On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks
Brauer had already introduced the defect of a block and opened the way towards a classification by solving all the problems in defects zero and one, and by providing some evidence for the finiteness of the set of blocks with a given defect. In 1959 he dis
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Series Editors H. Bass
J. Oesterle
A. Weinstein
Lluis Puig
On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks
Springer Basel AG
Author: Lluis Puig Carreres CNRS, Institut de Math6natiques de Jussieu Universite Denis Diderot (paris VII) F· 75251 Paris Cedex 05 1991 MBthematics Subject Classification 20CII
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek CataJoging-in-Publication Data
Pulg, Lluis: On the local structure of Morita and Rickard equivalences between Brauer blocks I
Lluis Puig. - Basel ; Boston; Berlin : Birkhäuser, 1999 (Progress in mathematics ; Vol. 178) ISBN 978-3-0348-9732-7 ISBN 978-3-0348-8693-2 (eBook) DOI 10.1007/978-3-0348-8693-2
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, 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, pennission from the copyright owna must be obtained. CI 1999 Springer Basel AG Originally published by Birkhäuser Verlag in 1999 Softcovcr reprint of the hardcover Ist edition 1999
Printed on acid-free paper produced of chlorine-free pulp. TCF ISBN 978-3-0348-9732-7 98765432 1
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Contents
1 2
Introduction ................................................... . General notation, terminology and quoted results .................. .
1 9
3 4 5 6 7 8 9 10 11 12 13 14 15
Noninjective induction of G-interior algebras
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113
.... .. .. ... . ... .. .... .... ... .... 123
Brauer sections in basic induced .2>G-interior algebras .............. 133 Pointed groups on .2>G-interior algebras and Higman embeddings 151 Hecke .2>G-interior algebras and noninjective induction
. . . . . . . . . . . .. 175
16
On the local structure ofHecke DG-interior algebras
.. .. .. .. .... ... 181
17
Brauer sections in basic Hecke DG-interior algebras
........ ... . ... 199
18
Rickard equivalences between Brauer blocks
19
Basic Rickard equivalences between Brauer blocks
. . . . . . . . . . . . . . . . . . . . . .. 215 . . . . . . . . . . . . . . . .. 229
Appendix 1 A proof of Weiss' criterion for permutation (A,»))K 19K 19
which, up to identify G / K to cp (G), is actually an {9 ( G / K )-interior algebra homomorphism as it is easily checked, and is clearly bijective whenever either A' is {9 -free or A is a 1 x I-projective {9(K x K)-module(sincewehave ({9 019K IndfxxIK (X)( ~ X for any (9-module X.) On the other hand, by2.6.8 we have the canonical (9G'-interior algebra isomorphism 3.16.4
Ind~~G)({9 0 A)K) 0 A' ~ Ind~~G)({9 0 A)K 0 Res~~G)(A'») 19K
19
19K
(9
Hence, inducing homomorphism 3.16.3 from cp(G) to G' and composing the induced homomorphism with isomorphism 3.16.4, we get from 3.3.1 the announced homomorphism 3.16.1, which is indeed bijective whenever
