) ∼ RWGk (F, aut Gk W = lim RW(•) ˆ ←−

16.35.2.

On the other hand, isomorphism 16.32.2 determines a functor tri from F i ¯ ¯ to W C mapping any subgroup Ri of Qi on the unique C-stable subgroup R

312

Frobenius categories versus Brauer blocks

of Q fulﬁlling R ∩ Qi = Ri , where we identify Qi with the obvious subgroup ¯ of Q , and mapping any F i -morphism ψi : Ti → Ri on the unique C-stable ¯ C W -morphism ψ : T → R such that T ∩ Qi = Ti , that R ∩ Qi = Ri and that the restriction of ψ to Ti coincides with ψi ; then, it is quite clear that the i sc restriction trsc i of tri to (F ) induces a new functor i sc ¯ sc ∗ ch∗ (trsc −→ ch∗C¯ (W C ) i ) : ch (F )

16.35.3.

16.36 The key point, proved below, is that there is a natural map C¯ ˆ (•) ◦ ch∗ (trsc ) (F i )sc −→ R ˆ Gk W τi : gk ◦ aut i W(•)

16.36.1

¯ inducing an RGk (C)-module isomorphism ¯ ⊗O Gk (F i ,aut (F i )nc ) RGk (C) C¯ ∼ ˆ (•) ◦ ch∗ (trsc ) Gk W = lim RW(•) ˆ i

16.36.2,

←−

and that the functor ch∗ (trsc i ) induces an O-module isomorphism C¯ C¯ ˆ (•) ∼ ˆ (•) ◦ ch∗ (trsc ) lim RW(•) Gk W Gk W = lim RW(•) ˆ ˆ i ←−

←−

16.36.3.

Then, from isomorphism 16.35.2 we get ¯ ⊗O Gk (F i , aut (F i )nc ) ∼ F nc ) RGk (C) = RWGk (F, aut

16.36.4.

Firstly, we prove the second isomorphism. 16.37 Recall that, by the very deﬁnition of the inverse limit, up to suitable identiﬁcations we have C¯ ˆ (•) ⊂ ˆ lim RW(•) Gk W RW(r) Gk F(r) ˆ ˆ ←−

16.37.1

r

sc sc where r : ∆n → F runs over the set of F -chains such that C¯r = C¯ (cf. 15.33), whereas the right-hand member in 16.36.3 is contained in the ˆ by isomorphism 16.32.2, subproduct where, identifying C¯ with its image in G sc sc ¯ r : ∆n → F runs over the set of C-stable F -chains; moreover, it is clear that the projection map from the whole direct product to this subproduct induces an O-module homomorphism

C¯ C¯ ˆ (•) −→ lim R ˆ (•) ◦ ch∗ (trsc ) lim RW(•) Gk W Gk W ˆ ˆ i W(•) ←−

←−

16.37.2.

16. The local-global question: reduction

313

ˆ b stabilizes the normal k ∗ -sub16.38 On the other hand, since Outk∗ (G) ˆ , it follows from 16.3 and 16.4 that this group acts on the above group W C¯ ˆ (•) . Actually, since C¯ acts regudirect product stabilizing lim RW(•) Gk W ˆ ←−

ˆ b on the above subproduct larly on I , we still can deﬁne an action of Outk∗ (G) C¯ ∗ sc ˆ stabilizing lim RW(•) Gk W (•) ◦ ch (tri ) and it is easily checked that hoˆ ←−

ˆ b -module homomorphism. momorphism 16.37.2 becomes an OOutk∗ (G) Proposition 16.39 With the notation and the hypothesis above, the functor ˆ ch∗ (trsc i ) induces an OOutk∗ (G)b -module isomorphism C¯ C¯ ˆ (•) ∼ ˆ (•) ◦ ch∗ (trsc ) lim RW(•) Gk W Gk W = lim RW(•) ˆ ˆ i ←−

←−

sc

16.39.1. sc

Proof: It follows from Lemma 15.51 that any (F i ) -chain ri : ∆n → (F i ) sc sc can be extended to a suitable (F i ) -chain ˆri : ∆n+1 → (F i ) such that we have Op F˜ i ˆri (n + 1) = {1} 16.39.2 (F i )sc (cf. Theby applying this lemma to ri (n) ; then, applying the functor aut ∗ i orem 11.32) to the ch (F )-morphism n (idri , δn+1 ) : (ˆri , ∆n+1 ) −→ (ri , ∆n )

16.39.3,

we get a k ∗ -group isomorphism (cf. 15.51.1) Fˆ i (ˆri ) ∼ = Fˆ i (ri )

16.39.4.

Consequently, denoting by Ri the set of F i -selfcentralizing subgroups Ri of Qi fulﬁlling Op F˜ i (Ri ) = {1} 16.39.5 Ri

i and by trR the restriction of the functor tri to the full subcategory (F i ) i sc i of (F ) over Ri , it is not diﬃcult to check that we have a canonical ˆ b -module isomorphism OOutk∗ (G)

C¯ ˆ (•) ◦ ch∗ (trsc ) lim RW(•) Gk W ˆ i ←− C¯ ∼ ˆ (•) ◦ ch∗ (trRi) Gk W = lim RW(•) ˆ i

16.39.6.

←−

Similarly, it follows from Lemma 15.51 and Lemma 16.40 below that any sc sc F -chain r : ∆n → F such that C¯r = C¯ (cf. 15.33) can be extended to a

314

Frobenius categories versus Brauer blocks sc

sc

suitable F -chain ˆr : ∆n+1 → F such that we have rˆ(n + 1) = tri (Ri ) for F sc (cf. Theorem 11.32) to some Ri ∈ Ri and that, applying the functor aut the ch∗ (F)-morphism n (idri , δn+1 ) : (ˆr, ∆n+1 ) −→ (r, ∆n )

16.39.7,

we still get a k ∗ -group isomorphism ˆ r) ∼ ˆ F(ˆ = F(r)

16.39.8.

As above, it is not diﬃcult to check that, in order to compute the inverse C¯ ˆ (•) , it suﬃces to consider its restriction limit of the functor RW(•) Gk W ˆ i) tr (R tri (Ri ) ¯ ¯ tri (Ri ) i to ch∗C¯ (W C ) (cf. 16.35) where (W C ) =F denotes the full ¯

sc

subcategory of (W C ) = F

sc

over the set tri (Ri ) (cf. Proposition 12.12). Ri tri (Ri ) ⊂F Moreover, considering the obvious image category tri (F i ) , tri (Ri ) tri (Ri ) ¯ ¯ it is easily checked that any F -chain r : ∆n → F such that Cr = C i tri (Ri ) ∗ i R is ch F -isomorphic to a tri ((F ) )-chain, and it is clear that two i Ri i R tri ((F ) )-chains are ch∗ tri ((F i ) ) -isomorphic if (and only if) they are tri (Ri ) ˆ ch∗ F -isomorphic. Then, since F(r) acts trivially on Gk F(r) , it is not diﬃcult to check that, in order to compute the inverse limit of the C¯ ˆ (•) , it suﬃces to consider its restriction to the whole functor RW(•) Gk W ˆ Ri proper category of chains ch∗ tri ((F i ) ) . But, it is clear that tri induces Ri

an equivalence of categories between (F i )

Ri

and tri ((F i ) ) . We are done.

Lemma 16.40 With the notation and the hypothesis above, any F-selfcentralizing subgroup R of Q fulﬁlling F(R)/W(R) ∼ = C¯

and

˜ Op F(R) = {1}

16.40.1

is F-isomorphic to tri (Ri ) where Ri is the image of R in Qi . Proof: For any j ∈ I , denote by Rj the image of R in Qj and by T the direct product j∈I Rj ; then, since F(R)/W(R) ∼ = C¯ , the image of NT (R) in F(R) ˜ is a normal p-subgroup and therefore, since we assume that Op F(R) = {1} and R is F-selfcentralizing, we have NT (R) = R which forces T = R . Moreover, if gj is the block of CGˆ j (Rj ) such that (Qj , fj ) contains (Rj , gj ) for any j ∈ I , and we set g = ⊗j∈I gj , it is clear that NW ˆ (R, g) =

j∈I

NGˆ j (Rj , gj )

16.40.2;

16. The local-global question: reduction

315

∗ ∼ ¯ thus, since NGˆ (R, g)/NW ˆ (R, g) = C , as in 16.31.1 above we have a k -group isomorphism (cf. 1.32)

NG (R, g) ∼ = NGi (Ri , gi ) C¯

16.40.3

and, since C¯ is cyclic, it is easily checked that a complement of NW ˆ (R, g) ¯ in NGˆ (R, g) is G-conjugate to the image of C by isomorphism 16.32.2; consequently, R is F-isomorphic to tri (Ri ) . We are done. 16.41 Now, let us discuss the ﬁrst isomorphism in 16.36. First of all, we have to deﬁne a natural map C¯ ˆ (•) ◦ ch∗ (trsc ) (F i )sc −→ R ˆ Gk W gk ◦ aut i W(•) sc

16.41.1;

sc

that is to say, for any (F i ) -chain ri : ∆n → (F i ) , we have to deﬁne an ˆ O-module homomorphism from Gk Fˆ i (ri ) to RW ˆ (r) Gk F(r) where we set r = tri ◦ ri . But, it is quite clear that (cf. 1.32) F(r) ∼ = F i (ri ) C¯

ˆ ∼ ˆ and F(r) C¯ = W(r)

16.41.2

ˆ ˆ = Fˆ i (ri ) C¯ , we get (cf. 16.32) and therefore, setting F(r) ˆ ˆˆ ∼ ∼ ˆ ˆ ˆ F(r)/Ker(∇ k∗ ) = F(r) and k ⊗k(k∗ )I k F(r) = k∗ F(r)

16.41.3.

Thus, as in 16.33 above, for any j ∈ I denoting by rj the conjugate of ri via the corresponding element of C¯ , and setting ˆ ˆ W(r) = Fˆ j (rj )

16.41.4,

j∈I

ˆˆ any k∗ Fˆ i (ri )-module Mi can be considered as a k W(r)-module and we conˆ ˆ F (r)

sider the tensor induction Ten ˆˆ

W(r)

ˆ (Mi ) which becomes a k∗ F(r)-module.

Proposition 16.42 With the notation and the hypothesis above, the corsc sc respondence mapping any (F i ) -chain ri : ∆n → (F i ) and any k∗ Fˆ i (ri )-moˆ

ˆ i ◦ ri )-module TenFˆ (tri ◦r ) (Mi ) determines a natural dule Mi on the k∗ F(tr ˆ i ˆ W(tr i ◦r ) map C¯ ˆ (•) ◦ ch∗ (trsc ) (F i )sc −→ R ˆ Gk W τi : gk ◦ aut 16.42.1 i W(•) i

ˆ b -module isomorphism which induces an OOutk∗ (G) C¯ ¯ ⊗O Gk (F i , aut ˆ (•) ◦ ch∗ (trsc ) 16.42.2 (F i )nc ) ∼ RGk (C) Gk W = lim RW(•) ˆ i ←−

316

Frobenius categories versus Brauer blocks

Proof: As we remark in 16.31, our arguments above apply to a more general situation; in particular, our arguments in the proof of Proposition 16.34 can sc sc be translated here proving that, for any (F i ) -chain ri : ∆n → (F i ) , we get an O-module homomorphism i ˆ (τi )r : Gk Fˆ i (ri ) −→ RW(tr i Gk F(tri ◦ r ) ˆ i ◦r )

16.42.3

ˆ i ◦ ri ) -module isomorphism which actually induces an OOutk∗ F(tr i ¯ ⊗O Gk k∗ Fˆ i (ri ) ∼ ˆ RGk (C) = RW(tr i Gk F(tri ◦ r ) ˆ i ◦r )

16.42.4.

(F i )sc to a ch∗ (F i )sc -morOn the other hand, applying the functor aut phism (µi , δ) : (ti , ∆m ) → (ri , ∆n ) , we get a k ∗ -group homomorphism (F i )sc (ν i , δ) : Fˆ i (ti ) −→ Fˆ i (ri ) µ ˆi = aut

16.42.5

which clearly induces (k ∗ )I -extension homomorphisms (cf. 16.41) ˆ ˆˆ ˆ ˆˆ i i i ˆ ˆ i ◦ ti ) −→ F(tr ˆ µ ˆ : F(tr i ◦ r ) and W(tri ◦ t ) −→ W(tri ◦ r )

16.42.6.

Moreover, it is easily checked that we have a canonical isomorphism ˆ ˆ F (tri ◦ti )

Ten ˆˆ

W(tri ◦ti )

ˆ ˆ F (tr ◦ri ) Res µˆi (Mi ) ∼ = Resµˆˆ Ten ˆˆ i i (Mi ) W(tri ◦r )

16.42.7

ˆ and, since µ ˆ is clearly compatible with the k ∗ -group homomorphism ˆ i ◦ ti ) −→ F(tr ˆ i ◦ ri ) F sc (tri ∗ µi , δ) : F(tr aut

16.42.8,

isomorphism 16.42.7 shows the naturality of the correspondence mapping r on (τi )r . Then, it is quite clear that the corresponding naturality of isomorphisms 16.42.4 proves isomorphism 16.42.2. We are done. Proposition 16.43 With the notation and the hypothesis in 16.26 above, ˆ W ˆ acts regularly on I and let us choose an assume that the group C¯ = G/ ˆ b -module isomorphisms element i ∈ I . We have OOutk∗ (G) ¯ ⊗O Gk (G ˆ i , bi ) ∼ ˆ b) Gk (C) = Gk (G, ¯ ⊗O Gk (F i , aut (F i )nc ) ∼ F nc ) Gk (C) = Gk (F, aut ˆ i ) then it holds for (b, G) ˆ . In particular, if (Q) holds for (bi , G

16.43.1.

16. The local-global question: reduction

317

¯ of C¯ , it follows from Propositions 16.34, 16.39 Proof: For any subgroup D ˆ D¯ )b -module isomorphisms and 16.42 that we have OOutk∗ (W ¯ ⊗O Gk (G ˆ i , bi ) ∼ ˆD RGk (D) = RW ˆ Gk (W , b) ¯

¯ D ¯ ⊗O Gk (F i , aut (F i )nc ) ∼ (W D¯ )nc ) RGk (D) = RW ˆ Gk (W , aut

16.43.2

ˆ D¯ denotes the converse image of D ¯ in G ˆ ; in particular, since C¯ acts where W ¯ D ˆ on W stabilizing b and we have a canonical group homomorphism ¯

ˆ b −→ OOutk∗ (W ˆ D )b OAutk∗ (G)

16.43.3

ˆ b -module containing and normalizing the image of C¯ , we still have OOutk∗ (G) isomorphisms ¯ C ⊗O Gk (G ˆ i , bi ) ∼ ˆD C RGk (D) = RW ˆ Gk (W , b) ¯

¯

¯

¯ D ¯ C¯ ⊗O Gk (F i , aut (F i )nc ) ∼ (W D¯ )nc )C¯ RGk (D) = RW ˆ Gk (W , aut

16.43.4.

ˆ b -module isomorphisms Then, considering the direct sum of these OOutk∗ (G) ¯ runs over the set of subgroups of C¯ , isomorphisms 16.43.1 follow when D from isomorphisms 15.23.4 and 15.38.1. ˆ i )b -module isomorphism In particular, if we have an OOutk∗ (G i

ˆ i , bi ) (F i )nc ) ∼ Gk (F , aut = Gk (G i

16.43.5,

then, from isomorphisms 16.43.1 and via the canonical group homomorphism ˆ b → Outk∗ (G ˆ i )b (cf. 16.33.1), we get an OOutk∗ (G) ˆ b -module isoOutk∗ (G) i morphism ˆ b) F nc ) ∼ Gk (F, aut 16.43.6. = Gk (G, 16.44 In conclusion, putting all together Propositions 16.6, 16.7, 16.8, 16.9, 16.15, 16.19, 16.23, 16.25, 16.28, 16.30 and 16.43, we have proved the following reduction result. ˆ having a normal sub-block Theorem 16.45 Assume that any block (c, H) ∗ ˆ (d, S) of positive defect such that the k -quotient of Sˆ is simple, that we have ˆ = k ∗ , that H/ ˆ Sˆ is a p -group and that H/ ˆ S·C ˆ ˆ (Q, f ) is cyclic where CHˆ (S) H ˆ (Q, f ) is a maximal Brauer (d, S)-pair, fulﬁlls the following three conditions: 16.45.1 16.45.2

Out(S) is solvable. ˆ = S·C ˆ ˆ (Q, f ) , the canonical k ∗ -group homomorphism Setting K H ∗ CSˆ (Q) −→ k∗ C¯Sˆ (Q)f¯

ˆ (Q,f ) -stable k ∗ -group homomorphism can be extended to an Autk∗ (K) ∗ NKˆ (Q, f ) −→ k∗ C¯Sˆ (Q)f¯ .

318 16.45.3

Frobenius categories versus Brauer blocks ˆ c -module isomorphism There is an OOutk∗ (H) ˆ c) . (F nc ) ∼ Gk (F(c,H) = Gk (H, ˆ , aut ˆ ) (c,H)

ˆ there is an OOutk∗ (G) ˆ b -module isomorphism Then, for any block (b, G) ˆ b) (F nc ) ∼ Gk (F(b,G) = Gk (G, ˆ , aut ˆ ) (b,G)

16.45.4.

Chapter 17

Localities associated with a Frobenius P-category 17.1 Let us come back to our abstract setting: let P be a ﬁnite p-group and F a P -category (cf. 2.2). In this chapter, we introduce the localities associated with F — which are extensions of the category F and generalize the “localit´es” associated with a ﬁnite group G having P as a Sylow p-subgroup, introduced in [35]. As we explain in the Introduction (cf. I 41), we are mainly interested in the possible existence and uniqueness of the perfect locality — deﬁned in 17.13 below — for a Frobenius P -category, which to some extent generalizes the O-localit´e introduced in [35, Ch. VI]. But, we ﬁnd other meaningful localities canonically associated with any Frobenius P -category, as the basic locality in chapter 22. 17.2 A possible motivation to consider extensions of F could be supplied by the case where a ﬁnite group G acts on a set Ω ; then, it seems reasonable to consider the category FG,Ω where the objects are all the p-subgroups of G and the morphisms are the pairs formed by an FG -morphism (cf. 1.8) and by a permutation of Ω both determined by the same element of G , the composition of morphisms being induced by the product in G . As usual, we identify FG,Ω with its equivalent full subcategory over all the subgroups of a Sylow p-subgroup P . In particular, the regular action of G over G deﬁnes the transporter category TG = FG,G . 17.3 Denote by κ : TP → F the canonical functor determined by the conjugation (cf. 1.8). A locality L associated with F — an F-locality for short — is a category where the objects are all the subgroups of P , endowed with two functors τ : TP −→ L and π : L −→ F

17.3.1

which are the identity on the set of objects and fulﬁll π ◦ τ = κ , π being full; as usual, for any pair of subgroups Q and R of P , we denote by L(Q, R) the set of L-morphisms from R to Q , by · the composition law and by τQ,R : TP (Q, R) −→ L(Q, R) and πQ,R : L(Q, R) −→ F(Q, R)

17.3.2

the corresponding maps; often, we will write πx and τu instead of πQ,R (x) and τQ,R (u) , where x and u are L- and TP -morphisms respectively; moreover, Q

we set iR = τQ,R (1) whenever R ⊂ Q and, in all these notations, we write Q only once if Q = R (cf. 1.8).

320

Frobenius categories versus Brauer blocks

17.4 As for the P -categories, it is handy to consider partially deﬁned F-localities: if X is a nonempty set of subgroups of P , which contains any subgroup Q of P such that F(Q, R) = ∅ for some R ∈ X , an (F, X)X X or an F -locality L is a category where the objects are the elements of X , endowed with two functors as above, fulﬁlling the analogous condiX tions where we respectively replace TP and F by the full subcategories TP X X of TP and F of F over X ; note that L can always be extended to an ordinary F-locality L by setting L(R, T ) = F(R, T ) for any pair of subgroups R and T of P such that T ∈ X , and x·ψ = πx ◦ ψ for any x ∈ L(Q, R) where Q is another subgroup of P . In particular, if L is an F-locality, we denote sc by L the full subcategory of L over the set of F-selfcentralizing subgroups of P , endowed with the corresponding two functors. 17.5 Let L be an F-locality, Q a subgroup of P and K a subgroup of Aut(Q) , and assume in F (cf. 2.6); then, that Q is fully K-normalized denoting by NLK (Q) (R, T ) the converse image of NFK (Q) (R, T ) in L(R, T ) for any pair of subgroups R and T of NPK (Q) , it is quite clear that we obtain a subcategory NLK (Q) of L , where the objects are the subgroups of NPK (Q) , endowed with two functors TNPK (Q) −→ NLK (Q) and NLK (Q) −→ NFK (Q)

17.5.1

induced by the functors above; that is to say, we obtain a locality associated with the NPK (Q)-category NFK (Q) that we call K-normalizer of Q in L — or centralizer and normalizer in the cases where K = {idQ } and K = Aut(Q) . 17.6 As we will see in Proposition 18.4 below, it is useful to deﬁne the NFK (Q)-locality NFK,Q (Q) (cf. 17.2) endowed with the functors TNPK (Q) −→ NFK,Q (Q) and NFK,Q (Q) −→ NFK (Q)

17.6.1,

where the objects in NFK,Q (Q) are, once again, all the subgroups of NPK (Q) , where the morphisms are the pairs formed by an NFK (Q)-morphism and an element of K both determined by the same morphism in F , and where the composition and the two structural functors are the obvious ones. If we X X consider a partially deﬁned F -locality L , it suﬃces that NPK (Q) belongs X to X to be able to deﬁne the corresponding K-normalizer of Q in L — an K NFK (Q)NX (Q) -locality, where NXK (Q) denotes the obvious set. 17.7 As in 2.3 above, we say that an F-locality L is divisible whenever it fulﬁlls the condition 17.7.1 If Q , R and T are subgroups of P , for any x ∈ L(Q, R) and any y ∈ L(Q, T ) such that πy (T ) ⊂ πx (R) there is a unique z ∈ L(R, T ) such that x·z = y .

17. Localities associated with a Frobenius P -category

321

Actually, all the localities we will consider are divisible; in this case, for any x ∈ L(Q, R) , the equality |Q| = |R| implies that L(R, Q) has an element x−1 such that x·x−1 and x−1 ·x are the unity elements in L(Q) = L(Q, Q) Q and L(R) respectively; further, setting T = πx (R) , we have iT = x·(x∗ )−1 for a suitable x∗ ∈ L(T, R) . 17.8 Obviously, the divisibility of L forces the divisibility of F and then Ker(πR ) acts regularly on the “ﬁbers” of the maps πQ,R : L(Q, R) −→ F(Q, R)

17.8.1;

conversely, if F is divisible and, for any pair of subgroups Q and R of P , Ker(πR ) acts regularly on the “ﬁbers” of πQ,R then L is clearly divisible. In this case, the correspondence mapping Q on Ker(πQ ) can be extended to a contravariant functor Ker(π) : L −→ Gr 17.8.2 mapping x ∈ L(Q, R) on the group homomorphism Ker(πQ ) → Ker(πR ) sending z ∈ Ker(πQ ) to the unique element w ∈ Ker(πR ) fulﬁlling x·w = z·x (cf. 17.7.1). It is clear that if L is divisible then, for any subgroup Q of P and any subgroup K of Aut(Q) such that Q is fully K-normalized in F , the NFK (Q)-locality NLK (Q) is divisible too. 17.9 Let L be a divisible F-locality; if Q and R are subgroups of P , for any u ∈ TP (Q, R) and any v ∈ R ⊂ TP (R, R) we obviously have uv = (uvu−1 )u = κu (v)u

17.9.1,

so that τQ,R (u)·τR (v) = τQ κu (v) ·τQ,R (u) . We say that L is coherent whenever we still have† x·τR (v) = τQ πx (v) ·x 17.9.2 for any pair of subgroups Q and R of P , any x ∈ L(Q, R) and any v ∈ R . Once again, if L is coherent then, for any subgroup Q of P and any subgroup K of Aut(Q) such that Q is fully K-normalized in F , NLK (T ) is also coherent. If F is a Frobenius P -category, in order to get coherence it suﬃces to restrict condition 17.9.2 to the automorphism group of suitable F-objects; precisely, we have the following statement, which remains true for divisible (F, X)-localities whenever X contains all the F-essential subgroups of P (cf. 5.8). †

This corresponds to condition (C) in [13, Deﬁnition 1.6] which is equivalent to condi-

tion 1.8.2 in [34] for suitable groups L(Q) , as shown in Proposition 17.10 and Remark 17.11.

322

Frobenius categories versus Brauer blocks

Proposition 17.10 If F is a Frobenius P -category, a divisible F-locality L is coherent whenever we have Ker(πQ ) ⊂ CL(Q) τQ (Q) for any subgroup Q of P , and if either Q = P or Q is F-essential then, for any u ∈ Q and any x ∈ L(Q) , we have x·τQ (u)·x−1 = τQ πx (u) 17.10.1. Proof: Since for any pair of subgroups Q and R of P , for any x ∈ TP (Q, R) and for any v ∈ R we clearly have P P iQ ·τQ πx (v) ·x = τP,Q πP,Q (iQ ·x) (v) ·x P = τP πiP ·x (v) ·iQ ·x

17.10.2,

Q

it suﬃces to prove equality 17.9.2 when Q = P . We argue by induction on the length D of πP,R (x) (cf. 5.15); if D = 0 then P

πP,R (x) = πP (n) ◦ πP,R (iR )

17.10.3

P

for some n ∈ L(P ) , and therefore we get x = n·iR ·z for some z ∈ L(R) such that πR (z) = idR ; thus, since z centralizes τR (v) , we can write P P x·τR (v) = n·iR ·τR πz (v) ·z = n·τP πiP ·z (v) ·iR ·z R = τP πx (v) ·x

17.10.4.

If D ≥ 1 then we have P

πP,R (x) = πP,Q (iQ ) ◦ πQ (m) ◦ πQ,R (y)

17.10.5

for some F-essential subgroup Q of P , some m ∈ L(Q) and some y ∈ L(Q, R) P such that πP,R (iQ ·y) has length D − 1 (cf. 5.15); up to a suitable choice of y , P

we may assume that x = iQ ·m·y and, by the induction hypothesis, we still have P P P iQ ·y·τR (v) = τP πiP ·y (v) ·iQ ·y = iQ ·τQ πy (v) ·y 17.10.6, Q

so that we get y·τR (v) = τQ πy (v) ·y (cf. 17.7.1), which ﬁnally implies that

P P x·τR (v) = iQ ·m·τQ πy (v) ·y = iQ ·τQ πm·y (v) ·m·y = τP πx (v) ·x We are done.

17.10.7.

17. Localities associated with a Frobenius P -category

323

Remark 17.11 Note that if L is a coherent F-locality, Q is a subgroup of P , R and T are subgroups of NP (Q) containing Q , and x is an element of L(R, T ) fulﬁlling πx (Q) = Q , then, on the one hand there is y ∈ L(Q) such R T that iQ ·y = x·iQ (cf. 17.7.1), so that we have πy (u) = πx (u) for any u ∈ Q ; on the other hand, for any w ∈ T we have (cf. 17.9.2) T R T iQ ·y·τQ (w) = x·τT (w)·iQ = τR πx (w) ·x·iQ R = iQ ·τQ πx (w) ·y

17.11.1,

so that we get y·τQ (w)·y −1 = τQ πx (w) (cf. 17.7.1); in other words, 17.11.2 The group homomorphisms ψ in F(R, T ) which stabilize Q induce group homomorphisms τQ (T ) → τQ (R) already induced by elements of L(Q) lifting the action of ψ over Q . 17.12 Assume that F is a Frobenius P -category; as a matter of fact, we are mostly interested in F-localities which only depend on the knowledge of P and F , and thus the groups Ker(πQ ) , when Q runs over the set of subgroups of P , should only depend on P . We say that a coherent F-locality L is P -bounded if, for any subgroup Q of P fully normalized in F , we have Ker(πQ ) ⊂ τQ NP (Q) 17.12.1; in this case, since FP (Q) is a Sylow p-subgroup of F(Q) (cf. Proposition 2.11), τQ NP (Q) is a Sylow p-subgroup of L(Q) . It is clear that, for any subgroup P of P and any Frobenius P -subcategory F of F , a coherent F -sublocality L of L is P -bounded too; in particular, for any subgroup Q of P and any subgroup K of Aut(Q) such that Q is fully K-normalized in F , NLK (Q) is NP (Q)-bounded . 17.13 On the other hand, if Ker(πQ ) is a p-group for any subgroup Q of P , it follows from Remark 17.11 that, if Q is fully centralized in F and T is a subgroup of CP (Q) , all the elements in CF (Q) CP (Q), T induce group homomorphisms from τQ (T ) to τQ CP (Q) which are already induced by inner automorphisms of Ker(πQ ); thus, denoting by HCF (Q) the corresponding CF (Q)-hyperfocal subgroup (cf. 13.2), in this case we get HCF (Q) ⊂ Ker(τQ )

17.13.1.

Let us say that a coherent F-locality L is perfect when L is P -bounded and all these inclusions are equalities†. As above, it is not diﬃcult to prove that if L is perfect then, for any subgroup Q of P and any subgroup K of Aut(Q) such that Q is fully K-normalized in F , NLK (Q) is perfect too. †

Called a centric linking system in [13].

324

Frobenius categories versus Brauer blocks

17.14 When F is a Frobenius P -category, we already know from Proposition 12.3 that, for any F-stable subgroup U of P , there exists a U -quotient of F , which is a Frobenius category F¯ over P¯ = P/U ; similarly, if we have ¯ a perfect F-locality L, it makes sense to look for a perfect F-locality L¯ as a kind of U -quotient of L . In this last part of the chapter, we describe the construction of such a quotient. 17.15 From the deﬁnition of perfect localities, it is quite easy a priori to foresee what should be the kernel UL (Q) of the canonical group homomor¯ Q) ¯ for any subgroup Q of P , where Q ¯ denotes the image phism L(Q) → L( ¯ of Q in P ; indeed, since Ker(πQ ) should be a p-group (cf. 17.13), we have the following necessary condition ¯ ¯ ∼ ¯ ¯ ¯ ¯ ¯ Q)/O ¯ 17.15.1 L( p L(Q) = F(Q)/Op F(Q) and therefore, denoting by UF (Q) the kernel of the canonical group homo¯ Q) ¯ , we necessarily have morphism F(Q) → F( −1 UF (Q) 17.15.2. Op UL (Q) = Op πQ 17.16 Moreover, if ϕ : Q → P is an F-morphism such that Q = ϕ(Q) is fully normalized in F (cf. Proposition 2.7), Q is also fully UF (Q )-normalized in F (cf. Proposition 2.11) and then it is easily checked that the Frobenius ¯ ¯ ) is the NU (Q )-quotient of N UF (Q ) (Q ) ; thus, acCP¯ (Q)-category CF¯ (Q F cording to the perfectness condition (cf. 17.13) and to Proposition 13.10, it is quite clear that, setting F

Q

U (Q )

= NF F

(Q )

and P

Q

U (Q )

= NP F

(Q )

17.16.1,

the image of NU (Q )·HF Q in L(Q ) should be a Sylow p-subgroup of UL (Q ) . 17.17 Consequently, choosing x ∈ L(Q, Q ) fulﬁlling πx ϕ(u) = u for any u ∈ Q , we set −1 UL (Q) = Op πQ UF (Q) ·x·τQ NU (Q )·HF Q ·x−1 17.17.1 which does not depend on our choices of ϕ and x ; indeed, if Q is a subgroup of P which is F-isomorphic to Q and fully UF (Q )-normalized in F then, according to condition 2.8.2, there is an F-morphism U (Q )

ζ : NP F

U (Q )

(Q ) −→ NP F

(Q )

17.17.2

such that ζ(Q ) = Q ; since Q is fully UF (Q )-normalized in F , we actually have U (Q ) U (Q ) ζ NP F (Q ) = NP F (Q ) 17.17.3

17. Localities associated with a Frobenius P -category

325

and, with the notation in 17.16.1 above, it is easily checked that the isomorQ Q Q Q phism P ∼ determined by ζ is (F , F )-functorial (cf. 12.1); hence, =P we still we have (cf. 13.2 and 12.2) ζ(HF Q ) = HF Q and ζ NU (Q ) = NU (Q ) 17.17.4. Note that our next result remains true for perfect partially deﬁned F-localities. Theorem 17.18 Assume that F is a Frobenius P -category. Let L be a perfect F-locality and U an F-stable subgroup of P , set P¯ = P/U and denote ¯ by F¯ the U -quotient of F . Then, we have a perfect F-locality L¯ where, for ¯ ¯ ¯ any pair of subgroups Q = Q/U and R = R/U of P , the set of morphisms ¯ to Q ¯ is the quotient set from R ¯ Q, ¯ R) ¯ = L(Q, R)/UL (R) L(

17.18.1

and the structural functors are induced by the structural functors of L . Moreover, there is a unique functor L → L¯ compatible with the structural functors which, for any pair of subgroups Q and R of P containing U , maps ¯ Q, ¯ R) ¯ . y ∈ L(Q, R) on its class y¯ ∈ L( Proof: By the deﬁnition above it is clear that, for any pair of mutually F-isomorphic subgroups R and R of P , and any x ∈ L(R , R) , we have UL (R ) = x·UL (R)·x−1

17.18.2.

Moreover, we claim that 17.18.3 If Q is a subgroup of P containing R and R contains Q∩U then any Q Q x ∈ UL (Q) fulﬁlls πx (R) = R and the element y ∈ L(R) such that x·iR = iR ·y belongs to UL (R) . Indeed, arguing by induction on |Q : R| , we may assume that R is normal ¯ , it is clear that in Q ; since R contains Q ∩ U and πx acts trivially on Q ¯ πx (R) = R and that πy acts trivially on R ; furthermore, since we still have Q Q −1 xn ·iR = iR ·y n for any n ∈ N , if x is a p -element of πQ UF (Q) then y is a p -element of πR−1 UF (R) (cf. 17.7.1). That is to say, denoting by L(Q)R the stabilizer of R in L(Q) , the divisibility of L determines a group homomorphism Q gR : L(Q)R −→ NL(R) τR (Q) 17.18.4 and actually we have proved above that we have Q UL (Q) ⊂ L(Q)R and gR Op UL (Q) ⊂ Op UL (R)

17.18.5;

Q thus, in order to prove that gR UL (Q) ⊂ UL (R) , it suﬃces to prove that the image of a Sylow p-subgroup of UL (Q) is contained in UL (R) .

326

Frobenius categories versus Brauer blocks

It follows from Proposition 2.7 that there is an F-morphism ϕ : Q → P such that R = ϕ(R) is fully normalized in F ; moreover, since NF (R ) is a Frobenius NP (R )-category (cf. Proposition 2.16) and Q = ϕ(Q) is contained in NP (R ) , there is an NF (R )-morphism ν : Q → NP (R ) such that Q = ν (Q ) is fully normalized in NF (R ) (cf. Proposition 2.7) and then, since U (Q )

NP F

U (R )

(Q ) ⊂ NP F

(R ) = P

R

17.18.6,

it follows from Proposition 2.11 and Lemma 2.17 that Q is fully UF (Q )-norQ

R

malized in F ; but, with the notation in 17.16.1, F is a subcategory of F and therefore we get HF Q ⊂ HF R (cf. 13.2); moreover, it is quite clear that NU (Q ) ⊂ NU (R ) . Hence, we get Q gR τQ NU (Q )·HF Q ⊂ τR NU (R )·HF R

17.18.7

and, in order to prove our claim, it suﬃces to consider z ∈ L(Q, Q ) such that πz ν ϕ(u) = u for any u ∈ Q (cf. 17.16). Consequently, denoting by X the set of subgroups of P containing U and X by L the full subcategory of L over X , the correspondence UL deﬁned over X X induces a contravariant functor from L to the category Gr (cf. 1.1) and therefore it is easily checked that there is a category L¯ where the objects are ¯ to Q ¯ is the quotient the subgroups of P¯ , where the set of morphisms from R set L(Q, R)/UL (R) for any Q, R ∈ X , and where the composition is induced by the composition in L . At the same time, we get an evident functor from X X L to L¯ and the structural functors of the X-locality L determine structural ¯ functors π ¯ : L¯ → F¯ and τ¯ : TP¯ → L¯ , so that L¯ becomes an F-locality. Since L is divisible, Ker(πR ) acts regularly on the “ﬁbers” of the map πQ,R (cf. 17.3.2) and clearly UF (R) acts regularly on the “ﬁbers” of the map ¯ Q, ¯ R) ¯ F(Q, R) −→ F(

17.18.8;

¯ R) ¯ → F( ¯ R) ¯ still it easily follows that the kernel of the homomorphism π ¯R : L( acts regularly on the “ﬁbers” of the map ¯ Q, ¯ R) ¯ −→ F( ¯ Q, ¯ R) ¯ π ¯Q,R : L( ¯

17.18.9;

thus, since F¯ is divisible, L¯ is divisible too. Moreover, it is quite clear that X the coherence of L forces the coherence of L¯ .

17. Localities associated with a Frobenius P -category

327

On the other hand, since from the very deﬁnition of UF (R) we have the isomorphism ¯ R) ¯ L(R)/πR−1 UF (R) ∼ 17.18.10, = F( it is easily checked that L¯ is P¯ -bounded; similarly, if R is fully normalized ¯ is fully normalized in F¯ , R is also UF (R)-normalized in F (cf. in F then R Proposition 2.11) and HCF (R) ¯ is the image of HF R (cf. Proposition 13.10), ¯ is perfect too. so that we have Ker(¯ τR ) = HCF (R) ¯ . In conclusion, L Finally, we prove that the canonical functor L → L¯ has a unique extension to a functor λ : L → L¯ which is the identity map over the set of objects. Consider a pair of mutually F-isomorphic subgroups Q and Q of P , assume ˆ = NQ·U (Q) and Q ˆ = NQ ·U (Q ) , and that Q is fully normalized in F , set Q ˆ , Q) ˆ Q ,Q the set of ϕ ∈ F(Q ˆ , Q) ˆ such that ϕ(Q) = Q and denote by F(Q ˆ ˆ ˆ , Q) ˆ ; these sets are by L(Q , Q)Q ,Q the corresponding converse image in L(Q nonempty since there is an F-morphism ζ : NP (Q) → P such that ζ(Q) = Q ˆ ⊂Q ˆ . Once again, the divisibility (cf. condition 2.8.2) and then we have ζ(Q) of L determines an evident map X

ˆ , Q) ˆ Q ,Q −→ L(Q , Q) gQ ,Q : L(Q

17.18.11.

Moreover, since τQ NP (Q ) is a Sylow p-subgroup of L(Q ) (cf. 17.12), by the Frattini argument we get L(Q ) = Op UL (Q ) ·NL(Q ) UL (Q ) ∩ τQ NP (Q )

17.18.12

ˆ ) , the second factor and, since U is F-stable and UL (Q ) contains τQ (Q ˆ ; hence, we still get stabilizes Q ˆ) L(Q ) = Op UL (Q ) ·NL(Q ) τQ (Q

17.18.13.

But, since Q is fully centralized in F (cf. Proposition 2.11), it follows from statement 2.10.1 that the homomorphism ˆ Q ˆ )Q −→ NL(Q ) τ (Q ˆ) gQ : L(Q Q

17.18.14

is surjective. Consequently, since the map gQ ,Q is not empty and L(Q ) acts regularly on L(Q , Q) , gQ ,Q induces a surjective map (cf. 17.18.2) ˆ , Q) ˆ Q ,Q −→ L(Q , Q) Op UL (Q) ¯Q ,Q : L(Q g

17.18.15.

328

Frobenius categories versus Brauer blocks

ˆ containing Q , the divisibility Furthermore, for any subgroup R of Q of L determines as above a group homomorphism R gQ : L(R )Q −→ NL(Q ) τQ (R )

17.18.16

R (gQ )−1 Op UL (Q ) ⊂ UL (R )

17.18.17.

and we claim that

¯ Q ¯ ) is trivial and Q ¯ = R ¯ , it is clear Indeed, since the image of UL (Q ) in F( that R (gQ )−1 UL (Q ) ⊂ πR−1 UF (R ) 17.18.18 and therefore we have R R (gQ )−1 Op UL (Q ) ⊂ Op πR−1 UF (R ) ·Ker(gQ )

17.18.19;

R hence, since Ker(gQ ) ⊂ πR−1 UF (R ) , it suﬃces to prove that UL (R ) con R

tains a Sylow p-subgroup of Ker(gQ ) . We already know that τQ NP (Q ) is a Sylow p-subgroup of L(Q ) and may assume that it contains a Sylow p-subgroup of NL(Q ) τQ (R ) ; moreover, since L is perfect (cf. 17.13) and Q is fully centralized in F , we actually have τQ NP (Q ) ∼ 17.18.20; = NP (Q )/HCF (Q ) then, it is easily checked that τR NHC (Q ) (R ) is a Sylow p-subgroup of F

R

Ker(gQ ) . But, the inclusions U (Q )

CP (Q ) ⊂ NP F

(Q ) = P

Q

and CF (Q ) ⊂ F

Q

17.18.21

imply that (cf. 13.2) HCF (Q ) ⊂ HF Q moreover, an F-morphism ξ : P

Q

17.18.22;

→ P such that R = ξ(R ) is fully normal-

ized in F (cf. Proposition 2.7) determines a group homomorphism from P to P

R

which is clearly (F

Q

,F

R

Q

)-functorial and therefore we still have

ξ(HF Q ) ⊂ HF R

17.18.23.

Hence, since UL (R ) contains τR (HF R ) , UL (R ) contains indeed a Sylow R

p-subgroup of Ker(gQ ) .

17. Localities associated with a Frobenius P -category

329

ˆ implies U ⊂ Q , arguing by induction on Since the equality Q = Q |P : Q| , we may assume that the functor we are looking for is already deﬁned ˆ , Q) ˆ , so that it determines a map over L(Q ˆ , Q) ˆ −→ L( ¯Q ¯ , Q) ¯ λQˆ ,Qˆ : L(Q

17.18.24

and then, according to the surjectivity of the map 17.18.15 and to our claim above, we get a new map ¯Q ¯ , Q) ¯ λQ ,Q : L(Q , Q) −→ L(

17.18.25

ˆ , Q) ˆ Q ,Q and any y ∈ Op UL (Q) , sends g (x)·y which, for any x ∈ L(Q Q ,Q to λQˆ ,Qˆ (x) . Note that, if we already have a functor λ : L → L¯ as above and we want to prove that it is unique, arguing by induction on |P : Q| we may assume that λQˆ ,Qˆ = λQˆ ,Qˆ and then it is easily checked that λQ ,Q = λQ ,Q . Now, if Q is a subgroup of P which is F-isomorphic to Q and Q , and fully normalized in F , we have the maps λQ ,Q , λQ ,Q and λQ ,Q , and we claim that, for any x ∈ L(Q , Q) and any x ∈ L(Q , Q ) , we get λQ ,Q (x ·x) = λQ ,Q (x )·λQ ,Q (x)

17.18.26.

ˆ = NQ ·U (Q ) and choosing elements x ˆ , Q) ˆ and Indeed, setting Q ˆ ∈ L(Q ˆ ,Q ˆ ) such that x ˆ ∈ L(Q x−1 ·gQ ,Q (ˆ x) ∈ Op UL (Q) x−1 ·gQ ,Q (ˆ x ) ∈ Op UL (Q )

17.18.27,

it is easily checked that (x ·x)−1 ·gQ ,Q (ˆ x ·ˆ x) belongs to Op UL (Q) and therefore we get λQ ,Q (x ·x) = λQˆ ,Qˆ (ˆ x ·ˆ x) = λQˆ ,Qˆ (ˆ x )·λQˆ ,Qˆ (ˆ x) = λQ ,Q (x )·λQ ,Q (x) .

17.18.28.

In particular, if we modify our hypothesis and Q is no more fully normalized in F , then equality 17.18.26 supplies a deﬁnition for λQ ,Q ; explicitly, consider a triple of F-isomorphic subgroups Q , Q and Q of P , assume that Q is fully normalized in F , and choose x ∈ L(Q , Q ) ; then, we deﬁne a map ¯Q ¯ , Q) ¯ λQ ,Q : L(Q , Q) −→ L( 17.18.29 sending x ∈ L(Q , Q) to λQ ,Q (x )−1 ·λQ ,Q (x ·x) .

330

Frobenius categories versus Brauer blocks

This deﬁnition does not depend on our choices since, if Q is a subgroup of P which is F-isomorphic to Q and fully normalized in F , and x is an element of L(Q , Q ) , we have (cf. 17.18.26) λQ ,Q (x )−1 ·λQ ,Q (x ·x)

= λQ ,Q (x )−1 ·λQ ,Q (x ·x−1 )·(x ·x) = λQ ,Q (x )−1 ·λQ ,Q (x ·x−1 )·λQ ,Q (x ·x)

17.18.30.

= λQ ,Q (x )−1 ·λQ ,Q (x ·x) . More generally, if we remove the hypothesis that Q is fully normalized, it follows from deﬁnition 17.18.29 above that, since x ·x−1 belongs to L(Q , Q ) , for any x ∈ L(Q , Q ) and any x ∈ L(Q , Q) we still have λQ ,Q (x ·x) = λQ ,Q (x ·x−1 )−1 ·λQ ,Q (x ·x) = λQ ,Q (x ·x−1 )−1 ·λQ ,Q (x )·λQ ,Q (x )−1 ·λQ ,Q (x ·x) = λQ ,Q (x ·x−1 )−1 ·λQ ,Q (x ·x−1 )·x ·λQ ,Q (x)

17.18.31.

= λQ ,Q (x )·λQ ,Q (x) . On the other hand, if R is a subgroup of Q , the compatibility with the structural functors forces Q

Q

λQ,R (iR ) = ¯ıR¯ = τ¯Q, ¯ R ¯ (1)

17.18.32;

then, the uniqueness of λ is clear and it is not diﬃcult to see that, in order to complete its construction, it suﬃces to prove that, for any x ∈ L(Q , Q) , setting R = πx (R) and denoting by y the element of L(R , R) such that Q

Q

x·iR = iR ·y , we have Q

Q

λQ ,Q (x)·¯ıR = ¯ıR ·λR ,R (y)

17.18.33.

We argue by induction on |P : Q| and on |Q : R| ; we clearly may assume ˆ = NR·U (R) . that |Q : R| = p ; in this case, either Q ∩ U ⊂ R or Q ⊂ R ˆ ˆ ˆ → P be Firstly assume that Q ⊂ R ; setting R = NR ·U (R ) , let ψ : R an F-morphism such that R = ψ (R ) is fully normalized in F (cf. Proposition 2.7), set Q = ψ (Q ) , choose x ∈ L(Q , Q ) and denote by y the Q

Q

element of L(R , R ) such that x ·iR = iR ·y ; then, it suﬃces to prove equality 17.18.33 for x ·x and for x . That is to say, we may assume that R is fully normalized in F ; then, ˆ such that y −1 ·g (ˆ ˆ , R) y ) belongs to Op UL (R) ; hence, there is yˆ ∈ L(R R ,R

17. Localities associated with a Frobenius P -category

331 ˆ R

ˆ R

denoting by x ˆ the element of L(Q , Q) such that yˆ·iQ = iQ ·ˆ x , we get (cf. 17.18.16) Q gR (x−1 ·ˆ x) = y −1 ·gR ,R (ˆ y ) ∈ Op UL (R) 17.18.34 Q and, since UL (Q) contains (gR )−1 Op UL (R) , x−1 ·ˆ x belongs to UL (Q) , so ¯

Q ¯ Q) ¯ . Consequently, according to that λQ (x−1 ·ˆ x) is the trivial element ¯ıQ¯ of L( ¯=Q ¯ and R ¯ = Q ¯ , our deﬁnition, we have λR ,R (y) = λ ˆ ˆ (ˆ y ) and, since R

R ,R

y ) = λQ ,Q (ˆ x) ; since by our induction hypothesis we may assume that λRˆ ,Rˆ (ˆ ¯ Q

λQ (x−1 ·ˆ x) = ¯ıQ¯ , we obtain λR ,R (y) = λQ ,Q (x) . Secondly, assume that Q ∩ U ⊂ R , so that we have Q ∩ U ⊂ R too; let ψ : Q → P be an F-morphism such that Q = ψ (Q ) is fully normalized in F , set R = ψ (R ) , choose x ∈ L(Q , Q ) and denote by y the element Q

Q

of L(R , R ) such that x ·iR = iR ·y ; once again, it suﬃces to prove equality 17.18.33 for x ·x and for x . That is to say, in this case we may assume that Q is fully normalized in F ; then, according to our deﬁnition, there is ˆ such that ˆ , Q) x ˆ ∈ L(Q x−1 ·gQ ,Q (ˆ x) ∈ Op UL (Q) and λQ ,Q (x) = λQˆ ,Qˆ (ˆ x)

17.18.35. ˆ Q

ˆ Q

Moreover, denoting by yˆ the element of L(R , R) such that x ˆ·iR = iR ·ˆ y , it is easily checked that −1 Q Q x ·gQ ,Q (ˆ x) ·iR = iR ·(y −1 ·ˆ y)

17.18.36

and, since Q ∩ U ⊂ R and x−1 ·gQ ,Q (ˆ x) ∈ UL (Q) , it follows from statement 17.18.3 that y −1 ·ˆ y belongs to UL (R) so that we get λR ,R (y) = λR ,R (ˆ y) . Consequently, by the induction hypothesis, we have ¯ Q

¯ Q

¯ Q

¯ Q

x)·¯ıR = ¯ıR ·λR ,R (ˆ y ) = ¯ıR¯ ·λR ,R (y) λQ ,Q (x)·¯ıR = λQˆ ,Qˆ (ˆ We are done.

17.18.37.

Chapter 18

The localizers in a Frobenius P-category 18.1 Let P be a ﬁnite p-subgroup and F a Frobenius P -category. As a matter of fact, independently of the possible existence of a perfect F-locality L (cf. 17.13), the groups L(Q) where Q runs over the set of subgroups of P have a direct characterization in terms of F — we call them the F-localizers — and their existence and uniqueness admits a direct proof. In this chapter we give this characterization and such a direct proof; this allows us, in the next chapter, to determine the (p-)solvable Frobenius P -categories. Moreover, we show the functorial nature of the F-localizers — provided we replace F by the proper category of F-chains ch∗ (F) of F (cf. A2.8) — by constructing with them a suitable functor from ch∗ (F) — the localizing functor of F — which actually holds some kind of “universal property” (cf. 18.20 below). 18.2 Let Q be a subgroup of P and K a subgroup of Aut(Q) such that Q is fully K-normalized in F , so that it is fully centralized too (cf. Proposition 2.11); recall that then NFK (Q) is a Frobenius NPK (Q)-category (cf. Proposition 2.16). It is clear that CP (Q) is an NFK (Q)-stable subgroup of NPK (Q) (cf. 12.2) and that the Frobenius CP (Q)-category CF (Q) is normal in NFK (Q) (cf. 12.6); moreover, it follows from Proposition 13.9 that the hyperfocal subgroup HCF (Q) is also a NFK (Q)-stable subgroup of NPK (Q) and therefore, by Proposition 12.3, we have the Frobenius category NFK (Q)/HCF (Q) over the p-group NPK (Q)/HCF (Q) . 18.3 We are interested in an extension of this category, which actually will be a quotient of the NFK (Q)-locality NFK,Q (Q) introduced in 17.6; precisely, let us denote by NFK,Q (Q)/HCF (Q) the category where the objects are just the NFK (Q)/HCF (Q) -objects and the morphisms are the pairs formed by an NFK (Q)/HCF (Q) -morphism and by an element of K both determined by the same NFK,Q (Q)-morphism, the composition being induced by the composition in NFK (Q)/HCF (Q) and the product in K . On the other hand, if L is an F-locality then L(Q) acts on Q throughout F(Q) and, considering Q just as a set, recall that in 17.2 we have deﬁned the FL(Q) -locality FL(Q),Q . Proposition 18.4 Assume that L is a perfect F-locality. For any subgroup Q of P fully normalized in F , the group homomorphisms τQ and πQ induce an equivalence of categories NF ,Q (Q)/HCF (Q) ∼ = FL(Q),Q

18.4.1.

334

Frobenius categories versus Brauer blocks

Proof: Since FP (Q) is a Sylow p-subgroup of F(Q) (cf. Proposition 2.11), the image of the homomorphism τQ is a Sylow p-subgroup of L(Q) (cf. 17.12); thus, since we have (cf. 17.13.1) HCF (Q) = Ker(τQ )

18.4.2,

we may consider the Frobenius category FL(Q) associated with L(Q) (cf. 1.8) as a Frobenius category over the p-group NP (Q)/HCF (Q) . Then, if R and T are subgroups of NP (Q) containing HCF (Q) , for any F-morphism ϕ : Q·T −→ Q·R

18.4.3

stabilizing Q and inducing some F-morphism ψ :T → R , it follows from Remark 17.11 that there is x ∈ TL(Q) τQ (R), τQ (T ) inducing ψ and fulﬁlling πx (u) = ϕ(u) for any u ∈ Q ; this proves the existence of a faithful functor NF ,Q (Q)/HCF (Q) −→ FL(Q),Q

18.4.4.

Moreover, any FL(Q),Q -morphism from τQ (T ) to τQ (R) is determined by a suitable x ∈ L(Q) fulﬁlling τQ (T ) ⊂ τQ (R)x and, according to statement 2.10.1, there is an F-morphism ϕ : Q·T −→ Q·R

18.4.5

extending πx ; then, it follows from Remark 17.11 that there is an element y ∈ L(Q) determining the FL(Q),Q -morphism τQ (T ) → τQ (Q·R) induced by ϕ and fulﬁlling πy = πx ; hence, y −1 ·x belongs to (cf. 17.13.1) Ker(πQ ) = τQ CP (Q) 18.4.6, which proves the fullness. We are done. 18.5 Let L be a perfect F-locality and Q an F-selfcentralizing subgroup of P fully normalized in F ; since τQ NP (Q) is a Sylow p-subgroup of L(Q) (cf. 17.12) and we have Ker(πQ ) = Z(Q) (cf. 13.2.2 and 17.13.1), the isomorphism class of the group L(Q) is certainly determined by the element of 2 H F(Q), Z(Q) which once restricted to NP (Q)/Z(Q) coincides with the element of H2 NP (Q)/Z(Q), Z(Q) determined by NP (Q) . Conversely, this remark shows a way to construct the group L(Q) from the corresponding Frobenius P -category, and it turns out that this group already fulﬁlls condition 17.11.2† and therefore the equivalence of categories 18.4.1. Actually, as †

In our old work [46], we proved the equivalence between this condition and the fact that a

suitable 1-cocycle was a 1-coboundary; recently, we found that Jackowski-McClure proved in [32] a general result which actually implies that all the n-cocycles are n-coboundaries for n≥1; but, here we give a direct proof in our situation. In an earlier version of [13], the authors proved the existence of the localizers from their obstruction theory and from the same result of JackowskiMcClure for n=3 ; presently, their result appears in [15].

18. The localizers in a Frobenius P -category

335

we show in our next result, for any subgroup Q of P we can construct an essentially unique extension L(Q) — called the F-localizer of Q — of F(Q) by chF (Q) (cf. Proposition 13.14), fulﬁlling the equivalence of categories 18.4.1. Theorem 18.6 For any subgroup Q of P fully normalized in F , there is a triple formed by a ﬁnite group L(Q) and by two group homomorphisms τQ : NP (Q) −→ L(Q)

and

πQ : L(Q) −→ F(Q)

18.6.1

fulﬁlling the two conditions

h 18.6.2 Ker(τQ ) = HCF (Q) , Ker(πQ ) = τQ CP (Q) ∼ = cF (Q) , πQ is surjective and we have πQ ◦ τQ = κQ . 18.6.3

τQ and πQ induce an equivalence of categories NF ,Q (Q)/HCF (Q) ∼ = FL(Q),Q .

Moreover, for another such a triple L , τQ and πQ , there is a group isomorh ∼ phism λ : L(Q) = L , unique up to c (Q)-conjugation, fulﬁlling λ ◦ τ = τ and πQ ◦ λ = πQ .

F

Q

Q

Proof: By Proposition 2.11, we already know that FP (Q) is a Sylow p-subgroup of F(Q) . Assume ﬁrst that Z = Z(Q) = CP (Q)

18.6.4;

then we have HCF (Q) = {1} (cf. 13.2.2) and we claim that the element ν¯ of H2 FP (Q), Z induced by N = NP (Q) belongs to the image of H2 (F(Q), Z) . Indeed, according to Theorem 10.1 in [18, Ch. XII], it suﬃces to prove that, for any subgroup R of NP (Q) containing Z(Q) and any ϕ ∈ F(Q) such that ϕ ◦ FR (Q) ◦ ϕ−1 ⊂ FP (Q)

18.6.5,

the restriction resϕ (¯ ν ) of ν¯ via the conjugation by ϕ coincides with the ele ment of H2 FR (Q), Z determined by R ; but, since Q is fully normalized in F , it is also fully ϕ FR (Q)-normalized (cf. 2.10) and therefore, by condition 2.8.2, there is an F-morphism ζ : Q·R → N extending ϕ ; now, the existence of a group homomorphism R → N which coincides with ϕ over Z(Q) and with the conjugation by ϕ over FR (Q) proves the claim. In particular, the corresponding element of H2 F(Q), Z determines a group extension π 1 −→ Z −→ L −→ F(Q) −→ 1 18.6.6

336

Frobenius categories versus Brauer blocks

and an injective group extension homomorphism τ : N → L such that Im(τ ) is a Sylow p-subgroup of L ; moreover, since τ is injective, we may choose π in such a way that, for any x ∈ L and any u ∈ Q , we have xτ (u)x−1 = τ π(x) (u) 18.6.7; then, we claim that, up to a suitable modiﬁcation of our choice of τ , the group L endowed with τ fulﬁlls condition 18.6.3. In any case, since Q is F-selfcentralizing, for any pair of subgroups R and T of N containing Q and any ϕ ∈ F(R, T ) such that ϕ(Q) = Q , the element ψ ∈ F(Q) determined by ϕ fulﬁlls ψ ◦ FT (Q) ◦ ψ −1 ⊂ FR (Q)

18.6.8;

conversely, it follows from statement 2.10.1 that in this way we obtain all the elements of F(Q) fulﬁlling this inclusion. Thus, for any x ∈ L such that τ (T ) ⊂ τ (R)x there is ϕx ∈ F(R, T ) lifting π(x) ; that is to say, for any w ∈ T , we have x τ w θx (w) = τ ϕx (w) 18.6.9 for a suitable θx (w) ∈ Z ; note that, for any u ∈ Q , we have θx (u) = 1 , and whenever x ∈ τ (R) we may assume that θx (w) = 1 . Now, for any w, w ∈ T , we get x x x τ ww θx (ww ) = τ ϕx (ww ) = τ ϕx (w) τ ϕx (w ) = τ w θx (w) τ w θx (w ) 18.6.10 w = τ w θx (w) w θx (w ) = τ ww θx (w) θx (w ) and therefore we still get

θx (ww ) = θx (w)w θx (w )

18.6.11;

hence, the map θx determines a Z-valued 1-cocycle from the image T˜ of T in (cf. 1.3) + Aut(Q) = Out(Q) 18.6.12. Actually, the cohomology class θ¯x of this 1-cocycle does not depend on the choice of ϕx ; indeed, if another choice ϕx determines θx then, for a suitable z ∈ Z , we have (cf. Proposition 4.6) x x τ w θx (w) = τ ϕx (w) = τ ϕx (wz ) = τ wz θx (wz )

18.6.13

for any w ∈ T , and therefore we get θx (w)θx (w)−1 = w−1 wz = (z −1 )w z

18.6.14.

18. The localizers in a Frobenius P -category

337

Consequently, this correspondence determines a map (cf. 17.2) θ¯R,T : TL τ (R), τ (T ) −→ H1 (T˜, Z)

18.6.15

sending x ∈ TL τ (R), τ (T ) to the cohomology class θ¯x of θx . Moreover, if U is a subgroup of N containing Q and y an element of L fulﬁlling τ (U ) ⊂ τ (T )y then, for a choice of ϕy ∈ F(T, U ) lifting π(y) , ϕx ◦ ϕy lifts π(xy) and, for any w ∈ U , we may assume that xy y τ w θxy (w) = τ (ϕx ◦ ϕy )(w) = τ ϕy (w) θx ϕy (w) y = τ w θy (w) τ θx ϕy (w) −1 θx ϕy (w) = τ w θy (w) π(y)

18.6.16;

+ hence, denoting by y˜ the image of y in Aut(Q) = Out(Q) (cf. 1.3) and by ∼ ˜ ˜ ϕy˜ : U → T and Z(˜ y ) : Z = Z the corresponding group homomorphisms, we get the 1-cocycle condition (cf. A3.11) θ¯xy = θ¯y + H1 ϕy˜, Z(˜ y ) (θ¯x )

18.6.17;

in particular, it is easily checked from this condition that θ¯x only depends on the class of x in the exterior quotient (cf. 1.3) T˜L τ (R), τ (T ) = τ (R)\TL τ (R), τ (T )

18.6.18.

˜, R ˜ and N ˜ the images of L , R and N in Thus, respectively denoting by L ¯ + Aut(Q) , the map θR,T admits the factorization ˜˜ (R, ˜ T˜) = R ˜ \T˜ (R, ˜ T˜) −→ H1 T˜, Z θ¯˜R, ˜ T ˜ : TL L

18.6.19.

That is to say, considering the exterior quotient T˜L˜ (cf. 1.3) of the category TL˜ (cf. 17.2), it is clear that we have a contravariant functor h1Z from this category to Ab mapping T˜ on H1 T˜, Z and then, identifying the ˜ T˜) with the obvious T˜˜ -chain ∆1 −→ T˜˜ , the family T˜L˜ -morphism x ˜ ∈ T˜L˜ (R, L L θ¯ = {θ¯x˜ }x˜ deﬁnes a 1-cocycle from T˜L˜ to h1Z (cf. A3.8) since equalities 18.6.17 guarantee that the diﬀerential of θ¯ is zero (cf. A3.11.2). As in [32], we claim that this 1-cocycle is a 1-coboundary; indeed, for any ˜ of N ˜ , choose a set of representatives X ˜˜ ⊂ L ˜ for N ˜ \L/ ˜ R ˜ and, subgroup R R n ˜ ˜ ˜ , set R ˜ n˜ = R ˜ ∩ N , consider the T˜˜ -morphisms n ˜ n˜ → N ˜ for any n ˜∈X ˜ :R R

L

338

Frobenius categories versus Brauer blocks

˜ ˜ n˜ → R ˜ respectively determined by n and ˜ıR :R ˜ and by the trivial element ˜n R ˜ ˜ of L (cf. 17.2), and denote by

˜ ˜ n˜ , Z −→ H1 R, ˜ Z (h1Z )◦ (˜ıR ) : H1 R ˜n R ˜

18.6.20

the transfer homomorphism (cf. §8 in [18, Ch. XII]); then, setting σ ¯R˜ =

|N | 1 ◦ R˜ ¯ · (hZ ) (˜ıR˜ n˜ ) (θn˜ ) |L|

18.6.21,

˜˜ n ˜ ∈X R

˜ T˜) , we have we claim that, for any x ˜ ∈ T¯L˜ (R, θ¯x˜ = σ ¯T˜ − h1Z (˜ x) (¯ σR˜ )

18.6.22.

Indeed, note that h1Z (˜ x) is the composition of the restriction via the ˜ TL˜ -morphism ˜ ˜ ˜ıR :x ˜T˜x ˜−1 −→ R 18.6.23 x ˜T˜ x ˜−1 with the conjugation determined by x ˜ , which we denote by h1Z (˜ x∗ ) ; thus, by the corresponding Mackey equalities (cf. Proposition 9.1 in [18, Ch. XII]), we get ˜ ¯ h1Z (˜ (h1Z )◦ (˜ıR x) ) ( θ ) n ˜ ˜n R ˜ ˜R n ˜ ∈X

˜T˜ x ˜−1 1 ¯ = h1Z (˜ x∗ ) ) ◦ h (˜ r ) ( θ ) (h1Z )◦ (˜ıxN n ˜ n˜ ˜ r −1 ˜ ∩˜ Z xT˜ x ˜ ˜ ˜ r˜∈Y˜n n ˜ ∈X ˜ R

=

18.6.24

˜ 1 rx ˜) (θ¯n˜ ) (h1Z )◦ (˜ıTN˜ n˜ ˜ rx ˜ ∩T ˜ ) ◦ hZ (˜

˜ ˜ r˜∈Y˜n n ˜ ∈X ˜ R

˜ is a set of representatives for R ˜ n˜ \R/ ˜ x where, for any n ˜ ∈ XR˜ , Y˜n˜ ⊂ R ˜T˜x ˜−1 and, for any r˜ ∈ Y˜n˜ , we consider the T˜L˜ -morphisms ˜ n˜ r˜ ∩ x ˜ n˜ r˜ : N ˜T˜x ˜−1 −→ R

˜ n˜ r˜x˜ ∩ T˜ −→ R ˜ n˜ and r˜x ˜:N

18.6.25.

˜ ˜ and r˜ ∈ Y˜n˜ , it follows from Moreover, setting m ˜ = n ˜ r˜x ˜ for n ˜ ∈ X R equality 18.6.17 that

1 T˜ ¯ ˜ = θ¯m h1Z (˜ rx ˜) (θ¯n˜ ) = θ¯m ıT˜m˜ ) (θ¯x˜ ) ˜ − θr˜x ˜ − hZ (˜

18.6.26

18. The localizers in a Frobenius P -category

339

˜˜ = ) ˜ n since we assume that θr˜ = 0 ; thus, choosing X ˜ Y˜n˜ x ˜ , we get (cf. T n ˜ ∈XR ˜ equality (6) in [18, Ch. XII, §8]) σ ¯T˜ − h1Z (˜ x) (¯ σR˜ ) =

1 |N | 1 ◦ T˜ ¯ (hZ ) (˜ıT˜m˜ ) θm rx ˜) (θ¯n˜ ) · ˜ − hZ (˜ |L| ˜˜ m∈ ˜ X T

=

|N | 1 ◦ T˜ 1 T˜ ¯ hZ (˜ıT˜m˜ ) (θx˜ ) (hZ ) (˜ıT˜m˜ ) · |L|

18.6.27.

˜˜ m∈ ˜ X T

=

|T˜/T˜m ˜| ¯ ·θx˜ = θ¯x˜ ˜ ˜ | L/ N | ˜

m∈ ˜ XT˜

˜ of N ˜ , we get In particular, for any subgroup R ˜ σ ¯R˜ = h1Z (˜ıN σN˜ ) ˜ ) (¯ R ˜ , Z) can be lifted and the element σ ¯N˜ ∈ H1 (N which determines a group automorphism σ : N on uσN˜ (˜ u) where u ˜ denotes the image of u in equality 18.6.22, in 18.6.9 we may choose

18.6.28 ˜ → Z to a 1-cocycle σN˜ : N ∼ = N mapping u ∈ NP (Q) ˜ ; moreover, according to N

−1 −1 θx (w) = σN˜ (w) σN˜ ϕ ˜ π(x) x (w)

18.6.29.

Hence, replacing τ by τˆ = τ ◦ σ , the maps π and τˆ still fulﬁll condition 18.6.2 and in equality 18.6.9 we get x τ ϕx (w) = τ w θx (w) −1 −1 = τ w w−1 σ(w) π(x) ϕx (w)−1 σ ϕx (w) −1 −1 = τ σ(w) π(x) σ ϕx (w) ϕx (w)

18.6.30,

x −1 = τˆ(w)τ σ ϕx (w) ϕx (w) x x = τˆ(w)ˆ τ ϕx (w)−1 τ ϕx (w) so that we get x τˆ ϕx (w) = τˆ(w)

and ϕx (u) = π(x) (u) for any u ∈ Q

18.6.31.

340

Frobenius categories versus Brauer blocks

Consequently, we get a functor from FL,Q to NF ,Q (Q) (cf. 17.2 and 17.6) which, for any pair of subgroups R and T of N , maps the FL,Q -morphism from τˆ(T ) to τˆ(R) determined by an element x ∈ L such that τˆ(T ) ⊂ τˆ(R)x on the NF ,Q (Q)-morphism from T to R determined by the corresponding F-morphism ϕx : Q·T → Q·R , and this functor is necessarily faithful. Moreover, for any F-morphism ψ : Q·T → Q·R fulﬁlling ψ(Q) = Q and ψ(T ) ⊂ R , there is y ∈ L such that π(y) coincides with the restriction of ψ and therefore, according to Proposition 4.6, we have ψ = ϕy ◦ κz for some z ∈ Z , so that we still have ψ = ϕyτˆ(z) . In conclusion, the triple formed by L , τˆ and π fulﬁlls conditions 18.6.2 and 18.6.3. This completes the proof in the F-selfcentralizing case. Otherwise, choose an F-morphism ξ : Q·CP (Q) → P such that the image ˆ = ξ Q·CP (Q) is fully normalized in F (cf. Proposition 2.7); thus, we have Q ˆ FP Q·CP (Q) ⊂ FP (Q)

ξ

18.6.32

and, in particular, it follows from statement 2.10.1 that there is an F-mor ˆ ; morephism ζ : N → P extending ξ and therefore fulﬁlling ζ N ⊂ NP (Q) ˆ = Z(Q ) and therefore there is a triple formed by a over, we have CP (Q) ˆ and by two group homomorphisms group L ˆ −→ L ˆ and π ˆ −→ F(Q) ˆ τˆ : NP (Q) ˆ:L

18.6.33

ˆ. which fulﬁlls conditions 18.6.2 and 18.6.3 with respect to Q Now, setting U = τˆ ζ(Q) , we claim that the triple formed by the group L = NLˆ (U ) Op CLˆ (U ) ·ˆ τ ζ(HCF (Q) )

18.6.34

and by the two group homomorphisms τ : N −→ L and π : L −→ F(Q)

18.6.35

respectively mapping u ∈ N on the image of τˆ ζ(u) in L , and the image ˆ on the automorphism ϕ of Q such that ζ ϕ = π x ∈ L of x ˆ∈L ˆ (ˆ x) , fulﬁlls the conditions above with respect to Q . ˆ endowed with τˆ and π Indeed, since we assume that the group L ˆ fulﬁlls ˆ condition 18.6.3, for any ϕ ∈ F(Q) clearly there is x ˆ ∈ L such that −1 τˆ ζ ϕ(u) = x ˆτˆ ζ(u) x ˆ = τˆ π ˆ (ˆ x) ζ(u)

18.6.36

for any u ∈ Q , and therefore, since τˆ is injective, we get ϕ = π(x) where x is the image of x ˆ in L . From the same condition, it is not diﬃcult to prove

18. The localizers in a Frobenius P -category 341 that τˆ ζ NP (Q) is a Sylow p-subgroup of NLˆ (U ) , so that τˆ ζ CP (Q) is a Sylow p-subgroup of CLˆ (U ) , and then it is easy to check that Op CLˆ (U ) ∩ τˆ ζ CP (Q) ⊂ τˆ ζ(HCF (Q) )

18.6.37.

On the other hand, the kernel of the homomorphism NLˆ (U ) → F(Q) induced by ζ and τˆ clearly coincides with CLˆ (U ) . Now, it is clear that the triple L , π and τ fulﬁlls condition 18.6.2. Moreover, let R and T be subgroups of N containing HCF (Q) ; it follows from Proposition 13.12 that any F-morphism ϕ : Q·T → Q·R fulﬁlling ϕ(Q) = Q can be extended to an F-morphism ψ : CP (Q)·Q·T −→ P

18.6.38

and then we have ψ CP (Q) = CP (Q) since Q is fully centralized in F ˆ = ζ(R) and Tˆ = ζ(T ) , there is an (cf. Proposition 2.11). Thus, setting R F-morphism ˆ Tˆ −→ Q· ˆR ˆ ψˆ : Q· 18.6.39 ˆ Q) ˆ =Q ˆ and ψˆ ζ(w) = ζ ϕ(w) for any w ∈ T ; hence, there is such that ψ( ˆ such that x ˆ∈L

ˆ u) and x ˆ w) π ˆ (ˆ x) (ˆ u) = ψ(ˆ ˆτˆ(w)ˆ ˆ x−1 = τˆ ψ( ˆ

18.6.40

ˆ and any w for any u ˆ∈Q ˆ ∈ Tˆ ; in particular, x ˆ normalizes U and, denoting by x the image of x ˆ in L , we get

π(x) (u) = ϕ(u) and xτ (w)x−1 = τ ϕ(w)

18.6.41

for any u ∈ Q and any w ∈ T . That is to say, we get a functor NF ,Q (Q)/HCF (Q) −→ FL,Q

18.6.42

which is necessarily faithful. ˆ and τˆ(Tˆ) are Conversely, if x ∈ L fulﬁlls τ (T ) ⊂ τ (R)x then, since τˆ(R) ˆ Sylow p-subgroups of the respective converse images of τ (R) and τ (T ) in L ˆ such that τˆ(Tˆ) ⊂ τˆ(R) ˆ xˆ and therefore (cf. 18.6.37), we can lift x to x ˆ∈L ˆ ˆ ˆ ˆ ˆ there is an F-morphism ψ : Q·T → Q·R still fulﬁlling equalities 18.6.40; now, the fullness of the functor above follows from considering the F-morphism ϕ : T → R such that ψˆ ζ(w) = ζ ϕ(w) for any w ∈ T .

342

Frobenius categories versus Brauer blocks

Finally, assume that there is a triple formed by a group L and by two group homomorphisms τ : N −→ L

and π : L −→ F(Q)

18.6.43

fulﬁlling conditions 18.6.2 and 18.6.3; we will apply 18.8 below to L , Lemma to L , to the normal subgroup Ker(π ) = τ CP (Q) of L , and to the group homomorphism ρ : τ NP (Q) → L mapping τ (u) on τ (u) for any u ∈ NP (Q) . We claim that they fulﬁll condition 18.8.1 below. Indeed, if R is a subgroup of NP (Q) , any x ∈ L such that τ NP (Q) contains τ (R)x also fulﬁlls x τ Q·CP (Q)·R ⊂ τ NP (Q) 18.6.44 and therefore, by condition 18.6.3, there is an F-morphism ϕ : Q·CP (Q)·R −→ NP (Q)

18.6.45

such that, for any u ∈ Q and for any w ∈ Q·CP (Q)·R , we have

−1 π(x) (u) = ϕ(u) and x−1 τ (w)x = τ ϕ(w)

18.6.46;

in particular, we still have ϕ(Q) = Q and therefore, by condition 18.6.3 again, there is an element y ∈ L such that we have −1 −1 π (y) (u) = ϕ(u) = π(x) (u) 18.6.47 for any u ∈ Q , so that π (y) = π(x) , and that, for any w ∈ R , we still have y −1 τ (w)y = τ ϕ(w) = ρ τ ϕ(w) = ρ x−1 τ (w)x 18.6.48. Consequently, it follows from the lemma below that there is a group homomorphism λ : L → L fulﬁlling λ◦τ = τ and π ◦λ = π , which is actually an isomorphism moreover, since since both groups fulﬁll condition 18.6.2; Ker(π ) ⊂ τ NP (Q) , another group homomorphism λ : L → L fulﬁlling the corresponding equalities induces the same L-action over Ker(π ) as λ and therefore, according to the lemma below, it is Z Ker(π ) -conjugate of λ . We are done. Remark 18.7 If Q is F-nilcentralized (cf. 4.3) then τQ is injective (cf. 13.2.2) and the categories NF ,Q (Q) and FL(Q),Q are equivalent; conversely, in this case the existence of an injective group homomorphism τQ : NP (Q) −→ L(Q)

18.7.1

which induces an equivalence of categories NF ,Q (Q) ∼ = FL(Q),Q can replace both conditions above.

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Lemma 18.8 Let L and M be ﬁnite groups, P a Sylow p-subgroup of L , Q ¯ = M/Q a group homomorphism. a normal p-subgroup of M and σ ¯ :L → M Assume that there is a group homomorphism τ : P → M lifting the restriction of σ ¯ and fulﬁlling the condition 18.8.1 For any subgroup R of P and any x ∈ L such that Rx ⊂ P , there is y ∈ M such that σ ¯ (x) = y¯ and τ (ux ) = τ (u)y for any u ∈ R . Then, there is a group homomorphism σ : L → M lifting σ ¯ and extending τ . Moreover, if σ : L → M is a group homomorphism which lifts σ ¯ , extends τ and induces the same L-action over Q as σ†, then there is z ∈ Z(Q) such that σ (x) = σ(x)z for any x ∈ L . ˜ = M/Z Proof: First of all, we prove the existence of σ ; set Z = Z(Q) , M ˜ and Q = Q/Z ; a group homomorphism τ as above determines a group homo˜ which still fulﬁlls the corresponding condition 18.8.1; morphism τ˜ : P → M ˜ thus, arguing by induction on |Q| , there is a group homomorphism σ ˜ :L → M lifting σ ¯ and extending τ˜ ; in particular, σ ˜ determines an action of L on Z and it makes sense to consider the cohomology groups Hn (L, Z) and Hn (P, Z) . ˜ , Z) and if there is a group But, M determines an element µ ˜ of H2 (M homomorphism τ : P → M lifting the restriction of σ ˜ then the corresponding image of µ ˜ in H2 (P, Z) has to be zero; thus, since the restriction map H2 (L, Z) −→ H2 (P, Z) is injective (cf. Theorem 10.1 in [18, Ch. XII]), we also get 2 H (˜ σ , idZ ) (˜ µ) = 0

18.8.2

18.8.3

and therefore there is a group homomorphism σ : L → M lifting σ ˜. At this point, the diﬀerence between τ and the restriction of σ to P deﬁnes a 1-cocycle θ : P → Z and, for any subgroup R of P and any x ∈ L such that Rx ⊂ P , it follows from condition 18.8.1 that, for a suitable y ∈ M fulﬁlling y¯ = σ ¯ (x) , for any u ∈ R we have θ(ux ) = τ (ux )−1 σ(ux ) = τ (u−1 )y σ(u)σ(x) = τ (u−1 )y τ (u)σ(x) θ(u)σ(x) σ(x) −1 τ (u) = yσ(x)−1 θ(u) yσ(x)−1

18.8.4;

−1 τ (u) consequently, since the map sending u ∈ R to yσ(x)−1 yσ(x)−1 is a 1-coboundary, the cohomology class θ¯ of θ is L-stable and it follows again from Theorem 10.1 in [18, Ch. XII] that it is the restriction of a suitable element η¯ ∈ H1 (L, Z) ; then, it suﬃces to modify σ from a representative of η¯ to get a new group homomorphism σ : L → M lifting σ ¯ and extending τ . †

We thank Peter Symonds for helping us in the uniqueness of the noncommutative case.

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Frobenius categories versus Brauer blocks

Now, if σ : L → M is a group homomorphism which lifts σ ¯ , extends τ and induces the same L-action on Q as σ , the element σ (x)σ(x)−1 belongs to Z(Q) for any x ∈ L and thus, we get a 1-cocycle λ : L → Z(Q) mapping x ∈ L on σ (x)σ(x)−1 , which vanish over P ; hence, it is a 1-coboundary (cf. Theorem 10.1 in [18, Ch. XII]) and therefore there exists z ∈ Z(Q) such that λ(x) = z −1 σ(x)zσ(x)−1 18.8.5 so that we have σ (x) = σ(x)z for any x ∈ L . We are done. Remark 18.9 Whenever Q is Abelian the L-actions over Q induced by σ and σ only depend on σ ¯ and therefore always coincide. 18.10 This lemma allows us to exhibit the functorial nature of the F-localizers mentioned above; but, the lack of uniqueness in the general situation forces us to replace the F-localizer L(Q) of a subgroup Q of P fully normalized in F by the quotient (cf. Lemma 13.3) ' ( ¯ L(Q) = L(Q) τQ FCF (Q) = L(Q) Ker(πQ ), Ker(πQ ) 18.10.1; coherently, we denote by ¯ ¯ τ¯Q : NP (Q) −→ L(Q) and π ¯Q : L(Q) −→ F(Q)

18.10.2

the group homomorphisms induced by the structural ones. Note that if Q ¯ is F-selfcentralizing then L(Q) = L(Q) since Ker(πQ ) ∼ = Z(Q) (cf. condition 18.6.2). 18.11 Recall that the objects of the proper category of chains ch∗ (F) of F (cf. A2.8) are the pairs (q, ∆n ) where n ∈ N and q : ∆n → F is a functor — called an (n, F)-chain — and the morphisms from (q, ∆n ) to another object (r, ∆m ) are the pairs (ν, δ) where δ : ∆m → ∆n is a functor or, equivalently, an order-preserving map, and ν : q ◦ δ ∼ = r is a natural isomorphism, the composition with another morphism (µ, ε) : (r, ∆m ) → (t, ∆6 ) being deﬁned by (cf. A2.6.3) (µ, ε) ◦ (ν, δ) = µ ◦ (ν ∗ ε), δ ◦ ε 18.11.1. Then, it follows from Proposition A2.10 that we have a functor autF : ch∗ (F) −→ Gr

18.11.2

mapping any object (q, ∆n ) on its group of automorphisms in ch∗ (F) — denoted by F(q) (cf. 2.19) — which is isomorphic to Nat(q, q)∗ (cf. 1.5). 18.12 Moreover, let us denote by Loc the category where the objects are the pairs (L, Q) formed by a ﬁnite group L and a normal p-subgroup Q of L and where the morphisms from (L, Q) to (L , Q ) are the group homomorphisms f : L → L fulﬁlling f (Q) ⊂ Q . This category has an inner structure (cf. 1.3) mapping any object (L, Q) on the subgroup of the group

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+ of automorphisms determined by the Q-conjugation, and we denote by Loc the corresponding exterior quotient (cf. 1.3); that is to say, the category + has the same objects as Loc and the morphisms from (L, Q) to (L , Q ) Loc are the Q-conjugacy classes of group homomorphisms f : L → L fulﬁlling + contain Gr as a full subcategory and f (Q) ⊂ Q . Obviously, Loc and Loc + → Gr mapping (L, Q) on L/Q . Our we consider the Levi functor lv : Loc purpose is to exhibit a functor + locF : ch∗ (F) −→ Loc

18.12.1

— called the F-localizing functor — lifting autF (cf. 18.11.2) and mapping the (0, F)-chain determined by a subgroup Q of P fully normalized in F on ¯ the pair L(Q), Ker(¯ π ) (cf. Theorem 18.6). Q

18.13 Explicitly, for any (n, F)-chain q : ∆n → F and any i ∈ ∆n we have an evident group homomorphism F(q) −→ F q(i) 18.13.1 and actually we can identify F(q) with its image in F q(n) ; if q is fully normalized in F (cf. 2.18) then Theorem 18.6 supplies an essentially unique triple τq(n) , L q(n) , πq(n) formed by a group L q(n) and two group homomorphisms πq(n) τq(n) NP q(n) −−−→ L q(n) −−−→ F q(n) 18.13.2, and we denote by L(q) the converse image of F(q) in L q(n) , by NP (q) the converse image of FP (q) in NP q(n) (cf. 2.19) and by τq

πq

NP (q) −→ L(q) −→ F(q)

18.13.3

the corresponding restrictions of homomorphisms 18.13.2. 18.14 Since any F-chain admits a naturally isomorphic chain which is fully normalized in F (cf. 2.18), in order to exhibit the F-localizing funcfn tor we can restrict ourselves to the full subcategory ch∗ (F) of ch∗ (F) over the set of fully normalized chains of F . Let (ν, δ) : (r, ∆m ) → (q, ∆n ) be a fn ch∗ (F) -morphism; by the very deﬁnition of the proper category of chains of F (cf. A2.8), we have a group isomorphism νn : r δ(n) ∼ = q(n) and therefore we get a group homomorphism (cf. 18.13.1) F(r) −→ F r δ(n) ∼ 18.14.1; = F q(n) it is easy to check that the image of this group homomorphism is contained in F(q) and, actually, this inclusion determines the functor 18.11.2 above autF (ν, δ) : F(r) −→ F(q)

18.14.2.

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Frobenius categories versus Brauer blocks

18.15 In particular, since FP (q) is a Sylow p-subgroup of F(q) (cf. 2.18 and Proposition 2.11), there is ρ ∈ F(q) such that autF (ν, δ) FP (r) ⊂ FP (q)ρ 18.15.1; that is to say, the composition with ρ ◦ νn : r δ(n) ∼ = q(n) of the inverse of the group isomorphism induced by r(δ(n) • m) r(δ(n) • m)∗ : r δ(n) ∼ 18.15.2 = Im r(δ(n) • m) maps the action of NP (r) over Im r(δ(n) • m) into the action of NP (q) over q(n) . Consequently, since CP q(n) ⊂ NP (q) , it follows from Proposition 2.11 that there is an F-morphism ζ : NP (r)·Im r(δ(n) • m) −→ NP (q)·q(n) 18.15.3 fulﬁlling ζ r(δ(n) • m)(v) = ρ νn (v) for any v ∈ r δ(n) and therefore, according to inclusion 18.15.1, ζ maps NP (r) into NP (q) . Proposition 18.16 With the notation above, for any ch∗ (F) -morphism (ν, δ) : (r, ∆m ) → (q, ∆n ) , there is a group homomorphism fn

λ : L(r) −→ L(q)

18.16.1

πq ◦ λ = autF (ν, δ) ◦ πr

18.16.2

such that and that, for some r ∈ L(q) lifting ρ and any v ∈ NP (r) , we have r λ τr (v) = τq ζ(v)

18.16.3.

Moreover, these group homomorphisms induce mutually Ker(¯ πq )-conjugate homomorphisms ¯ : L(r) ¯ −→ L(q) ¯ λ 18.16.4. Proof: Set Q = q(n) , R = r(m) and ϕ = r(δ(n) • m)∗ (cf. 18.15.2); ﬁrst of all, note that we have ζ CP (R) ⊂ CP (Q) and ζ(v) = ρ νn ϕ−1 (v) if v ∈ Im(ϕ) 18.16.5; moreover, as in the proof of Proposition 13.14, it is easily checked that ζ induces a CF (R), CF (Q) -functorial homomorphism from CP (R) to CP (Q) (cf. 12.1), so that we get (cf. 13.2) ζ(HCF (R) ) ⊂ HCF (Q)

and ζ(FCF (R) ) ⊂ FCF (Q)

18.16.6.

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Denoting by κρ : F(q) ∼ = F(q) the inner automorphism determined by ρ , on the one hand consider the composed homomorphism κρ

autF (ν,δ) πr L(r) −→ F(r) −−−−−−→ F(q) ∼ 18.16.7; = F(q) on the other hand, consider the homomorphism η : τr NP (r) → L(q) map ping τr (v) on τq ζ(v) for any v ∈ NP (r) , which makes sense by the left-hand inclusion in 18.16.6. We claim that these group homomorphisms together with the groups L(r) and L(q) , the Sylow p-subgroup τr NP (r) of L(r) , and the normal subgroup Ker(πq ) of L(q) fulﬁll condition 18.8.1.

Indeed, if T is a subgroup of NP (r) containing HCF (R) and y an element of L(r) such that τr (T )y ⊂ τr NP (r) , it is clear that y determines a mor phism τr (T ) → τr NP (r) in the FL(R) -locality FL(R),R (cf. 17.11.2); then, it follows from condition 18.6.3 that there is an F-morphism ψ : R·T → R·NP (r) fulﬁlling ψ(v) = πR (y) (v) and τR ψ(w) = τR (w)y 18.16.8 for any v ∈ R and any w ∈ T . Hence, since πR (y) stabilizes the image in R of r(j • m) for any j ∈ ∆m , the restriction of ψ gives an Im(ϕ)·T → Im(ϕ)·NP (r) (cf. 2.4) F-morphism and then, since we have ζ Im(ϕ) = Q , the “translation” via ζ induces an F-morphism θ : Q·ζ(T ) −→ Q·NP (q) 18.16.9 fulﬁlling θ(Q) = Q and θ ζ(T ) ⊂ NP (q) (cf. 2.4); in other words, θ determines a morphism ζ(T ) → NP (q) in the NF (Q)-locality NF ,Q (Q) (cf. 17.6) and therefore, it follows again from condition 18.6.3 that there is x ∈ L(Q) fulﬁlling x θ(u) = πQ (x) (u) and τQ θ ζ(w) = τQ ζ(w) 18.16.10 for any u ∈ Q and any w ∈ T . Hence, setting ξ = ϕ ◦ (νn )−1 ◦ ρ−1 , for any u ∈ Q we get (cf. 18.15 and 18.16.5) κρ ◦ autF (ν, δ) ◦ πR (y) (u) = ρ ◦ autF (ν, δ) πR (y) ◦ ρ−1 (u) 18.16.11, = ζ πR (y) ξ(u) = ζ ψ ξ(u) = θ ζ ξ(u) = θ(u) = πQ (x) (u) so that x ∈ L(q) is a lifting of the image of y by homomorphism 18.16.7.

348

Frobenius categories versus Brauer blocks Moreover, for any w ∈ T we still get η τR (w)y = η τr ψ(w) = τq ζ ψ(w) x = τq θ ζ(w) = τQ ζ(w)

18.16.12.

At this point, it follows from Lemma 18.8 that there is a group homomorphism λ : L(r) → L(q) fulﬁlling πq ◦ λ = κρ ◦ autF (ν, δ) ◦ πr

and λ τr (v) = τq ζ(v)

18.16.13

for any v ∈ NP (r) ; in particular, for any r ∈ L(q) lifting ρ , the group homomorphism λr : L(r) → L(q) fulﬁlls the announced condition. Thus, it follows from the right-hand inclusion in 18.16.6 that λr τr (FCF (R) ) ⊂ τq (FCF (Q) )

18.16.14

and therefore λr induces a group homomorphism ¯ −→ L(q) ¯ λr : L(r)

18.16.15.

Similarly, it is clear that any group homomorphism λ : L(r) → L(q) fulﬁlling condition 18.16.13 still fulﬁlls inclusion 18.16.14 and therefore it induces also ¯ → L(q) ¯ ; then, it follows from Lemma 18.8 a group homomorphism (λ )r : L(r) r r and Remark 18.9 that λ and (λ ) are Ker(¯ πq )-conjugate. We are done. Remark 18.17 Our condition above does not depend on the choices of ρ and ζ . Indeed, for another ρ ∈ F(q) fulﬁlling inclusion 18.15.1, it is clear that ρ ◦ ρ−1 maps the action of ζ NP (r) over q(n) into the action of NP (q) and therefore it follows again from Proposition 2.11 that there is an F-morphism θ : q(n)·ζ NP (r) −→ q(n)·NP (q)

18.17.1

−1 extending ρ ◦ ρ ; at this point, it follows from Theorem 18.6 that there is t ∈ L q(n) fulﬁlling

πq(n) (t) = ρ ◦ ρ−1

t and τq θ ζ(v) = τq ζ(v) if v ∈ NP (r)

18.17.2;

then, setting r = t r and ζ = θ ◦ ζ∗ where ζ∗ denotes the F-isomorphism r for any v ∈ NP (r) . induced by ζ , we still have λ(ν,δ) τr (u) = τq ζ (v) Similarly, for another choice ζ of ζ , we still can deﬁne an F-morphism θ fulﬁlling ζ = θ ◦ ζ∗ but this time θ induces the identity on q(n) , so that t belongs to Ker(πq ) .

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18.18 Thus, with the notation above, it follows from this proposition fn and from Lemma 18.8 and Remark 18.9 that, for any ch∗ (F) -morphism + (ν, δ) : (r, ∆m ) → (q, ∆n ) , there is a unique Loc-morphism ¯ ¯ loc(ν,δ) : L(r), Ker(¯ πr ) −→ L(q), Ker(¯ πq ) 18.18.1 ¯ : L(r) ¯ → L(q) ¯ ¯ admitting a representative λ which, for some r¯ ∈ L(q) lifting ρ (cf. 18.15), fulﬁlls ¯ = autF (ν, δ) ◦ π ¯ τ¯r (v) = τ¯q ζ(v) r¯ π ¯q ◦ λ ¯r and λ 18.18.2 + → Gr the functor for any v ∈ NP (r) . Recall that we denote by lv : Loc + mapping any Loc-object (L, Q) on L/Q (cf. 18.12). Proposition 18.19 There is a unique isomorphism class of functors + locF : ch∗ (F) −→ Loc 18.19.1 fn fn ¯ mapping any ch∗ (F) -object (q, ∆n ) on L(q), Ker(¯ πq ) and any ch∗ (F) morphism (ν, δ) : (r, ∆m ) → (q, ∆n ) on loc(ν,δ) . In particular, we have lv ◦ locF ∼ = autF

18.19.2.

Proof: We already know that we can restrict ourselves to the full subcatefn fn gory ch∗ (F) ; thus, let (η, ε) : (t, ∆6 ) → (r, ∆m ) be a second ch∗ (F) -mor¯ ¯ phism and denote by µ ¯ : L(t) → L(r) a representative of loc(η,ε) . Setting Q = q(n) , R = r(m) , T = t(D) , ϕ = r(δ(n) • m)∗ and ψ = t(ε(m) • D)∗ (cf. 18.15.2), we already know that (cf. 18.18) π ¯r ◦ µ ¯ = autF (η, ε) ◦ π ¯t

18.19.3

and that there are an F-morphism (cf. 18.15) ξ : Im(ψ)·NP (t) −→ R·NP (r)

18.19.4

¯ fulﬁlling and an element s¯ in L(r) s¯ µ ¯ τ¯t (u) = τ¯r ξ(u) and ξ ψ(v) = π ¯r (¯ s) ηm (v) 18.19.5 for any u ∈ NP (t) and any v ∈ t ε(m) . Since (ν, δ) ◦ (η, ε) = ν ◦ (η ∗ δ), ε ◦ δ (cf. A2.6.3), it is clear that autF ν ◦ (η ∗ δ), ε ◦ δ ◦ π ¯t ¯t = autF (ν, δ) ◦ autF (η, ε) ◦ π = autF (ν, δ) ◦ π ¯r ◦ µ ¯ ¯◦µ =π ¯q ◦ (λ ¯)

18.19.6.

350

Frobenius categories versus Brauer blocks On the other hand, considering the F-morphisms ξ

Im(ψ)·NP (t) −→

R·NP (r) 18.19.7 ∪ ζ Im(ϕ)·NP (r) −→ Q·NP (q) and setting ω = t (ε ◦ δ)(n) • D ∗ (cf. 18.15.2), we get the F-morphism θ : Im(ω)·NP (t) −→ Q·NP (q) 18.19.8 mapping w ∈ Im(ω)·NP (t) on ζ ξ(w) . In particular, since we have t (ε ◦ δ)(n) • D = t(ε(m) • D) ◦ t (ε ◦ δ)(n) • ε(m) 18.19.9, = t(ε(m) • D) ◦ (ηm )−1 ◦ r(δ(n) • m) ◦ ηδ(n) if v ∈ t (ε ◦ δ)(n) setting w = ϕ ηδ(n) (v) then we get ω(v) = ψ (ηε(m) )−1 (w) 18.19.10 and therefore, setting σ = π ¯r (¯ s) we still have −1 ζ ξ ω(v) = ζ ξ ψ (ηε(m) ) (w) = ζ σ(w)

18.19.11;

thus, denoting by σδ(n) ∈ F r δ(n) the element induced by σ , for any v ∈ t (ε ◦ δ)(n) from the naturality of η we get (cf. 18.15) θ ω(v) = ζ ϕ (σδ(n) ◦ ηδ(n) )(v) = ρ νn (σδ(n) ◦ ηδ(n) )(v) 18.19.12 . = ρ ◦ autF (ν, δ)(σ) (νn ◦ ηδ(n) )(v) = ρ ◦ autF (ν, δ)(σ) ◦ ν ◦ (η ∗ δ) n (v) That is to say, since we have (cf. 18.15.1) autF (ν, δ) ◦ autF (η, ε) FP (t) ⊂ autF (ν, δ) FP (r)σ autF (ν,δ)(σ) = autF (ν, δ) FP (r) ⊂ FP (q)ρ◦autF (ν,δ)(σ)

18.19.13,

the F-morphism θ fulﬁlls the last condition in 18.15 with respect to the element ρ ◦ autF (ν, δ)(σ) ∈ F(q) ; moreover, it is quite clear that, for any u ∈ NP (t) , we get (cf. 18.19.5 and 18.18.2) ¯ ¯◦µ ¯ τ¯r ξ(u) s¯ = τ¯q θ(u) r¯λ(¯s) (λ ¯) τ¯t (u) = λ 18.19.14.

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Consequently, λ ◦ µ fulﬁlls condition 18.18.2 with respect to the ch∗ (F)-mor + phism ν◦(ω∗δ), ε◦δ and therefore it is a representative of the Loc-morphism loc(ν◦(ω∗δ),ε◦δ) , so that locF (ν, δ) ◦ (η, ε) = locF (ν, δ) ◦ locF (η, ε)

18.19.15.

We are done. 18.20 Finally, let us discuss a particular kind of “universality” of the F-localizing functor locF . Let Lˆ be a coherent F-locality, denote by τˆ : TP −→ Lˆ and π ˆ : Lˆ −→ F

18.20.1

the structural functors and assume that, for any subgroup Q of P , Ker(ˆ πQ ) is an Abelian p-group; it follows from Proposition A2.10 that we have a functor ˆ −→ Gr autLˆ : ch∗ (L)

18.20.2

ˆ ˆ mapping any ch∗ (L)-object (ˆq, ∆n ) on the group of all the ch∗ (L)-automorˆ phisms of (ˆq, ∆n ) ; since L is divisible (cf. 17.7), this group can be identiﬁed ˆ q) . Then, it with the converse image of F(ˆ π ◦ ˆq) in L ˆq(n) — denoted by L(ˆ is clear that autLˆ induces an obvious functor + locLˆ : ch∗ (F) −→ Loc

18.20.3

ˆ q), Ker(ˆ mapping any ch∗ (F)-object (q, ∆n ) on L(ˆ πq ) , where ˆq : ∆n → Lˆ ˆ q) ; ˆq is the restriction of π ˆq(n) to L(ˆ is a functor lifting q : ∆n → F and π ˆ q) does not depend on the choice of ˆq . In particular, we have note that L(ˆ (cf. 18.12) lv ◦ locLˆ = autF 18.20.4. Proposition 18.21 With the notation above, there is a unique natural map λLˆ : locF −→ locLˆ

18.21.1

such that lv∗λLˆ = idautF and that, for any chain q : ∆n → F fully normalized in F , we have (λLˆ)(q,∆n ) ◦ τ¯q = τˆq . Proof: First of all, for any subgroup Q of P fully normalized in F , choose an F-localizer L(Q) of Q and denote by τQ : NP (Q) −→ L(Q) and πQ : L(Q) −→ F(Q)

18.21.2

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Frobenius categories versus Brauer blocks

the structural homomorphisms; we will apply Lemma 18.8 to the groups ¯ ˆ L(Q) (cf. 18.10) and L(Q) , to the quotient F(Q) of both, and to the group homomorphism ˆ η : τ¯Q NP (Q) −→ L(Q) 18.21.3 mapping τ¯Q (u) on τˆQ (u) for any u ∈ NP (Q) , which makes sense since Ker(ˆ πQ ) is an Abelian p-group, so that FCF (Q) ⊂ Ker(ˆ τQ ) (cf. Lemma 13.3 and 17.13.1) ¯ We know that τ¯Q NP (Q) is a Sylow p-subgroup of L(Q) (cf. 17.12) and it is clear that the restriction of π ¯Q to τ¯Q NP (Q) coincides with the composition π ˆQ ◦ η . Let R be a subgroup of NP (Q) and x an element of L(Q) such that τQ (R)x ⊂ τQ NP (Q) ; since we still have τQ (Q·R)x ⊂ τQ NP (Q)

18.21.4,

it follows from Theorem 18.6 that there is an F-morphism ϕ : Q·R → NP (Q) such that we have ϕ(u) = πx (u) and τQ ϕ(v) = τQ (v)x 18.21.5 for any u ∈ Q and any v ∈ Q·R . ˆ At this point, it follows from statement 17.11.2 that there is x ˆ ∈ L(Q) fulﬁlling ϕ(u) = π ˆxˆ (u) and τˆQ ϕ(v) = τˆQ (v)xˆ 18.21.6 for any u ∈ Q and any v ∈ Q·R . Consequently, denoting by x ¯ the image of x ¯ in L(Q) , we get π ˆQ (ˆ x) = π ¯Q (¯ x)

xˆ and η τ¯Q (v)x¯ = τˆQ (v)xˆ = η τ¯Q (v)

18.21.7

for any v ∈ Q·R . Now, it follows from Lemma 18.8 that there is a unique Ker(ˆ πQ )-conjugacy class of group homomorphisms ¯ ˆ λQ : L(Q) −→ L(Q)

18.21.8

fulﬁlling π ˆ Q ◦ λQ = π ¯Q and λQ ◦ τ¯Q = τˆQ . Then, it is clear that, for any chain q : ∆n → F fully normalized in F , the corresponding group homomorphism λq(n) : L¯ q(n) −→ Lˆ q(n) 18.21.9 ¯ into L(q) ˆ maps L(q) = (ˆ πq )−1 F(q) and Ker(¯ πq ) , and therefore πq ) into Ker(ˆ + determines a unique Loc-morphism ˜ q : locF (q, ∆n ) −→ loc ˆ(q, ∆n ) λ L

18.21.10.

18. The localizers in a Frobenius P -category

353

Now, we claim that the correspondence λLˆ which maps any ch∗ (F) -object ˜ q deﬁnes a natural map over ch∗ (F)fn and therefore, since this (q, ∆n ) on λ subcategory is equivalent to ch∗ (F) , it can be uniquely extended to the announced natural map. fn

Actually, up to equivalence, we can restrict ourselves to the full subfni fn fn category ch∗ (F) of ch∗ (F) over the ch∗ (F) -objects (q, ∆n ) such that q(n) we have q(D) ⊂ q(n) and q(D • n) = ιq(i) for any D ∈ ∆n ; in this case, ˆ we denote by ˆq : ∆n → Lˆ the L-chain mapping D ∈ ∆n on q(D) and D • n on ˆıq() = τˆq(n),q() (1) . Let (ν, δ) : (r, ∆m ) → (q, ∆n ) be a ch∗ (F) -morphism and denote by (ˆ ν , δ) : (ˆr, ∆m ) → (ˆq, ∆n ) a lifting of (ν, δ) to the corresponding ˆ ; it is quite clear that the group homomorphism full subcategory of ch∗ (L) (cf. 18.20.2) ˆ r) −→ L(ˆ ˆ q) ⊂ Lˆ q(n) autLˆ(ˆ ν , δ) : L(ˆ 18.21.11 fni

q(n)

is induced by the composition ˆ r) −→ Lˆ r δ(n) ∼ L(ˆ = Lˆ q(n)

18.21.12

where the right isomorphism is determined by νˆn : r δ(n) ∼ = q(n) and the left ˆ r) ⊂ Lˆ r(m) on the element yˆ ∈ Lˆ r δ(n) homomorphism maps x ˆ ∈ L(ˆ fulﬁlling r(m)

r(m)

ˆır(δ(n)) ·ˆ y=x ˆ·ˆır(δ(n))

18.21.13;

thus, for any v ∈ NP (ˆr) , this homomorphism maps τˆr(m) (v) on τˆr(δ(n)) (v) . On the other hand, if we have νn ◦ FP (r) ◦ (νn )−1 ⊂ FP (q)ρ

18.21.14

for some ρ ∈ F(q) (cf. 18.15) then, since q(n) is fully centralized in F (cf. Proposition 2.11), there is an F-morphism (cf. condition 2.10.1) ζ : r δ(n) ·NP (r) −→ q(n)·NP (q)

18.21.15

fulﬁlling ζ(w) = (ρ ◦ νn )(w) for any w ∈ r δ(n) ; moreover, since we have CP q(n) ⊂ NP (q) , it is easily checked that ζ NP (r) ⊂ NP (q) . Hence, there are zˆ ∈ Lˆ q(n), r δ(n) lifting the element of F q(n), r δ(n) determined ¯ by ζ and r¯ ∈ L(q) lifting ρ such that, for any v ∈ NP (r) , we have zˆ = λq (¯ r)·ˆ νn

and

λq (¯r) autLˆ(ˆ ν , δ) τˆr (v) = τˆq ζ(v)

18.21.16.

354

Frobenius categories versus Brauer blocks But, according to Proposition 18.16, we have a group homomorphism ¯ (ν,δ) : L(r) ¯ −→ L(q) ¯ λ

18.21.17

¯ (ν,δ) = autF (ν, δ) ◦ π such that π ¯q ◦ λ ¯r and that, for some r¯ ∈ L(q) lifting ρ , r¯ ¯ (ν,δ) τ¯r (v) = τ¯q ζ(v) we have λ for any v ∈ NP (r) ; moreover, we always

can modify our lifting νˆ of ν in such a way that r¯ = r¯ . In this case, for any v ∈ NP (r) we get λq (¯r) ¯ (ν,δ) τ¯r (v) autLˆ(ˆ ν , δ) τˆr (v) = τˆq ζ(v) = λq λ 18.21.18;

¯ (ν,δ) that is to say, the two group homomorphisms autLˆ(ˆ ν , δ) ◦ λr and λq ◦ λ ¯ ˆ from L(r) to L(q) coincide over τ¯r NP (r) which is a Sylow p-subgroup ¯ , and induce the same homomorphism from L(r) ¯ to F(q) . of L(r) Consequently, it follows from the uniqueness part of Lemma 18.8 that ¯ (ν,δ) are mutually the group homomorphisms autLˆ(ˆ ν , δ) ◦ λr and λq ◦ λ + Ker(ˆ πq )-conjugate and therefore we obtain the following equality of Loc-morphisms ˜r = λ ˜ q ◦ locF (ν, δ) locLˆ(ν, δ) ◦ λ 18.21.19 which proves the announced naturality. Finally, the uniqueness of the natural map λLˆ follows from the uniqueness of the Ker(ˆ πQ )-conjugacy classes of the group homomorphisms 18.21.8 above. We are done. 18.22 Naturally, this proposition raises a question: does there exist an F-sublocality L of Lˆ such that the corresponding subfunctor locL ⊂ locLˆ is just the image of λLˆ ? Note that, restricting our attention to the full subsc sc categories L and Lˆ over the set of F-selfcentralizing subgroups Q of P and sc sc assuming that the restricted functor τˆsc : TP → Lˆ (cf. 17.4) is faithful , in sc sc this case L is a perfect F -locality since L(Q) is indeed the F-localizer of Q (cf. 17.13 and 18.5). As a matter of fact, these questions on the full subcatesc gory Lˆ — on this existence and on the corresponding uniqueness — admit + π sc ) ¯˜zF sc -valued 2- and cohomological answers in terms of two stable Ker(ˆ sc sc sc ˆ sc : Lˆ → F denotes the restriction 1-cocycles over F˜ (cf. A3.18), where π + π sc ) is the factorization throughout F˜ sc of the contravariant funcof π ˆ , Ker(ˆ sc sc tor Ker(ˆ π sc ) : Lˆ → Ab (cf. 17.8 and Proposition 17.10), and ¯˜zF sc : F˜ → Ab is the corresponding image of the contravariant functor deﬁned by the centers (cf. 13.13). ¯ 18.23 Let us exhibit these cocycles. First of all note that, denoting by Lˆ sc sc + π sc ) ¯˜zF sc , for the obvious quotient F -locality Lˆ /zF sc and setting ¯˜k = Ker(ˆ any F-selfcentralizing subgroup Q of P we have ¯ˆ ¯ ∼ L(Q) = ˜k(Q) F(Q)

18.23.1

18. The localizers in a Frobenius P -category

355

ˆ since, identifying the F-localizer L(Q) to its image in L(Q) by a representative of (λLˆ)Q (cf. Proposition 18.21), we have ˆ L(Q) = Ker(πQ )·L(Q)

18.23.2.

sc

sc

Secondly, any F -morphism ϕ : R → Q clearly determines a (1, F )-chain sc oϕ : ∆1 → F mapping 0 on R , 1 on Q and the ∆1 -morphism 0 • 1 on ϕ , sc and then we consider the obvious ch∗ (F )-morphism (idR , δ10 ) : (oϕ , ∆1 ) −→ (oR , ∆0 )

18.23.3,

sc

where oR : ∆0 → F maps 0 on R ; the functor autF sc (cf. Proposition A2.10) sc maps this ch∗ (F )-morphism on a group homomorphism autF sc (idR , δ10 ) : F(Q)R −→ F(R)

18.23.4,

where we set R = ϕ(R) and, as usual, F(Q)R denotes the stabilizer of R in F(Q) . ¯ : L¯ˆ → F sc induced by π 18.24 Moreover, the functor π ˆ ˆ sc determines a natural map (cf. Proposition A2.10) autπ¯ˆ : autL¯ˆ −→ autF sc

18.24.1

+ over the objects and actually, identifying Gr with the full subcategory of Loc (G, {1}) (cf. 18.12) where G runs over the ﬁnite groups, it still determines a natural map (cf. 18.20) locπ¯ˆ : locL¯ˆ −→ autF sc

18.24.2;

¯ ˆ : locF sc → loc ¯ determined then, this natural map and the natural map λ ˆ L L sc ∗ by λLˆ (cf. 18.21.1), both applied to the ch (F )-morphism 18.23.3, yield the + commutative Loc-diagram

locF

locF sc (R) & 0 sc (idR ,δ1 ) locF sc (oϕ )

¯ ˆ )R (λ

L −−− −→

locL¯ˆ(R) & loc ¯ˆ (idR ,δ10 ) L

−→

locL¯ˆ(oϕ )

(loc ¯ )R

−−−πˆ−→ autF

autF sc (R) & 0 sc (idR ,δ1 )

−→

autF sc (oϕ )

18.24.3.

18.25 In this diagram, since the right-hand vertical arrow is simply a ¯ ˆ) we can choose a group homomorphism, ﬁxing a representative λR for (λ L R representative for the middle vertical arrow in such a way that we still have the commutative diagram of group homomorphisms λ

L(R) & 

−−R→

L(Q)R

−→

¯ˆ L(R) &  ¯ˆ L(Q) R

(aut ¯ )R

ˆ −−−π− →

autF

F(R) & 

0 sc (idR ,δ1 )

−→

F(Q)R

18.25.1.

356

Frobenius categories versus Brauer blocks

That is to say, by the very deﬁnition of locL¯ˆ(idR , δ10 ) (cf. 18.20), we have ¯ˆ ¯ˆϕ ∈ L(Q, found x R) which lifts ϕ ∈ F(Q, R) ; more precisely, considering ¯ˆ ¯ˆ ¯ ˆoϕ : ∆1 → L¯ˆ and ¯ˆoR : ∆0 → L¯ˆ as above the (1, L)- and the (0, L)-chains ¯ˆ ¯ˆϕ and R , and the obvious ch∗ (L)-morphism respectively deﬁned by x

ˆoϕ , ∆1 ) −→ (¯ˆoR , ∆0 ) τ¯ˆR (1), δ10 : (¯

18.25.2,

we are just putting autLˆsc τˆR (1), δ10 in the middle vertical arrow of diagram 18.25.1. 18.26 In particular, note that the commutativity of the left-hand square ¯ˆ in diagram 18.25.1 implies that autLˆsc τˆR (1), δ10 maps the image in L(Q) R ¯ ˆ of L(Q)R on the image in L(R) of L(R) ; but, by the very deﬁnition of isomorphisms 18.23.1, these images respectively coincide with the images of F(Q)R and F(R) ; consequently, considering the action of F(Q) × F(R) ¯ˆ on L(Q, R) deﬁned by the composition on the left and on the right via isomor¯ˆϕ coincides phisms 18.23.1, we claim that the stabilizer F(Q) × F(R) x¯ˆ of x ϕ with the stabilizer F(Q) × F(R) ϕ of ϕ with respect to the corresponding action on F(Q, R) . ¯ˆϕ lifts ϕ , the inclusion 18.27 Indeed, since x

F(Q) × F(R) x¯ˆ ⊂ F(Q) × F(R) ϕ ϕ

18.27.1

is clear; conversely, for any (σ, τ ) ∈ F(Q) × F(R) ϕ , we have σ ◦ ϕ = ϕ ◦ τ and, in particular, σ belongs to F(Q)R and we get

autF sc (idR , δ10 ) (σ) = τ

18.27.2;

hence, by our remark above and the commutativity of the right-hand square in diagram 18.25.1, autLˆsc τˆR (1), δ10 still maps the image of σ on the image ¯ˆϕ = x ¯ˆϕ ·τ . In conclusion, we can choose the lifting of τ and we still have σ·x ¯ ¯ˆϕ ∈ L(Q, ˆ x R) of ϕ ∈ F(Q, R) in such a way that ¯ˆϕ ·τ for any σ ∈ F(Q) and any τ ∈ F(R) . ¯ˆσ◦ϕ◦τ = σ·x 18.27.3 We have x X

sc

Now, considering the full subcategory F of F over a set of representatives sc X X for the set of isomorphism classes of F -objects, for all the F -morphisms sc we make a choice as above and then, for any F -morphism ϕ : R → Q , we ¯ˆϕ = σ ·x ¯ˆϕ ·τ where σ : Q ∼ deﬁne x = R are F-isomorphisms, = Q and τ : R ∼ Q and R belong to X and ϕ = σ ◦ ϕ ◦ τ .

18. The localizers in a Frobenius P -category

357

Proposition 18.28 With the notation above, the correspondence mapping sc ¯ any pair of F -morphisms ψ : T → R and ϕ : R → Q on the element k˜ϕ,ψ ˜k(T ) = Ker(π )/Z(T ) such that of ¯ T

¯ ¯ˆϕ ·x ¯ˆψ = x ¯ˆϕ◦ψ ·k˜ x ϕ,ψ

18.28.1

˜k-valued 2-cocycle over F˜ sc . Moreover, this 2-cocycle is determines a stable ¯ sc ¯ ¯ a 2-coboundary if and only if the functor π ˆ : Lˆ → F admits a functorial sc sc sc section. In this case, there is an F -sublocality L of Lˆ such that locLsc is the image of λLˆsc . ¯ˆϕ ·x ¯ˆψ and x ¯ˆϕ◦ψ have the same image in F(Q, T ) , the divisibility Proof: Since x ¯ˆ ¯ of L guarantees the existence and the uniqueness of k˜ϕ,ψ ∈ ¯˜k(T ) ; that is sc sc to say, we have a correspondence mapping any (2, F )-chain q : ∆2 → F ¯ ∼ Q , τ : R = ∼ R on k˜q(0•1),q(1•2) . Moreover, for any F-isomorphisms σ : Q = and ω : T ∼ = T , setting ϕ = σ ◦ ϕ ◦ τ −1 and ψ = τ ◦ ψ ◦ ω −1 and assuming that Q , R and T belong to X , we get ¯ˆϕ ◦ψ ·k¯ ˜ϕ ,ψ = x ¯ˆϕ ·x ¯ˆψ = (σ·x ¯ˆϕ ·τ −1 )·(τ ·x ¯ˆψ ·ω −1 ) x ¯ ¯ −1 ¯ˆϕ◦ψ ·k˜ ¯ˆϕ◦ψ ·ω −1 · ¯˜k(˜ = σ·(x = σ·x ω ) (k˜ϕ,ψ ) ϕ,ψ )·ω ¯ ¯ ¯ˆϕ ◦ψ · k(˜ ˜ ω ) (k˜ϕ,ψ ) =x

18.28.2

¯ ¯ ˜k(˜ and therefore we obtain k˜ϕ ,ψ = ¯ ω ) (k˜ϕ,ψ ) . Hence, the same equality is true for any choice of the F-isomorphisms σ:Q ∼ = Q , τ : R ∼ = R and ω : T ∼ = T , proving the stability of the above correspondence (cf. A3.18). In particular note that, for any u ∈ Q and any v ∈ R , we obtain ¯ ¯ k˜κQ (u)◦ϕ,κR (v)◦ψ = k˜ϕ,ψ 18.28.3, sc ¯ proving that k˜ϕ,ψ only depends on the classes ϕ˜ and ψ˜ in F˜ of ϕ and ψ ; that is to say, the above correspondence factorizes throughout the set of sc sc (2, F˜ )-chains ˜q : ∆2 → F˜ . sc

Moreover, for a third F -morphism η : U → T , it is clear that ¯ ¯ˆϕ ·x ¯ˆψ ·x ¯ˆη = x ¯ˆϕ◦ψ ·k¯ ˜ϕ,ψ ·x ¯ˆη = x ¯ˆϕ◦ψ ·x ¯ˆη · ¯˜k(˜ x η ) (k˜ϕ,ψ ) ¯ ¯ˆϕ◦ψ◦η ·k¯ ˜ϕ◦ψ,η · ¯ ˜k(˜ =x η ) (k˜ϕ,ψ )

18.28.4

and, similarly, we have ¯ˆϕ ·x ¯ˆψ ·x ¯ˆη = x ¯ˆϕ◦ψ◦η ·k¯ ˜ϕ,ψ◦η ·k¯˜ψ,η x

18.28.5;

358

Frobenius categories versus Brauer blocks

hence, we get the 2-cocycle condition ¯ ¯ ¯ ¯ ˜k(˜ k˜ϕ◦ψ,η · ¯ η ) (k˜ϕ,ψ ) = k˜ϕ,ψ◦η ·k˜ψ,η

18.28.6.

In conclusion, the above correspondence determines indeed a stable ¯˜k-valued sc 2-cocycle over F˜ . sc ¯ ¯ On the other hand, if the functor π ˆ : Lˆ → F admits a functorial section, ¯ it is clear that such a section provides a choice fulﬁlling k˜ϕ,ψ = 1 for any ϕ ¯ and ψ . Conversely, if there exists a family of elements D˜ϕ˜ ∈ ¯˜k(R) , where sc ϕ˜ : R → Q runs over the set of F˜ -morphisms, fulﬁlling ¯ ˜ (D¯˜ϕ˜ )·(D¯˜ ˜ )−1 ·D¯˜˜ ˜k(ψ) k˜ϕ,ψ = ¯ ϕ◦ ˜ ψ ψ

18.28.7

then equality 18.28.1 becomes

¯ˆϕ ·(D¯˜ϕ˜ )−1 · x ¯ˆψ ·(D¯˜˜ )−1 = x ¯ˆϕ◦ψ ·(D¯˜ ˜ )−1 x ψ ϕ◦ ˜ ψ

18.28.8.

Thus, we can modify our starting choice of liftings, obtaining the announced sc functorial section; the converse image in Lˆ of the image of this section yields sc the announced F -sublocality. We are done. Proposition 18.29 With the notation above, assume that there are two sc ¯ ¯ functorial sections s and s of π ˆ : Lˆ → F such that s F(Q) = s F(Q) sc sc for any F -object Q . Then, the correspondence mapping any F -morphism ¯ ˜k(T ) = Ker(π )/Z(T ) such that ϕ : R → Q on the element D˜ϕ of ¯ T ¯ s (ϕ) = s(ϕ)·D˜ϕ

18.29.1

˜k-valued 1-cocycle over F˜ sc . Moreover, this 1-cocycle is determines a stable ¯ a 1-coboundary if and only if we have a natural isomorphism s ∼ = s. Proof: Since s (ϕ) and s(ϕ) have the same image in F(Q, R) , the divisibility ¯ ¯ of Lˆ guarantees the existence and the uniqueness of D˜ϕ ∈ ¯˜k(R) . Moreover, for ∼ ∼ any F-isomorphisms σ : Q = Q and τ : R = R , setting ϕ = σ ◦ ϕ ◦ τ −1 we have ¯ ¯ s(σ)·s(ϕ)·s(τ )−1 ·D˜ϕ = s(ϕ )·D˜ϕ = s (ϕ ) = s (σ)·s (ϕ)·s (τ )−1 ¯ = s (σ)·s(ϕ)·D˜ϕ ·s (τ )−1

18.29.2;

but, since s F(Q) = s F(Q) , s(σ)−1 ·s (σ) belongs to s F(Q) and therefore we easily get s (σ) = s(σ) (cf. 18.23.1); similarly, we get s (τ ) = s(τ )

18. The localizers in a Frobenius P -category

359

¯ ¯ ¯ and thus equality 18.29.2 forces D˜ϕ = ˜ k(˜ τ ) (D˜ϕ ) , proving the stability of the above correspondence. In particular note that, for any u ∈ Q , we obtain ¯ ¯ D˜κQ (u)◦ϕ = D˜ϕ

18.29.3,

sc ¯ proving that D˜ϕ only depends on the class ϕ˜ in F˜ of ϕ ; thus, the above sc sc correspondence factorizes throughout the set of F˜ -chains ˜r : ∆1 → F˜ . sc

Moreover, for a second F -morphism ψ : T → R , we have ¯ ¯ s(ϕ)·s(ψ)·D˜ϕ◦ψ = s(ϕ ◦ ψ)·D˜ϕ◦ψ = s (ϕ ◦ ψ) = s (ϕ)·s (ψ) ¯ ¯ ˜ (D¯˜ϕ )·D¯˜ψ = s(ϕ)·D˜ϕ ·s(ψ)·D˜ψ = s(ϕ)·s(ψ)· ¯˜k(ψ)

18.29.4

and therefore we get the 1-cocycle condition ¯ ¯ ¯ ¯ ˜ (D˜ϕ )·D˜ψ D˜ϕ◦ψ = ˜ k(ψ)

18.29.5.

In conclusion, the above correspondence determines indeed a stable ¯˜k-valued sc 1-cocycle over F˜ . On the other hand, if we have a natural isomorphism ν : s ∼ = s then the naturality of ν yields νQ ·s (ϕ) = s(ϕ)·νR ; but, according to decom¯ ¯˜ for suitable position 18.23.1, we have νQ = s(σQ )·m ˜ Q and νR = s(σR )·m ˜k(Q) and m ˜k(R) ; consequently, we get ¯ ¯ σQ ∈ F(Q) , σR ∈ F(R) , m ˜ ∈¯ ˜ ∈¯ ¯ ¯˜ D¯˜ϕ νQ ·s (ϕ) = s(σQ )·m·s ˜ (ϕ) = s(σQ ◦ ϕ)· ¯˜k(ϕ) ˜ (m)· ¯ ¯˜ s(ϕ)·νR = s(ϕ)·s(σR )·m ˜ = s(ϕ ◦ σR )·m

18.29.6

¯ implies that σQ ◦ ϕ = ϕ ◦ σR , we still get and therefore, since the image by π ˆ (cf. 18.23.1) ¯ ˜k(ϕ) ¯ ¯˜ D˜ϕ = ¯ ˜ (m) ˜ −1 ·m 18.29.7. sc

Conversely, it is clear that a correspondence which maps any F -object Q on ˜k(Q) and fulﬁlls equality 18.29.7 for any F sc -morphism ϕ , determines a ¯ m ˜ ∈¯ sc natural isomorphism s ∼ = s (with σQ = idQ for any F -object Q). We are done.

Chapter 19

Solvability for Frobenius P-categories 19.1 As we explain in the Introduction (cf. I 28), one of the positive features of the Frobenius categories over ﬁnite p-groups is the fact that they admit a reasonable deﬁnition of solvability which forces a solvable one to come from a ﬁnite group, showing that our abstract setting lives not so far from the true ﬁnite groups. In this chapter we expose these facts as a consequence of our results in chapters 12, 13 and 18. 19.2 Let P be a ﬁnite p-group and F a Frobenius P -category; consider the following two sequences of F-stable subgroups Pi of P and of normal Frobenius Pi -subcategories Fi of F (cf. 12.6); we set P0 = P and F0 = F , and for any i ∈ N we inductively deﬁne (cf. Corollary 12.17 and 13.8) P2i+1 = P2i

and F2i+1 = (F2i )a

P2i+2 = HF2i+1

and F2i+2 = (F2i+1 )h

19.2.1;

indeed, assuming that Pi is F-stable and Fi is normal in F , it follows either from Corollary 12.17 or from Proposition 13.9 that Pi+1 is F-stable and Fi+1 is normal in F . Then, we say that F is (p-)solvable if, for some n ∈ N , we have Pn = {1} . 19.3 From Corollary 12.17 and Proposition 13.9, it is not diﬃcult to see that this deﬁnition does not change if we consider the sequences of F-stable subgroups Pi of P and normal Frobenius Pi -subcategories Fi of F starting on P and F , and exchanging the indices a and h in equalities 19.2.1; indeed, from those statements, for any i ∈ N we get Fi+1 ⊂ Fi

and Fi+1 ⊂ Fi

19.3.1.

By similar arguments, if F is solvable then, for any F-stable subgroup P of P , any normal Frobenius P -subcategory F of F is also solvable and the P -quotient of F is solvable too; moreover, in this case the groups F(Q) are p-solvable for any subgroup Q of P (cf. Corollary 12.17 and Theorem 13.6). 19.4 In order to give an alternative deﬁnition of solvability for F , recall that a subgroup N of P is normal in F or F-normal whenever it is fully normalized in F and we have NF (N ) = F (cf. 12.6); according to the next proposition, there is a biggest F-normal subgroup of P — denoted by Op (F) . Proposition 19.5 A subgroup N of P is F-normal if and only if it is F(P )-stable and, for any F-essential subgroup Q of P , Q contains N and F(Q) stabilizes N . Moreover, if N is F-normal and F-nilcentralized then F coincides with the Frobenius category FL associated with the F-localizer L of N .

362

Frobenius categories versus Brauer blocks

Proof: If N is fully normalized in F and we have NF (N ) = F then N is clearly F(P )-stable and, for any F-essential subgroup Q of P (cf. 5.8), it follows from the very deﬁnition of NF (N ) (cf. 2.14) that any element of F(Q) is the restriction of an element of F(N ·Q) stabilizing N ; in particular, FN (Q) is normal in F(Q) and therefore it follows from Theorem 5.11 that FN (Q) is contained in FQ (Q) , and more precisely, that NN (Q) ⊂ Q . Consequently, we get N ⊂ Q and F(Q) stabilizes N . Conversely, it suﬃces to prove that, for any subgroup R of P , any F-morphism ψ : R → P can be extended to N ·R , stabilizing N ; we argue by induction on the length D(ψ) of ψ (cf. 5.15). If ψ belongs to F(P ) ◦ ιP R, then the statement is clear; otherwise, we have ψ = ιP ◦ σ ◦ η where Q Q is an F-essential subgroup of P , σ an element of XF (Q) , η : R → Q an F-morphism and we have D(ιP Q ◦ η) = D(ψ) − 1 (cf. 5.15); hence, by the induction hypothesis, there is an F-morphism ηˆ : N ·R → P extending ιP Q ◦η and fulﬁlling ηˆ(N ) = N ⊂ Q , so that ηˆ deﬁnes an F-morphism µ : N ·R → Q extending η and fulﬁlling µ(N ) = N ; then, ψˆ = ιP Q ◦ σ ◦ µ extends ψ and ˆ fulﬁlls ψ(N ) = N . Moreover, if N is F-nilcentralized then we have HCF (N ) = {1} and NF ,N (N ) = NF (N )

19.5.1;

similarly, denoting by L the F-localizer of N (cf. 18.5), we get FL,N = FL ; now, the last statement follows from Theorem 18.6 and Remark 18.7. Corollary 19.6 If P is an F-stable subgroup of P , F a normal Frobenius P -subcategory of F and N an F-normal subgroup of P , then N is normal in F . Moreover, Op (F ) is normal in F . Proof: Since we assume that NF (N ) = F , F(P ) stabilizes N and, for any subgroup Q of P , FN (Q) is normal in F(Q) ; in particular, if Q is an F -essential subgroup of P , the image of NN (Q) in F (Q) is contained in FQ (Q) by condition 5.11.2, and therefore Q contains NN (Q) by condition 5.11.1; thus, Q contains N and then N is F(Q)-stable; at this point it suﬃces to apply Proposition 19.5 to prove the ﬁrst statement. Moreover, set T = Op (F ) ; since F(P ) stabilizes F , it stabilizes T and, in particular, F(P ) still stabilizes T (cf. Proposition 12.8); on the other hand, if R is an F-essential subgroup of P , any σ ∈ F(R) stabilizes R = R ∩ P and, denoting by ϕ : R → P the homomorphism mapping u ∈ R on σ(u) , it follows from Proposition 12.8 that ϕ = τ ◦ ϕ for suitable τ ∈ F(P ) and ϕ ∈ F (P , R ) . Then, since T is normal in F , there exists an F -morphism we ψ : T ·R → P fulﬁlling ψ (T ) = T and inducing ϕ on R and therefore get an F-morphism ψ : T ·R → P mapping u ∈ T ·R on τ ψ (u) ; thus, ψ fulﬁlls ψ(T ) = T and extends ϕ .

19. Solvability for Frobenius P -categories

363

Consequently, there exists a group automorphism η : T ·R ∼ = T ·R mapping v·u on ψ(v)·σ(u) for any v ∈ T and any u ∈ R ; then, since R is F-selfcentralizing, it follows from statement 2.10.1 and from the existence of η that σ can be extended to an F-automorphism of NT (R)·R and, in particular, σ normalizes the image of NT (R)·R in F(R) . That is to say, F(R) normalizes its own p-subgroup FT ·R (R) ; then it follows from Theorem 5.11 that FT ·R (R) = FR (R) and therefore that NT ·R (R) = R , which forces T ⊂ R ; now, the last statement follows from Proposition 19.5. We are done. 0

19.7 We are ready to prove our main result on solvability. Set P = P , F = F and, for any i ∈ N , inductively deﬁne 0

P where F

i

i

i+1

=P

i

i

Op (F ) and F

i+1

i

=F

i

i

Op (F )

19.7.1

i

Op (F ) denotes the Op (F )-quotient of F .

Theorem 19.8 The following statements on the Frobenius P -category F are mutually equivalent: 19.8.1 F is solvable. n 19.8.2 There is n ∈ N such that P = {1} . 19.8.3 There is a chain {Qi }0≤i≤n of F-stable subgroups of P such that Q0 = {1} , Qn = P and, for any 1 ≤ i ≤ n , Qi /Qi−1 is normal in the Qi−1 -quotient of F . 19.8.4 F = FG for a p-solvable ﬁnite group G . 19.8.5 There is an F-normal F-selfcentralizing subgroup Q of P such that the group F(Q) is p-solvable. Moreover, in this case G/Op (G) is isomorphic to the F-localizer of Op (F) . Proof: We may assume that P = {1} . If F is solvable and n ∈ N is the ﬁrst integer fulﬁlling Pn+1 = {1} (cf. 19.2) then n is odd and Fn necessarily coincides with the Frobenius category FPn associated with the group Pn ; but, Pn is clearly F(P )-stable (cf. 19.2); moreover, for any F-essential subgroup Q of P , since Fn = FPn is normal in F (cf. 19.2), we have FPn (Q) B F(Q)

19.8.6

which forces NPn (Q) ⊂ Q (cf. Theorem 5.11), and therefore Pn is contained in Q . That is to say, Pn is F-normal and, in particular, Op (F) is not trivial; then, since the Op (F)-quotient of F is also solvable (cf. 19.3), to get statement 19.8.2 it suﬃces to argue by induction on |P | . It is obvious that statement 19.8.2 implies statement 19.8.3. Statement 19.8.4 implies statement 19.8.5 since it is easily checked from Theorem 3.2 in [28, Ch. 6] that, in statement 19.8.4, Op (F) coincides with a

364

Frobenius categories versus Brauer blocks

Sylow p-subgroup of Op ,p (G) and that it is F-selfcentralizing; moreover, in this case Proposition 19.5 implies the last statement. Similarly, applying Proposition 19.5 again, statement 19.8.5 implies that we have F = FL for an F-localizer L of Q (cf. 18.5) and, since L is then p-solvable, it is very easy to prove statement 19.8.1. From now on, assume that statement 19.8.3 holds and set P = Op (F)·CP Op (F) and U = FOp (F ) Op (F) 19.8.7; it is not diﬃcult to check that P is F-stable (cf. Corollary 12.5) and that the Frobenius P -subcategory F = Op (F)·CF Op (F) = NFU Op (F) 19.8.8 is normal in F ; it is easily checked from Corollary 19.6 that F still fulﬁlls the corresponding statement 19.8.3. Hence, arguing by induction on |P | and on |F(P, Q)| where Q runs over the set of subgroups of P , if either P = P , Q or P = P and F = F , then F is the Frobenius category FG associated with a p-solvable ﬁnite group G , and we may assume that Op (G ) = 1 ; thus, according to Corollary 19.6, Op (F ) is normal in F , so that it is contained in Op (F) , and, conversely, Op (F) is normal in F , so that we get the equality Op (F ) = Op (F) ; moreover, since F = FG , it follows from Theorem 3.2 in [28, Ch. 6] that Op (G ) = Op (F ) and then, by the very deﬁnition of F , that CF Op (F) = CF Op (F) = CF Op (G ) = FZ(Op (G )) 19.8.9. In this case, by the very deﬁnition of F , we simply have G = Op (F) and therefore Op (F) is F-selfcentralizing; hence, according to Proposition 19.5, F is the Frobenius category associated with the F-localizer L of Op (F) ; moreover, it is quite clear that F¯ = F Op (F) still fulﬁlls condition 19.8.3 and therefore, by the induction hypothesis, F¯ is the Frobenius category as¯ ; in particular, F¯ is solvable which sociated with a p-solvable ﬁnite group G clearly implies that F is also solvable, so that F Op (F) is a p-solvable group (cf. 19.3); thus, since L/Op (F) ∼ = F Op (F) the group L is p-solvable too. Finally, assume that P = P and F = F ; in this case, we prove that P = Op (F) and F = FOp (F )

19.8.10.

Clearly, the Q1 -quotient F¯ of F over P¯ = P/Q1 still fulﬁlls the corresponding statement 19.8.3 and therefore, arguing by induction on |P | , F¯ is the ¯ , and we Frobenius category FG¯ associated with a ﬁnite p-solvable group G ¯ may assume that Op (G) = 1 . Then, we claim that the converse image T ¯ in P is normal in F . of Op (G)

19. Solvability for Frobenius P -categories

365

It follows from Corollary 12.5 that T is F(P )-stable. On the other hand, let R be an F-essential subgroup of P fully normalized in F and denote by ¯ its image in P¯ ; since Q1 ⊂ Op (F) ⊂ R , by the very deﬁnition of F we R have F(Q1 ) = F (Q1 ) = FOp (F ) (Q1 ) 19.8.11 and therefore any p -subgroup K of F (R) = F(R) centralizes Q1 ; thus, ¯ , it is trivial (cf. Theorem 7.2 in [28, Ch. 5]). if moreover K centralizes R ¯ R) ¯ is a p-group Hence, the kernel of the canonical homomorphism F(R) → F( and therefore it is contained in FR (R) (cf. Theorem 5.11); in particular, ¯ in P is contained in R and therefore R ¯ is the converse image of CP¯ (R) ¯ F-selfcentralizing. Moreover, since Q1 ⊂ Op (F) ⊂ R

19.8.12,

¯ R) ¯ is surjective (cf. Proposition 12.3). Conthe homomorphism F(R) → F( ¯ is F-essential ¯ sequently, it follows from Theorem 5.11 that R and therefore, ¯ ¯ according to Proposition 19.5, it contains Op (F) = Op (G) which implies that R contains T . At this point, it follows from Proposition 19.5 again that T is F-normal and therefore it is contained in Op (F) ; but, the image of Op (F) in P¯ is clearly normal in F¯ and, since its normalizer in F¯ is the Frobenius category ¯ (cf. Corollary 3.6), it follows from the associated with its normalizer in G equality (cf. Theorem 3.2 in [28, Ch. 6]) ¯ = Z Op (G) ¯ CG¯ Op (G)

19.8.13

¯ ; hence, we get T = Op (F) and therefore we that this image is normal in G have F(T ) = F (T ) = FOp (F ) (T ) 19.8.14, ¯ centralizes Op (G) ¯ ; by equality 19.8.13, this so that any p -subgroup of G ¯ = Op (G) ¯ and therefore F is the Frobenius category associated forces G with Op (F) . We are done. 19.9 This result leads us to consider the following class of subgroups of P . We say that a subgroup Q of P is F-solvcentralized whenever, for some ϕ ∈ F(P, Q) such that ϕ(Q) is fully centralized in F , the Frobenius CP ϕ(Q) -category CF ϕ(Q) is solvable; in this case, it is clear that the Frobenius CP ψ(Q) -category CF ψ(Q) is solvable for any ψ ∈ F(P, Q) such that ψ(Q) is fully centralized in F . Note that 19.9.1 If Q contains a F-solvcentralized subgroup R of P then it is F-solvcentralized too.

366

Frobenius categories versus Brauer blocks

Indeed, we may assume that R B Q and then that both are fully centralized in F (cf. Corollary 2.21); in this case, Q is also fully centralized in the solvable Frobenius Q·CP (R)-category F (R)

Q·CF (R) = NF Q

(R)

19.9.2,

CP (Q) = CQ·CP (R) (Q)

19.9.3

we clearly have and the Frobenius categories CF (Q) and CQ·CF (R) (Q) coincide. Our last result in this chapter gives several characterizations of the F-solvcentralized subgroups of P ; as usual, if R is a subgroup of Q , we denote by F(Q)R the stabilizer of R in F(Q) . Proposition 19.10 Let Q be a subgroup of P fully centralized in F . Then the following statements on Q are mutually equivalent: 19.10.1 Q is F-solvcentralized. 19.10.2 U = Q·Op CF (Q) is F-selfcentralizing and the kernel of the restriction map F(U )Q → F(Q) is p-solvable. 19.10.3 There exists an F-selfcentralizing subgroup R of Q·CP (Q) such that we have Q ⊂ R B Q·CF (Q) and that the kernel of the restriction map F(R)Q → F(Q) is p-solvable. Moreover, if F Q·CP (Q) Q stabilizes R , the restriction map F(R)Q → F(Q) is surjective. Proof: If Q is F-solvcentralized then Op CF (Q) is CF (Q)-selfcentralizing by statement 19.8.5; moreover, according to Corollary 2.21, we may assume that U = Q·Op CF (Q) is also fully centralized in F ; then, since CP (U ) is contained in CP (Q) and centralizes Op CF (Q) , U is F-selfcentralizing. On the other hand, we are assuming that the Frobenius Q·CP (Q)-category H = Q·CF (Q) is solvable and, by the very deﬁnition of CH (Q) (cf. 2.14), the kernel of the restriction map F(U )Q → F(Q) coincides with CH (Q) (U ) . It is clear that statement 19.10.2 implies statement 19.10.3. Assume that statement 19.10.3 holds; since R is also H-selfcentralizing and the kernel of the restriction homomorphism F(R)Q → F(Q) coincides with CH (Q) (R) (cf. 2.14), statement 19.10.1 follows from 19.8.5. Moreover, since we have FQ·CP (Q) (Q) = FQ (Q)

19.10.4,

it follows from statement 2.10.1 that any ϕ ∈ F(P, Q) can be extended toQ·CP (Q) ; in particular, any σ ∈ F(Q) can be lifted to some element τ in F Q·CP (Q) Q and if F Q·CP (Q) Q stabilizes R then τ induces ρ ∈ F(R) lifting σ . We are done.

19. Solvability for Frobenius P -categories

367

Remark 19.11 From the arguments in [35, Ch. V], it is not diﬃcult to prove that if p = 2 and the subgroup L(P ) of P deﬁned there is not normal in F then there is some F-essential subgroup Q of P such that F(Q) is of even order — the proof of this statement actually need not quote Thompson’s Classiﬁcation of the so-called Quadratic pairs. In other words, we have 19.11.1 If p = 2 and F(Q) has odd order for any subgroup Q of P then F is solvable. In general, assuming that p ≥ 5 and quoting Thompson’s Classiﬁcation of the Quadratic pairs (cf. [58]), the arguments developed in [35, Ch. V] remain essentially valid, providing a step towards an eventual classiﬁcation of the simple Frobenius P -categories deﬁned in 12.19.

Chapter 20

A perfect F-locality sc from a perfect F -locality 20.1 Let P be a ﬁnite p-group and F a Frobenius P -category; recall that sc we denote by F the full subcategory of F over the set of F-selfcentralizing subgroups of P (cf. 6.1) and that in chapter 17 we have introduced the Fsc and F -localities (cf. 17.4), introducing in 17.13 the perfect ones. In this sc sc chapter, we prove that any perfect F -locality L can be extended to a unique perfect F-locality L . Our argument depends on a precise consequence of Theorem 17.18 (cf. 20.9); the remainder of our long proof is just routine, yet we have not found a shorter method. As Theorem 4.12, this result illustrates sc the fact that the full subcategory F “determines” what happens in F . sc

sc

20.2 From now on, we assume that there exists a perfect F -locality L and denote by sc

sc

τ sc : TP −→ L

sc

and π sc : L −→ F

sc

20.2.1

the structural functors (cf. 17.3 and 17.4); recall that, for any pair of subQ Q groups Q and R of P such that R ⊂ Q , we set iR = (τ sc )R (1) (cf. 17.3). In order to construct L , we start by deﬁning the L-isomorphisms and the restriction maps between them. Explicitly, let Q and Q be F-isomorphic subgroups of P , R a subgroup of Q and R a subgroup of Q , and let us assume that the set F(Q , Q)R ,R of ϕ ∈ F(Q , Q) fulﬁlling ϕ(R) = R is not empty; since F is divisible, there is a unique restriction map Q ,Q

fR ,R : F(Q , Q)R ,R −→ F(R , R)

20.2.2

Q sending ϕ ∈ F(Q , Q)R ,R to ψ ∈ F(R , R) such that ιQ R ◦ ψ = ϕ ◦ ιR , and Q ,Q

we start by deﬁning the sets L(Q , Q)R ,R and L(R , R) , and by lifting fR ,R to a map Q ,Q

gR ,R : L(Q , Q)R ,R −→ L(R , R)

20.2.3.

Whenever all these subgroups are F-selfcentralizing, the existence of all the sc items in 20.2.3 will be an easy consequence of the existence of L ; in a second step, we will consider the case where all these subgroups are fully centralized in F ; ﬁnally, in the general case, we argue by induction on |P : Q| . 20.3 If Q and Q are F-selfcentralizing, we simply set L(Q , Q) = L (Q , Q) sc

20.3.1

370

Frobenius categories versus Brauer blocks

and denote by L(Q , Q)R ,R the converse image of F(Q , Q)R ,R in L(Q , Q) via the structural homomorphisms (π sc )Q ,Q . Moreover, if R and R are sc

F-selfcentralizing too, it follows from the divisibility of L that there exists a Q ,Q

map gR ,R from L(Q , Q)R ,R to L(R , R) such that, for any x ∈ L(Q , Q)R ,R , we have Q Q ,Q Q iR ·gR ,R (x) = x·iR 20.3.2; actually, it follows from Proposition 4.6 and from the injectivity of (τ sc )R Q ,Q

(cf. 17.13) that gR ,R is injective. In particular, for any x ∈ L(Q , Q)R ,R and any u ∈ TP (Q , Q) ∩ TP (R , R) , we have Q ,Q Q ,Q (π sc )Q ,Q gR ,R (x) = fR ,R (π sc )R ,R (x) 20.3.3 Q ,Q gR ,R (τ sc )Q ,Q (u) = (τ sc )R ,R (u) and we simply set πQ ,Q = (π sc )Q ,Q and τQ ,Q = (τ sc )Q ,Q . 20.4 Note that the family of these maps, for the diﬀerent choices of sc F-selfcentralizing subgroups, agrees with the composition of L -morphisms and fulﬁlls a transitivity condition. Namely, if Q is an F-selfcentralizing subgroup of P which is F-isomorphic to Q and Q , R is an F-selfcentralizing subgroup of Q , and x is an element of L(Q , Q )R ,R , then we clearly have Q ,Q

Q ,Q

Q ,Q

gR ,R (x ·x) = gR ,R (x )·gR ,R (x)

20.4.1.

Moreover, if T and T are F-selfcentralizing subgroups of P which are respectively contained in R and R , and x ∈ L(Q , Q)R ,R ∩ L(Q , Q)T ,T , then it is easily checked that Q ,Q R ,R Q ,Q gT ,T (x) = gT ,T gR ,R (x) 20.4.2. ˆ = Q·CP (Q) 20.5 Now, assume that Q is fully centralized in F and set Q ˆ = L(Q, ˆ Q) ˆ is already which is clearly F-selfcentralizing; in particular, L(Q) sc sc sc deﬁned and the structural functor τ : TP → L determines a group hoˆ → L(Q) ˆ ; then, we need to consider the following momorphism τQˆ : NP (Q) ˆ (cf. 13.2) subgroup of L(Q) O(Q) = Op CL(Q) (Q) ·τQˆ (HCF (Q) ) 20.5.1. ˆ τQ ˆ ˆ = Q ·CP (Q ) , it follows If Q is also fully centralized in F and we set Q from statement 2.10.1 that any ϕ ∈ F(Q , Q) can be extended to a suitable ˆ , Q) ˆ ; thus, the map ϕˆ ∈ F(Q ˆ ˆ

Q ,Q ˆ , Q) ˆ Q ,Q −→ F(Q , Q) fQ ,Q : F(Q

is surjective.

20.5.2

sc

20. A perfect F-locality from a perfect F -locality

371

ˆ , Q) ˆ is already deﬁned and the groups 20.6 Moreover, in this case, L(Q ˆ and L(Q ˆ ) act on this set by composition on the right and on the left L(Q) respectively; moreover, it is quite clear that the respective subgroups O(Q) ˆ , Q) ˆ Q ,Q and that the corresponding quotient sets and O(Q ) stabilize L(Q ˆ , Q) ˆ Q ,Q /O(Q) and O(Q )\L(Q ˆ , Q) ˆ Q ,Q coincide. Thus, we deﬁne L(Q ˆ , Q) ˆ Q ,Q /O(Q) = O(Q )\L(Q ˆ , Q) ˆ Q ,Q L(Q , Q) = L(Q and denote by

20.6.1

ˆ ˆ

Q ,Q ˆ , Q) ˆ Q ,Q −→ L(Q , Q) gQ ,Q : L(Q

20.6.2

the canonical map. 20.7 It is clear that there is a unique map πQ ,Q : L(Q , Q) −→ F(Q , Q)

20.7.1

ˆ , Q) ˆ Q ,Q , fulﬁlls which, for any x ∈ L(Q ˆ ,Q ˆ Qˆ ,Qˆ Q πQ ,Q gQ ,Q (x) = fQ ,Q πQˆ ,Qˆ (x)

20.7.2;

ˆ , Q) ˆ too, and we similarly, if u belongs to TP (Q , Q) then it belongs to TP (Q consider the map τQ ,Q : TP (Q , Q) → L(Q , Q) deﬁned by ˆ ,Q ˆ Q τQ ,Q (u) = gQ ,Q τQˆ ,Qˆ (u)

20.7.3. sc

Furthermore, it is clear that the composition in L deﬁnes a compatible composition between these morphisms; explicitly, if Q is a subgroup of P which is F-isomorphic to Q and Q , and is fully centralized in F , setting ˆ = Q ·CP (Q ) we get a composition map Q L(Q , Q ) × L(Q , Q) −→ L(Q , Q)

20.7.4

ˆ , Q) ˆ Q ,Q and any x ∈ L(Q ˆ , Q ˆ )Q ,Q — denoted by · — that for any x ∈ L(Q fulﬁlls ˆ ,Q ˆ ˆ ,Q ˆ ˆ ,Q ˆ Q Q Q gQ ,Q (x ·x) = gQ ,Q (x )·gQ ,Q (x) 20.7.5 sc

and the associativity of the composition in L forces the obvious associativity here. 20.8 Assume that R and R are also fully centralized in F and normal ˆ and Q ˆ respectively normalize R and R , in Q and Q respectively; thus, Q and we set ˆ P (R) = Q·R ˆ Tˆ = Q·C

ˆ ·CP (R ) = Q ·R ˆ and Tˆ = Q

20.8.1

372

Frobenius categories versus Brauer blocks

which respectively coincide with the converse images of FQ (R) in NP (R) and of FQ (R ) in NP (R ) ; since R is also fully FQ (R)-normalized in F (cf. 2.10) F (R) and we have NP Q (R) = Tˆ , we get the Frobenius Tˆ-category and the associated partial perfect locality (cf. 17.4 and 17.5) R

F (R)

F = NF Q

F (R)

R

Q (R) and L = NLsc

(R)

20.8.2

R deﬁned over the set X of F-selfcentralizing subgroups of P contained in Tˆ ; R R sc sc we identify F and L with their canonical image in F and L respectively. R 20.9 Then, since HF R is an F -stable subgroup of Tˆ (cf. Lemma 13.3), it follows from Theorem 17.18 that we have the quotient partial perfect R R F /HF R -locality L /HF R ; but, it is easily checked from Lemma 13.3 that R

L /HF R can be identiﬁed to the full subcategory of TTˆ/H

F

R

(cf. 17.2) over

R

the image in Tˆ/HF R of the set X ; hence, we have a canonical functor R

tR : L −→ TTˆ/H

F

20.9.1

R

compatible with the structural functors, and, since HF R centralizes R , it R ˆ fulﬁlling tR (x) = 1 , follows from Theorem 17.18 that, for any x ∈ L (Tˆ, Q) πx acts trivially on R . Proposition 20.10 With the notation and the hypothesis above, for any ˆ , Q) ˆ Q ,Q ∩ L(Q ˆ , Q) ˆ R ,R there is w element x ˆ ∈ L(Q ˆ ∈ L(Tˆ , Tˆ)R ,R such that ˆ T

ˆ w ˆ −1 ·iQˆ ·ˆ x ∈ L (Tˆ, Q) R

and

ˆ T

tR (w ˆ −1 ·iQˆ ·ˆ x) = 1

20.10.1.

ˆ ,Q ˆ ˆ ,R ˆ T ˆ ,T ˆ Q R Moreover, the correspondence sending gQ ,Q (ˆ x) to gR ,R gRˆ ,Rˆ (w) ˆ determines a unique map Q ,Q

gR ,R : L(Q , Q)R ,R −→ L(R , R)

20.10.2.

Proof: By the very deﬁnition of L(Q , Q) (cf. 20.6.1), it is clear that ˆ ,Q ˆ Q ˆ , Q) ˆ Q ,Q ∩ L(Q ˆ , Q) ˆ R ,R = L(Q , Q)R ,R gQ ,Q L(Q

20.10.3; ˆ Q

ˆ Q

moreover, denoting by ψ ∈ F(R , R) the element fulﬁlling πxˆ ◦ιR = ιR ◦ψ , we clearly have ψ FQ (R) = FQ (R ) ; then, since R is normal in Tˆ and we have FTˆ (R) = FQ (R) , it follows from statement 2.10.1 that ψ can be extended to an F-morphism ω : Tˆ → P , and, since Tˆ coincides with the converse image

sc

20. A perfect F-locality from a perfect F -locality

373

of FQ (R ) in NP (R ) , we clearly have ω(Tˆ) = Tˆ ; in particular, there is ˆ T R ˆ , and actually we w ˆ ∈ L(Tˆ , Tˆ)R ,R such that w ˆ −1 ·iQˆ ·ˆ x belongs to L (Tˆ, Q) can modify our choice of w ˆ in such a way that ˆ

tR (w ˆ −1 ·iTQˆ ·ˆ x) = 1

20.10.4.

ˆ ,R ˆ T ˆ ,T ˆ ˆ ,Q ˆ R Q Moreover, we claim that gR ,R gRˆ ,Rˆ (w) ˆ only depends on x = gQ ,Q (ˆ x) , ˆ ˆ deﬁning the announced map. Indeed, if x ˆ ∈ L(Q , Q)Q ,Q is another lifting of x then, by deﬁnition 20.6.1 we have x ˆ = x ˆ·z·τQˆ (u) for suitable elements z ∈ Op CL(Q) (Q) ˆ τQ ˆ

and u ∈ HCF (Q) F (R)

R

ˆ =N Q in particular, note that z and τQˆ (u) belong to L (Q) ˆ L(Q) the inclusion R ⊂ Q forces p Op CL(Q) C (Q) ⊂ O (R) τ τ ˆ ˆ ˆ ˆ L(Q) Q Q

20.10.5; τQˆ (R) ; but,

20.10.6;

hence, we have tR (z) = 1 . Similarly, the inclusions CP (Q) ⊂ CP (R) and CF (Q) ⊂ CF (R)

20.10.7

force HCF (R) ⊂ HCF (R) (cf. Proposition 13.9) and therefore we still have ˆ we can make the same choice w ˆ. tR τQˆ (u) = 1 . Consequently, for x On the other hand, if w ˆ ∈ L(Tˆ , Tˆ)R ,R is another choice such that ˆ

ˆ

R T T ˆ and tR (w w ˆ −1 ·iQˆ ·ˆ x ∈ L (Tˆ, Q) ˆ −1 ·iQˆ ·ˆ x) = 1

20.10.8,

R then the diﬀerence w ˆ −1 ·w ˆ belongs to L (Tˆ) and we have tR (w ˆ −1 ·w ˆ ) = 1 ; but, according to Theorem 17.18 and deﬁnition 20.8.2, it is easily checked R that the kernel of the homomorphism L (Tˆ) → Tˆ/HF R determined by the functor tR is equal to R Op L (Tˆ) ·(τ R )Tˆ (HF R ) = Op CL(Tˆ) τTˆ (R) ·τTˆ (HCF (R) ) 20.10.9 R

R

R

where τ : TTˆ → L denotes the structural functor; consequently, since ˆ ,T ˆ T p gR, O C (R) ⊂ Op CL(R) (R) τ ˆ τR ˆ L(Tˆ ) Tˆ ˆ R ˆ ˆ ,T ˆ T

gR, ˆ R ˆ ˆ ,T ˆ T

τTˆ (HCF (R) ) = τRˆ (HCF (R) )

the element gR, (w ˆ −1 ·w ˆ ) belongs to O(R) . We are done. ˆ R ˆ

20.10.10,

374

Frobenius categories versus Brauer blocks

Lemma 20.11 With the notation and the hypothesis above, we have Q ,Q Q ,Q πR ,R gR ,R (x) = fR ,R πQ ,Q (x) Q ,Q gR ,R τQ ,Q (u) = τR ,R (u)

20.11.1.

for any x ∈ L(Q , Q)R ,R and any u ∈ TP (Q , Q) ∩ TP (R , R) . Proof: With the notation and the hypothesis above, let x ˆ be an element ˆ ,Q ˆ Q ˆ , Q) ˆ Q ,Q fulﬁlling x = g of L(Q (ˆ x) (cf. 20.5.2), and w ˆ an element of Q ,Q ˆ

T R ˆ and fulﬁlls the equalL(Tˆ , Tˆ)R ,R such that w ˆ −1 ·iQˆ ·ˆ x belongs to L (Tˆ, Q) ˆ T

ˆ T

ity tR (w ˆ −1 ·iQˆ ·ˆ x) = 1 ; in particular, note that πTˆ,Qˆ (w ˆ −1 ·iQˆ ·ˆ x) acts trivially Q ,Q

on R (cf. deﬁnition 20.8.2). Then, according to the deﬁnition of gR ,R above, we have ˆ ,R ˆ T ˆ ,T ˆ Q ,Q R gR ,R (x) = gR ,R gRˆ ,Rˆ (w) ˆ 20.11.2 ˆ T

and, since πTˆ,Qˆ (w ˆ −1 ·iQˆ ·ˆ x) acts trivially on R , it follows from equalities 20.7.2 and 20.10.3 that we still have Rˆ ,Rˆ Tˆ ,Tˆ Q ,Q πR ,R gR ,R (x) = πR ,R gR ,R gRˆ ,Rˆ (w) ˆ ˆ ,R ˆ ˆ ,T ˆ Tˆ ,Tˆ R T 20.11.3. = fR ,R πRˆ ,Rˆ gRˆ ,Rˆ (w) ˆ = fR ,R πTˆ ,Tˆ (w) ˆ ˆ ,Q ˆ Q Q ,Q = fR ,R πQˆ ,Qˆ (ˆ x) = fR ,R πQ ,Q (x)

Similarly, by equalities 20.3.3 and 20.7.3, since u still belongs to TP (Tˆ , Tˆ) ˆ ,Q ˆ Q

and we have τQ ,Q (u) = gQ ,Q (τQˆ ,Qˆ (u)) , by the proposition above we get ˆ ,R ˆ T ˆ ,T ˆ Q ,Q R gR ,R τQ ,Q (u) = gR ,R gRˆ ,Rˆ (τTˆ ,Tˆ (u)) = τR ,R (u)

20.11.4.

We are done. Q ,Q

20.12 Now, we claim that the family of maps gR ,R obtained in Proposition 20.10, for the diﬀerent choices of the subgroups of P fully centralized in F , is compatible with the composition maps deﬁned in 20.7.4 and fulﬁlls the corresponding transitivity condition. Lemma 20.13 With the notation and hypothesis above, let Q be a subgroup of P which is F-isomorphic to Q and Q , and R a normal subgroup of Q . Assume that both are fully centralized in F . Then, for any (x , x) ∈ L(Q , Q )R ,R × L(Q , Q)R ,R , we have Q ,Q

Q ,Q

Q ,Q

gR ,R (x ·x) = gR ,R (x )·gR ,R (x)

20.13.1.

sc

20. A perfect F-locality from a perfect F -locality

375

ˆ and choose an element x Proof: With the notation above, set Tˆ = Q ·R ˆ in ˆ ,Q ˆ Q ˆ , Q ˆ )Q ,Q fulﬁlling x = g L(Q (ˆ x ) , and an element w ˆ in L(Tˆ , Tˆ )R ,R Q ,Q

such that ˆ

ˆ

T T R ˆ ) and tR (w w ˆ −1 ·iQˆ ·ˆ x ∈ L (Tˆ , Q ˆ −1 ·iQˆ ·ˆ x ) = 1

20.13.2;

then, the product x ˆ = x ˆ ·ˆ x lifts x = x ·x and, setting w ˆ = w ˆ ·w ˆ , we claim that ˆ T

ˆ T

ˆ and tR (w w ˆ −1 ·iQˆ ·ˆ x ∈ L (Tˆ, Q) ˆ −1 ·iQˆ ·ˆ x ) = 1 R

20.13.3. ˆ T

We argue by induction on the length D of πTˆ ,Qˆ (w ˆ −1 ·iQˆ ·ˆ x ) as an R

F -morphism (cf. 5.15 and 20.8.2); if D = 0 then we have ˆ T

ˆ T

w ˆ −1 ·iQˆ ·ˆ x = yˆ·iQˆ R

20.13.4

R

for some yˆ ∈ L (Tˆ ) fulﬁlling t (ˆ y ) = 1 and therefore we easily get ˆ T

ˆ T

w ˆ −1 ·iQˆ ·ˆ x = (w ˆ −1 ·ˆ y ·w)·( ˆ w ˆ −1 ·iQˆ ·ˆ x)

20.13.5;

since w ˆ −1 ·ˆ y ·w ˆ ∈ L (Tˆ) and tR (w ˆ −1 ·ˆ y ·w) ˆ = 1 , in this case we are done. If D ≥ 1 then we have R

ˆ T

ˆ T

w ˆ −1 ·iQˆ ·ˆ x = iUˆ ·ˆ y ·ˆ v

20.13.6

R ˆ of Tˆ , some p -element yˆ of the converse for some F -essential subgroup U R ˆ ) in L (U ˆ ) and some element vˆ ∈ LR (U ˆ , Q ˆ ) , in such image of XF R (U ˆ T

a way that πTˆ ,Qˆ (iUˆ ·ˆ v ) has length D − 1 (cf. 5.15); thus, by the induction hypothesis, we already know that ˆ −1 Tˆ T R ˆ and tR w w ˆ −1 ·(iUˆ ·ˆ v )·ˆ x ∈ L (Tˆ, Q) v )·ˆ x =1 ˆ ·(iUˆ ·ˆ

20.13.7

ˆ = (πwˆ )−1 (U ˆ ) and denoting by u ˆ , U ˆ ) the and therefore, setting U ˆ ∈ L(U ˆ T

ˆ T

element fulﬁlling w ˆ −1 ·iUˆ = iUˆ ·ˆ u−1 , the divisibility in L implies that u ˆ−1 ·ˆ v ·ˆ x −1 R ˆ , Q) ˆ and then we have tR u belongs to L (U v ·ˆ x = 1 ; consequently, we ˆ ·ˆ still get ˆ T

R

ˆ T

ˆ T

w ˆ −1 ·iQˆ ·ˆ x = w ˆ −1 ·(iUˆ ·ˆ y ·ˆ v )·ˆ x = iUˆ ·(ˆ u−1 ·ˆ y ·ˆ u)·(ˆ u−1 ·ˆ v ·ˆ x)

20.13.8

ˆ ) , we have tR (ˆ and, since u ˆ−1 ·ˆ y ·ˆ u is a p -element of L (U u−1 ·ˆ y ·ˆ u) = 1 , which proves the claim. R

376

Frobenius categories versus Brauer blocks Now, according to 20.4, to 20.7 and to Proposition 20.10, we have ˆ ,Q ˆ ˆ ,R ˆ T ˆ ,T ˆ R Q ,Q Q ,Q Q gR ,R (x ) = gR ,R gQ ,Q (ˆ x ) = gR ,R gRˆ ,Rˆ (w ˆ ) ˆ ,R ˆ T ˆ ,T ˆ ˆ ,T ˆ R T = gR ,R gRˆ ,Rˆ (w ˆ )·gRˆ ,Rˆ (w) ˆ 20.13.9. ˆ ,R ˆ T ˆ ,T ˆ Rˆ ,Rˆ Tˆ ,Tˆ R = gR ,R gRˆ ,Rˆ (w ˆ ) ·gR ,R gRˆ ,Rˆ (w) ˆ Q ,Q

Q ,Q

= gR ,R (x )·gR ,R (x) We are done. Lemma 20.14 With the notation and the hypothesis above, let U be a normal subgroup of Q which contains R and is fully centralized in F . Then, for any x ∈ L(Q , Q)R ,R , setting U = πx (U ) we have U ,U Q ,Q Q ,Q gR ,R gU ,U (x) = gR ,R (x) 20.14.1. ˆ = U ·CP (U ) normalizes R and we set Proof: As above, U ˆ =U ˆ ·CP (R) = U ·R ˆ W

20.14.2

ˆ normalizes U , it normalwhich clearly is contained in Tˆ ; moreover, since Q ˆ and we analogously set izes U ˆ = Q·C ˆ P (U ) Vˆ = Q·U

20.14.3;

ˆ = U ·R ˆ and Vˆ = Q ·U ˆ . If x mutatis mutandis, we also consider W ˆ is an ˆ ˆ ˆ ˆ element of L(Q , Q)Q ,Q lifting x then there is vˆ ∈ L(V , V )U ,U such that ˆ V

ˆ V

ˆ and tU (ˆ vˆ−1 ·iQˆ ·ˆ x ∈ L (Vˆ , Q) v −1 ·iQˆ ·ˆ x) = 1 U

and moreover we have (cf. Proposition 20.10) ˆ ,U ˆ V ˆ ,V ˆ Q ,Q U gU ,U (x) = gU ,U gUˆ ,Uˆ (ˆ v)

20.14.4

20.14.5. ˆ

V Similarly, since we have Tˆ = Vˆ ·CP (R) and vˆ−1 ·iQˆ ·ˆ x centralizes R ⊂ U ˆ ˆ (cf. deﬁnition 20.8.2), so that vˆ also belongs to L(V , V )R ,R , there exists an element tˆ ∈ L(Tˆ , Tˆ)R ,R such that ˆ T

ˆ T

tˆ−1 ·iVˆ ·ˆ v ∈ L (Tˆ, Vˆ ) and tR (tˆ−1 ·iVˆ ·ˆ v) = 1 R

20.14.6.

On the other hand, with the notation in 20.8 above applied to R and to U in Q , it is clear that the inclusion F (U ) F (R) Vˆ = NP Q (U ) ⊂ Tˆ = NP Q (R)

20.14.7

sc

20. A perfect F-locality from a perfect F -locality U

377

R

is (F , F )-functorial; thus, we have an obvious faithful functor and an inclusion (cf. Proposition 13.9) U

R

L −→ L

and HF U ⊂ HF R

20.14.8.

Moreover, we claim that we have the commutative diagram tR

R

L

−−−−→

TTˆ/H

↑

↑ tU

U

−−−−→

L

F

R

20.14.9

TVˆ /H

F

U

U Indeed, it is clear that Vˆ ∩ HF R is an F -stable subgroup of Vˆ (cf. 13.4) and then, it follows from the uniqueness of the functor in Theorem 17.18 that the composition of tU with the canonical functor

TVˆ /H

F

−→ TVˆ /Vˆ ∩H

U

F

U

coincides with the functor L → TVˆ /Vˆ ∩H

F

U

R

20.14.10

R

induced by the composition of

R

the functors L → L and tR . In particular, ˆ

ˆ

R V V ˆ and tR (ˆ x ∈ L (Vˆ , Q) v −1 ·iQˆ ·ˆ x) = 1 vˆ−1 ·iQˆ ·ˆ ˆ

ˆ

20.14.11; ˆ

T V T consequently, the composition (tˆ−1 ·iVˆ ·ˆ v )·(ˆ v −1 ·iQˆ ·ˆ x) = tˆ−1 ·iQˆ ·ˆ x belongs ˆ

R ˆ and fulﬁlls tR (tˆ−1 ·iT ·ˆ to L (Tˆ, Q) x) = 1 , and therefore we get ˆ Q ˆ ,R ˆ T ˆ ,T ˆ Q ,Q R gR ,R (x) = gR ,R gRˆ ,Rˆ (tˆ)

20.14.12.

ˆ ,V ˆ V

Q ,Q

Finally, by equality 20.14.5, u ˆ = gUˆ ,Uˆ (ˆ v ) is a lifting of gU ,U (x) in ˆ

W ˆ , U ˆ )U ,U ; thus, if we choose w ˆ , W ˆ )R ,R such that w L(U ˆ ∈ L(W ˆ −1 ·iUˆ ·ˆ u ˆ W

ˆ ,U ˆ ) and fulﬁlls tR (w belongs to L (W ˆ −1 ·iUˆ ·ˆ u) = 1 , we have R

ˆ ,R ˆ W ˆ ,W ˆ R U ,U Q ,Q gR ,R gU ,U (x) = gR ,R gRˆ ,Rˆ (w) ˆ

20.14.13.

Moreover, we still have (cf. 20.3.2) ˆ T

ˆ V

ˆ T

ˆ T

ˆ V

(tˆ−1 ·iVˆ ·ˆ v )·iUˆ = (tˆ−1 ·iVˆ )·(ˆ v ·iUˆ ) = tˆ−1 ·iUˆ ·ˆ u ˆ T

ˆ W

= (tˆ−1 ·iWˆ ·w)·( ˆ w ˆ −1 ·iUˆ ·ˆ u)

20.14.14

378

Frobenius categories versus Brauer blocks ˆ

R T ˆ ) and we already know and therefore, since tˆ−1 ·iWˆ ·w ˆ belongs to L (Tˆ, W that ˆ ˆ W T tR (w ˆ −1 ·iUˆ ·ˆ u) = 1 = tR (tˆ−1 ·iVˆ ·ˆ v) 20.14.15, ˆ T

we get tR (tˆ−1 ·iWˆ ·w) ˆ = 1. Now, we claim that the element ˆ ,T ˆ T

ˆ ,W ˆ W

y = gRˆ ,Rˆ (tˆ)−1 ·gRˆ ,Rˆ (w) ˆ

20.14.16

belongs to O(R) ; indeed, since we have (cf. 20.3.2) ˆ T

ˆ W

ˆ T

ˆ ,W ˆ W

ˆ T

(tˆ−1 ·iWˆ ·w)·i ˆ Rˆ = tˆ−1 ·iRˆ ·gRˆ ,Rˆ (w) ˆ = iRˆ ·y R

20.14.17,

F (R)

ˆ = N Q (R) and we still have tR (y) = 1 ; but, it is easily y belongs to L (R) ˆ L(R) checked that R ˆ Op O(R) = Op L (R)

and O(R) ∩ τRˆ (Tˆ) = τRˆ (HF R )

20.14.18

which proves the claim. Consequently, we obtain ˆ ,R ˆ T ˆ ,T ˆ ˆ ,R ˆ W ˆ ,W ˆ Q ,Q U ,U Q ,Q R R gR ,R (x) = gR ,R gRˆ ,Rˆ (tˆ) = gR ,R gRˆ ,Rˆ (w) ˆ = gR ,R gU ,U (x) 20.14.19.

We are done. 20.15 We are ready to deﬁne the set L(Q , Q) for any pair of F-isomorphic subgroups Q and Q of P ; we proceed by induction on |P : Q| and, obviously, our deﬁnition will extend the deﬁnitions in the previous situations; moreover, since L(P ) is already deﬁned, we may assume that Q = P . Then, we have N = NP (Q) = Q and it follows from Corollary 2.21 that there is an F-morphism ν : N → P such that ν(N ) and ν(Q) are both fully centralized in F ; in this situation, according to our induction hypothesis, we may assume that the set L ν(N ), N and the canonical map πν(N ),N : L ν(N ), N −→ F ν(N ), N

20.15.1

are already deﬁned and may choose n ∈ L ν(N ), N lifting ν . That is to say, we may assume that 20.15.2 There is a pair (N, n) formed by a subgroup N of P which strictly contains and normalizes Q , and by n ∈ L ν(N ), N lifting ν for a suitable F-morphism ν : N → P such that ν(N ) and ν(Q) are both fully centralized in F .

sc

20. A perfect F-locality from a perfect F -locality

379

20.16 We denote by N(Q) the set of such pairs and often we write n instead of (N, n) , setting n N = ν(N ) and πn = πν(N ),N (n) . For another pair ¯, n ¯ , n N ) , it fulﬁlls (N ¯ ) ∈ N(Q) , note that the element n ¯ ·n−1 belongs to L(n¯ N πn¯ ·n−1 (n Q) = n¯ Q and, since n Q and n¯ Q are fully centralized in F , we get the element n ¯ n , N n·n−1 ) ∈ L(n¯ Q, n Q) 20.16.1. gn¯ ,n = gn N Q,n Q (¯ ¯,n Then, for any pair (N ¯ ) ∈ N(Q) , it follows from Lemma 20.13 that gn¯ ,¯n ·gn¯ ,n = gn¯ ,n

20.16.2.

On the other hand, if Q is fully centralized in F¯ then N = NP (Q) is F-selfcentralizing, so that it is fully centralized too, and therefore the pair N (N, iN ) belongs to N(Q) . 20.17 Now, for any pair of mutually F-isomorphic subgroups Q and Q of P , we denote by L(Q , Q) the subset of ˆ , Q) = L(n Q , n Q) 20.17.1 L(Q n∈N(Q) n ∈N(Q )

formed by the families {xn ,n }n∈N(Q),n ∈N(Q ) fulﬁlling gn¯ ,n ·xn ,n = xn¯ ,¯n ·gn¯ ,n

20.17.2.

In other words, the set L(Q , Q) is the inverse limit of the family formed by the sets L n Q , n Q and by the bijections between them induced by gn¯ ,n and gn¯ ,n . 20.18 Note that, according to equalities 20.16.2, the projection map onto the factor labelled by the pair (N, n), (N , n ) induces a bijection n n Q, Q

20.18.1;

πQ ,Q : L(Q , Q) −→ F(Q , Q)

20.18.2

nn ,n : L(Q , Q) ∼ =L moreover, we have a map

sending x ∈ L(Q , Q) to (cf. 20.5) nN ,N

πQ ,Q (x) = fn

Q ,Q

n N,N (πn )−1 ◦ πnQ ,n Q nn ,n (x) ◦ fn Q,Q (πn )

20.18.3;

then, from equality 20.17.2, it is not diﬃcult to prove that this map does not depend on the choice of (N, n) and (N , n ) . In particular, if Q and Q are already fully centralized in F , setting N = NP (Q) and N = NP (Q ) , L(Q , Q) N N can be identiﬁed with the factor labelled by the pair (N , iN ), (N, iN ) throughout niN ,iN . N

N

380

Frobenius categories versus Brauer blocks

20.19 On the other hand, if Q is a subgroup of P , F-isomorphic to Q and Q , the composition map in 20.7.4 determines a new composition map L(Q , Q ) × L(Q , Q) −→ L(Q , Q)

20.19.1

sending (x , x) ∈ L(Q , Q ) × L(Q , Q) to x ·x = (nn ,n )−1 nn ,n (x )·nn ,n (x)

20.19.2

for a choice of (N, n) ∈ N(Q) , (N , n ) ∈ N(Q ) and (N , n ) ∈ N(Q ) ; ¯, n ¯ , n ¯ ) ∈ N(Q ) and indeed, for another choice of pairs (N ¯ ) ∈ N(Q) , (N ¯ , n (N ¯ ) ∈ N(Q ) , we get (cf. 20.16.2) gn ,¯n · nn¯ ,¯n (x )·nn¯ ,¯n (x) = nn ,n (x )·gn ,¯n ·nn¯ ,¯n (x) 20.19.3. = nn ,n (x )·nn ,n (x)·gn,¯n = nn ,n (x ·x)·gn,¯n If Q is a subgroup of P , F-isomorphic to Q , Q and Q , and x an element of L(Q , Q ) , it is quite clear that (x ·x )·x = x ·(x ·x)

20.19.4.

20.20 From now on, for any pair of subgroups R of Q and R of Q such that F(Q , Q)R ,R is not empty we will construct the restriction map Q ,Q

gR ,R : L(Q , Q)R ,R −→ L(R , R)

20.20.1

announced in 20.2. Firstly, we assume that R is normal in Q and may assume that R = Q ; then, it follows from Corollary 2.21 that there exists an F-morphism ν : Q → P such that the subgroups ν(Q) and ν(R) are fully centralized in F and we may choose n ∈ L ν(Q), Q lifting ν (cf. 20.18.2); that is to say, we have the pair (Q, n) in N(R) (cf. 20.16) and, since nR is fully centralized in F , we can consider the element nR

nR = (ninQ ,n )−1 (inR ) ∈ L(nR, R)

20.20.2.

nQ

Note that, up to our identiﬁcation, for another pair (Q, n ¯ ) ∈ N(R) we have (cf. Proposition 20.10 and 20.16.1) n Q,n ¯Q

n−1 ) = nR ·(¯ nR )−1 gn,¯n = gnR,nR (n·¯

20.20.3;

indeed, according to equality 20.17.2 and deﬁnition 20.19.2, we have ninQ ,inQ nR ·(¯ nR )−1 = ninQ ,n (nR )·nn,in¯Q (¯ nR )−1 nQ

nQ

= nn,in¯Q

n ¯Q

nQ

n ¯Q

(¯ nR )−1 = gn,¯n ·nn¯ ,in¯Q (¯ nR )−1 = gn,¯n n ¯Q

20.20.4.

sc

20. A perfect F-locality from a perfect F -locality

381

Q ,Q

We are ready to deﬁne the map gR ,R in the normal case. Lemma 20.21 With the notation and the hypothesis above, the restriction Q ,Q

map fR ,R admits a lifting Q ,Q

gR ,R : L(Q , Q)R ,R −→ L(R , R)

20.21.1

sending x ∈ L(Q , Q)R ,R to the element nQ ,n Q

Q ,Q

gR ,R (x) = (nR )−1 ·gn

R ,n R

n ·x·n−1 ·nR

20.21.2

for any choice of pairs (Q , n ) in N(R ) and (Q, n) in N(R) . Moreover, for any subgroup Q of P F-isomorphic to Q and Q , any normal subgroup R of Q such that F(Q , Q )R ,R is not empty and any x ∈ L(Q , Q )R ,R we have Q ,Q Q ,Q Q ,Q gR ,R (x ·x) = gR ,R (x )·gR ,R (x) 20.21.3. Proof: We ﬁrstly prove that our deﬁnition does not depend on the choices ¯ ) ∈ N(R) , we have (cf. Lemma 20.13 of (Q, n) and (Q , n ) ; indeed, if (Q, n and equality 20.20.3) nQ ,n ¯Q

(nR )−1 ·gn

n−1 ·¯ n ·x·¯ nR

R ,nR

¯Q nQ ,n

= (nR )−1 ·gn

R ,nR

nQ ,u Q

= (nR )−1 ·gn

R ,nR

nQ ,n Q

= (nR )−1 ·gn

R ,nR

n−1 ) ·¯ nR (u ·x·n−1 )·(n·¯

20.21.4. (n ·x·n−1 )·gn·¯n ·¯ uR (n ·x·n−1 )·nR

Similarly, if (Q , n ¯ ) ∈ N(R ) , we get the “symmetric” equalities. Now, we prove equality 20.21.3; according to 20.21.2 and Lemma 20.13, choosing (Q, n) ∈ N(R) , (Q , n ) ∈ N(R ) and (Q , n ) ∈ N(R ) , we get Q ,Q

Q ,Q

gR ,R (x )·gR ,R (x) nQ ,nQ

= (nR )−1 ·gn

R ,n R

nQ ,nQ

= (nR )−1 ·gn

R ,n R

We are done.

nQ ,n Q

(n ·x ·n−1 )·gn

R ,n R

n ·x·n−1 )·nR Q ,Q

(n ·x ·x·n−1 )·nR = gR ,R (x ·x) .

20.21.5.

382

Frobenius categories versus Brauer blocks

20.22 Moreover, for any mutually F-isomorphic subgroups Q and Q of P , the elements considered in 20.20.2 allow us to deﬁne a map τQ ,Q : TP (Q , Q) −→ L(Q , Q)

20.22.1

extending the corresponding deﬁnition in 20.7 when Q and Q are fully centralized in F . For any v ∈ TP (Q , Q) and any pair (N, n) in N(Q) , setting ˆ = nN and N ˆ = vN ˆ , it is obvious that v still belongs to the N = vN , N intersection ˆ , N ˆ) TP (N , N ) ∩ TP (N 20.22.2 and, arguing by induction on |P : Q| , we may assume that the maps τN ,N and τNˆ ,Nˆ are already deﬁned; then, setting n = τNˆ ,Nˆ (v)·n·τN ,N (v)−1 it is quite

clear that (N , n ) belongs to N(Q ) and that v still belongs to TP (n Q , n Q) ; thus, since τnQ ,n Q (v) is already deﬁned (cf. deﬁnition 20.7.3), we set τQ ,Q (v) = (nQ )−1 ·τnQ ,n Q (v)·nQ

20.22.3

and it is easily checked from Lemma 20.11 and equality 20.20.3 that τQ ,Q (v) does not depend on our choice of (N, n) . Note that, always arguing by induction on |P : Q| , these maps are clearly compatible with the composition Q ,Q

maps. We are ready to deﬁne gR ,R in the general case. Proposition 20.23 For any pair of mutually F-isomorphic subgroups Q and Q of P , any subgroup R of Q and any subgroup R of Q such that F(Q , Q)R ,R = ∅ , there exists a unique map Q ,Q

gR ,R : L(Q , Q)R ,R −→ L(R , R)

20.23.1

Q ,Q

lifting fR ,R and fulﬁlling the two conditions 20.23.2 For any subgroup T of R , any subgroup T of R such that the set F(R , R)T ,T is not empty, and any x ∈ L(Q , Q)R ,R ∩L(Q , Q)T ,T , we have R ,R Q ,Q Q ,Q gT ,T gR ,R (x) = gT ,T (x) . 20.23.3 If R and R are respectively normal in Q and Q then, for any x ∈ L(Q , Q)R ,R and any pairs (Q, n) ∈ N(R) and (Q , n ) ∈ N(R ) , we have nQ ,n Q Q ,Q gR ,R (x) = (nR )−1 ·gn n n ·x·n−1 ·nR . R , R

Moreover, the following two statements hold:

sc

20. A perfect F-locality from a perfect F -locality

383

20.23.4 For any subgroup Q of P which is F-isomorphic to Q and Q , any subgroup R of Q such that F(Q , Q )R ,R = ∅ , any x ∈ L(Q , Q )R ,R and any x ∈ L(Q , Q)R ,R , we have Q ,Q

Q ,Q

Q ,Q

gR ,R (x ·x) = gR ,R (x )·gR ,R (x) . 20.23.5 have

For any x ∈ L(Q , Q)R ,R and any v ∈ TP (Q , Q) ∩ TP (R , R) , we Q ,Q Q ,Q πR ,R gR ,R (x) = fR ,R πQ ,Q (x) Q ,Q gR ,R τQ ,Q (v) = τR ,R (v) .

Proof: We ﬁrstly consider all the pairs Q and R where R is a normal proper subgroup of Q ; then R is normal in Q and we have all the liftings deﬁned above, which fulﬁll conditions 20.23.4 and 20.23.3; we claim that they also fulﬁll conditions 20.23.2 and 20.23.5. Indeed, in this case, it follows from Lemma 20.11 that nQ ,nQ Q ,Q πR ,R gR ,R (x) = πR ,R (nR )−1 ·gn n (n ·x·n)·nR R , R

nQ ,nQ

= πR ,nR (nR )−1 ◦ fn

R ,nR

πnQ ,nQ (n ·x·n) ◦ πnR,R (nR )

20.23.6

Q ,Q = (δn )−1 ◦ fR ,R πQ ,Q (x) ◦ δn where we set nQ,Q −1 δn = fnR,R πnQ,Q (n) ·πnR,R (nR ) nQ ,Q

δn = fn

R ,R

−1 ·πnR ,R (nR ) πnQ ,Q (n )

20.23.7;

but, according to equality 20.18.3, we get nQ,nQ nQ,Q πnR,R (nR ) = fnR,nR πnQ,nQ (1) ◦ πnR,nR (1) ◦ fnR,R πnQ,Q (n) nQ,Q = fnR,R πnQ,Q (n)

20.23.8,

so that δn = idR and similarly δn = idR . Let T and T be respective proper subgroups of R and R , normal in Q and Q , and assume that L(Q , Q)R ,R ∩ L(Q , Q)T ,T is not empty; then, for any x ∈ L(Q , Q)R ,R ∩ L(Q , Q)T ,T , we claim that R ,R Q ,Q Q ,Q gT ,T gR ,R (x) = gT ,T (x)

20.23.9;

384

Frobenius categories versus Brauer blocks

indeed, it follows from Corollary 2.21 that we can choose pairs (Q, n) ∈ N(R) and (Q , n ) ∈ N(R ) which still belong to N(T ) and N(T ) respectively; hence, we have (cf. 20.20.2 and 20.23.3) nQ ,n Q

Q ,Q

gR ,R (x) = (nR )−1 ·gn

R ,n R

nQ ,n Q

Q ,Q

gT ,T (x) = (nT )−1 ·gn

T ,n T

(n ·x·n−1 )·nR 20.23.10; (n ·x·n−1 )·nT

but, according to equality 20.23.8, we have nR R = n R and nR T = n T , and therefore the pair (R, nR ) belongs to N(T ) and it is easily checked that (nR )T = nT (cf. 20.20.2); similarly, (R , nR ) belongs to N(R ) and we have (nR )T = nT (cf. 20.20.2); consequently, we still have nR ,n R R ,R Q ,Q Q ,Q gT ,T gR ,R (x) = (nT )−1 ·gn n nR ·gR ,R (x)·(n−1 ) ·nT R T , T

20.23.11

and then it follows from equalities 20.23.10 and Lemma 20.14 that nR ,n R nQ ,n Q R ,R Q ,Q gT ,T gR ,R (x) = (nT )−1 ·gn n gn n (n ·x·n−1 ) ·nT T , T

nQ ,n Q

= (nT )−1 ·gn

T ,n T

R , R

(n ·x·n−1 )·nT

20.23.12.

Q ,Q

= gT ,T (x) Now, we can complete the proof of condition 20.23.5 in the normal case; according to Lemma 20.11, to 20.20 and to statement 20.23.3, we have Q ,Q πR ,R gR ,R (x) nQ ,n Q = πnR ,R (nR )−1 ◦ πnR ,nR gn n (n ·x·n−1 ) ◦ πnR,R (nR ) R , R

nQ ,n Q

= πnR ,R (nR )−1 ◦ fn

R ,nR

πnQ ,nQ (n ·x·n−1 ) ◦ πnR,R (nR )

20.23.13;

Q ,Q = fR ,R πnQ ,Q (n )−1 ◦ πnQ ,nQ (n ·x·n−1 ) ◦ πnQ,Q (n) Q ,Q = fR ,R πQ ,Q (x) similarly, according to Lemma 20.11, to deﬁnition 20.22.3 and to statement 20.23.3, we still have nQ ,n Q Q ,Q gR ,R τQ ,Q (v) = (nR )−1 ·gn n τnQ ,n Q (v) ·nR R , R

= (nR )−1 ·τnR ,n R (v)·nR = τR ,R (v)

20.23.14.

sc

20. A perfect F-locality from a perfect F -locality

385

In the not normal case, we argue by induction on |Q : R| ; it is clear that, setting N = NQ (R) and N = NQ (R ) , we may assume that N = Q and Q ,Q

N ,N

N = Q , and therefore that the maps gN ,N and gR ,R are already deﬁned; then, for any element x ∈ L(Q , Q)R ,R , we deﬁne Q ,Q N ,N Q ,Q gR ,R (x) = gR ,R gN ,N (x)

20.23.15;

Q ,Q

N ,N

note that, in the case that R is normal in Q , gR ,R = gR ,R is already deﬁned Q ,Q

and gN ,N = idL(Q ,Q) , so that equality 20.23.15 still holds. In order to prove statement 20.23.3, let T and T be respectively subgroups of R and R , assume that F(R , R)T ,T = ∅ and set M = NQ (T ) and M = NQ (T ) . For any element x ∈ L(Q , Q)R ,R ∩ L(Q , Q)T ,T , we argue by induction on |Q : R||Q : T | . First of all, if M = Q , we get R ,R Q ,Q R ,R Q ,Q M ∩R ,M ∩R gT ,T gR ,R (x) = gT ,T gM ∩R ,M ∩R gR ,R (x)

Q ,Q gM ∩R ,M ∩R (x) M ,M Q ,Q M ∩R ,M ∩R = gT ,T gM ∩R ,M ∩R gM ,M (x) M ∩R ,M ∩R

= gT ,T

M ,M

= gT ,T

20.23.16.

Q ,Q Q ,Q gM ,M (x) = gT ,T (x)

Secondly, if M = Q we may assume that N = Q and we still get (cf. 20.23.15) N ,N Q ,Q R ,R Q ,Q R ,R gT ,T gR ,R (x) = gT ,T gR ,R gN ,N (x) N ,N Q ,Q Q ,Q = gT ,T gN ,N (x) = gT ,T (x)

20.23.17.

Finally, statements 20.23.4 and 20.23.5 follow from statement 20.23.2, from our induction hypothesis and from the corresponding statement in the normal case (cf. Lemmas 20.11 and 20.13). We are done. sc

sc

Theorem 20.24 The partial perfect F -locality L unique perfect F-locality L .

can be extended to a

Proof: For any pair of subgroups Q and R of P , we deﬁne the set of morphisms in L from R to Q by the disjoint union L(Q, R) =

L(R , R)

20.24.1

R

where R runs over the set of subgroups of Q which are F-isomorphic to R . If x is an L-morphism from R to Q , T a subgroup of P and y an L-morphism

386

Frobenius categories versus Brauer blocks

from T to R , then, respectively denoting by R and T the subgroups of Q and R labelling the corresponding terms and by x∗ and y∗ the corresponding elements in L(R , R) and L(T , T ) , and setting T = πR ,R (x) (T ) , the composition x·y of x and y in L is deﬁned by R ,R

(x·y)∗ = gT ,T (x∗ )·y∗

20.24.2.

This makes sense since all the sets L(R , R) are already deﬁned, together with the maps πR ,R : L(R , R) → F(R , R) , and therefore x∗ obviously belongs to L(R , R)T ,T , so that, by Proposition 20.23, we get the R ,R

element gT ,T (x∗ ) in L(T , T ) ; moreover, in 20.19 we already have deﬁned R ,R

R ,R

the composition of gT ,T (x∗ ) and y∗ . Since gR ,R = idL(R ,R) (cf. Proposition 20.10 and statement 20.23.3), the new composition extends the composition maps 20.19.1. In order to prove the associativity of the new composition, consider a subgroup U of P and an L-morphism z from U to T ; that is to say, denoting by U the subgroup of T labelling the corresponding term and by z∗ the corresponding element in L(U , U ) , and analogously setting U = πy (U ) and U = πx (U ) , we have

(x·y)·z

∗

T ,T T ,T R ,R = gU ,U (x·y)∗ ·z∗ = gU ,U gT ,T (x∗ )·y∗ ·z∗ R ,R

T ,T

R ,R

= gU ,U (x∗ )·gU ,U (y∗ )·z∗ = gU ,U (x∗ )·(y·z)∗ = x·(y·z) ∗

20.24.3.

We have an evident functor π : L → F which is the identity over the objects and, for any pair of subgroups Q and R of P , deﬁnes the map πQ,R : L(Q, R) −→ F(Q, R)

20.24.4

sending x ∈ L(Q, R) to ιQ R ◦πR ,R (x∗ ) , where R is the subgroup of Q labelling the corresponding term (cf. 20.18.2) and x∗ is the corresponding element of L(R , R) ; indeed, with the notation above, we get (cf. Lemma 20.11)

R ,R πQ,T (x·y) = ιQ T ◦ πT ,T gT ,T (x∗ )·y∗ R ,R = ιQ T ◦ fT ,T πR ,R (x∗ ) ◦ πT ,T (y∗ ) R = ιQ R ◦ πR ,R (x∗ ) ◦ ιT ◦ πT ,T (y∗ ) = πQ,R (x) ◦ πR,T (y)

20.24.5.

sc

20. A perfect F-locality from a perfect F -locality

387

We also have a canonical functor τ : TP → L which is the identity over the objects and deﬁnes the map τQ,R : TP (Q, R) → L(Q, R)

20.24.6

sending v ∈ TP (Q, R) to the element x in L(Q, R) fulﬁlling x∗ = τvR,R (v) ; indeed, with the notation above, if w ∈ TP (R, T ) then, setting R = vR , T = w T and T = v T , we have (cf. 20.23.3)

R ,R τQ,R (v)·τR,T (w) ∗ = gT ,T τR ,R (v) ·τT ,T (w) = τT ,T (v)·τT ,T (w) = τT ,T (vw) = τQ,T (vw) ∗

20.24.7.

Now, the category L endowed with the functors τ : TP → L and π : L → F is clearly an F-locality (cf. 17.3). We claim that L is divisible; indeed, if Q , R and T are subgroups of P , x is an element of L(Q, R) and y ∈ L(Q, T ) fulﬁlls T = πy (T ) ⊂ πx (R) = R , then x∗ belongs to L(R , R) , y∗ belongs R ,R

to L(T , T ) and, setting T = (πx )−1 (T ) , the element gT ,T (x∗ ) belongs R ,R

to L(T , T ) ; moreover, the composition gT ,T (x∗ )−1 ·y∗ belongs to L(T , T ) and determines an element z ∈ L(R, T ) such that x·z = y ; thus, L is indeed divisible (cf. 17.7). sc

Moreover, since L is coherent and any F-essential subgroup of P is F-selfcentralizing, it is clear that L still fulﬁlls the conditions in Proposition 17.10 and therefore it is coherent too. Finally, by deﬁnition 20.6.1, for any subgroup Q of P fully centralized in F , we have Ker(τQ ) = HCF (Q)

20.24.8,

so that L is perfect. The uniqueness of L follows easily from the uniqueness of our construction and from Lemma 20.21.

Chapter 21

Frobenius P-categories: the second deﬁnition 21.1 Let P be a ﬁnite p-group; in this chapter we give an equivalent deﬁnition of a Frobenius P -category in terms of a P × P -set; as usual, we call P × P -set a set Ω endowed with a P × P -action, but we write u·ω·v instead of (u, v)·ω for any (u, v) ∈ P × P and any ω ∈ Ω . Although this deﬁnition is surprisingly simple — at least from our point of view — it does not reﬂect our motivation for considering such an object; however, this new deﬁnition is very suitable for the construction of localities (cf. 17.3) associated with a Frobenius P -category, as we show in chapters 22 and 23 below. 21.2 Let Ω be a ﬁnite nonempty P × P -set such that {1} × P acts freely on Ω and assume that Ω◦ ∼ =Ω

and |Ω|/|P | ≡ 0 (mod p)

21.2.1

where Ω◦ denotes the symmetric P ×P -set, namely the P ×P -set obtained exchanging both factors; denote by G the automorphism group of Res{1}×P (Ω) or, equivalently, the centralizer of {1} × P in the symmetric group SΩ of Ω (cf. 1.32), and identify P with the image of P × {1} in G . It is clear that, for any pair of subgroups Q and R of P , the image of TG (Q, R) (cf. 17.2) Ω in Hom(R, Q) — denoted by F (Q, R) — is determined by the set of group homomorphisms ϕ : R → P such that ϕ(R) ⊂ Q and Res ϕ×idP (Ω) ∼ = Res ιPR ×idP (Ω)

21.2.2

and moreover that this correspondence deﬁnes a divisible P -category (cf. 2.3) — denoted by F Ω . 21.3 On the other hand, according to our hypothesis, the stabilizer in P × P of any element of Ω intersects P × {1} and {1} × P trivially and therefore it has the form $ % ∆ϕ (Q) = u, ϕ(u) u∈Q 21.3.1 for some subgroup Q of P and a suitable injective group homomorphism ϕ : Q → P ; with this in mind, for any subgroup Q of P , let us denote by HomΩ (Q, P ) the set of injective group homomorphisms ϕ : Q → P such that the set of ﬁxed points Ω∆ϕ (Q) of ∆ϕ (Q) in Ω is nonempty; note that Ω condition 21.2.1 forces HomΩ (P, P ) = ∅ . Then, for any ψ ∈ F (Q, R) , it is quite clear that HomΩ (Q, P ) ◦ ψ ⊂ HomΩ (R, P ) 21.3.2

390

Frobenius categories versus Brauer blocks

and, since HomΩ (P, P ) = ∅ , we get Ω

|F (P, Q)| ≤ |HomΩ (Q, P )|

21.3.3;

more precisely, for any σ ∈ HomΩ (P, P ) , we still get F (P, Q) ⊂ σ −1 ◦ HomΩ (Q, P ) = HomResidP ×σ (Ω) (Q, P ) Ω

21.3.4.

21.4 We are ready to make our deﬁnition. We say that a P × P -set Ω is basic if {1} × P acts freely on Ω , Ω fulﬁlls condition 21.2.1 and, for any subgroup Q of P , we have Ω

F (P, Q) = HomΩ (Q, P )

21.4.1;

moreover, we say that a basic P ×P -set is thick if the multiplicity of the indeΩ composable P × P -set (P × P )/∆ϕ (Q) is at least two for any ϕ ∈ F (P, Q) . 21.5 Note that, if F is a divisible P -category and Ω is a P × P -set fulﬁlling all the conditions in 21.2 in such a way that, for any subgroup Q of P , we have Ω

HomΩ (Q, P ) ⊂ F(P, Q) and F(P, Q) ⊂ F (P, Q)

21.5.1, Ω

then it follows from inequality 21.3.3 that Ω is basic and that F = F ; in this case we say that Ω is an F-basic P × P -set. We always can embed the disjoint union of two F-basic P × P -sets Ω and Ω into a third F-basic P × P -set Ω , namely the disjoint union of Ω with p copies of Ω . Actually, we prove below that F admits an F-basic P × P -set if and only if F is a Frobenius P -category. 21.6 Equivalently, a P × P -set Ω is F-basic if and only if, setting ∆ϕ,ϕ (Q) = {(ϕ(u), ϕ (u))}u∈Q

21.6.1

for any subgroup Q of P , it fulﬁlls condition 21.2.1 and the statement: 21.6.2 The stabilizer of any element of Ω coincides with ∆ψ,ψ (R) for some subgroup R of P and suitable ψ, ψ ∈ F(P, R) , and we have |Ω∆ϕ,ϕ (Q) | = |Ω∆(Q) | for any subgroup Q of P and any ϕ, ϕ ∈ F(P, Q) . Indeed, it easily follows from its very deﬁnition that an F-basic P × P -set fulﬁlls this statement; conversely, if Ω fulﬁlls this statement then it is clear that {1} × P acts freely on Ω and that we have HomΩ (Q, P ) ⊂ F(P, Q) for any subgroup Q of P ; moreover, for any ϕ ∈ F(P, Q) and any subgroup W of Q × P , Res ϕ×idP (Ω)W and ΩW are empty unless W = ∆ψ (R) for some subgroup R of Q and a suitable ψ ∈ F(P, R) , and then we have Res ϕ×id (Ω)∆ψ (R) = |Ω∆ϕ ,ψ (R) | = |Ω∆(R) | = |Ω∆ψ (R) | 21.6.3, P

21. Frobenius P -categories: the second deﬁnition

391

where ϕ ∈ F(P, R) is the restriction of ϕ , which proves that Resϕ×idP (Ω) ∼ = ResιPQ ×idP (Ω)

21.6.4,

Ω

so that F(P, Q) ⊂ F (P, Q) . 21.7 This leads us to introduce the following relative situation which is useful in some inductive arguments. If X is a set of subgroups of P such that any subgroup Q of P fulﬁlling F(Q, R) = ∅ for some R ∈ X belongs X to X , let us say that a P × P -set Ω is F -basic if it is either empty or fulﬁlls condition 21.2.1 and the statement 21.7.1 The stabilizer of any element of Ω coincides with ∆ψ,ψ (R) for some R ∈ X and suitable ψ, ψ ∈ F(P, R) , and we have |Ω∆ϕ,ϕ (Q) | = |Ω∆(Q) | for any Q ∈ X and any ϕ, ϕ ∈ F(P, Q) . Moreover, we say that an F-basic P × P -set Ω is thick outside X if the multiplicity of the indecomposable P × P -set (P × P )/∆ϕ (Q) is at least 2 for any subgroup Q ∈ X of P and any ϕ ∈ F(P, Q) . Lemma 21.8 If Ω is a basic P × P -set then, for any subgroup Q of P and Ω any ϕ ∈ F (P, Q) , the number of Q×P -orbits isomorphic to (Q×P )/∆ϕ (Q) in Ω is equal to Ω∆(Q) CP ϕ(Q) . Moreover, if CP ϕ(Q) is maximal $ % in CP ϕ (Q) ϕ ∈F Ω (P,Q) then we have ∆(Q) CP ϕ(Q) ≡ 0 (mod p) Ω

21.8.1.

Proof: any ϕ ∈ HomΩ (P, Q) , it is easily checked that the group ∆ϕ (Q) For has CP ϕ(Q) ﬁxed elements in (Q × P )/∆ϕ (Q) and no ﬁxed element in the Q × P -orbits of Ω which are not isomorphic to (Q × P )/∆ϕ (Q) ; thus, the number kϕ of Q × P -orbits in Ω isomorphic to (Q × P )/∆ϕ (Q) is clearly equal to Ω∆ϕ (Q) CP ϕ(Q) . But, since we are assuming that Ω

HomΩ (P, Q) = F (P, Q)

21.8.2, Ω

it follows from the symmetry of Ω and the very deﬁnition of F that we have ResιPQ ,ιPQ (Ω) ∼ = ResιPQ ,ϕ (Ω)

21.8.3

and therefore we get |Ω∆ϕ (Q) | = |Ω∆(Q) | , so that kϕ = Ω∆(Q) CP ϕ(Q)

21.8.4.

392

Frobenius categories versus Brauer blocks

$ % Moreover, if CP ϕ(Q) is maximal in CP ϕ (Q) ϕ ∈F Ω (P,Q) then pkϕ obviously divides kϕ for any ϕ ∈ F Ω (P, Q) such that CP ϕ (Q) is not maximal; since all the other Q × P -orbits in Ω have a cardinal divisible by p|P | , we easily get the announced congruence from the congruence in condition 21.2.1. Proposition 21.9 If Ω is a basic P × P -set, F

Ω

is a Frobenius P -category.

Proof: The group FP (P ) of inner automorphisms of P clearly acts on the set HomΩ (P, P ) and there is a natural bijection between the quotient set FP (P )\HomΩ (P, P ) and the set of isomorphism classes of P × P -orbits of Ω cardinal |P | in Ω ; hence, since F (P ) = HomΩ (P, P ) , it follows from isomorphism 21.2.2 that all these P × P -orbits have the same multiplicity in Ω . Moreover, by condition 21.2.1, the number of P × P -orbits of cardinal |P | Ω in Ω is prime to p ; consequently, |F (P )/FP (P )| is also prime to p . Let Q be an F Ω -selfcentralizing subgroup of P ; since F Ω is divisible (cf. 21.2), according to Theorem 4.12 it suﬃces to prove that, for any F-morphism ϕ : Q → P and any subgroup R of NP (Q) such that Q ⊂ R and ϕ FR (Q) ⊂ FP ϕ(Q) 21.9.1, Ω

there is ρ ∈ F (P, R) such that ρ(u) = ϕ(u) for any u ∈ Q . Note that an element (v, w) ∈ P × P normalizes ∆ϕ (Q) if and only if we have ϕ(u)w = ϕ(uv ) for any u ∈ Q and therefore R is contained in the image T of NP ×P ∆ϕ (Q) in NP (Q) by the ﬁrst projection. On the other hand, since CP ϕ(Q) = Z ϕ(Q) (cf. 4.8) and the group {1} × Z ϕ(Q) is normal in the normalizer NP ×P ∆ϕ (Q) , it is clear that this normalizer acts on the quotient set Ω∆ϕ (Q) Z ϕ(Q) which, according to Lemma 21.8, has a cardinal the group prime to p ; consequently, ∆ϕ (Q) NP ×P ∆ϕ (Q) stabilizes ω·Z ϕ(Q) for some ω ∈ Ω , and therefore the Ω stabilizer of ω in NP ×P ∆ϕ (Q) has the form ∆ρ (T ) for some ρ ∈ F (P, T ) ; since ∆ϕ (Q) ⊂ ∆ρ (T ), for any u ∈ Q ⊂ T , we get ϕ(u) = ρ(u) and, since F Ω is divisible, the restriction of ρ to R belongs to F (P, R) . We are done.

Ω

Remark 21.10 Once we have this proposition, it follows from statement 2.10.1 that our maximality condition in Lemma 21.8 amounts to saying that Q is fully centralized in F Ω . Proposition 21.11 Assume that Ω is a basic P ×P -set. Let Q be a subgroup Ω of P and K a subgroup of Aut(Q) such that Q is fully K-normalized in F . , ∆χ (Q) Then, the union ΩK is a basic NPK (Q) × NPK (Q)-set and we Q = χ∈K Ω have F

ΩK Q

= NFKΩ (Q) .

21. Frobenius P -categories: the second deﬁnition

393

Proof: Let us ﬁrstly prove that NPK (Q) × NPK (Q) stabilizes ΩK Q ; indeed, if ∆χ (Q) K χ∈K, ω ∈Ω and v, w ∈ NP (Q) then, for any u ∈ Q , we have v −1 ·ω·w = v −1 ·u·ω·χ(u)−1 ·w = (uv )·(v −1 ·ω·w)·χ(u−1 )w = (uv )·(v −1 ·ω·w)·χ (uv )−1

21.11.1

where χ = κw−1 ◦ χ ◦ κv−1 still belongs to K . Moreover, for any χ ∈ K , we know that the stabilizer in P × P of any element ω ∈ Ω∆χ (Q) coincides with ∆ψ (R) for some subgroup R of P and some ψ ∈ F(P, R) , so that R contains Q and ψ(u) = χ(u) for any u ∈ Q . Thus, on the one hand, ω determines χ and therefore the union ΩK Q =

Ω∆χ (Q)

21.11.2

χ∈K

is a disjoint union; on the other hand, the stabilizer of ω in NPK (Q) × NPK (Q) Ω is equal to ∆ψQ NRK (Q) , where ψQ : NRK (Q) → NPK (Q) is the F -morphism K determined by ψ which actually is an NF Ω (Q) -morphism (cf. 2.14); but, it is easily checked that

ΩK Q NFKΩ (Q) NPK (Q), NRK (Q) ⊂ F NPK (Q), NRK (Q)

21.11.3.

If s : Ω ∼ = Ω is a bijection such that, for any ω ∈ Ω and any u, v ∈ P , we have s(u·ω·v −1 ) = v·s(ω)·u−1 , then s maps Ω∆χ (Q) on Ω∆χ−1 (Q) for any χ ∈ K and therefore it stabilizes ΩK Q . Finally, it follows from Lemma 21.8 that Ω Ω∆χ (Q) = |K ∩ F (Q)||Ω∆(Q) | 21.11.4 χ∈K

and it is quite clear that |NPK (Q)| = |K ∩ FP (Q)||CP (Q)| ; but, according to Ω Proposition 21.9, F is a Frobenius P -category and therefore, by ProposiΩ tion 2.11, K ∩ FP (Q) is a Sylow p-subgroup of K ∩ F (Q) . Consequently, ΩK Q also fulﬁlls condition 21.2.1. We are done. Proposition 21.12 Let F be a Frobenius P -category, X a set of subgroups of P such that any subgroup Q of P fulﬁlling F(Q, R) = ∅ for some R ∈ X X belongs to X , and Ω an F -basic P × P -set. There is an F-basic P × P -set Ω containing Ω which is thick outside X and fulﬁlls Ω∆ϕ (Q) = Ω∆ϕ (Q) for any Q ∈ X and any ϕ ∈ F(P, Q) .

21.12.1

394

Frobenius categories versus Brauer blocks

Proof:† Let {Ωn }n∈N be a family of P × P -sets inductively deﬁned as follows; if X = ∅ then we set Ω0 = Ω ; if X = ∅ then Ω is also empty and, choosing k0 ∈ N − {0, 1} prime to p , the P × P -set Ω0 is the disjoint union of k0 copies of the P × P -sets (P × P )/∆σ (P ) where σ runs over a set of representatives for F(P )/FP (P ) in F(P ) ; note that, for any Q ∈ X ∪ {P } and any elements ϕ, ϕ ∈ F(P, Q) , we have (Ω0 )∆(Q) = (Ω0 )∆ϕ,ϕ (Q)

21.12.2

and that the P × P -sets Ω0 and (Ω0 )◦ are isomorphic. For any n ∈ N , we choose a set of representatives Rn for the F-isomorphism classes of subgroups of P of index pn which do not belong to X in such a way that any element of Rn is fully normalized in F ; we argue by induction on n , so that, if n ≥ 1 and m < n , we assume that the P × P -sets Ωm and (Ωm )◦ are isomorphic, that {1} × P acts freely on Ωm and that, for any Q ∈ X ∪ Rm and any ϕ, ϕ ∈ F(P, Q) , we have (Ωm )∆(Q) = (Ωm )∆ϕ,ϕ (Q)

21.12.3

Now, choose kR ∈ N − {0, 1} such that (Ωn−1 )∆(R) + kR (P × P )/∆(R) ∆(R) ∆ (R) ≥ (Ωn−1 )∆ψ,ψ (R) + 2 (P × P )/∆ψ,ψ (R) ψ,ψ

21.12.4

for any ψ, ψ ∈ F P, R , and denote by kψ,ψ the positive rational number fulﬁlling (Ωn−1 )∆(R) + kR (P × P )/∆(R) ∆(R) ∆ (R) 21.12.5; = (Ωn−1 )∆ψ,ψ (R) + kψ,ψ (P × P )/∆ψ,ψ (R) ψ,ψ then, provided we prove that kψ,ψ is an integer for any ψ, ψ ∈ F P, R , we will choose Ωn as the disjoint union of Ωn−1 with the disjoint union of kψ,ψ copies of (P × P )/∆ψ,ψ (R) where R runs over Rn and ψ and ψ run over F P, R ; at that point, it will be clear that |(Ωn )∆(Q) | = |(Ωn )∆ϕ,ϕ (Q) | and Ωn ∼ = (Ωn )◦ In order to prove that kψ,ψ is an integer, note that ∆ (R) ¯P ×P ∆ψ,ψ (R) (P × P )/∆ψ,ψ (R) ψ,ψ = N †

This proof follows the arguments of the proof of Proposition 5.5 in [13].

21.12.6

21.12.7

21. Frobenius P -categories: the second deﬁnition

395

¯P ×P ∆ψ,ψ (R) acts on (Ωn−1 )∆ψ,ψ (R) ; the key point and that the quotient N ∆(R) of the argument in [13] is that this quotient and that still actson (Ωn−1 ) ¯ the “diﬀerence” between these two NP ×P ∆ψ,ψ (R) -sets is a multiple of the regular orbit. Indeed, since R is fully normalized in F , considering the inverse of the isomorphism ψ : R ∼ = ψ(R) determined by ψ , it follows from condition 2.8.2 that there are an F-morphism ζ : NP ψ(R) → P and an element χ ∈ F(R) such that ζ ψ(v) = χ(v) for any v ∈ R ; moreover, according to Proposition 2.11, R is also fully centralized in F and therefore, considering the group homomorphism

η : ψ (R) −→ P mapping ψ (v) on χ(v) for any v ∈ R , and the image T = π NP ×P ∆ψ,ψ (R) ⊂ NP ψ (R)

21.12.8

21.12.9

where π : P × P → P is the secondprojection, there is ξ ∈ F(P, T ) extend ing η and, in particular, fulﬁlling ξ ψ (v) = χ(v) for any v ∈ R (cf. statement 2.10.1); thus, ζ × ξ determines a group homomorphism NP ×P ∆ψ,ψ (R) −→ P × P 21.12.10 which maps ∆ψ,ψ (R) onto ∆(R) and therefore it fulﬁlls (ζ × ξ) NP ×P ∆ψ,ψ (R) ⊂ NP ×P ∆(R)

21.12.11.

¯P ×P ∆ψ,ψ (R) Hence, the homomorphism ζ × ξ induces an action of N on (Ωn−1 )∆(R) and from equality 21.12.7 we get (P × P )/∆(R) ∆(R)

∆ψ,ψ (R) NP ×P ∆(R) = (P × P )/∆ψ,ψ (R) NP ×P ∆ψ,ψ (R)

21.12.12.

Moreover, at this point it follows from our induction hypothesis that we have ¯ ¯ (Ωn−1 )∆(R) Q = (Ωn−1 )∆ψ,ψ (R) Q

21.12.13

¯ of N ¯P ×P ∆ψ,ψ (R) ; this is equivalent to for any nontrivial subgroup Q ¯P ×P ∆ψ,ψ (R) -sets obtained removing all the regular saying that the two N orbits from (Ωn−1 )∆(R) and (Ωn−1 )∆ψ,ψ (R) are isomorphic, which proves our claim.

396

Frobenius categories versus Brauer blocks

Finally, we claim that the union Ω = indeed, it is clear that

, n∈N

Ωn is an F-basic P × P -set;

|Ω|/|P | ≡ |Ω0 |/|P | (mod p)

21.12.14

and we have either Ω0 = Ω or |Ω0 | = k0 |F(P )||Z(P )| , so that |Ω|/|P | is prime to p ; now, it easily follows from , 21.12.6 that Ω fulﬁlls conditions 21.2.1 and 21.6.2; moreover, for any R ∈ n∈N Rn , it is clear that the multiplicity kψ,ψ of (P × P )/∆ψ,ψ (R) is at least 2 , so that Ω is thick outside X , and that ∆(Q) ﬁxes no element in (P ×P )/∆ψ,ψ (R) for any Q ∈ X which implies equalities 21.12.1. We are done.

Chapter 22

The basic F-locality 22.1 Let P be a ﬁnite p-group, Ω a thick basic P × P -set (cf. 21.4) and Ω F = F the corresponding Frobenius P -category (cf. Proposition 21.9). In b this chapter we exhibit a coherent F-locality L — called the basic F-locality — which, although constructed from this P × P -set Ω , actually does not depend on the chosen thick F-basic P × P -set (cf. Proposition 22.12 below). 22.2 Denote by G the group of automorphisms of Res{1}×P (Ω) and identify the p-group P with the image of P × {1} in G , so that from now b on P acts freely on Ω (cf. 21.2). In order to construct L , we ﬁrstly have to discuss the behaviour of the centralizers in G of the subgroups Q of P ; as a matter of fact, the thickness condition will guarantee a “stable” description Ω — with respect to the diﬀerent choices of Ω such that F = F — of these centralizers and of their Abelian quotients. 22.3 Let Q be a subgroup of P ; clearly, the centralizer CG (Q) coincides with the group of automorphisms of ResQ×P (Ω) ; moreover, any orbit of this Q × P -set is isomorphic to the quotient set (Q × P )/∆η (T ) (cf. 21.3) for some subgroup T of Q and some η ∈ F(P, T ) and, by the thickness, all these Q × P -sets are isomorphic to orbits of ResQ×P (Ω) (cf. 21.4). Note that, denoting by Aut (Q × P )/∆η (T ) the group of automorphisms of this Q × P -set, the multiplication by NQ×P ∆η (T ) on the right induces a group isomorphism (cf. 1.9) ¯Q×P ∆η (T ) Aut (Q × P )/∆η (T ) ∼ =N

22.3.1

∆η (T ) and that this group acts regularly on (Q × P )/∆η (T ) . 22.4 Actually, the isomorphism class of this Q × P -set (Q × P )/∆η (T ) only depends on the conjugacy class of T in Q and on the class η˜ of η ˜ in F(P, T ) (cf. 1.3); in other words, by the very deﬁnition of the product ˜ (cf. A2.7) of the exterior quotient F˜ of F (cf. 1.3), the pair cover pc(F) ˜ ˜ (cf. A2.7) and the set of isomorphism (Q, P ) deﬁnes the pc(F)-object Q×P classes of Q × P -orbits in ResQ×P (Ω) corresponds bijectively with the set ˜ ˜ of isomorphism classes of objects in the full subcategory F˜Q×P of pc(F) ˜ Q×P ˜ ˜ over the set of pc(F)-morphisms ˜ (cf. 1.7) — the full subcategory of pc(F) Q×P

˜ when T runs over the set of subgroups of P . η˜ : T → Q×P 22.5 Let us denote by OQ a set of representatives for the set of isomorphism classes of F˜Q×P ˜ -objects; actually, we may assume that, for any

398

Frobenius categories versus Brauer blocks

˜ ˜ in OQ , T is contained in Q and the composipc(F)-morphism η˜ : T → Q×P ˜ ˜ → Q coincides with ˜ιQ tion of η˜ with the structural pc(F)-morphism Q×P T (cf. 1.9); then, we denote by ηˆ ˜ the composition of η˜ with the structural ˜ ˜ → P . Moreover, we denote by Q ×η˜ P the correspc(F)-morphism Q×P ponding indecomposable Q × P -set, by kη˜ the multiplicity of this Q × P -set in Ω and by Skη˜ the corresponding symmetric group (cf. 1.32). With all this notation, it is quite clear that we have a canonical Gr-isomorphism (cf. 1.3 and 1.32) ω ˜ : CG (Q) ∼ Aut(Q ×η˜ P ) Sk 22.5.1. = η ˜

Q

η˜∈OQ

22.6 Let us exhibit a representative of ω ˜ Q ; it is clear that the correspondence mapping any injective Q × P -set homomorphism h : Q ×η˜ P −→ ResQ×P (Ω)

22.6.1

on its image induces a bijection between the set of Aut(Q ×η˜ P )-orbits in the set of these injective Q × P -set homomorphisms, and the set of Q × P -orbits isomorphic to Q ×η˜ P in ResQ×P (Ω) ; thus, denoting by Kη˜ a set of representatives for the ﬁrst set of orbits, we have |Kη˜| = kη˜ and we can identify Skη˜ with the group of permutations SKη˜ of Kη˜ (cf. 1.32); up to this identiﬁcation, for any z ∈ CG (Q) and any h ∈ Kη˜ , there are unique elements σz ∈ SKη˜ and az,h ∈ Aut(Q ×η˜ P ) fulﬁlling z ◦ h = σz (h) ◦ az,h

22.6.2

and a representative θQ of the Gr-isomorphism 22.5.1 maps z ∈ CG (Q) on the element of the direct product having (az,h )h∈Kη˜ , σz in its η˜-component. Proposition 22.7 For any subgroup Q of P , denote by S(Q) the minimal normal subgroup of CG (Q) containing (ωQ )−1 S for a represenk η ˜ η˜∈OQ tative ωQ of ω ˜ Q . Then ω ˜ Q induces an Ab-isomorphism ω ¯ Q : CG (Q)/S(Q) ∼ =

ab Aut(Q ×η˜ P )

22.7.1.

η˜∈OQ

Proof: It follows from the thickness condition and from the general lemma below. Lemma 22.8 If n ∈ N − {0, 1} , H is a group and N is the minimal normal subgroup containing Sn in the wreath product H Sn , then we have (H Sn )/N ∼ = ab(H)

22.8.1.

22. The basic F-locality

399

Proof: Firstly assume that H is Abelian; then, we have an obvious group homomorphism H Sn → H (cf. 1.32) mapping an element

n (hi )1≤i≤n , σ ∈ H Sn

22.8.2

i=1

n on the product i=1 hi in H , which is clearly surjective; moreover, denoting n by h(i) the element of i=1 H having h ∈ H in the entry i and 1 elsewhere, and by τi the transposition (i, i + 1) for any 1 ≤ i < n , the kernel of this homomorphism is generated by the elements (h(i+1) )−1 h(i) = [τi , h(i) ]

22.8.3.

In the general case, for any h, g ∈ H , the elements (h(i+1) )−1 h(i) , −1 (g (i+1) )−1 g (i) and (hg)(i+1) (hg)(i) still belong to N and therefore the product −1 (h(i) )−1 h(i+1) (g (i) )−1 g (i+1) (hg)(i+1) (hg)(i) = [h(i) , g (i) ] also belongs to N ; hence, the n-direct product We are done.

n

i=1 [H, H]

22.8.4

is contained in N .

22.9 Let Q and R be subgroups of P such that R ⊂ Q ; in order to determine the inclusion CG (Q) ⊂ CG (R) in terms of the right-hand members of the corresponding isomorphisms 22.5.1, we need some general nota tion. If m ≤ n , let us denote by ˜ιnm : Sm → Sn the natural Gr-inclusion; it is clear that, if i∈I mi ≤ n then we can ﬁnd representatives of the n Gr-morphisms ˜ι (cf. 1.3), where i runs over I , determining a unique injecmi

tive Gr-morphism

˜ιn{mi }i∈I :

Smi −→ Sn

22.9.1.

i∈I

In the case where mi = m for any i ∈ I and n = |I|m , we denote by |I| δ˜m : Sm −→ S|I|m

22.9.2

the composition of the diagonal map Sm → i∈I Sm with the correspond ing Gr-morphism 22.9.1; more generally, if i∈I Di mi ≤ n then we denote by ˜ιn{(6i ,mi )}i∈I the composition i∈I

˜ ιn {

i mi }i∈I

Smi −−−−−−−→

i∈I

δ˜mi i

S6i mi −−−−−−−→ Sn i∈I

22.9.3.

400

Frobenius categories versus Brauer blocks

22.10 Moreover, for any η˜ ∈ OQ and any ν˜ ∈ OR , denote by kνη˜˜ the multiplicity of R ×ν˜ P in ResR×P (Q ×η˜ P ) and choose a set of representatives Kνη˜˜ for the set of Aut(R ×ν˜ P )-orbits of injective R × P -set homomorphisms g : R ×ν˜ P −→ ResR×P (Q ×η˜ P )

22.10.1,

so that Kνη˜˜ = kνη˜˜ ; note that the composition induces a bijection η˜ Kη˜ × Kνη˜˜ ∼ = Kη˜ ◦ Kν˜

22.10.2

and, since {Im(h◦g)}h∈Kη˜ , g∈Kη˜ is a family of mutually disjoint subsets of Ω , ν ˜ ) we may assume that Kν˜ coincides with the disjoint union η˜∈OQ Kη˜ ◦ Kνη˜˜ , for any representative ν˜ ∈ OR , Proposition 22.11 With the notation above, let Q and R be subgroups of P such that R ⊂ Q . Then, the inclusion CG (Q) ⊂ CG (R) and the ωR )−1 and Gr-morphisms (˜ ωQ )−1 , (˜ ν ˜ ˜ιk{(k

η ˜ )}η∈O η ˜ ,kν ˜ ˜ Q

:

Skη˜ −→ Skν˜

22.11.1

η˜∈OQ

where ν˜ runs over OR , determine the following commutative Gr-diagram CG (Q) &  Skη˜

−→ −→

η˜∈OQ

CG (R) &  Skν˜

22.11.2.

ν ˜∈OR

In particular, we have S(Q) ⊂ S(R) . Proof: First of all note that, considering the decomposition of ResQ×P (Ω) in R × P -orbits, for any ν˜ ∈ OR we clearly have kν˜ =

kνη˜˜ kη˜

22.11.3.

η˜∈OQ

Moreover, for any η˜ ∈ OQ , any σ ∈ Skη˜ and any h ∈ Kη˜ , let us identify, up to the choice of a representative ωQ of ω ˜ Q , Skη˜ with SKη˜ acting on Ω just permuting the Q × P -orbits isomorphic to Q ×η˜ P ; then, according to our deﬁnition in 22.6.2, we have (ωQ )−1 (σ) ◦ h = σ(h) and therefore, considering (ωQ )−1 (σ) as an element of CG (R) , for any g ∈ Kνη˜˜ our deﬁnition in 22.6.2 simply becomes (ωQ )−1 (σ) ◦ (h ◦ g) = σ(h) ◦ g 22.11.4.

22. The basic F-locality

401

Now, it is straightforward to check that the map sending σ to the permutation over Kη˜ ◦ Kνη˜˜ mapping h ◦ g on σ(h) ◦ g coincides with the group homomorphism SKη˜ −→ SKη˜ ×Kη˜ ∼ 22.11.5 = SKη˜ ◦Kη˜ ν ˜

ν ˜

coming from the evident diagonal map

SKη˜ −→

SKη˜ ×{g} ⊂ SKη˜ ×Kη˜

22.11.6;

ν ˜

η ˜ g∈Kν ˜

)

Kη˜ ◦ Kνη˜˜ , it is quite clear that the direct product of these maps when η˜ runs over OQ coincides with the Gr-morphism 22.11.1. Finally, for any ν˜ ∈ OR , any τ ∈ Skν˜ and any D ∈ Kν˜ , identifying as above Skν˜ with SKν˜ acting on Ω just permuting the R × P -orbits isomorphic to Q ×ν˜ P , we still have (ωR )−1 (τ ) ◦ D = τ (D) by our deﬁnition 22.6.2. We are done. then, assuming that Kν˜ =

η˜∈OQ

Proposition 22.12 The correspondence mapping any pair of subgroups Q and R of P on the quotient set b

L (Q, R) = TG (Q, R)/S(R)

22.12.1,

endowed with the natural maps b

b τQ,R : TP (Q, R) → L (Q, R)

and

b

b πQ,R : L (Q, R) → F(Q, R)

22.12.2,

b

deﬁnes a coherent F-locality (τ b , L , π b ) which does not depend on the chosen thick F-basic P × P -set Ω . Proof: It follows from the last statement in Proposition 22.11 that, for any triple of subgroups Q , R and T of P , the product in G induces a map b

b

b

L (Q, R) × L (R, T ) −→ L (Q, T )

22.12.3 ; b

then, it is quite clear that these maps determine a composition in L † and that the natural maps in 22.12.2 deﬁne the structural functors (cf. 17.3) b

τ b : TP −→ L

b

and π b : L −→ F

22.12.4; b

moreover, the divisibility (cf. 17.7) and the coherence (cf. 17.9) of L are easy consequences of the fact that G is a group. †

It corresponds to the very deﬁnition of “localit´ e” in [35, Ch.I]

402

Frobenius categories versus Brauer blocks

On the other hand, for another choice of a thick F-basic P × P -set Ω , we already know that we can embed their disjoint union in a third thick F-basic P × P -set Ω (cf. 21.5); thus, setting G = Aut{1}×P (Ω )

and G = Aut{1}×P (Ω )

22.12.5

and denoting by (G )Ω,Ω the stabilizer in G of the images of Ω and Ω , we have surjective canonical group homomorphisms G ←− (G )Ω,Ω −→ G

22.12.6

mapping P ⊂ (G )Ω,Ω onto both P ⊂ G and P ⊂ G . More precisely, for any pair of subgroups Q and R of P , it is quite clear that we still have surjective maps (cf. 17.2) TG (Q, R) ←− T(G )Ω,Ω (Q, R) −→ TG (Q, R)

22.12.7;

then, denoting by S (R) and S (R) the respective subgroups of CG (R) and CG (R) analogous to S(R) , it is easily checked that we get TG (Q, R) = T(G )Ω,Ω (Q, R)·S (R)

22.12.8

and that this equality, together with the surjective maps 22.12.7, induces bijections TG (Q, R)/S(R) ∼ = TG (Q, R)/S (R) ∼ = TG (Q, R)/S (R)

22.12.9

which are clearly compatible with the corresponding natural maps 22.12.2. Consequently, the three F-localities we obtain are mutually equivalent. We are done. 22.13 Since the basic F-locality is divisible and, for any subgroup Q b of P , the group Ker(πQ ) = CG (Q)/S(Q) is Abelian, Ker(π b ) induces a contravariant functor (cf. 17.8) ˜k : F˜ −→ Ab

22.13.1

which, according to Proposition 22.7, maps Q on ˜k(Q) =

ab Aut(Q ×η˜ P )

22.13.2.

η˜∈OQ

˜ In particular, for any F-morphism ϕ˜ : R → Q , it is clear that we have ˜k(ϕ) ˜ =

η˜∈OQ , ν ˜∈OR

˜k(ϕ) ˜ ην˜˜

22.13.3

22. The basic F-locality

403

for suitable group homomorphisms ˜k(ϕ) ˜ ην˜˜ : ab Aut(Q ×η˜ P ) −→ ab Aut(R ×ν˜ P )

22.13.4

and we will make them explicit. 22.14 It suﬃces to consider the case where R is a subgroup of Q and ϕ˜ the class of the inclusion map, and then we write ˜kην˜˜ instead of ˜k(˜ιQ )η˜˜ . R ν For any η˜ ∈ OQ and any ν˜ ∈ OR , it is clear that Aut(Q ×η˜ P ) acts on the set of R × P -orbits in Q ×η˜ P , preserving their isomorphism class, and therefore, for any identiﬁcation of Skη˜ with SKη˜ , we get a group homomorphism ν ˜

ν ˜

ανη˜˜ : Aut(Q ×η˜ P ) −→ Skη˜

22.14.1

ν ˜

which, for any a ∈ Aut(Q ×η˜ P ) and any g ∈ Kνη˜˜ , fulﬁlls a ◦ g = ανη˜˜ (a) (g) ◦ ba,g

22.14.2

for a suitable ba,g ∈ Aut(R ×ν˜ P ) ; it is clear that the set of these group homomorphisms form a unique Gr-morphism α ˜ νη˜˜ . Proposition 22.15 With the notation above, let Q and R be subgroups of P such that R ⊂ Q . Then, the inclusion CG (Q) ⊂ CG (R) and the Gr-morphisms (˜ ωQ )−1 , ω ˜ R and α ˜ νη˜˜ , where η˜ and ν˜ respectively run over OQ and OR , determine the commutative Gr-diagram CG (Q) &  kη˜

Aut(Q ×η˜ P )

η˜∈OQ i=1

−→

−→

CG (R)  # 22.15.1.

Skν˜ ν ˜∈OR

kη˜ Proof: With the notation in 22.6 above, identify i=1 Aut(Q ×η˜ P ) with Aut(Q × P ) and, for any h ∈ K , identify any a ∈ Aut(Q ×η˜ P ) η ˜ η ˜ h∈Kη˜ in the factor indexed by h with its image in CG (Q) via a representative ωQ of ω ˜ Q ; then, it is clear that a ◦ h = h ◦ a (cf. 22.6.2); thus, since we may assume that h ◦ g ∈ Kν˜ for any g ∈ Kνη˜˜ (cf. 22.10.2), from deﬁnition 22.14.2 above we get a ◦ (h ◦ g) = h ◦ ανη˜˜ (a) (g) ◦ ba,g 22.15.2 for a suitable ba,g ∈ Aut(R ×ν˜ P ) .

404

Frobenius categories versus Brauer blocks

For any ν˜ ∈ OR , on the one hand we consider the direct product of the homomorphisms h∈Kη˜ ανη˜˜ when η˜ runs over OQ (cf. 22.14.1)

Aut(Q ×η˜ P ) −→

η˜∈OQ h∈Kη˜

Skη˜

η˜∈OQ h∈Kη˜

22.15.3;

ν ˜

on the other hand, we consider the family of integers (˜ η ,h)

Kν˜ = {kν˜ (˜ η ,h)

where kν˜

}(˜η,h)∈OQ ×Kη˜

22.15.4

= kνη˜˜ for any h ∈ Kη˜ , and the Gr-morphism (cf. 22.9.1 and 22.10)

˜ιkKν˜ν˜ :

η˜∈OQ h∈Kη˜

Skη˜ −→ Skν˜

22.15.5;

ν ˜

thus, the composition of the Gr-morphisms 22.15.3 and 22.15.5 determines a Gr-morphism Aut(Q ×η˜ P ) −→ Skν˜ 22.15.6. η˜∈OQ h∈Kη˜

Finally, from equality 22.15.2 it is easy to check that the Gr-morphism

η˜∈OQ h∈Kη˜

Aut(Q ×η˜ P ) −→

Skν˜

22.15.7

ν ˜∈OR

obtained from the Gr-morphisms 22.15.6 when ν˜ runs over OR yields the commutativity of diagram 22.15.1. We are done. 22.16 Now, for any pair of subgroups Q and R of P such that R ⊂ Q , any η˜ ∈ OQ and any ν˜ ∈ OR , we are ready to describe the group homomorphism above ˜kη˜ : ab Aut(Q ×η˜ P ) −→ ab Aut(R ×ν˜ P ) 22.16.1. ν ˜ It is clear that the direct product Aut(Q ×η˜ P ) × Aut(R ×ν˜ P ) acts, by composition on the left and on the right, on the set of injective R × P -set homomorphisms g : R ×ν˜ P −→ ResR×P (Q ×η˜ P ) 22.16.2 and let us consider a set of representatives Iν˜η˜ for the corresponding set of orbits; moreover, for any g ∈ Iν˜η˜ , denote by Aut(Q ×η˜ P )g the stabilizer of g ◦ Aut(R ×ν˜ P ) in Aut(Q ×η˜ P ) and by ιg : Aut(Q ×η˜ P )g −→ Aut(Q ×η˜ P )

22.16.3

22. The basic F-locality

405

the inclusion map; then, we clearly have an injective group homomorphism βg : Aut(Q ×η˜ P )g −→ Aut(R ×ν˜ P )

22.16.4

such that a ◦ g = g ◦ βg (a) for any a ∈ Aut(Q ×η˜ P )g . Proposition 22.17 With the notation above, for any pair of subgroups Q and R of P such that R ⊂ Q , any η˜ ∈ OQ and any ν˜ ∈ OR , we have ˜kη˜ = ν ˜

ab(βg ) ◦ ab◦ (ιg )

22.17.1.

g∈Iν˜η˜

Proof: As above, let us identify Skη˜ with SKη˜ and Skν˜ with SKν˜ ; it follows from Propositions 22.11 and 22.15 that the inclusion CG (Q) ⊂ CG (R) maps an element (cf. 22.5.1)

(ah )h∈Kη˜ , ση˜ ∈ Aut(Q ×η˜ P ) SKη˜

22.17.2

h∈Kη˜

on the element $

(b6 )6∈Kν˜ , τν˜

% ν ˜∈OR

∈

ν ˜∈OR

Aut(R ×ν˜ P ) SKν˜

22.17.3

6∈Kν˜

where, identifying the set Kνη˜˜ × Kη˜ with the corresponding subset of Kν˜ (cf. 22.10), τν˜ maps (g, h) ∈ Kνη˜˜ × Kη˜ on ανη˜˜ (ah ) (g), ση˜(h) and ﬁxes all the other elements in Kν˜ , and where bh◦g fulﬁlls

ανη˜˜ (ah ) (g) ◦ bh◦g = ah ◦ g

22.17.4,

whereas b6 = {1} otherwise. Then, denoting by a ¯h and ¯b6 the respective classes of ah in ab Aut(Q ×η˜ P ) and of b6 in ab Aut(R ×ν˜ P ) , it is clear ¯h on 6∈Kν˜ ¯b6 . that kην˜˜ maps h∈Kη˜ a More explicitly, we may choose the set of representatives Iν˜η˜ inside the set of representatives Kνη˜˜ ; moreover, since Aut(Q×η˜ P ) acts freely on Q×η˜ P , it still acts freely on the set of injective Q×P -set homomorphisms 22.16.2 and therefore, for any g ∈ Iν˜η˜ , choosing a set of representatives Ag ⊂ Aut(Q×η˜ P ) for Aut(Q ×η˜ P )/Aut(Q ×η˜ P )g , we may assume that Kνη˜˜ =

g∈Iν˜η˜

Ag ◦ g

22.17.5.

406

Frobenius categories versus Brauer blocks

With these choices in hand, for any g ∈ Iν˜η˜ and any ag ∈ Ag , equality 22.17.4 becomes η˜ αν˜ (ah ) (ag ◦ g) ◦ bh◦ag ◦g = ah ◦ ag ◦ g 22.17.6 η˜ and, since the element αν˜ (ah ) (ag ◦g) of Kνη˜˜ belongs to the Aut(Q×η˜ P )-orbit of g , for some a ˆg ∈ Ag we have

ανη˜˜ (ah ) (ag ◦ g) = a ˆg ◦ g

22.17.7;

now, equality 22.17.6 implies that g ◦ bh◦ag ◦g = (ˆ ag )−1 ◦ ah ◦ ag ◦ g

22.17.8

and therefore, by the very deﬁnition of βg , ﬁnally we get bh◦ag ◦g = βg (ˆ ag )−1 ◦ ah ◦ ag

22.17.9.

Consequently, it follows from the argument above and from all the equalities 22.17.9 that for any a ∈ Aut(Q ×η˜ P ) we have ανη˜˜ (a) =

βg (ˆ ag )−1 ◦ a ◦ ag

g∈Iν˜η˜ ag ∈Ag

=

g∈Iν˜η˜

ab(βg ) (ˆ ag )−1 ◦ a ◦ ag

22.17.10

ag ∈Ag

where a ¯ is the image of a in ab Aut(Q ×η˜ P ) and, for any ag ∈ Ag , we denote by βg (ag ) the image of βg (ag ) in ab Aut(R ×ν˜ P ) and by a ˆg the element of Ag such that a ◦ ag belongs to a ˆg ◦ Aut(Q ×η˜ P )g (cf. 22.17.7). But, by the very deﬁnition of ab◦ (ιg ) , for any g ∈ Iν˜η˜ we have

ab◦ (ιg ) (¯ a) = (ˆ ag )−1 ◦ a ◦ ag

22.17.11.

ag ∈Ag

We are done. 22.18 Finally, let us be more explicit on homomorphisms 22.16.2. For ˜ ˜ in OQ , choose a any subgroup Q of P and any pc(F)-morphism η˜ : T → Q×P ˜ ˆ representative η of η˜ ; then, for any F-morphism ϕ : R → Q , any pc(F)-mor˜ in OR and any representative ν of νˆ˜ , it is clear that an phism ν˜ : U → R×P injective R × P -set homomorphism g : (R × P )/∆ν (U ) −→ Resϕ×idP (Q × P )/∆η (T )

22.18.1

22. The basic F-locality

407

is determined by the image (w, w)∆ ˆ η (T ) of the class (1, 1)∆ν (U ) where (w, w) ˆ is an element of Q × P fulﬁlling ϕ(U )w = ϕ(R)w ∩ T and η ϕ(v)w = ν(v)wˆ 22.18.2 for any v ∈ U ; in this case, we denote by ψw : U → T the F-morphism mapping v ∈ U on ϕ(v)w . Lemma 22.19 With the notation above, the correspondence mapping any injective R × P -set homomorphism g : (R × P )/∆ν (U ) −→ Resϕ×idP (Q × P )/∆η (T ) 22.19.1 such that g (1, 1)∆ν (U ) = (w, w)∆ ˆ η (T ) on ϕ(R)wT induces an injective map to ϕ(R)\Q/T from the set of Aut(R ×ν˜ P )-orbits in the set of these homomorphisms. Moreover, a double class in ϕ(R)\Q/T belongs to the image of this map if and only if it admits a representative w ∈ Q fulﬁlling ϕ(U )w = ϕ(R)w ∩ T

ηˆ˜ ◦ ψ˜w = νˆ˜

and

Proof: Let g be such an injective R × P -set homomorphism, set g (1, 1)∆ν (U ) = (w , w ˆ )∆η (T )

22.19.2.

22.19.3

and, with the notation above, assume that w = ϕ(r)wt for some r ∈ R and some t ∈ T ; thus, w and w ˆ fulﬁll equalities 20.18.2 and in particular we have −1 −1 ϕ(r)−1 −1 ϕ(U ) = ϕ(R) ∩ T w = ϕ(R) ∩ T w = ϕ(U r ) 22.19.4, so that r normalizes U . Moreover, for any v ∈ U we still have η(t) ˆ ν(v)wˆ = η ϕ(v)w = η ϕ(v r )w = ν(v r )wη(t)

22.19.5,

so that the element (r, w ˆ η(t)−1 w ˆ −1 ) normalizes ∆v (U ) ; hence, since (w , w ˆ t, η(t) 22.19.6, ˆ ) = ϕ(r), w ˆ η(t)−1 w ˆ −1 (w, w) g belongs to the Aut(R ×ν˜ P )-orbit of g (cf. 22.3.1). Conversely, to the Aut(R ×ν˜ P )-orbit of g , then there are if g belongs (r, rˆ) ∈ NR×P ∆ν (U ) and t ∈ T fulﬁlling (w , w ˆ ) = ϕ(r), rˆ (w, w) ˆ t, η(t) 22.19.7, so that w = ϕ(r)wt . Finally, it is clear that equalities 22.18.2 imply equalities 22.19.2; conversely, the equality ηˆ ˜ ◦ ψ˜w = νˆ ˜ forces the existence of w ˆ ∈ P which fulﬁlls equalities 22.18.2 and, in particular, deﬁnes an injective Q × P -set homomorphism g as above mapping on w . We are done.

Chapter 23

Narrowing the basic F -locality sc

23.1 Let P be a ﬁnite p-group and F a Frobenius P -category. As in sc chapter 4, denote by F the full subcategory of F over the set of F-selfcensc tralizing subgroups of P and by F˜ its exterior quotient (cf. 1.3). In this b,sc chapter, we exhibit some signiﬁcant quotients of the full subcategory L b of L over the set of F-selfcentralizing subgroups of P — namely, the polysc central and the reduced F -localities — which are necessary candidates to sc contain any perfect F -locality, as we show in the last chapter; in particusc lar, we give a suﬃcient condition for the existence of a perfect F -locality (cf. 17.4 and 17.13). We freely use the notation introduced in the previous chapter and abundantly employ the terminology and the results of chapter 6. 23.2 Let Ω be a thick F-basic P × P -set and set G = Aut{1}×P (Ω) (cf. 21.2); recall that, for any subgroup Q of P , we denote by OQ a set of representatives for the set of isomorphism classes of objects in the full sub˜ ˜ over the set of pc(F)-morphisms ˜ ˜ category F˜Q×P of pc(F) η˜ : T → Q×P ˜ Q×P ˜ in OQ , we where T is a subgroup of P (cf. 22.5). For any η˜ : T → Q×P assume that T is contained in Q and that the composition of η˜ with the ˜ ˜ → Q coincides with ˜ιQ structural pc(F)-morphism Q×P T (cf. 1.9). Then, denoting by ˜k the factorization of Ker(π b ) throughout F˜ (cf. 22.13.1), we have (cf. isomorphism 22.7.1) ˜k(Q) = CG (Q)/S(Q) ∼ ab Aut(Q ×η˜ P ) 23.2.1 = η˜∈OQ

˜ and, for any F-morphism ϕ˜ : R → Q , any η˜ ∈ OQ and any ν˜ ∈ OR , the evident homomorphisms (cf. 22.13.4) ˜k(ϕ) ˜ ην˜˜ : ab Aut(Q ×η˜ P ) −→ ab Aut(R ×ν˜ P ) 23.2.2 ˜ U ) = ∅ (cf. 22.18). are trivial unless ˜k(ϕ) ˜ ην˜˜ = 0 which forces F(T, 23.3 Thus, if X is a nonempty set of subgroups of P containing any subgroup Q such that F(Q, R) = ∅ for some R ∈ X then, with the notation above, we have 23.3.1 ˜k(ϕ) ˜ η˜ = 0 and U ∈ X implies that T ∈ X . ν ˜

˜ ˜ That is to say, denoting by OX ˜ : T → Q×P Q the set of pc(F)-morphisms η X

in OQ such that T ∈ X , and by S (Q) the subgroup of CG (Q) deﬁned by X S (Q)/S(Q) ∼ ab Aut(Q ×η˜ P ) 23.3.2, = η˜∈OQ −OX Q

410

Frobenius categories versus Brauer blocks

so that we have CG (Q)/S (Q) ∼ = X

ab Aut(Q ×η˜ P )

23.3.3,

η˜∈OX Q

it is easily checked from this remark that, for any F-morphism ϕ : R → Q , X X we still have S (Q)x ⊂ S (R) for any x ∈ G inducing ϕ . b,X

X

23.4 Consequently, respectively denoting by F and L the full subb X b,X categories over X of F and L , we obtain a new coherent F -locality L¯ b,X (cf. 17.4) which is the quotient of L deﬁned by b,X X L¯ (Q, R) = TG (Q, R)/S (R)

23.4.1

for any pair of elements Q and R of X , together with the induced functors X

b,X

τ¯b,X : TP → L¯

b,X

and π ¯ b,X : L¯

→F

X

23.4.2.

We are particularly interested in the case where X is the set of F-selfcenb,sc tralizing subgroups of P and then we denote by (¯ τ b,sc , L¯ , π ¯ b,sc ) the corressc ponding F -locality. 23.5 First of all, let us give an explicit description of the contravariant functor ˜ksc : F˜ sc −→ Ab 23.5.1 sc sc induced by Ker(¯ π b,sc ) (cf. 17.8). Consider the additive cover ac(F˜ ) of F˜ sc ◦ ◦ (cf. 6.2) which coincides with pc (F˜ ) (cf. A2.7.3); the existence of the sc ˜ exterior intersection in ac(F ) (cf. 6.8) allows us to deﬁne a distributive functor sc sc pc(F˜ ) −→ ac(F˜ ) 23.5.2 sc which is the identity over the full subcategory F˜ of both covers and, for any sc ˜ on Q ∩ ˜ R (cf. 22.4); the fact that pair of F˜ -objects Q and R , maps Q×R sc the exterior intersection is a direct product in ac(F˜ ) (cf. Proposition 6.14) guarantees the existence and uniqueness of such a functor.

23.6 Actually, this functor induces an equivalence of categories between sc sc sc sc the subcategories (F˜ )Q×R of pc(F˜ )Q×R and (F˜ )Q ∩˜ R of ac(F˜ )Q ∩˜ R ˜ ˜ Q,sc

(cf. 23.2); coherently, we denote by O the corresponding set of represensc tatives for the set of isomorphism classes of objects in (F˜ )Q ∩˜ P and, for any sc ˜ P , we still denote by Q ×η˜ P the corresponding (F˜ )Q ∩˜ P -object η˜ : T → Q ∩ Q,sc

Q × P -set. Moreover, let us denote by Ω the union of all the Q × P -orbits Q,sc ; note that, since all in Ω which are isomorphic to Q ×η˜ P for some η˜ ∈ O Q,sc the Q × P -orbits outside Ω have a cardinal divisible by p|P | , we still have |Ω

Q,sc

|/|P | ≡ |Ω|/|P | = 0 (mod p)

23.6.1.

sc

23. Narrowing the basic F -locality

411

23.7 More explicitly, for any F-selfcentralizing subgroup Q of P , choose a set of representatives S˜Q for the set of Q-conjugacy classes of the F-selfcentralizing subgroups Q of Q such that (cf. 6.4.1) ˜ F(P, Q )˜ιQ = ∅

23.7.1

Q

˜ Q for the set of and, for any Q ∈ S˜Q , choose a set of representatives M Q ˜ Q ) Q . Then, according to our deﬁnition of the exteF˜Q (Q )-orbits in F(P, ˜ ιQ

rior intersection (cf. 6.11), we may assume that ˜P = Q∩ Q

23.7.2;

˜Q Q ∈S˜Q η˜ ∈M Q

˜ Q determine an (F˜ sc )Q ∩˜ P -object in particular any Q ∈ S˜Q and any η˜ ∈ M Q ˜P θ˜Q ,˜η : Q −→ Q ∩

23.7.3

sc

which is clearly maximal (cf. A5.1); these (F˜ )Q ∩˜ P -objects are mutually Q,sc

Q,sc

nonisomorphic and we denote by Omax ⊂ O

the subset of them.

sc

˜ P determines some 23.8 Conversely, any (F˜ )Q ∩˜ P -object η˜ : T → Q ∩ Q ˜ (cf. 23.7.2); moreover, up to an (F˜ sc )Q ∩˜ P -isoQ ∈ S˜Q and some η˜ ∈ M Q

morphism, we may assume that Q contains T and that ˜ιQ ˜ ◦˜ιQ T coincide T and η sc with the composition of η˜ with the respective structural ac(F˜ )-morphisms ˜ P −→ Q and Q ∩ ˜ P −→ P Q∩

23.8.1;

Q we choose a representative η of η˜ ◦ ˜ιQ T and set η = η ◦ ιT . Then, it is easily checked that we have NQ×P ∆η (T ) = Z(T ) × {1} ∆η NQ (T ) 23.8.2

¯Q (T ) is isomorphic to the group of (F˜ sc )Q ∩˜ P -automorphisms and that N sc ˜ P , which actually coincides with the of the (F˜ )Q ∩˜ P -object η˜ : T → Q ∩ intersection (cf. Proposition 6.14) sc ∗ Aut(˜ η ) = (F˜ )Q ∩˜ P (˜ η ) = F˜Q (T ) ∩ η F˜P η(T ) 23.8.3 where η ∗ denotes the inverse of the isomorphism T ∼ = η(T ) induced by η ; in particular, setting Z(˜ η ) = Z(T ) and considering the evident action of Aut(˜ η) on Z(˜ η ) , from isomorphism 22.3.1 we get Aut(Q ×η˜ P ) ∼ η ) Aut(˜ η) = Z(˜ ' ( ¯ η ) = Z(˜ and therefore, setting Z(˜ η ) Z(˜ η ), Aut(˜ η ) , we still get ¯ η ) × ab Aut(˜ ab Aut(Q ×η˜ P ) ∼ η) = Z(˜

23.8.4

23.8.5.

412

Frobenius categories versus Brauer blocks

23.9 At this point, for any F-selfcentralizing subgroup Q of P , from isomorphism 23.3.3 we get ˜ksc (Q) = CG (Q)/Ssc(Q) ∼ ¯ η ) × ab Aut(˜ Z(˜ η) 23.9.1. = Q,sc

η˜∈O

Q,sc

Moreover, for any F-selfcentralizing subgroup R of Q , any η˜ ∈ O and any R,sc ν˜ ∈ O , it follows from Proposition 22.17 that the group homomorphism ˜ksc (˜ιQ ) : Z(˜ ¯ η ) × ab Aut(˜ ¯ ν ) × ab Aut(˜ η ) −→ Z(˜ ν) 23.9.2 R is equal to ˜kη˜ = ν ˜

ab(βg ) ◦ ab◦ (ιg )

23.9.3,

g∈Iν˜η˜

where Iν˜η˜ is a set of representatives for the set of orbits of the direct product Aut(Q×η˜ P )×Aut(R×ν˜ P ) over the set of injective Q×P -set homomorphisms g : R ×ν˜ P −→ ResR×P (Q ×η˜ P )

23.9.4,

and where we denote by Aut(Q ×η˜ P )g the stabilizer of g ◦ Aut(R ×ν˜ P ) in Aut(Q ×η˜ P ) , by ιg : Aut(Q ×η˜ P )g −→ Aut(Q ×η˜ P )

23.9.5

the inclusion map and by βg : Aut(Q ×η˜ P )g −→ Aut(R ×ν˜ P )

23.9.6

the group homomorphism fulﬁlling a◦g = g◦βg (a) for any a ∈ Aut(Q×η˜ P )g . sc c,sc We are ready to exhibit the polycentral F -locality (τ c,sc , L , π c,sc ). Proposition 23.10 With the notation above, we have a natural direct desc ˜k where ˆ ˜k are contravariant functors from F˜ sc ˜z × ˆ ˜z and ˆ composition ˜k ∼ =ˆ to Ab mapping any F-selfcentralizing subgroup Q of P on ˆ ˜k(Q) = ¯ η ) and ˆ ˜z(Q) = Z(˜ ab Aut(˜ η) 23.10.1. Q,sc

Q,sc

η˜∈O

η˜∈O sc

c,sc

In particular, there is a coherent F -locality (τ c,sc , L c,sc

L

b,sc (Q, R) = L¯ (Q, R) ˆ˜k(R)

, π c,sc ) where 23.10.2

for any pair of F-selfcentralizing subgroups Q and R of P , and where the structural functors are induced by τ¯b,sc and π ¯ b,sc . Proof: With the notation in 23.9 above, it is quite clear that it suﬃces ¯ η ) on Z(˜ ¯ ν) to show that the group homomorphism ˜kην˜˜ in 23.9.2 maps Z(˜ and ab Aut(˜ η ) on ab Aut(˜ ν) .

sc

23. Narrowing the basic F -locality

413

First of all, it is easily checked that the image of Z(˜ η ) in Aut(Q ×η˜ P ) stabilizes g ◦ Aut(R ×ν˜ P ) and therefore, denoting by Aut(˜ η )g the corresponding stabilizer in Aut(˜ η ) , we have Aut(Q ×η˜ P )g ∼ 23.10.3; η ) Aut(˜ η )g = Z(˜ then, it is quite clear that ιg decomposes on the direct product of the identity over Z(˜ η ) and the inclusion map ˆιg : Aut(˜ η )g −→ Aut(˜ η)

23.10.4;

coherently, it is easily checked that ab◦ (ιg ) still decomposes on the direct product of the homomorphism ab◦ (ˆιg ) and the group homomorphism ' ( ¯ η ) −→ Z(˜ Z(˜ η ) Aut(˜ η )g , Z(˜ η) 23.10.5 mapping the class of z ∈ Z(˜ η ) in the left-hand term on the class of u uzu−1 in the right-hand term where, with the notation in 23.7 above, u ∈ NQ (T ) runs over a set of representatives for Aut(˜ η )/Aut(˜ η )g . On the other hand, denoting by T and U the respective origins of η˜ and ν˜ , and by ψg : U → T the group homomorphism induced by g (cf. 22.18), ˜ ˆ we already know that ˜ιQ ιQ ˜ ◦ ψ˜g = νˆ˜ (cf. 22.19.2) and therefore T ◦ ψg = ˜ U and η Q ˜ P ) ◦ ν˜ ; that is to say, with the notation in 23.7 above, ˜ id we get η˜ ◦ ψ˜g = (˜ιR ∩ ˜ R , and then ˜ P determines some R ∈ S˜R and some ν˜ ∈ M ν˜ : U → R ∩ R ˜ ψ˜g : U → T induces an F-morphism ψ˜g : R → Q fulﬁlling ˜ ˜ιQ ιQ Q ◦ ψg = ˜ R

and η˜ ◦ ψ˜g = ν˜

23.10.6;

thus, choosing a coherent family of representatives η of η˜ ◦ ˜ιQ T and ν of

R ν˜ ◦ ˜ιR η = η ◦ ιQ U , and setting T and ν = ν ◦ ιU , we may assume that we have η ψg (v ) = ν (v ) for any v ∈ NR (U ) , and then we still have (ψg × idP )−1 ∆η NQ (T ) ⊂ ∆ν NR (U ) 23.10.7.

Consequently, up to a suitable choice of isomorphisms 23.10.3, βg induces an injective group homomorphism βˆg : Aut(˜ η )g −→ Aut(˜ ν)

23.10.8

mapping any element of Aut(˜ η )g , which is necessarily determined by ∆η (u ) for some u ∈ NQ (T ) in the image of ψg , on the element of Aut(˜ ν ) deter mined by the element v ∈ NR (U ) fulﬁlling ψg (v ) = u ; with this choice, throughout isomorphism 23.10.3 βg also decomposes on the direct product of zF sc (ψg ) : Z(T ) → Z(U ) (cf. 13.13.2) and βˆg . Finally, independently of any choice, ab(βg ) decomposes on the direct product of ab(βˆg ) and the group homomorphism ' ( ¯ ν) Z(˜ η ) Aut(˜ η )g , Z(˜ η ) −→ Z(˜ 23.10.9 induced by zF sc (ψg ) : Z(˜ η ) → Z(˜ ν ) . We are done.

414

Frobenius categories versus Brauer blocks

sc sc 23.11 As a matter of fact, for a suitable representation F˜ → CC of F˜ endowed with a suitable weak structure (cf. A2.3), the corresponding stable adjoin image introduced in A2.14 supplies a good description of the functor ˆ˜z above; then, this description provides a signiﬁcant vanishing cohomology property (cf. Theorem 23.20 and equality 23.23.5 below). Explicitly, let us consider the functor sc sc intP : F˜ −→ ac(F˜ ) 23.11.1 sc sc P ; ˜ P and any F˜ -morphism ϕ˜ on ϕ˜ ∩ ˜ id mapping any F˜ -object Q on Q ∩ sc then, the representation of F˜ we have to consider is the composed functor (cf. A2.2 and A2.4) sc sc rn = rgac(F˜ sc ) ◦ intP : F˜ −→ ac(F˜ ) −→ CC

23.11.2

sc sc which maps any F˜ -object Q on the category ac(F˜ )Q ∩˜ P , and the weak structure (cf. A2.3) we need sc rni = rgiac(F˜ sc ) ◦ intP : F˜ −→ CC

23.11.3

sc sc sc maps any F˜ -object Q on the subcategory ac(F˜ )iQ ∩˜ P of ac(F˜ )Q ∩˜ P with sc the same objects and only with the ac(F˜ )Q ∩ P -morphisms deﬁned by an sc injective map and F˜ -isomorphisms (cf. 1.7 and 6.2). sc 23.12 The point is that, for any F˜ -morphism ϕ˜ : R → Q , the functor (cf. 1.7) sc sc sc ac(F˜ )ϕ˜ ∩˜ id : ac(F˜ )R ∩˜ P −→ ac(F˜ )Q ∩˜ P 23.12.1 P

admits a right adjoin a sc sc sc : ac(F˜ )Q ∩˜ P −→ ac(F˜ )R ∩˜ P ac(F˜ )ϕ˜ ∩˜ id P

23.12.2

sc

deﬁned by the ac(F˜ )-pull-backs (cf. Proposition 6.21) ˜P R∩ ν ˜ ↑

˜ i ϕ ˜∩ dP −−−−−→

˜P Q∩ ↑ η˜

˜ ψ

−−−−−→

U

23.12.3

T

sc

and by the ac(F˜ )-morphisms between them ˜ ψ

U

& / ˜P ω ˜ R∩ 1ν˜ ν ˜

U

−−→ −→ ˜ ψ

−−→

η˜

T

0 & θ˜ ˜P Q∩ η˜3 T

23.12.4;

sc

23. Narrowing the basic F -locality

415

a sc sc namely, ac(F˜ )ϕ˜ ∩˜ id sends the ac(F˜ )Q ∩˜ P -object η˜ in diagram 23.12.3 P sc sc to the ac(F˜ )R ∩ P -object ν˜ , and the ac(F˜ )Q ∩ P -morphism θ˜ in diagram sc ˜ . Moreover, it is easily checked that 23.12.4 to the ac(F˜ )R ∩˜ P -morphism ω sc a preserves the weak structure ac(F˜ )ϕ˜ ∩˜ id P

a

sc

ac(F˜ )ϕ˜ ∩˜ id

sc sc ac(F˜ )iQ ∩˜ P ⊂ ac(F˜ )iR ∩˜ P

P

23.12.5.

23.13 Consequently, according to A2.14, we have the rni -stable adjoin image functor sc sc (prn,rni )a : Fct (rn F˜ )◦ , Ab −→ Fct (F˜ )◦ , Ab

23.13.1;

sc sc moreover, for any F˜ -object Q , it is clear that the category ac(F˜ )Q ∩˜ P sc Q ∩˜ P ; hence, according has a ﬁnal object, namely the ac(F˜ )-morphism id to A2.17, we still have another functor and a natural map

sc sc orn : Fct (rn F˜ )◦ , Ab −→ Fct (F˜ )◦ , Ab ωrn,rni : orn −→ (prn,rni )a

23.13.2.

sc 23.14 More explicitly, note that for any F˜ -morphism ϕ˜ : R → Q , sc ˜ P → R and Q ∩ ˜ P → Q yield the the structural ac(F˜ )-morphisms R ∩ sc ac(F˜ )-pull-back

sc

ϕ ˜

R ↑

−−−−−→

˜P R∩

−−−−−→

˜ i ϕ ˜∩ dP

Q ↑

23.14.1;

˜P Q∩ Q,sc

˜ P in O hence, for any ac(F˜ )-morphism η˜ : T → Q ∩ — where we are assuming that T ⊂ Q and that the composition of η˜ with the ﬁrst structural map is deﬁned by this inclusion (cf. 23.8) — it follows from the transitivity sc of pull-backs and from Proposition 6.19 that the ac(F˜ )-pull-back 23.12.3 is given by ˜ i ϕ ˜∩ dP ˜P ˜P R∩ −−−−−→ Q∩ (˜ νw )w∈Wη˜ ↑ ↑ η˜ 23.14.2 ˜ ) ( ψ

w w∈Wη ˜ −−−−−−→ T w∈Wη˜ Uw − where, for a choice of ϕ ∈ ϕ˜ , Wη˜ is a set of representatives for the set of double classes ϕ(R)wT ∈ ϕ(R)\Q/T such that Uw = ϕ(R)w ∩ T is F-selfcensc tralizing, ψ˜w : Uw → T is the F˜ -morphism induced by the inclusion, and ν˜w is deﬁned by the F-morphism Uw → R sending ϕ(u)w ∈ Uw to u ∈ R and by sc ˜P →P. the composition of η˜ ◦ ψ˜w with the structural F˜ -morphism Q ∩

416

Frobenius categories versus Brauer blocks 23.15 Moreover, for any contravariant functor b : rn F˜

sc

−→ Ab

23.15.1

sc ∼ η˜ of η , Q) (cf. A2.7), an automorphism σ ˜ : η˜ = and any rn F˜ -object (˜ sc sc Q ) of ˜ ˜ the ac(F )Q ∩˜ P -object η˜ determines the rn F -automorphism (˜ σ , id η , Q) and therefore Aut(˜ η ) acts on b(˜ η , Q) (cf. 23.8.3). In order to compute (˜ (prn,rni )a (b) (Q) (cf. 23.13.1) we have to consider the quotient

' ( ¯ η , Q) = b(˜ b(˜ η , Q) b(˜ η , Q), Aut(˜ η)

23.15.2

sc ˜ P isomorphic to η˜ , and note that, for any ac(F˜ )Q ∩˜ P -object η˜ : T → Q ∩ ¯ η , Q) since any ac(F˜ sc )Q ∩˜ P -isomorphism ¯ η , Q) with b(˜ we may identify b(˜ ¯ η , Q) . ¯ η , Q) ∼ between them determines the same isomorphism b(˜ = b(˜ sc 23.16 Now, for any F˜ -object Q , we claim that

(prn,rni )a (b) (Q) ∼ =

η˜∈O

¯ η , Q) b(˜

23.16.1;

Q,sc

indeed, denoting by (bQ )i the restriction of the contravariant functor bQ to sc the subcategory ac(F˜ )iQ ∩˜ P (cf. A2.15), it is quite clear that the direct limit

¯ b(˜ η , Q) where η˜ runs over a set lim(bQ )i is a quotient of the direct sum η˜

−→

sc

of representatives for the set of isomorphism classes of ac(F˜ )Q ∩ P -objects; sc then, for any ac(F˜ )Q ∩˜ P -object

η˜ :

˜P Ti −→ Q ∩

23.16.2,

i∈I sc it suﬃces to consider the ac(F˜ )iQ ∩˜ P -morphisms obtained from the obvi

sc ous ac(F˜ )-morphisms Ti → i ∈I Ti where i runs over I , to easily get isomorphism 23.16.1. sc 23.17 Similarly, for any F˜ -morphism ϕ˜ : R → Q , choose ϕ ∈ ϕ˜ and a set of representatives Wη˜ for the set of double classes ϕ(R)wT ∈ ϕ(R)\Q/T sc such that Uw = ϕ(R)w ∩ T is F-selfcentralizing; then, for any ac(F˜ )-morQ,sc ˜ P in O phism η˜ : T → Q ∩ , with the notation in 23.14 it is easily checked from deﬁnition A2.16.2 that the group homomorphism

˜ : (prn,rni )a (b) (ϕ)

Q,sc

η˜∈O

¯ η , Q) −→ b(˜

ν ˜∈O

R,sc

¯ ν , R) b(˜

23.17.1

sc

23. Narrowing the basic F -locality

417

¯ η , Q) on maps ¯b ∈ b(˜ (prn,rni )a (b) (ϕ) ˜ (b) = ˜ (b) b(ψ˜w , ϕ)

23.17.2

w∈Wη˜

where b ∈ b(˜ η , Q) is a representative for ¯b and b(ψ˜w , ϕ) ˜ (b) is the image sc ¯ νw , R) of b(ψ˜w , ϕ) in b(˜ ˜ (b) ∈ b(˜ νw , R) . Finally, for any F˜ -object Q , it follows from our deﬁnition in A2.17 that we have the equality Q ∩˜ P , Q) orn (b) (Q) = b(id 23.17.3 and a natural group homomorphism (cf. 23.13.2) Q ∩˜ P , Q) −→ ωrn,rni (b)Q : b(id

¯ η , Q) b(˜

23.17.4

Q,sc

η˜∈O

Q ) : (˜ Q ∩˜ P , Q) when determined by the rn F˜ -morphisms (˜ η , id η , Q) → (id sc

Q,sc

η˜ runs over O

.

23.18 In order to state the announced vanishing cohomology criterion, we have to consider the functor sc sc im : rgac(F˜ sc ) ac(F˜ ) −→ rgac(F˜ sc ) ac(F˜ )

23.18.1

sc deﬁned as follows. Recall that an rgac(F˜ sc ) ac(F˜ )-object is a pair (˜ η , Q)

sc sc formed by an ac(F˜ )-object Q = Qi and an ac(F˜ )-morphism (cf.

i∈I

A2.7) η˜ : T =

T6 −→ Q =

Qi

23.18.2

i∈I

6∈L

sc given by a map g : L → I and by F˜ -morphisms η˜6 : T6 → Qg(6) where D runs over L (cf. 6.2); then, the functor im maps (˜ η , Q) on the pair formed by

sc sc the ac(F˜ )-object Q = i∈g(L) Qi and by the ac(F˜ )-morphism η˜ : T → Q sc given by the induced map and the same F˜ -morphisms. Similarly, the functor sc im maps any rg ˜ sc ac(F˜ )-morphism

ac(F

)

˜ ϕ) (ψ, ˜ : (˜ ν , R) −→ (˜ η , Q)

23.18.3

sc

on the rgac(F˜ sc ) ac(F˜ )-morphism ˜ ϕ˜ ) : (˜ (ψ, ν , R ) −→ (˜ η , Q )

23.18.4

sc where ϕ˜ is given by the induced map and the same F˜ -morphisms. Note that we have an evident natural map

ι : im → idrg

˜ ac(F

sc

˜ sc ) ac(F

)

23.18.5

418

Frobenius categories versus Brauer blocks

and that we still have im ◦ im = im and ι ∗ im = idim = im ∗ ι

23.18.6;

moreover, recall that, according to Proposition A2.20, we have a functor sc sc idrn intP : rn F˜ −→ rgac(F˜ sc ) ac(F˜ )

23.18.7.

23.19 To prove our announced vanishing cohomology criterion, we still sc sc need the following notation. For any F˜ -chain q : ∆n → F˜ (cf. A2.8), let sc sc us denote by Xq the set of pairs (µ, q ) formed by an F˜ -chain q : ∆n → F˜ q (i)

such that q (i − 1) ⊂ q (i) and that q (i−1 • i) = ˜ιq (i−1) for any 1 ≤ i ≤ n , for and by a natural map µ : q → q such that ˜ιP belongs to F˜ P, q (i) q (i)

µ ˜i

any i ∈ ∆n (cf. 6.4.1); then, if u ∈ P , we denote by (µ, q )u the element sc of Xq formed by the F˜ -chain and by the natural map respectively sending i ∈ ∆n to (cf. 1.8) qu (i) = q (i)u Recall that we denote by (cf. A3.17).

and

(µu )i = µ ˜i ◦ κ ˜ q (i),q (i)u (u)

sc Hn∗ (F˜ , •)

23.19.1.

sc the stable n-cohomology groups of F˜

Theorem 23.20 With all the notation above, choose b = a◦im◦(idrnintP ) for a contravariant functor sc a : rgac(F˜ sc ) ac(F˜ ) −→ Ab sc Then, for any n ≥ 1 we have Hn∗ F˜ , (prn,rni )a (b) = {0} .

23.20.1.

sc sc Proof: For any F˜ -chain q : ∆n → F˜ and any pair (µ, q ) in the set Xq sc sc deﬁned above, let us consider the F˜ -chain qP : ∆n+1 → F˜ deﬁned by

n qP ◦ δn+1 = q

,

qP (n + 1) = P

and qP (n • n+1) = ˜ιP q (n) 23.20.2

sc sc and moreover, for any D ∈ ∆n , the F˜ -chain hn6 (µ) : ∆n+1 → F˜ deﬁned ˆ = (prn,rni )a (b) and let b = {bq } in Lemma A4.2. Set b ˜ sc ) be a q∈Fct(∆n ,F ˆ stable b-valued n-cocycle (cf. A3.18); thus, considering the components of

the diﬀerential of b labelled by qP and by hn6 (µ) for any D ∈ ∆n , we get (cf. A3.11.2) n ˆ (0•1)) (bqP ◦δn ) + 0 = b(q bqP ◦δin + (−1)n+1 bq 0 i=1 n+1 ˆ (0•1)) (bhn (µ)◦δn ) + 0 = b(q (−1)i bhn (µ)◦δin 0 i=1 n+1 ˆ µ0 ) (bq ) + 0 = b(˜ (−1)i bhn0 (µ)◦δin i=1

if 1 ≤ D ≤ n

23.20.3.

sc

23. Narrowing the basic F -locality

419

Then, setting a(µ,q ),i =

n−1

(−1)6 bhn−1 (µ∗δn−1 ) + (−1)n b(q ◦δn−1 )P i

i

23.20.4

6=0

ˆ q (1) if i = 0 and to b ˆ q (0) if 1 ≤ i ≤ n , it follows which belongs to b from Lemma A4.2 that the “alternating sum” of equalities 23.20.3 yields

n ˆ µ0 ) (bq ) = b(q ˆ (0•1)) (a(µ,q ),0 ) + b(˜ (−1)i a(µ,q ),i

23.20.5;

i=1

indeed, the last terms in the top equality and in the n-th middle equality cancel with each other in the alternating sum; the same happens with the second terms in the ﬁrst middle equality and in the bottom equality, and with the (D + 2)-th terms in the D-th and (D + 1)-th middle equalities. Moreover, since (µ, q ) belongs to Xq , it is quite clear that, for any 0 ≤ i ≤ n , the pair (µ ∗ δin−1 , q ◦ δin−1 ) belongs to Xq◦δn−1 and that we have i

q

P

◦

δin

= (q ◦ δin−1 )P

23.20.6.

sc On the other hand, for any F˜ -object Q , recall that (cf. isomorphism 23.16.1) ˆ ¯ γ , Q) b(Q) = b(˜ 23.20.7 Q,sc

γ ˜ ∈O

Q,sc ˆ and let us denote by xγ˜ the γ˜ -component of x ∈ b(Q) for any γ˜ ∈ O ; thus, equality 23.20.5 becomes

n ˆ µ0 ) (bq )η˜ = b(q ˆ (0•1)) (a(µ,q ),0 )η˜ + b(˜ (−1)i (a(µ,q ),i )η˜

23.20.8

i=1 q (0),sc

for any η˜ ∈ O

. Moreover, setting (cf. Proposition 6.14)

P ) ◦ η˜ : T −→ q (0) ∩ ˜ P −→ q(0) ∩ ˜P ˜ id η˜ = (˜ µ0 ∩

23.20.9,

we claim that Aut(˜ η ) = Aut(˜ η) . Indeed, according to deﬁnition 23.20.9, it is quite clear that Aut(˜ η ) P ˜ is contained in Aut(˜ η ) and, since ˜ιq (0) belongs to F q(0), q (0) µ˜ , from 0 statement 6.5.1 it is not diﬃcult to check that we get the equality; actually, it sc ˜P follows from Lemma 23.21 below that any ac(F˜ )-morphism η˜ : T → q(0) ∩ determines a P -orbit of pairs (q , µ) ∈ Xq such that η˜ factorizes as in 23.20.9 q (0),sc

for a suitable η˜ ∈ O , and that we have ˜ P = η˜ , q (0) = im η˜ , q (0) ∩ ˜P im η˜, q(0) ∩

23.20.10,

420

Frobenius categories versus Brauer blocks

so that η˜ and η˜ mutually determine each other. Consequently, it follows from 23.18.6 that we have (cf. 23.15.2) ¯ η˜, q(0) = b ¯ η˜ , q (0) b

23.20.11

T,µ and that b(id ˜0 ) is equal to the identity on it. ˆ We are ready to show that b is a stable b-valued n-coboundary (cf. A3.18); sc sc ˜ ˜ assuming that for any F -chain q : ∆n → Fsc and any (F˜ )q(0) ∩˜ P -object q(0),sc

˜ P in O such that |P : U | < pm we have (bq )ε˜ = 0 , we ε˜ : U → q(0) ∩ argue by induction on m ∈ N . Denoting by {dn }n∈N the diﬀerential map ˆ ◦ v ˜ sc ) (cf. A3.11 and A3.18), it determined by the functor (pch∗ (F˜ sc ) )∗ (b F ˆ suﬃces to exhibit a stable b-valued n-coboundary dn−1 (a) fulﬁlling (dn−1 (a)q )ε˜ = 0

and

(ˆbq )η˜ − (dn−1 (a)q )η˜ = 0

23.20.12

sc

˜ P such that |P : U | < pm and for any (F˜ )q(0) ∩˜ P -objects ε˜ : U → q(0) ∩ q(0),sc

˜ P such that |P : T | = pm , both in O ; indeed, the eleη˜ : T → q(0) ∩ ˆ ment b − dn−1 (a) is then a stable b-valued n-coboundary by our induction hypothesis. sc sc In order to exhibit a , for any F˜ -chain r : ∆n−1 → F˜ , let us consider ˆ r(0) which, for any (F˜ sc )r(0) ∩ P -object ε˜ : U → r(0) ∩ ˜P the element ar ∈ b r(0),sc

in O

, has the following ε˜-component (cf. 23.20.7 and 23.20.11) (ar )ε˜ =

n−1

(−1)6 (bhn−1 (ν) )ε˜ + (−1)n (brP )ε˜

23.20.13,

6=0

˜ P are respectively a pair in Xr and an where (ν, r ) and ε˜ : U → r (0) ∩ sc ˜ (F )r (0) ∩ P -object determined, up to a P -action, by ε˜ (cf. Lemma 23.21 ε) and, for any 0 ≤ i ≤ n , below); in particular, we have Aut(˜ ε ) = Aut(˜ n−1 n−1 considering the pair (µ ∗ δi , q ◦ δi ) , we get (cf. 23.20.4) (aq◦δn−1 )ε˜ = (a(µ,q ),i )ε˜

23.20.14.

a = {ar }r∈Fct(∆n−1 ,F˜ sc )

23.20.15;

i

Finally, we set note that our deﬁnition does not depend on the choice of the pair (ν, r ) in its P -orbit since, for any u ∈ P , considering (ν, r )u = (ν ◦ κu , ru ) sc (ν ◦ κu ) are re(cf. 23.19), the corresponding F˜ -chains (ru )P and hn−1 6 sc spectively ch∗ (F˜ )-isomorphic to rP and to hn−1 (ν) for any D ∈ ∆n−1 , and 6 then it suﬃces to quote the stability of b (cf. A3.18).

sc

23. Narrowing the basic F -locality

421

sc We claim that the family a is also stable by ch∗ (F˜ )-isomorphisms; insc sc sc deed, if ¯r : ∆n−1 → F˜ is an F˜ -chain ch∗ (F˜ )-isomorphic to r and λ : r ∼ = ¯r sc sc ∗ ˜ ˜ is a ch (F )-isomorphism then, for the (F )¯r(0) ∩˜ P -object

˜0 ∩ P ) ◦ ε˜ : U −→ r(0) ∩ ˜ id ˜ P −→ ¯r(0) ∩ ˜P (λ

23.20.16,

˜0 ∩ P ) ◦ ε˜-component of a¯r from the pair (λ ◦ ν, r ) ; ˜ id we can deﬁne the (λ sc ˜ in this case, the F -chain rP does not change and, for any D ∈ ∆n−1 , the sc sc corresponding F˜ -chain hn−1 (λ ◦ ν) is ch∗ (F˜ )-isomorphic to hn−1 (ν) ; it 6 6 suﬃces again to quote the stability of b . sc sc sc ˜ P an (F˜ )q(0) ∩˜ P Let q : ∆n → F˜ be an F˜ -chain and ε˜ : U → q(0) ∩ object such that |P : U | < pm ; by the very deﬁnition of the diﬀerential map, we have (cf. A3.11) n ˆ (dn−1 (a)q )ε˜ = b(q(0•1)) (aq◦δn−1 )ε˜ + (−1)i (aq◦δn−1 )ε˜ 0

i

i=1

23.20.17.

But, according to equality 23.20.14 and to deﬁnition 23.20.13, we obtain (aq◦δn−1 )γ˜ = (a(µ,q ),0 )γ˜ = 0 0

23.20.18

(aq◦δn−1 )ε˜ = (a(µ,q ),i )ε˜ = 0 i

sc

˜ P and any 1 ≤ i ≤ n ; moreover, it for any (F˜ )q(1) ∩˜ P -object γ˜ : U → q(1) ∩ is clear that ˆ ˆ b(q(0•1)) (aq◦δn−1 )ε˜ = b(q(0•1)) (aq◦δn−1 )β˜ ε˜ 23.20.19 0

0

β˜

˜ P runs over the set of elements of O where β˜ : V → q(1) ∩ such that |U | ≤ |V | (cf. 23.17). Consequently, this sum is also equal to zero and we get indeed (dn−1 (a)q )ε˜ = 0 . sc ˜ P be an (F˜ )q(0) ∩˜ P -object such On the other hand, let η˜ : T → q(0) ∩ q(1),sc

that |P : T | = pm ; once again, it follows from Lemma 23.21 below that η˜ sc determines a P -orbit of pairs (µ, q ) ∈ Xq and also an (F˜ )q (0) ∩˜ P -object P ) ◦ η˜ . As above, we have ˜ id ˜ P such that η˜ = (µ0 ∩ η˜ : T → q (0) ∩ ˆ µ0 ) (bq )η˜ = ˆ µ0 ) (bq,˜γ )η˜ b(˜ 23.20.20 b(˜ q(0),sc

γ ˜ ∈O

q(0),sc

˜ P runs over the set of elements in O where γ˜ : V → q(0) ∩ admitting an sc sc ˜ ˜ ˜ F -morphism ψ : T → V such that the following ac(F )-diagram is commutative (cf. 23.17) γ ˜ ˜P V −→ q(0) ∩ ˜ ψ ↑ ↑ µ˜0 ∩˜ idP 23.20.21;

T

η˜ ˜P −→ q (0) ∩

422

Frobenius categories versus Brauer blocks

consequently, all the terms in this sum are equal to zero unless ψ˜ is an sc F˜ -isomorphism and then we necessarily have V = T and γ˜ = η˜ ; thus, we get (cf. 23.20.11) ˆ µ0 ) (bq )η˜ = (bq )η˜ b(˜ 23.20.22. Moreover, according to Lemma 23.21 below, for any 1 ≤ i ≤ n , the P -orbit of (µ ∗ δin−1 , q ◦ δin−1 ) in Xq◦δn−1 actually coincides with the P -orbit i

determined by η˜ , whereas the composition q(0•1) ◦ η˜ determines the P -orbit sc of (µ ∗ δ0n−1 , q ◦ δ0n−1 ) in Xq◦δn−1 , together with the (F˜ )q (1) ∩˜ P -object 0

˜P q(0•1) ◦ η˜ : T −→ q (1) ∩

23.20.23;

hence, arguing as in 23.20.19 above, we still have (cf. 23.20.14) ˆ b(q(0•1)) (aq◦δn−1 )η˜ = (aq◦δn−1 )q(0•1)◦˜η = (a(µ,q ),0 )q(0•1)◦˜η 0 0 ˆ (0•1)) (a(µ,q ),0 )η˜ = b(q

23.20.24.

In conclusion, from equalities 23.20.8, 23.20.14, 23.20.22 and 23.20.24, we get ˆ µ0 ) (bq )η˜ (bq )η˜ = b(˜ n ˆ (0•1)) (a(µ,q ),0 )η˜ + = b(q (−1)i (a(µ,q ),i )η˜ i=1 n ˆ = b(q(0•1)) (aq◦δn−1 )η˜ + (−1)i (aq◦δn−1 )η˜

0

i=1

23.20.25

i

= (dn−1 (b)q )η˜ which completes our induction argument. We are done. sc sc Lemma 23.21 With the notation above, for any F˜ -chain q : ∆n → F˜ we have ˜P ∼ q(0) ∩ q (0) 23.21.1 =

(µ,q )

where the pair (µ, q ) runs over a set of representatives in Xq for the set of sc ˜ P , there P -orbits. In particular, for any (F˜ )q(0) ∩˜ P -object η˜ : T → q(0) ∩ is a unique P -orbit of pairs (µ, q ) in Xq such that η˜ factorizes throughsc ˜ P determined by µ out the ac(F˜ )-morphism q (0) → q(0) ∩ ˜0 : q (0) → q(0) P and ˜ιq (0) : q (0) → P . Proof: Recall that, for any F-selfcentralizing subgroup Q of P , we may assume that (cf 23.17.2) ˜P = Q∩ Q 23.21.2 ˜Q Q ∈S˜Q δ˜ ∈M Q

sc

23. Narrowing the basic F -locality

423

where S˜Q is a set of representatives for the set of Q-conjugacy classes of the F-selfcentralizing subgroups Q of Q such that (cf. 6.4.1) ˜ F(P, Q )˜ιQ = ∅

23.21.3

Q

˜ Q is a set of representatives for the set of F˜Q (Q )and, for any Q ∈ S˜Q , M Q ˜ ˜ orbits in F(P, Q )˜ιQ ; but, according to statement 6.5.2, δ˜ ∈ F(Q, Q )˜ιP Q

Q

˜ ˜ sc is equivalent to ˜ιP Q ∈ F(P, Q )δ˜ and we know that an ac(F )-morphism sc ˜ P is deﬁned by an F˜ -morphism η˜ : T → Q for some Q ∈ S˜Q η˜ : T → Q ∩ ˜ Q (cf. 6.2). This proves the lemma for n = 0 . and some δ ∈ M Q

Assume that n ≥ 1 ; for any (µ, q ) ∈ Xq it is clear that (µ ∗ δ0n , q ◦ δ0n ) still belongs to Xq◦δ0n ; but, for any (ν, r ) ∈ Xq◦δ0n , by deﬁnition 6.18.2, we have ˜ q(1) r (0) ∼ q(0) ∩ R 23.21.4 = r (0)

(˜ ε ,R ,˜ ιR

)

where R runs over a set of representatives for the set of r (0)-conjugacy classes of F-selfcentralizing subgroups of r (0) such that r (0) F˜ q(0), R ˜ιr (0) = ∅ and q(0 • 1) ◦ γ˜ = ν˜0 ◦ ˜ιR R

23.21.5

for some γ˜ ∈ F˜ q(0), R ˜ιr (0) , and, for such a R , ε˜ runs over a set of R representatives for F˜ q(0), R ˜ιr (0) F˜r (0) (R ) fulﬁlling R

r (0)

q(0 • 1) ◦ ε˜ = ν˜0 ◦ ˜ιR

23.21.6.

r (0)

Moreover, for such a triple (˜ ε , R , ˜ιR ) , it is easily checked that the pair sc (µ, q ) formed by the functor q : ∆n → F˜ deﬁned by q (0) = R

,

q ◦ δ0n = r

r (0)

and q (0 • 1) = ˜ιR

23.21.7

and by the natural map µ : q → q fulﬁlling µ ˜0 = ε˜

and µ ∗ δ0n = ν

23.21.8,

belongs to Xq . On the other hand, arguing by induction on n , we may assume that ˜P ∼ q(1) ∩ =

(ν,r )

q (1)

23.21.9

424

Frobenius categories versus Brauer blocks

where the pair (ν, r ) runs over a set of representatives in Xq◦δ0n for the set of P -orbits; but, it easily follows from Propositions 6.14 and 6.21 that we have ˜P ∼ ˜ q(1) q(1) ∩ ˜ q(1) q (1) ˜P = q(0) ∩ q(0) ∩ 23.21.10 = q(0) ∩ (ν,r )

where (ν, r ) runs over the same set of representatives; consequently, it suﬃces to apply isomorphism 23.21.4 to any summand, and then the argument above proves isomorphism 23.21.1. sc Finally, for any F-selfcentralizing subgroup T of P and any ac(F˜ )-mor˜ P , it follows from our induction hypothesis that the phism η˜ : T → q(0) ∩ composed morphism P ◦ η˜ : T −→ q(0) ∩ ˜ id ˜ P −→ q(1) ∩ ˜P q(0•1) ∩ 23.21.11 determines the P -orbit of a suitable pair (ν, r ) in isomorphism 23.21.9 above sc P ◦ η˜ has the an˜ id in such a way that the (F˜ )q(1) ∩˜ P -object q(0 • 1) ∩ nounced factorization; but, η˜ clearly determines a term of the sum in the right-hand member of equality 23.21.10 (cf. 6.2), and then a term of the sum in the right-hand member of the corresponding isomorphism 23.21.4 (cf. 6.2); once again, by the argument above we get the announced factorization. We are done. 23.22 We are ready to exhibit our announced description of the functor ˆ˜z (cf. Proposition 23.10). Setting ˜z = ˜zF˜ sc (cf. 13.18) and considering the covering functor (cf. A2.18) sc sc cac(F˜ sc ) : rgac(F˜ sc ) ac(F˜ ) −→ ac(F˜ )

23.22.1

sc and the natural transformation (intP , idrn ) from the pair (F˜ , rn) to the pair sc ac(F˜ ), rgac(F˜ sc ) (cf. A2.19), we get the contravariant functor (cf. 23.18.7 and A4.10)

c

sc

˜ ) sc id intP sc sc ac(˜ z) ac(F rn F˜ −−rn −−−−→ rgac(F˜ sc ) ac(F˜ ) −−−−−→ ac(F˜ ) −−−→ Ab

and then it is easy to check that ˆ ˜z = (prn,rni )a ac(˜z) ◦ cac(F˜ sc ) ◦ (idrn intP )

23.22.2

23.22.3.

Moreover, it follows from our deﬁnition of im (cf. 23.18) that we have cac(F˜ sc ) ◦ im = cac(F˜ sc ) and therefore we still have ˆ ˜z = (prn,rni )a ac(˜z) ◦ cac(F˜ sc ) ◦ im ◦ (idrn intP )

23.22.4

23.22.5,

sc

23. Narrowing the basic F -locality

425

so that Theorem 23.20 guarantees that the positive stable ˆ˜z-valued cohomology sc of F˜ vanish. But, for instance, in the last member of these equalities we can replace cac(F˜ sc ) by pac(F˜ sc ) (cf. A2.18) obtaining a new vanishing result. 23.23 More precisely, we are especially interested in the vanishing result corresponding to the following quotient of ac(˜z)◦cac(F˜ sc ) . Consider the natural map deﬁned in A2.14 κac(F˜ sc ) : cac(F˜ sc ) −→ pac(F˜ sc )

23.23.1

which determines a new natural map ac(˜z) ∗ κac(F˜ sc ) : ac(˜z) ◦ pac(F˜ sc ) −→ ac(˜z) ◦ cac(F˜ sc )

23.23.2,

and, setting ac(˜z) ◦ pac(F˜ sc ) = Im ac(˜z) ∗ κac(F˜ sc ) , we consider the quotient sc (ac(˜z) ◦ cac(F˜ sc ) ) (ac(˜z) ◦ pac(F˜ sc ) ) : rgac(F˜ sc ) ac(F˜ ) −→ Ab

23.23.3.

In order to simplify notation, let us set sc m = ac(˜z) ◦ cac(F˜ sc ) ◦ (idrn intP ) : rn F˜ −→ Ab

23.23.4,

sc n = (ac(˜z) ◦ pac(F˜ sc ) ) ◦ im ◦ (idrn intP ) : rn F˜ −→ Ab

˜z = (prn,rni )a (m) and we will consider ˜y = (prn,rni )a (n) ; then so that we have ˆ Theorem 23.20 still guarantees that, for any n ≥ 1 , we have sc sc ˜z/˜y) = {0} = Hn∗ (F˜ , ˜y) Hn∗ (F˜ , ˆ

23.23.5.

sc

Corollary 23.24 With the notation above, there exists an F -sublocality d,sc c,sc (τ d,sc , L , π d,sc ) of L , unique up to natural isomorphisms, such that Ker(π d,sc ) = ˜y ◦ π ˜ c,sc

23.24.1

c,sc sc where π ˜ c,sc : L → F˜ denotes the composition of π c,sc with the canonical sc sc functor F → F˜ . sc

c,sc

Proof: We consider the evident quotient F -locality L that the canonical functor c,sc

π ¯ c,sc : L¯

c,sc

=L

/(˜y ◦ π ˜ c,sc ) −→ F

/(˜y ◦ π ˜ c,sc ) and claim

sc

23.24.2

426

Frobenius categories versus Brauer blocks

admits a functorial section s , unique up to natural isomorphisms; then, the c,sc sc converse image in L of the image of s supplies the announced F -subd,sc locality L . The existence and the uniqueness of s is a consequence of the equalities (cf. 23.23.5) sc

sc

˜z/˜y) = {0} = H1∗ (F˜ , ˆ˜z/˜y) H2∗ (F˜ , ˆ

23.24.3.

Indeed, ﬁrst of all note that the obvious inclusion Z(Q) ⊂ CG (Q) for sc sc any F˜ -object Q , determines injective natural maps from ˜z to ˜k and to ˆ˜z (cf. Proposition 23.10), and that its image in ˆ˜z is contained in ˜y ; morec,sc over, since Ker(πQ ) is an Abelian p-group, Proposition 18.21 applies to the sc

c,sc

F -locality L

and in particular we have (cf. 18.23.1) c,sc ˜z/˜y)(Q) F(Q) L¯ (Q) ∼ = (ˆ

23.24.4.

Consequently, it follows from 18.27 that there exists a correspondence which sc c,sc maps any F -morphism ϕ : R → Q on an element x ¯ϕ ∈ L¯ (Q, R) in such a way that, for any F-isomorphisms σ : Q ∼ = Q and τ : R ∼ = R , we have x ¯σ◦ϕ◦τ −1 = σ·¯ xϕ ·τ −1

23.24.5

c,sc where F(Q) and F(R) act on L¯ (Q, R) via isomorphisms 23.23.4.

Now, it follows from Proposition 18.28 that the correspondence mapping sc any pair of F -morphisms ψ : T → R and ϕ : R → Q on the element z˜ϕ,ψ ˜z/˜y)(T ) such that of (ˆ x ¯ϕ ·¯ xψ = x ¯ϕ◦ψ ·˜ zϕ,ψ 23.24.6 sc ˜z/˜y-valued 2-cocycle over F˜ , which is a 2-coboundary determines a stable ˆ sc ˜z/˜y) = {0} ; as in Proposition 18.28, this easily proves the since H2∗ (F˜ , ˆ existence of the announced functorial section s . sc c,sc Similarly, if we have two functorial sections s and s from F to L¯ , it follows part of Proposition 18.21 that we may assume from theuniqueness sc that s F(Q) = s F(Q) for any F -object Q . Then, it follows from Proposc sition 18.29 that the correspondence mapping any F -morphism ϕ : R → Q ˜z/˜y)(T ) fulﬁlling on the element D˜ϕ ∈ (ˆ

s (ϕ) = s(ϕ)·D˜ϕ

23.24.7

sc ˜z/˜y-valued 1-cocycle over F˜ , which is a 1-coboundary determines a stable ˆ sc ˜z/˜y) = {0} ; as in Proposition 18.29, this easily proves that s since H1∗ (F˜ , ˆ and s are naturally isomorphic. We are done.

sc

23. Narrowing the basic F -locality sc

427 d,sc

23.25 This F -locality (τ d,sc , L , π d,sc ) still can be “narrowed” till the sc r,sc reduced F -locality (τ r,sc , L , π r,sc ) below where the contravariant functor sc orn (n) = ac(˜z) ◦ intP : F˜ −→ Ab

23.25.1

sc determines the kernel of π r,sc . For any F˜ -object Q , recall that (cf. 23.7)

˜P = intP (Q) = Q ∩

Q

23.25.2

˜Q Q ∈S˜Q η˜ ∈M Q

˜ Q determine a maximal (F˜ sc )Q ∩˜ P -oband that any Q ∈ S˜Q and any η˜ ∈ M Q ˜ P (cf. 23.7.3), getting a bijection ject θ˜Q ,˜η : Q → Q ∩

Q,sc ˜ Q ∼ M Q = Omax

23.25.3

Q ∈S˜Q

which, with evident notation, allows us to write

˜P = intP (Q) = Q ∩

Qθ˜

23.25.4;

Q,sc ˜ θ∈O max

Q,sc Q,sc ˜ P in Omax moreover, any η˜ ∈ O determines an element θ˜η˜ : Qθ˜η˜ → Q ∩ and, in particular, we have (cf. 23.18)

˜ P) Qθ˜η˜ = pac(F˜ sc ) im(˜ η, Q ∩

23.25.5.

Q,sc

˜ P in O Hence, for any η˜ : T → Q ∩ , setting Z(˜ η ) = Z(T ) and (cf. deﬁnition 23.15.2) η ˜ Z(θ˜η˜) = Z(θ˜η˜) Z(θ˜η˜) ∩ [Z(T ), Aut(˜ η )] 23.25.6, ˜ = {id Q } for any θ˜ ∈ OQ,sc (cf. 23.8) we get (cf. 23.16.1) since Aut(θ) max ˜ θ ˜ P) = Z(Q ∩

˜ and ˜y(Q) = Z(θ)

Q,sc ˜ θ∈O max

η ˜

Z(θ˜η˜)

23.25.7

Q,sc

η˜∈O

and the natural map in 23.17.4 applied to n gives a natural map sending Q to the group homomorphism ωr,ri (n)Q :

Q,sc ˜ θ∈O max

˜ −→ Z(θ)

Q,sc

η˜∈O

η ˜

Z(θ˜η˜)

23.25.8

428

Frobenius categories versus Brauer blocks η ˜

determined by all the canonical homomorphisms Z(θ˜η˜) → Z(θ˜η˜) ; it is clear Q,sc that the projection over the factors in Omax deﬁnes a section of ωrn,rni (n)Q but these sections need not deﬁne a natural section of ωrn,rni (n) ; however, we have the following result. Proposition 23.26 With the notation above, for any n ∈ N the corresponsc dence sending any F˜ -object Q to the k-module homomorphism n

n

˜ P )p k ⊗Z ˜y(Q)p −→ k ⊗Z Z(Q ∩ Q,sc

Q,sc

deﬁned by the projection over the factors in Omax ⊂ O n of k ⊗Z ωrn,rni (np ) .

23.26.1 is a natural section

sc Proof: It suﬃces to prove that, for any F˜ -morphism ϕ˜ : R → Q , any η˜ in Q,sc Q,sc O − Omax and any n ∈ N , ˜y(ϕ) ˜ = (prn,rni )a (n) (ϕ) ˜ (cf. 23.23) maps the η ˜ n p ˜ elements of Z(θη˜) on the elements of the product

n+1

Z(P) ˜ p

R,sc M∈O ˜ max

×

R,sc

ν ˜∈O

Z(P ˜ ν˜ )pn

ν ˜

23.26.2

R,sc −Omax

R,sc

R,sc

where P ˜ ν˜ denotes the element of Omax determined by ν˜ ∈ O . pn ˜ ˜ But it follows from 23.14 and 23.17 that, for any z ∈ Z(θη˜) , y(ϕ) ˜ maps η ˜

its image z¯ in Z(θ˜η˜)pn on

(prn,rni )a (n) (ϕ) ˜ (¯ z) = ˜ (z) n(ψ˜w , ϕ)

23.26.3

w∈Wη˜

ν ˜ where n(ψ˜w , ϕ) ˜ (z) denotes the image in Z(P ˜ ν˜ )pn of n(ψ˜w , ϕ) ˜ (z) ; moreover, for any w ∈ Wη˜ , it follows from our deﬁnition of n (cf. 23.23.4) that, sc P ) between ˜ id denoting by ϕ˜w the F˜ -morphism determined by im(ψ˜w , ϕ˜ ∩ the origins of P ˜ ν˜w and θ˜η˜ , and setting Z(ϕ˜w ) = ac(˜z) (ϕ˜w ) , we have n(ψ˜w , ϕ) ˜ = Z(ϕ˜w ) : Z(θ˜η˜) −→ Z(P ˜ ν˜w )

23.26.4.

On the other hand, respectively denoting by T and Q the origins of η˜ ¯Q (T ) (cf. 23.8) and therefore, since η˜ ∈ OQ,sc , η) ∼ and θ˜η˜ , recall that Aut(˜ =N max we have T = Q and in particular Aut(˜ η ) is a nontrivial p-group. Moreover, with our notation in 23.14, it is clear that NQ (T ) acts on the set of double classes ϕ(R)\Q/T by the multiplication on the right and therefore it acts on the set of representatives Wη˜ ; but, for any w ∈ Wη˜ and any u ∈ NQ (T ) , since Uwu = ϕ(R)wu ∩ T = (Uw )u 23.26.5,

sc

23. Narrowing the basic F -locality R,sc

it is clear that the elements of O ˜P ν˜wu : Uwu −→ R ∩

429 sc which are (F˜ )R ∩˜ P -isomorphic to

˜P or ν˜w : Uw −→ R ∩

23.26.6

determine the same element P ˜ ∈ Omax ; ﬁnally, denoting by R the origin of P ˜ , we may assume that ϕ(R )w ⊂ Q and that the corresponding group homomorphism determines ϕ˜w ; consequently, since u acts trivially on Z(θ˜η˜) = Z(Q ) , we get Z(ϕ˜wu ) = Z(ϕ˜w ) . ¯Q (T ) in Wη˜ and any w ∈ O , In conclusion, for any orbit O of Aut(˜ η) ∼ =N we actually get ν˜ ν˜w Z(ϕ˜w ) (z) w = |O|· Z(ϕ˜w ) (z) 23.26.7 R,sc

w ∈O ν ˜w

and therefore if O = {w} then this sum belongs to Z(R )pn+1 ; thus, if NQ (T ) has no ﬁxed points in Wη˜ , we are done. But, if NQ (T ) ﬁxes some w ∈ Wη˜ then, for any u ∈ NQ (T ) , we have wu = ϕ(v)wt for suitable v ∈ R and t ∈ T , and therefore u t−1 belongs to ϕ(R)w ∩ Q = ϕ(R )w

23.26.8

and normalizes ϕ(R)w ∩ T = Uw , namely u t−1 determines an element ¯ϕ(R )w (Uw ) ; ﬁnally, we get a group homomorphism of Aut(˜ νw ) ∼ =N Aut(˜ η ) −→ Aut(˜ νw )

23.26.9

which is injective by Corollary 4.9 and, in particular, it forces Aut(˜ νw ) = {1} , R,sc so that ν˜w ∈ Omax . We are done. Corollary 23.27 With the notation above, for any m ∈ N and any n ≥ 1 , we have sc m Hn∗ F˜ , k ⊗Z (ac(˜z) ◦ intP )p = {0} 23.27.1. sc m m Hn∗ F˜ , k ⊗Z ˜yp /(ac(˜z) ◦ intP )p = {0} Proof: It follows from Proposition 23.26 that, for any m ∈ N , we have m m m m 23.27.2; k ⊗Z ˜yp ∼ = k ⊗Z (ac(˜z) ◦ intP )p ⊕ k ⊗Z ˜yp /(ac(˜z) ◦ intP )p but, according to Theorem 23.20 applied to the contravariant functor pm sc k ⊗Z ac(˜z) ◦ pac(F˜ sc ) : rgac(F˜ sc ) ac(F˜ ) −→ k-mod

23.27.3,

for any n ≥ 1 we still have (cf. 23.23) sc

m

Hn∗ (F˜ , k ⊗Z ˜yp ) = {0} hence, we get equalities 23.27.1.

23.27.4;

430

Frobenius categories versus Brauer blocks sc

Corollary 23.28 With the notation above, there exists an F -sublocality r,sc d,sc (τ r,sc , L , π r,sc ) of L , unique up to natural isomorphisms, such that Ker(π r,sc ) = ac(˜z) ◦ intP ◦ π ˜ d,sc

23.28.1

d,sc sc where π ˜ d,sc : L → F˜ denotes the composition of π d,sc with the canonical sc sc functor F → F˜ .

Proof: The proof follows the same pattern as the proof of Corollary 23.24 except that the graded equalities 23.27.1 force us to argue by induction on m . d,sc d,sc Set (τ0d,sc , L0 , π0d,sc ) = (τ d,sc , L , π d,sc ) , a˜0 = ac(˜z) ◦ intP and ˜y0 = ˜y ; for any 1 ≤ i ≤ m , inductively deﬁne a˜i and ˜yi as the respective p-th powers sc of a˜i−1 and ˜yi−1 and assume that there exists a coherent F -sublocality d,sc d,sc d,sc d,sc (τid,sc , Li , πid,sc ) of (τi−1 , Li−1 , πi−1 ) fulﬁlling (˜yi + a˜0 ) ◦ π ˜id,sc = Ker(πid,sc ) and ˜yi ∩ a˜0 = a˜i

23.28.2.

Then, considering the obvious surjective functor d,sc d,sc d,sc d,sc rm : Lˆm = Lm (˜ym+1 + a˜m ) ◦ π −→ Lm /(˜ym ◦ π ˜m ˜m ) = Lm it is clear that

d,sc Ker(rm ) = ˜ym /(˜ym+1 + a˜m ) ◦ π ˜m

23.28.3,

23.28.4.

˜ym = ˜ym /(˜ym+1 + a˜m ) , it follows from Corollary 23.27 that, But, setting ¯ for any n ≥ 1 , we have sc ˜ym ) = {0} Hn∗ (F˜ , ¯ 23.28.5. At this point, we claim that the particular equalities sc sc ˜ym ) = {0} = H1∗ (F˜ , ¯˜ym ) H2∗ (F˜ , ¯

23.28.6

force the existence of a functorial section sm of rm , unique up to natural d,sc isomorphisms; then, the converse image in Lm of the image of sm in Lˆm sc d,sc d,sc d,sc d,sc d,sc d,sc deﬁnes a coherent F -sublocality (τm+1 , Lm+1 , πm+1 ) of (τm , Lm , πm ) fulﬁlling condition 23.28.2. sc Indeed, for any F˜ -object Q , since Ker(π d,sc ) is an Abelian p-group, Q

sc Proposition 18.21 applies to the F -locality Lˆm and in particular we get

˜ym (Q) Lm (Q) Lˆm (Q) ∼ =¯

23.28.7.

Consequently, arguing as in 18.23, 18.24, 18.25, 18.26 and 18.27 above, it is not diﬃcult to prove that there exists a correspondence which maps any

sc

23. Narrowing the basic F -locality

431

Lm -morphism x : R → Q on an element x ˆ ∈ Lˆm (Q, R) in such a way that, ∼ for any Lm -isomorphisms s : Q = Q and t : R ∼ = R , we have - = s·ˆ s·x·t x·t

23.28.8

where Lm (Q) and Lm (R) act on Lˆm (Q, R) via isomorphisms 23.27.7. Now, arguing as in Proposition 18.28 it is easy to check that the correspondence mapping any pair of Lm -morphisms y : T → R and x : R → Q ˜ym (T ) such that on the element a ˜x,y of ¯ ax,y x ˆ·ˆ y = x·y·˜

23.28.9

sc ˜ym -valued 2-cocycle over L˜m = F˜ (cf. 1.3), which is a determines a stable ¯ sc ˜ym ) = {0} ; as in Proposition 18.28, this easily 2-coboundary since H2∗ (F˜ , ¯ proves the existence of the announced functorial section sm . Similarly, if we have two functorial sections sm and sm from Lm to Lˆm , it follows from the uniqueness part of Proposition 18.21 that we may assume that sm Lm (Q) = sm Lm (Q) 23.28.10

for any Lm -object Q . Then, arguing as in Proposition 18.29 it is easily checked that the correspondence mapping any Lm -morphism x : R → Q on ˜ym (T ) fulﬁlling the element ˜bx ∈ ¯ sm (x) = sm (x)·˜bx

23.28.11

sc ˜ym -valued 1-cocycle over L˜m = F˜ (cf. 1.3), which is a determines a stable ¯ sc ˜ym ) = {0} ; as in Proposition 18.29, this easily 1-coboundary since H1∗ (F˜ , ¯ proves that sm and sm are naturally isomorphic. This completes our induction argument. But, for m big enough, the contravariant functor ˜ym is trivial, so that we d,sc d,sc have Lm = Lm+1 ; in this situation, it follows from condition 23.28.2 that d,sc

d,sc d,sc (τm , Lm , πm ) fulﬁlls condition 23.28.1; moreover, the uniqueness follows from the uniqueness in each step of our induction. We are done. sc

23.29 At this point, the main diﬃculty to obtain a perfect F -locality r,sc as a quotient of L is the lack of naturality of the splitting of the exact sequences ˜ P ) −→ Z(Q ∩ ˜ P )/Z(Q) −→ 1 1 −→ Z(Q) −→ Z(Q ∩

23.29.1

sc when Q runs over the set of F˜ -objects. More precisely, respectively denoting by sc sc j ˜ sc : F˜ → ac(F˜ ) and γP : intP → j ˜ sc 23.29.2

F

F

432

Frobenius categories versus Brauer blocks

the canonical functor and the natural map obtained from the ﬁrst structural morphism (cf. 23.11.1), the corresponding injective natural map idac(˜z) ∗ γP : ˜z = ac(˜z) ◦ jF˜ sc −→ ac(˜z) ◦ intP

23.29.3

only admits an “almost” natural section in the sense of the following result. For this result, we need the functor we have deﬁned in 13.18 sc ˜z◦ = ˜z◦F˜ sc : F˜ −→ Ab

23.29.4.

Proposition 23.30 For any F-selfcentralizing subgroup Q , the structural ˜ )| (mod p) ˜ P ) raised to a suitable power cQ ≡ |F(P injection Z(Q) → Z(Q ∩ admits a section ˜ P) = σQ : Z(Q ∩

Q ∈S˜Q

˜ M ˜Q θ∈ Q

Z(Q ) −→ Z(Q)

23.30.1

˜ ˜ ˜Q sending z ∈ Z(Q ) to ˜z◦ (˜ιQ Q ) (z ) for any Q ∈ SQ and any θ ∈ MQ . sc Moreover, for any F˜ -morphism ϕ˜ : R → Q , denote by Z(R) the quotient of Z(R) by the sum ˜◦ ιQ ˜z◦ (δ˜ϕ ) ◦ ˜z(˜ιQ Q ) Z(Q ) Q ∩ϕ(R) ) ◦ z (˜

23.30.2

(ϕ,Q ) Q

where ϕ ∈ ϕ˜ and Q ∈ S˜Q run over the pairs such that Q ∩ ϕ(R) is not F-selfcentralizing and where, for such a pair, Q runs over the set of minimal subgroups of Q containing Q such that Q ∩ ϕ(R) is F-selfcentralizing, and δϕ : Q ∩ ϕ(R) → R maps ϕ(v) on v ∈ R . Then, we have the commutative diagram σQ ˜ P ) −→ Z(Q ∩ Z(Q)   # ˜z(ϕ) ˜ (ac(˜ z)◦intP )(ϕ) ˜ # 23.30.3. ˜ P) Z(R ∩

σ

R −→

Z(R)

˜ P ) of Proof: It follows from Lemma 13.19 that σQ maps the image in Z(Q ∩ |Q :Q | z ∈ Z(Q) on Q ∈S˜Q θ∈ , and from Proposition 6.7 that ˜ M ˜Q z Q

Q ∈S˜Q

˜ M ˜Q θ∈

Q

˜ ˜ )| ≡ 0 (mod p) |Q : Q | ≡ |F(P, Q)| ≡ |F(P

23.30.4.

sc

23. Narrowing the basic F -locality

433

sc Let ϕ˜ : R → Q be an F˜ -morphism; it follows from 23.14 that, for any sc Q,sc sc ˜ P in Omax , we have the ac(F˜ )-pull-back (F˜ )Q ∩˜ P -object θ˜ : Q → Q ∩

⊕w∈W˜ θ

θ˜

Q & ϕ ˜w 

w∈Wθ˜

Rw

−→

˜P Q∩ & ϕ˜ ∩ idP

˜w ⊕w∈W˜ M

−−−−−θ−−→

23.30.5

˜P R∩

for a choice of an element ϕ ∈ ϕ˜ and of a set of representatives Wθ˜ in Q for ϕ(R)\Q/Q , in such a way that for any w ∈ Wθ˜ we have w ϕ(Rw ) = ϕ(R)w ∩ Q sc then, it is quite clear that the ac(F˜ )-pull-back 23.14.1 forces ∼ ˜P Rw = R∩

23.30.6;

23.30.7

Q,sc w∈W˜ ˜ θ∈O θ max

or, equivalently, up to suitable choices it implies that R,sc {P ˜ w }w∈Wθ˜ = Omax

23.30.8.

Q,sc ˜ θ∈O max sc

Then, we claim that the following ac(F˜ )-diagram is commutative σQ ˜ ˜ P) = Q,sc Z(θ) Z(Q ∩ −→ Z(Q) ˜ θ∈O max   #(ac(˜z))(ϕ˜ ∩˜ idP ) # ˜z(ϕ) ˜ σR ˜ P) = R,sc Z(P) Z(R ∩ ˜ −→ Z(R) M∈O ˜

23.30.9;

max

indeed, it is quite clear that it suﬃces to prove the commutativity of the sc restricted ac(F˜ )-diagrams Z(Q )  ˜ z(ϕ ˜w )#

˜ z◦ (˜ ιQ ) Q

−−−−−→ w∈W˜

w∈Wθ˜

Z(Q)  # ˜z(ϕ) ˜

Rw

θ Z(Rw ) −−−−−− −−−−→

Z(R)

but, since we have (cf. Proposition 6.19 and equality 23.30.6) ˜ Q Q ) Z(Rw )∼ Z ϕ(R)w ∩ Q = Z(R ∩ = w∈Wθ˜

23.30.10;

˜ z◦ (˜ ιR )

w∈Wθ˜

the commutativity follows from Lemma 13.19. We are done.

23.30.11,

434

Frobenius categories versus Brauer blocks

23.31 A suﬃcient (strong!) condition to overcome this diﬃculty is to assume that 23.31.1 For every F-selfcentralizing subgroup Q of P and every nontrivial ¯ of N ¯P (Q) , the k R-module ¯ subgroup R k ⊗Z Z(Q) has no nonzero projective factors. This condition is fulﬁlled if P has no p-elementary Abelian subgroups of 3 2 order pp which, as a bound, is stronger than the bounds pp and pp respectively guaranteeing the existence and the uniqueness of the perfect F-locality in [13]; but, condition 23.31.1 can be fulﬁlled independently of such a bound. Corollary 23.32 With the notation above, assume that for every F-self¯ of N ¯P (Q) , centralizing subgroup Q of P and every nontrivial subgroup R ¯ the k R-module k ⊗Z Z(Q) has no nonzero projective factors. Then, for any m ∈ N , the correspondence sending any F-selfcentralizing subgroup Q of P to the k-module homomorphism m

˜ P )p −→ k ⊗Z Z(Q)p k ⊗Z Z(Q ∩

m

23.32.1

induced by σQ is a natural map. In particular, for any n ≥ 1 we have sc m m Hn∗ F˜ , k ⊗Z (ac(˜z) ◦ intP )p /˜zp = {0} 23.32.2. sc m Hn∗ F˜ , k ⊗Z ˜zp = {0} ¯P (Q)-module homomorphism Proof: Since we have an obvious surjective k N pm ¯ of N ¯P (Q) from k ⊗Z Z(Q) to k ⊗Z Z(Q) , for every nontrivial subgroup N pm ¯ the k N -module k ⊗Z Z(Q) has no nonzero projective factors and therefore, according to Lemma 23.33 below, we have m+1 ¯ pm TrN ⊂ Z(Q)p 23.32.3. 1 Z(Q) sc

Consequently, for any F˜ -morphism ϕ˜ : R → Q , since in the sum 23.30.2 we always have Q = Q , we get ◦ Q m m+1 ˜z (˜ιQ ) Z(Q )p ⊂ Z(Q )p 23.32.4 and therefore the corresponding sum (cf. 23.30.2) pm ˜◦ ιQ ˜z◦ (δ˜ϕ ) ◦ ˜z(˜ιQ Q ) Z(Q ) Q ∩ϕ(R) ) ◦ z (˜

23.32.5

(ϕ,Q ) Q

is contained in Z(Q )p

m+1

; thus, we have the commutative diagram

˜ P )p k ⊗Z Z(R ∩

k⊗Z σQ

m

−−−−→

m

k⊗Z σR

˜ P )p k ⊗Z Z(Q ∩  k⊗Z (ac(˜ z)◦intP )(ϕ) ˜ #

−−−−→

m

k ⊗Z Z(Q)p  # k⊗Z˜z(ϕ) ˜ k ⊗Z Z(R)p

m

23.32.6.

sc

23. Narrowing the basic F -locality

435

Now, according to Proposition 23.30, we have m m m m k⊗Z (ac(˜z)◦ intP )p ∼ = (k⊗Z ˜zp ) ⊕ k⊗Z (ac(˜z)◦ intP )p /˜zp

23.32.7

and therefore the last equalities are a consequence of the top equality in 23.27.1. We are done. Lemma 23.33 Let H be a ﬁnite group and M a kH-module. The restriction from M to M1H = TrH 1 (M ) induces a surjective homomorphism H Endk (M )H 1 −→ Endk (M1 )

23.33.1.

In particular, if M has no nonzero projective factors then M1H = {0} . H Proof: Setting A = Endk (M ) , it is easily checked that AH = M1H ; in 1 ·M H particular, the restriction from M to M1 induces a homomorphism from AH 1 to Endk (M1H ) . Moreover, if g ∈ Endk (M1H ) and {mi }i∈I is a basis of M1H , it is quite clear that, completing {mi }i∈I till to a basis of M , we can deﬁne a k-linear map f : M → M fulﬁlling

TrH 1 f (mi ) = g(mi )

23.33.2

for any i ∈ I and then homomorphism 23.33.1 sends TrH 1 (f ) to g ; that is to H say, we get a surjective homomorphism AH → End (M k 1 1 ). In particular, by the so-called Higman Criterion (cf. [29]), if M has no nonzero projective factors then AH 1 has no nonzero idempotents and therefore it is nilpotent (cf. Corollary 2.14 in [51]), which forces M1H to be zero. Corollary 23.34 Assume that for every F-selfcentralizing subgroup Q of P ¯ of N ¯P (Q) , the k N ¯ -module k ⊗Z Z(Q) has and every nontrivial subgroup N sc sc no nonzero projective factors. Then, there is a perfect F -sublocality L r,sc of L , unique up to conjugation by an element of lim (π t,sc ) . ←−

Proof: The proof follows the same pattern as in the proof of Corollary 23.28, replacing equalities 23.27.1 by equalities 23.32.2. Remark 23.35 According to our hypothesis, it follows from Theorem 4.13 sc in [52] that Hn (F˜ , ˜z) = {0} for any n ≥ 1 and therefore Proposition 3.1 in [13] already proves the existence and the uniqueness of a partial perfect sc F -sublocality. Here we provide another proof of this fact since this corollary r,sc r,sc guarantees the existence and the uniqueness of L inside L and in the sc r,sc next chapter we show that any perfect F -locality lies inside L .

Chapter 24

Looking for a perfect F -locality sc

24.1 Let P be a ﬁnite p-group and F a Frobenius P -category. In the previous chapter, we already have given a suﬃcient condition for the existence sc sc and the uniqueness of a perfect F -locality L as a sublocality of the reduced sc r,sc F -locality L ; as we point out in Remark 23.35, in this case Proposition 3.1 sc in [13] already provides a complete proof of the uniqueness of L . In this last sc r,sc chapter, we prove that any perfect F -locality has to be a sublocality of L independently of any condition; in particular, when our suﬃcient condition is sc fulﬁlled, this provides another proof of the uniqueness of L . From now on, sc sc sc (τ sc , L , π sc ) is a perfect F -locality L ; recall that the structural functor sc sc π sc : L → F induces an equivalence of categories sc sc L˜ ∼ = F˜

24.1.1

between their respective exterior quotients (cf. 1.3); we write τ and π for sc sc short. We will consider the additive cover ac(L ) of L (cf. A2.7 and A4.10). sc

Proposition 24.2 Any morphism x : R → Q in the category L morphism and an epimorphism.

is a mono-

Proof: Let T be an F-selfcentralizing subgroup of P and y, y ∈ L (R, T ) two morphisms such that x·y = x·y ; in particular, we have πx ◦ πy = πx ◦ πy and therefore, since πx is injective, we get πy = πy ; thus, we have y = y·z for some z ∈ τT Z(T ) and therefore we still have x·y = x·y = x·y·z ; now, sc the divisibility of L forces z = τT (1) , so that y = y . sc

On the other hand, if w, w ∈ L (P, Q) fulﬁll w·x = w ·x , we have again πw ◦ πx = πw ◦ πx and therefore, by Proposition 4.6, we get πw = πw ◦ κπx (z) for some z ∈ Z(R) ; since Z(Q) ⊂ πx Z(R) , up to a suitable choice of z , we actually may assume that w = w·τQ πx (z) and then, by the coherence sc in L , we still get (cf. 17.9.2) sc

w·x = w ·x = w·τQ πx (z) ·x = w·x·τR (z)

24.2.1;

sc

once again the divisibility of L , together with the injectivity of τR (cf. 17.13 and 13.2.2), forces z = 1 , so that w = w . sc

24.3 The fact that any morphism in L is an epimorphism allows us to develop in this category the same constructions as, in chapter 6, we do sc in F˜ . Explicitly, for any triple of F-selfcentralizing subgroups Q , R and T ,

438

Frobenius categories versus Brauer blocks sc

sc

any morphism x : Q → R in L induces an injective map from L (T, R) sc to L (T, Q) and, as in 6.4.1, we set sc

sc

L (T, Q)x = L (T, Q) −

L (T, Q )·z sc

24.3.1

z

where z runs over the set of nonisomorphisms z : Q → Q from Q in L sc — the set of nonﬁnal objects in (L )Q (cf. 1.7) — fulﬁlling x .z = x for sc some x ∈ L (R, Q ) ; then, x is uniquely determined by this equality and we simply say that z divides x setting x = x/z . Note that the existence sc of x for some z ∈ L (Q , Q) is equivalent to the existence of a subgroup of R which is F-isomorphic to Q and contains πx (Q) . Thus, it is quite clear that sc sc ˜ 24.3.2 L (T, Q)x is the converse image of F(T, Q)π˜x in L (T, Q) . sc

sc

sc

In particular, we have L (T, Q)x = L (T, Q) if and only if πx is an isomorsc phism, and moreover, for any y ∈ L (T, Q) we have (cf. 6.5.2) sc

sc

24.3.3 y ∈ L (T, Q)x is equivalent to x ∈ L (R, Q)y . Proposition 24.4 For any triple of F-selfcentralizing subgroups Q , R and T sc of P and any x ∈ L (R, Q) , we have sc

L (T, Q) =

L (T, Q )x/z ·z sc

24.4.1

z

where z : Q → Q runs over a set of representatives for the isomorphism sc classes of objects in (L )Q dividing x . Proof: It easily follows from Proposition 6.7 and statement 24.3.2 above. sc

sc

24.5 Let us consider the additive cover ac(L ) of L (cf. A4.10); recall sc that the ac(L )-objects are the ﬁnite sequences Q = {Qi }i∈I — denoted

sc by i∈I Qi — of F-selfcentralizing subgroups Qi of P , and an ac(L )-mor

sc phism from another ac(L )-object R = j∈J Rj to Q = i∈I Qi is a sc

pair (x, f ) formed by a map f : J → I and a family x = {xj }j∈J of L -morsc phisms xj : Rj → Qf (j) ; the composition of (x, f ) with another ac(L )-morphism (y, g) : T = T6 −→ R 24.5.1

6∈L

is the pair (x ∗ g) ◦ y, f ◦ g (cf. A2.6.3) where (x ∗ g) ◦ y is the family of sc L -morphisms xg(6) ◦ y6 : T6 −→ Rg(6) −→ Q(f ◦g)(6) 24.5.2 where D runs over L .

sc

24. Looking for a perfect F -locality

439 sc

24.6 Now, the decomposition 24.4.1 leads us to consider in ac(L ) the sc L -intersection of two F-selfcentralizing subgroups of P ; explicitly, if R and T are two F-selfcentralizing subgroups of P , we consider the set YR,T of triples (x, Q, y) where Q is an F-selfcentralizing subgroup of P , x belongs sc to L(R, Q)y and y belongs to L (T, Q)x or, equivalently, we have (cf. 6.5.1) ∗ FR πx (Q) ∩ (πy ) FT πy (Q) = FQ (Q)

(πx )∗

24.6.1

where (πx )∗ : πx (Q) ∼ = Q and (πy )∗ : πy (Q) ∼ = Q are the inverse of the respective isomorphisms determined by πx and πy ; note that, in our present sc sc situation, we can identify L (R, Q) and L (T, Q) with the respective subsets sc

sc

sc

τP,R (1)·L (R, Q) ⊂ L (P, Q) ⊃ τP,T (1)·L (T, Q)

24.6.2.

It is clear that, for any v ∈ R and any w ∈ T , the triple v·(x, Q, y)·w−1 = τR (v)·x, Q, τT (w)·y

24.6.3

still belongs to YR,T and that the quotient set (R × T )\YR,T coincides with the set of triples TR,T introduced in 6.9. 24.7 As in 6.9, we say that two triples (x, Q, y) and (x , Q , y ) are equisc valent if there exists an L -isomorphism z : Q ∼ = Q fulﬁlling x ·z = x and y ·z = y

24.7.1;

R,T the set of equivalence classes of such triples; since then, we denote by Y sc sc we always L is divisible, such an L -isomorphism z is unique; in particular, may assume that Q ⊂ R and that x = τR,Q (1) , and then τR,Q (1), Q, y is the unique element of this form in its equivalence class. As above, note that R,T coincides with the set of classes T R,T in 6.9. the quotient set (R × T )\Y sc Then, in ac(L ) we set

sc

R ∩L T =

Q

24.7.2

ˇ R,T (x,Q,y)∈Y

ˇ R,T of Y R,T in YR,T , and we have for a choice of a set of representatives Y sc canonical ac(L )-morphisms sc

R ←− R ∩L T −→ T

24.7.3

respectively determined by x : Q → R and y : Q → T . Note that, for another choice of the set of representatives, we get an isomorphic object and a unique sc ac(L )-isomorphism which is compatible with the canonical morphisms.

440

Frobenius categories versus Brauer blocks sc

Proposition 24.8 With the notation above, the category ac(L ) admits a distributive direct product mapping any pair of selfcentralizing subgroups R sc sc and T of P on their L -intersection R ∩L T . Proof: With the notation above, in order to discuss the functorial nature of sc the L -intersection, consider an F-selfcentralizing subgroup U of P and two sc sc morphisms a ∈ L (R, U ) and b ∈ L (T, U ) ; it follows from Proposition 24.4 sc that b determines an isomorphism class of (L )U -objects z : U → U dividing sc a such that, setting a = a/z , we have b = b ·z for a suitable b ∈ L (T, U )a and, once again, b is uniquely determined; that is to say, the pair (a, b) R,T and, once we have chosen determines an equivalence class of triples in Y ˇ a set of representatives YR,T , it determines a unique triple (a , U , b ) and a sc unique L -morphism z : U → U fulﬁlling a = a ·z and b = b ·z ; hence, the map sc sc sc L (Q, U ) −→ L (R, U ) × L (T, U ) 24.8.1. ˇ R,T (x,Q,y)∈Y

) sc sending the element z ∈ (x,Q,y)∈Y ˇ R,T L (Q, U ) in the term labelled by (x, Q, y) to (x·z, y·z) is bijective. In particular, considering two L -morphisms a : R → R and b : T → T sc and a triple (x , Q , y ) ∈ YR ,T , we get the L -morphisms sc

b·y

a·x

R ←−−− Q −−−→ T

24.8.2 sc

ˇ R,T and an L -morphism and therefore we still get a triple (x, Q, y) in Y z : Q → Q fulﬁlling a·x = x·z and b·y = y·z 24.8.3. In conclusion, we have obtained a map a,b : Y R ,T −→ Y R,T Y

24.8.4

ˇ R,T and Y ˇ R ,T for and, for any respective choices of sets of representatives Y R ,T , this map induces a new map R,T and Y Y ˇ a,b : Y ˇ R ,T −→ Y ˇ R,T Y

24.8.5

which, together with the L -morphisms z : Q → Q and the isomorphisms sc coming from the equivalences, determines an ac(L )-morphism sc

sc

sc

sc

a ∩L b : R ∩L T −→ R ∩L T

24.8.6.

Finally, always from the bijection 24.8.1, it is not diﬃcult to check that, for sc two other L -morphisms a : R → R and b : T → T , we get sc

sc

sc

(a ∩L b)·(a ∩L b ) = (a·a ) ∩L (b·b )

24.8.7.

sc

24. Looking for a perfect F -locality

441 sc

By distributivity, we can extend the L -intersection to the whole cate

sc gory ac(L ) , namely if R = i∈I Ri and T = j∈J Tj are two objects in this category, where Ri and Tj are F-selfcentralizing subgroups of P , then we set sc sc R ∩L T = Ri ∩L Tj 24.8.8. (i,j)∈I×J sc

Similarly, if we have two ac(L )-morphisms (a, f ) : R −→ R = Ri and (b, g) : T −→ T = Tj i ∈I

24.8.9

j ∈J

where f : I → I and g : J → J are maps and where a and b are respecsc tive families of L -morphisms ai : Ri → Rf (i) and bj : Tj → Tg(j) , i and j sc

respectively running over I and J , then we have ac(L )-morphisms sc

sc

sc

ai ∩L bj : Ri ∩L Tj −→ Rf (i) ∩L Tg(j)

24.8.10

which, together with the map f × g : I × J → I × J , clearly deﬁne a new sc ac(L )-morphism sc

sc

sc

(a, f ) ∩L (b, g) : R ∩L T −→ R ∩L T

24.8.11.

Finally, it is quite clear that the bijections 24.8.1 imply the bijections sc sc sc sc ac(L ) (R ∩L T, U ) ∼ 24.8.12 = ac(L ) (R, U ) × ac(L ) (T, U ) sc

sc

for any ac(L )-object U ; this proves that the L -intersection is a direct sc product in the category ac(L ) . We are done. sc

24.9 Here, we are particularly interested in the L -intersection of P with itself; more explicitly, denoting by Ω the set of pairs (Q, y) formed by sc an F-selfcentralizing subgroup Q of P and an element y of L (P, Q)τP,Q (1) , we have (cf. 24.7) sc P ∩L P = Q 24.9.1 (Q,y)∈Ω

since the set of triples {(τP,Q (1), Q, y)}(Q,y)∈Ω is a set of representatives ˜ P,P ; moreover, since P × P acts on Y ˜ P,P (cf. 24.6 and 24.7), this group for Y acts on Ω ; explicitly, for any u, v ∈ P , we have u·(τP,Q (1), Q, y)·v −1 = (τP,Q (u), Q, τP (v)·y) which is equivalent to the triple −1 τ u−1 (1), Qu , τP (v)·y·τ P,Q

Q,Qu

−1

(u−1 )

24.9.2

24.9.3;

442

Frobenius categories versus Brauer blocks

that is to say, we have proved that 24.9.4

−1 (u, v) ∈ P × P maps (Q, y) ∈ Ω on Qu , τP (v)·y·τ

−1

Q,Qu

(u−1 ) .

In particular, {1} × P acts freely on Ω . On the other hand, it is clear that the map sending a triple (x, Q, y) in YP,P to (y, Q, x) induces a P × P -set isomorphism Ω ∼ = Ω◦ . Proposition 24.10 With the notation above, the stabilizer of (Q, y) ∈ Ω in P × P coincides with ∆πy (Q) . In particular, we have |Ω|/|P | ≡ 0 (mod p) Proof: On the one hand, if (u, v) ∈ ∆πy (Q) then u ∈ Q , so that Qu sc and from the coherence of L (cf. 17.9.2) we get y·τ

−1

Q,Qu

(u−1 ) = τP (v −1 )·y

24.10.1. −1

= Q,

24.10.2,

so that ∆πy (Q) ﬁxes (Q, y) . On the other hand, if (u, v) ﬁxes (Q, y) then it follows from statement 24.9.4 that u normalizes Q and that y = τP (v)·y·τQ (u−1 ) ; in particular, v normalizes πy (Q) and πy maps the action of u over Q on the action of v sc over πy (Q) . But, since y ∈ L (P, Q)τP,Q (1) (cf. 24.9), we have (cf. equality 6.5.1 and statement 24.3.2) ∗ F˜P (Q) ∩ (πy ) F˜P πy (Q) = {1}

24.10.3. sc

Consequently, u belongs to Q and, once again from the coherence of L (cf. 17.9.2), we get y = τP (v)·y·τQ (u−1 ) = τP (v)·τP πy (u−1 ) ·y = τP vπy (u−1 ) ·y

24.10.4;

then, Proposition 24.2 and the injectivity of τP (cf. 17.13 and 13.2.2) force v = πy (u) , so that (u, v) ∈ ∆πy (Q) . In particular, if Q = P then p|P | divides the cardinal of the P × P -orbit of (Q, y) and therefore, considering the subset Ω of Ω determined by the sc pairs (P, x) where x runs over L (P ) , it is quite clear that |Ω|/|P | ≡ |Ω |/|P | = |L (P )|/|P | (mod p) sc

24.10.5;

sc but, we know that τP (P ) ∼ = P is a Sylow p-subgroup of L (P ) (cf. 17.12). We are done.

sc

24. Looking for a perfect F -locality

443

Corollary 24.11 With the notation above, for any F-selfcentralizing subgroup Q of P and any ϕ, ϕ ∈ F(P, Q) , we have |Ω∆ϕ,ϕ (Q) | = |Z(Q)| = |Ω∆(Q) |

24.11.1.

In particular, there is an F-basic P × P -set Ω which contains Ω , fulﬁlls Ω∆ϕ (Q) = Ω∆ϕ (Q)

24.11.2

for any F-selfcentralizing subgroup Q of P and any ϕ in F(P, Q) , and is thick outside the set of them. Proof: According to Proposition 24.10, an element (R, y) of Ω belongs to Ω∆ϕ,ϕ (Q) if and only if ∆ϕ,ϕ (Q) is contained in ∆πy (R) , namely if and only if ϕ(Q) ⊂ R and πy ϕ(u) = ϕ (u) for any u ∈ Q ; then, the element x = y·τR,ϕ(Q) (1) still fulﬁlls πx ϕ(u) = ϕ (u) for any u ∈ Q . Conversely, any sc x ∈ L P, ϕ(Q) fulﬁlling this equality, together with τP,ϕ(Q) (1) , determines sc an ac(L )-morphism (cf. Proposition 24.8) sc

ϕ(Q) −→ P ∩L P

24.11.3;

that is to say, it determines (R, y) ∈ Ω such that R contains ϕ(Q) and x = y·τR,Q (1) (cf. deﬁnition 24.9.1); hence, we get |Ω∆ϕ,ϕ (Q) | = |Z ϕ(Q) | = |Z(Q)|

24.11.4.

Then, the last statement follows from Proposition 21.12. Remark 24.12 Actually, this argument and Proposition 24.4 determine a canonical bijection sc L (P, Q) ∼ Ω∆ψ (R) 24.12.1 = R ψ∈F (P,R)ιP

R

where R runs over the subgroups of P containing Q . 24.13 Thus, the P × P -set Ω is contained in an F-basic P × P -set Ω fulﬁlling Ω∆(Q) = Ω∆(Q) for any F-selfcentralizing subgroup Q of P ; since we cannot guarantee the thickness, we consider the disjoint union Ω of Ω with |P | copies of a thick F-basic P × P -set Ω . Now, Ω is thick and, as in 21.2, we denote by G the automorphism group of Res {1}×P (Ω) and identify P with the image of P × {1} in G ; recall that, from the group G , b we can construct the basic F-locality L (cf. Proposition 22.12) and then the sc c,sc d,sc r,sc F -localities L (cf. Proposition 23.10), L (cf. Corollary 23.24) and L (cf. Corollary 23.28).

444

Frobenius categories versus Brauer blocks

24.14 On the other hand, for any F-selfcentralizing subgroup Q of P , it follows from Proposition 24.8 that the inclusion Q ⊂ P determines an sc ac(L )-morphism sc

sc

sc

τP,Q (1) ∩L τP (1) : Q ∩L P −→ P ∩L P

24.14.1

and in particular, denoting by ΩQ the set of indices in the structural decomposition of the left-hand term, it determines a map ΩQ → Ω ; actually, according to 24.7, we may identify ΩQ with the set of pairs (T, z) formed by sc an F-selfcentralizing subgroup T of Q and an element z of L (P, T )τQ,T (1) . sc

Proposition 24.15 The map ΩQ → Ω determined by the ac(L )-morphism sc

τP,Q (1) ∩L τP (1) sends (T, z) ∈ ΩQ to (R, y) ∈ Ω if and only if we have T = Q ∩ R and z = y·τR,T (1) . In particular, this map is injective and induces an injective Q × P -set homomorphism ΩQ −→ Res Q×P (Ω)

24.15.1.

sc

Proof: If the map associated with τP,Q (1) ∩L τP (1) sends the pair (T, z) sc in ΩQ to (R, y) ∈ Ω , then it induces an L -morphism x : T → R fulﬁlling τP,T (1) = τP,R (1)·x and z = y·x

24.15.2

sc

and therefore, by the divisibility of L , we have x = τR,T (1) , so that T is contained in Q ∩ R and z = y·τR,T (1) ; in particular, y·τR,Q∩R (1) extends z , sc which forces T = Q ∩ R since z belongs to L (P, T )τQ,T (1) (cf. 24.3.1); the converse is clear. sc

In particular, if the map associated to τP,Q (1) ∩L τP (1) also sends the pair (T , z ) ∈ ΩQ to (R, y) , we have T = Q ∩ R = T and z = y·τR,T (1) = z . We are done. sc 24.16 Now, for any L -isomorphism x : Q ∼ = Q , it follows again from sc Proposition 24.8 that we have an ac(L )-isomorphism

sc

sc

sc

x ∩L τP (1) : Q ∩L P ∼ = Q ∩L P

24.16.1

and therefore a bijection between the sets of indices ΩQ and ΩQ , which is compatible via πx with the actions of Q × P and Q × P on both sets; that is to say, we get Q × P -set isomorphism fx : ΩQ ∼ = Resπx ×idP (ΩQ )

24.16.2.

sc

24. Looking for a perfect F -locality

445

sc Proposition 24.17 For any L -isomorphism x : Q ∼ = Q , the Q × P -set isomorphism fx : ΩQ ∼ 24.17.1 = Resπx ×idP (ΩQ )

ˆ can be extended to an element fˆx of TG (Q , Q) which stabilizes any copy of Ω and induces the same automorphism in all of them. Moreover, the image c,sc ˆ of fˆx in L (Q , Q) is uniquely determined by x . Proof: Since the Q × P -sets Res Q×P (Ω) and Res πx ×idP Res Q ×P (Ω) are isomorphic to each other (cf. 21.2.2 and 21.4), and the Q × P - and Q × P -set homomorphisms ΩQ −→ Res Q×P (Ω) and

ΩQ −→ Res Q ×P (Ω)

24.17.2

are injective (cf. Proposition 24.15), fx can be extended to a Q × P -set isomorphism fˆx : Res Q×P (Ω) ∼ 24.17.3; = Res πx ×idP Res Q ×P (Ω) moreover, since the Q × P -sets Res Q×P (Ω ) and Res πx ×idP Res Q ×P (Ω ) are mutually isomorphic too, fˆx can be extended to a Q × P -set isomorphism ˆ fˆx : Res Q×P (Ω) ∼ = Res πx ×idP Res Q ×P (Ω)

24.17.4

which stabilizes any copy of Ω and induces the same automorphism in all of ˆ them; that is to say, we get an element fˆx of TG (Q , Q) . c,sc ˆ Then, we claim that the image of fˆx in L (Q , Q) is independent of our choices; indeed, for another choice gˆ ˆx ∈ TG (Q , Q) fulﬁlling the above condiˆ ˆx belongs to CG (Q) and induces the identions, the composed map (fˆx )−1 ◦ gˆ tity in ΩQ ; but, it follows from Corollary 24.11 that, for any F-selfcentralizing subgroup R of P and any ψ ∈ F(P, R) , we have

Ω∆ψ (R) = Ω∆ψ (R)

24.17.5

sc

˜ in OQ (cf. 22.5) such in particular, for any pc(F˜ )-morphism η˜ : T → Q×P that T is F-selfcentralizing, any Q × P -orbit Γ in Res Q×P (Ω) isomorphic to Q×η˜ P is contained in Ω since, for any representative η of the composition ηˆ˜ ˜ ˜ → P (cf. 22.5), ∆η (T ) has a of η˜ with the structural pc(F)-morphism Q×P ﬁxed point in Γ . Moreover, it follows from Propositions 24.10 and 24.15 that Γ is actually contained in ΩQ . Consequently, since we have (cf. 22.5.1) CG (Q) ∼ =

η˜∈OQ

Γη˜

Aut(Γη˜) ×

|P | i=1 Γη˜

Aut(Γη˜ ) Skη˜

24.17.6

446

Frobenius categories versus Brauer blocks

where Γη˜ and Γη˜ respectively run over the set of Q × P -orbits isomorphic to Q ×η˜ P in Res Q×P (Ω) and in Res Q×P (Ω ) , and, according to our choices, ˆ ˆ (fˆx )−1 ◦ gˆ ˆx stabilizes Ω and any copy of Ω , this element (fˆx )−1 ◦ gˆˆx has a trivial image in (cf. 23.9) ˜ksc (Q) =

ab Aut(Q ×η˜ P )

24.17.7.

Q,sc

η˜∈O Q,sc

Indeed, for any η˜ ∈ O , this element acts trivially on the union of the Q × P -orbits Γη˜ above and therefore, denoting by hη˜ the automorphism of ˆ the union of the Q × P -orbits Γη˜ above induced by (fˆx )−1 ◦ gˆˆx , its action on the union of all the Q × P -orbits in Ω isomorphic to Q ×η˜ P coincides with |P | the product i=1 hη˜ ; but, according to our choices, all the factors have the same image in ab Aut(Q ×η˜ P ) and we clearly have (cf. 22.3.1) |P | ab Aut(Q ×η˜ P ) = {1}

24.17.8.

We are done. sc

r,sc

Corollary 24.18 There is a faithful functor λ : L → L which is compatible with the structural functors and, up to a suitable choice of the inclur,sc c,sc sc ˆ sion L ⊂ L , maps any L -isomorphism x : Q ∼ = Q on the image of fˆx c,sc

in L

sc c,sc (Q , Q) , and τP,Q (1) on τP,Q (1) .

c,sc ˆ Proof: Let us denote by λ(x) the image of fˆx in L (Q , Q) ; ﬁrst of all, let sc x : Q ∼ = Q be a second L -isomorphism; it is clear that the automorphism ˆ ˆ Res πx ×idP (fˆx ) ◦ fˆx of Ω extends Resπx ×idP (fx ) ◦ fx and stabilizes any copy of Ω , inducing the same automorphism in all of them; consequently, by the argument above, we get

λ(x ·x) = λ(x )·λ(x) sc

24.18.1.

sc

Secondly, by the divisibility of L , any L -morphism z : R → Q is the sc sc composition of τQ,R (1) with an L -isomorphism z∗ : R ∼ = R = πz (R) and we simply deﬁne r,sc λ(z) = τQ,R (1)·λ(z∗ ) 24.18.2. Now, in order to prove that this correspondence deﬁnes a functor, it suﬃces sc to show that, for any L -isomorphism x : Q ∼ = Q and any subgroup R of Q ,

sc

24. Looking for a perfect F -locality

447

sc setting R = πx (R) and denoting by y : R ∼ = R the L -isomorphism induced by x (cf. 17.7), we still have c,sc λ(x)·τQ,R (1) = τQc,sc (1)·λ(y) ,R

24.18.3. sc

But, it is quite clear that the commutative ac(L )-diagram (cf. Proposition 24.8) sc

sc

L

R∩ sc L y∩ τP (1) ↓

P

sc

R ∩L P

τQ,R (1) ∩L τP (1)

−−−−−−−−−−−−→ τ

Q ,R

sc L

(1) ∩

τP (1)

−−−−−−−−−−−−−→

sc

Q ∩L P sc ↓ x ∩L τP (1)

24.18.4

sc

Q ∩L P

determines a commutative diagram of R × P -sets ΩR fy

↓

−→

ResQ×P R×P (ΩQ ) Q×P

Resπy ×idP (ΩR ) −→ ResR×P

↓ ResQ×P (fx ) R×P

Resπx ×idP (ΩQ )

24.18.5.

ˆ ˆ Consequently, we can choose fˆy = fˆx . Moreover, it is easily checked that this functor is faithful. In particular, it follows from Proposition A2.10 that we have a natural map autλ : autLsc −→ autLc,sc 24.18.6 which, according to 18.20 and to a suitable partial version of Propositions 18.19 and 18.21, induces a new natural map locλ : locF sc ∼ = locLsc −→ locLc,sc

24.18.7.

Furthermore, it follows again from this partial version of Proposition 18.21 that we have a natural map λLr,sc : locF sc −→ locLr,sc r,sc

24.18.8 c,sc

and that, considering the inclusion functor icr : L → L , the natural map loc icr ◦ λLr,sc coincides with locλ ; that is to say, up to a suitable choice of sc sc r,sc c,sc r,sc of L , we may assume that λ L (Q) ⊂ L (Q) the F -sublocality L for any F-selfcentralizing subgroup Q of P , and then it follows from Corolr,sc lary 5.14 that the image of the functor λ is contained in L . We are done. Corollary 24.19 Assume that for every F-selfcentralizing subgroup Q of P ¯ of N ¯P (Q) , the k R-module ¯ and every nontrivial subgroup R k ⊗Z Z(Q) has no sc sc nonzero projective factors. Then, there is a unique perfect F -locality L . Proof: It is an immediate consequence of Corollaries 23.34 and 24.18.

Appendix

Simpliciality In this Appendix we collect all the generalities on categories and on their cohomological groups we need in this book. As we already mention in chapter 1, in general we only consider the small categories, namely the categories over a set of objects. Note that all the results below are wellknown and we have no claim of originality; our purpose in stating them is both, to provide a handy reference for the reader and to develop our own point of view and our uniﬁed notation. When dealing with categories and their cohomology, it is a wide experience that, although sometimes quite long, many proofs are straightforward. Thus, without pushing the argument as far as Serge Lang did in his wellknown book Algebra†, we partially follow Lang’s point of view and state our results without proofs when considering that they are straightforward. A1 On the 2-categories A1.1 Recall that a 2-category is a category X where, for any pair of objects X and Y , the set of X-morphisms from X to Y is endowed with a structure of category — denoted by X(Y, X) (cf. 1.2) — and where, for any triple of X-objects X , Y and Z , the left-hand composition is extended to a functor lX A1.1.1 Y,Z : X(Z, Y ) −→ Fct X(Y, X), X(Z, X) in such a way that, for any quadruple of X-objects X , Y , Z and U , any pair of pairs of X-morphisms g , g from Y to Z , and h , h from Z to U , and any respective X(Z, Y )- and X(U, Z)-morphisms β : g → g and γ : h → h , it fulﬁlls the associative conditions X X lX h ◦ lg = lh◦g

,

X X lX γ ∗ lg = lγ∗g

X X and lX h ∗ lβ = lh∗β

A1.1.2

where ∗ denotes the “composition” between functors and natural maps and X X X Y Y we set lX g = lY,Z (g) , lβ = lY,Z (β) , γ ∗ g = lγ (g) and h ∗ β = lh (β) . A1.2 To avoid confusion, if f and f are two X-morphisms from X to Y , we denote by NatX (f , f ) the set of X(Y, X)-morphisms from f to f , and call them X-natural maps. We denote by X◦ the 2-category over the opposite category, by XN the 2-subcategory of X formed by the same objects, the same morphisms and the natural X-isomorphisms, and by Xc the underlying category. †

In EXERCISES at the end of Chapter IV, Serge Lang write “Take any book on homological

algebra, and prove all the theorems without looking at the proofs given in that book.”

450

Frobenius categories versus Brauer blocks

A1.3 Note that, if f is an X-morphism from X to Y , the left-hand equality in A1.1.2 implies the usual associative condition h ◦ (g ◦ f ) = (h ◦ g) ◦ f

A1.3.1;

more generally, if f is a second X-morphism from X to Y and α : f → f a X-natural map, the left-hand equality in A1.1.2 forces h ∗ (g ∗ α)) = (h ◦ g) ∗ α

A1.3.2;

moreover, if f is a third X-morphism from X to Y and α ∈ NatX (f , f ) , the functoriality of lX g implies that g ∗ (α ◦ α) = (g ∗ α ) ◦ (g ∗ α)

A1.3.3.

A1.4 Similarly, if g is a second X-morphism from Y to Z and β a X-natural map from g to g , the right-hand equality in A1.1.2 forces h ∗ (β ∗ f ) = (h ∗ β) ∗ f

A1.4.1;

moreover, if g is a third X-morphism from Y to Z and β ∈ NatX (g , g ) , the functoriality of lX Y,Z implies that (β ◦ β) ∗ f = (β ∗ f ) ◦ (β ∗ f )

A1.4.2;

ﬁnally, the middle equality in A1.1.2 implies that γ ∗ (g ◦ f ) = (γ ∗ g) ∗ f

A1.4.3.

A1.5 Moreover, the naturality of lYβ : lYg → lYg applied to the X-natural map α : f → f implies that we have (g ∗ α) ◦ (β ∗ f ) = (β ∗ f ) ◦ (g ∗ α)

A1.5.1.

Conversely, it is not diﬃcult to see that all these associative and distributive conditions on the composition of morphisms and natural maps allow the (re)construction of functors lX Y,Z as above fulﬁlling conditions A1.1.2. Note that, since they are symmetric, they still allow us to construct the corresponding right-hand composition functors rZ X,Y , which actually commute with the left-hand ones. A1.6 The standard example of a 2-category is the category of small categories CC where the objects are the small categories (cf. 1.2) and the morphisms between two small categories C and D are all the functors Fct(C, D) from C to D , which form a set†. Indeed, Fct(C, D) has a natural structure of small category since between two functors f and g from C to D we still have the set of natural maps Nat(f, g) , and the natural maps can be composed. †

A contrario, there is no category of categories.

A. Simpliciality

451

A1.7 Besides CC , the 2-category which motivates our interest is the simplicial 2-category ∆ , usually considered as a simple category in the literature. The ∆-objects are the ordered sets ∆n = {i | 0 ≤ i ≤ n}

A1.7.1

for any n ∈ N ; then, for any n, m ∈ N , the ∆-morphisms from ∆n to ∆m , are the order-preserving maps with the usual composition; moreover, for any pair of morphisms δ : ∆n → ∆m and ε : ∆n → ∆m in ∆ , we denote by Nat∆ (ε, δ) a set with one element δ • ε if δ(i) ≤ ε(i) for any i ∈ ∆n and the empty set otherwise, and we have an obvious composition of these elements. A1.8 For any 2-category X , we consider the category of connected components π0 (X) = X of X †, namely the quotient category of X by the minimal equivalence relation between the X-morphisms such that, for any pair of X-objects X and Y , and any pair of X-morphisms f and g from X to Y , if NatX (f, g) = ∅ then f and g are equivalent. The existence of the functors lX Y,Z fulﬁlling conditions A1.1.2 guarantees that the composition in X induces indeed a composition in X . Note that any category C admits a trivial 2-category structure where, for any pair of objects A and B , the set C(A, B) has the trivial category structure, namely it has no morphism diﬀerent from the identity of any object. A1.9 If Y is another 2-category, recall that a 2-functor x : X → Y is a functor between the underlying categories, together with a natural family of functors xX ,X : X(X , X) −→ Y x(X ), x(X) A1.9.1 where X and X run over the collection of X-objects, which coincide with x over the corresponding morphisms and which, for any triple of X-objects X , X and X , any pair of X-morphisms f , f from X to X , any pair of X-morphisms g , g from X to X , and any natural maps α ∈ NatX (f , f ) and β ∈ NatX (g , g) , fulﬁll xX ,X (g ◦ f ) = x(g) ◦ x(f ) xX ,X (g ∗ α) = x(g) ∗ xX ,X (α)

A1.9.2.

xX ,X (β ∗ f ) = xX ,X (β) ∗ x(f ) Moreover, in this case x induces a functor x : X → Y . We denote by idX the identity 2-functor of X . A1.10 If Z is a third 2-category and u : Y → Z is a 2-functor, it is easily checked that we have a 2-functor u ◦ x from X to Z such that its underlying †

Alberto Arabia helped us to understand the interest of this deﬁnition.

452

Frobenius categories versus Brauer blocks

functor coincides with the composition of the underlying functors of x and u , and that it fulﬁlls (u ◦ x)X ,X = ux(X ),x(X) ◦ xX ,X A1.10.1 for any pair of X-objects X and X ; then, we have u ◦ x = u ◦ x . A1.11 Recall that if y : X → Y is a second 2-functor, a (strict) 2-natural map µ : x → y is a correspondence mapping any X-object X on a Y-morphism µX : x(X) → y(X) in such a way that, for any pair of X-objects X and X we have (cf. A1.5) y(X ) lx(X) ◦ yX ,X A1.11.1; µX ◦ xX ,X = rµX in this case, µ clearly induces a natural map µ : x → y . If t : X → Y is a third 2-functor and ν : y → t a 2-natural map, we have a composed 2-natural map ν ◦ µ from x to t . A1.12 It is clear that the composition of 2-natural maps is associative and that if X , Y and Z are 2-categories, x : X → Y , x : X → Y , y : Y → Z and y : Y → Z are 2-functors, and µ : x → x and ν : y → y are 2-natural maps, then we have 2-natural maps ν ∗ x : y ◦ x −→ y ◦ x and y ∗ µ : y ◦ x −→ y ◦ x

A1.12.1.

With the same notation, if x : X → Y and y : Y → Z are 2-functors, and µ : x → x and ν : y → y are 2-natural maps, then we have (ν ◦ ν)∗ x = (ν ∗ x)◦(ν ∗ x) and y ∗ (µ ◦µ) = (y ∗ µ )◦(y ∗ µ)

A1.12.2.

Moreover, if U is a 2-category, t : Z → U and t : Z → U are 2-functors and η : t → t is a 2-natural map, then we still have (η ∗ y) ∗ x = η ∗ (y ◦ x) and t ∗ (y ∗ µ) = (t ◦ y) ∗ µ

A1.12.3.

A1.13 A meaningful example of a 2-functor and a 2-natural map comes from the simplicial 2-category ∆ . The shifting functor sh : ∆ −→ ∆

A1.13.1

is a 2-functor mapping any ∆n on ∆n+1 , mapping any order-preserving map δ : ∆n → ∆m on the order-preserving map ∆n+1 → ∆m+1 sending 0 to 0 and i ∈ ∆n+1−{0} to δ(n−1)+1 and, for any order-preserving map ε : ∆n → ∆m such that δ(i) ≤ ε(i) for any i ∈ ∆n , mapping the natural map δ •ε on the natural map sh(δ)•sh(ε) , which makes sense. The shifting natural map ς : id∆ −→ sh

A1.13.2

is a 2-natural map which sends ∆n to the order-preserving map ∆n → ∆n+1 sending i ∈ ∆n to i+ 1 .

A. Simpliciality

453

A1.14 Although barely needed in this book, let us recall the deﬁnition of the supranatural maps. If X and Y are 2-categories, x : X → Y and y : X → Y are 2-functors and λ : x → y and µ : x → y are 2-natural maps, a supranatural map σ : λ → µ is a correspondence mapping any X-object X on a Y-natural map σX ∈ NatY (µX , λX ) in such a way that any X-morphism f : X → X fulﬁlls the equality y(f ) ∗ σX = σX ∗ x(f ) A1.14.1. Moreover, if ν : x → y is a third 2-natural map and τ : µ → ν a supranatural map, the correspondence mapping X on τX ◦ σX deﬁnes a supranatural map τ ◦ σ : λ → ν ; this composition is associative. A1.15 Finally, if t : X → Y is a 2-functor, λ : y → t and µ : y → t are 2-natural maps, and σ : λ → µ is a supranatural map, then we have the supranatural maps λ ∗ σ : λ ◦ λ −→ λ ◦ µ and σ ∗ λ : λ ◦ λ −→ µ ◦ λ

A1.15.1.

Similarly, if Z is a third 2-category, t : Y → Z and u : Y → Z are 2-functors, ν : t → u and η : t → u are 2-natural maps, and τ : ν → η a supranatural map, then we have the supranatural maps t < σ : t ∗ λ −→ t ∗ µ and τ < x : ν ∗ x −→ η ∗ x

A1.15.2.

A2 Representations and semidirect products A2.1 The formalism of the semidirect product† we introduce here uniﬁes several constructions of categories employed in this book just by mimicking the deﬁnition of the semidirect product of a group by one of its linear representations. A2.2 Let X be a 2-category; we call representation of X a 2-functor m : X → CC from X to the category of small categories CC (cf. A1.6); for any item • in X , let us denote by m • its image. Our main example is the standard representation of the simplicial 2-category (cf. A1.7) st : ∆ −→ CC

A2.2.1

mapping any n-simplex ∆n on the category — still denoted by ∆n — where the objects are all the elements i ∈ ∆n and the set of ∆-morphisms from i ≤ n to j ≤ n is not empty only if i ≤ j , and then has a unique element, denoted by i • j ; moreover, st maps any ∆-morphism from ∆n to ∆m on the obvious functor and any ∆-natural map on the obvious natural map. †

Some people pointed out to us that, although with another name, this construction already

has been introduced by Alexander Grothendieck.

454

Frobenius categories versus Brauer blocks

A2.3 Since any set can be considered as a category with no other morphisms than the identity for any element, and then the maps become functors, denoting by ℵ the category of ﬁnite sets (cf. 1.1), we have the identity representation i : ℵ −→ CC A2.3.1. Note that any representation m of X induces functors m∗ and mo from the underlying category Xc to CC respectively mapping any X-object X on the subcategories (mX)∗ and (mX)o of mX (cf. 1.5), where the respective morphisms are the mX-isomorphisms and the mX-identity of any object (cf. A2.2), and any X-morphism f : X → Y on the corresponding functors (m f)∗ : (mX)∗ −→ (mY )∗

and

(m f)o : (mX)o −→ (mY )o

A2.3.2.

More generally, we call weak structure of m a subfunctor mw of m mapping w any X-object X on a subcategory m X of mX with the same objects. A2.4 Any small category C considered as a 2-category admits the regular representation rgC : C −→ CC A2.4.1 deﬁned as follows. Firstly, for any C-object C , we consider the category CC (cf. 1.7); recall that the CC -objects are the C-morphisms f : A → C , the CC -morphisms from f : A → C to another CC -object f : A → C are the C-morphisms h : A → A fulﬁlling f ◦ h = f , and the composition comes from the composition in C . Secondly, for any C-morphism D : C → C , we have a functor C6 : CC −→ CC A2.4.2 mapping any CC -object f : A → C on the composition D ◦ f : A → C , which is indeed a CC -object, and any CC -morphism h from f : A → C to another CC -object f : A → C — namely any C-morphism from A to A fulﬁlling the equality f ◦ h = f in C — on the same h considered as a CC -morphism from D ◦ f : A → C to D ◦ f : A → C , since we still have D ◦ f ◦ h = D ◦ f . Then, the regular representation rgC maps any C-object C on the category CC and any C-morphism D : C → C on the functor C6 : CC → CC . A2.5 Moreover, for any small category C , we have the C-dual of a representation m of X , which is a representation of X◦ Fct(m, C) : X◦ −→ CC

A2.5.1

mapping any X-object X on the category of functors Fct(mX, C) , any X-morphism f : Y → X on the functor Fct(m f, C) : Fct(mX, C) −→ Fct(m Y, C)

A2.5.2

A. Simpliciality

455

determined by the composition with µ : f → g on the natural map

m

f : m Y → mX , and any X-natural map

Fct(m µ, C) : Fct(m f, C) −→ Fct(m g, C)

A2.5.3

sending any functor h : mX → C to the natural map h ∗ m µ : h ◦ m f −→ h ◦ m g

A2.5.4.

The weak structure Fct∗ (m, C) of Fct(m, C) maps any X-object X on the category of functors Fct(mX, C)∗ with natural isomorphisms. Proposition A2.6 For any 2-category X and any representation m : X → CC of X , there is a 2-category m X fulﬁlling: A2.6.1 The m X-objects are the pairs (A, X) formed by an X-object X and a mX-object A . A2.6.2 The m X-morphisms from (A, X) to another m X-object (B, Y ) are the pairs (f, f) formed by an X-morphism f : X → Y and a m Y -morphism f : m f (A) −→ B . A2.6.3 The composition of the m X-morphism (f, f) : (A, X) → (B, Y ) with another m X-morphism (g, g) : (B, Y ) → (C, Z) is given by (g, g) ◦ (f, f) = g ◦ m g (f ), g ◦ f . A2.6.4 The m X-natural maps from (f, f) to another m X-morphism (f , f ) : (A, X) → (B, Y ) are the X-natural maps α : f → f fulﬁlling f ◦ (mα)A = f . A2.6.5 The compositions between m X-morphisms and m X-natural maps, and between m X-natural maps are induced by the corresponding compositions in X . A2.7 We call semidirect product of X by m the 2-category m X ; note that we have an obvious 2-functor pX,m : m X −→ X

A2.7.1;

moreover, a weak structure mw of m determines a subcategory mw X of m X with the same objects. Our ﬁrst example is the product cover of a small category: for any small category C , we consider the C-dual representation Fct(i, C) (cf. A2.5) of the identity representation of ℵ (cf. A2.3) and then the product cover of C is the category pc(C) = Fct(i, C)ℵ◦

A2.7.2.

456

Frobenius categories versus Brauer blocks

Explicitly, a pc(C)-object is just a ﬁnite sequence {Ci }i∈I of objects in C — denoted by i∈I Ci , or i∈I Ci to avoid confusion with a possible direct product in C — and a pc(C)-morphism from i∈I Ci to another pc(C)-object j∈J Dj is formed by a map f : J → I and by a C-morphism αj : Cf (j) → Dj for any j ∈ J . In this book, we widely employ the additive cover of C , introduced by S. Jackowski and J. McClure in [32], which coincides with (cf. A4.10 below) ac(C) = pc(C◦ )◦ A2.7.3. A2.8 Our second example of a semidirect product is the category of chains of a small category: for any small category C , we consider the C-dual representation Fct(st, C) (cf. A2.5) of the standard representation st of ∆ (cf. A2.2) and then the category of C-chains is the 2-category ch(C) = Fct(st, C)∆◦

A2.8.1.

Actually, in this book we are only interested in the 2-subcategory ch∗ (C) of ch(C) — called proper category of C-chains — coming from the weak structure Fct(st, C)∗ of Fct(st, C) (cf. A2.5); explicitly, ch∗ (C) is formed by the same objects (q, ∆n ) , where q : ∆n → C is a functor, by the morphisms (ν, δ) : (r, ∆m ) −→ (q, ∆n )

A2.8.2,

where δ : ∆m → ∆n is an order-preserving map and where — writing δ for st δ — ν :r ∼ = q ◦ δ is a natural isomorphism instead of a simple natural map, and by the same ch(C)-natural maps between these ch(C)-morphisms. A2.9 Note that ch∗ (C) need not be the semidirect product of ∆◦ by the representation of ∆◦ mapping ∆n on the category Fct(∆n , C)∗ , where the morphisms are the natural isomorphisms, since the ch∗ (C)-natural maps need not be isomorphisms; this is meaningful when dealing with the category of connected components ch∗ (C) (cf. Proposition A5.4). The main reason to consider the proper category of chains comes from the following fact which has to be understood as a generalization of the “barycentric decomposition”. Proposition A2.10 For any small category C , the correspondence mapping any ch∗ (C)-object (q, ∆n ) , where q : ∆n → C is a functor, on its ch∗ (C)-automorphism group Nat(q, q)∗ can be extended to a functor autC : ch∗ (C) −→ Gr

A2.10.1

mapping any ch∗ (C)-morphism (ν, δ) : (r, ∆m ) → (q, ∆n ) on the group homomorphism sending ρ ∈ Nat(r, r)∗ to ν ◦ (ρ ∗ δ) ◦ ν −1 ∈ Nat(q, q)∗ . A2.11 Let us illustrate the usefulness of the semidirect product in the usual construction of the direct image of an Ab-valued contravariant functor

A. Simpliciality

457

(cf. 1.6). Let B and C be small categories and f : B → C ; the starting point is the graph representation of f (cf. 1.5) grf : C◦ × B −→ CC

A2.11.1

mapping any C◦ × B-object (C, B) on the set C C, f(B) considered as a trivial category and any C◦ × B-morphism on the obvious map; it is quite clear that we can identify this functor with a new functor grf : C◦ −→ Fct(B, CC)

A2.11.2.

A2.12 That is to say, for any C-object C we get a representation grf (C) : B −→ CC

A2.12.1

of B and therefore we still get a category grf (C) B , actually endowed with a structural functor (cf. A2.7) pC = pB,grf (C) : grf (C) B −→ B

A2.12.2.

Analogously, for any C-morphism g : C → C we have a natural map grf (g) : grf (C ) −→ grf (C)

A2.12.3

and therefore we still have a functor (cf. Proposition 2.20 below) grf (g) idB : grf (C ) B −→ grf (C) B

A2.12.4;

moreover, it is clear that pC ◦ (grf (g) idB ) = pC

A2.12.5.

A2.13 At this point, with evident terminology, it is clear that we have obtained a pointed representation of C◦ (cf. 1.7) prf : C◦ −→ (CC)B

A2.13.1

mapping any C-object C on the (CC)B -object pC deﬁned in A2.12.2 above and any C-morphism g : C → C on the (CC)B -morphism grf (g) idB . On the other hand, we know that any contravariant functor a : B → Ab and the inverse limit deﬁne two functors (cf. 1.7) (CC)a : (CC)B −→ (CC)Ab◦

and

lim : (CC)Ab◦ −→ Ab −→

A2.13.2.

Finally, it is easily checked that f∗ (a) = lim ◦(CC)a ◦ prf −→

A2.13.3.

458

Frobenius categories versus Brauer blocks

A2.14 But in this book, we are also interested in another kind of image involving the semidirect product. Namely, we are interested in the particular situation where, for a representation m of a 2-category X , endowed with a weak structure mw , the functor m f : mX → m Y admits a right adjoin (m f)a (cf. 1.6) such that w w (m f)a (m X) ⊂ m Y A2.14.1 for any X-morphism f : X → Y . In this case, let us denote by αf : m f ◦ (m f)a −→ idm Y

A2.14.2

the corresponding adjoinness natural map and by w

w

(m f)a,w : m X → m Y

A2.14.3

the corresponding restriction. The point is the existence of a functor (pm,mw )a : Fct (m X)◦ , Ab −→ Fct(X◦ , Ab) A2.14.4 — the mw -stable adjoin image — deﬁned as follows. A2.15 First of all, for any contravariant functor b : m X −→ Ab

A2.15.1

and any X-object X , we consider the contravariant functor bX : mX −→ Ab

A2.15.2

mapping any mX-object A on b(A, X) and any mX-morphism f : A → A w on b(f, idX ) , and denote by (bX )w the restriction of bX to m X ; note that, for any X-morphism f : X → Y , the adjoinness αf determines a natural map bf : bY −→ bX ◦ (m f)a

A2.15.3

sending any m Y -object B to b (αf )B , f (cf. A2.6.2). Moreover, for any natural map µ : b → b from b to another contravariant functor b : m X → Ab and any X-object X , we have an obvious natural map µX : bX → bX . A2.16 Then, (pm,mw )a maps b on the contravariant functor (pm,mw )a (b) : X −→ Ab

A2.16.1

sending X to the direct limit lim (bX )w and any X-morphism f : X → Y to −→

the group homomorphism lim (bY )w −→ lim (bX )w ◦ (m f)a,w −→ lim (bX )w −→

−→

−→

A2.16.2

A. Simpliciality

459

which is the composition of lim bf with the corresponding canonical homo−→

morphism. Moreover, (pm,mw )a maps µ : b → b on the natural map (pm,mw )a (µ) : (pm,mw )a (b) −→ (pm,mw )a (b )

A2.16.3

sending any X-object X to lim µX : lim (bX )w −→ lim (bX )w −→

−→

−→

A2.16.4.

A2.17 In this situation, note that if the category mX has a ﬁnal object OX for any X-object X then we have (m f)a (OY ) = OX for any X-morphism f : X → Y and therefore we get a contravariant functor and a natural map om : Fct (m X)◦ , Ab −→ Fct(X◦ , Ab) A2.17.1 ωm,mw : om −→ (pm,mw )a deﬁned as follows; the functor om maps b on the contravariat functor sending any X-object X to b(OX , X) and any X-morphism f : X → Y to b(f, f) where f : m f(OX ) → OY is the unique m Y -morphism, and µ : b → b on the natural map om (µ) : om (b) → om (b ) sending X to the group homomorphism µ(OX ,X) : b(OX , X) −→ b (OX , X)

A2.17.2;

the natural map ωm,mw sends b to the natural map ωm,mw (b) : om (b) −→ (pm,mw )a (b)

A2.17.3

sending X to the obvious group homomorphism b(OX , X) −→ lim (bX )w −→

A2.17.4.

A2.18 Let us come back to the general situation. For any small category C considered as a 2-category, it is also useful to consider the semidirect product rgC C of C by its regular representation; in this case, besides the canonical functor pC : rgC C → C mentioned in A2.7 above, we have the covering functor and the covering natural map cC : rgC C −→ C and κC : cC −→ pC

A2.18.1

sending any pair (f, C) , where C is a C-object and f : A → C a C-morphism, to A and to f : A → C respectively, and any rgC C-morphism (h, D) : (f, C) −→ (f , C )

A2.18.2,

where C is a C-object and f : A → C a C-morphism fulﬁlling f ◦ h = D ◦ f , to h : A → A .

460

Frobenius categories versus Brauer blocks A2.19 Let X and Y be two 2-categories and consider representations m : X −→ CC and n : Y −→ CC

A2.19.1

of X and Y ; we call natural transformation from (X, m) to (Y, n) any pair (x, α) formed by a 2-functor and a 2-natural map x : X −→ Y and α : m −→ n ◦ x

A2.19.2;

in particular, we have the natural transformation (x, idn◦x ) : (X, n ◦ x) −→ (Y, n)

A2.19.3.

Analogously, for another representation m : X → CC and any natural map µ : m → m , we get the natural transformation (idX , µ) : (X, m) −→ (X, m )

A2.19.4.

For instance, it is quite clear that any functor f : C → C between two categories induces a 2-natural map Fct(st, f) : Fct(st, C) −→ Fct(st, C )

A2.19.5

and therefore we get the natural transformation id∆◦ , Fct(st, f) ; thus, according to the next proposition, we have a 2-functor ch(f) = Fct(st, f)id∆◦ : ch(C) −→ ch(C )

A2.19.6.

Proposition A2.20 Let X and Y be two 2-categories, m : X → CC and n : Y → CC respective representations of X and Y , and (x, α) a natural transformation from (X, m) to (Y, n) . Then, we have a 2-functor α x : m X −→ n Y

A2.20.1

mapping

A2.20.2 any m X-object (A, X) on the n Y-object αX (A), x(X) , A2.20.3 any mX-morphism (f, f) : (A, X) → (B, Y ) on the nY-morphism

αY (f ) ◦ xX,Y (f)−1 A , x(f) : αX (A), x(X) −→ αY (B), x(Y ) ,

A2.20.4 any m X-natural map µ : (f, f) → (f , f ) , where µ ∈ NatX (f , f) , on the n Y-natural map determined by xY,X (µ) .

A. Simpliciality

461

A2.21 With the same notation, if Z is a third 2-category and p : Z → CC a representation of Z , any natural transformation (y, β) : (Y, n) → (Z, p) can be composed with (x, α) : (X, m) → (Y, n) to get the natural transformation y ◦ x, (β ∗ x) ◦ α : (X, m) −→ (Z, p) A2.21.1 and we have

(β ∗ x) ◦ α (y ◦ x) = (β y) ◦ (α x)

A2.21.2.

Moreover, if (x , α ) : (X, m) → (Y, n) is another natural transformation, we call supranatural transformation from (x, α) to (x , α ) any pair (θ, σ) formed by a 2-natural map and a supranatural map θ : x −→ x

and σ : (n ∗ θ) ◦ α −→ α

A2.21.3.

For instance, it is quite clear that any natural map µ : f → f , where f and f are two functors from a category C to a category C , induces a supranatural map Fct(st, µ) : Fct(st, f) −→ Fct(st, f ) A2.21.4 sending any C-chain q : ∆n → C to the natural map µ ∗ q : f ◦ q → f ◦ q ; thus, according to the next proposition, we have a 2-natural map ch(µ) = Fct(st, µ)idid∆◦ : ch(f) −→ ch(f )

A2.21.5.

Proposition A2.22 Let (x, α) and (x , α ) be natural transformations from (X, m) to (Y, n) . A supranatural transformation (θ, σ) from (x, α) to (x , α ) determines a 2-natural map σ θ : α x −→ α x mapping any m X-object (A, X) on the n Y-morphism ((σX )A , θX ) : αX (A), x(X) −→ αX (A), x (X)

A2.22.1

A2.22.2

and any m X-morphism (f, f) : (A, X) → (B, Y ) on the n Y-natural isomorphism θY,X (f) : (σY )B , θY ◦ αY (f ) ◦ xY,X (f)−1 A , x(f) A2.22.3. ∼ = α (f ) ◦ x (f)−1 , x (f) ◦ (σX )A , θX Y

Y,X

A

Corollary A2.23 With the same notation, let (α , x ) natural transforma tions from (X, m) to (Y, n) and (θ , σ ) : (x , α ) → (x , α ) a supranatural transformation. Then, the pair θ ◦ θ, σ ◦ (n ∗ θ ) ∗ σ is a supranatural transformation from (x, α) to (x , α ) and we have (σ θ ) ◦ (σ θ) = σ ◦ (n ∗ θ ) ∗ σ (θ ◦ θ)

A2.23.1.

462

Frobenius categories versus Brauer blocks

Moreover, let Z be a 2-category, p : Z → CC a representation of Z , (y, β) and (y , β ) natural transformations from (Y, n) to (Z, p) , and (ω, τ ) a supranatural transformation from (y, β) to (y , β ). Then, we have (β y) ∗ (σ θ) = (β ∗ x ) ∗ σ ◦ (β < θ)−1 ∗ α (y ∗ θ) A2.23.2. (τ ω) ∗ (α x) = (τ < x) ∗ α (ω ∗ x) A2.24 Although not needed in this book, let us mention the existence of semidirect products between representations, between natural transformations and between supranatural transformations. Let X be a 2-category, m : X → CC a representation of X and q : m X → CC a representation of m X ; we obviously can consider the semidirect product q (m X) and the point is that we can deﬁne another representation of X q m : X −→ CC

A2.24.1

— called semidirect product of m by q — in such a way that we have a natural isomorphism (q m) X ∼ A2.24.2. = q (m X) A2.25 Similarly, let Y be another 2-category, n : Y → CC a representation of Y and r : n Y → CC a representation of n Y ; we already know that a natural transformation (x, α) from (X, m) to (Y, n) determines a 2-functor α x from m X to n Y and then, a 2-natural map τ : q → r ◦ (α x) builds a new natural transformation (α x, τ ) from (m X, q) to (n Y, r) and therefore determines a 2-functor τ (α x) : q (m X) −→ r (n Y)

A2.25.1.

Once again, the point is that we can deﬁne a 2-natural map τ α : q m −→ (r n) ◦ x

A2.25.2

— called semidirect product of α by τ — deﬁning a natural transformation (x, τ α) from (X, q m) to (Y, r n) in such a way that the 2-functors τ (α x) and (τ α) x coincide. A2.26 Finally, with the same notation, if we have a supranatural transformation (θ, σ) from (x, α) to another natural transformation (x , α ) from (X, m) to (Y, n) , we get a 2-natural map (cf. A2.22.1) σ θ : α x −→ α x

A2.26.1;

mutatis mutandis, a supranatural transformation (σ θ, ω) from (α x, τ ) to a second natural transformation (α x , τ ) from (m X, q) to (n Y, r) still determines a 2-natural map (cf. A2.20.1) ω (σ θ) : τ (α x) −→ τ (α x )

A2.26.2.

A. Simpliciality

463

Once again, we can deﬁne a supranatural map ω σ : (r n) ∗ θ ◦ (τ α) −→ τ α

A2.26.3

in such a way that, up to suitable identiﬁcations, we have ω (σ θ) = (ω σ) θ

A2.26.4.

A3 Cohomology of the small categories A3.1 As usual, in the simplicial 2-category ∆ (cf. A1.7) we denote by δin : ∆n → ∆n+1 the injective order-preserving map which does not cover i : ∆n+1 → ∆n the surjective order-preserving map in ∆n+1 , and by εn+1 i which covers i twice in ∆n ; it is well-known that all the nonidentity morphisms in ∆ are suitable compositions of these morphisms and that, for any i, j ∈ ∆n , we have * n+1 n if j ≤ i εi+1 ◦ δj δjn−1 ◦ εni = n+1 n εi ◦ δj+1 if i < j A3.1.1. n εn+1 ◦ δin = id∆n = εn+1 ◦ δi+1 i i

A3.2 Recall that, for any functor a : ∆ → Ab from the simplicial 2-category to the category of Abelian groups Ab , the graded homology group of a H(a) = Hn (a) A3.2.1 n∈N

is the graded homology group of the diﬀerential complex formed by the abelian groups a(∆n ) and the diﬀerential maps dan =

n+1

(−1)i a(δin ) : a(∆n ) −→ a(∆n+1 )

A3.2.2

i=0

where n runs over N ; explicitly, setting da−1 : {0} → a(∆0 ) , we have Hn (a) = Ker(dan )/Im(dan−1 )

A3.2.3.

A3.3 Note that if b : ∆ → Ab is another functor and µ : a → b a natural map, for any n ∈ N we get a group homomorphism Hn (µ) : Hn (a) −→ Hn (b)

A3.3.1.

and then, if c : ∆ → Ab is a third functor and λ : b → c a natural map, we obviously have Hn (λ ◦ µ) = Hn (λ) ◦ Hn (µ) A3.3.2.

464

Frobenius categories versus Brauer blocks

Moreover, it is well-known that whenever we have an exact sequence of such functors µ λ E : 0 −→ a −→ b −→ c −→ 0 A3.3.3, setting H−n (a) = H−n (b) = H−n (c) = {0} for any n > 0 , there is a 1-graded Z-module homomorphism δE :

Hn (c) −→

n∈Z

Hn (a)

A3.3.4

n∈Z

— called a connecting homomorphism — such that we have the following exact triangle (cf. §2 in [18, Ch. III] and Theorem 1.1 in [18, Ch. IV])

n∈Z

δE

Hn (c)

⊕n∈Z Hn (λ)

3

−→

n∈Z

0

n∈Z

Hn (a)

⊕n∈Z Hn (µ)

A3.3.5.

Hn (b)

In order to exploit this exactness, it is useful to employ the following lemma which is an easy consequence of equalities A3.1.1. Lemma A3.4 Let a : ∆ → Ab be a functor. For any n ∈ N we have n+1 ida(∆n+1 ) = a(εn+2 ) ◦ da◦sh + da◦sh ) n n−1 ◦ a(ε0 0

A3.4.1

0 where we set da◦sh −1 = a(δ1 ) . In particular, we have

H0 (a ◦ sh) = Im a(δ10 )

and

Hn (a ◦ sh) = {0} for any n ≥ 1

A3.4.2.

A3.5 An immediate consequence of this lemma is: A3.5.1 Any functor i : ∆ → Ab which is an injective object in the category of functors Fct(∆, Ab) fulﬁlls Hn (i) = {0} for any n ≥ 1 . Indeed, since i(εn+1 ) ◦ i(δ0n ) = idi(∆n ) , the natural map (cf. A1.13) 0 i ∗ ς : i −→ i ◦ sh

A3.5.2

is injective, so that i is a direct summand of i ◦ sh and it suﬃces to apply Lemma A3.4. A3.6 From equality A3.2.3 it is easily checked that H0 (a) coincides with the inverse limit lim a and then, this remark and the exact triangle A3.3.5 ←−

shows that Hn is the n-th right derived functor (cf. §3 in [18, Ch. V]) of the functor lim : Fct(∆, Ab) −→ Ab A3.6.1. ←−

A. Simpliciality

465

On the other hand, denoting by t the trivial functor from ∆ to Ab mapping the ∆-objects on Z and the ∆-morphisms on idZ , it is clear that lim a ∼ A3.6.2 = Nat(t, a) ←−

and therefore, for any projective resolution of t ... → pn+1 → pn → ... → p1 → t → 0

A3.6.3,

H(a) still coincides with the graded homology group of the diﬀerential complex (cf. Proposition 1.1 in [18, Ch. VI]) ... ←− Nat(pn+1 , a) ←− Nat(pn , a) ←− ... ←− Nat(p1 , a)

A3.6.4.

A3.7 Let C be a small category; in order to recall the deﬁnition of the cohomology groups of a contravariant functor from C to Ab , we have to consider the following subcategory — called the naive category of chains of C — of the category of chains ch(C) (cf. A2.2, A2.3, A2.5 and A2.7) cho (C) = Fct(st, C)o ∆◦

A3.7.1

with its structural functor (cf. A2.7.1) loC = p∆◦ ,Fct(st,C)o : cho (C) −→ ∆◦

A3.7.2,

and the evaluation functor vC : ch(C) −→ C

A3.7.3

mapping any ch(C)-object (q, ∆n ) on the C-object q(0) , and any ch(C)-morphism (µ, δ) : (r, ∆m ) → (q, ∆n ) on the composition of C-morphisms µ0 r(0•δ(0)) r(0) −−−−−→ r δ(0) −−→ q(0) A3.7.4; note that this functor factorizes throughout ch(C) = π0 ch(C) (cf. A1.8). We denote by v∗C and voC the respective restrictions of vC to ch∗ (C) and cho (C) . A3.8 But, as explained in the Introduction (cf. I 33), in this book we have to consider slightly more general cohomology groups, namely the cohomology groups of contravariant functors from ch∗ (C) to Ab . More generally, any contravariant functor ao : cho (C) → Ab determines a functor from ∆ to Ab via the direct image throughout loC (cf. 1.6 and A2.13.3) and for any n ∈ N we simply deﬁne Hno (C, ao ) = Hn (loC )∗ (ao ) A3.8.1 — called the naive n-cohomology group of C over ao . Since Hn and (loC )∗ are true functors (lo )∗ Hn Fct cho (C)◦ , ∆ −−C−→ Fct(∆, Ab) −→ Ab A3.8.2, we actually get a functor

Hno (C, •) : Fct cho (C)◦ , Ab −→ Ab

A3.8.3.

466

Frobenius categories versus Brauer blocks

A3.9 Usually, we only consider contravariant functors a∗ : ch∗ (C) → Ab which factorize throughout ch∗ (C) (cf. A1.8) and then, denoting by ao the restriction of a∗ to cho (C) , it is not diﬃcult to check that we have a natural isomorphism lim a∗ ∼ A3.9.1. = H0o (C, ao ) ←−

Moreover, if we start from an ordinary contravariant functor a : C → Ab , in order to apply our previous deﬁnition it suﬃces to consider the composed functor a ◦ voC : cho (C) → C → Ab and, for any n ∈ N , we simply set Hn (C, a) = Hno (C, a ◦ voC ) = Hn (loC )∗ (a ◦ voC ) A3.9.2 — called the n-cohomology group of C over a. Once again, we get a functor Hn (C, •) : Fct(C◦ , Ab) −→ Ab

A3.9.3.

A3.10 More generally, the correspondence mapping any small category C on the small category Fct cho (C)◦ , Ab , mapping any functor f : C → C between two small categories on the obvious functor Fct cho (f), Ab : Fct cho (C )◦ , Ab −→ Fct cho (C)◦ , Ab A3.10.1 which sends any contravariant functor ao : cho (C ) → Ab to ao ◦ cho (f) , and mapping any natural map µ : f → f between two such functors on the natural map sending ao to ao ∗ cho (µ) , actually deﬁnes a representation of (CC)◦ Fct cho (•), Ab◦ : (CC)◦ −→ CC A3.10.2 and then it is easily checked that the naive n-cohomology groups deﬁne a functor Hno : Fct cho (•), Ab◦ (CC)◦ −→ Ab A3.10.3; mutatis mutandis, the ordinary n-cohomology groups deﬁne a functor Hn : Fct(•, Ab◦ ) (CC)◦ −→ Ab

A3.10.4.

A3.11 Explicitly, the direct image (loC )∗ (ao ) (cf. 1.6) maps ∆n on

(loC )∗ (ao ) (∆n ) =

ao (q, ∆n )

A3.11.1

q∈Fct(∆n ,C)

and, according to deﬁnition A3.2.2, diﬀerential map sends the corresponding an element a = (aq )q∈Fct(∆n ,C) of (loC )∗ (ao ) (∆n ) on (lo )∗ (ao )

dn C

(a) =

n+1 i=0

(−1)i ao (idr◦δin , δin ) (ar◦δin )

r∈Fct(∆n+1 ,C)

A3.11.2.

A. Simpliciality

467

In particular, (loC )∗ maps an exact sequence of contravariant functors from cho (C) to Ab µo

λo

E o : 0 −→ ao −→ bo −→ co −→ 0

A3.11.3

on an exact sequence of functors (loC )∗ (E o ) from ∆ to Ab and therefore, setting o −n o −n o again H−n o (C, a ) = Ho (C, b ) = Ho (C, c ) = {0} for any n > 0 , we have the exact triangle (cf. A3.3.5)

δ

Hno (C, co )

(lo )∗ (E o ) C

−−−−−−→

n∈Z

Hno (C, ao )

n∈Z

3

0

A3.11.4.

Hno (C, bo )

n∈Z

A3.12 As above, this exact triangle shows that Hn (C, •) is the n-th right derived functor of the functor lim : Fct(C◦ , Ab) −→ Ab ←−

A3.12.1,

provided we prove a statement analogous to A3.5.1. This depends on the covering functor cC and the covering natural map κC introduced in A2.18; for any contravariant functor a : C → Ab , we have to consider the new contravariant functor (pC )∗ (a ◦ cC ) : C −→ Ab A3.12.2 and the injective natural map (pC )∗ (a ∗ κC ) ◦ ιa : a −→ (pC )∗ (a ◦ pC ) −→ (pC )∗ (a ◦ cC )

A3.12.3

where ιa denotes the image of ida◦pC by the adjoinness isomorphism (cf. 1.6) Nat(a ◦ pC , a ◦ pC ) ∼ = Nat a, (pC )∗ (a ◦ pC )

A3.12.4.

Indeed, with them we can develop a similar argument since we have the following result (cf. Lemma A3.4 and deﬁnition A3.8.1). Proposition A3.13 With the notation above, we have a natural isomorphism (loC )∗ (pC )∗ (a ◦ cC ) ◦ voC ∼ A3.13.1. = (loC )∗ (a ◦ voC ) ◦ sh In particular, we have Hn C, (pC )∗ (a ◦ cC ) = {0} for any n ≥ 1 .

468

Frobenius categories versus Brauer blocks

A3.14 As above, an immediate consequence of this proposition is: A3.14.1 Any contravariant functor i : C → Ab which is an injective object in the category of functors Fct(C◦ , Ab) fulﬁlls Hn (C, i) = {0} for any n ≥ 1 . Moreover, it is easy to see that any functor f : B → C such that the direct image f∗ (cf. 1.6) preserves monomorphisms, composed with an injective object in the category Fct(C◦ , Ab) , becomes an injective object in Fct(B◦ , Ab) ; hence, from the Isomorphism Criterion (cf. Proposition 4.4 in [18, Ch. V]) we get the following result. Proposition A3.15 For any pair of small categories B and C , any contravariant functor a : C → Ab and any functor f : B → C , if f∗ preserves monomorphisms and f induces a group isomorphism H0 (C, a) ∼ = H0 (B, a ◦ f) then, for any n ≥ 1 , f induces a group isomorphism Hn (C, a) ∼ = Hn (B, a ◦ f)

A3.15.1.

Corollary A3.16 For any small category C , any contravariant functor a : C → Ab and any n ∈ N , the evaluation functor vC : ch(C) → C induces a group isomorphism ¯C Hn (C, a) ∼ A3.16.1. = Hn ch(C), a ◦ v A3.17 We also have to consider the stable cohomology groups of C or, more generally, the D-stable cohomology groups of C for a subcategory D of C , with the same set of objects, over a contravariant functor a∗ : ch∗ (C) −→ Ab

A3.17.1.

In order to deﬁne it, we consider the weak structure of Fct(st, C) (cf. A2.3 and A2.5) Fct(st, C)∗D : ∆◦ −→ CC A3.17.2 mapping ∆n on the subcategory Fct(∆n , C)∗D of the category of functors Fct(st, C) formed by all the functors and, for any pair of them q : ∆n → C and q : ∆n → C , by the natural maps ν : q → q such that ν˜i : q(i) → q (i) is a D-isomorphism; we consider the semidirect product (cf. A2.7) ch∗D (C) = Fct(st, C)∗D ∆◦

A3.17.3

which is clearly contained in the proper category of C-chains ch∗ (C) (cf. A2.8) and contains the naive category of C-chains cho (C) (cf. A3.7). ∗ ◦ ∗ A3.18 Then, denoting by lD C : chD (C) → ∆ the restriction to chD (C) of the structural functor p∆◦ ,Fct(st,C) (cf. A2.7.1) and by aD the restriction of a∗ to ch∗D (C) , for any n ∈ N we also consider D HnD (C, a∗ ) = Hn (lD A3.18.1 C )∗ (a )

A. Simpliciality

469

— called the D-stable n-cohomology group of C over a∗ ; if D = C , we write n ∗ l∗C and Hn∗ (C, a∗ ) instead of lD C and HD (C, a ) respectively, and call stable ∗ n-cohomology group of C over a this group Hn∗ (C, a∗ ) . More explicitly, the D direct image (lD C )∗ (a ) (cf. 1.6) maps ∆n on the subgroup

D (lD C )∗ (a ) (∆n ) ⊂

a∗ (q, ∆n )

A3.18.2

q∈Fct(∆n ,C)

formed by the D-stable elements, namely the elements a = (aq )q∈Fct(∆n ,C) ful ﬁlling aq = a∗ (ν, id∆n ) (aq ) for any ch(D)-isomorphism (ν, id∆n ) between two ch(C)-objects (q, ∆n ) and (q , ∆n ) . Since the direct image (lD C )∗ (cf. 1.6) need not preserve the surjectivity, these groups need not provide any (exact) triangle (cf. A3.11.4). A4 Basic results on the cohomology of small categories A4.1 Let C be a small category; here we collect some basic results on the naive and the stable cohomology of C over a contravariant functor a∗ : ch∗ (C) −→ Ab

A4.1.1

and we denote by ao the restriction of a∗ to cho (C) . Most of these results depend on the following lemma and proposition†. Lemma A4.2 Let q and r be functors from ∆n to C and µ : q → r a natural map. For any i ∈ ∆n , denote by hni (µ) : ∆n+1 → C the functor which coincides with q over ∆i , coincides with r ◦ εn+1 over ∆n+1 − ∆i , and maps i i • (i+1) on µi : q(i) → r(i) , and by κni (µ) : q ◦ εn+1 −→ hni (µ) i

A4.2.1

the natural map sending j ≤ i to idq(j) , and i < j ≤ n + 1 to µj−1 . Then, for any i ∈ ∆n and any j ∈ ∆n−1 , we have n n hnj+1 (µ) ◦ δj+1 = hnj (µ) ◦ δj+1 n n κnj+1 (µ) ∗ δj+1 = κnj (µ) ∗ δj+1 * n hj+1 (µ) ◦ δin n−1 hn−1 (µ ∗ δ ) = n j i hnj (µ) ◦ δi+1 * n κj+1 (µ) ∗ δin κn−1 (µ ∗ δin−1 ) = n j κnj (µ) ∗ δi+1

†

We learned them from Alberto Arabia.

if i ≤ j if j < i . if i ≤ j if j < i

A4.2.2.

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Frobenius categories versus Brauer blocks

n n Proof: Indeed, hnj+1 (µ) ◦ δj+1 and hnj (µ) ◦ δj+1 coincide with q over ∆i and with q over ∆n − ∆i , and both map j • (j +1) on

µj+1 ◦ q (j • (j +1)) = q(j • (j +1)) ◦ µi

A4.2.3;

n n then, κnj+1 (µ) ∗ δj+1 and κnj (µ) ∗ δj+1 coincide with the natural map from q n n to the functor hj (µ) ◦ δj+1 , sending j ≤ i to idq(j) , and i < j ≤ n to µj . Moreover, for any D ≤ j we have

hn−1 (µ ∗ δin−1 ) (D) = (q ◦ δin−1 )(D) j

A4.2.4

and therefore we get

n  n  q (D+1)= hj+1 (µ) ◦δi (D) if i ≤ D if D < i ≤ j A4.2.5. hn−1 (µ ∗ δin−1 ) (D) = q (D) = hnj+1 (µ) ◦ δin (D) j n  n q (D) = hj (µ) ◦ δi+1 (D) if j < i On the contrary, if j < D ≤ n then we have

hn−1 (µ ∗ δin−1 ) (D) = (q ◦ δin−1 )(D−1) j

A4.2.6

and therefore we get

n  n if i ≤ j  q(D) = hj+1 (µ) ◦ δi (D) n−1 n n q(D) = h (µ) ◦ δ (D) if j < i < D A4.2.7. hn−1 (µ ∗ δ ) (D) = j i+1 j i  q(D−1) = hnj (µ) ◦ dni+1 (D) if D ≤ i

More precisely, it is easily checked that the announced equalities still hold over the ∆i - and the “∆n −∆i ”-morphisms; ﬁnally, we have

(µ ∗ δin−1 ) (j • j +1) = µδn−1 (j) hn−1 j i * n h (µ) ◦ δin (j • j +1) if i ≤ j = j+1 n hnj (µ) ◦ δi+1 (j • j +1) if j < i

A4.2.8.

Mutatis mutandis, for any D ≤ j we have

κn−1 (µ ∗ δin−1 ) (D) = id(q◦δn−1 )(6) j i

A4.2.9

and therefore we get

n  κj+1 (µ) ∗ δin (D) if i ≤ D  idq(6+1) = if D < i ≤ j A4.2.10. κn−1 (µ∗δin−1 ) (D) = idq(6) = κnj+1 (µ) ∗ δin (D) j  n idq(6) = κnj (q) ∗ δi+1 (D) if j < i

A. Simpliciality

471

On the contrary, if j < D ≤ n then we have

κn−1 (µ ∗ δin−1 ) (D) = µδn−1 (6−1) j j

A4.2.11

and therefore we get

n  n if i ≤ j  µ6 = κj+1 (µ) ∗ δi (D) n−1 n n µ = κ (µ) ∗ δ (D) if j < i < D κn−1 (µ ∗ δ ) (D) = 6 i+1 j i j  n µ6−1 = κnj (µ) ∗ δi+1 (D) if D ≤ i

A4.2.12.

We are done. A4.3 Let B be a second small category and f : B → C a functor, so that we have a contravariant functor a∗ ◦ ch∗ (f) : ch∗ (B) −→ Ab

A4.3.1,

where ch∗ (f) denotes the functor ch∗ (B) → ch∗ (C) obtained by the restriction of ch(f) (cf. A2.19.6); recall that we set (cf. A3.7.2) loC = p∆◦ ,Fct(st,C)o

and loB = p∆◦ ,Fct(st,B)o

A4.3.2

and, denoting by cho (f) the corresponding restriction of ch(f) , it is clear that we have loB = loC ◦ cho (f) . Then, considering the natural map (loC )∗ (ao ) ◦ loC −→ ao

A4.3.3

determined by id(loC )∗ (ao ) and by the adjoinness natural isomorphism (cf. 1.6), the composition with cho (f) yields a new natural map (loC )∗ (ao ) ◦ loB −→ ao ◦ cho (f)

A4.3.4

and ﬁnally, by applying again the adjoinness natural isomorphism, we get the natural map ωf : (loC )∗ (ao ) −→ (loB )∗ ao ◦ cho (f)

A4.3.5;

for any n ∈ N note that, by its very deﬁnition, Hn (ωf ) coincides with the image — denoted by Hno (f, a∗ ) — of the Fct cho (•), Ab◦ (CC)◦ -morphism (idao ◦cho (f) , f) by the functor Hno (cf. A3.10.3). A4.4 Let g : B → C be a second functor and µ : f → g a natural map; on the one hand, we have a second natural map ωg : (loC )∗ (ao ) −→ (loB )∗ ao ◦ cho (g)

A4.4.1.

472

Frobenius categories versus Brauer blocks

On the other hand, assume either that µ is a natural isomorphism — so that a∗ ∗ ch(µ) makes sense — or that a∗ can be extended to a contravariant functor a from ch(C) to Ab — so that a ∗ ch(µ) is deﬁned; in both cases, we get a natural map a? ∗ ch(µ) : ao ◦ cho (g) −→ ao ◦ cho (f)

A4.4.2

where a? coincides with a∗ or a accordingly, which determines a third natural map ωµ : (loB )∗ ao ◦ cho (g) −→ (loB )∗ ao ◦ cho (f) A4.4.3 and in the next proposition we prove that the corresponding triangle is homotopically commutative. For any n ∈ N , let us set (cf. A3.11) ∗

(lo )∗ (ao )

(d a )n = dn C

(lo )∗ (ao ◦cho (f))

∗

(d a ,f )n = dn B

and

A4.4.4

and let us consider the group homomorphism hn : (loC )∗ (∆n+1 ) −→ (loB )∗ (∆n ) mapping a = (ar )r∈Fct(∆n+1 ,C) ∈ hn (a) =

n

(loC )∗ (∆n+1 )

A4.4.5

on (cf. Lemma A4.2)

(−1)i a? (κni (µ ∗ q), εn+1 ) (ahni (µ∗q) ) i

i=0

q∈Fct(∆n ,B)

A4.4.6.

Proposition A4.5 Let B and C be two small categories, f : B → C and g : B → C two functors, µ : f → g a natural map and a∗ : ch∗ (C) → Ab a contravariant functor. With the notation and the hypothesis above, for any n ∈ N we have ∗

∗

(ωf − ωµ ◦ ωg )∆n = (d a ,f )n−1 ◦ hn−1 + hn ◦ (d a )n

A4.5.1.

In particular, the following diagram is commutative: Hno B, a? ◦ ch∗ (g)

Hn (B,a? ∗ch(µ))

o −− −−−−−−−→

∗ Hn o (g,a )

3

1

Hno B, a? ◦ ch∗ (f) A4.5.2.

∗ Hn o (f,a )

Hno (C, a∗ ) Proof: For any a = (aq )q∈Fct(∆n ,C) in (loC )∗ (∆n ) we have ∗ (d a ,f )n−1 hn−1 (a) =

n−1 i=0

=

∗

(−1)i (d a ,f )n−1

n n−1 j=0 i=0

n n−1 a? (κn−1 (µ ∗ t), ε ) (a ) i i h (µ∗t) i

A4.5.3,

t

(−1)i+j a? (κn−1 (µ ∗ qj ), δjn−1 ◦ εni ) (ahn−1 (µ∗qj ) ) i i

q

A. Simpliciality

473

where t and q respectively run over Fct(∆n−1 , B) and Fct(∆n , C) , and we set qj = q ◦ δjn−1 ; similarly, the second term of the right-hand member maps a on n+1 ∗ hn (d a )n (a) = (−1)j hn a? (idr◦δjn , δjn ) (ar◦δjn ) j=0

=

n n+1

r

(−1)i+j a? (κni (µ ∗ q) ◦ δjn , εn+1 ◦ δjn ) (ahni (µ∗q)◦δjn ) i

j=0 i=0

A4.5.4, q

where r and q respectively run over Fct(∆n+1 , C) and Fct(∆n , C) . But, according to Lemma A4.2, for any j ∈ ∆n and any i ∈ ∆n−1 we know that n n hni+1 (µ ∗ q) ◦ δi+1 = hni (µ ∗ q) ◦ δi+1 n n κni+1 (µ ∗ q) ∗ δi+1 = κni (µ ∗ q) ∗ δi+1 * n hi+1 (µ ∗ q) ◦ δjn if j ≤ i hn−1 (µ ∗ qj ) = n i hni (µ ∗ q) ◦ δj+1 if i < j . * n n κi+1 (µ ∗ q) ∗ δj if j ≤ i κn−1 (µ ∗ qj ) = n i κni (µ ∗ q) ∗ δj+1 if i < j

A4.5.5.

Consequently, in the bottom member of equality A4.5.4, the terms where j = i and j = i + 1 cancel with each other for any 1 ≤ j ≤ n ; moreover, the term (i, j) in the bottom member of equality 4.5.3 cancels either with the term (i + 1, j) if 1 ≤ j ≤ i ≤ n − 1 , or with the term (i, j + 1) in the bottom member of equality 4.5.4 if 0 ≤ i < j ≤ n . Finally, the term (i, 0) in the bottom member of equality 4.5.3 cancels with the term (i + 1, 0) in the bottom member of equality 4.5.4 for any 0 ≤ i ≤ n − 1 , whereas the terms (0, 0) and (n, n + 1) respectively coincide with −(ωµ ◦ ωg )∆n (a) and (ωf )∆n (a) which proves equality A4.5.1. We are done. Corollary A4.6 With the notation and the hypothesis above, for any n ∈ N , hn preserves stability sending (l∗C )∗ (∆n+1 ) to (loB )∗ (∆n ) . In particular, we have Hn∗ (f, a∗ ) = Hn∗ B, a? ∗ ch(µ) ◦ Hn∗ (g, a∗ ) A4.6.1. Proof: For any a = (ar )r∈Fct(∆n+1 ,C) ∈ (l∗C )∗ (∆n+1 ) , any isomorphic functors q and q from ∆n to B , any natural isomorphism ν : q ∼ = q and any i ∈ ∆n , it is quite clear that we have a natural isomorphism hni (µ, ν) : hni (µ ∗ q) ∼ = hni (µ ∗ q )

A4.6.2

474

Frobenius categories versus Brauer blocks

sending j ∈ ∆n+1 to

hni (µ, ν)j =

 = (f ◦ q )(j)  f(νj ) : (f ◦ q)(j) ∼ 

if j ≤ i

g(νj ) : (g ◦ q)(j − 1) ∼ = (g ◦ q )(j − 1) if j > i

A4.6.3

and therefore we get ahni (µ∗q ) = a? hni (µ, ν), idid∆n+1 (ahni (µ∗q) ) , so that

a? (κni (µ ∗ q ), εn+1 ) (ahni (µ∗q ) ) i = a? hni (µ, ν) ∗ κni (µ ∗ q ), εn+1 (ahni (µ∗q) ) i = a? κni (µ ∗ q) ◦ (f ∗ ν), εn+1 (ahni (µ∗q) ) i n (µ∗q) ) = a? (f ∗ ν, idid∆n ) a? (κni (µ ∗ q), εn+1 ) (a h i i

4.6.4.

Consequently, it follows from the very deﬁnition of hn (cf. A4.4.6) that hn (a) belongs to (loB )∗ (∆n ) and then equality 4.6.1 follows from equality 4.5.1. Proposition A4.7 Let B and C be two small categories, f : B → C and g : C → B two functors, µ : idC → f ◦ g and ν : idB → g ◦ f two natural maps such that µ ∗ f = f ∗ ν , and a∗ : ch∗ (C) → Ab a contravariant functor. Assume either that µ and ν are natural isomorphisms, or that a∗ can be extended to a contravariant functor a from ch(C) to Ab , and denote by a? either a∗ or a accordingly. Then, for any n ∈ N , we have the group isomorphisms Hno (f, a∗ ) : Hno (C, a∗ ) ∼ = Hno B, a? ◦ ch(f) Hn (f, a∗ ) : Hn (C, a∗ ) ∼ = Hn B, a? ◦ ch(f) ∗

∗

A4.7.1.

∗

Proof: Applying Proposition A4.5 and Corollary A4.6 to the functors idC and f ◦ g , for any n ∈ N we get Hno C, a? ∗ ch(µ) ◦ Hno (f ◦ g, a∗ ) = Hno (idC , a∗ ) = idHno (C,a∗ ) Hn∗ C, a? ∗ ch(µ) ◦ Hn∗ (f ◦ g, a∗ ) = Hn∗ (idC , a∗ ) = idHn∗ (C,a∗ )

A4.7.2

and therefore the group homomorphisms (cf. A3.10) Hno (f ◦ g, a∗ ) = Hno g, a? ◦ ch(f) ◦ Hno (f, a∗ ) Hn∗ (f ◦ g, a∗ ) = Hn∗ g, a? ◦ ch(f) ◦ Hn∗ (f, a∗ ) are injective, so that Hno (f, a∗ ) and Hn∗ (f, a∗ ) are injective too.

A4.7.3

A. Simpliciality

475

Mutatis mutandis, applying Proposition A4.5 and Corollary A4.6 to the functors idB and g ◦ f , for any n ∈ N we get Hno B, a? ◦ ch(f) ∗ ch(ν) ◦ Hno g ◦ f, a? ◦ ch(f) = idHno (B,a? ◦ch(f)) Hn∗ B, a? ◦ ch(f) ∗ ch(ν) ◦ Hn∗ g ◦ f, a? ◦ ch(f)

A4.7.4,

= idHn∗ (B,a? ◦ch(f)) so that the group homomorphisms (cf. A3.10) Hno B, a? ◦ ch(f) ∗ ch(ν) ◦ Hno f, a? ◦ ch(f ◦ g) Hn∗ B, a? ◦ ch(f) ∗ ch(ν) ◦ Hn∗ f, a? ◦ ch(f ◦ g)

A4.7.5

are surjective; but, since we are assuming that µ ∗ f = f ∗ ν , we get (cf. A3.10) Hno (f, a∗ ) ◦ Hno C, a? ∗ ch(µ) = Hno B, a? ◦ ch(f) ∗ ch(ν) ◦ Hno f, a? ◦ ch(f ◦ g) Hn∗ (f, a∗ ) ◦ Hn∗ C, a? ∗ ch(µ) = Hn∗ B, a? ◦ ch(f) ∗ ch(ν) ◦ Hn∗ f, a? ◦ ch(f ◦ g)

A4.7.6;

hence, Hno (f, a∗ ) and Hn∗ (f, a∗ ) are surjective too. We are done. Corollary A4.8 Let C be a small category with a ﬁnal object and a : C → Ab a contravariant functor. For any n ≥ 1 we have Hn (C, a) = {0} . Proof: We apply Proposition A4.7 to the subcategory B of C over the ﬁnal object, to the inclusion functor i : B → C , to the unique functor j : C → B , to the unique natural map µ : idC → i ◦ j , which actually fulﬁlls µ ∗ i = idi , and to the contravariant functor a ◦ vC : ch(C) → Ab ; then, the corollary follows. Corollary A4.9 Let B and C be two small categories, f : B → C an equivalence of categories and a∗ : ch∗ (C) → Ab a contravariant functor. Then, for any n ∈ N , Hno (f, a∗ ) and Hn∗ (f, a∗ ) are group isomorphisms. Proof: It is well-known that there is a functor g : C → B such that we have two natural isomorphisms µ : idC ∼ = f ◦ g and ν : idB ∼ =g◦f and then the corollary follows from Proposition A4.7.

A4.9.1

476

Frobenius categories versus Brauer blocks

A4.10 Consider the additive cover ac(C) = pc(C◦ )◦ of C deﬁned in A2.7.3 above. That is to say, according to Proposition A2.6,

the ac(C)-objects are the ﬁnite sequences {Ci }i∈I — denoted by C = C — of C-objects

i∈I i

and an ac(C)-morphism from another object D = j∈J Dj to C = i∈I Ci is a pair (α, f ) formed by a map f : J → I and by a family α = {αj }j∈J of C-morphisms αj : Dj → Cf (j) ; the composition of (α, f ) with another morphism (β, g) : E = E6 −→ D A4.10.1, 6∈L

is the pair (α ∗ g) ◦ β, f ◦ g where, if β is the family {β6 }6∈L , (α ∗ g) ◦ β is the family {αg(6) ◦ β6 }6∈L of composed C-morphisms αg(6) ◦ β6 : E6 −→ Dg(6) −→ C(f ◦g)(6)

A4.10.2.

In particular, we have an obvious functor jC : C → ac(C) , and any contravariant functor a : C → Ab can be extended to a contravariant functor ac(a) : ac(C) −→ Ab mapping C on

i∈I

A4.10.3

a(Ci ) and (α, f ) on the group homomorphism i∈I

a(Ci ) −→

a(Dj )

A4.10.4

j∈J

sending i∈I ai , where ai ∈ a(Ci ) , to j∈J a(αj ) (af (j) ) . In [32], S. Jackowski and J. McClure prove the following result. Proposition A4.11 With the notation above, ac(a) is the direct image of a throughout jC and, for any n ∈ N , jC induces a group isomorphism Hn ac(C), ac(a) ∼ = Hn (C, a)

A4.11.1.

Proof: For any contravariant functor b : ac(C) → Ab and any natural map µ : b ◦ jC → a , we have an obvious natural map ac(µ) : ac(b ◦ jC ) −→ ac(a)

A4.11.2.

Moreover, for any ac(C)-object C = ⊂∈I Ci , the evident ac(C)-morphisms Ci → C determine group homomorphisms b(C) → b(Ci ) for any i ∈ I and therefore we get a group homomorphism b(C) → ac(a) (C) ; as a matter of fact, this correspondence deﬁnes a natural map λb : b → ac(b ◦ jC ) which

A. Simpliciality

477

fulﬁlls λb ∗ jC = idb◦jC . Consequently, we have obtained an injective group homomorphism Nat(b ◦ jC , a) −→ Nat b, ac(a) A4.11.3 mapping µ on ac(µ) ◦ λb , which is easily checked to be surjective, proving that (jC )∗ (a) = ac(a) . Moreover, we have ac(a) ◦ jC = a , ac clearly preserves the monomorphisms and jC induces a group isomorphism H0 ac(C), ac(a) ∼ = lim a ∼ = H0 (C, a) = lim ac(a) ∼ ←−

←−

A4.11.4;

hence, isomorphism A4.11.1 follows from Proposition A3.15. A4.12 Finally, for any small category C , any subcategory D over the same set of objects and any contravariant functor a∗ : ch∗ (C) → Ab , let us discuss the relationship between the groups Hno (C, a∗ ) and HnD (C, a∗ ) (cf. A3.18). Since we have a natural inclusion (cf. A3.7, A3.11.1 and A3.18) D o o ι(aD ) : (lD C )∗ (a ) −→ (lC )∗ (a )

A4.12.1,

we always have a group homomorphism HnD (C, a∗ ) → Hno (C, a∗ ) for any n ∈ N and, for n = 0 , it is easily checked that H0D (C, a∗ ) ∼ = H0o (C, a∗ )

A4.12.2.

We are interested in the case where the group of D-automorphisms of any object is ﬁnite and has an order which is invertible in a suitable ring R , and we consider contravariant functors from ch∗ (C) to the category R-mod of ﬁnitely generated R-modules. Proposition A4.13 Let C be a small category, D a subcategory over the same set of objects, where the group of D-automorphisms of any C-object is ﬁnite, and R a ring such that, for any C-object C , |D(C)| is invertible in R . Then, for any contravariant functor m∗ : ch∗ (C) → R-mod and any n ∈ N , D o o the inclusion (lD C )∗ (m ) ⊂ (lC )∗ (m ) induces a group isomorphism HnD (C, m∗ ) ∼ = Hno (C, m∗ )

A4.13.1.

Proof: According to Corollary A4.9, we may assume that two C-isomorphic C-objects always coincide; for any n ∈ N , let us denote by o D o ι(mD )n : (lD C )∗ (m ) (∆n ) −→ (lC )∗ (m ) (∆n )

A4.13.2

478

Frobenius categories versus Brauer blocks

the corresponding inclusion map; in our situation, it admits a section σ(mD )n mapping any v = (vq )q∈Fct(∆n ,C) , where vq ∈ m(q, ∆n ) , on σ(mD )n (v) =

1 · b(ν, id∆n )(vq ) zq q∈Fct(∆n ,C)

A4.13.3,

q ,ν

where q runs over the set of functors from ∆n to C which are D-isomorphic to q , ν runs over the set of natural D-isomorphisms q ∼ = q , and we set D q(i) zq = A4.13.4, i∈∆n

which coincides with the cardinal of the set of pairs (q , ν) . Indeed, it such D D is easily checked that σ(m )n (v) belongs to (lC )∗ (mD ) (∆n ) and that we have σ(mD )n ◦ ι(mD )n = id((lD )∗ (mD ))(∆n ) A4.13.5. C

Moreover, for any i ≤ n + 1 , it is straightforward to check the commutativity of the diagram

σ(mD )n+1 o D (lD −−−−−−− (lC )∗ (mo ) (∆n+1 ) C )∗ (m ) (∆n+1 ) ← ↑ ((loC )∗ (mo ))(δin ) D σ(m )n (lC )∗ (mD ) (∆n ) ←−−−−−−− (loC )∗ (mo ) (∆n )

D n ((lD C )∗ (m ))(δi )

↑

A4.13.6.

D

In particular, the family of sections σ(mD ) = {σ(mD )n }n∈N still determines a homomorphism between the diﬀerential complexes associated with the functors (cf. A3.2) D (lD C )∗ (m ) : ∆ −→ R-mod and

(loC )∗ (mo ) : ∆ −→ R-mod

so that, for any n ∈ N , we get a group homomorphism D Hn σ(mD ) : Hn (loC )∗ (mo ) −→ Hn (lD C )∗ (m ) + + Hno (C, m∗ ) HnD (C, m∗ ) moreover, it follows from equalities A4.13.5 that we have Hn σ(mD ) ◦ Hn ι(mD ) = idHn (C,m∗ ) D

A4.13.7,

A4.13.8;

A4.13.9

and we claim that we also have Hn ι(mD ) ◦ Hn σ(mD ) = idHno (C,m∗ ) . Once again using Lemma A4.2, for any n ∈ N we consider the group homomorphism hn : (loC )∗ (∆n+1 ) −→ (loC )∗ (∆n ) A4.13.10

A. Simpliciality

479

mapping v = (vr )r∈Fct(∆n+1 ,C) ∈ (loC )∗ (∆n+1 ) on n (−1)i ∗ n hn (v) = · m κi (ν), εn+1 (vhni (ν) ) i zr q∈Fct(∆n ,C) i=0

A4.13.11

q ,ν

where q runs over the set of functors ∆n → C which are D-isomorphic to q , and ν over the set of natural D-isomorphisms q ∼ = q . Now, we claim that id(loC )∗ (∆n ) − ι(mD )n ◦ σ(mD )n = dn−1 ◦ hn−1 + hn ◦ dn

A4.13.12,

where dn : (loC )∗ (∆n ) → (loC )∗ (∆n+1 ) denotes the corresponding diﬀerential map (cf. A3.11). As in the proof of Proposition A4.5, the ﬁrst term of the right-hand member maps v = (vq )q∈Fct(∆n ,C) on dn−1 hn−1 (v) =

n−1 i=0

=

(−1)i dn−1

1 ·m∗ κn−1 (λ), εni (vhn−1 (λ) ) i i zt t t ,λ

A4.13.13

n n−1 (−1)i+j n−1 · wi,j (ν ∗ δjn−1 ) zq q i=0 j=0 q ,ν

where (t, t ) runs over the set of pairs of D-isomorphic functors from ∆n−1 to C , λ runs over the set of natural D-isomorphisms t ∼ = t , (q, q ) runs over the set of pairs of D-isomorphic functors from ∆n to C , ν runs over the set of natural D-isomorphisms q ∼ = q , and we set n−1 wi,j (ν ∗ δjn−1 ) = m∗ (κn−1 (ν ∗ δjn−1 ), δjn−1 ◦ εni ) (vhn−1 (ν∗δn−1 ) ) i i

A4.13.14.

j

Similarly, the second term of the right-hand member maps the same v on

n+1 hn dn (v) = (−1)j hn m∗ (idr◦δn−1 , δjn )(vr◦δjn ) r j=0

j

A4.13.15

n n+1 (−1)i+j n = · ui,j (ν) zq q j=0 i=0 q ,ν

where r runs over the set of functors from ∆n+1 to C , (q, q ) runs over the set of D-isomorphic pairs of functors from ∆n to C , ν runs over the set of natural D-isomorphisms q ∼ = q , and we set uni,j (ν) = m∗ (κni (ν) ∗ δjn−1 , εn+1 ◦ δjn ) (vhn (ν)◦δn−1 ) A4.13.16. i i

j

480

Frobenius categories versus Brauer blocks

Then, it follows from Lemma A4.2 that, in the last members of equalities A4.13.13 and A4.13.15, comparing the components corresponding to a functor q : ∆n → C , the term (i, j) in the ﬁrst sum cancels either with the term (i + 1, j) in the second sum if j ≤ i , or with the term (i, j + 1) if i < j , the terms (i, i + 1) and (i + 1, i + 1) of the second sum cancel with each other, and it remains the last and the ﬁrst terms of the second sum, which 1 respectively coincide with vq and with − · q ,ν m∗ (ν, id∆n )(vq ) , where q zq runs over the set of functors ∆n → C which are D-isomorphic to q and ν over the set of natural D-isomorphisms q ∼ = q . We are done. A5 Regular cohomology in ordered categories A5.1 Let us call ordered category any small category C such that 45.1.1 Any pair of C-objects C and C which admit C-morphisms C → C and C → C are C-isomorphic and then all the C-morphisms C → C are C-isomorphisms. Clearly, for any ﬁnite p-group P , any Frobenius P -category F is an ordered category. The point is that, for any ordered category C and any contravariant functor a∗ : ch∗ (C) → Ab which factorizes via ch∗ (C) (cf. A1.8), on the one hand the stable cohomology groups Hn∗ (C, a∗ ) can be computed from the regular C-chains deﬁned below, and on the other hand if C is ﬁnite then there are only ﬁnitely many such C-chains. A5.2 Let C be an ordered category; we say that a C-chain q : ∆n → C is regular if, for any 0 ≤ i < j ≤ n , the C-morphism q(i • j) is not a C-isomorphism; actually, q is regular if and only if q(i − 1) ∼ q(i) for any = 1 ≤ i ≤ n ; moreover, for any injective order-preserving map δ : ∆m → ∆n , if q is regular then q ◦ δ is regular too. This last fact leads us to consider the 2-subcategory ∆r of ∆ formed by the same objects, the injective order-preserving maps and the corresponding natural maps. Actually, since the deﬁnition of the diﬀerential map (cf. A3.2.2) only needs injective orderpreserving maps, we consider that the functors Hn (cf. A3.2), where n runs over N , have been deﬁned over the category of functors Fct(∆r , Ab) . A5.3 Coherently, we denote by str : ∆r → CC the restriction to ∆r of the standard representation of ∆ (cf. A2.2), by Fctr (str , C) : (∆r )◦ −→ CC

A5.3.1

the subrepresentation of Fct(str , C) mapping ∆n on the full subcategory Fctr (∆n , C) of Fct(∆n , C) over the regular C-chains q : ∆n → C , and by chr (C) = Fctr (str , C) (∆r )◦

A5.3.2

the corresponding category of regular C-chains, which is obviously a 2-subcategory of ch(C) ; moreover, we call regular any ch(C)-object (q, ∆n ) such

A. Simpliciality

481

that q is regular. Similarly, we still consider the proper category of regular C-chains ch∗r (C) which is the full 2-subcategory of ch∗ (C) over the regular ch(C)-objects. To some extent, the next result mutually justiﬁes the introduction of the category of connected components (cf. A1.8) and of the regularity. Proposition A5.4 Let C be an ordered category. The inclusion 2-functor ch∗r (C) → ch∗ (C) induces an equivalence of categories ch∗r (C) ∼ = ch∗ (C) . Proof: If (q, ∆n ) and (r, ∆m ) are regular ch∗ (C)-objects, if (ν, δ) : (r, ∆m ) → (q, ∆n ) and

(ν , δ ) : (r, ∆m ) → (q, ∆n )

A5.4.1

are ch∗ (C)-morphisms and if we have Natch∗ (C) (ν , δ ), (ν, δ) = ∅, then we have δ(i) ≤ δ (i) for any i ∈ ∆n and, since C is an ordered category, the isomorphisms νi : r δ(i) ∼ = q(i) and νi : r δ (i) ∼ = q(i)

A5.4.2

force r δ(i) • δ (i) to be an isomorphism (cf. A5.1); hence, since r is regular, we have δ(i) = δ (i) and then, according to deﬁnition A2.6.4, we neces sarily have νi = νi ; thus, if two ch∗r (C)-morphisms (ν, δ) and ν , δ ) have the same class in ch∗ (C) , arguing by induction on the length of a sequence of natural maps connecting them, we conclude that they coincide, proving the faithfulness of the functor ch∗r (C) → ch∗ (C) . On the other hand, if q : ∆n → C and r : ∆m → C are two functors and (ν, δ) : (r, ∆m ) → (q, ∆n ) is a ch∗ (C)-morphism, it follows from Lemma A5.5 below that it suﬃces to replace (r, ∆m ) and (q, ∆n ) by ch∗ (C)-isomorphic ch∗ (C)-objects to obtain that q(i • i+1) and r(j • j+1) are not isomorphisms for any i ∈ ∆n and any j ∈ ∆m , and then q and r are regular. In this case, if i < i and we assume that δ(i ) = δ(i) then the isomorphisms νi : r δ(i ) ∼ = q(i ) and νi : r δ(i) ∼ = q(i)

A5.4.3

force q(i ) ∼ = q(i) and therefore q(i • i) is an isomorphism (cf. A5.1), a contradiction. We are done. Lemma A5.5 Let C be a small category. For any C-chain q : ∆n → C and any i ∈ ∆n such that q(i • i+1) is a C-isomorphism, the ch∗ (C)-morphisms (idq◦δn−1 , δin−1 ) : (q, ∆n ) −→ (q ◦ δin−1 , ∆n−1 ) i

n−1 n−1 (idq◦δn−1 , δi+1 ) : (q, ∆n ) −→ (q ◦ δi+1 , ∆n−1 ) i+1

determine isomorphisms in ch∗ (C) .

A5.5.1

482

Frobenius categories versus Brauer blocks

n−1 Proof: Set r = q◦δin−1 and t = q◦δi+1 ; on the one hand, it follows from A3.1.1 that (idr , δin−1 ) ◦ (µ, εni ) = (idr , id∆n−1 ) A5.5.2. n−1 (idt , δi+1 ) ◦ (ν, εni ) = (idt , id∆n−1 )

On the other hand, we have evident natural ∆◦ -maps αn : id∆n −→ δin−1 ◦ εni

n−1 and βn : δi+1 ◦ εni −→ id∆n

A5.5.3

which determine natural maps (cf. deﬁnition A2.5.3) Fct(αn , C) : Fct(id∆n , C) −→ Fct(δin−1 ◦ εni , C) n−1 Fct(βn , C) : Fct(δi+1 ◦ εni , C) −→ Fct(id∆n , C)

A5.5.4;

−1 thus, setting ν = Fct(βn , C)q and µ = Fct(αn , C)q which makes sense since q(i • i + 1) is a C-isomorphism, we get the ch∗ (C)-natural maps (cf. A2.6.4) βn

α

n n−1 (ν, δi+1 ◦ εni ) −→ (idq , id∆n ) −→ (µ, δin−1 ◦ εni ) + + n−1 (ν, εni ) ◦ (idt , δi+1 ) (µ, εni ) ◦ (idr , δin−1 )

A5.5.5,

n−1 so that the classes of (µ, εni ) ◦ (idr , δin−1 ) and (ν, εni ) ◦ (idt , δi+1 ) in ch∗ (C) coincide with the class of (idq , id∆n ) . We are done.

A5.6 Let C be an ordered category and a∗ : ch∗ (C) → Ab a contravariant functor which factorizesvia ch∗ (C) ; for any n ∈ N , we say that an element ∗ ∗ a = (aq )q∈Fct(∆n ,C) in (lC )∗ (a ) (∆n ) is regular or singular according to aq = 0 for any nonregular or for any regular C-chain q : ∆n → C respectively. Then, denoting by lrC : ch∗r (C) −→ (∆r )◦ A5.6.1 ∗ ∗ the structural functor and by a∗r the restriction r ∗ of∗ a to chr (C) , we can ∗ identify (lC )∗ (ar ) (∆n ) with the quotient of (lC )∗ (a ) (∆n ) by the subgroup of all the singular elements; actually, denoting by ir : ∆r → ∆ the inclusion 2-functor, we have an evident natural map

(l∗C )∗ (a∗ ) ◦ ir −→ (lrC )∗ (a∗r )

A5.6.2.

Finally, for any n ∈ N , we deﬁne the n-th regular cohomology group of C over a∗ by Hnr (C, a∗ ) = Hn (lrC )∗ (a∗r ) A5.6.3.

A. Simpliciality

483

Proposition A5.7 Let C be an ordered category and a∗ : ch∗ (C) → Ab a contravariant functor which factorizes via ch∗ (C) ; denote by a∗r the restriction of a∗ to ch∗r (C) . For any n ∈ N , the natural map (l∗C )∗ (a∗ ) ◦ ir −→ (lrC )∗ (a∗r )

A5.7.1

induces a group isomorphism Hn∗ (C, a∗ ) ∼ = Hnr (C, a∗ )

A5.7.2.

Proof: It is clear that, for any n ∈ N , the subgroup of (l∗C )∗ (a∗ ) of all the (l∗ )∗ (a∗ )

regular elements maps bijectively onto (lrC )∗ (a∗r ) ; set dn = dn C drn

=

(lr ) (a∗ ) dn C ∗

for short. For n = 0 , we actually have the equality ∗ (lC )∗ (a∗ ) (∆0 ) = (lrC )∗ (a∗r ) (∆0 )

and

A5.7.3

and moreover, for any a = (aq )q∈Fct(∆0 ,C) in (l∗C )∗ (a∗ ) (∆0 ) and any nonregular C-chain t : ∆1 → C , the t-component d0 (a)t of the element d0 (a) vanishes since t(0•1) is an isomorphism; consequently, we get H0∗ (C, a∗ ) = Ker(d0 ) = Ker(dr0 ) = H0r (C, a∗ )

A5.7.4.

r From now on, we assume that n ≥ 1 ; it is clear o that, for any a r ∗ in Ker(dn ) , there is a unique regular element a ∈ (lC )∗ (a ) (∆n ) lifting ar (cf. A3.11.1) and it is easily checked that a belongs to (l∗C )∗ (a∗ ) (∆n ) (cf. A3.18); moreover, for any C-chain t : ∆n+1 → C we have

dn (a)t =

n+1

a∗ (idt◦δin , δin ) (at◦δin )

A5.7.5

i=0

and, if t is nonregular, there is i ≤ n such that, for a suitable C-chain t n isomorphic to t , we have t ◦ δin = t ◦ δi+1 and t ◦ δjn remains nonregular for any j ∈ ∆n − {i, i+1} ; thus, since a belongs to (l∗C )∗ (a∗ ) (∆n ) , we get

n n n , δi+1 ) (at◦δi+1 ) a∗ (idt◦δin , δin ) (at◦δin ) = a∗ (idt◦δi+1

A5.7.6

and, since a is regular, we still get at◦δjn = 0 for any j ∈ ∆n − {i, i+1} , so that we ﬁnally get dn (a)t = 0 ; but, if t is regular, it is clear that dn (a)t = drn (ar )t = 0

A5.7.7;

consequently, we have dn (a) = 0 which shows the surjectivity of homomorphism A5.7.2.

484

Frobenius categories versus Brauer blocks

On the other hand, it follows from Lemma A5.9 below that, for any element a in Ker(dn ) there exists b ∈ (l∗C )∗ (a∗ ) (∆n−1 ) such that the diﬀerence a − dn−1 is regular ; hence, denoting by ar and br the respective images (b) of a in (lrC )∗ (a∗r ) (∆n ) and of b in (lrC )∗ (a∗r ) (∆n−1 ) , if we assume that for some cr ∈ (lrC )∗ (a∗r ) (∆n−1 ) we have ar − drn−1 (br ) = drn−1 (cr )

A5.7.8

then, for the unique regular element c ∈ (l∗C )∗ (a∗ ) (∆n−1 ) lifting cr we still have a − dn−1 (b) = dn−1 (c) A5.7.9. We are done.

A5.8 Note that an element a = (aq )q∈Fct(∆n ,C) of (l∗C )∗ (a∗ ) (∆n ) (cf. A3.18) is regular if and only if we have aq = 0 for any C-chain q : ∆n → C such that for some i ∈ ∆n−1 we have q(i) = q(i+1) and q(i • (i+1)) = idq(i)

A5.8.1

and, of fact, in the next lemma we can deal with any element as a matter of (loC )∗ (a∗ ) (∆n ) (cf. A3.7) fulﬁlling this condition, without any hypothesis on C . Thus, let us say that a C-chain q : ∆n → C is redundant whenever it fulﬁlls condition A5.8.1 for some i ∈ ∆n−1 , and then, that an element a = (aq )q∈Fct(∆n ,C) in (loC )∗ (a∗ ) (∆n ) is regular if aq = 0 for any redundant C-chain q : ∆n → C . Lemma A5.9 Let ao : cho (C) → Ab be a contravariant functor which fac (lo ) (ao ) torizes via cho (C) and set dn = dn C ∗ . For any a ∈ (loC )∗ (ao ) (∆n ) such o that the element dn (a) is regular, there is b ∈ (lC )∗ (ao ) (∆n−1 ) such that the diﬀerence a − dn−1 (b) is regular too. Moreover, if ao can be extended functor a∗ : ch∗ (C) → Ab and the element a belongs to to ∗a contravariant ∗ (lC )∗ (a ) (∆n ) then b can be chosen in (l∗C )∗ (a∗ ) (∆n−1 ) . Proof: If a is not regular, then there is q ∈ Fct(∆n , C) such that aq = 0 while, for some i < n , it fulﬁlls condition A5.8.1; that is to say, we have q = r◦εni for some r ∈ Fct(∆n−1 , C) . We argue by induction on the minimal i fulﬁlling condition A5.8.1 and follow Eilenberg–MacLane’s construction (cf. Theorem 4.1 in [23]); in particular, we are assuming that ar◦εn = 0 for any D < i and any r ∈ Fct(∆n−1 , C) and that the element dn (a) =

n+1 j=0

is regular.

(−1)j ao (idt◦δjn , δjn ) (at◦δjn )

t∈Fct(∆n+1 ,C)

A5.9.1

A. Simpliciality

485

But, for any j < i we have εn+1 ◦ δjn = δjn−1 ◦ εni−1 (cf. A3.1.1) and i therefore, according to the minimality of i , for any q ∈ Fct(∆n , C) , we still have aq◦(εn+1 ◦δn ) = 0 , so that we get i

j

0=

n+1

(−1)j ao (idq◦(εn+1 ◦δn ) , δjn ) (aq◦(εn+1 ◦δn ) ) i

j=i

i

j

A5.9.2;

j

n moreover, we have εn+1 ◦ δin = id∆n = εn+1 ◦ δi+1 (cf. A3.1.1), so that i i equality A5.9.2 becomes n (−1)i ao (idq , δin ) − ao (idq , δi+1 ) (aq )

=

n

(−1)j ao (idq◦(εn+1 ◦δn

j=i+1

i

j+1

n ) , δj+1 )

(aq◦(εn+1 ◦δn i

j+1

))

Now, in (loC )∗ (ao ) (∆n−1 ) consider the element b = ao (idr◦εni , εni )(ar◦εni ) r∈Fct(∆ ,C) n−1

A5.9.3.

A5.9.4;

note that, if we assume that ao can be extended to a contravariant functor a∗ : ch∗ (C) → Ab and then that a belongs to (l∗C )∗ (a∗ ) (∆n ) (cf. 3.18), for any r ∈ Fct(∆n−1 , C) isomorphic to r , and any natural isomorphism µ : r ∼ = r , we have the natural isomorphism µ ∗ εni : r ◦ εni ∼ = r ◦ εni

A5.9.5

and therefore we get a∗ (µ, id∆n−1 ) a∗ (idr ◦εni , εni ) (ar ◦εni ) = a∗ (µ ∗ εni , εni ) (ar ◦εni ) A5.9.6, = a∗ (idr◦εni , εni ) a∗ (µ ∗ εni , id∆n ) (ar ◦εni ) = a∗ (idr◦εni , εni ) (ar◦εni ) , so that b belongs to (l∗C )∗ (a∗ ) (∆n−1 ) . In any case, for any q ∈ Fct(∆n , C) , the q-component of dn−1 (b) is dn−1 (b)q =

n j=0

(−1)j ao (idq◦(δn−1 ◦εn ) , δjn−1 ◦ εni ) (aq◦(δn−1◦εn ) ) j

i

j

i

A5.9.7.

First of all, let us consider the (r ◦ εn6 )-component of dn−1 (b) for any D < i and any r ∈ Fct(∆n−1 , C) ; for any D < i , it is easily checked that ! n δjn−2 ◦ εn−1 if j < D i−1 ◦ ε6−1 n−1 n n ε6 ◦ δ j ◦ ε i = A5.9.8, n−2 n−1 n δj−1 ◦ εi−1 ◦ ε6 if j > D + 1

486

Frobenius categories versus Brauer blocks

and therefore, according to the minimality of i , for any j ∈ ∆n − {D, D+1} we get ar◦(εn ◦δn−1 ◦εn ) = 0 ; hence, in this case this component, up to a sign,

j

i

becomes (cf. A5.9.7 ) o n−1 a (idr◦εni , δ6n−1 ◦ εni ) (ar◦εni ) − ao (idr◦εni , δ6+1 ◦ εni ) (ar◦εni )

A5.9.9.

n−1 Moreover, since εn6 ◦ δ6n−1 = id∆n−1 = εn6 ◦ δ6+1 (cf. A3.1.1), we have the o following ch (C)-morphisms

(idr , δ6n−1 ) : (r ◦ εn6 , ∆n ) −→ (r, ∆n−1 )

A5.9.10

n−1 (idr , δ6+1 ) : (r ◦ εn6 , ∆n ) −→ (r, ∆n−1 )

n−1 which have the same class in cho (C) ; indeed, the ∆-natural map δ6+1 •δ6n−1 n−1 from δ6+1 to δ6n−1 determines a natural map (cf. A2.5.3) n−1 n−1 Fct(δ6+1 • δ6n−1 , C) : Fct(δ6+1 , C) −→ Fct(δ6n−1 , C)

A5.9.11

n−1 n−1 which sends r ◦ εn6 to idr and therefore, by deﬁnition A2.6.4, δ6+1 •δ6 is also o a ch (C)-natural map between the morphisms A5.9.10. A ﬁrst consequence of this fact is that, since we are assuming that ao factorizes via cho (C) , the diﬀerence A5.9.9 is equal to zero and therefore the (r◦εn6 )-component of dn (b) n+1 n vanishes for any D < i . A second consequence is that, setting εn+1 , i,i = εi ◦εi n equality A5.9.3 applied to q = r ◦ εi becomes n

0=

(−1)j ao (idr◦(εn+1 ◦δn i,i

j=i+1

n ) , δj+1 ) j+1

(ar◦(εn+1 ◦δn i,i

j+1

))

A5.9.12.

Secondly, let us consider the (r ◦ εni )-component of dn−1 (b) ; it is easily checked that (cf. A3.1.1) ! δjn−1 ◦ εni−1,i−1 if j < i n−1 n n εi ◦ δ j ◦ ε i = A5.9.13 n εn+1 if j > i + 1 i,i ◦ δj+1 n−1 and recall again that εni ◦ δin−1 = id∆n−1 = εni ◦ δi+1 ; thus, according to equality A5.9.7, this component becomes n

(−1)j ao (idr◦(εn ◦δn−1◦εn ) , δjn−1 ◦ εni ) (ar◦(εn ◦δn−1◦εn ) )

j=i+2

=

n j=i+2

i

j

i

(−1)j ao (idr◦(εn+1 ◦δn i,i

j+1

i

n+1 n ◦ δj+1 ) ) , εi

j

i

(ar◦(εn+1 ◦δn i,i

j+1

))

A5.9.14,

n−1 = (−1)i ao (idr◦εni , δi+1 ◦ εni ) (ar◦εni ) since the second member of the these equalities coincides with the image via ao (idr◦εn+1 , εn+1 ) of the sum in A5.9.12 without the ﬁrst summand. i i,i

A. Simpliciality

487

Finally, we claim that the class of the cho (C)-morphism n−1 (idr◦εni , δi+1 ◦ εni ) : r ◦ εni −→ r ◦ εni

A5.9.15

in the category cho (C) coincides with the corresponding identity morphism; n−1 indeed, we have a unique natural map ν : δi+1 ◦ εni → id∆n sending i + 1 to i • (i+1) , which determines a natural map (cf. A2.5.3) n−1 Fct(ν, C) : Fct(δi+1 ◦ εni , C) −→ idFct(∆n ,C)

A5.9.16

sending r ◦ εni to idr◦εni , and therefore, according to deﬁnition A2.6.4, ν is n−1 also a cho (C)-natural map from (idr◦εni , δi+1 ◦ εni ) to (idr◦εni , id∆n ) . In conclusion, for any D ≤ i and any r ∈ Fct(∆n−1 , C) the (r ◦ εn6 )-component of the element a − (−1)i dn−1 (b) vanishes and, since dn a − (−1)i dn−1 (b) = dn (a)

A5.9.17

is regular, it follows from our induction hypothesis that there is an element c = (cr )r∈Fct(∆n−1 ,C) in (loC )∗ (ao ) (∆n−1 ) such that the diﬀerence a − dn−1 (−1)i b + c is regular. Moreover, if ao can be extended to a con travariant functor a∗ : ch∗ (C) → Ab and a belongs to (l∗C )∗ (a∗ ) (∆n ) , we al ∗ ready know that b belongs to (lC )∗ (a∗ ) (∆n−1 ) and then it follows from our induction hypothesis that c can be chosen in (l∗C )∗ (a∗ ) (∆n−1 ) too. We are done.

References [1] Jon Alperin, Sylow intersections and fusion, Journal of Algebra, 1 (1964), 110-113. [2] Jon Alperin, Weights for ﬁnite groups, in Proc. Symp. Pure Math., 47 (1987), 369-379, Amer. Math. Soc., Providence. [3] Jon Alperin and Michel Brou´e, Local methods in block theory, Annals of Math., 110 (1979), 143-157. [4] Michael Aschbacher, Normal subsystems of fusion systems, preprint. [5] Dave Benson, Cohomology of sporadic groups, ﬁnite loop spaces and the Dickson invariants, Proc. 1994 Durham Conference, London Math. Soc., Springer Lecture Notes, 252 (1998), 10-23. [6] Serge Bouc, The Dade group of a p-group, Inventiones mathematicæ 164 (2006), 189-231. [7] Serge Bouc and Jacques Th´evenaz, Gluing torsion endo-permutation modules, preprint, 2006. [8] Richard Brauer, Investigations on group characters, Annals of Math., 42 (1941), 937-958. [9] Richard Brauer, On the structure of blocks of characters of ﬁnite groups, Proc. Second International Conference on Groups, Camberra, 1973, 103-130. [10] Michel Brou´e, Les D-blocs des groupes GL(n, q) et U (n, q 2 ), et leurs structures locales, Ast´erisque, 133/134 (1986), 159-188. [11] Michel Brou´e and Llu´ıs Puig, Characters and Local Structure in G-algebras, Journal of Algebra, 63 (1980), 306-317. [12] Michel Brou´e and Llu´ıs Puig, A Frobenius theorem for blocks, Inventiones mathematicæ, 56 (1980), 117-128. [13] Carles Broto, Ran Levi and Bob Oliver, The homotopy theory of fusion systems, Journal of Amer. Math. Soc. 16 (2003), 779-856. [14] Carles Broto, Ran Levi and Bob Oliver, The theory of p-local groups, a survey, Contemp. Math., 346 (2004), Amer. Math. Soc., Providence. [15] Carles Broto, Nat` alia Castellana, Jesper Grodal, Ran Levi and Bob Oliver, Subgroup families controlling p-local ﬁnite groups, Proc. Lond. Math. Soc., 91 (2005), 325-254. [16] Jon Carlson, The dimension of modules and their restriction over modular group algebras, Journal of Algebra 69 (1981), 95-105. [17] Jon Carlson and Jacques Th´evenaz, The classiﬁcation of torsion endo-trivial modules, Annals of Math., 162 (2005), 823-883.

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[18] Henri Cartan and Samuel Eilenberg, “Homological Algebra”, Princeton Math. 19, 1956, Princeton University Press. [19] Everett Dade, Endopermutation modules over p-groups, I and II, Annals of Math. 107 (1978), 459-494, and 108 (1978), 317-346. [20] Everett Dade, A correspondence of characters, in Proc. Symp. Pure Math. 37 (1980), 401-403, Amer. Math. Soc., Providence. [21] Everett Dade, Counting characters in Blocks, I , Inventiones mathematicæ 109 (1992), 187-210. [22] Andreas Dress, Contributions to the theory of induced representations, in “Algebraic K-theory II ”, Lecture Notes in Math. 342 (1973), 183-240. [23] Samuel Eilenberg and Saunders Mac Lane, On the groups H(Π, n), I , Annals of Math., 58 (1953), 55-106. [24] Yun Fan and Llu´ıs Puig, On blocks with nilpotent coeﬃcient extensions, Algebras and Representation Theory, 1 (1998), 27-73, and Publisher revised form, 2 (1999), 209. [25] Walter Feit and John Thompson, Solvability of groups of odd order , Paciﬁc Journal of Math., 13 (1963), 775-1029. ¨ [26] Georg Frobenius, Uber auﬂ¨ asbare Gruppen, V , Sitzungs. der K¨ onig. Preu. Akad. der Wissen. zu Berlin, 1324-1329 (1901). [27] George Glauberman, A characteristic subgroup of a p-stable group, Can. J. Math., 20 (1968), 1101-1135. [28] Daniel Gorenstein, “Finite groups” Harper’s Series, 1968, Harper and Row. [29] James Green, Some remarks on defect groups, Math. Zeit., 107 (1968), 133-150. [30] James Green, Functors on categories of ﬁnite group representations, Journal of Pure and Applied Algebra, 37 (1985), 265-298. [31] Bertram Huppert, “Endliche Gruppen I”, Die Grundlehren der math. Wissenschaften, 134 (1967), Springer-Verlag, Berlin. [32] Stefan Jackowski and James McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology, 31 (1992), 113-132. [33] Reinhard Kn¨ orr and Geoﬀrey Robinson, Some remarks on a conjecture of Alperin, Journal of London Math. Soc. 39 (1989), 48-60. [34] Burkhard K¨ ulshammer and Llu´ıs Puig, Extensions of nilpotent blocks, Inventiones mathematicæ, 102 (1990), 17-71. [35] Llu´ıs Puig, Structure locale dans les groupes ﬁnis, Bulletin Soc. Math. France, M´emoire 47 (1976).

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[36] Llu´ıs Puig, The source algebra of a nilpotent block , preprint, Mathematisches Forschungsinstitut Oberwolfach, June 1981. [37] Llu´ıs Puig, Pointed groups and construction of characters, Math. Zeit. 176 (1981), 265-292. [38] Llu´ıs Puig, La classiﬁcation des groupes ﬁnis simples: bref apper¸cu et quelques cons´equences, Ast´erisque, 92/93 (1982), 101-128. [39] Llu´ıs Puig, Local fusions in block source algebras, Journal of Algebra, 104 (1986), 358-369. [40] Llu´ıs Puig, Local extensions in endopermutation modules split: a proof of Dade’s theorem, in “Seminaire sur les Groupes Finis III”, Publications Math. de Paris VII, 25 (1986). [41] Llu´ıs Puig, Nilpotent blocks and their source algebras, Inventiones mathematicæ, 93 (1988), 77-116. [42] Llu´ıs Puig, Pointed groups and construction of modules, Journal of Algebra, 116 (1988), 7-129. [43] Llu´ıs Puig, Aﬃrmative answer to a question of Feit, Journal of Algebra, 131 (1990), 513-526. [44] Llu´ıs Puig, On the general structure of a block , manuscript, MSRI, 1990. [45] Llu´ıs Puig, Une correspondance de modules pour les blocks a ` groupes de d´efaut ab´eliens, Geometriæ Dedicata, 37 (1991), 9-43. [46] Llu´ıs Puig, Brauer-Frobenius categories, manuscript, Paris, 1992. [47] Llu´ıs Puig, On Joanna Scopes’ Criterion of Equivalence for Blocks of Symmetric Groups, Algebra Colloquium 1 (1994), 25-55. [48] Llu´ıs Puig, “On the Morita and Rickard equivalences between Brauer blocks”, Progress in Math., 1999, Birkh¨ auser, Basel. [49] Llu´ıs Puig, The hyperfocal subalgebra of a block, Inventiones mathematicæ, 141 (2000), 365-397. [50] Llu´ıs Puig, Source algebras of p-central group extensions, Journal of Algebra, 235 (2001), 359-398. [51] Llu´ıs Puig, “Blocks of Finite Groups”, Springer Monographs in Mathematics, 2002, Springer-Verlag, Berlin, Barcelona. [52] Llu´ıs Puig, Frobenius categories, Journal of Algebra, 303 (2006), 309-357. [53] Llu´ıs Puig, Block Source Algebras in p-Solvable Groups, to appear in Michigan Journal of Mathematics, 2009. [54] Jean-Pierre Serre, “Linear Representations of Finite Groups”, Graduate Texts in Mathematics 42, 1977, Springer-Verlag, Berlin.

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[55] Jean-Pierre Serre, “Local Fields”, Graduate Texts in Mathematics 67, 1979, Springer-Verlag, Berlin, Barcelona. [56] Ronald Solomon, Finite groups with Sylow 2-subgroup of type .3 , Journal of Algebra 28 (1974), 182-198. [57] John Thompson , Finite groups with ﬁxed-point-free automorphisms of prime order , Proc. Nat. Acad. Sci., 45 (1959), 578-581. [58] John Thompson , Quadratic pairs, in “Actes du Congr´es International de Math., 1970”, Gauthier-Villars, Paris, 1971.

Index a Aα . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A(Hα ) . . . . . . . . . . . . . . . . . . . . . . . . . 18 Ab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ab , ab◦ . . . . . . . . . . . . . . . . . . . . . . . . 25 ac(C) . . . . . . . . . . . . . . . . . . . . . . . . . 456 P αQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 (Q,f )

α ˆ (R,g) . . . . . . . . . . . . . . . . . . . . . . . . 166 autC . . . . . . . . . . . . . . . . . . . . . . . . . . 456 F nc . . . . . . . . . . . . . . . . . . . . . . . . 211 aut (F nc . . . . . . . . . . . . . . . . . . . 171 aut (b,G) ) ℵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 identity representation i . . . . 454 adjoin subcategory F a . . . . . . . . 182 additive cover . . . . . . . . . . . . . . . . . 456 (b, G)-admissible integer . . . . . . 152 algebra . . . . . . . . . . . . . . . . . . . . . . . . 17 G-algebra. . . . . . . . . . . . . . . . . . . .17 p-permutation G-algebra . . . . . 19 primitive G-algebra . . . . . . . . . . 17 radical. . . . . . . . . . . . . . . . . . . . . . .18 source algebra . . . . . . . . . . . . . . . 18 Alperin F-fusion . . . . . . . . . . . . . . . 57 b BrA Q , BrQ . . . . . . . . . . . . . . . . . . . . . . 18 P . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Br Q (P, ¯s)-balanced function . . . . . . . 145 basic F-locality . . . . . . . . . . . . . . . 397 basic P × P -set . . . . . . . . . . . . . . . 390 thick basic P × P -set . . . . . . . 390 F-basic P × P -set . . . . . . . . . . 390 (b, G)-admissible integer . . . . . . 152 P -bounded F-locality . . . . . . . . . 323 Brauer block . . . . . . . . . . . . . . . . . . . 17 normal sub-block . . . . . . . . . . . 241 Brauer homomorphism . . . . . . . . . 18 Brauer (α, G)-pair . . . . . . . . . . . . . 19 Brauer (b, G)-pair . . . . . . . . . . . . . . 19 Brauer quotient . . . . . . . . . . . . . . . . 18

c cC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 chF , cfF . . . . . . . . . . . . . . . . . . . . . . . . 205 ch(C) . . . . . . . . . . . . . . . . . . . . . . . . . 456 ch∗ (C) . . . . . . . . . . . . . . . . . . . . . . . . 456 fn ch∗ (F) . . . . . . . . . . . . . . . . . . . . . . 345 Fˆ CG ˆ (R, g) . . . . . . . . . . . . . . . . . . . . . 242 ˆ

G CH ˆ (Rε ). . . . . . . . . . . . . . . . . . . . . . .251 ˜ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 CC. . . . . . . . . . . . . . . . . . . . . . . . . . . .450 category . . . . . . . . . . . . . . . . . . . . . . . 15 C(X, Y ) , C(X) . . . . . . . . . . . . . . 15 C◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 C∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 CX . . . . . . . . . . . . . . . . . . . . . . . . . . 16 K C . . . . . . . . . . . . . . . . . . . . . . . . . . 73 direct product . . . . . . . . . . . . . . . 16 intersection . . . . . . . . . . . . . . . . . . 16 opposite category . . . . . . . . . . . . 15 small category . . . . . . . . . . . . . . . 15 full subcategory . . . . . . . . . . . . . 15 regular representation . . . . . . 454 2-category X . . . . . . . . . . . . . . . . . . 449 π0 (X) , X . . . . . . . . . . . . . . . . . . . 451 representation of X . . . . . . . . . 453 k ∗ -category . . . . . . . . . . . . . . . . . . . 151 P -category . . . . . . . . . . . . . . . . . . . . . 27 divisible P -category. . . . . . . . . .27 category of chains . . . . . . . . . . . . . 456 evaluation functor . . . . . . . . . . 465 proper category of chains . . . 456 autC . . . . . . . . . . . . . . . . . . . . . . . . 456 category of K-objects . . . . . . . . . . 73 category of small categories . . . 450 centric linking system . . . . . . . . . 323 centric orbit category . . . . . . . . . . . 73 F-centric subgroup . . . . . . . . . . . . . 49 F-chain . . . . . . . . . . . . . . . . . . . . . . . . 36 normal F-chain . . . . . . . . . . . . . . 37 coherent F-locality . . . . . . . . . . . 321

494

Frobenius categories versus Brauer blocks

cohomology groups . . . . . . . . . . . 465 naive cohomology groups . . . 465 stable cohomology groups . . . 468 D-stable cohomology . . . . . . . 468 D-stable elements . . . . . . . . . . 437 covering functor cC . . . . . . . . . . . .459 covering natural map κC . . . . . . 459 d dan

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 ∆ , ∆n . . . . . . . . . . . . . . . . . . . . . . . 451 st . . . . . . . . . . . . . . . . . . . . . . . . . . 453 standard representation . . . . . 453 δin , εn+1 . . . . . . . . . . . . . . . . . . . 463 i diﬀerential maps . . . . . . . . . . . . 463 homology groups . . . . . . . . . . . 463 ∆r . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 str . . . . . . . . . . . . . . . . . . . . . . . . . 480 ∆P,S,Q . . . . . . . . . . . . . . . . . . . . . . . .122 ∆ϕ (Q) . . . . . . . . . . . . . . . . . . . . . . . . 389 ∆ϕ,ϕ (Q) . . . . . . . . . . . . . . . . . . . . . 390 Dk . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Dk (P ) . . . . . . . . . . . . . . . . . . . . . . . . . 21 Dk (P )p . . . . . . . . . . . . . . . . . . . . . . . 109 n Dk (P )p . . . . . . . . . . . . . . . . . . . . . . . 137 Dade P -algebra . . . . . . . . . . . . . . . . 21 defect of a block . . . . . . . . . . . . . . . 20 defect group . . . . . . . . . . . . . . . . . . . 20 defect pointed group . . . . . . . . . . . 19 F-dimorphism . . . . . . . . . . . . . . . . . 57 direct image . . . . . . . . . . . . . . . . . . . 16 divisible P -category . . . . . . . . . . . . 27 divisible F-locality . . . . . . . . . . . . 320 e EG (Pγ ) . . . . . . . . . . . . . . . . . . . . . . . . 23 ˆG (Pγ ) . . . . . . . . . . . . . . . . . . . . . . . . 23 E embedding . . . . . . . . . . . . . . . . . . . . . 18 F-essential subgroup . . . . . . . . . . . 59 F-essential normal chain . . . . . . . 67 evaluation functor . . . . . . . . . . . . 465 exterior intersection . . . . . . . . . . . . 77

exterior quotient . . . . . . . . . . . . . . . 15 f FA (Pγ ) , FˆA (Pγ ) . . . . . . . . . . . . . . . 29 FS (P ) , FˆS (P ) . . . . . . . . . . . . . . . . 117 Ff (π) , Fˆf (π) . . . . . . . . . . . . . . . . . 118 ˆ FˆS (P ) . . . . . . . . . . . . . . . 119 FˆS (P ) ∩ Fct(A, B) . . . . . . . . . . . . . . . . . . . . . . 16 Fct(m, C) . . . . . . . . . . . . . . . . . . . . . 454 F a . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 F h . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 FG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 F(b,G) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 F(α,G) . . . . . . . . . . . . . . . . . . . . . . . . . 43 e F(α,G) . . . . . . . . . . . . . . . . . . . . . . . . . 43 FG,Ω . . . . . . . . . . . . . . . . . . . . . . . . . . 319 F/U . . . . . . . . . . . . . . . . . . . . . . . . . .179 nc (F(b,G) ) . . . . . . . . . . . . . . . . . . . . . 151 nc F . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 sc F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 F H . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 ˆ F(q) , F(q) . . . . . . . . . . . . . . . . . . . 211 ˜ q) , . . . . . . . . . . . . . . . . . . . . . . . . . 216 F(˜ ˆ ˜ q) . . . . . . . . . . . . . . . . . . . . . . . . . . 217 F(˜ φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Φ(P ) . . . . . . . . . . . . . . . . . . . . . . . . . . 25 FctUh . . . . . . . . . . . . . . . . . . . . . . . . . 222 ˆh , G), ˆ O . . . . . . 219 FctUh Monk∗ (U F-basic P × P -set . . . . . . . . . . . . 390 F-dimorphism . . . . . . . . . . . . . . . . . 57 F-essential subgroup . . . . . . . . . . . 59 F-essential normal chain . . . . . . . 67 F-hyperfocal subgroup . . . . . . . . 195 F-intersected subgroup . . . . . . . . 49 F-irreducible . . . . . . . . . . . . . . . . . . 59 F-localizer . . . . . . . . . . . . . . . . . . . . 335 F-localizing functor . . . . . . . . . . . 345 F-maximal normal chain . . . . . . . 67 F-nilcentralized subgroup . . . . . . 47 F-normal . . . . . . . . . . . . . . . . . . . . . 182 F-reducible . . . . . . . . . . . . . . . . . . . . 59

Index

495

Frattini subgroup . . . . . . . . . . . . . . 25 Frobenius category FG . . . . . . . . . 16 Frobenius category over P . . . . . 29 Frobenius P -category F . . . . . . . . 29 adjoin subcategory F a . . . . . . 182 hyperfocal subcategory F h . . 201 Frobenius functor . . . . . . . . . . . . . 179 F-selfcentralizing subgroup. . . . .49 F-solvcentralized subgroup . . . . 365 F-stable subgroup . . . . . . . . . . . . 179 F(β,H,X) -stable element . . . . . . . 152 fully centralized subgroup . . . . . . 28 fully highnormalized subgroup . . . . . 184 fully normalized subgroup . . . . . . 28 fully K-normalized subgroup . . . 28 fully normalized F-chain . . . . . . . 36 fully conormalized F-chain . . . . . 37 functor . . . . . . . . . . . . . . . . . . . . . . . . 15 mw -stable adjoin image . . . . . 458 contravariant functor . . . . . . . . 15 direct image . . . . . . . . . . . . . . . . . 16 2-functor . . . . . . . . . . . . . . . . . . . . . 451 shifting functor . . . . . . . . . . . . . 452 (F, F )-functorial homomorphism . . . . 179 A-fusion . . . . . . . . . . . . . . . . . . . . . . . 23 fusion system . . . . . . . . . . . . . . . . . . 27 g ˆ× ˆ . . . . . . . . . . . . . . . . . . . . . . . . . 22 ˆG G ˆ∗G ˆ . . . . . . . . . . . . . . . . . . . . . . . . . 22 G F nc ) . . . . . . . . . . . . . . . . . 212 Gk (F, aut ˆ Gk (G) . . . . . . . . . . . . . . . . . . . . . . . . 211 ˆ b) . . . . . . . . . . . . . . . . . . . . . . 241 Gk (G, gk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 k∗ G k ∗ -group . . . . . . . . . . . . . . . . . . . . . . . 22 Gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 k ∗ -Gr . . . . . . . . . . . . . . . . . . . . . . . . . 118 generator family . . . . . . . . . . . . . . . 59 (X, n)-gluing element. . . . . . . . . .152 (P, X, n)-gluing family . . . . . . . . 144

h HF . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 h0 (m) . . . . . . . . . . . . . . . . . . . . . . . . . 223 Hn (a) . . . . . . . . . . . . . . . . . . . . . . . . 463 Hn (C, a) . . . . . . . . . . . . . . . . . . . . . . 466 Hno (C, ao ) . . . . . . . . . . . . . . . . . . . . . 465 HnD (C, a∗ ) . . . . . . . . . . . . . . . . . . . . . 468 Hn∗ (C, a∗ ) . . . . . . . . . . . . . . . . . . . . . 469 Hnr (C, a∗ ) . . . . . . . . . . . . . . . . . . . . . 482 hyperfocal subcategory . . . . . . . . 201 F-hyperfocal subgroup . . . . . . . . 195 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ∩ T , β˜ ∩ β˜ . . . . . . . . . . . . . . . . . . . . . . 82 ∩ sc

∩L . . . . . . . . . . . . . . . . . . . . . . . . . . 439 ιG H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 iGr , k ∗ -iGr . . . . . . . . . . . . . . . . . . . 219 p-iGr . . . . . . . . . . . . . . . . . . . . . . . . . 206 ˆ

IndG ˆ (B) . . . . . . . . . . . . . . . . . . . . . . . 24 H ˆ induced G-interior algebra . . . . . 24 ˆ G-interior algebra . . . . . . . . . . . . . . 23 interior structure in a category . . . . . . . 15 F-intersected subgroup . . . . . . . . 49 sc L -intersection . . . . . . . . . . . . . . . 439 R-intersectional family . . . . . . . . . 81 F-irreducible . . . . . . . . . . . . . . . . . . 59 ˆ b) . . . . . . . . . . . . . . . . . . . . . . 22 Irrk (G, j J(A). . . . . . . . . . . . . . . . . . . . . . . . . . .18 P jQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 jh . . . . . . . . . . . . . . . . . . . . . . . . . . . . .225 k k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 κC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 κQ,R (x) , κQ (x) , κx . . . . . . . . . . . . 20 kη˜ , Kη˜ . . . . . . . . . . . . . . . . . . . . . . . 398 kνη˜˜ , Kνη˜˜ . . . . . . . . . . . . . . . . . . . . . . . 399

496

Frobenius categories versus Brauer blocks

k , k(ϕ) ˜ ην˜˜ , kην˜˜ . . . . . . . . . . . . . . . . . . . 402 k-mod . . . . . . . . . . . . . . . . . . . . . . . . . 85 k ∗ -Gr . . . . . . . . . . . . . . . . . . . . . . . . . 118 k ∗ -iGr . . . . . . . . . . . . . . . . . . . . . . . . 219 k ∗ -category . . . . . . . . . . . . . . . . . . . 151 k ∗ -group . . . . . . . . . . . . . . . . . . . . . . . 22 K-objects . . . . . . . . . . . . . . . . . . . . . . 73 l b

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 c,sc L . . . . . . . . . . . . . . . . . . . . . . . . . . 412 d,sc L . . . . . . . . . . . . . . . . . . . . . . . . . . 425 r,sc L . . . . . . . . . . . . . . . . . . . . . . . . . . 426 sc L . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 LPA (P ) . . . . . . . . . . . . . . . . . . . . . . . . 18 loC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 lD C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 l∗C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 lrC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 + , lv . . . . . . . . . . . . . . . . . . 344 Loc , Loc locF . . . . . . . . . . . . . . . . . . . . . . . . . . 344 locLˆ , λLˆ . . . . . . . . . . . . . . . . . . . . . . 351 length of ϕ. . . . . . . . . . . . . . . . . . . . .66 sc L -intersection . . . . . . . . . . . . . . . 439 local point . . . . . . . . . . . . . . . . . . . . . 18 local pointed group . . . . . . . . . . . . 18 F-locality L . . . . . . . . . . . . . . . . . . 319 basic F-locality . . . . . . . . . . . . . 397 coherent F-locality . . . . . . . . . 321 divisible F-locality . . . . . . . . . 320 P -bounded F-locality . . . . . . . 323 perfect F-locality . . . . . . . . . . . 323 polycentral F-locality . . . . . . . 412 reduced F-locality . . . . . . . . . . 427 X F - , (F, X)-locality . . . . . . . . . . 320 locality associated with F . . . . . 319 F-localizer . . . . . . . . . . . . . . . . . . . . 335 F-localizing functor . . . . . . . . . . . 345 m Mackey functor . . . . . . . . . . . . . . . . 85 F-maximal normal chain . . . . . . . 67 k-mod , O-mod . . . . . . . . . . . . . . . . . 85 M¨ obius function, µX (x) . . . . . . . . 25

Mon(Uh , G) . . . . . . . . . . . . . . . . . . . 219 ˆh , G) ˆ . . . . . . . . . . . . . . . . . 218 Monk∗ (U n NFK (Q) . . . . . . . . . . . . . . . . . . . . . . . . 34 NFK,Q (Q) . . . . . . . . . . . . . . . . . . . . . 320 ¯ X (H) . . . . . . . . . . . . . . . . . . . . . . . . 17 N G ¯ˆ G (Pγ ). . . . . . . . . . . . . . . . . . . . . . . .23 N νˆP,S,S . . . . . . . . . . . . . . . . . . . . . . . . 126 N(Q) . . . . . . . . . . . . . . . . . . . . . . . . . 379 Nat(f, g) , Nat(f, g) . . . . . . . . . . . . . 16 naive cohomology groups . . . . . . 465 natural map . . . . . . . . . . . . . . . . . . . 16 2-natural map . . . . . . . . . . . . . . . . 452 shifting natural map . . . . . . . . 452 natural transformation . . . . . . . . 460 F-nilcentralized subgroup . . . . . . 47 nilpotent block . . . . . . . . . . . . . . . . . 20 normal P -subcategory . . . . . . . . 182 F-normal subgroup . . . . . . . . . . . 182 X-normalizer . . . . . . . . . . . . . . . . . . 17 o O , O-mod . . . . . . . . . . . . . . . . . . . . . 85 K-objects . . . . . . . . . . . . . . . . . . . . . . 73 Ω1 (P ) . . . . . . . . . . . . . . . . . . . . . . . . . 25 Op (G) , Op (G) . . . . . . . . . . . . . . . . . 25 Op (G) , Op (G) . . . . . . . . . . . . . . . 25 Op (F) . . . . . . . . . . . . . . . . . . . . . . . . 361 OQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Q,sc O . . . . . . . . . . . . . . . . . . . . . . . . . 410 ω ˜ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 orbit category . . . . . . . . . . . . . . . . . . 73 ordered category . . . . . . . . . . . . . . 480 regular chain . . . . . . . . . . . . . . . 480 category of regular chains . . . 480 chr (C) . . . . . . . . . . . . . . . . . . . . . . 480 ch∗r (C) . . . . . . . . . . . . . . . . . . . . . . 481 Hnr (C, a∗ ) . . . . . . . . . . . . . . . . . . . 482 p p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 P(A) , PA (H) . . . . . . . . . . . . . . . . . . 18

Index

497

ψ

K , ψχ . . . . . . . . . . . . . . . . . . . . . . . . 28 pX,m . . . . . . . . . . . . . . . . . . . . . . . . . . 455 P PQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Ph,Gˆ . . . . . . . . . . . . . . . . . . . . . . . . . 207 partially deﬁned F-locality . . . . 300 perfect F-locality . . . . . . . . . . . . . 303 point . . . . . . . . . . . . . . . . . . . . . . . . . . 18 pointed group . . . . . . . . . . . . . . . . . . 18 proper category of chains . . . . . 456 (P, ¯s)-balanced function . . . . . . . 145 (P, X, n)-gluing family . . . . . . . . 144 q qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 k ∗ -quotient . . . . . . . . . . . . . . . . . . . . 22 r

rh,h . . . . . . . . . . . . . . . . . . . . . . . . . . 224 r1P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 rdP Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 rgC . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 RDk (P ) , R◦ Dk (P ) . . . . . . . . . . . 105 RGk (C) . . . . . . . . . . . . . . . . . . . . . . 240 ˆ C ) . . . . . . . . . . . . . . . . . . . . 240 RHˆGk (H (HC )nc ) . . . . . . . . . 250 RHGk (HC , aut radical . . . . . . . . . . . . . . . . . . . . . . . . . 18 F-reducible . . . . . . . . . . . . . . . . . . . . 59 reduced F-locality . . . . . . . . . . . . 427 redundant chain . . . . . . . . . . . . . . 484 regular chain . . . . . . . . . . . . . . . . . 480 regular representation of a category . . . . . . . . . . . 454 relative exterior intersection . . . . 82 representation of a 2-category . . . . . . . . . 453 weak structure. . . . . . . . . . . . . .454 residual Dade group . . . . . . . . . . 108 C-residual Grothendieck group . . . . 256 s Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 S(Q) . . . . . . . . . . . . . . . . . . . . . . . . . 398

X

sc

S (Q) , S (Q) . . . . . . . . . . . . . . . 409 sh . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 sh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 ς . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 saturated fusion system. . . . . . . . .29 selfcentralizing Brauer pair . . . . . 96 selfcentralizing subgroup . . . . . . . 17 F-selfcentralizing subgroup. . . . .49 semidirect product . . . . . . . . . . . . 455 pX,m . . . . . . . . . . . . . . . . . . . . . . . .455 shifting functor . . . . . . . . . . . . . . . 452 shifting natural map . . . . . . . . . . 452 similar Dade P -algebras . . . . . . . . 21 simplicial 2-category . . . . . . . . . . 451 solvable Frobenius P -category . . . 361 F-solvcentralized subgroup . . . . 365 F-stable subgroup . . . . . . . . . . . . 179 F(β,H,X) -stable element . . . . . . . 152 stable cohomology groups . . . . . 468 D-stable cohomology groups . . 468 D-stable elements . . . . . . . . . . . . . 469 mw -stable adjoin image . . . . . . . 458 standard representation . . . . . . . 453 strict semicovering . . . . . . . . . . . . . 18 supranatural map . . . . . . . . . . . . . 453 supranatural transformation . . 461 symmetric group . . . . . . . . . . . . . . . 25 t tdP Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 tenP Q . . . . . . . . . . . . . . . . . . . . . . . . . . 108 TG (K, H) . . . . . . . . . . . . . . . . . . . . . 25 TenG H (B) . . . . . . . . . . . . . . . . . . . . . 103 TG . . . . . . . . . . . . . . . . . . . . . . . . . . . .319 X TP . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 θˆ(Q,f ) . . . . . . . . . . . . . . . . . . . . . . . . . 164 R,T , T ˇ R,T . . . . . . . . . . . . . 77 TR,T , T ˜ ˜ , T ˇ ˜ ˜ . . . . . . . . . . . . . 80 T ˜ ˜ , T β,β

β,β

β,β

tensor induction . . . . . . . . . . . . . . 103 thick basic P × P -set . . . . . . . . . 390 triples . . . . . . . . . . . . . . . . . . . . . . . . . 77

498

Frobenius categories versus Brauer blocks u

x

ˆh . . . . . . . . . . . . . . . . . . . . . . . . 219 uh , u

XF˜ (Q) , XF (Q) . . . . . . . . . . . . . . . . 64

v vC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 w ˆ h . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 w ˆ aut w h . . . . . . . . . . . . . . . . . . . . . . . . . . 224 weak structure . . . . . . . . . . . . . . . . 454

z zF sc . . . . . . . . . . . . . . . . . . . . . . . . . . 205 z◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 z◦F sc . . . . . . . . . . . . . . . . . . . . . . . . . . 207 zch∗ (F sc ) . . . . . . . . . . . . . . . . . . . . . . 217 ˆ ˜z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412