Serge Bouc
Green Functors and G-sets
~ Springer
Author Serge Bouc Equipe des groupes finis CNRS UMR 9994 UFR de Mathdmatiques Universit6 Paris 7 - Denis Diderot 2, Place Jussieu F-75251 Paris, France e-mail:
[email protected], fr
Cataloging-in-Publication Data applied for
D i e D e u t s c h e B i b l i o t h e k - CIP-Einheitsaufnahme
B o u t , Serge: Green functors and G-sets / Serge Bouc. - Berlin ; Heidelberg ; N e w York ; Barcelona ; Budapest ; H o n g K o n g ; L o n d o n ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1997 (Lecture notes in mathematics ; 1671) ISBN 3-540-63550-5
Mathematics Subject Classification (1991): 19A22, 20C05, 20J06, 18D35
ISSN 0075- 8434 ISBN 3-540-63550-5 Springer-Verlag Berlin Heidelberg New York 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, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10553356 46/3142-543210 - Printed on acid-free paper
Contents Mackey functors 1.1 E q u i v a l e n t definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 D e f i n i t i o n in t e r m s of s u b g r o u p s . . . . . . . . . . . . . . . . . . 1.1.2 D e f i n i t i o n in t e r m s of G-sets . . . . . . . . . . . . . . . . . . . . 1.1.3 D e f i n i t i o n as m o d u l e s over t h e M a c k e y a l g e b r a . . . . . . . . . . 1.2 T h e M a c k e y f u n c t o r s M ~ M y . . . . . . . . . . . . . . . . . . . . . . 1.3 C o n s t r u c t i o n of H ( M , N ) and M(~N . . . . . . . . . . . . . . . . . . 1.4 I d e n t i f i c a t i o n of H ( M , N ) . . . . . . . . . . . . . . . . . . . . . . . . 1.5 I d e n t i f i c a t i o n of M @ N . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 A n o t h e r i d e n t i f i c a t i o n of M Q N . . . . . . . . . . . . . . . . . . . . . 1.7 F u n c t o r i M i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 n-fold t e n s o r p r o d u c t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Universal property . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 C o m m u t a t i v i t y a n d a s s o c i a t i v i t y . . . . . . . . . . . . . . . . . . . . . 1.10 A d j u n c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 5 5 6 7 8 .
9
.
10 12 16 24 25 25 29 38 38
. .
Green functors 2.1 D e f i n i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 D e f i n i t i o n in t e r m s of G-sets . . . . . . . . . . . . . . . . . . . . . . . . 2.3 E q u i v a l e n c e of t h e two definitions . . . . . . . . . . . . . . . . . . . . . 2.4 T h e B u r n s i d e f u n c t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 T h e B u r n s i d e f u n c t o r as M a c k e y f u n c t o r . . . . . . . . . . . . . 2.4.2 T h e B u r n s i d e f u n c t o r as G r e e n f u n c t o r . . . . . . . . . . . . . . 2.4.3 T h e B u r n s i d e f u n c t o r as initial o b j e c t . . . . . . . . . . . . . . . 2.4.4 T h e B u r n s i d e f u n c t o r as u n i t . . . . . . . . . . . . . . . . . . .
41 41 46 48 52 52 55 57 59
The category associated to a Green functor 3.1 E x a m p l e s of m o d u l e s over a G r e e n f u n c t o r . . . . . . . . . . . . . . . . 3.2 T h e c a t e g o r y CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A - m o d u l e s a n d r e p r e s e n t a t i o n s of CA . . . . . . . . . . . . . . . . . . .
61 61 65 71
The 4.1 4.2 4.3 4.4 4.5
81 81 82 84 85 94
algebra associated to a Green functor The evaluation functors . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation and equivalence . . . . . . . . . . . . . . . . . . . . . . . . . T h e a l g e b r a A(f/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P r e s e n t a t i o n by g e n e r a t o r s a n d r e l a t i o n s . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI
CONTENTS
4.5.1 4.5.2
The Mackey algebra The Yoshida algebra
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94 95
Morita equivalence and relative projectivity 5.1 M o r i t a e q u i v a l e n c e of a l g e b r a s A ( X 2) . . . . . . . . . . . . . . . . . . . 5.2 R e l a t i v e p r o j e c t i v i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 C a r t e s i a n p r o d u c t in CA . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 C a r t e s i a n p r o d u c t in CA x CA . . . . . . . . . . . . . . . . . . . 5.4 M o r i t a e q u i v a l e n c e a n d r e l a t i v e p r o j e c t i v i t y . . . . . . . . . . . . . . . 5.5 P r o g e n e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Finitely generated modules ..................... 5.5.2 Idempotents and progenerators ..................
99 99 100 103 103 107 109 112 114 114 115
Construction of Green functors 6.1 T h e f u n c t o r s H ( M , M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The product 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 T h e o p p o s i t e f u n c t o r of a G r e e n f u n c t o r . . . . . . . . . . . . . . . . . 6.2.1 Right modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 T h e d u a l of a n A - m o d u l e . . . . . . . . . . . . . . . . . . . . . . 6.3 T e n s o r p r o d u c t of G r e e n f u n c t o r s . . . . . . . . . . . . . . . . . . . . . 6.4 B i m o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 C o m m u t a n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 T h e f u n c t o r s M | N . . . . . . . . . . . . . . . . . . . . . . . . . . .
123 123 125 127 129 130 134 141 143 146
A Morita theory 7.1 C o n s t r u c t i o n of b i m o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 M o r i t a c o n t e x t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 C o n v e r s e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A r e m a r k on b i m o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . . .
153 153 154 160 163
Composition
167
8,1 8.2 8.3 8.4 8.5 8.6 8.7
167 168 170 173 175 177 180
Bisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition and tensor product ...................... Composition and Green functors ...................... C o m p o s i t i o n a n d a s s o c i a t e d categories . . . . . . . . . . . . . . . . . . Composition and modules . . . . . . . . . . . . . . . . . . . . . . . . . Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: induction and restriction ....................
Adjoint constructions 9.1 A left a d j o i n t to t h e f l m c t o r Z ~-+ U OH Z . . . . . . . . . . . . . . . . . 9.2 T h e categories D u ( X ) . . . . . . . . . . . . . . . . . . . 9.3 T h e f u n c t o r s Q u ( M ) . . . . . . . . . . . . . . . . . . . . . 9.4 T h e f u n c t o r s L u ( M ) . . . . . . . . . . . . . . . . . . . . 9.5 Left a d j u n c t i o n . . . . . . . . . . . . . . . . . . . . . . . . 9.6 T h e f u n c t o r s S u ( M ) . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
183 183 186 188 193 196 205
CONTENTS 9.7 9.8 9.9
T h e f u n c t o r s Ru(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Right adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 Induction and restriction ...................... 9.9.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.3 Coinflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Adjunction and Green functors 10.1 F r o b e n i u s m o r p h i s m s . . . . . . . . . . . . . . . . . . . . 10.2 Left a d j o i n t s a n d t e n s o r p r o d u c t . . . . . . . . . . . . . . . . . . . . . . 10.3 T h e G r e e n f u n c t o r s L u ( A ) . . . . . . . . . . . . . . . . . 10.4 Lu(A)-modules a n d a d j u n c t i o n . . . . . . . . . . . . . . . . . . . . . . 10.5 R i g h t a d j o i n t s a n d t e n s o r p r o d u c t . . . . . . . . . . . . . . . . . . . . . 10.6 R u ( M ) as L u ( A ) - m o d u l e . . . . . . . . . . . . . . . . . . 10.7 Lu(A)-modules a n d r i g h t a d j o i n t s . . . . . . . . . . . . . . . . . . . . . 10.8 E x a m p l e s a n d a p p l i c a t i o n s . . . . . . . . . . . . . . . . . 10.8.1 I n d u c t i o n a n d r e s t r i c t i o n . . . . . . . . . . . . . . . . . . . . . . 10.8.2 T h e case U/H = 9 . . . . . . . . . . . . . . . . . 10.8.3 A d j u n c t i o n a n d M o r i t a c o n t e x t s . . . . . . . . . . . . . . . . . . 11 T h e 11.1 11.2 11.3 11.4
VII 207 209 215 215 217 217
223 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
simple modules Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of t h e s i m p l e m o d u l e s . . . . . . . . . . . . . . . . . . . . T h e s t r u c t u r e of algebras / i ( H ) . . . . . . . . . . . . . . . . . . . . . . T h e s t r u c t u r e of' s i m p l e m o d u l e s . . . . . . . . . . . . . . . . . . . . . .
11.4.1 T h e i s o m o r p h i s m SH,v(X) ~-- H o m ( [ X H ] , V)~ G(H) . . . . . . . . 11.4.2 T h e A - m o d u l e s t r u c t u r e of SH,V . . . . . . . . . . . . . . . . . . 11.5 T h e s i m p l e G r e e n f u n c t o r s . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 S i m p l e f u n c t o r s a n d e n d o m o r p h i s m s . . . . . . . . . . . . . . . . . . . .
223 227 231 234 242 250 255 264 264 264 266 275 275 275 278 282 282 289 291 295
12 C e n t r e s 12.1 T h e c e n t r e of a G r e e n f u n c t o r . . . . . . . . . . . . . . . . . . . . . . . 12.2 T h e f u n c t o r s CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 A n o t h e r a n a l o g u e of t h e c e n t r e . . . . . . . . . . . . . . . . . . 12.2.2 E n d o m o r p h i s m s of t h e r e s t r i c t i o n f u n c t o r . . . . . . . . . . . . . 12.2.3 I n d u c t i o n a n d inflation . . . . . . . . . . . . . . . . . . . . . . . 12.3 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.] T h e f u n c t o r s FPB . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 T h e blocks of M a c k e y algebra . . . . . . . . . . . . . . . . . . . .
305
Bibliography
337
Index
339
305 315 315 323 329 332 332 335
Introduction The theory of Mackey functors has been developed during the last 25 years in a series of papers by various authors (J.a. Green [8], a. Dress [5], T. Yoshida [17], J. Th~venaz and P. Webb [13],[15],[14], G. Lewis [6]). It is an attempt to give a single framework for the different theories of representations of a finite group and its subgroups. The notion of Mackey functor for a group G can be essentially approached from three points of view: the first one ([8]), which I call "naive", relics on the poset of subgroups of G. The second one ([5],[17]) is more "categoric", and relies on the category of G-sets. The third one ([15]) is "algebraic", and defines Mackey functors as modules over the Mackey algebra. Each of these points of view induces its own natural definitions, and the reason why this subject is so rich is probably the possibility of translation between them. For instance, the notion of minimal subgroup for a Mackey functor comes from the first definition, the notion of induction of Mackey functors is quite natural with the second, and the notion of projective Mackey functor is closely related to the third one.
The various rings of representations of a group (linear, pernmtation, p-permutation...), and cohomology rings, are important examples of Mackey functors, having moreover a product (tensor product or cup product). This situation has been axiomatized, and those functors have been generally called C-functors in the literature, or Green functors. This definition of a Green functor t o r a group G is a complement to the "naive" definition of a Mackey fnnctor: to each subgroup of G corresponds a ring, and the various rings are connected by operations of transfer and restriction, which are compatible with the product through Frobenius relations. The object of this work is to give a definition of Green functors in terms of Gsets, and to study various questions raised by this new definition. From that point of view, a Green fimctor is a generalized ring, in the sense that the theory of Green functors for the trivial group is the theory of ordinary rings. Now ring theory gives a series of directions for possible generalizations, and I will treat some cases here (tensor product, bimodnles, Morita theory, commutants, simple modules, centres). The first chapter deals only with Mackey functors: my purpose was not to give a full exposition of the theory, and I just recall the possible equivalent definitions, as one can find for instance in the article of Thevenaz and Webb ([15]). I show next how to build Mackey functors "with values in the Mackey functors", leading to the functors 7-{(M, N) and M@N, which will be an essential tool: they are analogous to the homomorphisms modules and tensor products for ordinary modules. Those constructions already appear in Sasaki ([12]) and Lewis ([6]). Thc notion of r~dinear map can be generalized in the form of r~-linear morphism of Mackey functors. The
2
INTROD UCTION
reader may find that this part is a bit long: this is because I have tried here to give complete proofs, and as the subject is rather technical, this requires many details. Chapter 2 is devoted to the definition of Green functors in terms of G-sets, and to the proof of the equivalence between this definition and the classical one. It is then possible to define a module over a Green functor in terms of (-;-sets. I treat next the fundamental case of the Bm-nside functor, which plays for Green functors the role of the ring Z of integers. In chapter 3, I build a category CA associated to a Green functor ,4, and show that the category of A-modules is equivalent to the category of representations of CA. This category is a generalization of a construction of Lindner ([9]) for Mackey functors, and of the category of permutation modules studied by Yoshida ([17]) for cohomological Mackey functors. Chapter 4 describes the algebra associated to a Green functor: this algebra enters the scene if one looks %r G-sets ~ suct~ that the evaluation functor at ft is an equivalence of categories between the category of representations of Cn and the category of Endc~(f~)-modules. This algebra generalizes the Mackev algebra defined by Thevenaz and Webb ([1.5]) and the Hecke algebra, of Yoshida ([17]). It is possible to give a definition of this algebra by generators and relations. This algebra depends on the set f/, but only up to Morita equivalence. Chapter 5 is devoted to the relation between those Morita equivalences and the classical notion of relative projectivity of a Green functor with respect to a G-set (see for instance the article of Webb [16]). More generaliy, I will deduce some progenerators for the category of A-modules. Chapter 6 introduces some tools giving new Green functors from known ones: after a neat description of the Green functors ~(/11,/1I), I define the opposite functor of a Green flmctor, which leads to the notion of right module over a Green functor. A natural example is the dual of a left module. The notion of tensor product of Green functors leads naturally to the definition of bimodule, and the notion of comnmtant to a definition of the Mackey functors 7t.4(M, N) and M(~,4N. Those constructions are the natural framework for Morita contexts, in chapter 7. The usual Morita theory can be generalized without difficulty to the case of Green functors for a given group G. The chapters S,9, and 10 examine the relations between Green functors and bisets: this notion provides a single framework for induction, restriction, inflation, and coinflation of Mackey functors (see [2]). In chapter 8, I show how the composition with U, if U is a G-set-H, gives a Green functor A o U for the group H starting with a Green functor A for the group G. This construction passes down to the associated categories, so there is a corresponding functor from CAoU to Ca. This gives a functor between the categories of representations, which can also be obtained by composition with U. I study next the functoriality of these constructions with respect to U, and give the example of induction and restriction. Chapter 9 is devoted to the construction of the associated adjoint functors: I build a left and a right adjoint to the functors of composition with a biset /14 ~ 114 o U for Mackey fnnctors, and I give the classical examples of induction, restriction and inflation, and also the less well-known example of coinfiation. Chapter 10 is the most technical of this work: I show how the previous left adjoint
INTRODUCTION
3
functors give rise to Green functors, and I study the associated functors and their adjoints between the corresponding categories of modules. An important consequence of this is the compatibility of left adjoints of composition with tensor products, which proves that if there is a surjective Morita context for two Green functors A and B for the group G, then there is one for all the residual rings A(H) and B(H), for any subgroup H of G. In chapter 11, I classify the simple modules over a Green functor, and describe their structure. Applying those results to the Green functor A@A ~ I obtain a new proof of the theorem of Th4venaz classifying the simple Green functors. Finally, I study how the simple modules (or similarly defined modules) behave with respect to the constructions ~ ( - , - ) and - Q - . Chapter 12 gives two possible generalizations of the notion of centre of a ring, one in terms of commutants, the other in terms of natural transformations of functors. The first one gives a decomposition of any Green functor using the idempotents of the Burnside ring, and shows that up to (usual) Morita equivalence, it is possible to consider only the case of Green functors which are projective relative to certain sets of solvable rr-subgroups. The second one keeps track of the blocks of the associated algebras. Then I give the example of the fixed points functors, and recover the isomorphism between the center of Yoshida algebra and the center of the group algebra. Next, the example of the Burnside ring leads to the natural bijection between the p-blocks of the group algebra and the blocks of the p-part of the Mackey algebra.
Chapter 1 Mackey functors All the groups and sets with group action considered in this book will be finite.
1.1
Equivalent definitions
Throughout this section, I denote by G a (finite) group and R a ring, that m a y be non-commutative. First I will recall briefly the three possible definitions of Mackey functors: the first one is due to Green ([8]), the second to Dress ([5]), and the third to Th6venaz and Webb ([15]).
1.1.1
D e f i n i t i o n in t e r m s o f s u b g r o u p s
One of the possible definitions of Mackey functors is the following: A Mackey functor for the group G, with values in the category R - M o d of R-modules, consists of a collection of R-modules M ( H ) , indexed by the subgroups H of G, togerber with maps t H : M ( K ) --+ M ( H ) and r K ll : M ( H ) --+ M ( K ) whenever Ir is a subgroup of H, and maps Cc,H : M ( H ) , M ( ~ H ) for x 6 G, such that: 9 If L C_ t ( C_ H, then t hH- Kt L
A ' rHK = r H . t H and r L
:
9 If x, y E G and H G G, then CyjHCx, H : Cyx,H. 9 If x E G and H C_ G, t h e n Cx,H tH = e~,H = I d if x E H.
t .XH KCx, K
H and cx,icr K
=
7,~H xKceGH.
Moreover
9 (Mackey axiom) If L C H _D K , then H H
FL tA"
=
E
L
K
~LnxKCx,LXAA-FLxnA.
xEL\H/K
H are called restrictions. The maps tK H are called transfers or traces, and the maps r K A morphism 6 from a Mackey functor M to a Mackey functor N consists of a collection of morphisms of R-modules OH : M ( H ) --+ N ( H ) , for H C_ G, such that if
6
CHAPTER I. M A C K E Y FUNCTORS
K C_ H and x E G, the squares
M(I,,d,))
. . . .
which proves that f ' = f, and completes the proof of the proposition.
,.
Let f E s ,M.,~; P) be an n-linear morphism, associated by the previous proposition to a unique morphism of Mackey functors ffrom MI~... ~M~ to P. The formula
.fx ([grZl @... @ WtnJ(Y,~)) z P.(~)P*(5~,y)fy,...,y(ml,..., rn,~) shows that f is entirely determined by the knowledge, for any G-set Y, of the s-linear map fz : M~(Y) x . . . x M,~(Y) ~ P(Y) defined by
/r(-~,...,-~,)
= P'( YX'Y' ( yx'y' ~ I \g(y)x'y'] )
kfx'y') is cartesian. So
op
(x'y'~
(x'y
x~y
(a •
yx,g, f JI
A . ( g ) ( a ) or, ~ ' = M. t x' ) M. \ x , g ( y ) ) M *
7321)
It follows that ((M~162
x
A.(g)(a))(m') . . . .
. . . . CM*(f)M. \ x' ] M. \x,g(y)j M* \yx'g(y)J (a x mr) . . . . =r The square
XY X
(x'Y~ M. ( x'y ~ (a
Sg(y))
\ ~')
(,/%) I
x m')
X'Y
ix,)
)
X ~
f
is cartesian. So
((M~162
x A.(g)(a))(m')
.... +~./~)
f(x)y M* \yx'g(y)) (a x m ' )
~" (x"//o~'~,x~'~/=.x~ +/oo~*~,~'~/ . . . . (r x a)M*(f x g)(m') = (M~
and this proves that the product is covariant. If now qY E M~ if a' E A(Y'), and if m E
((M~162
') x A*(g)(a'))(m) = (M~162
M(X
x g)(r x
x Y), then
m) . . . .
....
....
a)(m')
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
132
=r
p m)
Moreover
A*(g)(a') oF m = M.
yxy
g(y)xy
(a' •
So
....
f(x)
r
g(y)xy
(a' • m)
On the other hand
(M~
• g)(r • a')(m) = (r • a')M.(f • g)(m) = r
oy, ~ M . ( f • g)(m))
But
a oy, M . ( f •
= M. \ x' ] M* \y'x'y']
....
(x'y'~ M* ( x'y' ~ ( y'xy M. \ x' ] \y'x'y'] M. \y'f(x)g(y)] ( a ' • m)
As the square
g(s)xy XY
YIXY
)
\y'x'y'] is cartesian, this is also
a oy, M . ( f •
\ x' / M . ( f •
....
g(y)xy
M.
f(x)
(a'xm) ....
g(y)xy
(a' x m)
So I have
(M~
• g)(r • a')(m) = r
f(x)
g(y)xy
(a' • m) . . . .
6.2. THE OPPOSITE FUNCTOR OF A GREEN FUNCTOR ....
((M~162
133
') x A*(g)(a'))(m)
which proves that the product is also contravariant. I must still prove that the product is associative and unitary. So let X, Y, and Z be G-sets. If r E M~ if a e A(Y), and b E A(Z), then for m e M ( X • Y x Z), I have ( (r • a) x b) (m) = (r x a)(b oz~ ~ ) = r ( ~ 0 7 (b ozo~ m)) But
a~ (b~176 = M* (XY)M* ( xY ) (a x (b~176 Moreover
a x (bo~ m) = a x M. (xYZl M* ( xYZ t ( b x m ) . . . . \ xy ] \zxyz/ ....
M,(y'xy2zlM*(y'xy2Zl(axbxm \ ylxy2 ] \ylzxy2z/
As the square
yxyz/ XYZ
)
YXYZ
~
YXY
(x::) 1
1( :::: )
XY yxy is cartesian, I have
yxy
\ ylxy2 ]
\ xy /
kyxyz/
and then
\ xy ]
\yxyz/
.... On the other hand
(~ •
\ylzxy2z]
M.(x:Z) M.(
xyz ~ ( a • 2 1 5 \yzxyz/
(a • b))(~): ~((a x b/o~z~ ~)
Since (a• b)op
Ovz m = M.
the product is associative. Moreover, if r C M~
(x;z)
114.
( xyz ~ (a x b x m) \yzxyz/
and if m E M(X), then
(r • ~ ) ( ~ ) = r
o:~ ~)
As
(?)M" .x. the product is unitary, which completes the proof of the proposition.
)
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
134
6.3
Tensor product of Green functors
If A and B are Green functors for the group G, then their tensor product A@B can be given a structure of Green functor:
If X and Y are G-sets, if ( U,r ) is a G-set over X and (V,r ) is a G-set over Y, ira E A(U), b E B(U), c E A(V) and d E B(V), I define the product of the dement [a @ b](u,r of A@B(X) by the dement [c | d](v,r of A@B(Y) by setting Definition:
[a | b](u.r x [c | d](v.r = [(a x c) | (b x d)](vxv,r215162 E ( A + B ) ( X x Y)
If CA and r
are the units A and B, I set
P r o p o s i t i o n 6.3.1: L e t A a n d B b e G r e e n f u n c t o r s for t h e g r o u p G. T h e n t h e p r o d u c t x t u r n s A@B into a G r e e n functor, w i t h unit e. Proof: First I must check that this product is well defined. So let f : (U, r be a morphism of G-sets over X, let a E A(U) and b' E B(U'). Then
. . . .
[(A.U x ' V
> y g
6.3. TENSOR PRODUCT OF GREEN FUNCTORS
135
so that
(A@B)*(f)(u) = [A*(c~)(a)@B*(c~)(b)](p,;) (A~B)*(g)(v) = [A*('7)(c)@B*(7)(d)](Q,~) The product (A@B)*(I)(u) x (A@B)*(g)(v)is then equal to
which can also be written as [(A*(c~ x @ ( a x c))|
(B*(a • 7)(bx d))J(pxQ,Zx5 )
As the square
PxQ
~x~
, U•
,8x51
[r
X'xX
~ XxY fxg
is also cartesian, I have
• g)( x
)(ax c))|
)(bx
)
which proves that the product x on A@B is contravariant, hence bifunctoriah It is clear from the definitions that the product x is associative. Finally, the element c is a unit, since [a @ b](g,r • [CA | eB](o,Id) = [(a • CA) @ (b x eB)l(uxo,r
= [a | b](g,r
[CA @ eB](o,za) x [a | b](v,r = [(EA • a) | (eB • b)](.•162
= [a | b](u,r
and this completes the proof of the proposition.
"
The tensor product of Green functors is functoriah Lemma6.3.2: : If f : A ~ A' and g : B ~ B' are m o r p h i s m s o f G r e e n f u n e t o r s for t h e g r o u p G, t h e n f @ g : A @ B ~ A @ B ~ is a m o r p h i s m o f G r e e n f u n c t o r s . If f a n d g are u n i t a r y , so is f @ g .
Proof: The morphism f (resp. the morphism g) is determined by morphisms fu (resp. gu) from A(U) to At(U) (resp. from B(U) to B'(U)), for any G-set U. Moreover, the image of the element [a | b](a,r of (A4B)(X) under (f@9)x is given by
(f@g)x ([a | bl(u,r
= [f~(a) | gu(b)l(u,r
The lemma follows easily, since if f and g are morphisms of Green functors, then
fu•
• c) = fu(a) x fv(c)
gu•
• d) = gv(b) • gv(d)
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
136
A special case of this lemma is the case when f is the identity morphism of A, and g is the unique (unitary) morphism of Green functors from b to B. since A@b ~- A, what I get is a unitary morphism ~A of Green functors from A to A@B, and it is easy to see that this morphism is given by
a 9 A ( X ) ~ "TA,X(a) = In @ r
a @ B*
(r
(X,Id)
Similarly, there is a morphism ?B from B to A@B defined by bE
(Xdd)
A@B(X)
The notion of opposite functor of a Green functor leads to the following definition: Definition: Let A be a Green functor for the group G. If X and Y are G-sets, I will say that the element a 9 A ( X ) commutes with/3 9 A(Y) ira x fl = a x ~ /3, i.e.
xy
a x/3=A,
Similarly, I will say that a subset P C A(X) commutes with a subset Q c_ A ( Y ) if any element of Y commutes with any element of Q. If M is a sub-Mackey functor of A, and N a sub-Mackey functor of B, I will say that M commutes with N if M ( X ) commutes with N ( Y ) for any G-sets X and Y. It is clear that if a commutes with fl, then fl commutes with a, since taking the image of the above equality under A. ( ~ ) exchanges the roles of X and Y and of a and/3. I will also say that in a more symmetric way that a and/3 commute. L e m m a 6.3.3: T h e image of 7A in
A~B
c o m m u t e s with the image of "~B.
P r o o f i Let X and Y be G-sets. If a 9 A ( X ) and b 9 B(Y), then
7A,X(a) • 7B,g( b) = [a @ r
x [CA,y @ 5](Y,Id) = [(a X ~ A,Y ) @ (CB,X X b)](XxY, id)
Moreover
aXCA,y=axA*(Y)(r Similarly, I have eB,X x b = B* (~Y)(b). Then
~/A,X(a) X'TB,Y(b)= [ A * ( X ; J ) ( a ) | 2 1 5
)
On the other hand
"/By(b) • ~/A,X(a) ---- [gAy @ b](Y,Id) • [a @ gB,X](X,ld) = [(CA,y • a) | (b | s or
7S,y(b)•
[A*(YxX)(a)|215
)
6.3. TENSOR PRODUCT OF GREEN FUNCTORS
137
The image under ( A ~ B ) . (~;) of this element is
Since moreover
it is also
[A* (?) (a)@'* (::) ~" (Y:) (b)](XxY,id) I have also
B.
xy
yx
y
so finally ~A,x(a) • ~/.,y(b) = ( A 6 B ) .
•
xy
7A,X(a))
which proves the lemma.
9
This temma leads naturMIy to the universal property of the tensor product of Green functors: P r o p o s i t i o n 6.3.4: Let A, B and C be G r e e n f u n c t o r s for t h e g r o u p G. I f f (resp. g) is a m o r p h i s m of G r e e n f u n c t o r s f r o m A to C (resp. f r o m B to C), a n d if t h e i m a g e of f c o m m u t e s with t h e i m a g e of g, t h e n t h e r e exists a u n i q u e m o r p h i s m of G r e e n f u n c t o r s h f r o m AQB to C such t h a t f = ho"/A andg=hoTB. C o n v e r s e l y , if h is a m o r p h i s m of G r e e n f u n c t o r s f r o m A@B to C, t h e n f = hOTA (resp. g = h o ~ B ) is a m o r p h i s m of G r e e n f u n c t o r f r o m A to C (resp. f r o m B to C), and t h e i m a g e s of f a n d g c o m m u t e . M o r e o v e r h is u n i t a r y if and only if f and g are. Proof: If (U,r
is a G-set over X, if a E A(U) and b E B(U), then
Moreover
k Ul /
\ 71"2/
(UxU,Id)
The image of this element under (A@B)* (~\), which is the product 7A(a)."/B(b)in A@B(U), is obtained using the cartesian square
U
~ UxU
U
, UxU
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
138 So it is equal to
vA(a)."/B(b) =
(ulu2~ (a)~) B* ( u ) B* (ulu~
[A*( u ) A * UU
\
Now if h is a morphism of
hx([a *
b](u,r
:
~t 1 /
A~B
C.(r
UU
\
U2
(b)]
/
=
[a|
zd)
(U, Id)
to C, I have
* b](u,id)) =
C.(r
(uL ) hu•
•
VB(b))
Whence
hx([a |
b](u,r
(uL) "hu'TA(a) • huvB(b))
= C.(r
and h is determined by h o 7A and h o "~B. Then if f is a morphism from A to C and g is a morphism from B to C, there is at most one morphism h from A~B to C such that f = h o ~'A and g = h o 7B: the morphism h is given by
hx([a|162
= C.(r
(uL) (fv(a) • gu(b))
Now proposition 1.8.3 shows that a morphism h from A~B to C is determined by the bilinear morphisms ]~u from A(U) • B(U) to C(U), defined for any G set U by (a,b) E
A(U) x B(U) H C* ( uuu)hU([a|
and h is a morphism of Mackey functors if and only if the morphisms ]~u satisfy conditions i), ii) and iii) of proposition 1.8.3. But here
hu(a, b) = fu(a).gu(b) morphism of G-sets, and let a E A(Y), b E B(Y), a' C A(U')
So let k : U' ---+ U be a and b' C B(U'). Condition i) can be written as
]~u(A.(k)(a'),b) = C.(k)fu,(a',B*(k)(b)) or
fvA.(k)(a').gu(b)
C.(k)(fu,(a').gu, B*(k)(b))
=
Since f is a morphism of Mackey functors, the left hand side is also equal to
C.(k)fu,(a').gu(b) But the product "." on C is the map associated by the proposition 1.8.3 to the bilinear morphism from C, C to C defined by the product of C. So relation i) holds for this product, which gives
C.(k)fv,(a').gu(b)
=
C.(k)(fv,(a').B*(k)gv(b))
and this proves that i) holds for h. A similar argument proves relation relation iii) can be written as
]~u,(A*(k)(a), B*(k)(b)) = C*(k)]tu(a, b)
ii).
Now
6.3. TENSOR PRODUCT OF GREEN FUNCTORS
139
which gives here
fu, A*(k)(a).gu,B*(k)(b) = C*(k)(fu(a).gu(b)) The left hand side is also
C*(k)fu(a).C*(k)gu(b) and since relation
iii) holds for the product ".', I have C*(k)fv(a).V*(k)gu(b) = C*(k)(fu(a).gu(b))
(in other words, the maps C*(k) are ring homomorphisms for the product ".", which is also a consequence of lemma 5.2.2). Thus the maps ] satisfy i), ii), and iii), and it follows that h is a morphism of Mackey functors. To prove that h is a morphism of Green functors, I must check that
The left hand side is
hx•
• c)|
(bx
d)](uxv,r162
= C.(r
r
x c).gu•
x d))
(6.1)
and the right hand side is
C.(r
x C.(r
= C.(r x
r
x fy(c).gy(d)) (6.2)
Equality of (6.1) and (6.2) for all r and r is equivalent to equality fv•
• e)mv•
• d) = fv(a)mv(b) • f v ( c ) m v ( d )
The left hand side is
and the right hand side is
~'(~~,~)(~o)~ g~/~/) ~ ~" ( ~ ) ( s ~ ) ~
~/~) ....
.... C*(uuvvUV ) (fv(a)• gu(b)•
fv(c)•
gv(d))
(6.4)
If the images of g and f commute, I have
Then
fu(a) x gu(b) • fv(c) • gv(d) -- C. (u,v~u2v2~ (fu(a) x fy(c) • gu(b) x gv(d)) k Ul ~,t2'U1~2 /
But since
C, (UlVlU2V2~ = C. (UlU2VlV2~ \Ul~d2vlv2 / \?.llVll~2v2/
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
140
the right hand side of (6.4) is equal to
C*( uv )~y,(?~lU,2ylv2~ (fu(a) • fv(c) xgu(D)).(gv(d) ) .... ~ ?//OV
\ ?~1 Vlt/21)2 /
which is the right hand side of (6.3). Thus h is a morphism of Green functors. Conversely, if h is a morphism of Green functors from A@B to C, the images of f = h o 7A and 9 = h o 7B must commute: indeed, the images of 7A and 7B commute, and moreover, I have the following lemma L e m m a 6.3.5: L e t X a n d Y be G-sets, a n d 0 be a m o r p h i s m of G r e e n f u n c t o r s f r o m D t o C. I f a E D(X) c o m m u t e s w i t h ~ C D(Y), t h e n Ox(a) c o m m u t e s w i t h 0y(fl). P r o o f : It suffices to write
0r(Z) x 0x(~) = 0yxx(Z x ~) = 0y•
....
C,
xy
yx) xy (~ • 9) . . . .
xy
The lemma follows.
9
To complete the proof of the proposition, I must still observe that if f and g are unitary, then so is h: indeed, the unit of A ~ B is leA | eB](oJd), and its image under h is
( '..) which is equal to the unit of C if f and g are unitary. Conversely, if h is unitary, then f = h o "/A and 9 = h o 7B are composition products of unitary morphisms, so they are unitary. 9 P r o p o s i t i o n 6.3.6: L e t A a n d B be G r e e n f u n c t o r s for t h e g r o u p be an A - m o d u l e a n d N b e a B - m o d u l e . T h e n M ~ N has a n a t u r a l of A Q B - m o d u l e , d e f i n e d as follows: if X a n d Y a r e G-sets, if (U, r o v e r X , a n d (V,r is a G-set o v e r Y, if a E A(U), if b ~ B(U), if and n C N(V), then
[a | b](~,~) • [~ e ~](.,r
G. L e t M structure is a G-set rn C M(V)
= [(a x ~ ) | (b • ")](~xv,+•
P r o o f : To prove that this product is well defined, associative, and unitary, one just has to mimic the proof of proposition 6.3.1, replacing A by M and B by N in suitable places. 9
6.4. BIMODULES
6.4
141
Bimodules
The notion of right-moduie over a Green functor leads naturally to the notion of bimodule:
Definition: Let A and B be Green functors for the group G. If M is an A-module, which is also a module-B, [ will say that M is an A-module-B if for any G-sets X , Y, and Z, and any elements a 9 A(X), m 9 M ( Y ) and b r B(Z), I have (a•
xb=ax(m•
in M ( X x Y x Z )
A morphism of A-modules-B from M to N is a morphism of Mackey functors from M to N, which is also a morphism of A-modules and a morphism of modules-B. With those definitions, I can speak of the category of A-modules-B, that I will denote by A - M o d - B .
Proposition 6.4.1: The category A-Mod-B is equivalent to the category A@B~ Proof: To give M a structure of AQB~
is equivalent (see proposition 2.1.2) to give ~ unitary morphism of Green functors fl'om AQB ~ to ~ ( M , M). By proposition 6.3.4, this is equivalent to give unitary morphisms from A and B ~ to 7~(M,M), the images of which commute. In particular, the module M is an A-module, and a B~ i.e. a module-B. If X is a G-set, and if a E A(X), then a defines an element Aa ~ HomM~k(G)(M, Mx) by m
E M(Y)
;%(m)
/\ M. [xy } (a • m ) E M ( Y • X ) = M x ( Y ) \]yx
If Z is a (;-set, and if b r B(Z), then b determines an element Pb of ~ ( M , M ) ( Z ) = HomM~ck(a)(M, Mz), by
m c M ( Y ) ~ - ~ P b ( m ) = M * ( z Y()b • 1 7 6
....
Now Aa and Pb commute if and only if
Pb X Aa = ~ ( M ' M ) * ( x z ) (A~ • where the products x are in C = ~ ( M , M). But if a is an element of C(X), determined by morphisms av : M ( Y ) ~ M ( Y X ) = Mx(Y) and if/3 is an element of C(Z), determined by morphisms
5y : M ( Y ) ~ M ( Y Z ) = M z ( Y )
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
142
then the product c~ x fl is the element of C(X x Y) determined by the morphisms
(o~ x ,~)z = M. (YZXl o azz o ~z \YXZl The product A~ x Pb is then determined by the morphisms (A~ • Pb)Z = M, \(yzx~ yxz / o /~a,Y Z 0 Pb,Y :
\yxz/
\yzx
\yxz/
On the other hand, the product pb x A~ is determined by :
\yzx/
0 Pb,YX o Aa,y:?:Q ~
M.
\yzx/
(.]~/I.
yx
((t, X 7"ft) X b/)
Now 1a and Pb commute if and only if (Pb • :~o)Y('~)= M. ( y x z ] (Ao • pb)~,(-~)
\yzx)
or equivalently
M*CYxzl ( M * ( xyx Y)(a•215
= M * ( yzx) y x z l ( M * (\ xy yx z l)
This is also
\yzx)
\yxz)
\yzx)
\yxz/
thus, since M. (,:yz] \ yzx ] is bijective
a x (m• b) = (a x m ) • and this proves the proposition.
,'
P r o p o s i t i o n 6.4.2: Let A and B be Green functors for the group G. If M is an A - m o d u l e , and N a B - m o d u l e , then the product 5 and the m o r p h i s m s from A to H(M, M) and from B to H ( N , N ) induce a natural structure of B - m o d u l e - A on ~ ( M , N). Proof." By proposition 2.1.2, to say that M is an A-module (resp. that N is a B-module) is equivalent to give a unitary morphism of Green functors from A to 7-/(M, M) (resp. from B to ~ ( M , M)). So it suffices to give ~ ( M , N) a structure of 7-/(N, N)-module-7"/(M, M). So let X, Y, and Z be G-sets. If a C "H(M,M)(X), if f E 7t(M,N)(Y), and b E 7~(N,N)(Z), I have seen that if U is a G-set, then b 5 f 5 a is the element of ~(M, P ) ( Z Y X ) defined on the G-set U by
I
(b 5 f 5 a)u = IV. \uzyx ) o buxz o fux o au
6.5.
COMMUTANTS
143
The product 5 turns "H(M,N) into a bimodule, because it is bifunctorial and associative (proposition 6.1.2), and also unitary: if b is the unit of "H(N, N), i.e. the identity morphism from N to N. = N, then it is clear that =
o IdN(ux) o f u = f u k'~ 9 X /
Similarly, if a is the identity of M, then m 5 a = m. This completes the proof of the proposition. ..
6.5
Commutants
D e f i n i t i o n : Let A be a Green flmctor for the group G, and M an A-module-A. I f U is a G-set, and o~ 6 A(U), [ set for any G-set X
\ ~/x /
Similarly, if P is a subset of
A(U), I set
Va6P}
CM(P)(X) = {m e M ( X ) I~ • m = M* (XU) (m • More generally, if L is a sub-Mackey functor of A, I set
CmL)(X)= {m9
I~•
(x~,](m•
VU, W e L ( U ) }
KUX/
L e m m a 6.5.1: Let < P > be t h e s u b - M a c k e y f u n c t o r of A g e n e r a t e d by P (i.e. t h e i n t e r s e c t i o n of t h e s u b - M a c k e y f u n c t o r s L of A such that L(U) D_P). Then for any X, I have CM(P)(X) = CM(
)(X). Proof: It is clear that CM(< P > ) ( X ) _C C M ( P ) ( X ) . Conversely, it is easy to see that for any G-set X, I have
(Y)=
~
pepCA(Y)
]:U~Z
9:Z--Y
so that any element c~ of < P >
(Y) can be written as c~ = E A.(gi)A*(fi)(p{) i
for suitable elements g{, fi and Pi E P. But if m 6 C M ( P ) ( X ) , then o~ • m = ~ d . ( g i ) g * ( f i ) ( p i ) • m = ~ M.(gi • I d ) M * ( f i • Id)(pi • m) . . . . i
i • i
tlx
And for f : U ---+ Z and g : Z --+ Y, I have
KJUX
KX~L/
p{)
CHAPTER 6. CONSTRUCTION OF GREEN FUNCTORS
144
. . . . M*(gxld)M* (ZX) M*(Idx f)
M*(gxId)M* (XZ) M*(Id•
. . . . . 'l'(XY) M*(Id•
....
f)
Thus
i
• gi)M*(Id
yx
x f/)(m •
X
yx
Pi) . . . .
M * (g)M*(fi)(p i i ))
=M,
zy) (m x c~) yx
which proves the lemma.
P r o p o s i t i o n 6.5.2: T h e p r e v i o u s d e f i n i t i o n s t u r n C M ( L ) i n t o a s u b - M a c k e y f u n c t o r of M, c a l l e d t h e c o m m u t a n t of L in M. Proof: If f : X ~ Y is morphism of G-sets, and if m r I have
~ • V.(f)(-O = M.(ld • f)(a •
CM(X),
M.(IU • f ) M . ( x ~
then for a r
L(U),
(~•
\ ~x J
. . . . M. uy and this shows that
uy
M~(f)(CM(L)(X)) C C . ( L ) ( Y ) .
a x M*(f)(m) = M*(Id x f)(a • m) = M*(Id x f)M.
....
M.
Similarly, if m E
M(Y)
uy
M*._. ~
•
X?2
which shows that M*(f)(CM(L)(Y)) Mackey functor of M. k
/
c_ CM(L)(X),
hence that
CM(L)
is a sub,,
In the special case when M is the functor A, viewed as an A-module-A, there is a little more:
P r o p o s i t i o n 6.5.3: Let L be a s u b - M a e k e y f u n c t o r of t h e G r e e n f u n c t o r A. T h e n CA(L) is a s u b - G r e e n f u n c t o r of A. Proof: I must check that if X, Y and (J" are G-sets, if a E A(X) and /? E A(Y) commute with l E L(U), then ~ • also commutes with I. But
\uxy/
\uxy/
6.5. COMMUTANTS =
\uxy/
145
~ • A.
(1 • ~)
yu
\uxy/ \uxy/
\xyu/
)
Ix
)(u2, Y:)/
~(tzi , x, >)(u,,
....
N.
(~,.T, y)(~, y)
... --
(~,x,y)(~,y)
and finally c~ x U n is equal to the image of c~ x n under the m a p (~,x,y,)
X.
\(~,x,y)(~,y)
o...
8.6. FUNCTORIALITY
177
oN'*( (~,x,y) )N.((Ul,X, yl)('U,2,Y2))_~ "'"
(72, X, y)(tt, y)
k
(721, X)(2s Y2) )
''"
which proves the claimed formula
8.6
Functoriality
If G and H are finite groups, and U is an H-set-G, then I have a functorial construction X ~ U oa X from G - s e t to H - s e t . This construction is not quite functorial in U: if f : U --+ V is a morphism of H-sets-G, there is in general no associated morphism U oa X --+ V oa X: this is because if (u, x) E U oa X, i.e. if the right stabilizer of u is contained in the left stabilizer of x, generally, the right stabilizer of f(u) is not. I have studied this question in [2], and showed that it is natural to ask moreover that f is injective when restricted to each orbit of G (or equivalently to ask that the right stabilizer of f(u) is equal to the right stabilizer of u, for any u C U). Then, there is a morphism of functors f oa - from U oa - to V oa - defined on the G-set X by In those conditions (see [2] prop. 10 and 11), if M is a Mackey functor for H, then M o U and M o V are Mackey functors for G, and the morphism f induces two morphisms of Mackey functors (denoted by jr. and f* in [2], but differently here to avoid confusion): a morphism Mf from M o U to M o V, and a morphism M / from M o V to M o U. Those morphisms are defined for a G-set X by
Mf,x = M . ( f oa X) : (M o U)(X) ~ (M o V)(X) M~ = M*(f oc X ) : (M o V)(X) ~ (M o U)(X) W i t h those notations: P r o p o s i t i o n 8.6.1: L e t G a n d H b e f i n i t e g r o u p s , a n d l e t A b e a G r e e n f u n c t o r for H. L e t m o r e o v e r U a n d V b e H - s e t s - G , a n d f : U --* V b e a m o r p h i s m o f H - s e t s - G , w h i c h is i n j e c t i v e on e a c h r i g h t o r b i t . 9 I f A is a G r e e n f u n c t o r for H , t h e n A f is a u n i t a r y m o r p h i s m o f G r e e n f u n c t o r s f r o m A o V t o A o U. 9 I f M is a n A - m o d u l e , t h e n r e s t r i c t i o n a l o n g A f g i v e s M o V a n d M o
U structures of A o V-modules, and the morphisms morphisms of A o V-modules. Proof:
Mi and M f are
For the first assertion, I must show that if X and Y are G-sets, if a E
(A o V)(X) and b E (A o V)(Y), then A]x(a) •
A~(b) = A]x•
•
b)
CHAPTER 8. COMPOSITION
178 The left hand side is equal to
A ( 6Ux y ) ( A * ( f oG X)(a) • A*Cf oc Y)(b)) = A'(6~#y)A*((f oo X) x (fom Y))(a • b) and the
right hand side to A * ( f oc~ (X x Y))A*(6~y)(a x b)
Now equality follows from
since for ( u , x , y ) 6 U oG ( X x Y)
((lo~x) x (,focy))o4,y(~,x,~)= ((,-ocx) xisoc,,-)) ((~,=), (~,~))
. . . .
.... ((s(=), =), (s(~), ~)) : ~.L.-(,f(,,),.~,,~) = 4,~o(so~(x • Y))(.,,,,,~) Moreover
A{@AoV) = A*(f oc ")A*(py/a)(Cm) = A*(pu/c)@A) = CAoU since PU/G = Pv/c; o ( f oG .). So the morphism A I is a unitary morphism of Green funetors. For the second assertion, I must show that if a e (A o V)(X) and m 6 (M o U)(Y), then a x v M ! y ( m ) = My,xy(Af(a) x U m) (8.3) and that if m' C (M o V)(Y), then
MY(a x y m') : AY(a) x u MY(m ')
(8.4)
But
axVMym (= )M6('~Y':a()xM*X (~
'
(v,x)(f(u),y)](v'x)(u'Y) ](axrn.)
The square
U oG (XY) O~,x,y) V oa ( X Y )
(u,x,y) (f(~,), z)(,,, v)) )
((~,x)(~,y))
(V oo X) x (U oc Y)
\(~, x)(f(u), y)] (v oa x) • (v oc Y)
(c)
("'~'Y) /
is cartesian: if ((v,x), (u, y)) E (V oG X) x (U oo Y)and (v', x', y') 9 I/oG ( X Y )
such that (v',x') = (v,x)
(v',y') = (f(u),y)
are
8.6.
FUNCTORIALITY
179
then there exists s and t in G such that ?2t 7--- ~)S
SX t z X
v'= f(u)t
ty'= y
In those conditions, the element ( u , t x ' , y ) is in U oa (XY): indeed, if r E G is such that ur = u, then as (u,y) E U oa Y, I have ry = y. Moreover
v ' t - l r t = f ( u ) r t = f ( u r t ) -- f ( u t ) = f ( u ) t - v' and as (v', x/) E V oo X, I have t - % t x ' = ix', or r.tx' = tx'. Moreover
( f ( u ) , tx', y) = (v't -1, tx', y) = (v', x', t - l y ) = (v', x', y') and ((f(u),~Xt),('U,,,y))
~- ( ( V t ~ - i , t x t ) , ( ? j , ~ ] ) )
: ((Vt, x t ) , ( u , y ) )
....
Conversely, if (Ul, Xl, Yl) E U oa ( X Y ) is such that :
:
then the last equality proves that replacing (Ul, Xl, Yl) by (uls -1 , SXl, syl) for a suitable s E G, I can suppose Ul = u and y~ = y. Then (f(u),Xm) = ( v , x ) = ( f ( u ) , t x l ) . So there exists r E G such that
f(u)r = f(u)
r-ix1 = tx'
As f is injective on the right orbits, the first equality shows that ur = u, and since (u, Xl, y) E U oa ( X Y ) , I have rxl = Xl = tx', and
(u,, x,, y, ) : (u, tx', y) which proves that (C) is cartesian. It follows that
\(f(u),x,y)]
\(f(u),x)(u,y)]
But the right hand side of (8.3) is equal to
M f . x y ( A f ( a ) xU m) : M . ( f oa ( X Y ) ) M ' ( 5 ~ ; , y ) ( A * ( f o a X ) ( a ) • m) . . . .
"'"
(f(~l), X), ('U2, y)
9 which proves equality (8.3).
..=M,((u,~-,y)~ M" ( (f(u),x,y)]
(u,x,y)
\(f(u),x)(u,y)
) (a • ~)
CHAPTER 8. COMPOSITION
180 Similarly, the left hand side of (8.4) is
M/(axVm')=M*(f~
( ~z)(f(~), ';'Y)
y) )
(a •
whereas the right hand side is
A/(a) xVM/(m ') = M *(hx,y)(A ~ "(foaX)(a) " xUM*(foa r)(~')) . . . . .. : i*((hU,y)M * ( ( ~ l ' X ) ( ~ 2 ' ~ / ) ~ \ (f(ul), z)(f(u2), y)]
((/x;'7"/): ]~/./"(
(7.l,x,y) ) (a)<mt) \ (f(u), x)(f(u), y)
9
This proves equality (8.4), and the proposition.
8.7
Example:
induction
and
"
restriction
Let O be a finite group, and H be a subgroup of G. Let U be the set G, viewed as a G-set-H, and V be the set G, viewed as ai1 H-set-G. If X is an H-set, then U oa X identifies with I n d e X , functorially in X. It follows that if M is a Mackey functor for G, then M o U is isomorphic to Res~M. If Y is a G-set, then V oG Y identifies with Res~Y, functorially in Y, by the map (g, Y) ~ gY. Thus if N is a Mackey functor for H, then N o V identifies with Ind~N. So if A is a Green functor for G, then Res~A is a Green functor for H, and this is not surprising. The case of induction is less clear, but corresponds to what Th6venaz calls coinduction (see [13]):
P r o p o s i t i o n 8.7.1: L e t H b e a s u b g r o u p of t h e g r o u p G, a n d B b e a G r e e n f u n c t o r for H. then: 9 T h e f u n c t o r I n d ~ B is a G r e e n f u n c t o r for G. I f K is a s u b g r o u p of
G, t h e n there is an isomorphism of rings (with unit)
(In4B)(I,') =
[I
xEH\GII(
B(H n ~I~)
9 T h e f u n c t o r B H IndCHB, f r o m t h e c a t e g o r y Green(H) of G r e e n funet o r s for H, t o Green(G), is r i g h t a d j o i n t to the f u n c t o r A ~ Res~A f r o m Green(G) to Green(H). P r o o f : The formula giving ( I n d ~ B ) ( K ) is well known for Mackey functors. point is that is is still true for Green functors. By definition of induction (Ind~B)(h')
=
B
(Res~(G/K))
But
Res~(G/K) ~_
I_I H/(H ~eHW/I
M(Y)
M.(uz)
M.(uy)
, M(UoHG\U.Z)
Oa\g.z , N(G\U.Z)
, M(UoHG\U.Y)
, N(G\U.Y) Oa\u.y
The left square is commutative because gc~ = f. The middle square is commutative by lemma 9.3.2. The right square is commutative because 0 is a morphism of Mackey fimctors. So I have a morphism e x from M(X) to Tgu(N)(X). If 4) : X + X ' is a morphism of G-sets, and if rn C M(X), then ~x(rn) is the sequence indexed by the objects (Y, f) o f / ? u ( X ) defined by ~bx(m)(y,f) = Oa\urM.(ur)M*(fx)(m) Then 7~v(a).(4))Ox(m) is the sequence rn{r,,y, ) indexed by the objects of D u ( X ' ) , and defined by filling the cartesian square g
Y
,
y'
X
,
X'
More precisely
m}y,,f,) = M.(G\U.a)(~bx (re)if, i)) = N.(G\U.a)OG\uyM.(py)M*(fx)(m) On the other hand, the element r
is the sequence n(y,],) defined by *
!
n(y,,f,) = Oa\u.y,M.(uy,)M (fx,)M.(4)) As the square a
Y
X
~
r
y'
, X'
is cartesian, I have M*(f~,)M.(r
= M.(a)M*(fx)
Moreover as ~ , o a ; (U o~ ( G W . ~ ) ) o ~Y, Ihave M . ( ~ , ) M . ( a ) = M.(U o . ( C \ U . a ) ) M . ( ~ ) Finally as 0 is a morphism of Mackey functors, I have
Oa\v.y,M. (U OH (G\U.a)) = N.(G\U.a)Oa\u.y
CHAPTER 9. ADJOINT CONSTRUCTIONS
212 and finally
n(v,,l,) = N.(G\U.a)Oa\u.vM.(uv)M*(fx)(m) = m}v,4, ) This equality proves that 7~v(M).(r
= ~,x,M.(r
Now if rn' E M(X'), then ~'x,(m') is the sequence indexed by the objects (Y', f ' ) of Du(X'), defined by
~x,(rn')(v, f,) = Oa\v.y,M.(uy,)M*(f~x,)(rrz ') is the sequence indexed by the objects (Y, f) of Du(X),
Its image under 7r162 defined by
, =
But (~f)x, = efx, and then
(T@( M)*( r
(y,f) = Oaxu.yM.(uy )M*(fx )M'( r
= ~x (V*(r
which proves that
7r162
= $xM*(r
Thus ~b is a morphism of Mackey functors from M to ~u(M). Conversely, if ~b is a morphism of Mackey functors from M to 7~u(N), I have for any G-set X a morphism %bx from M(X) to "fCu(N)(X). In particular, if Z is an H-set, I hame a morphism ~ v o . z : M(U o . Z) -+ Tau(N)(U o . Z)
An element of TQs(N)(U OH Z) is a sequence n(r,f) indexed by the objects of 7)u(U OH Z), with n(v,f) E N(G\U.Y). I can then consider the element
and its image under N.OTz), which is an element of N(Z). I get a morphism Oz from M(U oH Z) = (M o U)(Z) to N(Z), defined by
To prove that 0 is a morphism of Mackey functors, it suffices to observe that 0 is composed of the morphism ~/Jo U from M o U to ~ u ( M ) o U deduced from ~b, and of the morphism O from 7r o U to N, defined on the set Z by
,~ ~ (T~u(X) o V)(Z) = 7r
o . Z) ~ Oz(~) = X.(~z)(n(uo.a.~) )
It suffices then to prove that this is a morphism of Mackey functors. So let r : Z --* Z ~ be a morphism of H-sets. The element
( ~ ( N) o u).( r )( ,, )(Vo,)
: CzIz, this is also = N.(r
thus Oz, (7~u(N) o U),(r = N,(r Conversely, if n' E Tgu(N)(U OH Z'), then
N*(r
(,n(uouZ,,~,z))
= N (r
is the sequence indexed by the objects (Y, f)
On the other hand (TZu(N) o U)*(r of Du(U OH Z), defined by *
t
(Y,(Uonr It follows that *
!
!
But I have already observed in the proof of theorem 9.5.2 that the morphism U OH r is a morphism in 7?u(U OH Z') from (U OH Z, ((U OH r in Tgu(N)(U OH zr), I have
n'(Uo.Z,(UoHr ~
to (U OH Z',Tcz,). Then
: N.(U oH r
SO that *
l
r
I have again the commutative diagram a\u.(u o. z)
o G\U.(U oH Z')
Vz,
' Z'
CHAPTER 9. ADJOINT CONSTRUCTIONS
214
and lemma 9.5.3 shows that i is injective, and that if II = Ira(i) [I 11', denoting by j the injection from 11' into II, the set (II', bj) is v-disjoint. I have then
N*(r
Sz,) = N.(a)N*(b)(nluouz,,r
....
) ....
....
As n' E Tgu(N)(U oH Z'), and as (II',bj) is v-disjoint, the second term is zero, and this gives
N*(r
= N.(ai)N*(bi)(nluo,Z,fz)) . . . . ....
N.(..)N'(GW
(Vo.
Finally, I have proved that
and O is a morphism of Mackey functors. To complete the proof of the theorem, it remains to state that the correspondences A : 0 F-+ ~b and B : ~b ~-+ 0, which are clearly functorial in M and N, are inverse to each other. It suffices to check that if 0 is the identity, then so is (B o A)(O), and that if ~b is the identity, so is (A o B)(r Let 0 be the identity endomorphism of M o U, and ~b = A(O). If X is a G-set, the morphism ~bx from M(X) to 7r o U)(X) maps the element m C M(X) to the sequence '~bx(rn)(rj) defined by
~x(rn)(r,]) = M.(vy)M*(fx)(m) e M(U OH (G\U.Y)) = (M o U)(GkU.Y) \ I have for any H-set Z and any m E M(U oH Z)
Then if O' = B ( r
O'z(m ) = ( M o U).(7?z ) (ZbUo.z(m )(yo.z,,~z) ) which gives
e'z(m) : M.(U oH 7lz)M.(vUouz)M* ((~rz)x)(m) But I have seen that (U OH rlz)VUo.Z is the identity map, as well as (Trz)x. Thus O} is the identity, and so is 0'. Now if ~b is the identity endomorphism of TCu(M), and if 0 = B(~b), then for any H-set Z and any
m' C (Tiu(M) o U)(Z) = ~u(M)(U OH Z) I have If r = A(O), if m E R u ( M ) ( X ) , then r (I/, f ) of Du(X) defined by r
is the sequence indexed by the objects
= Oa\u.yT~u(M).(vy)TCu(M)*(fx)(m)
9.9. EXAMPLES Setting m ' =
But by
215
T~u(M).(uz)Tiu(M)*(fx)(m),
nu(M)*(fx)(m)
Moreover rn' is the
I have then
n(E,e)indexedby
is the sequence
n(E,e) = m(E,~xe) image of n under Tiu(M).(uy). The
the objects of
component
T)u(Y),
defined
m}uo~(a\u.z),,~a\uy )
is then obtained by observing that if e is the morphism from Y to Y defined by e(y) = ((y, fu(Y)), then the square fly
Y
,
U o.
e!
(u\u.~)
~rCaiu.Y
Y
, U OH (a\U.Y) Py
is cartesian. Then
,
m(uoH(G\U.y),rCc\uy ) =
As moreover
fxe
M.(G\U.uy)( m(y,~xe ) )
= f , I have then r
=
M.(rla\u.y )M.( G\ U.uy )(m(zj) )
As finally rla\u.z(G\U.uz ) is the identity map, I have r is the identity, which completes the proof of the theorem.
= re(y j ) . Thus ~ ' 9
R e m a r k : Let X be a G-set. The expression of the limit over :Du(X) shows that if M is a Mackey functor for G, then Tiu(M)(X) is the set of sequences m(gj), indexed by G-sets (Y, f ) over X x (U/H), such that rn(y,f) E M(G\U.Y), and M*(a)(-~(y,s)) = 0
whenever (Z, a) is a r-disjoint H-set over m(zj) =
G\U.Y,
and moreover
M*(G\U.a)(m(y,j,))
whenever a : (Y, f ) --+ (Y', f ' ) is a rnorphism of G-sets over X • injective on each G-orbit.
9.9 9.9,1
(U/H)
which is
Examples Induction and restriction
Let G be a group, and H be a subgroup of G. If U is the set G, viewed as a G-set-H, then the fimctor N ~ N o U is the restriction functor for Mackey functors from G to H.
CHAPTER 9. ADJOINT CONSTRUCTIONS
216
As U/H = G/H, an object ( Y , f ) over U/H is of the form I n d ~ Z , with Z = f - l ( H ) (see lemma 2.4.1). An object of 7?u(X) is then an H-set Z, with a morphism from Ind/~Z to X , i.e. a morphism from Z to Res/~X. Moreover, the group G acts freely on U, so on U.Y. Thus if c~ is a morphism of G-sets over X x (U/H), then U.c~ is injective on each left orbit. In other words, the category 7?u(X) identifies with
H-set~aes~X. Moreover, the set G\U.Y identifies with Z, by the map
G(u,y) E G\U.Y H u-ly E g This map is indeed surjective, because if z E Z, then z is the image of G(1,z). Conversely, if u-ly = u'-ly ', then G(u, y) = G ( u , u u ' - l y ') = G(1, u'-~y ') = G(u', y') It follows that
U OH (G\U.Y) ~- Ind~(G\U.Y) ~_ Ind/~Z ~_ Y and that vg is an isomorphism. Then if the H-set (T, a) over G\U.Y is v-disjoint, the image of U OH a is disjoint of the image of yr. As vy is surjective, I have U OH T ----0. But U OH T = Ind~rT, and then T = 0. In particular, I see that
Qu(M)(Y, f) = M(G\U.Y) = M ( Z ) = Sv(M)(Y, f) As ReSaHX is a final object of 7?u(X), I see that
Cu(M)(X) =
lim
M(Z) = M(Res~X)
ze~u(x) As ReSaHX is an initial object of :Du(X) ~ I have also
T@(M)(X) = M(ReSUHX ) and the following isomorphisms follow easily:
s
~- I n d , ( M ) _~ Tiu(M)
I recover that way the adjunction properties of induction and restriction. Now switching the roles of H and G, I consider V = G as an H-set-G. The functor N o V is then the induction functor for Mackey functors from H to G. As V/G = . , an H-set (]I, f ) over V / G is just an H-set, and V.Y = V • Y. The group H acts freely on V, so if a is a morphism of H-sets from Y to Z, then V.c~ is injective on each left orbit of H on U.Y (because h.u = u implies h = 1). The category :Dv(X) identifies then with H-setJ, x, and has a final object X. If Y is an H-set, then H \ V . Y identifies with Ind~Y. Let ( T , a ) be a v-disjoint G-set over H \ V . Y . If t E T and a(t) = H(v, y), then as G acts freely on V, I have (v, t) C V oG T, which contradicts the hypothesis on (T, a). So T = 13. In those conditions, it is clear that
Lv(M) ~ Res~,(M) _~ 7~v(M) and I recover once again the adjunction properties of induction and restriction.
9.9. EXAMPLES
9.9.2
217
Inflation
Let N be a normal subgroup of the group G, and H be the quotient G/N. If U is the set H, viewed as an H-set-G, then the functor M ~ M o U is the inflation functor for Mackey functors: indeed, if X is a G-set, then U oa X = X N. As U/G = *, an object of 7?u(X) is just an H-set over X. If (!/, f ) is such a set, then U.Y = U • Y, and H\U.Y identifies with I n k Y . As H acts regularly on U, if ~: (Y, f) ~ (Z,g) is a morphism of sets over X, then U.a is injective on each orbit of H on U.Y. So X is a final object in 7Pu(X), and then for any X, I have
s
= Qu(M)(X)
~ u ( M ) ( X ) = Su(M)(X)
Furthermore as U oc (G\U.Y) = ( I n ~ Y ) N _~ Y, the morphism uy is an isomorphism for any Y. So an object (T,a) over H\U.Y = In~HY is ,-disjoint if and only if U oa T = T N = O. Finally: P r o p o s i t i o n 9.9.1: L e t N b e a n o r m a l s u b g r o u p o f t h e g r o u p G, a n d H = GIN. I f M is a M a c k e y f u n c t o r for G, a n d X is a n H - s e t , I s e t
M N ( x ) = M(In~HX)/ ~ M.(a)M(T) (T,a)
Mar(X) = ~ KerM*(a) (T.~)
w h e r e t h e s u m a n d i n t e r s e c t i o n r u n o v e r t h e G - s e t s ( T , a ) o v e r InfaHX s u c h t h a t T N = O. I f ~ : X --~ X ' is a m o r p h i s m o f H - s e t s , t h e n t h e m a p s M . ( I n ~ r and M*(In~r i n d u c e m o r p h i s m s b e t w e e n MN(X) a n d MN(x'), a n d b e t w e e n Mar(X) a n d Mar(X'), w h i c h t u r n M N a n d MN i n t o M a c k e y f u n c t o r s for H. T h e f u n c t o r M ~-* M N is left a d j o i n t t o t h e f u n c t o r L ~-~ I n , L, a n d t h e f u n c t o r M ~ MN is r i g h t a d j o i n t t o it. R e m a r k : W i t h the notations of Th~venaz and Webb (see [14], [15]), it is easy to identify M x with M +, and MN with M - : any set T such that Tar = 0 is indeed isomorphic to a disjoint union of sets of the form G/K, for N g K , and it follows easily that if L/N is a subgroup of H = G/N
MN(L/N) = M ( L ) / ~ tL-M(I()
MN(L/N) = A Ker ri K
where the sum and intersection run on the subgroups K of L (which give morphisms from G/K to G/L) not containing N.
9.9.3
Coinflation
Let N be a normal subgroup of G, and H = G/N. Let V be the set H, viewed as a G-set-H. Then if Z is an H-set, the set V on Z identifies with I n , Z, and the functor M H M o V is the functor that I have denoted by p~ (and denoted by fl! by Th6venaz and Webb see[15].5). Here again, the set V/H is trivial, so if X is a G-set, an object (Y,f) of Dr(X) is just a G-set over X. The set V.Y is the product 1/ x Y. Let c~ : (Y,f) ---, (Z,g) be a
CHAPTER 9. ADJOINT CONSTRUCTIONS
218
morphism of sets over X. Then V.a is injective on the left orbits of G on V.Y if and only if the hypothesis imply gy = y. But the stabilizer in G of any point v of V is equal to N. Thus V.a is injective on the left orbits of G on V.Y if and only if c~ is injective on the orbits of N on Y. a morphism in l ) v ( X ) from ( Y , f ) to (Z,g) is then a morphism a of sets over X , which is moreover injective on each orbit of N. Now the set G \ V . Y identifies with N \ Y . Then V OH (G\V.Y) identifies with InfaH(N\Y). The morphism uy maps y to its orbit Ny by N. In particular, it is surjective. So an object ( T , a ) over G\V.Y is ~,-disjoint if and only if V OH T = In~HT = ~, i.e. if T = ~. It follows that
Qv(M)(Y, f) = M ( N \ Y ) = Sv(M)(Y, f) Finally: P r o p o s i t i o n 9.9.2: L e t N b e a n o r m a l s u b g r o u p of G, and H = G/N. I f V is t h e s e t H , v i e w e d as a G - s e t - H , a n d if X is a G - s e t , t h e n :Dv(X) is i s o m o r p h i c t o t h e c a t e g o r y w h i c h o b j e c t s are t h e G - s e t s o v e r X , a n d t h e m o r p h i s m s are t h e m o r p h i s m s of s e t s o v e r X w h i c h are m o r e o v e r i n j e c t i v e on e a c h o r b i t of N . I f M is a M a c k e y f u n c t o r for H, t h e n
s
=
M(N\Y)
lira
Tiv(M)(X) :
(vJ )~Vv(X )
lim
M(N\Y)
(v,f )~Vv(X )~
Let K be a subgroup of G. I denote by K, ordered by the following relation
Notations:
wN(K) the
set of subgroups of
LCL' L A N = L'NN
LZL'~{
If M is a Mackey functor for the group H = G/N, I denote by lim M ( L N / N ) L6wN(K) the quotient of OLc_KM(LN/N) by the submodule generated by the elements of the .L'N/N form ~ L N / N m - - m , for L ~_ L' and m 6 M ( L N / N ) . The group K acts on lim M ( L N / N ) , and I denote by LEwN(K) lira M ( L N / N ) ) ~ . L6wN(K)
~)~~w.~((~(~,s).~)(~,~)) . . . .
10.7. Lu(A)-MODULES AND RIGHT ADJOINTS
257
As r is a morphism of Mackey functors, it is also
G.u.t ] r
[A" \ G.u.z ) (a).m(Z~T,lbg)]
Finally as r is a morphism of A-modules, I have
TgU(r )( a(zj).m )(r,g) . . . . 9
\
c.u.z
]
(a(z,j).T~u(r
. . . .
which proves that T@(r
. . . .
s
is a morphism of
Let 0 be a m o r p h i s m o f A-modules from NOUIA to M. As NOUIA is a direct summand of N o U as Mackey fnnctor, I have a morphism O' of Mackey functors from N o U to M, defined on the H-set Z by
0 (m) =
• m)
By adjunction, it correspond to 0' a morphism r of Mackey functors from N to as follows: if X is a C-set and (r,g) an object of 7)u(X), and if
Tiu(M), defined n E N(X), then
r
= O~\u.TN.(vT)N*(gx)(n) 6 M ( G \ U . T )
Thus ~(n)(T,~) = 0a\U.T(AA,.(CA) • The previous lemma shows that setting T' =
AA,.(eA) X U N.(vT)N*(gx)(n) But as N is an
=
N.(.~)N*(gx)(n))
G\U.T
A*(pc\u.(Uo.T,))(CA)(Uo.T,,~r,).N.(~'T)N*(gx)(n)
s
A*(pa\u.(Uo,T,))(C a)Wo,T,,~,).N.(,T)N*(gx )(n) . . . .
Let
dr
be the map from T to 2F defined by
T
l/T
T
, UoHT ~
, UoHT' PT
is cartesian. So
dT(t) = (t,gg(t)).
The square
CHAPTER 10. ADJUNCTION AND GREEN FUNCTORS
258
....
A*( G\ U.~'T)A*(pG\u.(UoHT'))(CA )(T,dr) . . . . ....
A*(PG\U.T)(eA)(T,dT)
This gives finally
~(n)(T,g) = Oa\u.TN,(vT)(A (PG\U.T)(~A)(T,dT).N
(~X)(/7))
I must show that ~b is a morphism of s from N to g u ( M ) . So let (Z, f) be an object of 9 and a E A(G\U.Z). Then by definition of the product on Tgu(M), I have
(oCreen functor of A, which is clearly
Definitions: If A and B are Green flmctors fox" the group G, then the direct sum A | B of A and B is the direct sum of A and B as Mackey flmetors, with the product defined for O-sets X and Y, and elements a E A ( X ) , b E B ( X ) , c E A(}") and
(z ~ B(Y) bx (,~ r
b) • (~ ~ d) = (a • c) r (t, • d) c ( A ~ B ) ( X
• Y)
The unit of A @ t3 is the element CA @ CB of ( A @ B )( . ). If M is an A-module, and if z E Z ( A ) ( . ) , I denote by z x M the A-module defined for a G-set X by
(~ • M)(X)
= ~ • M(X)
1Te is an idempotent of Z ( A ) ( . ) , I denote by e x A the subfunctor of A defined for a O-set X by
(~
• A)(X)
-
~: • A ( X )
c_ A ( X )
Then e x A is a sub-Green functor of A (the inclusion being not unitary in general), with e = e x CA C (e X A ) ( . ) as unit. Moreover, if M is an A-module, then ( x M is a e x A-module T h e m o d u l e z • M is an A - m o d u l e , because if X and Y are G-sets, if c~ E A ( X ) and rn E M ( Y ) , t h e n o~ x ~ x m = (disjoint, 188 u(yj), 185 ~N(K), 218 v-perfect, 308 rrR(G), 307 ~z, 199 M ( H ) , 273
M(H), 27a ~ ( H ) , 27s r XUy 168 x ~ 127 • U, 171 xH, 181 U• 231 CA, 134 5AoU, 171 s 232 X, 194 (A(H), 315
(A(X), 316
16
a.m, 49 a ox m, 72 a oz a', 65 a x b, 46 a x r n , 47,76 b, 52 b 5 a, 125 e~, 308
INDEX
340
f.,f*, 68 fHa, 308 kl(L), 183 k2(L), 183 rn(g.~), t94 pI(L), 183
p~(L), ls3
q(L), 183 Du(X), 186 ~(M, N), 9
~A(M, N),
145
Zx, 316
~:(M,,..., ,%; P), s 193 s 196
29
Qv(M), 188 7~(M), 2o7
T~u(O), 209 8u(M), 205 adjunction between ~ and ~ , 38 between @B and "HA, 148 co-unit, 185 unit, 185 algebra associated to a Green functor, 84, 99, 164 Alperin's conjecture, 304 balanced, 99, 154 bifunctoriality, 46, 47 bimodule, 141 construction, 15a structure on 7-{(M, N), 142 biproducts, 68 biset, 167 composition with, 168 blocks of Mackey algebra, aa5 Burnside functor as Green functor, ,55 as initial object, 57 as Mackey functor, 52 as unit, 59 cartesian product in CA, 103 in CA • CA, 109
category Du(X), 186 adding direct snmmands to a, 310 associated to a Green functor, 67 equivalence of, 79, 310 .312, 314 representation of, 71, 81 centre, 305 of Yoshida algebra, a35 coinduction. I80 coinfiation, 168, 217 commutant, l,t3 commutative Green functor, 110, 129, 305 commute, 136 composition and associated categories, 173 and Green functors, 170 and modules, 175 and tensor product, 168 with a biset, 168 direct summand in CA, 82 divides in C.4, 82 dual of a module, 130 embedding of a.Lgebras, 112 endosimple A module, 304 equivalence of categories, 79, 84, 112, 129, 141, 155, 310-312, 314 of the definitions of (-been functors, 48 examples of algebras A(fT~), 94, 95 of composition with a biset, 168 of Frobenius morphisms, 227 of functors (,4, 3a2 of functors s and •u(M), 215 of Green functors s 264 finitely generated module over a. Green functor, 114 Frobenius nlorl)hisn~s, 223 functor from G-sets to CA, 68 of evaluation, 81
INDEX functorial ideal, 291 functoriality of M@BN, 147
of U ~ U oa X, 177 of "H(M, N) and MQN, 24 of 7fA(M, N), 146 generators and relations for A(f~2), 85, 88 for the Mackey algebra, 7 Green functors and solvable re-subgroups, 314 centre, 305 composition with a biset, 170 definition in terms of G-sets, 46 definition in terms of subgroups, 41 direct sum, 305 tensor product, 134 identification of ~A, 320, 323, 327 of "]~A(M,_N), 145 of internal homomorphisms, 11 of tensor product (first), 15 of tensor product (second), 23 injective on the orbits, 177, 186, 192 left adjoint and Morita contexts, 273 and tensor product, 272 toM~MoU, 197 to Z ~-+ U OH Z, 183 Lindner construction, 94 Mackey algebra, 7 anti-automorphism, 10 axiom, 5 functors composition with a biset, 168 definition as modules, 7 definition in terms of G-sets, 6 definition in terms of subgroups, 5 internal homomorphisms, 9 tensor product of, 9 module dual, 130
341 of finite type, 114 over a Green functor, 41, 47 examples, 61 over the Burnside functor, ,59 right, 129 Morita context, 100, 154 equivalence, 99 of algebras A(X2), 99, 100 theory, 1,55, 160 morphism n-linear, 29 universal property, 29 bilinear, 37, 127 Probenius, 223 of bimodules, 141 of Green functor, 41 of Mackey functors, 5, 6 of modules over a Green functor, 42 multiple of in CA, 82 natural transformation, 316 non-commutative tensor product, 164 opposite Green fimctor, 127 product OH, 167 8 , 125 • for A~B, 134 • for CA, 317 x for ~ ( M , M), 123 • for s 231 progenerator, 115, 117, 118, 155, 161, 321 projective G-algebra, 292 projective relative to, 102 solvable ~r-subgroups, 314 relative projectivity, 100 representation of a category, 71, 81 residual rings, 274 restriction, 5 for M~N, 15, 23 for ~ ( M , N), 10
342 right adjoint to M ~ - + M o U , 210 right modules, 129 same stabilizers, 100 simple Green functors, 291 classification, 295 simple modules, 275 classification, 276 structure, 282 source algebra, 114 surjective Morita context, 154 tensor product 7~-fold, 25 universal property, 29 and composition with a biset, 168 and left adjoints, 227 and residues, 298 and right adjoints, 242 associativity, 38 commutativity, 38 of "algebras" over b(ft2), 164 of Green functors, 134 universal property, 137 of modules over Green functors, 140 of simple modules, 298 Th~venaz's theorem, 295 trace, see transfer transfer, 5 for M Q N , 15, 23 for 7-{(M,N), 10 unitary Green functor, 46 morphism of Green functors, 41, 43 Yoshida algebra, 95 centre, 335
INDEX