Mackey Functors and Related Structures in Representation ... · Mackey Functors and Related Structures in Representation Theory and Number Theory Robert Boltje Universit at Augsburg

Mackey Functors and Related Structures in

Representation Theory and Number Theory

Robert Boltje

Universitat AugsburgInstitut fur Mathematik

Universitatsstr. 286 135 Augsburg

Germany

February 1995

Contents

Introduction 5

Notation 15

1 The Language 17

1.1 The categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 The functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3 The mark morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Adjointness properties . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.5 Scalar extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6 Invertible group order . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Canonical Induction Formulae 43

2.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Invertible group order . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3 The standard situation . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Examples 61

3.1 The character ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 The ring of Brauer characters . . . . . . . . . . . . . . . . . . . . . . 73

3.3 Projective characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4 Trivial and linear source rings . . . . . . . . . . . . . . . . . . . . . . 84

3.5 The Green ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.7 Extraspecial p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 Applications 111

4.1 An extension teorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3 Adams operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.5 Virtual extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.6 Glauberman correspondence . . . . . . . . . . . . . . . . . . . . . . . 132

4.7 Projectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3

4 CONTENTS

5 Monomial Resolutions 1415.1 kG-monomial modules . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.2 The monomial resolution . . . . . . . . . . . . . . . . . . . . . . . . . 1615.3 Monomial resolutions as projective resolutions . . . . . . . . . . . . . 179

6 Class Group Relations 1896.1 Cohomological Mackey functors . . . . . . . . . . . . . . . . . . . . . 1896.2 Application to class groups . . . . . . . . . . . . . . . . . . . . . . . 191

A The Burnside Ring 197

B Posets and Mobius Inversion 201

Bibliography 205

List of Symbols 209

Introduction

Let G be a finite group and H a subgroup of G. In many areas we have the notion ofrestriction of a G-action to an H-action and induction of an object with H-actionwhich results in an object with G-action. The objects we talk about may be fi-nite sets, abelian groups, vector spaces, fields, manifolds, topological spaces, vectorbundles, etc. Besides the usual transitivity properties of restriction and induc-tion, G. Mackey discovered a formula (the Mackey-decomposition formula) whichallows to express the composition of restriction after induction as a combination ofcompositions of inductions after restrictions. By the construction of Grothendieckgroups, the objects we consider can be regarded as elements of an abelian group,or sometimes even as elements of a commutative ring. The functors of restrictionand induction are reflected in maps between these groups, and what we obtain isa Mackey functor (see Definition I.1.1), named after G. Mackey, since the Mackeyaxiom (see I.1.1 (M4)) reflects the Mackey decomposition theorem. Note that in thedefinition of a Mackey functor, there is a third structure map called conjugation,which is always so obviously present that it tends to be overlooked.

A precise definition of a Mackey functor is due to Green (cf. [Gr71]) and to Dressin the early seventies. Particularly Dress developed a theory of Mackey functors inthe most general and abstract setup (cf. [Dr73]), which made it difficult to be appliedby others not willing to follow through a dense jungle of abstractness and notation.The definition of a Mackey functor given here is only a special case of Dress’ generaldefinition, and understandable without algebraic background. In contrast to thisdefinition the rest of the language we introduce in Chapter I deserves again to becompared with a jungle of notation. But there is no remedy for this dilemma.Once we have seen that certain complicated structures arise naturally in a vastvariety of situations the mere self-understanding of mathematics forces us to give aname to these structures and to prove theorems about them, independently from aspecial situation. The alternative would be to repeat more or less identical proofsin different situations, or to prove theorems in special situations by a mixture ofgeneral arguments and arguments only possible in this situation, which would hidethe real reasons for the proven result, namely the presence of a general structure.

The general study of Mackey functors was continued by Yoshida in the earlyeighties and by Webb and Thevenaz from the late eighties to now. Other namesthat should be mentioned here are tom Dieck (cf. [tD87]) and Sasaki.

Let us indicate the importance of Mackey functors in two different areas. Oneof the most fascinating conjectures in modular representation theory is Alperin’sconjecture (cf. [Al87]). Thevenaz and Webb showed in [TW90] that this conjecture

5

6 INTRODUCTION

is equivalent to the existence of a virtual Mackey functor with certain properties. Innumber theory Kato developed a class field theory for local fields of higher dimen-sion. His complicated proofs were simplified considerably by Fesenko who replacedthe groups Kato used by groups carrying the right additional (Mackey functor)structure, cf. [Ne92, IV.6, Exercise 20]. Note that the usual class field theory canbe formulated and proved very clearly in the language of Mackey functors, cf. [Ne92,IV.6, Exercise 9 ff.].

In this thesis the language of Mackey functors will be used for two purposes. InChapter VI we apply a theorem of Yoshida on Mackey functors to derive explicitrelations for the class groups of intermediate fields of a finite Galois extension ofnumber fields. The amazing aspect is that number theory is only used for the(almost obvious) fact that the class groups form a Mackey functor. The rest followsfrom the general theory of Mackey functors, the particular role the Burnside ringplays for Mackey functors, and explicit formulae for the idempotents of the Burnsidering. This should be compared with a classical result by Brauer (cf. [Bra51]) whoobtains relations between class numbers (not class groups) up to a rational factorwhich may assume only finitely many values. Brauer used comparatively deepnumber theoretic arguments, as for example the theory of Artin L-functions andthe analytic class number formula. In the meanwhile we learnt that Roggenkampand Scott used in [RS82] a very similar approach to the one we give in ChapterVI. One of the consequences we obtain in Chapter VI is the following: Let L/Kbe a Galois extension of number fields with finite Galois group G. Then the classgroup of L is uniquely determined by the class groups of the other intermediatefields through an explicit relation which depends only on G, provided that G is nothypo-elementary, i.e. G/Op(G) is not cyclic for any prime p, and provided that thereduced Euler characteristic of the poset of proper non-trivial subgroups of G isnot zero. This applies for example for the symmetric group on four letters and thedihedral group of order 12.

The other and principal aim of this thesis is to construct canonical inductionformulae in various representation theoretic situations.

Recall that Brauer showed in [Bra47] that each character of a finite group isan integral linear combination of induced one-dimensional characters. In [Bo89] wegave a canonical version of Brauer’s induction theorem. Let us explain what thismeans. Note that the character rings R(H) for the subgroups H of a finite groupG form a Mackey functor with the usual conjugation, restriction, and inductionmaps. We constructed another Mackey functor Rab

+ (H), H ≤ G, which may bethought of the set of all integral linear combinations of induced one-dimensionalcharacters, where we consider two such linear combinations as the same, if onearises from the other by replacing subgroups and one-dimensional characters withconjugate subgroups and conjugate characters. Note that the induction of such alinear combination and also the restriction (by the Mackey-axiom) results again ina certain integral linear combination of induced one-dimensional characters. Inter-preting such a linear combination again as a virtual character yields a (surjective)map

bG : Rab+ (G)−→R(G)

such that the collection of maps bH , H ≤ G, is a morphism of Mackey functors,

INTRODUCTION 7

i.e. commutes with conjugation, restriction, and induction. In [Bo89] we constructeda map

aG : R(G)−→Rab+ (G)

such that the composition

R(G)aG−−→Rab

+ (G)bG−−→R(G)

is the identity map. This justifies to call aG a Brauer induction formula. Moreover,the maps aH , H ≤ G, were defined in a way that they commute with restriction andconjugation, what induced us to call them a canonical induction formula. It can beshown (cf. Remark II.1.2) that there can’t exist a collection aH , H ≤ G, of sectionsfor bH , H ≤ G, which commute with conjugation, restriction, and induction. So weobtained the best we could hope for. Another canonical but non-additive Brauerinduction formula on the semiring of genuine characters was given by Snaith in[Sn88]. A comparison of these two formulae can be found in [BSS92]. For a geomet-ric interpretation of the canonical induction formula in [Bo89] given by Symondsconsult [Sy91].

The notation Rab+ (G) needs some explanation. Let Rab(G) denote the subring

of the character ring R(G) which is generated by one-dimensional characters. Thenthe groups Rab(H), H ≤ G, do no longer form a Mackey functor, since induced one-dimensional characters don’t generally decompose into one-dimensional characters.However, the groups Rab(H), H ≤ G, are stable under conjugation and restriction,so that they form what we call a restriction functor, a Mackey functor withoutinduction. The construction of the groupsRab

+ (H), H ≤ G, from the groupsRab(H),H ≤ G, is in fact a functor from the category of restriction functors to the categoryof Mackey functors. More precisely, the functor−+ is the left adjount of the forgetfulfunctor from the category of Mackey functors to the category of restriction functors.Hence we might call it ‘Mackeyfication’ of a restriction functor.

One can show that an arbitrary morphism aH : R(H) → Rab+ (H), H ≤ G, of

restriction fuctors, is uniquely determined by its residue πH aH : R(H)→ Rab(H),H ≤ G, where πH : Rab

+ (H) → Rab(H) maps the linear combination of inducedone-dimensional characters to the part which induces from H, i.e. induces trivially.The residue of our canonical induction formula consists of the maps pH : R(H) →Rab(H), H ≤ G, wich are induced by taking fixed points under the commutatorsubgroup, i.e. pH is the identity on one-dimensional characters and trivial on otherirreducible characters. Since the maps pH , H ≤ G, do not commute with restric-tion, but still with conjugations, we define a third category whose objects we callconjugation functors, namely restriction functors without restriction. This categoryis very important for the theory of Mackey functors, since there is another functor−+ from the category of conjugation functors to the category of Mackey functors,which has useful adjointness properties.

Most of Chapter I is dedicated to the definition of the three categories of Mackeyfunctors, restriction functors, and conjugation functors, and to the functors −+

and −+ between them. There is also a variety of natural transformations involved.So, for instance, if A is a restriction functor, then there are the so-called markhomomorphisms ρH : A+ → A+, H ≤ G, which generalize the mark homomorphismon the Burnside ring, cf. Example I.2.3.

8 INTRODUCTION

In Chapter II we define the notion of a canonical induction formula for a Mackeyfunctor M from a restriction subfunctor A ⊆ M . The reader should think of Mreplacing the character ring R and A replacing the abelianized character ring Rab.A canonical induction formula is a morphism

a : M −→A+

of restriction functors such that composition with a certain natural morphism

bM,A : A+−→M

of Mackey functors is the identity. The idea to generalize the canonical Brauerinduction formula was already explained in [Bo89, Chapt. III] and incorporated inthe notion of an induction triple. Since then we obtained examples which do notfit into the framework of an induction triple so that we present a slightly differentdefinition of a canonical induction formula in Section II.1. In Section II.2 we studythe case where the abelian groups M(H) and A(H), H ≤ G, carry the structure ofa k-module, where |G| is invertible in k. In most of the applications we may think ofk being Q. In this situation the morphisms a : M → A+ of k-restriction functors arein bijective correspondence to the morphisms p : M → A of k-conjugation functorsby associating to a its residue p = πA a in a similar way as explained for thecharacter ring. We can relate properties of a to those of p. In Section II.3 werestrict the situation further to the case where A(H), H ≤ G, has a Z-basis whichis stable under conjugations, and whose positive span is stable under restriction.This is the case in all examples we will consider. In this standard situation alsoA+(H), H ≤ G, is free and we may embed the whole setup injectively into theQ-tensored version and apply the results of Section II.2. Now the main difficultyarises: The morphism

aM,A,p : Q⊗M −→Q⊗A+

of restriction functors corresponding to p : M → A is in general not integral,i.e. M(H) is in general not mapped to A+(H) but only to Q⊗A+(H) for H ≤ G.For a long time we had no criterion to decide for given p whether aM,A,p is integral,i.e. aM,A,p(M) ⊆ A+. In Section II.4 we state a condition (∗π) for p which impliesthat |H|π′ · aM,A,p

H (M(H)) ⊆ A+(H) for H ≤ G, where π is any set of primes (seeTheorem II.4.5). In particular, if p satisfies condition (∗π) for the set π of all primes,then aM,A,p is integral.

The condition (∗π) is applied in Chapter III, where we give various examples forM , A, and p. More precisely we consider the situations where M is the characterring, the Brauer character ring, and the group of p-projective characters or π-projective characters for a set π of primes. In these situations there is not verymuch to do except referring to the general theory developed in Chapter II, andintegral canonical induction formulae drop like ripe fruit in Sections III.1–III.3. InSection III.4 we consider the Mackey functor of trivial source modules and linearsource modules over a complete discrete valuation ring O. We show that there is anintegral canonical induction formula in both cases which induces only modules ofrank one. Here the proof of the integrality is divided into two steps. In a first stepwe show that condition (∗π) holds for the set π of primes different from the residue

INTRODUCTION 9

characteristic p of O. In the second step we define a different canonical inductionformula whose residue satisfies condition (∗p) and we show that the first formulais the composition of the second one with an integral morphism. Having first onlyconsidered the trivial source ring after a suggestion of Kulshammer, we realized thatalmost all proofs work without any changes also for the linear source ring, a ringwhich hasn’t been studied as extensively in the literature as the trivial source ring,although it shares many nice properties. For example it is semisimple after tensoringwith C so that there is a kind of character theory for it. Moreover tensoring withthe quotient field and the residue field of O yields a surjection onto the characterring and the Brauer character ring. In Section II.5 we consider the Green ring as acandidate for M . But here we have to leave things in a most unsatisfactory state.In particular we have no integral canonical induction formula except one wherewe allow to induce all modules from all solvable subgroups. Section III.6 givesa summary of the different induction formulae and the relations between them.Finally we calculate in Section III.7 the canonical Brauer induction formula for thecase of an extraspecial p-group of odd exponent p.

Chapter IV is devoted to applications of canonical induction formulae. Assumethat we have two Mackey functors M and N together with restriction subfunctorsA ⊆ M and B ⊆ N . Assume further that we have a canonical induction formulaa : M → A+ for the first pair. In Section IV.1 we show that each morphism f : A→B of restriction functors can be extended to a morphism F : M → N of restrictionfunctors

MF−−→ N⋃ ⋃

Af−−→ B,

(1)

and we give conditions on M , A, and N which imply that F is unique. For instance,in the case where M = N = R is the character ring Mackey functor and A = B =Rab is the abelianized character ring restriction functor, it is clear that F extendsuniquely. In general F is given by the composition

Ma−−→A+

f+−−→B+bN,B−−→N,

in other words, if

χ =∑H≤G

indGH(aH)

describes the canonical induction formula for χ ∈ M(G) with aH ∈ A(H) forH ≤ G, then FG(χ) is defined by

F (χ) =∑H≤G

indGH(fH(aH)

).

This is surprising, since it means that∑H≤G

indGH(FH(aH)

)=∑H≤G

FG(indGH(aH)

)even if F does not commute with induction.

10 INTRODUCTION

In Sections IV.2–IV.4 we apply this extension theorem in the cases where f isthe determinant, the Adams operation, and the total Chern class, and where M isthe character ring, the Brauer character ring, the (π-)projective character group,the linear source ring and the trivial source ring Mackey functor. Note that in everycombined case for M and f , it is very easy to define f on A, but it is not alwaysclear how to extend f to M . In particular, we don’t know about Chern classesdefined on the Brauer character ring, the projective character group, the trivialsource ring or the linear source ring. Note that for these applications it is crucialto have an integral canonical induction formula, since N may be a torsion group,as for example in the cases of cohomology groups.

A speculation should be allowed here. It is straightforward to extend everythingwe mentioned so far to profinite groups G, if we only consider open subgroups. Fixa prime p and consider the absolute Galois group GQp of the the p-adic completionQp of Q. For a finite extension K/Qp we define M(GK) = R(GK) as the direct limitof all R(GK/GL), where L runs over all finite Galois extensions of K, and we defineA(GK) = Rab(GK) as the direct limit of all Rab(GK/GL), both limits taken withrespect to the inflation maps. This defines a Mackey functor M and a restrictionsubfunctor A ⊆ M . Moreover, for a finite extension K/Qp, we define B(GK) asthe free abelian group on the set of homomorphisms from K× to C× with kernel offinite index. Then B is a restriction functor with restrictions given by compositionwith the norm maps. Local class field theory establishes an isomorphism f : A→ Bof restriction functors. One special case of the Langlands conjecture predicts a‘canonical’ isomorphism between M(GK) and certain representations of the variousgroups GLn(K) for each finite extension K of Qp. Unfortunately it is not clearhow these representations give rise to a Mackey functor structure. At this state theextension theorem of Section IV.1 just waits for someone to define the right Mackeyfunctor N in order to give a canonical Langlands correspondence between M andN .

Let G and S be finite groups of coprime order and assume that S acts on G sothat we can form the semidirect product GS. In Section IV.5 we use the canonicalinduction formula in the various situations of Chapter III in order to define mapswhich associate to an S-fixed G-character or G-module a virtual GS-character orGS-module which extends the original character or module. Note that at least inthe situations for linear source rings and trivial source rings this seems to give newresults. Following a suggestion of Puig we define in Section IV.6 a map glG,S fromthe S-fixed points R(G)S of the character ring R(G) to the character ring R(GS) ofthe S-fixed points of G. We show that if S is a p-group, then the map glG,S is relatedto the Glauberman correspondence (cf. [Gla68]). It would be very interesting to seeif this map could give a direct definition for the Glauberman correspondence withoutgoing through a composition series of S in the case of general solvable S, which isnecessary if we use the descriptions which exist at the moment. Moreover, it mightbe that the map glG,S gives a unified definition of the Glauberman correspondence,which is defined for solvable S, and a correspondence of Isaacs, cf. [Is73], which isdefined for odd (and hence solvable) G. Moreover, the definition of glG,S is alsopossible for the other Grothendieck groups we considered in Chapter III.

for a character χ ∈ R(G) we define the p-projectification pr(χ) as the characterχ multiplied with the characteristic function of the set of p′-elements of G. In

INTRODUCTION 11

Section IV.7 we construct a canonical induction formula for pr(χ) which is a Q-linear combination of induced one-dimensional characters on p′-groups with thesame identification we explained at the beginning for Rab

+ (G). For an irreduciblecharacter χ of G the rational coefficients of the corresponding linear combinationhave p-power denominators, and the biggest occurring denominator is precisely pd(χ)

where d(χ) is the p-defect of χ, i.e. it satisfies |G|p = pd(χ)χ(1).

The Cartan map cG is an injective map from the p-projective character group tothe group of Brauer characters of G, which becomes an isomorphism after tensoringwith Q. For a Brauer character χ we define a canonical induction formula for c−1

G (χ)which is a Q-linear combination of induced one-dimensional Brauer characters onp′-subgroups. The residue of this induction formula satisfies condition (∗p′) of The-orem II.4.5 so that the rational coefficients in the above linear combination haveonly p-power denominators. This implies that the cokernel of cG is a finite p-group.

The projective resolution of a module is a familiar construction and leads tomany interesting invariants. In the case of the category kG−mod, of modulesover a group algebra kG, there is a variety of other interesting classes besides theprojective modules and one may also ask whether there is a reasonable theory of aresolution in terms of such modules. Brauer’s famous induction theorem emphasizesthe significance of one-dimensional modules for the representation theory of finitegroups over the field C of complex numbers. Also in number theory, one-dimensionalGalois representations have invariants (L-functions, conductors, etc.) which aremuch better understood (by using class field theory) than the general case. Sincethese number theoretical invariants behave well under induction and taking directsums, it is natural to consider the class of monomial modules, i.e. modules arisingas sums of induced one-dimensional modules.

Since Brauer’s induction theorem states that (over C) each module is a directsummand of a monomial module with a monomial complement, we can’t expectany interesting theory of resolutions by monomial modules in the classical sense.Therefore we consider in Chapter V the category kG−mon of kG-monomial mod-ules which consists of monomial kG-modules with the additional structure of a de-composition into submodules satisfying certain properties with respect to the groupaction. Here k may be any noetherian integral domain. A slightly different variantof this category has been considered in [Re71] and [Bo90] for k = C resp. k a field.We construct for each finitely generated kG-module a chain complex in kG−monwith certain natural properties, which form the axioms of a kG-monomial resolu-tion. The notion of a kG-monomial resolution has all the nice properties we knowfrom projective resolutions: It is unique up to homotopy equivalence and mor-phisms in kG−mod extend to morphisms between the kG-monomial resolutionswhich are unique up to homotopy. Hence, the kG-monomial resolution may beconsidered as an embedding of kG−mod into the homotopy category of chain com-plexes in kG−mon. If we view the objects in a kG-monomial resolution of someV ∈ kG−mod only as kG-modules, then we obtain a chain complex of monomialmodules whose homology is concentrated in degree zero and is isomorphic to V .

Interestingly the kG-monomial resolution yields non-trivial invariants even fork = C. We show that each CG-module V has a finite CG-monomial resolution andthat the Lefschetz invariant of this resolution as element in the Grothendieck group

12 INTRODUCTION

of CG−mon (which coincides with the ring Rab+ (G) introduced in Section III.1) is

just the canonical Brauer induction formula of Section III.1 applied to the characterof V . Hence, the functor of taking CG-monomial resolutions can be regarded as alift of the canonical Brauer induction formula on the level of Grothendieck groupsto a functor on the level of the underlying categories (cf. Remark V.2.17). Theminimial length of a CG-monomial resolution of a CG-module V is studied, andwe give upper bounds in terms of the rank of a finite poset associated to V . Usingagain properties of the canonical Brauer induction formula it is shown that V hasa CG-monomial resolution of length zero if and only if V is a direct sum of one-dimensional modules.

Chapter V is arranged as follows. In Section 1 we introduce the categorykG−mon of kG-monomial modules and study some basic properties. There is a fullclassification of isomorphism classes of objects and a description of the Grothendieckgroup in the case where k is local or a principal ideal domain. In Section 2 we de-fine the notion of a kG-monomial resolution and prove the existence, uniquenessand other properties mentioned above. Section 3 gives an interpretation of themonomial resolution as a projective resolution in a bigger category C by embeddingboth kG−mod and kG−mon into C such that the objects of kG−mon becomeprojective in C. We also describe C (by constructing equivalences of categories) asa module category and a category of sheaves on a poset.

It happens that the homotopy category of kG−mon shares many interestingproperties with the derived category of kG−mod, which in turn plays an impor-tant role in connection with recent conjectures in modular representation theory.Moreover, it seems that this category takes much more into account that we con-sider modules over group rings not just over arbitrary rings. But unfortunately wehave no block decomposition for kG−mon.

In Appendix A we collect the basic properties of the Burnside ring and fix thenotation used in connection with the Burnside ring. Appendix B provides the factsabout posets and Mobius functions which we need in this thesis.

References to items of other chapters are by triples. Theorem II.4.5 means thetheorem with the number 4.5 in Chapter II. Within Chapter II we would only refer toTheorem 4.5. Equation are numbered consequtively in each chapter. Equation (3.1)means the first equation in Chapter III. There is no indication of the section whichcontains the equation.

Acknowledgement

In alphabetical order I would like to thank Michel Broue for motivating me to lookfor a notion of a monomial resolution by his questions about chain complexes ofmonomial modules with certain properties, Hubert Fottner for suggesting Exam-ple IV.1.10, Burkhard Kulshammer for suggesting to construct a canonical inductionformula for the trivial source ring and for many fruitful discussions, Olaf Neiße forreading an earlier version of Chapter I and II, Lluis Puig for suggesting to definethe map gl (cf. Section IV.6), Jeremy Rickard for pointing out that the monomialresolution has an interpretation as projective resolution (cf. Section V.3), and AlWeiss for his interest in the methods and results of Chapter VI.

INTRODUCTION 13

Finally, I’m indebted to Jurgen Ritter for his support in so many ways duringthe last eight years.

14 INTRODUCTION

Notation

We assume the following notations throughout all chapters.

Numbers

N positive integers

N0 non-negative integers

Z integers

Q rational numbers

R real numbers

C complex numbersFp finite field with p elementsZp p-adic completion of Z at a prime pQp p-adic completion of Q at a prime p

Groups Let G be a group.

H ≤ G subgroup

H < G proper subgroup

H G normal subgroup

H / G proper normal subgroupgH gHg−1

(G : H) index of a subgroup H ≤ GHg g−1Hg

Gπ set of π-elements of G for a set π of primes

|G| order of G

Gab commutator factor groupOp(G) biggest normal p-subgroup for a prime p

K ≤G H K ≤ gKg−1 for some g ∈ G for K,H ≤ GK =G H K = gHg−1 for some g ∈ G, for K,H ≤ Gs =G t s = gt for some g ∈ G, for elements s, t of a G-set

stabG(s) the stabilizer of an element s in a G-set

R(G) character ring of G

Irr(G) set of irreducible characters of G

Categories Assume that A is a ring and G is a finite group.

A−Mod category of left A-modules

A−mod category of finitely generated left A-modules

G−set category of finite left G-sets

ab category of finite abelian groups

15

16 NOTATION

Rings Let k be a commutative ring and let G be a finite group.kG or k[G] group ring

k× multiplicative group of units of k

G(k) Hom(G, k×)

rkkV k-rank of a k-free k-module V

dimk V k-dimension of a k-vector space V , if k is a field

Miscellaneousf |B : B → C restriction of a map f : A→ C to a subset B ⊆ Anπ π-part of n ∈ N for a set π of primes#X or |X| cardinality of a finite set XM | N M is isomorphic to a direct summand of a module N

Chapter 1

The Language:Mackey Functors,Restriction Functors, andConjugation Functors

Throughout this chapter let G be a finite group and let k be a commutative ringwith unity. Unadorned tensor products will be taken over the ring Z of integers.

This chapter is supposed to serve as a dictionary for the language we use inlater chapters. From a logical point of view it can be read from Section 1 to Section6 continuously, but it seems more advisable to start out reading Chapter II andskip back whenever it is necessary to become acquainted with another part of thelanguage.

In Section 1 we introduce the notions of k-Mackey functors, k-restriction func-tors, and k-conjugation functors on G. Each of these notions has less structurethan its predecessor and consequently there are forgetful functors from each of theresulting categories to the next one. If G is the trivial group, the three categoriesare equivalent to the category of k-modules. Each of the three categories has ak-algebra variant, namely k-Green functors, k-algebra restriction functors, and k-algebra conjugation functors on G, and if G is the trivial group, they all coinciedwith the category of k-algebras. For the sake of consistency we might call a Greenfunctor also an algebra Mackey functor, but for historical reasons we stick to thenon-consistent terminology. The standard example of a k-Green functor on G con-sists of the Burnside rings of all subgroups of G tensored with k. Other examplesare provided by character rings, cohomology groups, ideal class groups, etc.

Mackey functors were introduced by Dress in [Dr73] and Green in [Gr71]. Somepapers of Thevenaz, Webb an Yoshida, who studied Mackey functors further, shouldalso be mentioned here: [Th88], [TW91], [We91], [Yo80], [Yo83b]. Some of theseauthors prefer the name ‘G-functor’ instead of ‘Green functor’ or ‘Mackey functor’.The category of restriction functors was introduced in [Bo89, III.1] where they werecalled ‘k-sheaves’. We introduce the notion of conjugation functors here, since manyMackey functors (even all of them, if the order of G is invertible in k) arise fromconjugation functors by the functor −+, which is introduced in Section 2. Also, the

17

18 CHAPTER 1. THE LANGUAGE

construction of the twin functor (see [Th88] or 2.8) factorizes through the category ofconjugation functors. The category of conjugation functors is indispensable for theanalysis of certain morphisms of restriction functors via their residues in Section 4.

In Section 2 we define various functors between the categories defined in Section1. The most important ones of them for the construction of canonical inductionformulae are the functors −+ and −+. The Burnside ring functor arises by thefunctor −+, and the mark homomorphism on the Burnside ring generalizes to anatural transformation ρA : A+ → A+ for a k-restriction functor A which is studiedin Section 3. The mark morphism ρA is injective in many cases (in fact in allcases that we consider in Chapter II) and it is an isomorphism if the order of Gis invertible in k (cf. Proposition 3.2). Since we can give an explicit inverse for ρA

in this case, we obtain explicit descriptions for the canonical induction formulaeconsidered in Chapter II.

In Section 4 we prove some adjointness properties of the functors −+ and −+.The functor −+ itself and the functor −+ followed by a forgetful functor are adjointfunctors to the two forgetful functors between the categories considered in Section1 (cf. Proposition 4.1). In Chapter II a canonical induction formula will be amorphism B → A+ of restriction functors for certain restriction functors A and B.The adjointness properties and the mark morphism allow us to set up a bijectionbetween such morphisms and morphisms B → A of the underlying conjugationfunctors, provided that the order of G is invertible in k (cf. Corollary 4.2). In moregeneral cases we still obtain an injective map.

In order to place ourselves in this situation we need to know how the functors−+ and −+ behave under scalar extension. This is examined in Section 5. Fi-nally, Section 6 summarizes miscallenous results in the case, where the order ofG is invertible in k. In this case the categories of Mackey functors and conjuga-tion functors are equivalent. Moreover there is a ‘split’ embedding of the categoryof conjugation functors into the category of restriction functors in general, whichinduces an isomorphism between the Grothendieck groups of these two categoriesunder certain finiteness conditions.

1.1 The categories

The following definition of a Mackey functor is the most convenient one, if one wantsto verify that a certain object is a Mackey functor. There are equivalent and moregeneral definitions which explain that a Mackey functor is indeed a functor or alsoa bifunctor. There is also an k-algebra (depending on G) whose module categoryis equivalent to the category of k-Mackey functors on G. See [Dr73] and [TW91,Sections 2 and 3] for such definitions.

1.1 Definition ([Dr73, Gr71]) A k-Mackey functor on G is a quadruple(M, c, res, ind) consisting of

• a family of k-modules M(H), one for each H ≤ G,

• a family of k-module homomorphisms cg,H : M(H) → M( gH), one for eachg ∈ G and H ≤ G, called conjugation maps,

1.1. THE CATEGORIES 19

• a family of k-module homomorphisms resHK : M(H) → M(K), one for eachpair K ≤ H of subgroups of G, called restriction maps, and

• a family of k-module homomorphisms indHK : M(K) → M(H), one for eachpair K ≤ H of subgroups of G, called induction maps,

which satisfy the following axioms:

(M1) (Triviality) ch,H = resHH = indHH = idM(H)

for H ≤ G, h ∈ H.

(M2) (Transitivity) cg′g,H = cg′, gH cg,HresHL = resKL resHK

indHL = indHK indKL

for L ≤ K ≤ H ≤ G, g, g′ ∈ G.

(M3) (G-equivariance of restriction and induction)

cg,K resHK = resgHgK cg,H

cg,H indHK = indgHgK cg,K

for K ≤ H ≤ G.

(M4) (Mackey axiom)

resHU indHK =∑

h∈U\H/K

indUU∩ hK

reshK

U∩ hK ch,K

for U,K ≤ H ≤ G.

We will often write M instead of (M, c, res, ind) when there is no need for specifyingconjugation, restriction and induction, and for H ≤ G, m ∈ M(H), g ∈ G we willabbreviate cg,H(m) by gm when there is no need for specifying H.

A morphism f : M → N between two k-Mackey functors M and N on G is afamily f =

(fH : M(H) → N(H)

)H≤G of k-module homomorphisms which com-

mute with conjugation, restriction and induction. The class of k-Mackey functorson G together with their morphisms form a category k−Mack(G).

A k-Mackey functor M on G is called cohomological, if the following axiomholds:

(M5) (Cohomologicality axiom)

indHK resHK = (H : K) · idM(H)

for K ≤ H ≤ G.

The cohomological k-Mackey functors on G form a full subcategory k−Mackc(G)of k−Mack(G).

A k-Green functor on G is a k-Mackey functor M on G together with a unitaryk-algebra structure on each M(H), H ≤ G, such that conjugations and restrictionsare unitary k-algebra homomorphisms and the following axiom is satisfied:


(M6) (Frobenius axiom)

m · indHK(n) = indHK(resHK(m) · n

)indHK(n) ·m = indHK

(n · resHK(m)

)for all K ≤ H ≤ G, m ∈M(H), n ∈M(K).

A morphism f : M → N between k-Green functors M,N on G is a morphismof the underlying Mackey functors such that each fH , H ≤ G, is a unitary k-algebra homomorphism. The category of k-Green functors on G will be denoted byk−Mackalg(G) and the full subcategory of cohomological k-Green functors on Gby k−Mackc

alg(G).

1.2 Remark (i) If M ∈ k−Mack(G) (resp. M ∈ k−Mackalg(G)), andN(H) ⊆ M(H) is a k-submodule (resp. ideal) for each H ≤ G such that thefamily

(N(H)

)H≤G is stable under conjugation, restriction and induction, then N

is called a subfunctor (resp. ideal) of M and we write N ⊆ M (resp. N M). Inthis case, the quotient functor M/N , with (M/N)(H) := M(H)/N(H) for H ≤ G,and conjugation, restriction and induction inherited from M , is again a k-Mackeyfunctor (resp. k-Green functor) on G. By a subfunctor N ⊆M of a k-Green functoron G we mean a family N(H) ⊆ M(H), H ≤ G, of unitary k-subalgebras whichare stable under conjugation, restriction and induction. It is clear that subfunctorsand quotient functors of cohomological Mackey functors and cohomological Greenfunctors are again cohomological.

(ii) Let M,N ∈ k−Mack(G) and f ∈ k−Mack(G)(M,N). The constructions(M ⊕ N)(H) := M(H) ⊕ N(H), ker(f)(H) := ker(fH) ⊆ M(H), im(f)(H) :=im(fH) ⊆ N(H) and coker(f)(H) := coker(fH) = N(H)/im(fH), for H ≤ G,which we call direct sum of M and N , kernel of f , image of f and cokernel of f ,inherit the structure of a k-Mackey functor on G. The direct sum M ⊕N is botha categorical product and coproduct of M and N , and the categories k−Mack(G)and k−Mackc(G) are abelian. Moreover, k−Mack(G)(M,N) is a k-module andcomposition of morphisms is k-bilinear. In case that M,N ∈ k−Mackalg(G) andf ∈ k−Mackalg(G)(M,N), ker(f) is an ideal in M , and M/ ker(f) is again a k-Green functor on G.

(iii) If k is a subring of the commutative ring k′ (with the same unity element),then the functor k′ ⊗k − : k−Mod → k′−Mod of scalar extensions of k-modulesinduces functors

k′ ⊗k − : k−Mack(G)→ k′−Mack(G)

andk′ ⊗k − : k−Mackalg(G)→ k′−Mackalg(G),

which map cohomological k-Mackey functors (resp. k-Green functors) to cohomo-logical k′-Mackey functors (resp. k′-Green functors).

1.3 Definition ([Dr73, Yo80, Th88]) (i) Let X,Y, Z ∈ k−Mack(G). A pair-

ing X ⊗ Y → Z is a family of k-module homomorphisms

X(H)⊗k Y (H)→ Z(H), x⊗k y 7→ x · y, H ≤ G,

such that the following axioms hold (cf. [Th88, Yo83a]):


(P1) (Compatibility with conjugation)

g(x · y) = gx · gy

for H ≤ G, x ∈ X(H), y ∈ Y (H), g ∈ G.

(P2) (Compatibility with restriction)

resHK(x · y) = resHK(x) · resHK(y)

for K ≤ H ≤ G, x ∈ X(H), y ∈ Y (H).

(P3) (Frobenius axioms)

(indHK(x)) · y = indHK(x · resHK(y))

x′ · indHK(y′) = indHK(resHK(x′) · y′)

for K ≤ H ≤ G, x ∈ X(K), y ∈ Y (H), x′ ∈ X(H), y′ ∈ Y (K).

We can define a k-Green functor on G alternatively as a k-Mackey functor M onG together with a pairing M ⊗kM →M which furnishes each M(H), H ≤ G withthe structure of a unitary k-algebra such that conjugations and restrictions preservethe unity elements.

(ii) Let A ∈ k−Mackalg(G). A k-Mackey functor M on G together with a pair-ing A⊗kM →M is called a left A-module, if the pairing furnishes each M(H), H ≤G, with the structure of a left A(H)-module. Let M and N be two left A-modules.An A-module homomorphism M → N is an element f ∈ k−Mack(G)(M,N)with the property that each fH : M(H) → N(H), H ≤ G, is an A(H)-modulehomomorphism. The category of left A-modules will be denoted by A−Mod.

(iii) Let A ∈ k−Mackalg(G) such that A(H) is commutative for all H ≤ G.A k-Green functor B on G together with a morphism i ∈ k−Mackalg(G)(A,B) iscalled an A-algebra, if each iH : A(H) → B(H), H ≤ G furnishes B(H) with thestructure of an A(H)-algebra, i.e. iH(A(H)) is contained in the center of B(H). Ifi : A→ B and j : A→ C define two A-algebras, then an A-algebra homomorphism

between B and C is an element f ∈ k−Mackalg(G)(B,C) with f i = j, i.e. eachfH : B(H) → C(H), H ≤ G, is a unitary A(H)-algebra hommomorphism. Thecategory of A-algebras will be denoted by A−Alg.

The standard example of a k-Green functor on G is the Burnside ring functork ⊗ Ω (see Appendix A for a definition and the basic properties). It is easilychecked that the commutative k-algebras k ⊗ Ω(H), H ≤ G, with conjugations,restrictions, inductions and multiplications defined as in Appendix A satisfy theaxioms (M1)–(M4) and (M6). The Burnside ring functor plays a distinguished rolefor the categories k−Mack(G) and k−Mackalg(G), namely the same role that Zplays for the category of abelian groups and the category of rings, as the nextproposition shows.

1.4 Proposition (cf. [Dr73, Prop. 4.2], [Th88, Prop. 6.1], [Yo80, Ex. 2.11])(i) Every k-Mackey functor M on G is in a unique way a k⊗Ω-module, namely

by the pairing defined by

(k ⊗ Ω(H))⊗kM(H)→M(H), [H/K]⊗k m 7→ indHK(resHK(m)),


for H ≤ G and K ≤ H. This pairing will be referred to as the canonical pairing.Moreover, each morphism of k-Mackey functors is a morphism of k ⊗ Ω-modules.

(ii) For every k-Green functor A on G there is a unique morphism i : k⊗Ω→ Aof k-Green functors on G, namely

iH : k ⊗ Ω(H)→ A(H), [H/K] 7→ indHK(1A(K)),

for H ≤ G and K ≤ H, i.e. k⊗Ω is an initial object in the category k−Mackalg(G).This morphism provides a unique k ⊗ Ω-algebra structure and the unique k ⊗ Ω-module structure on A. Moreover, each morphism of k-Green functors is a mor-phism of k ⊗ Ω-algebras.

Proof Both parts can be proved by straight forward verifications.

An example of a cohomological k-Mackey functor on G is provided by the coho-mology groups Hn(H,V ), H ≤ G, for a fixed kG-module V and a fixed non-negativeinteger n ∈ N0. Conjugation and restriction maps are the obvious ones, and induc-tion maps are defined as the corestriction maps of cohomology.

When we will work on induction formulae in Chapter II, our standard examplewill be the character ring Green functor R on G. There the subrings generatedby all one-dimensional characters will be of importance. These rings, however,do not form a subfunctor, since stability under induction fails. But still we havestability under conjugation and restriction. This leads to the following definition ofa restriction functor, a ‘Mackey functor without induction’.

1.5 Definition (cf. [Bo89, III.2]) A k-restriction functor on G is a triple(A, c, res) consisting of a family A(H), H ≤ G, of k-modules and k-module homo-morphisms cg,H : A(H)→ A( gH) (the conjugations) and resHK : A(H)→ A(K) (therestrictions) for K ≤ H ≤ G, g ∈ G, satisfying the following axioms:

(R1) (Triviality) ch,H = resHH = idA(H)

for H ≤ G.

(R2) (Transitivity) cg′g,H = cg′, gH cg,HresHL = resKL resHK

for L ≤ K ≤ H ≤ G, g, g′ ∈ G.

(R3) (G-equivariance of restriction)

cg,K resHK = resgHgK cg,H

for K ≤ H ≤ G, g ∈ G.

As in the case of Mackey functors we often write just A instead of (A, c, res) andalso just ga instead of cg,H(a) for g ∈ G, H ≤ G, a ∈ A(H). A morphism f : A→ Bbetween k-restriction functors A,B on G is a family

(fH : A(H) → B(H)

)H≤G of

k-module homomorphisms which commute with conjugations and restrictons. Wedenote the category of k-restriction functors on G by k−Res(G).


An k-algebra restriction functor on G is a k-restriction functor A on G to-gether with the structure of a unitary k-algebra on each A(H), H ≤ G, such thatconjugations and restrictions are unitary k-algebra homomorphisms. A morphism

between two k-algebra restriction functors on G is a morphism of the underlyingk-restriction functors on G which consists of unitary k-algebra homomorphisms.The category of k-algebra restriction functors on G is denoted by k−Resalg(G).

1.6 Remark Analogous to Remark 1.2 we may speak of subfunctors of re-striction functors, subfunctors and ideals of algebra restriction functors, quotientfunctors of (algebra) restriction functors, direct sums and products of restrictionfunctors and kernels, images and cokernels of morphisms of (algebra) restrictionfunctors. The category k−Res(G) is again abelian and the morphisms betweentwo objects form a k-module such that composition is k-bilinear. If k is a subringof the commutative ring k′, we have scalar extension functors

k′ ⊗k − : k−Res(G)→ k′−Res(G), k′ ⊗k − : k−Resalg(G)→ k′−Resalg(G).

In our standard example, the character rings R(H), H ≤ G, the spans of one-dimensional characters form a subfunctor Rab of the underlying Z-algebra restric-tion functor of R on G. The notation is explained by the obvious isomorphismof Z-algebra restriction functors Rab(H) ∼= R(Hab), H ≤ G, where Hab denotesthe commutator factor group of H. We will have to consider the family of mapspH : R(H) → Rab(H), H ≤ G, which are induced by taking the Hab-fixed pointsof a CH-module. Thus, pH is the identity on one-dimensional characters and zeroon all other irreducible characters. Since an irreducible character of degree higherthan one may have one-dimensional constituents after restriction to a subgroup, thefamily (pH)H≤G is in general not a morphism of Z-restriction functors on G. Butstill this family commutes with the conjugation maps. This leads to the notion ofa conjugation functor, a ‘restriction functor without restriction’.

1.7 Definition A k-conjugation functor on G (resp. k-algebra conjugation

functor on G) is a pair (X, c) consisting of k-modules (resp. unitary k-algebras)X(H), H ≤ G, and k-module homomorphisms (resp. unitary k-algebra homomor-phisms) cg,H : X(H)→ X( gH), the conjugations, for H ≤ G, g ∈ G, satisfying thefollowing axioms:

(C1) (Triviality) ch,H = idX(H)

for H ≤ G, h ∈ H.

(C2) (Transitivity) cg′g,H = cg′, gH cg,H

for H ≤ G, g, g′ ∈ G.

We will often write X instead of (X, c) and also gx instead of cg,H(x) for g ∈ G,H ≤ G, x ∈ X(H).

A morphism f : X → Y between k-conjugation functors (resp. k-algebra con-jugation functors) X,Y on G is a family

(fH : X(H) → Y (H)

)H≤G of k-module

homomorphisms (resp. unitary k-algebra homomorphisms) which commute with


conjugation maps. The category of k-conjugation functors (resp. k-algebra conju-gation functors) on G is denoted by k−Con(G) (resp. k−Conalg(G)).

1.8 Remark (i) The statements in Remark 1.2 hold also for the categoriesk−Con(G) and k−Conalg(G).

(ii) There is an obvious commutative diagram of forgetful functors:

k−Mackalg(G) −−−−→ k−Resalg(G) −−−−→ k−Conalg(G)y y yk−Mack(G) −−−−→ k−Res(G) −−−−→ k−Con(G).

In particular we will assume that each definition which will be made for one of thecategories in the above diagram is automatically a definition for all categories in thesame row on the left of this one. So we define for example (fH)

H≤G in k−Con(G) orin k−Conalg(G) to be injective (resp. surjective), if fH is injective (resp. surjective)for all H ≤ G.

(iii) Let X be an object in one of the six categories in the above diagram and letH ≤ G. Then, using the conjugation maps cn,H , n ∈ NG(H), the k-module X(H)is provided with the structure of a kNG(H)-module.

(iv) In a complete treatment of the sort of categories we have introduced by nowone should also define the notion of a k-induction functor on G as a ‘k-Mackeyfunctor without restriction’. While taking k-duals yields functors k−Mack(G) →k−Mack(G) and k−Con(G) → k−Con(G), the k-dual of a restriction functorwould be an induction functor and vice-versa. We don’t introduce this notion here,since it doesn’t play a role in the considerations of the following chapters. Moreover,it is not clear what the definition of an algebra induction functor should be. Withthis remark we want to explain the lack of symmetry when certain constructions willresult in restriction functors while the dual constructions result ‘only’ in conjugationfunctors. The truth is that these constructions would result in induction functors.

1.2 The functors

In this section we will define several functors between the categories defined in thelast section. Among them, the functors

−+ : k−Con(G)→ k−Mack(G)

and−+ : k−Res(G)→ k−Mack(G)

will be of particular interest in Chapter II.

2.1 The functor−+ : k−Con(G)→ k−Mack(G) is part of the construction ofthe ‘twin functor’ of a Mackey functor which was introduced by Thevenaz (cf. [Th88,Sec. 4]). Let (X, c) be a k-conjugation functor on G. We set

PX(H) :=∏K≤H

X(K),

1.2. THE FUNCTORS 25

for H ≤ G, and observe that NG(H) acts on PX(H) by the conjugation maps c,i.e. for n ∈ NG(H) and (xK)

K≤H ∈ PX(H) we set

n((xK)

K≤H

):=( n(xKn)

)K≤H

.

Thus, n ∈ NG(H) maps the K-component in PX(H) to the nK-component inPX( nH) = PX(H) via cn,K . In particular we may consider the H-fixed pointsPX(H)H of H on PX(H), and define

X+(H) := PX(H)H

=

(xK)K≤H ∈

∏K≤H

X(K)∣∣ h(xK) = x

(hK)

for K ≤ H, h ∈ H.

For K ≤ H ≤ G and g ∈ G we define a conjugation map

c+g,H : X+(H)→ X+( gH), (xL)

L≤H 7→( g(xLg))

L≤gH

,

a restriction map

res+HK : X+(H)→ X+(K), (xL)

L≤H 7→ (xL)L≤K ,

as the projection, and an induction map

ind+HK : X+(K)→ X+(H), (xL)

L≤K 7→∑

h∈H/K

c+h,K

((xL)

L≤K

),

as relative trace, where the last summands are considered as elements in the k-module X+(H) by filling the missing components with zero, and xL always denotesan element in X(L). It is easy to verify that these three maps are well defined andsatisfy the axioms (M1)–(M4) of Definition 1.1. Hence, (X+, c+, res+, ind+) is ak-Mackey functor on G.

For a morphism f : X → Y of k-conjugation functors on G we define

f+H : X+(H)→ Y +(H), (xK)

K≤H 7→(fK(xK)

)K≤H

,

for H ≤ G. Obviously f+H is well defined and f+ = (f+

H )H≤G is a morphism of

k-Mackey functors on G. Thus, the definition of the functor −+ : k−Con(G) →k−Mack(G) is complete.

We note that if X ∈ k−Conalg(G), then X+(H) is a k-subalgebra of the directproduct k-algebra

∏K≤H X(K) for all H ≤ G, and X+ is a k-Green functor on

G. Moreover, if f : X → Y is a morphism of k-algebra conjugation functors on G,then f+ is a morphism of k-Green functors on G. Hence we also obtain a functor−+ : k−Conalg(G)→ k−Mackalg(G) such that the diagram

k−Conalg(G)−+

−−−−→ k−Mackalg(G)y yk−Con(G)

−+

−−−−→ k−Mack(G)


is commutative, where the vertical arrows are the forgetful functors.

2.2 The construction of the functor −+ : k−Res(G) → k−Mack(G) is tosome extent dual to the construction of −+, although one needs a restriction functorto start with. This construction was introduced in [Bo89, III.2] where proofs forthe statements below can be found.

For (A, c, res) ∈ k−Res(G) we set

SA(H) :=⊕K≤H

A(K)

for H ≤ G. Note that, as k-modules, SA(H) and PA(H) are isomorphic. Nev-ertheless we prefer this formal distinction between direct sum and direct product,because of the dual nature of the construction, and also because in the algebra casewe will construct a multiplication which does not come from the direct product ofthe algebras A(K) as in (2.1). As before, the normalizer NG(H) acts on SA(H) by

n(aK) := cn,K(a) ∈ A( nK)

for n ∈ NG(H), K ≤ H, aK ∈ A(K). In contrast to the previous constructionwe now consider the H-cofixed points SA(H)H of the kH-module SA(H), i.e. thebiggest quotient of SA(H) by a kH-submodule that admits trivial H-action. Moreexplicitly, we define

A+(H) := SA(H)H := SA(H)/I(kH) · SA(H), H ≤ G,

where I(kH) is the augmentation ideal ∑

h∈H αhh |∑

h∈H αh = 0 of kG. Then

I(kH) ·SA(H) is the k-submodule of SA(H) generated by the elements aK − h(aK)for K ≤ H, aK ∈ A(K), h ∈ H. Clearly A+(H) is generated as a k-module,even as abelian group, by elements of the form aK + I(kH) · SA(H), with K ≤ H,aK ∈ A(K), and we introduce the following notation:

[K, a]H := a+ I(kH) · SA(H) for K ≤ H, a ∈ A(K).

In particular the symbol [K, a]H indicates that K ≤ H and a ∈ A(K). Obviouslywe have [ hK, ha]H = [K, a]H for K ≤ H, a ∈ A(K), h ∈ H. If RH is a setof representatives for the conjugacy classes of subgroups of H, we can write eachelement of A+(H) in the form∑

K∈RH

[K, aK ]H , aK ∈ A(K),

with the ambiguity∑K∈RH

[K, aK ]H =∑

K∈RH

[K, a′K ]H ⇐⇒ a′K = nK(aK) for all K ∈ RH and

some nK ∈ NH(K).

The conjugation maps of A induce conjugation maps on A+:

c+g,H : A+(H)→ A+( gH), [K, a]H 7→ [ gK, ga] gH ,


for H ≤ G, g ∈ G. For K ≤ H ≤ G we define a restriction map

res+HK : A+(H)→ A+(K), [L, a]H 7→

∑h∈K\H/L

[K ∩ hL, res

hL

K∩ hL( ha)

]K,

and an induction map

ind+HK : A+(K)→ A+(H), [L, a]K 7→ [L, a]H .

A straight forward (but not very short) verification shows that (A+, c+, res+, ind+)is a k-Mackey functor on G.

For a morphism f : A→ B of two k-restriction functors on G we define

f+H : A+(H)→ B+(H), [K, a]H 7→[K, fK(a)

]H, H ≤ G.

The family f+ = (f+H)H≤G then forms a morphism of k-Mackey functors on G andthis completes the definition of the functor −+ : k−Res(G)→ k−Mack(G).

If A is in k−Resalg(G), then we can define a k-algebra structure on A+(H) forH ≤ G by

[K, a]H · [L, b]H :=∑

h∈K\H/L

[K ∩ hL, resK

K∩ hL(a) · res

hL

K∩ hL( hb)

]H.

The element [H, 1A(H)]H is the unity in A+(H). It is easy to show that A+ is ak-Green functor on G. If A(H) is commutative for all H ≤ G, then the same istrue for A+(H), H ≤ G. Furthermore, if f : A → B is a morphism of k-algebrarestriction functors on G, then f+ : A+ → B+ is a morphism of k-Green functorson G, and we obtain a functor −+ : k−Resalg(G)→ k−Mackalg(G) which rendersthe diagram

k−Resalg(G)−+−−−−→ k−Mackalg(G)y y

k−Res(G)−+−−−−→ k−Mack(G)

commutative, where the vertical arrows are the forgetful functors.If A ∈ k−Resalg(G) is commutative, then A+(H) is an A(H)-algebra and in

particular an A(H)-module via the ring homomorphism A(H) → A+(H), a 7→[H, a]H , for H ≤ G.

2.3 Example We consider the constant Z-algebra restriction functor Z onG with Z(H) := Z for H ≤ G, and with conjugation and restriction maps beingthe identity. Then Z+ and the Burnside ring functor Ω are isomorphic as Z-Greenfunctors on G by the maps

fH : Ω(H)∼−→Z+(H), [H/K] 7→ [K, 1]H , H ≤ G.

On the other hand we have the Z-Green functor Z+ on G with Z(H) = (∏K≤H Z)H

for H ≤ G, and there are the well-known mark homomorphisms (cf. Appendix A)ρH : Ω(H)→ Z+(H), H ≤ G, which form a morphism of Z-Green functors on G, as


one can check easily. Identifying Ω(H) with Z+(H) for H ≤ G by the isomorphismfH we obtain the morphism

ρH : Z+(H)→ Z+(H), [K, 1]H 7→(|(H/K)U |

)U≤H , H ≤ G,

of Z-Green functors on G.In the same way we obtain from the constant k-algebra restriction functor k

the Burnside ring functor k ⊗ Ω over k with the same identification, and also themaps ρH , H ≤ G, generalize to a morphism ρH : k+ → k+ of k-Green functors. InSection 3 we will generalize this ‘mark homomorphism’ ρH , which is a main tool forthe study of the Burnside ring, to a morphism ρA : A+ → A+ of k-Mackey functors(resp. k-Green functors) on G for an arbitrary k-restriction functor (resp. k-algebrarestriction functor) A on G.

In Chapter II we will have to restrict our attention to k-restriction fuctors A onG with the property that A+(H) is k-free for all H ≤ G, since then certain mapsrelated to scalar extensions are injective (cf. Section 5). The following propositiongives a sufficient condition for this to hold.

We say that A ∈ k−Res(G) has a conjugation-stable basis (or just a stable

basis), if there exist k-bases B(H) of A(H), for H ≤ G, such that cg,H(B(H)) =B( gH) for H ≤ G, g ∈ G. Note that this is equivalent to the existence of a k-basisB(H) of A(H), where H runs through a set of representatives of the conjugacyclasses of subgroups of G, such that cn,H(B(H)) = B(H) for n ∈ NG(H), since suchbases can be transported to all A(H), H ≤ G, by the conjugation maps. For astable basis B(H) ⊆ A(H), H ≤ G, and a given subgroup H of G the disjoint union

BH :=.⋃K≤HB(K) is an H-set by the definition hb := ch,K(b) for h ∈ H, K ≤ H,

b ∈ B(K). Instead of BH we might as well consider the isomorphic H-set consistingof all pairs (K, b) with K ≤ H, b ∈ B(K), where H acts on the second componentas before and on the first component by conjugation.

2.4 Proposition Let A ∈ k−Res(G) with stable basis B(H) ⊆ A(H) for

H ≤ G. For H ≤ G let BH :=.⋃K≤HB(K). Then any set RH of representatives

for the H-orbits of the H-set of pairs (K, b), K ≤ H, b ∈ B(K), gives rise to ak-basis [K, b]H | (K, b) ∈ RH of A+(H).

Proof For H ≤ G let B′H ⊆ BH be the set of representatives for the H-orbits of BH which corresponds to RH . Clearly BH is a k-basis of the k-moduleSA(H) =

⊕K≤H A(K) and we have the following decomposition into k-submodules

SA(H) = kB′H ⊕ kb− hb | b ∈ B′H , h ∈ H.

It is obvious that the second summand equals I(kH) ·SA(H), and therefore A+(H)is free with basis [K, b]H , (K, b) ∈ RH .

2.5 For a k-Mackey functor M on G and a subgroup H ≤ G we consider thek-submodule

I(M)(H) :=∑K<H

indHK(M(K)) =∑K<H

im(indHK : M(K)→M(H)

)


of M(H). The G-equivariance of induction, cf. (M3), implies that I(M) is a k-conjugation subfunctor of M on G. Since morphisms of k-Mackey functors commutewith induction, these submodules are preserved under such morphisms, and weobtain a functor

I : k−Mack(G)→ k−Con(G).

Since the restriction maps don’t play a role in this construction, we could havestarted with an induction functor instead of a Mackey functor (cf. Remark 1.8 (iv)).Note that if M ∈ k−Mackalg(G), then I(M)(H) is an ideal in M(H) for H ≤ Gby the Frobenius axiom (M6).

Let M ∈ k−Mack(G). Following Thevenaz (cf. [Th88]) we call a subgroup Hof G primordial for M , if I(M)(H) 6= M(H), i.e. H is not primordial for M , if eachelement of M(H) can be obtained as a sum of properly induced elements. For thecharacter ring functor R for instance, the primordial subgroups are precisely theelementary subgroups (cf. [Se78, 11.3]). We denote the set of primordial subgroupsfor M by P(M). More about primordial subgroups can be found in [Th88]. Notethat we have

M(H) =∑K≤H

K∈P(M)

indHK(M(K))

for H ≤ G.

2.6 We dualize the construction in 2.5. For a k-restriction functor A on Gand a subgroup H ≤ G we consider the k-submodule

K(A)(H) :=⋂K<H

ker(resHK : A(H)→ A(K)

)of A(H). The k-submodules K(A)(H), H ≤ G, form a k-conjugation subfunctor ofA and they are preserved under morphisms of k-restriction functors on G. Hence,we obtain a functor

K : k−Res(G)→ k−Con(G).

Note that if A ∈ k−Resalg(G), then K(A)(H) is an ideal of A(H) for H ≤ G, sincethe restriction maps are k-algebra homomorphisms. Note also that K(A) is evena k-restriction subfunctor of A on G with trivial restrictions resHK , K < H ≤ G,but because of the triviality of the restrictions we prefer to view K only as a k-conjugation functor on G.

Let A ∈ k−Res(G). A subgroup H of G is called coprimordial for A (cf. [Bo89,II.1.9]), if K(A)(H) 6= 0, i.e. H is not coprimordial for A, if the elements of A(H)are separated by proper restrictions. For the character ring functor for instance,the coprimordial subgroups are precisely the cyclic subgroups. We denote the setof coprimordial subgroups for A by C(A). More about coprimordial subgroups andthe connection to primordial subgroups can be found in [Bo89, III.1.11-1.18]. Notethat for H ≤ G two elements x, y ∈ A(H) are equal if and only if resHK(x) = resHK(y)for all K ≤ H with K ≤ C(A).

2.7 For a k-Mackey functor M on G and H ≤ G the k-module

M(H) := M(H)/I(M)(H) = M(H)/∑K<H

indHK(M(K))


is called the residue of M at H. This notion was studied extensively by Puig in[Pu88] in the case of various representation rings and by Thevenaz in [Th88] inthe general case. Since morphisms of k-Mackey functors on G induce maps on theresidues, we obtain a functor ? : k−Mack(G) → k−Con(G). Since I(M)(H) isan ideal of M(H) for M ∈ k−Mackalg(G) and H ≤ G, we also obtain a functor? : k−Mackalg(G)→ k−Conalg(G) which renders the diagram

k−Mackalg(G)?−−−−→ k−Conalg(G)y y

k−Mack(G)?−−−−→ k−Con(G)

commutative, where the vertical arrows denote the forgetful functors.

2.8 In [Th88] Thevenaz defined the twin functor T (M) of a k-Mackey functor

M on G by T (M) := M+ ∈ k−Mack(G). This construction defines functors T

which are given as the composite functors

T : k−Mack(G)?−→ k−Con(G)

−+

−→ k−Mack(G)

and

T : k−Mackalg(G)?−→ k−Conalg(G)

−+

−→ k−Mackalg(G)

of the functors defined in (2.1) and (2.7) such that the diagram

k−Mackalg(G)T−−−−→ k−Mackalg(G)y y

k−Mack(G)T−−−−→ k−Mack(G)

is commutative, where the vertical arrows are the forgetful functors. FurthermoreThevenaz defined for M ∈ k−Mack(G) (resp. M ∈ k−Mackalg(G)) morphismsβM : M → T (M) of k-Mackey functors (resp. k-Green functors) on G by

βMH : M(H)→M+

(H), m 7→(resHK(m) + I(M)(K)

)K≤H ,

for H ≤ G, and showed that ker(βMH ) and coker(βMH ) are annihilated by a power of|H| (cf. [Th88, 3.2 and 12.2]). Moreover, the morphisms βM , for M ∈ k−Mack(G)(resp. M ∈ k−Mackalg(G)), form a natural transformation between the functorsId, T : k−Mack(G) → k−Mack(G) (resp. the functors Id, T : k−Mackalg(G) →k−Mackalg(G)).

1.3 The mark morphism

In this section we are going to generalize the mark homomorphism ρG : Ω(G) ∼=Z+(G)→ Z+(G) of the Burnside ring (cf. Appendix A and Example 2.3).

1.3. THE MARK MORPHISM 31

3.1 For A ∈ k−Res(G) (resp. A ∈ k−Resalg(G)) and H ≤ G, the inclusionA(H)→

⊕K≤H A(K) and the projection

⊕K≤H A(K)→ A(H) induce maps

ιAH : A(H)→ A+(H), a 7→ [H, a]H ,

and

πAH : A+(H)→ A(H), [K, a]H 7→a, if K = H,0, if K < H.

The maps ιAH , H ≤ G, are injective by axiom (R1) and form a morphism ιA : A→A+ of k-restriction functors (resp. k-algebra restriction functors) on G, which isnatural in A. The maps πAH , H ≤ G, are well-defined by axiom (R1). Theywill be called the Brauer maps (cf. [Th88, Section 1]), since πAH is surjective andvanishes exactly on elements which are induced from proper subgroups of H (cf. thedefinition of induction for A+). The Brauer maps form a morphism πA : A+ → A ofk-conjugation functors (resp. k-algebra conjugation functors) on G, which is naturalin A and a splitting morphism for ιA : A → A+, i.e. πA ιA = idA. Moreover theBrauer maps induce an isomorphism A+ → A of k-conjugation functors (resp. k-algebra conjugation functors) which is natural in A. Therefore (cf. 2.7) we callπAH(x) ∈ A(H) the residue of x for x ∈ A+(H), H ≤ G. Note that T (A+) =

A++ ∼= A+, i.e. A+ is the twin functor of A+, cf. 2.8.Using the Brauer maps we define the mark homomorphism:

ρAH :=(πAK res+

HK

)K≤H : A+(H)→ A+(H),

for H ≤ G. Note that ρAH takes values in the H-fixed points, since πAK and res+HK

commute with conjugations for K ≤ H ≤ G. Again it is easy to verify that themark homomorphisms ρAH , H ≤ G, form a morphism ρA : A+ → A+ of k-Mackeyfunctors (resp. k-Green functors) on G, which we call the mark morphism. Withthe above identification T (A+) ∼= A+, ρA : A+ → A+ coincides with Thevenaz’morphism βA+ (cf. [Th88, Section 3] and (2.8)). Note that ρA is natural in A,i.e. for A,B ∈ k−Res(G) (resp. A,B ∈ k−Resalg(G)) and f ∈ k−Res(G)(A,B)(resp. f ∈ k−Resalg(G)(A,B)) the following diagram is commutative:

A+f+−−−−→ B+

ρAy yρBA+ f+

−−−−→ B+.

If in particular, A = k as in Example 2.3, and f : k → B is the unique morphismof k-algebra restriction functors, namely fH(x) = x · 1B(H) for H ≤ G and x ∈ k,

then ρA is just the family of mark homomorphisms of the Burnside rings k⊗Ω(H),H ≤ G, f+ is the unique element in k−Mackalg(G)(k ⊗ Ω, B+), and the markmorphism ρB extends the mark morphism of the Burnside ring.

Next we investigate in which circumstances ρA is injective or even an isomor-phism. For H ≤ G we define a map

σAH : A+(H)→ A+(H), (aK)K≤H 7→

∑L≤K≤H

|L|µ(L,K)[L, resKL (aK)

]H,


where µ denotes the Mobius function on the poset of subgroups of G (cf. AppendixB).

The following proposition shows that σAH is almost an inverse of ρAH for H ≤ G.This property together with the explicit definition of σAH will be used in Chapter II togive explicit induction formulae. For the reader’s convenience we give a reformulatedproof from [Bo89, Prop. II.2.13] in our notation.

3.2 Proposition For A ∈ k−Res(G) (resp. A ∈ k−Resalg(G)) and H ≤ Gwe have

σAH ρAH = |H| · idA+(H) and ρAH σAH = |H| · idA+(H).

In particular the kernels and cokernels of ρAH and σAH are annihilated by |H|. If |G|is invertible in k, then ρA is an isomorphism of k-Mackey functors (resp. k-Greenfunctors) on G with inverse

(|H|−1 · σAH

)H≤G

. If, for H ≤ G, A+(H) has trivial

|H|-torsion, then ρAH is injective.

Proof For U ≤ H and a ∈ A(U) we have

ρAH([U, a]H

)=

((πAK res+

HK)([U, a]H)

)K≤H

=( ∑h∈K\H/U

πAK([K ∩ hU, res

hU

K∩ hU( ha)

]K

))K≤H

=( ∑h∈H/UK≤hU

reshUK ( ha)

)K≤H

,

since for K ≤ H the relation K ≤ hU implies KhU = hU . Applying σAH to thisfamily we obtain

(σAH ρAH)([U, a]H

)=

∑L≤K≤H

|L|µ(L,K)∑h∈H/UK≤hU

[L, res

hUL ( ha)

]H

=1

|U |∑h∈H

∑L≤K≤ hU

µ(L,K) |L|[L, res

hUL ( ha)

]H,

and considering for h ∈ H the map

f : L ≤ hU → A+(H), L 7→ |L|[L, res

hUL ( ha)

]H,

the inner sum collapses by Mobius inversion (see Proposition B.2 (i) in Appendix B)to | hU |[ hU, ha]H = |U |[U, a]H which gives

(σAH ρAH)([U, a]H

)=

1

|U |∑h∈H|U | [U, a]H = |H| [U, a]H .

Conversely let (aK)K≤H ∈ A+(H). Then the U -component, U ≤ H, of the

1.4. ADJOINTNESS PROPERTIES 33

element (ρAH σAH)((aK)

K≤H

)∈ A+(H) is given by(

πAU res+HU σH

)((aK)

K≤H

)=

∑L≤K≤H

|L|µ(L,K)πAU

(res+

HU

([L, resKL (aK)

]H

))=

∑L≤K≤H

|L|µ(L,K)∑h∈H/LU≤hL

reshLU

(h(

resKL (aK)))

=∑

L≤K≤Hµ(L,K)

∑h∈HU≤hL

reshKU

( h(aK))

=∑h∈H

∑Uh≤L≤K≤H

µ(L,K) reshKU ( h(aK)).

Considering for each h ∈ H the map f : Uh ≤ K ≤ H → A(U), K 7→ reshKU ( haK),

the inner sum collapses by Mobius inversion (see Proposition B.2 (ii) in AppendixB) to ha(Uh) = aU , and we have(

πAU res+HU σ

AH

)((aK)

K≤H

)=∑h∈U

aU = |H| aU ,

which completes the proof of the two equations. The other assertions of the propo-sition now follow immediately.

3.3 Remark (i) Note that the above proposition reproves the formula ofGluck (cf. Appendix A) for the primitive idempotents of Q ⊗ Ω(G) (and also ofk ⊗ Ω(G), if |G| is invertible in k),

e(G)H =

1

|NG(H)|∑K≤H

|K|µ(K,H) [G/K],

for H ≤ G.

(ii) If A+(H) has no |H|-torsion for all H ≤ G, then it follows immediatelyfrom Proposition 3.2 that with ρA also

((G : H) · σAH

)H≤G is a morphism of k-

Mackey functors (resp. k-Green functors) on G. We are convinced that this is truein general, but we don’t see any short proof. Since we will never need this result,it does no harm to leave the general case open.

1.4 Adjointness properties

The following adjointness properties illustrate the significance of the constructions−+ and −+. They will both be used in Chapter II for the construction of canonicalinduction formulae.

4.1 Proposition (i) The forgetful functor k−Mack(G)→ k−Res(G) is rightadjoint to −+ : k−Res(G) → k−Mack(G). More precisely, for A ∈ k−Res(G),


M ∈ k−Mack(G) we have inverse k-linear isomorphisms

κA,M : k−Mack(G)(A+,M) → k−Res(G)(A,M),

(fH)H≤G 7→ (fH ιAH)

H≤G ,([K, a]H 7→ indHK

(gK(a)

))H≤G

←7 (gH)H≤G ,

of k-modules which are natural in A and M . The same maps yield inverse natural k-linear isomorphisms between k−Mackalg(G)(A+,M) and k−Resalg(G)(A,M) forA ∈ k−Resalg(G) and M ∈ k−Mackalg(G).

(ii) The forgetful functor k−Res(G) → k−Con(G) is left adjoint to the com-position of the functor −+ : k−Con(G) → k−Mack(G) and the forgetful func-tor k−Mack(G) → k−Res(G). More precisely, for A ∈ k−Res(G) and X ∈k−Con(G) we have k-linear inverse isomorphisms

λA,X : k−Res(G)(A,X+) → k−Con(G)(A,X),

(fH)H≤G 7→ (prXH fH)

H≤G ,(A(H) 3 a 7→

(gK(resHK(a))

)K≤H

)H≤G

←7 (gH)H≤G ,

of k-modules which are natural in A and X, and where prXH : (∏K≤H X(K))H =

X+(H) → X(H) denotes the projection map onto the H-component. The samemaps yield inverse natural k-linear isomorphisms between k−Resalg(G)(A,X+) andk−Conalg(G)(A,X) for A ∈ k−Resalg(G) and X ∈ k−Conalg(G).

Proof All assertions in (i) and (ii) are immediate consequences of the verydefinitions of the categories and functors involved. The verifications are withoutexception straight forward.

Let A,B ∈ k−Res(G) and assume that for all H ≤ G, B+(H) has trivial |H|-torsion. Then

((G : H) · σBH

)H≤G

: B+ → B+ is a morphism of k-Mackey functors

on G (cf. Remark 3.3 (ii)), and we obtain k-homomorphisms

σA,B : k−Res(G)(A,B+) → k−Res(G)(A,B+),

(fH)H≤G 7→

((G : H) · σBH fH

)H≤G

,

and

ρA,B : k−Res(G)(A,B+)→ k−Res(G)(A,B+), f 7→ ρB f,

with σA,B ρA,B = id and ρA,B σA,B = id by Proposition 3.2. The compositionθA,B := λA,B ρA,B with λA,B from Proposition 4.1 and X replaced with B is thek-homomorphism

θA,B : k−Res(G)(A,B+)→ k−Con(G)(A,B), f 7→ πB f,

since prBH ρBH = πBH for H ≤ G. We call θA,B(f) = πB f the residue of f ∈k−Res(G)(A,B+); cf. 2.6 and 3.1 for the terminology.

Note that throughout the preceding paragraph we can replace k−Res(G) byk−Resalg(G) and k−Con(G) by k−Conalg(G).

1.4. ADJOINTNESS PROPERTIES 35

In Chapter II we will be interested in elements of k−Res(G)(A,B+) for A,B ∈k−Res(G). The following corollary allows us to analyze such elements by theirresidues.

4.2 Corollary Let A,B ∈ k−Res(G).

(i) Let f ∈ k−Res(G)(A,B+) and let g := θA,B(f) = πBf ∈ k−Con(G)(A,B)be the residue of f . Then the diagrams

A(H)fH−−→ B+(H)

(gKresHK)K≤H

yρBH=(πBKres+HK)

K≤H

B+(H),

H ≤ G, are commutative.

(ii) If B+(H) has trivial |G|-torsion for all H ≤ G, then θA,B is injective andfor f ∈ k−Res(G)(A,B+) we have

|G| · f =(σA,B λ−1

A,B

)(θA,B(f)

).

In particular, for H ≤ G and a ∈ A(H), we have

|H| · fH(a) =∑

L≤K≤H|L|µ(L,K)

[L, resKL

(θA,B(f)K(resHK(a))

)]H,

where µ denotes the Mobius function of the poset of subgroups of G.

(iii) If |G|−1 ∈ k, then θA,B is an isomorphism with inverse |G|−1 ·(σA,B λ−1A,B).

In particular, for g ∈ k−Con(G)(A,B), H ≤ G, and a ∈ A(H), we have

θ−1A,B(g)H(a) =

1

|H|∑

L≤K≤H|L|µ(L,K)

[L, resHK

(gK(resHK(a))

)]H,

where µ denotes the Mobius function of the poset of subgroups of G.

(iv) Assume that A and B are k-algebra restriction functors on G and thatρB is injective. Let f ∈ k−Res(G)(A,B+) and let g := θA,B(f) = πB f ∈k−Con(G)(A,B) be the residue of f . Then f ∈ k−Resalg(G)(A,B+) if and onlyif g ∈ k−Conalg(G)(A,B).

Proof (i) For K ≤ H ≤ G we have πBK res+HK fH = πBK fK resHK .

(ii) θA,B = λA,B ρA,B is injective, since λA,B is an isomorphism by Proposition4.1 (ii) and ρA,B is injective by the relation σA,B ρA,B = |G| · id. The two equationsfollow again from σA,B ρA,B = |G| · id and the definition of σBH , H ≤ G, from (3.1).

(iii) θA,B is an isomorphism, since λA,B and ρA,B are isomorphisms by Proposi-tion 4.1 (ii) and the relations ρA,B σA,B = |G| · id and σA,B ρA,B = |G| · id. Theexplicit equation for θ−1

A,B is immediate from part (i).

(iv) Let f ∈ k−Resalg(G)(A,B+). Since πBH is a k-algebra homomorphism forH ≤ G, so is gH = πBH fH . Conversely, let g ∈ k−Conalg(G)(A,B). Since ρBH isinjective and a k-algebra homomorphism for H ≤ G, it suffices to show that ρBH fHis a k-algebra homomorphism. But this follows from the diagram in part (i), sincerestrictions are k-algebra homomorphisms.


1.5 Scalar extension

In Proposition 3.2 and Corollary 4.2 we have seen that it is of advantage to have|G| invertible in k. We can place ourselves in this situation by considering the lo-calization k[|G|−1] of k at the multiplicative set 1, |G|, |G|2, . . . ⊆ k and assumingthat |G| is not a zero divisor in k, so that we also may consider k as a subring ofk[|G|−1]. Therefore we will have a closer look on the functor of scalar extension.Let k be a subring (with the same unity) of the commutative ring k′ throughoutthis section.

For X ∈ k−Con(G) (resp. X ∈ k−Conalg(G)) there is a canoncial morphism(i.e. natural in X) of k′-Mackey functors (resp. k′-Green functors) on G,

k′ ⊗k X+ → (k′ ⊗k X)+,

which maps α⊗k (xK)K≤H ∈ k′ ⊗k X+(H) to (α⊗k xK)

K≤H ∈ (k′ ⊗k X)+(H), forH ≤ G. Moreover there is a canonical morphism of k-Mackey functors (resp. k-Green functors) on G, X+ → k′ ⊗k X+, which maps (xK)

K≤H ∈ X+ for H ≤ G to1⊗k (xK)

K≤H ∈ k′ ⊗k X+(H).

For A ∈ k−Res(G) (resp. A ∈ k−Resalg(G)) there is a canonical morphism(i.e. natural in A) of k′-Mackey functors (resp. k′-Green functors) on G,

k′ ⊗k A+ → (k′ ⊗k A)+,

which maps α⊗k(a+I(kH)·SA(H)) ∈ k′⊗kA+(H) to (α⊗ka)+I(k′H)·Sk′⊗kA(H) ∈(k′ ⊗k A)+(H), for H ≤ G. Moreover there is a canonical morphism (natural in A)of k-Mackey functors (resp. k-Green functors) on G, A+ → k′ ⊗k A+, which mapsa+ I(kH) ·SA(H) ∈ A+(H) for H ≤ G to 1⊗k (a+ I(kH) ·SA(H)) ∈ k′⊗kA+(H).

It is easy to see that the diagram

A+ −−−−→ k′ ⊗k A+ −−−−→ (k′ ⊗k A)+

ρAy k′⊗kρA

y yρk′⊗kAA+ −−−−→ k′ ⊗k A+ −−−−→ (k′ ⊗k A)+

(1.1)

is commutative.

5.1 Lemma Let k ⊆ k′ be commutative rings with the same unity, X ∈k−Con(G), and A ∈ k−Res(G).

(i) If k′ is k-flat, then the canonical morphism k′ ⊗k X+ → (k′ ⊗k X)+ is anisomorphism.

(ii) The canonical morphism k′ ⊗k A+ → (k′ ⊗k A)+ is always an isomorphism.

(iii) If k′ is k-flat and X(H) is k-free for all H ≤ G, then the canonical morphismX+ → k′ ⊗k X+ is injective.

(iv) If A+(H) is k-free for all H ≤ G, then the canonical morphism A+ →k′ ⊗k A+ is injective. In particular, if A has a stable basis, then A+ → k′ ⊗k A+ isinjective.

Proof (i) Let M be any kG-module. The k-module MG of G-fixed points isnaturally isomorphic to HomkG(k,M), where k is the trivial kG-module. Moreover,

1.6. INVERTIBLE GROUP ORDER 37

k′ ∼= Homk′(k′, k′), and we obtain a sequence of natural isomorphisms

k′ ⊗kMG ∼= Homk′(k′, k′)⊗k HomkG(k,M) ∼= Homk′⊗kkG(k′ ⊗k k, k′ ⊗kM)

∼= Homk′G(k′, k′ ⊗kM) ∼= (k′ ⊗kM)G,

where the second map, given by f ⊗k g 7→ f ⊗k g for f ∈ Homk′(k′, k′), g ∈

HomkG(k,M), is an isomomorphism, since k′ is k-flat and k is finitely presentedas kG-module (cf. [KO74, Lemma I.4.1 (b)]). The resulting isomorphism mapsα⊗k m ∈ k′ ⊗kMG to α⊗k m ∈ (k′ ⊗kM)G. If we set M = PX(H) and replace Gwith H, for H ≤ G, these isomorphisms form the canonical morphism k′ ⊗k X+ →(k′ ⊗k X)+.

(ii) Let M be any kG-module. The k-module MG of G-cofixed points (which aredefined as MG := M/IM , where the IM := I(kG) ·M = m− gm | m ∈M , g ∈ G)is naturally isomorphic to k ⊗kGM , where k is the trivial kG-module. We obtaina sequence of natural isomorphisms

k′ ⊗kMG∼= k′ ⊗k (k ⊗kGM) ∼= k′ ⊗k′G (k′ ⊗kM) ∼= (k′ ⊗kM)G,

where the middle isomorphism is given by α′⊗k (α⊗kGm) 7→ α′⊗k′G (α⊗km) withinverse α′ ⊗k′G (β′ ⊗k m) 7→ α′β′ ⊗k (1⊗kG m) for α ∈ k, α′, β′ ∈ k′, m ∈ M . Theresulting isomorphism maps α′⊗k(m+IM ) to (α′⊗km)+Ik′⊗kM for α′ ∈ k′, m ∈M .If we set M = SA(H) and replace G with H, for H ≤ G, these isomorphisms formthe canonical morphism k′ ⊗k A+ → (A′ ⊗k A)+.

(iii) This follows from the commutativity of the diagram

X+(H) ∼= k ⊗k( ∏K≤H

X(K))H−−−−→ k′ ⊗k

( ∏K≤H

X(K))H

= k′ ⊗k X+(H)y yk ⊗k

∏K≤H

X(K) −−−−→ k′ ⊗k∏K≤H

X(K)

by the injectivity of the vertical maps (k′ is k-flat) and the injectivity of the lowerhorizontal map (

∏K≤H X(K) is k-flat and k → k′ is injective), for H ≤ G.

(iv) The first statement is obvious and the second follows from Proposition 2.4.

5.2 Corollary Let k ⊆ k′ be as in Lemma 5.1 and let A ∈ k−Res(G). If Ahas a stable basis and k′ is k-flat, then the left horizontal morphisms in diagram (1.1)are injective, the right horizontal morphisms are isomorphisms, and the verticalmorphisms are injective (cf. Propositions 2.4 and 3.2.)

1.6 Invertible group order

In this section we prove miscellaneous results about k-Mackey functors on G in thecase that |G| is invertible in k.

6.1 Let |G| be invertible in k. We identify the Burnside ring functor k⊗Ω withk+ as in Example 2.3. Since |G| is invertible in k we know from Proposition 3.2 that


the mark morphism ρk : k+ → k+ is an isomorphism of k-Green functors on G. ForH ≤ G and for any set RH of representatives for the conjugacy classes of subgroups

of H, the set e(H)K | K ∈ RH of mutually orthogonal primitive idempotents of

k ⊗ Ω(H), whose sum is the unity element, is also a k-basis of k ⊗ Ω(H). FromRemark 3.3 (i) we recall the explicit formula

e(H)K =

1

|NH(K)|∑L≤K

|L|µ(L,K) [H/L], (1.2)

for K ≤ H (see also Appendix A).In the sequel we will use for any k-Mackey functor M the canonical pairing

(k ⊗ Ω)⊗kM →M, [H/K]⊗k m 7→ indHK(resHK(m)),

for K ≤ H ≤ G, m ∈M(H), which was defined in Proposition 1.4.

6.2 Proposition ([Dr73, Theorem 2], [Yo83a, Theorem 4.1′]) Let |G| be in-vertible in k and let M ∈ k−Mack(G) (resp. M ∈ k−Mackalg(G)). For H ≤ Gwe have a decomposition into k-submodules (resp. ideals)

M(H) = e(H)H ·M(H)⊕ (1− e(H)

H ) ·M(H)

and the summands are given by (cf. 2.5 and 2.6)

e(H)H ·M(H) = K(M)(H)

(=⋂K<H

ker(resHK : M(H)→M(K)

))and

(1− e(H)H ) ·M(H) = I(M)(H)

(=∑K<H

im(indHK : M(K)→M(H)

)).

In particular, M decomposes as k-conjugation functor (resp. k-algebra conjugationfunctor) on G into M = K(M)⊕ I(M), and C(M) = P(M).

Moreover, for H ≤ G we have H ∈ C(M) if and only if e(H)H ·M(H) 6= 0, and

for an inclusion M ⊆ N of k-Green functors on G we have C(M) = C(N).

Proof Since the pairing (k⊗Ω)⊗kM →M provides M(H) with a k⊗Ω(H)-

module structure, we clearly have the decomposition M(H) = e(H)H ·M(H)⊕ (1−

e(H)H ) ·M(H) for H ≤ G, and the summands are ideals in M(H), if M is a k-Green

functor on G.For the proof of the remaining equations we first claim that for K < H ≤ G we

have resHK(e(H)H ) = 0. In fact, it is enough to show that (ρ

kK resHK)(e

(H)H ) ∈ k+(K) is

trivial. But ρkK commutes with restrictions, and the definition of e

(H)H via its image

under ρkH together with the definition of restriction for k+ proves the claim.

Using this we see that resHK(e(H)H ·M(H)) = resHK(e

(H)H ) · resHK(M(H)) = 0 for all

K < H ≤ G. Conversely, the explicit formula (1.2) yields

1− e(H)H = − 1

|H|∑K<H

|K|µ(K,H) [H/K],


and the definition of the canonical pairing shows that⋂K<H ker

(resHK : M(H) →

M(K))

is annihilated by 1 − e(H)H , and therefore contained in e

(H)H ·M(H) by the

decomposition of M(H). This proves the first remaining equation.

Again the definition of the canonical pairing and the above formula for 1− e(H)H

shows that (1−e(H)H )·M(H) is contained in I(M)(H) for H ≤ G. Conversely, for the

inclusion I(M)(H) ⊆ (1−e(H)H )·M(H), it suffices to show that e

(H)H ·indHK(M(K)) =

0 for K < H ≤ G. But this is immediate from the Frobenius axiom (P3) inDefinition 1.3 and the claim above:

e(H)H · indHK(M(K)) = indHK

(resHK(e

(H)H ) ·M(K)

)= 0.

Now it follows immediately that M decomposes into M = K(M) ⊕ I(M) ask-conjugation functor on G, that C(M) = P(M), and that H ≤ C(M) if and only

if e(H)H ·M(H) 6= 0.If M ⊆ N is an inclusion of k-Green functors on G and H ≤ G, then

H ∈ C(M) ⇐⇒ e(H)H ·M(H) 6= 0 ⇐⇒ e

(H)H · 1M(H) 6= 0

⇐⇒ e(H)H · 1N(H) 6= 0 ⇐⇒ e

(H)H ·N(H) 6= 0

⇐⇒ H ∈ C(N).

This completes the proof.

6.3 Corollary Let |G| be invertible in k, M ∈ k−Mack(G), H ≤ G, and

m ∈ M(H). Then m = 0 if and only if e(H)H · m = 0 and resHK(m) = 0 for all

K < H.

Proof One implication is trivial. So we assume that e(H)H · m = 0 and

resHK(m) = 0 for all K < H. Then by Proposition 6.2, the element m lies in

(1−e(H)H )·M(H), since m is annihilated by e

(H)H , and simultaneously in K(M)(H) =

e(H)H ·M(H). This implies m = 0.

6.4 Proposition Let |G| be invertible in k, and let M be a k-Green functoron G. Furthermore, let i ∈ k−Mackalg(G)(k ⊗ Ω,M) be the unique morphism ofk-Green functors on G (cf. Proposition 1.4).

(i) Assume that H ≤ G is not contained in any H ′ ∈ C(M). Then iG(e(G)H ) = 0.

(ii) Assume that M(H) is k-torsion free for H ≤ G. Then, the elements

iG(e(G)H ), where H runs through a set of representatives for the G-conjugacy classes

of C(M), are k-linearly independent in M(G).

Proof Note that by Proposition 1.4 we have

iH(x) ·m = x ·m,

for H ≤ G, x ∈ k⊗Ω(H), and m ∈M(H), where the first dot denotes multiplicationin M(H), and the second dot denotes the canonical pairing.

(i) Let H ≤ G. Since P(M) = C(M) by Proposition 6.2, we have

1M(G) ∈∑

K∈C(M)

indGK(M(K)).


The Frobenius axiom (P3) in Definition 1.3 implies

iG(e(G)H ) = e

(G)H · 1M(G) ⊆

∑K∈C(M)

e(G)H · indGK(M(K))

=∑

K∈C(M)

indGK(resGK(e

(G)H ) ·M(K)

)= 0,

since resGK(e(G)H ) = 0 by (A.4) in Appendix A.

(ii) Let CG be a set of representatives for the G-conjugacy classes of coprimordialsubgroups for M , and assume that∑

H∈CG

αH · iG(e(G)H ) = 0

for elements αH ∈ k, H ∈ CG. If not all elements αH , H ∈ CG, are zero, we chooseH0 ∈ CG of minimal order such that αH0 6= 0. Applying resGH0

to the above equation,we obtain

0 =∑H∈CG

αH · iH0

(resGH0

(e(G)H )

)= αH0 · iH0(e

(H0)H0

)

by (A.4) in Appendix A and by the minimality of |H0|. Since M(H0) is k-torsion

free, this implies iH0(e(H0)H0

) = 0, and we obtain by Proposition 6.2 that

K(M)(H0) = e(H0)H0·M(H0) = iH0(e

(H0)H0

) ·M(H0) = 0,

which contradicts H0 ∈ C(M).

6.5 Proposition Let |G|−1 ∈ k. Then the functors −+ : k−Con(G) →k−Mack(G) and ?: k−Mack(G) → k−Con(G) are inverse equivalences. Thesame functors are inverse equivalences between k−Conalg(G) and k−Mackalg(G).

Proof It has already been mentioned in 2.8 that Thevenaz proved in [Th88]

that there is a natural isomorphism βM : M →M+

of k-Mackey functors (resp. k-Green functors) on G for M ∈ k−Mack(G) (resp. M ∈ k−Mackalg(G)).

Conversely, let X ∈ k−Con(G) (resp. X ∈ k−Conalg(G)). By Proposition 6.2we have a natural isomorphism X+ = X+/I(X+) ∼= K(X+) of k-conjugation func-tors (resp. k-algebra conjugation functors) on G. Moreover, the definition of therestriction maps for M+ implies that

K(X+)(H) =

(xK)K≤H ∈

∏K≤H

X(K)∣∣xK = 0 for K < H

for H ≤ G. Hence, K(X+)(H) ∼= X(H), and K(X+) is naturally isomorphic toX as k-conjugation functor (resp. k-algebra conjugation functor) on G. AltogetherX+ is naturally isomorphic to X in k−Con(G) (resp. k−Conalg(G)).

6.6 Remark Let k be an arbitrary commutative ring (without assumptionon |G|).


(i) Let RG be a set of representatives for the conjugacy classes of subgroups ofG. Then the functor

k−Con(G)→∏

H∈RG

k[NG(H)/H]−Mod

which associates to X ∈ k−Con(G) the family X(H), H ∈ RG, endowed withk[NG(H)/H]-module structures via the conjugation maps, is obviously an equiv-alence. Hence, the category k−Mack(G) is semisimple by Proposition 6.4 in thecase that |G| is invertible in k.

Similarly the category k−Conalg(G) is equivalent to the direct product of thecategories of NG(H)/H-algebras over k, H ≤ RG, i.e. k-algebras A together witha group homoorphism NG(H)/H → Aut(A). Hence, in the case of invertible grouporder, the category of k-Green functors on G is exactly as complicated as the variouscategories of NG(H)/H-algebras over k.

(ii) There is a functor F : k−Con(G) → k−Res(G) which defines restrictionmaps on a k-conjugation functor X on G by resHH = idX(H) and resHK = 0 forK < H ≤ G. The composition of F with the forgetful functor k−Res(G) →k−Con(G) is equal to the identity functor. But in general these functors arenot inverse equivalences. If we restrict our attention to the full subcategories ofk−Con(G) (resp. k−Res(G)) consisting of objects X (resp. A) with the propertythat X(H) (resp. A(H)) has a finite composition series as k-module, for H ≤G, then the Grothendieck groups of these two subcategories are isomorphic viathe maps induced by the functors defined above. In fact, they induce a bijectionbetween the isomorphism classes of simple objects. Note that X ∈ k−Con(G)(resp. A ∈ k−Res(G)) is simple if and only if there is some H ≤ G such the X(H)(resp. A(H)) is a simple kNG(H)/H-module and X(H ′) = 0 (resp. A(H ′) = 0) forall H ′ ≤ G which are not conjugate to H.


Chapter 2

Canonical Induction Formulae

This chapter is arranged as follows. In Section 1 we define the general notion of acanonical induction formula a : M → A+ for a Mackey functor M on a finite groupG from a restriction subfunctor A ⊆ M . Moreover the residue p := πA a of acanonical induction formula a is defined. In Section 2 we consider the case where|G| is invertible in the base ring k. In this case we have a bijection between theset of morphisms a : M → A+ of k-restriction functors and the set of morphismsp : M → A of k-conjugation functors which is given by a 7→ p := πA a. Notethat it is much easier to define a morphism p : M → A of conjugation functorsthan a morphism a : M → A+ of restriction functors. We relate the properties ofa to properties of p. In Section 3 we consider for k = Z in Hypothesis 3.1 thesituation where M is Z-free and A has a stable basis B whose positive span isstable under restriction. This condition which we also call the standard situationwill always be satisfied in the examples we are interested in, and it implies thatA+(H) is Z-free for H ≤ G. Hence, we may imbed M , A, and A+ into Q ⊗M ,Q ⊗ A, and Q ⊗ A+ and apply the results obtained in Section 2 to the morphismaM,A,p : Q ⊗M → A+ of Q-restriction functors which is associated to a morphismp : Q ⊗M → Q ⊗ A of Q-conjugation functors. Using results from Chapter I weobtain an explicit formula for aM,A,p in terms of p, cf. Definition 3.4. In general,even if pH(M(H)) ⊆ A(H) for H ≤ G, the morphism aM,A,p is not necessarilyintegral, i.e. aM,A,p

H (M(H)) ⊆ A+(H) for H ≤ G. Section 4 is devoted to thisintegrality problem. Starting with the explicit formula for aM,A,p in terms of p andapplying suitable rearrangements of the summands in this formula, we succeed inderiving an expression for aM,A,p which allows to state a condition (∗π) on p inTheorem 4.5 such that this condition implies |H|π · aM,A,p

H (M(H)) ⊆ A+(H) forH ≤ G and any set π of primes. Luckily, condition (∗π) can be verified in almost allexamples we are interested in. Note that the proof of the integrality of the canonicalBrauer induction formula in [Bo89] or [Bo90] was quite elaborate and limited to thesituation of the character ring Mackey functor. Theorem 4.5 and the statement ofcondition (∗π) may be considered of the heart of Chapter I and II.

43

44 CHAPTER 2. CANONICAL INDUCTION FORMULAE

2.1 The definition

Throughout this section let k be a commutative ring, G a finite group and M ak-Mackey functor on G.

1.1 Definition For any k-restriction subfunctor A of M on G (i.e. A(H),H ≤ G, are stable under conjugation and restriction), the inclusion A ⊆ M is anelement of k−Res(G)(A,M) and corresponds via the adjointness in PropositionI.4.1 (i) to a morphism bM,A : A+ → M of k-Mackey functors on G which is givenfor H ≤ G by

bM,AH : A+(H)→M(H), [K, a]H 7→ indHK(a),

where K ≤ H and a ∈ A(K). We will call bM,A the induction morphism of Mfrom A.

1.2 Remark For k = Z, M the character ring functor R, and A the Z-restriction subfunctor Rab consisting of the Z-span of one-dimensional characters,the map bM,A

H : A+(H) → M(H) is surjective for H ≤ G by Brauer’s inductiontheorem (which explains the choice of the letter b, cf. [Se78, Theoreme 20]). Forfixed χ ∈M(H), H ≤ G, the ‘different ways’ of writing χ as a linear combination ofinduced one-dimensional characters correspond bijectively to the set of elements inA+(H) which are mapped to χ under bM,A

H . Here we identify two such linear com-binations, if one arises from the other by replacing subgroups and one-dimensionalcharacters with H-conjugate subgroups and the corresponding H-conjugate one-dimensional characters (which doesn’t change the induced one-dimensional charac-ter).

If we want to specify for each χ ∈M(H), H ≤ G, a preferred way of writing χ viaBrauer’s induction theorem, this amounts to defining a map aH : M(H)→ A+(H)for H ≤ G such that bM,A

H aH = idM(H) for H ≤ G. Knowing that A+ and M

are Mackey functors on G and bM,A is a morphism of Mackey functors, it is justnatural to require that a : M → A+ be also a morphism of Mackey functors on G.But – at least in this example – nature doesn’t allow the existence of such a sectiona for bM,A

H in the category of k-Mackey functors on G by the following reason:Assume that a : M → A+ is a morphism of k-Mackey functors on G with bM,A

a = idM . Since a then commutes with induction maps, Artin’s induction theorem(see [CR81, 15.4]), namely

|H| ·M(H) ⊆∑K≤HK cyclic

indHK(M(K)),

for H ≤ G, implies that

|H| · aH(M(H)) ⊆∑K≤HK cyclic

ind+HK(A+(K))

for H ≤ G. Recalling the definition of ind+HK for K ≤ H ≤ G from I.2.2 we observe

from this inclusion that |H| · aH(M(H)) is contained in the span of the elements[K,ϕ]H , where K ≤ H is a cyclic subgroup and ϕ is a one-dimensional character


of K. Since these elements are part of a Z-basis of A+(H) (cf. Lemma I.2.4), alsoaH(M(H)) is contained in this span. But then, bM,A

H aH = idM(H) implies that

M(H) =∑K≤HK cyclic

indHK(A(K)),

which is certainly not true for arbitrary finite groups H.

Therefore, in general, we cannot hope for a section of bM,A in the category ofMackey functors, even if bM,A is surjective. One obstruction, as the above exampleshows, is the commutativity of the section with induction maps. Consequently, wejust drop this requirement.

1.3 Definition Let A ⊆ M be a k-restriction subfunctor on G and letbM,A : A+ → M be the induction morphism of M from A (see Definition 1.1). Amorphism a ∈ k−Res(G)(M,A+) with bM,A a = idM is called a canonical induc-

tion formula for M from A. In that case, the morphism ρAa ∈ k−Res(G)(M,A+)corresponds by the adjointness in Proposition I.4.1 (b) to a morphism p := πA a =θM,A(a) ∈ k−Con(G)(M,A) which we call the residue of a (cf. the paragraphfollowing Proposition I.4.1).

2.2 The case of invertible group order

Throughout this section we assume that k is a commutative ring and G is a finitegroup whose order is invertible in k.

2.1 For M ∈ k−Mack(G) and a k-restriction subfunctor A ⊆ M on Gwe know from Corollary I.4.2 (iii) that the map a 7→ πA a induces a bijectionbetween k−Res(G)(M,A+) and k−Con(G)(M,A). If a and p = πA a are in cor-respondence, then a can be recovered from p by the the explicit formula (cf. Corol-lary I.4.2 (iii))

aH(m) =1

|H|∑

L≤K≤H|L|µ(L,K)

[L, (resKL pK resHK)(m)

]H, (2.1)

for H ≤ G and m ∈M(H), or by the commutative diagrams

M(H)aH−−→ A+(H)

(pKresHK)K≤H

yo ρAH=(πAKres+HK)

K≤H

A+(H)

(2.2)

for H ≤ G, since ρAH is an isomorphism by our assumptions on k and G (cf. Propo-sition I.3.2).

If a : M → A+ is a canonical induction formula for M from A and p : M → Ais its residue, we obtain from the explicit formula (2.1) in Remark 1.4 by applyingbM,AH the formula

m =1

|H|∑

L≤K≤H|L|µ(L,K) indHL

((resKL pK resHK)(m)

)(2.3)


for H ≤ G and m ∈M(H), where (resKL pK resHK)(m) is an element of A(L) forL ≤ K ≤ H ≤ G, m ∈M(H).

The following proposition determines the subset of k−Con(G)(M,A) whichconsists of the residues of canonical induction formulae for M from A, and in view ofthe above considerations this also determines the set of canonical induction formulaefor M from A. Recall the definition of the idempotent e

(H)H ∈ k ⊗Ω(H) for H ≤ G

from Appendix A and the decomposition M(H) = e(H)H ·M(H)⊕ (1− e(H)

H ) ·M(H)

from Proposition I.6.2 with e(H)H ·M(H) =

⋂K<H ker(resHK : M(H) → M(K)) and

(1− e(H)H ) ·M(H) =

∑K<H indHK(M(K)) for H ≤ G.

2.2 Proposition let M ∈ k−Mack(G), A ⊆ M a k-restriction subfunctoron G, a ∈ k−Res(G)(M,A+), and p := πA a ∈ k−Con(G)(M,A) the residue ofa. Then the following are equivalent:

(i) the morphism a is a canonical induction formula, i.e. bM,A a = idM .

(ii) We have e(H)H ·

(pH(m) −m

)= 0, i.e. pH(m) −m ∈

∑K<H indHK(M(K)),

for H ≤ G, m ∈M(H).

Proof Using Corollary I.4.2 we can express a in terms of p and obtain

e(H)H · bM,A

H

(aH(m)

)=

1

|H|e

(H)H ·

∑L≤K≤H

|L|µ(L,K) (indHL resKL pK resHK)(m)

= e(H)H · pH(m)

(2.4)

for H ≤ G, m ∈M(H), since e(H)H annihilates all summands with L < H.

If a is a canonical induction formula for M from A, then Equation (2.4) shows

that e(H)H ·m = e

(H)H · pH(m) for H ≤ G, m ∈M(H).

Conversely, let us assume that e(H)H ·m = e

(H)H · pH(m) for H ≤ G, m ∈M(H).

By Corollary I.6.3 it suffices to show that

eH · bM,AH

(aH(m)

)= e

(H)H ·m and resHK

(bM,AH

(aH(m)

))= resHK(m)

for all K < H ≤ G and m ∈ M(H). The first equation follows immediately fromEquation (2.4) and the assumption, and the second equation follows by induction on|H|, since bM,A and a commute with restrictions, and since the induction hypothesisfor K together with the first equation for K implies bM,A

K aK = idM(K).

We obtain as an easy corollary a necessary condition on the size of A(H), H ≤ G,for the existence of a canonical induction formula for M from A.

2.3 Corollary Let M and A be as in the above proposition. If there exists a

canonical induction formula for M from A, then e(H)H ·M(H) ⊆ e(H)

H ·A(H) for allH ≤ G.

We recall from I.(2.6) that the set C(M) of coprimordial subgroups of G with

respect to M consists of those subgroups H ≤ G with e(H)H ·M(H) 6= 0 which is

equivalent to (1− e(H)H ) ·M(H) 6= M(H) by Proposition I.6.2.


2.4 Corollary Let M ∈ k−Mack(G), and assume that A ⊆ M is a k-restriction subfunctor on G. Furthermore, let p ∈ k−Con(G)(M,A) with the prop-erty that A(H) = M(H) and pH = idM(H) for H ∈ C(H). Then p is the residue ofa canonical induction formula for M from A.

Proof Condition (ii) in Proposition 2.2 is satisfied for H ∈ C(M), since in thiscase pH(m)−m = 0 for m ∈M(H), and also for H /∈ C(M), since eH(H)·M(H) = 0in that case.

Next we determine the canonical induction formulae which are morphisms ofMackey functors; cf. Remark 1.2 for the impossibility of such formulae in an examplewith k = Z.

2.5 Lemma Let M be a k-Mackey functor on G, A ⊆ M a k-restrictionsubfunctor on G, a ∈ k−Res(G)(M,A+), and p := πA a ∈ k−Con(G)(M,A) theresidue of a. Then the following statements are equivalent:

(i) a ∈ k−Mack(G)(M,A+).(ii) pH

(indHK(M(K))

)= 0 for K < H ≤ G.

(iii) pH((1− e(H)

H ) ·M(H))

= 0 for H ≤ G.

Proof The statements (ii) and (iii) are equivalent by Proposition I.6.2. Wewill show that also (i) and (ii) are equivalent. The morphism a is an element ink−Mack(G)(M,A+) if and only if

ind+HK aK = aH indHK : M(K)→ A+(H)

for K ≤ H ≤ G. Since ρAH , H ≤ G is injective, this is equivalent to

πAU res+HU ind+

HK aK = πAU res+

HU aH indHK : M(K)→ A(U)

for H ≤ G and U,K ≤ G. We transform the left hand side by using the Mackeyaxiom and the commutativity of a with restrictions and conjugations, and we trans-form the right hand side by using the commutativity of a with restrictions, theMackey axiom, and the relation pU = πAU aU . Thus we obtain the equivalentcondition ∑

h∈U\H/K

πAU ind+U

U∩ hK a

U∩ hK reshK

U∩ hK ch,K

=∑

h∈U\H/K

pU indUU∩ hK

reshK

U∩ hK ch,K : M(K)→ A(U)

for H ≤ G and K,U ≤ H. Since πAU ind+U

U∩ hK= 0 unless U ≤ hK, this is

equivalent to ∑h∈U\H/K

U≤hK

pU reshKU ch,K

=∑

h∈U\H/K

pU indUU∩ hK

reshK

U∩ hK ch,K : M(K)→ A(U)

(2.5)


for H ≤ G and K,U ≤ H.Now Equation (2.5) implies 0 = pH indHK for K < H ≤ G by choosing U = H.

Conversely, if pH indHK = 0 for K < H ≤ G, then Equation (2.5) holds, since theright hand side reduces to the left hand side.

2.6 Remark In the proof of the equivalence of (i) and (ii) in Lemma 2.5 weactually don’t need the assumption that |G| is invertible in k, but only the fact thatρA is injective. Hence the equivalence of (i) and (ii) still holds when k is arbitraryand A has a stable basis (cf. Proposition I.2.4) or even weaker when A+(H) has no|H|-torsion for H ≤ G.

2.7 Corollary Let M , A, a, and p be as in the above lemmma. Then a is acanonical induction formula for M from A in the category of k-Mackey functors onG if and only if

(pH − idM(H))(M(H)) ⊆ (1− e(H)H ) ·M(H) ⊆ ker(pH)

for H ≤ G.

Proof This is an immediate consequence of Proposition 2.2 and Lemma 2.5.

2.8 Example For each k-Mackey functor M on G there is a canonical in-duction fromula for M of a minimal type (cf. Corollary 2.3), namely from A(H) :=

e(H)H ·M(H) for H ≤ G (which form a k-restriction subfunctor of M on G with trivial

restrictions resHK , K < H ≤ G) and with pH : M(H) → A(H) given by multiplica-

tion with e(H)H for H ≤ G. In fact, this choice of M , A, p satisfies the condition of

the above corollary. The corresponding induction formula a ∈ k−Mack(G)(M,A+)can be calculated explicitly from Equation (2.2) as

aH(m) =1

|H|∑K≤H

K∈C(M)

|K|[K, e

(K)K · resHK(m)

]H

for H ≤ G, m ∈M(H), by observing that resKL is trivial on im(pK) for L < K ≤ G,

and e(K)K ·M(K) = 0 for K /∈ C(M). Moreover, from Equation (2.3) and the explicit

formula (1.2) in I.6.1 for eK(K), K ≤ G, we derive

m =1

|H|∑

L≤K≤HK∈C(M)

|L|µ(L,K) indHL (resHL (m))

for H ≤ G, m ∈ M(H), which generalizes Brauer’s explicit version (cf. [Bra51] or[CR81, 15.4]) of Artin’s induction theorem for the character ring tensored with Q,where C(M) consists of the cyclic subgroups of G, and Conlon’s induction theo-rem for the Green ring and the ring of p-permutation modules tensored with Q,where C(M) consists of the p-hypo-elementary, i.e. cyclic mod p, subgroups of G(p denoting the prime characteristic of the base field), cf. [CR87, 80.61,81.32].

This kind of minimal example of a canonical induction formula was one of themain tools used in [BCS93] in the case of the character ring Mackey functor tensoredwith Q.

2.3. THE STANDARD SITUATION 49

2.9 Proposition Let M ∈ k−Mackalg(G), A ⊆ M a k-algebra restrictionsubfunctor on G, a ∈ k−Res(G)(M,A+), and p := πA a ∈ k−Con(G)(M,A) theresidue of a.

(i) The maps aH , H ≤ G, are k-algebra homomorphisms if and only if the mapspH , H ≤ G, are k-algebra homomorphisms.

(ii) The maps aH , H ≤ G, are A(H)-module homomorphisms if and only ifthe maps pH , H ≤ G, are A(H)-module homomorphisms, cf. the last paragraph inI.2.2.

Proof (i) This follows immediately from Corollary I.4.2 (iv).(ii) The maps aH are A(H)-linear for allH ≤ G if and only if aH(a·m) = [H, a]H ·

aH(m) for all H ≤ G, a ∈ A(H), m ∈M(H). By the injectivity and multiplicativityof ρAH this is equivalent to (ρAH aH)(a · m) = ρAH([H, a]H) · ρAH(aH(m)). SinceρAH = (πAK res+

HK)

K≤H , since a commutes with restriction and πAH aH = pH forH ≤ G, this is equivalent to

pK(resHK(a) · resHK(m)

)= resHK(a) · pK

(resHK(m)

)for K ≤ H ≤ G, a ∈ A(H), m ∈M(H). But this is equivalent to the A(H)-linearityof pH for all H ≤ G.

2.3 The standard situation

In this section we will restrict our attention to a less general situation, which willall the same cover the examples we are interested in. Throughout this section weassume the following hypothesis.

3.1 Hypothesis Let M be a Z-Mackey functor on a finite group G andA ⊆M a Z-restriction subfunctor of M on G such that

(i) M(H) is a free abelian group for H ≤ G;

(ii) A has a stable basis B(H) ⊆ A(H), H ≤ G, such that, for K ≤ H ≤ G andϕ ∈ B(H), the element resHK(ϕ) is a linear combination of the basis elementsin B(K) with non-negative coefficients.

This hypothesis is clearly designed for the various representation rings, as forexample character rings, Brauer character rings, projective character rings, trivialsource rings, Green rings, etc. as candidates for M , and certain subrings, whichwe will specify later for the different examples, as candidates for A. In our stan-dard example, where M is the character ring and A the span of one-dimensionalcharacters, we clearly stay within the limits of this hypothesis.

As a convention, the letters χ, ϑ, ξ, ζ will always denote elements in M(H),H ≤ G, and ϕ, ψ, λ will always denote elements in B(H) ⊆ A(H), H ≤ G, asalready done in Hypothesis 3.1. This should help the reader by referring to thestandard example mentioned above.

3.2 Definition A pair (H,ϕ) with H ≤ G and ϕ ∈ B(H) is called a monomial

pair. For H ≤ G, the set of monomial pairs (K,ψ) with K ≤ H will be denoted byM(H).


For K ≤ H ≤ G, ϕ ∈ B(H), ψ ∈ B(K) we define the multiplicity m(H,ϕ)(K,ψ) ∈ N

to be the coefficient of ψ in resHK(ϕ). For H ≤ G, we define the structure of anH-poset on M(H) by

(L, λ) ≤ (K,ψ) :⇐⇒ L ≤ K and m(K,ψ)(L,λ) 6= 0,

h(K,ψ) :⇐⇒ ( hK, hψ).

Note that for the transitivity axiom (P3) in Appendix B we need the positivitycondition in Hypothesis 3.1 (ii) We write (L, λ) =H (K,ψ) if (L, λ) and (K,ψ) liein the same H-orbit, and we denote by NH(L, λ) the H-stabilizer of (L, λ). Theposets M(H), H ≤ G, are subposets of the poset M(G).

In the following remark we recollect some results that have already been estab-lished in the situation of Hypothesis 3.1.

3.3 Remark For H ≤ G, A+(H) is a free Z-module with basis [K,ψ]H,where (K,ψ) runs through a set of representatives of the H-orbits H\M(H) ofM(H), cf. Proposition I.2.4. Therefore, Q ⊗ A+(H) has the same Q-basis, whenwe write again [K,ψ]H ∈ Q ⊗ A+(H) instead of 1 ⊗ [K,ψ]H . We will alwaysidentify (Q ⊗ A+)(H) with Q ⊗ A+(H) via the natural map, cf. Lemma I.5.1 (ii).These natural maps form an isomorphism between the Q-Mackey functors (Q ⊗A)+ and Q ⊗ A+ on G. Hence, conjugation, restriction and induction maps areidentified. Moreover the morphisms Q ⊗ πA and πQ⊗A are identified under thisisomorphism. Similarly we have a natural isomorphism Q ⊗ A+ → (Q ⊗ A)+ ofQ-Mackey functors on G such that the mark homomorphisms Q⊗ρA and ρQ⊗A areidentified, cf. Diagram I.(5.1). We will work with the Mackey functors Q⊗A+ andQ ⊗ A+ and denote their structure maps again by c+, res+, ind+, c+, res+, ind+,πA, ρA, etc. Note that the mark morphism ρA : A+ → A+ is injective and the markmorphism ρA : Q⊗A+ → Q⊗A+ is an isomorphism by Proposition I.3.2.

We have a commutative diagram

Q−Res(G)(Q⊗M,Q⊗A+)θQ⊗M,Q⊗A−−→ Q−Con(G)(Q⊗M,Q⊗A)⋃ ⋃

Z−Res(G)(M,A+)θM,A−−→ Z−Con(G)(M,A),

(2.6)

where the vertical inclusions are described by extensions of morphisms, Q ⊗ A+

is identified with (Q ⊗ A)+, and θM,A, θQ⊗M,Q⊗A are the maps of taking residues(cf. the paragraph preceding Corollary I.4.2). By Corollary I.4.2, θM,A is injectiveand θQ⊗M,Q⊗A is an isomorphism. The set of canonical induction formulae forM from A is a subset of Z−Res(G)(M,A+) and can be identified via θM,A witha subset of Z−Con(G)(M,A), namely the set of residues of canonical inductionformulae for M from A. We are going to find conditions on an arbitrary morphismp ∈ Z−Con(G)(M,A) which are equivalent to p being the residue of a canonicalinduction formula for M from A. Since we know an explicit formula for takingpreimages under θM,A (cf. Corollary I.4.2 (iii)), we are able to construct a canonicalinduction formula for M from A, whenever we find some p ∈ Z−Con(G)(M,A)satisfying these conditions.

2.3. THE STANDARD SITUATION 51

3.4 Definition For p ∈ Q−Con(G)(Q⊗M,Q⊗A) we define

aM,A,p := θ−1Q⊗M,Q⊗A(p) ∈ Q−Res(G)(Q⊗M,Q⊗A+)

(cf. Diagram (2.6)), i.e. aM,A,p makes the diagrams

Q⊗M(H)aM,A,p−−→ Q⊗A+(H)

(pKresHK)K≤H

yo ρQ⊗AH

Q⊗A+(H),

H ≤ G, commutative (cf. Diagram (2.2)), or more explicitly (cf. Equation (2.1)),

aM,A,pH (χ) :=

1

|H|∑

L≤K≤H|L|µ(L,K)

[L, (resKL pK resHK)(χ)

]H

for H ≤ G, χ ∈ Q ⊗ M(H). We call aM,A,p integral, if aM,A,pH (χ) ∈ A+(H) ⊆

Q⊗A+(H) for H ≤ G, χ ∈M(H). Since πAH aM,A,pH = pH for H ≤ G, a necessary

condition for aM,A,p to be integral is that p is the Q-extension of a morphism inZ−Con(G)(M,A).

3.5 Proposition Let p ∈ Z−Con(G)(M,A). Then p is the residue of (aunique) canonical induction formula for M from A, if and only if the following twoconditions are satisfied:

(i) aM,A,p is integral.

(ii) For H ≤ G and χ ∈M(H) we have

pH(χ)− χ ∈ Q⊗(∑K<H

indHK(M(K)))⊆ Q⊗M(H).

If conditions (i) and (ii) are satisfied, then aM,A,p restricted to M is the uniquecanonical induction formula for M from A with residue p.

Proof In view of Diagram (2.2), condition (i) is equivalent to p being in theimage of θM,A, i.e. p = πA a for some a ∈ Z−Res(G)(M,A+). This element anow is a canonical induction formula for M from A, if and only if bM,A a = idM ,which again is equivalent to Q ⊗ bM,A Q ⊗ a = idQ⊗M . But this means that theresidue of Q⊗a, which equals Q⊗p by the commutativity of Diagram (2.6), satisfiescondition (ii) for H ≤ G and χ ∈ Q ⊗M(H) by Proposition 2.2. Since M(H) isZ-free for H ≤ G, this is equivalent to condition (ii).

3.6 Remark In the later examples, typically condition (ii) is easy to verify,but the integrality condition (i) is by no means trivial. In order to attack condition(i) we have to go much deeper into the study of the nature of the alternating suminvolved. This will be done in the next section.

3.7 Lemma Let p ∈ Q−Con(G)(Q⊗M,Q⊗A), H ≤ G, and χ ∈ Q⊗M(H)with

resHK(pH(χ)

)= pK

(resHK(χ)

)


for all K ≤ H. Then aM,A,pH (χ) = [H, pH(χ)]H .

Proof In view of the explicit definition of aM,A,pH in Definition 3.4 we have to

show that

1

|H|∑

L≤K≤H|L|µ(L,K)

[L, resHL

(pH(χ)

)]H

= [H, pH(χ)]H .

But applying Mobius inversion in the second form of Corollary B.3 (i) to the function

K ≤ H → A+(H), K 7→ |K|[K, resHK

(pH(χ)

)]H

shows that the sum above collapses to |H|[H, pH(χ)]H , and the proof is complete.

3.8 Definition Let A,A′ ⊆ M be two k-restriction subfunctors on G andp : Q ⊗M → Q ⊗ A, p′ : Q ⊗M → Q ⊗ A′ morphisms of k-conjugation functorson G. We call the two canonical induction formulae a = aM,A,p and a′ = aM,A′,p′

equivalent, if they coincide as functions taking values in Q ⊗M+, i.e. if the twomaps

Q⊗M a−→Q⊗A+i+−→Q⊗M+ and Q⊗M a′−→Q⊗A′+

i′+−→Q⊗M+

coincide, where i : A→M and i′ : A′ →M denote the inclusion maps.

3.9 Lemma Let M , A, A′, p, p′, i, i′, a, a′ be as in the above definitionand assume that M has a stable basis. Then a and a′ are equivalent if and only ifi p : Q⊗M → Q⊗M and i′ p′ : Q⊗M → Q⊗M coincide.

Proof We first observe that πM i+ a = i p. In fact, since π is natural(cf. I.(3.1)), we have πM i+ = i πA : A+ → M , and the equation follows fromπA a = p. Similarly we obtain πM i′+ a′ = i′ p′. Hence i+ a = i′+ a′ impliesi p = i′ p′. Conversely, if i p = i′ p′, then πM i+ a = πM i′+ a′ andtherefore πMK i+K aK resHK = πMK i′+K a

′K resHK for all K ≤ H ≤ G. Since

a, a′, i+, i′+ commute with restrictions, we obtain ρM i+ a = ρM i′+ a′ andi+ a = i′+ a′ by the injectivity of ρM (cf. Propositions 2.4 and 3.2).

3.10 Remark Lemma 3.9 indicates that A may vary without changing theequivalence class of the canonical induction formula. One may always substitutep : Q⊗M → Q⊗A by p′ := ip : Q⊗M → Q⊗A→ Q⊗M whithout changing theequivalence class. Hence, not the choice of A ⊆M is important, but the choice of p,which we can always consider as an endomorphism in Q−Con(G)(Q⊗M,Q⊗M).The elements which are induced in the resulting canonical induction formula, arecontained in the image of p. Nevertheless it makes sense to choose a suitable A toindicate a priori the sort of elements which we allow to be induced. Moreover, wewill need to choose A as small as possible in Chapter IV, where we will compose awith functions on Q⊗A which are not defined on Q⊗M .

3.11 Proposition Let M ∈ Z−Mack(G) and let A′ ⊆M ′ be a Z-restrictionsubfunctor of M with stable basis B′, and assume that M ′, A′ and B′ satisfy Hy-pothesis 3.1. Furthermore let p : Q ⊗M → Q ⊗ A and p′ : Q ⊗M ′ → Q ⊗ A′ be

2.4. INTEGRALITY 53

morphisms of Q-conjugation functors on G and denote by a : Q ⊗M → Q ⊗ A+

and a′ : Q ⊗ M ′ → Q ⊗ A′+ the morphisms aM,A,p and aM′,A′,p′ respectively. If

f : Q⊗M → Q⊗M ′ is a morphism of Q-restriction functors with fH(Q⊗A(H)) ⊆Q⊗A′(H) for H ≤ G, then the diagram

Q⊗M a−−→ Q⊗A+

f

y yf+

Q⊗M ′ a′−−→ Q⊗A′+

is commutative if and only if the diagram

Q⊗M p−−→ Q⊗Af

y yfQ⊗M ′ p′−−→ Q⊗A′

is commutative.

Proof First we assume that the first diagram is commutative. ComposingπA′: Q⊗A′+ → Q⊗A′ with the first diagram we obtain

πA′ f+ a = πA

′ a′ f.

But πA′ a′ = p′, πA

′ f+ = f πA and πA a = p, so that we obtain f p = p′ f .Now we assume that f p = p′ f : Q⊗M → Q⊗A′. By the injectivity of ρA

′,

the first diagram commutes if and only if

pA′

K res+HK f+H aH = πA

′K res+

HK a

′H fH

for K ≤ H ≤ G. But the left hand side is equal to πA′

K f+K aK resHK = fK πAK aK resHK = fK pK resHK , and the right hand side is equal to πA

′K a′K fK resHK =

p′K fK resHK . Now the result follows.

2.4 Integrality

Throughout this section we assume Hypothesis 3.1 and fix a morphism p : M → Aof Z-conjugation functors on G.

4.1 Definition (i) For H ≤ G and χ ∈M(H) we write

pH(χ) =∑

ϕ∈B(H)

mϕ(χ) · ϕ

with mϕ(χ) ∈ Z. The number mϕ(χ) is called the multiplicity of ϕ in χ. Notethat these multiplicities depend on p : M → A. But in order to avoid overloadednotations we omit to indicate this.

(ii) Recall from Definition 3.2 that for H ≤ G, the setM(H) of monomial pairsis a poset. For any chain σ =

((H0, ϕ0) < . . . < (Hn, ϕn)

)∈ Γ(M(H)) ⊆ Γ(M(G))

we define the multiplicity mσ of σ by

mσ := m(H1,ϕ1)(H0,ϕ0) · . . . ·m

(Hn,ϕn)(Hn−1,ϕn−1).


Of course, this definition is independent of the choice of H.

4.2 Lemma For H ≤ G and χ ∈M(H) we have

aM,A,pH (χ) =

1

|H|∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈Γ(M(H))

(−1)n|H0|mσmϕn(resHHn(χ))[H0, ϕ0]H .

Proof We start with the explicit formula for aM,A,pH (χ) given in Definition 3.4

and use the identities

µ(L,K) =∑

L=H0<...<Hn=K

(−1)n

for L ≤ K ≤ H from (B.1) in Appendix B. This yields

aM,A,pH (χ) =

1

|H|∑

H0<...<Hn≤H(−1)n|H0|

[H0, (resHnH0

pHn resHHn)(χ)]H

=1

|H|∑

H0<...<Hn≤Hϕn∈B(Hn)

(−1)n|H0|mϕn(resHHn(χ))[H0, resHnH0

(ϕn)]H.

for H ≤ G and χ ∈M(H). For a fixed chain H0 < . . . < Hn of subgroups of H andϕn ∈ B(Hn) we have

resHnH0(ϕn) =

(resH1

H0 . . . resHnHn−1

)(ϕn)

=∑

ϕn−1∈B(Hn−1)

m(Hn,ϕn)(Hn−1,ϕn−1)

(resH1

H0 . . . res

Hn−1

Hn−2

)(ϕn−1)

...

=∑

ϕ0∈B(H0)

∑ϕ1∈B(H1)

. . .∑

ϕn−1∈B(Hn−1)

m(H1,ϕ1)(H0,ϕ0) . . .m

(Hn,ϕn)(Hn−1,ϕn−1) · ϕ0

=∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))mσ · ϕ0,

since m(Hi,ϕi)(Hi−1,ϕi−1) = 0 unless (Hi−1, ϕi−1) < (Hi, ϕi) for 1 ≤ i ≤ n. If we substitute

this expression for resHnH0(ϕ0) in the above formula for aM,A,p

H (χ), the result follows.

In 4.4 and Theorem 4.5 we will use the following definitions to modify the factor|H0| of the formula in the previous Lemma, and obtain integrality criteria for aM,A,p.

4.3 Let H ≤ G. For σ =((H0, ϕ0) < . . . < (Hn, ϕn)

)∈ Γ(M(H)) and a set

π of primes we define

NπH(σ) := s ∈ NH(σ) | sπ′ ∈ H0,

where NH(σ) denotes the stabilizer in H of the chain σ, i.e. the intersection of thestabilizers of the elements (Hi, ϕi), 0 ≤ i ≤ n, in the H-set M(H). Note that,

2.4. INTEGRALITY 55

for s ∈ NH(σ), the condition sπ′ ∈ H0 is equivalent to sH0 being a π-element inNH(σ)/H0. Hence, we have

|NπH(σ)| = |H0| · |(NH(σ)/H0)π| (2.7)

For H ≤ G and π as above we define

Γπ(M(H)) := (s, σ) ∈ H × Γ(M(H)) | s ∈ NπG(σ)

and give a partition

Γπ(M(H)) = Γπ0 (M(H)).∪ Γπ1 (M(H))

.∪ Γπ2 (M(H))

.∪ Γπ3 (M(H))

of Γπ(M(H)) into four subsets by defining for (s, σ) ∈ Γπ(M(H)) with σ =((H0, ϕ0) < . . . < (Hn, ϕn)

)that

(s, σ) ∈ Γπ0 (M(H)) :⇐⇒ s ∈ H0,

(s, σ) ∈ Γπ1 (M(H)) :⇐⇒ s /∈ Hn,

(s, σ) ∈ Γπ2 (M(H)) :⇐⇒ s ∈ Hi+1 \Hi and Hi(s) < Hi+1 for some

i ∈ 0, . . . , n− 1,(s, σ) ∈ Γπ3 (M(H)) :⇐⇒ s ∈ Hi+1 \Hi and Hi(s) = Hi+1 for some

i ∈ 0, . . . , n− 1,

where Hi(s) denotes the subgroup of H generated by Hi and s (s normalizes Hi).Note that the index i ∈ 0, . . . , n−1 in the definition of Γπ2 (M(H)) and Γπ3 (M(H))is uniquely determined by the condition s ∈ Hi+1\Hi. Note also, that the definitionof Γπ0 (M(H)) does not depend on π.

Next we consider the map

f : Γπ3 (M(H)) → Γπ1 (M(H)).∪ Γπ2 (M(H)),(

s, (H0, ϕ0) < . . . < (Hn, ϕn))7→

(s, (H0, ϕ0) < . . . (Hi+1, ϕi+1) . . . < (Hn, ϕn)

),

where i ∈ 0, . . . , n−1 is determined by s ∈ Hi+1 \Hi, and the pair (Hi+1, ϕi+1) =(Hi(s), ϕi+1) is omitted from σ.

For (s, σ) ∈ Γπ1 (M(H)) with σ =((H0, ϕ0) < . . . < (Hn, ϕn)

)we have

f−1((s, σ)) =(s, (H0, ϕ0) < . . . < (Hn, ϕn) < (Hn(s), ϕ)

)∣∣ϕ ∈ B(Hn(s)), m

(Hn(s),ϕ)(Hn,ϕn) 6= 0

,

and for (s, σ) ∈ Γπ2 (M(H)) with σ =((H0, ϕ0) < . . . < (Hn, ϕn)

)and i ∈

0, . . . , n− 1 with s ∈ Hi+1 \Hi we have

f−1((s, σ)) =(s, (H0, ϕ0) < . . . < (Hi, ϕi) < (Hi(s), ϕ) < (Hi+1, ϕi+1) < . . . < (Hn, ϕn)

)∣∣∣ϕ ∈ B(Hi(s)), m

(Hi(s),ϕ)(Hi,ϕi)

m(Hi+1,ϕi+1)(Hi(s),ϕ) 6= 0

.


4.4 If we define for H ≤ G, σ =((H0, ϕ0) < . . . < (Hn, ϕn)

)∈ Γ(M(H)),

and χ ∈M(H) the abbreviation,

g(χ, σ) := (−1)|σ|mσmϕn(resHHn(χ))[H0, ϕ0]H ∈ A+(H), (2.8)

we can rewrite Lemma 4.2 as

|H| · aM,A,pH (χ) =

∑(s,σ)∈Γπ0

g(χ, σ),

where the factor |H0| is replaced by summing over pairs (s, σ) with s ∈ H0. Hence,we may continue by

|H| · aM,A,pH (χ) =

∑(s,σ)∈Γπ(M(H))

g(χ, σ)−

−∑

(s,σ)∈Γπ1 (M(H)).∪Γπ2 (M(H))

.∪Γπ3 (M(H))

g(χ, σ).(2.9)

We examine the last sum in Equation (2.9) further using the function f :∑(s,σ)∈Γπ1 (M(H))

.∪Γπ2 (M(H))

.∪Γπ3 (M(H))

g(χ, σ)

=∑

(s,σ)∈Γπ1 (M(H))

(g(χ, σ) +

∑(s′,σ′)∈f−1((s,σ))

g(χ, σ′))

+

+∑

(s,σ)∈Γπ2 (M(H))

(g(χ, σ) +

∑(s′,σ′)∈f−1((s,σ))

g(χ, σ′)).

(2.10)

For (s, σ) ∈ Γπ1 (M(H)) with σ =((H0, ϕ0) < . . . < (Hn, ϕn)

), i.e. s /∈ Hn, we

have

g(χ, σ) +∑

(s′,σ′)∈f−1((s,σ))

g(χ, σ′) = (−1)|σ|mσ×

×(mϕn

(res

Hn(s)Hn

(ϑ))−

∑ϕ∈B(Hn(s))

m(Hn(s),ϕ)(Hn,ϕn) mϕ(ϑ)

)[H0, ϕ0]H

(2.11)

with ϑ := resHnHn(s)(χ) ∈M(Hn(s)), and for (s, σ) ∈ Γπ2 (M(H)) with σ as above and

i ∈ 0, . . . , n− 1 with s ∈ Hi+1 \Hi we have

g(χ, σ) +∑

(s′,σ′)∈f−1((s,σ))

g(χ, σ′)

= (−1)|σ|(mσmϕn

(resHHn(χ)

)−

−∑

ϕ∈B(Hi(s))

mσ

m(Hi+1,ϕi+1)(Hi,ϕi)

m(Hi(s),ϕ)(Hi,ϕi)

m(Hi+1,ϕi+1)(Hi(s),ϕ) mϕn

(resHHn(χ)

))[H0, ϕ0]H

= (−1)|σ|mσ

m(Hi+1,ϕi+1)(Hi,ϕi)

mϕn

(resHHn(χ)

)×

×(m

(Hi+1,ϕi+1)(Hi,ϕi)

−∑

ϕ∈B(Hi(s))

m(Hi(s),ϕ)(Hi,ϕi)

m(Hi+1,ϕi+1)(Hi(s),ϕ)

)[H0, ϕ0]H

= 0,

2.4. INTEGRALITY 57

sincem

(Hi+1,ϕi+1)(Hi,ϕi)

=∑

ϕ∈B(Hi(s))

m(Hi(s),ϕ)(Hi,ϕi)

m(Hi+1,ϕi+1)(Hi(s),ϕ)

by transitivity of restriction. Combining this with Equations (2.11), (2.10) and(2.9) we obtain

|H| · aM,A,pH (χ) =

∑(s,σ)∈Γπ(M(H))

g(χ, σ)−

−∑(

s,σ=(

(H0,ϕ0)<...<(Hn,ϕn)))∈Γπ1 (M(H))

(−1)|σ|mσ ×

×(mϕn

(res

Hn(s)Hn

(ϑ))−

∑ϕ∈B(Hn(s))


)[H0, ϕ0]Hq

(2.12)

for H ≤ G, χ ∈M(H) and ϑ := resHHn(s)(χ).

4.5 Theorem Let M and A satisfy Hypothesis (3.1) and let p : M → A be amorphism of Z-conjugation functors on G. Furthermore assume that for a set π ofprimes the following condition holds:

(∗π)

Let K H ≤ G be such that H/K is a cyclic π-group, let ψ ∈ B(K) be fixed under H (i.e. hψ = ψfor h ∈ H), and let χ ∈ M(H). Then the twoelements pK(resHK(χ)), resHK(pH(χ)) ∈ A(K) havethe same coefficient at the basis element ψ.

Then we have

aM,A,pH (χ) =

=1

|H|∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈Γ(M(H))

(−1)|σ||NπH(σ)|mσ ×

×mϕn

(resHHn(χ)

)[H0, ϕ0]H =

=∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)|σ||(NH(σ)/H0)π||NH(σ)/H0|

mσ ×

×mϕn

(resHHn(χ)

)[H0, ϕ0]H

for H ≤ G and χ ∈M(H), where the second sum runs over a set of representativesfor the H-orbits of Γ(M(H)).

Proof It is obvious that condition (∗π) implies

mϕn

(res

Hn(s)Hn

(ϑ))

=∑

ϕ∈B(Hn(s))


in Equation (2.12), since the two sides of the above equation are equal to the co-

efficients of ϕn in pHn(res

Hn(s)Hn

(ϑ))

and resHn(s)Hn

(pHn(s)(ϑ)

)respectively. Therefore,


the last sum in Equation (2.12) vanishes, and we obtain for H ≤ G and χ ∈M(H):

aM,A,pH (χ) =

1

|H|∑

(s,σ)∈Γπ(M(H))

g(χ, σ)

=1

|H|∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈Γ(M(H))

(−1)|σ||NπH(σ)|mσ ×

×mϕn

(resHHn(χ)

)[H0, ϕ0]H .

Since the summands in the last sum are constant on H-orbits of Γ(M(H)), wemay collect them and obtain further

aM,A,pH (χ) =

1

|H|∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)|σ||H|

|NH(σ)||Nπ

H(σ)|mσ ×

×mϕn

(resHHn(χ)

)[H0, ϕ0]H .

Now the second equation in the Theorem follows from Equation (2.7):

|NπH(σ)|

|NH(σ)|=|H0| · |(NH(σ)/H0)π||H0| · |NH(σ)/H0|

=|(NH(σ)/H0)π||NH(σ)/H0|

.

4.6 Remark Note that in the first formula for aM,A,pH (χ) in Theorem 4.5 the

factor |NπH(σ)| replaces the factor |H0| in the original formula in Lemma 4.2 without

changing the resulting sum. We obviously have H0 ⊆ NπH(σ), but in general not

equality. Equation (2.7) even shows that |H0| divides |NπH(σ)|. In the case of the

standard example, M(H) := R(H), A(H) := Rab(H), pH the canonical projection,for H ≤ G, the result of Theorem 4.5 was obtained in [Bo94, Theorem 3.2], whereit was proved by comparing Euler characteristics of two chain complexes that thereplacement of |H0| with |NH(σ)| does not change the alternating sum.

4.7 Corollary Let M , A, p be as in Theorem 4.5 and assume that (∗π) holdsfor a set of primes π. Then

|H|π′ · aM,A,pH (χ) ∈ A+(H)

for H ≤ G, χ ∈M(H), i.e. in aM,A,pH (χ) occur only π′-numbers as denominators.

Proof It is well-known (cf. [Hu67, V.19.14]) that |Gπ| is a multiple of |G|πfor any finite group G. Therefore we have in the second sum of Theorem 4.5:

|NH(σ)/H0|π′|(NH(σ)/H0)π||NH(σ)/H0|

=|(NH(σ)/H0)π||NH(σ)/H0|π

∈ Z,

and a fortiori |H|π′ · aM,A,pH (χ) ∈ A+(H) for H ≤ G, χ ∈M(H).

2.4. INTEGRALITY 59

4.8 Corollary Let M , A, p be as in Theorem 4.5 and assume that (∗π) holdsfor the set π of all primes. Then we have

aM,A,pH (χ) =

=∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)nmσmϕn

(resHHn(χ)

)[H0, ϕ0]H

for H ≤ G, χ ∈M(χ), in particular aM,A,p is integral.

Proof This is a trivial consequence of Theorem 4.5.

4.9 Corollary Let M , A, p be as in Theorem 4.5. Assume that (∗π) holdsfor the set π of all primes, A(H) = M(H) for all H ∈ C(M) (cf. I.2.6), andpH : M(H) → A(H) is the identity map for all H ∈ C(M). Then aM,A,p is acanonical induction formula for M from A.

Proof We have to show that condition (i) and (ii) in Proposition 3.5 aresatisfied. Condition (i) holds by Corollary 4.8.

In order to show that condition (ii) holds, we first note that aM,A,pH (χ) = [H,χ]H

for H ∈ C(M) and χ ∈ M(H) = A(H) by Lemma 3.7. This shows that bM,AH

aM,A,pH = idM(H) for H ∈ C(M).

Now we show that (bM,AH aM,A,p

H )(χ) = χ for arbitrary H ≤ G and χ ∈ M(H)by induction on the order of H. The trivial subgroup is always contained in C(M).Let H be non-trivial. Then either H is contained in C(M) (so there is nothing toshow) or elements of M(H) are uniquely determined by their restrictions to propersubgroups. But then we only have to show that (resHK b

M,AH aM,A,p

H )(χ) = χ forK < H and χ ∈ M(H). However, this equation holds by our induction hypothesisand commutativity of bM,A and aM,A,p with restrictions.


Chapter 3

Examples of CanonicalInduction Formulae

In this chapter we will give several examples of canonical induction formulae. Theywill all be of the form aM,A,p : Q⊗M → Q⊗A+ (cf. Definition II.3.4) for a Z-Mackeyfunctor M on a finite group G, a Z-restriction subfunctor A of M , a morphismp : M → A of Z-conjugation functors on G, and a stable basis B of A which satisfyHypothesis II.3.1 so that we are in the standard situation and can apply all resultsof Sections II.3 and II.4.

This chapter is divided into sections according to the different choices of theMackey functor M . More precisely, we will consider character rings (Section 1),Brauer character rings (Section 2), projective and π-projective character groups(Section 3), trivial source and linear source rings (Section 4), and Green rings(Section 5). Within each section we present for fixed M (or a family of related M ’s)different choices of A, p, and B as examples for canonical induction formulae for Mfrom A or for Q⊗M from Q⊗A. These examples are followed by propositions whichgive information about the main properties of the resulting canonical inductionformulae. We omit to mention in each example that Hypothesis II.3.1 is satisfied,since this is the case for all examples throughout this chapter.

For each choice of M , A, B, and p we have associated a G-poset M(G) ofmonomial pairs with H-subposetsM(H) for each H ≤ G, cf. Definition II.3.2, andmultiplicities mϕ(χ) and mσ for H ≤ G, ϕ ∈ B(H), m ∈ Q⊗M(H), σ ∈ Γ(M(H)),cf. Definition II.4.1. Moreover, for any H ≤ G, χ ∈ Q⊗M(H), (K,ψ) ∈M(H) wedefine a rational number αH(K,ψ)(χ) as the coefficient of the basis element [K,ψ]H of

Q⊗A+(H) in aM,A,pH (χ) so that we can write

aM,A,pH (χ) =

∑(K,ψ)∈H\M(H)

αH(K,ψ)(χ)[K,ψ]H , (3.1)

where (K,ψ) runs through a set of representatives of the H-orbits ofM(H). SinceaM,A,pH is a canonical induction formula, composition with bM,A

H is the identity, andwe obtain the formula

χ =∑

(K,ψ)∈H\M(H)

αH(K,ψ)(χ)indHK(ψ) (3.2)

61

62 CHAPTER 3. EXAMPLES

for H ≤ G, χ ∈ Q⊗M(H).

In order to avoid overloaded notation we will abbreviate aM,A,p and bM,A by aand b respectively, and we also use the same symbols M , A, B, p, M, m, and αin the different sections, regardless of the example we are considering at a givenmoment.

In Section 6 we give an overview over the Mackey functors M considered in thefive previous sections together with the relations between them and the canonicalinduction formulae for them, and we conclude this chapter with an explicit exampleof the canonical induction formula for the character ring of an extraspecial p-groupof exponent p for odd p in Section 7.

3.1 The character ring

Throughout this section let K be a field of characteristic zero which is a splittingfield for all finite groups occurring in this section. For a finite group G we denoteby RK(G) the Grothendieck ring of the category KG−mod of finite dimensionalleft KG-modules. It is well-known that RK(G) can be canonically identified withthe ring of virtual characters of KG-modules and that RK(G) is Z-free on theset IrrK(G) of the characters of irreducible KG-modules. We denote by G(K) ⊆IrrK(G) the set of one-dimensional characters, i.e. G(K) = Hom(G,K×), and byRabK (G) ⊆ RK(G) the span of G(K).

It is well-known that RK(G), RabK (G), IrrK(G), and G(K) do not really de-

pend on K. If K ′ is another splitting field for G of characteristic zero, then thecorresponding rings and sets can be (non-canonically) identified. Even better, theidentification can be chosen for all finite groups G simultaneous in a way that con-jugation, restriction and induction is respected. Therefore, we will omit the indexK in the sequel, which means that we work with the field of complex numbers, butwe keep in mind that we could as well work with K.

For a finite group G the rings R(H), H ≤ G, form a Z-Green functor on G withthe usual conjugation, restriction, induction and multiplication. It is obvious thatthe set C(R) of coprimordial subgroups for R consists of the set of cyclic subgroupsof G. The rings Rab(H), H ≤ G, form a Z-restriction subfunctor of R on G withstable basis B(H) := H for H ≤ G. By M(H), H ≤ G, we denote the associatedH-poset of monomial pairs, cf. Definition II.3.2.

For each finite group G there is a Galois action on RK(G) and RabK (G) in the

following sense: Let σ be a field automorphism of K, and let χ ∈ R(G). Thenσχ ∈ R(G) is defined by ( σχ)(g) := σ(χ(g)) (note that χ(g) ∈ K, since K is asplitting field for G). If we consider RK as Z-Green functor on G, then σ : RK →RK is an isomorphism of Z-Green functors with σ(Rab

K ) ⊆ RabK . This induces an

isomorphism σ : RabK + → Rab

K + of Z-Green functors given by σ[U, µ]H = [U, σµ]H ∈RabK +(H) for H ≤ G, (U, µ) ∈ M(H). Obviously this defines actions of the group

of automorphism of K on RK , RabK and Rab

K + which we all call Galois actions.

Now letG’ be another finite group and let f : G′ → G be a group homomorphism.Then there is an induced ring homomorphism resf : R(G)→ R(G′), χ 7→ χf , which

3.1. THE CHARACTER RING 63

maps Rab(G) to Rab(G′). Moreover there is a ring homomorphism

res+f : Rab+ (G)→ Rab

+ (G′),

[H,ϕ]G 7→∑

g∈f(G′)\G/H

[f−1( gH), resf : f−1(

gH)→ g

H( gϕ)]G′, (3.3)

with the property res+f ′ res+f = res+(ff ′) for another group homomorphismf ′ : G′′ → G′.

In the case of an inclusion f : H → G of a subgroup H of G, resf and res+f

coincide with resGH and res+GH . In the case of a cononical surjection f : G → G/N

for a normal subgroup N of G, the ring homomorphisms resf and res+f are calledinflation maps and will be denoted by infGG/N and inf+

GG/N respectively. We have

the explicit description

inf+GG/N : Rab

+ (G/N)→ Rab+ (G), [H/N,ϕ]G/N 7→ [H,ϕ]G, (3.4)

for N ≤ H ≤ G and ϕ the inflation of a homomorphism ϕ : H/N → C×.

Note that with these definitions, R, Rab, and Rab+ can be considered as con-

travariant functors from the category of finite groups to the category of abeliangroups or to the category of commutative rings. The defintion of resf is designed

to make bR,Rab

a natural transformation between the functors Rab+ and R. Another

motivation for the definition of res+f can be found in V.1.25.

Besides the well-known scalar product (−,−)G on R(G) there is a bilinear form[−,−]G on Rab

+ (G) given by[[K,ψ]G, [H,ϕ]G

]G

:= #g ∈ K\G/H | (K,ψ) ≤ g(H,ϕ)

for (K,ψ), (H,ϕ) ∈ M(G). Note that[[K,ψ]G, [H,ϕ]G

]G

equals the coefficient of

[K,ψ]G in res+GK([H,ϕ]G). For a motivation of this definition see V.1.27. This

bilinear form is non-degenerate, since the describing matrix is an upper triangularmatrix with the non-zero entries |NG(H,ϕ)/H| on the diagonal, if the basis ele-ments [H,ϕ]G of Rab

+ (G) are ordered in a way compatible with the partial order onG\M(G). For G 6= 1 this bilinear form is not symmetric.

We define a morphism ε : R → Rab of Z-algebra restriction functors on G byεH(χ) := χ(1) · 1 for H ≤ G, χ ∈ R(H). The restriction of ε to Rab induces amorphism ε+ ∈ Z−Mackalg(G)(Rab

+ , Rab+ ) with ε+H([K,ψ]H) = [K, 1]H for H ≤ G,

(K,ψ) ∈M(H).

For H ≤ G, the rings R(H) and Rab+ (H) are Rab(H)-algebras in a natural

way, cf. the last paragraph in I.2.2. The maps bH , H ≤ G, are Rab(H)-algebrahomomorphisms.

For a KG-module V and any monomial pair (H,ϕ) ∈ M(G) we define the(H,ϕ)-homogeneous part V (H,ϕ) of V as

V (H,ϕ) := v ∈ V | hv = ϕ(h)v for h ∈ H.

Thus, V (H,ϕ) is the sum of all submodules of resGH(V ) with character ϕ.

Note that mσ = 1 for all H ≤ G, σ ∈ Γ(M(H)), cf. Definition II.4.1.


The following Example is the standard example which was already consideredin [Bo89], [Bo90], [BSS92], [BCS93] and [Bo94].

1.1 Example Let G be a finite group, M := R ∈ Z−Mackalg(G), A :=

Rab ∈ Z−Resalg(G), B(H) := H for H ≤ G, and define p ∈ Z−Con(G)(M,A) forH ≤ G and χ ∈ Irr(H) by

pH(χ) =

χ, if χ ∈ H,0, otherwise.

Note that for H ≤ G, χ ∈M(H), and ϕ ∈ H we have mϕ(χ) = (χ, ϕ)H , cf. Defini-tion II.4.1.

Most parts of the following proposition have already been proved in [Bo89] and[Bo90], but some of them with different methods from the ones we apply here. Weinclude the full proof of the following proposition, since it illustrates how the verygeneral results of Chapter II can be applied in a familiar case. The old methodsin [Bo89] and [Bo90] do not apply to the various other examples in this chapter,but the new ones from Chapter II do. This is especially the case where integralityquestions are concerned.

1.2 Proposition Let G be a finite group, and let M , A, B, M, p, m, α, a,and b be defined according to Example 1.1.

(i) The morphism a is an integral canonical induction formula, i.e. it can beconsidered as a morphism a : M → A+.

(ii) For H ≤ G, the homomorphism aH is A(H)-linear, i.e. aH(ϕ · χ) =[H,ϕ]H · aH(χ) for ϕ ∈ H, χ ∈M(H).

(iii) For H ≤ G, the map aH is the right adjoint of the map bH with respectto the scalar product (−,−)H on M(H) and the bilinear form [−,−]H on A+(H)defined above.

(iv) The morphism a respects the Galois action.

(v) For H ≤ G and ϕ ∈ H we have aH(ϕ) = [H,ϕ]H .

(vi) The morphism a is given explicitly by

aH(χ) =∑(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n(resHHn(χ), ϕ)Hn [H0, ϕ0]H

for H ≤ G and χ ∈M(H)), and we have

V =∑(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)nindHH0(V (Hn,ϕn))

in M(H) for any finite dimensional CH-module V .


(vii) For H ≤ G, χ ∈M(H) and (K,ψ) ∈M(H) we have

αH(K,ψ)(χ) =∑(

(H0,ϕ0)<...<(Hn,ϕn)

)∈H\Γ(M(H))

(H0,ϕ0)=H (K,ψ)

(−1)n(resHHn(χ), ϕn)Hn

=|K|

|NG(K,ψ)|∑(

(H0,ϕ0)<...<(Hn,ϕn)

)∈Γ(M(H))

(H0,ϕ0)=(K,ψ)

(−1)n(resHHn(χ), ϕn)Hn .

(viii) For H ≤ G, χ ∈ Irr(H) and (K,ψ) ∈M(H) we have

(resHK(χ), ψ)K = 0 =⇒ αH(K,ψ)(χ) = 0.

(ix) We have b ε+ a = ε, or more explcitly∑(K,ψ)∈H\M(H)

αH(K,ψ)(χ)indHK(1) = χ(1) · 1

for H ≤ G, χ ∈M(H).(x) For H ≤ G and χ ∈M(H) we have∑

(K,ψ)∈H\M(H)

(H : K)αH(K,ψ)(χ) = χ(1) =∑

(K,ψ)∈H\M(H)

αH(K,ψ)(χ).

(xi) Let H ≤ G, χ ∈ Irr(H), and let (Z(χ), λ) ∈ M(H) be the central pair ofχ with Z(χ) denoting the centre of χ, i.e. the maximal subgroup of H such that therestriction of χ is a multiple of some one-dimensional character, and with λ beingexactly this one-dimensional character. Then we have for (K,ψ) ∈M(H):

(Z(χ), λ) 6≤ (K,ψ) =⇒ αH(K,ψ)(χ) = 0.

(xii) For H ≤ G, χ ∈M(H), and (K,ψ) ∈M(H) we have

Z(H) 6≤ K =⇒ αH(K,ψ)(χ) = 0.

(xiii) Let f : G′ → G be a group homoorphism. Then the diagram

R(G)aG−−−−→ Rab

+ (G)

resf

y yres+f

R(G′)aG′−−−−→ Rab

+ (G′)

commutes. In particular, the canonical induction formulae aG, where G runs overall finite groups, form a natural transformation between the contravariant functorsM and A+ with values in abelian groups.

Proof (i), (vi) First we show that condition (∗π) in Theorem II.4.5 is satisfiedfor the set π of all primes. Let KH ≤ G with H/K cyclic and let ψ ∈ K be fixedunder H. Let furthermore χ ∈ Irr(H). Then resHK(χ) is a sum of H-conjugatesof some ϑ ∈ Irr(K). If ϑ 6= ψ, then the multiplicities of ψ in resHK(pH(χ)) and


pK(resHK(χ)) are both zero. If ϑ = ψ, then χ is a constituent of indHK(ψ) by Frobeniusreciprocity. But indHK(ψ) is the sum of the (H : K) different extensions of ψ, sinceH/K is cyclic. Therefore, χ is an extension of ψ, and the multiplicities of ψ inresHK(pH(χ)) and pK(resHK(χ)) are both equal to 1. Now part (i) and part (vi)follow from Corollaries II.4.8 and II.4.9.

(ii) This follows from Proposition II.2.9 (ii), since pH is A(H)-linear for H ≤ G.

(iii) Let H ≤ G, χ ∈ Irr(H), and (K,ψ) ∈ M(H). Then[[K,ψ]H , aH(χ)

]H

is

the coefficient of [K,ψ]H in res+HK(aH(χ)) as mentioned in the introductory part

of this section. But res+HK(aH(χ)) = aK(resHK(χ)) and the coefficient of [K,ψ]K in

aK(resHK(χ)) equals (ψ, resHK(χ))K by part (vi).

(iv) This follows from Diagram (2.2), since πA, p, res and res+ respect the Galoisaction.

(v) This follows from Lemma II.3.7.

(vii) The first equation is immediate from part (vi) and the second equationfollows from Lemma II.4.2 by collecting the summands with [H0, ϕ0]H = [K,ψ]H ,i.e. (H0, ϕ0) =H (K,ψ), and observing that summing only over chains starting in(K,ψ) produces a factor |K|/|NG(K,ψ)|.

(viii) This follows immediately from part (vi), since (resHK(χ), ψ)K = 0 implies(resHK′(χ), ψ′)K′ = 0 for all (K ′, ψ′) ≥ (K,ψ).

(ix) Since b, ε+, a and ε commute with restrictions, it suffices to show theequality bH ε+H aH = εH for subgroups H ∈ C(M), i.e. cyclic subgroups H of G.By linearity it suffices then to prove equality, if both sides are applied to ϕ ∈ H,since H = Irr(H). Now part (v) yields the result.

(x) The first equation follows from evaluating formula (3.2) at the identity ele-ment of H, and the second equation follows from taking scalar products with thetrivial character on both sides of the equation in part (ix).

(xi) Let (K,ψ) ∈ M(H) with (Z(χ), λ) 6≤ (K,ψ). By the second equation inpart (vii) we have to show that∑

(H0,ϕ0)<...<(Hn,ϕn)(H0,ϕ0)=(K,ψ)

(−1)n(resHHn , ϕn)Hn = 0.

Let V be a CH-module affording the character χ. Note that Z(χ) is normal in H.First we observe that if (U, µ) ∈ M(H) with (Z(χ), λ) 6≤ (U, µ), then there is a

unique monomial pair (Z(χ)U, ν) with resZ(χ)UZ(χ) (ν) = λ and res

Z(χ)UU (ν) = µ, since

because both Z(χ) and U act on V (U,µ) = v ∈ V | uv = µ(u)v for u ∈ U viascalar multiplication so does Z(χ)U . We write λ ·µ instead of ν. Moreover we have(

µ, resHU (χ))U

= dimV (U,µ) = dim(V (U,µ) ∩ V (Z(χ),λ)

)=

= dimV (Z(χ)U,λ·µ) =(λ · µ, resHZ(χ)U (χ)

)Z(χ)U

.

For the moment we set

Γ :=(

(H0, ϕ0) < . . . < (Hn, ϕn))∈ Γ(M(H)) | n ∈ N0,

(H0, ϕ0) = (K,ψ), (resHHn(χ), ϕn)Hn 6= 0


and define a function f : Γ → Γ in the following way: Let σ =((H0, ϕ0) < . . . <

(Hn, ϕn))∈ Γ and let i ∈ 0, . . . , n be maximal such that (Z(χ), λ) 6≤ (Hi, ϕi). If

i = n we define

f(σ) =((H0, ϕ0) < . . . < (Hn, ϕn) < (Z(χ)Hn, λ · ϕn)

).

If i < n and Z(χ)Hi = Hi+1 we define

f(σ) =((H0, ϕ0) < . . . < (Hi+1, ϕi+1) < . . . (Hn, ϕn)

).

If i < n and Z(χ)Hi < Hi+1 we define

f(σ) =((H0, ϕ0) < . . . < (Hi, ϕi) < (Z(χ)Hi, λ · ϕi) <

< (Hi+1, ϕi+1) < . . . < (Hn, ϕn)).

It is easy to verify that f2 = idΓ, and that the parities of lengths of σ and f(σ)are different so that the summands belonging to σ and f(σ) in the first sum canceleach other by the above equation.

(xii) This follows immediately from part (xi), since Z(H) ≤ Z(χ) for all χ ∈Irr(H).

(xiii) Since f can be written as a composition G′f−→ f(G′) ≤ G of an epi-

morphism and an inclusion, and since the result is already proved for subgroupinclusions, we may assume by functoriality that f is surjective. We proceed byinduction on |G′|. If |G′| = 1, the result holds trivially. Now let |G′| > 1 andχ ∈ Irr(G). If χ ∈ G, then part (v) implies the result. So we can assume thatχ(1) > 1. By the injectivity of ρG, it suffices to show that

πAH′ res+G′

H′((aG′ resf )(χ)− (res+f aG)(χ)

)= 0

for all H ′ ≤ G′.For H ′ = G′ we obtain

(πAG′ aG′ resf )(χ) = pG′(resf (χ)) = 0,

since resf (χ) ∈ Irr(G′) is not one-dimensional, and on the other hand we obtain

(πAG′ res+f aG)(χ) = (πAG′ res+f )(∑ϕ∈G

αG(G,ϕ)(χ)[G,ϕ]G),

since (πAG′ res+f )([H,ϕ]G) = 0 for H < G. But αG(G,ϕ)(χ) = 0 for ϕ ∈ G by

part (viii).Now let H ′ < G′. Then the induction hypothesis for H ′ implies that

res+G′

H′ aG′ resf = aH′ resG′

H′ resf = aH′ resf : H′→f(H′) resGf(H′)

= res+f : H′→f(H′) af(H′) resGf(H′)

= res+f : H′→f(H′) res+Gf(H′) aG

= res+G′

H′ res+f aG,


and the result follows.

1.3 Remark (i) In general, the maps aH , H ≤ G, are not multiplicative(cf. Proposition II.2.9 (i)). Neither is a : M → A+ a morphism of Mackey functors,since condition (ii) in Lemmma II.2.5 fails.

(ii) If we define pH : M(H) → A(H) for solvable subgroups H ≤ G as in Ex-ample 1.1, and by pH = 0 for non-solvable subgroups H ≤ G, we obtain anotherintegral canonical induction formula for M from A which induces only from solvablesubgroups. The statements of Proposition 1.2 still hold except for the parts (iii),(v), (xi), (xii), (xiii). Part (v) still holds for solvable H. In parts (vi) and (vii) thesums have to run only over chains (H0, ϕ0) < . . . < (Hn, ϕn) with solvable Hn.

More generally we may define a similar integral canonical induction formula byusing instead of solvable subgroups a non-empty set T of subgroups of G, which isclosed under taking subgroups and conjugates, with the property that if KH ≤ G,K ∈ T , H/K cyclic, then H ∈ T . In fact, then condition (∗π) of Theorem II.4.5holds for the set π of all primes. On the other hand (∗π) is also a necessary condi-tion. It is shown in [Bo90] that the those sets T with the above property lead tointegral canonical induction formula, since otherwise already aG(1) is not integral.This comes from an interpretation of aG(1) as the idempotent in the Burnside ringassociated to the set T of subgroups of G. This is in contrast to Brauer’s inductiontheorem which needs only elementary subgroups. But there seems to be no integralcanonical induction formula which induces only from elementary subgroups.

The following example can be found in [BCS93].

1.4 Example Let G be a finite group, M := R ∈ Z−Mackalg(G), A := Rab ∈Z−Resalg(G), B(H) := H for H ≤ G, and define p ∈ Q−Con(G)(Q ⊗M,Q ⊗ A)by (cf. Proposition I.1.4 and Remark I.3.3)

pH(χ) := e(H)H · χ =

1

|H|∑K≤H

|K|µ(K,H)indHK(resHK(χ)

)for H ≤ G, χ ∈ Q ⊗M(H), and note that e

(H)H · χ ∈ Q ⊗ A(H), since e

(H)H · χ = 0

for non-cyclic H ≤ G, cf. Proposition I.6.2.


(i) The morphism a : Q⊗M → Q⊗A+ is a canonical induction formula.(ii) The morphism a is a morphism of Q-Green functors on G.(iii) For H ≤ G, aH is a morphism of Q⊗A(H)-modules.(iv) For H ≤ G and χ ∈ Q⊗M(H) we have the explicit formulae

aH(χ) =1

|H|∑

L≤K≤HK cyclic

|L|µ(|K/L|)[K, indKL

(resHL (χ)

)]H

and

χ =1

|H|∑

L≤K≤HK cyclic

|L|µ(|K/L|)indHL(resHL (χ)

),


where µ : N → −1, 0, 1 denotes the number theoretic Mobius function. In partic-ular, αH(K,ψ)(χ) = 0 for (K,ψ) ∈M(H) with K non-cyclic.

(v) For H ≤ G and χ ∈M(H) we have

αH(1,1)(χ) =χ(1)

|H|.

(vi) For H ≤ G and χ ∈M(H) we have∑(K,ψ)∈H\M(H)

αH(K,ψ)(χ)indHK(1) =χ(1)

|H|indH1 (1).

(vii) For H ≤ G and χ ∈M(H) we have∑(K,ψ)∈H\M(H)

1

|K|αH(K,ψ)(χ) =

χ(a)

|H|=

∑(K,ψ)∈H\M(H)

αH(K,ψ)(χ).

(viii) The morphism a is Galois invariant.(ix) For H ≤ G we have

aH(ker(εH : Q⊗M(H)→ Q⊗A(H))

)⊆ ker(ε+ : Q⊗A+(H)→ Q⊗A+(H)),

i.e. virtual characters of degree zero are mapped under aH to the span of [K,ψ]H −[K, 1]H , (K,ψ) ∈M(H).

Proof (i) This follows from Proposition II.2.2 (ii).(ii) It follows from Proposition II.2.9 (i) that the maps aH , H ≤ G, are ring

homomorphisms, and Lemma II.2.5 (iii) implies that a is a morphism of Mackeyfunctors on G.

(iii) This follows from Proposition II.2.9 (ii).(iv) By Definition II.3.4 we have

aH(χ) =1

|H|∑

L≤K≤H|L|µ(L,K)

[L, resKL

(e

(K)K · resHK(χ)

)]H.

Proposition I.6.2 implies that resKL(e

(K)K · resHK(χ)

)= 0 unless L = K and K is

cyclic. Hence we obtain

aH(χ) =1

|H|∑K≤HK cyclic

|K|[K, e

(K)K · resHK(χ)

]H

=1

|H|∑

L≤K≤HK cyclic

|L|µ(|K/L|)[K, indKL (resHL (χ))

]H,

since µ(L,K) = µ(|K/L|) for cyclic K (the posets of intermediate subgroups of Land K is isomorphic to the poset of divisors of |K/L|). The second equation followsby applying bH .

(v) This follows immediately from the first equation in part (iv) by evaluatingthe single summand with K = 1.


(vi) Using the first equation in part (iv) we have∑(K,ψ)∈H\M(H)

αH(K,ψ)(χ)indHK(1) =χ(1)

|H|∑

1≤L≤K≤HK cyclic

|L|µ(|K/L|)|K/L|indHK(1)

which equals |H|−1χ(1)indH1 (1) by Mobius inversion of the second form in Corol-lary B.3 (ii) in the poset of cyclic subgroups of H.

(vii) The first equation follows immediately from Equation (3.2) by countingdegrees. The second equation follows from taking scalar products with the trivialcharacter on both sides of the equation in part (vi).

(viii) This follows from the commutativity of Diagramm (2.2), since pK , πAK ,resHK , and res+

HK , K ≤ H ≤ G, respect the Galois action.

(ix) This follows immediately from part (iv), since indKL (resHL (χ)) has degreezero, if χ has.

1.6 Remark (i) Since e(H)H · χ = 0 for non-cyclic H ≤ G and χ ∈ M(H),

the following definition leads to an equivalent canonical induction formula: A(H)and pH are defined as in Example 1.4 for cyclic subgroups H ≤ G, and A(H) = 0,pH = 0 for non-cyclic subgroups H ≤ G. This is exactly the canonical inductionformula of Example II.2.8.

(ii) In [BCS93, Thm. 2.14] it is proved that the canonical induction formulaa : Q ⊗ M(H) → Q ⊗ A+(H) of Example 1.4 is the unique canonical inductionformula for Q ⊗M from Q ⊗ A which is a morphism of Q-Mackey functors on Gsuch that aH is Q ⊗ A(H)-linear for all H ≤ G. In [BCS93] the two canonicalinduction formulae of Example 1.1 and Example 1.4 where used together in orderto obtain criteria for compatibility with induction of functions on the character ringwhich are defined as extensions of functions on Rab using the canonical inductionformulae. This was applied to define a conductor in the non-separable residue fieldcase for arbitrary Galois representations.

(iii) The canonical induction formula of Example 1.4 does not commute withinflation or restriction along arbitrary group homomorphisms.

1.7 Example Let G be a finite group, M := R ∈ Z−Mackalg(G), A :=

Rab ∈ Z−Resalg(G), B(H) := H for H ≤ G, and let p ∈ Z−Con(G)(M,A) bydefined for H ≤ G and χ ∈ Irr(H) by

pH(χ) =

χ, if H is cyclic,

0, otherwise.

Note that for H ≤ G, ϕ ∈ H, and χ ∈ Q⊗M(H) we have mϕ(χ) = 0 unless H iscyclic, and for cyclic H we have mϕ(χ) = (ϕ, χ)H .


(i) The morphism a is a canonical induction formula.(ii) The morphism a is Galois invariant.(iii) For H ≤ G, aH is Q⊗A(H)-linear.


(iv) For H ≤ G, aH is multiplicative, but does not preserve the unity, if H isnon-cyclic.

(v) For H ≤ G and χ ∈M(H) we have

aH(χ) =1

|H|∑

L≤K≤HK cyclic

|L|µ(|K/L|)[L, resHL (χ)

]H

and

χ =1

|H|∑

L≤K≤HK cyclic

|L|µ(|K/L|)indHL (resHL (χ)).

(vi) For H ≤ G and χ ∈M(H) we have∑(K,ψ)∈H\M(H)

αH(K,ψ)(χ)indHK(1) = χ(1) · 1.

(vii) For H ≤ G and χ ∈M(H) we have∑(K,ψ)∈H\M(H)

(H : K)αH(K,ψ)(χ) = χ(1) =∑

(K,ψ)∈H\M(H)

αH(K,ψ)(χ).

(viii) If H is cyclic, then aH(ϕ) = [H,ϕ]H for ϕ ∈ H.

Proof (i) This follows from Corollary II.2.4.(ii) This follows from Diagram (2.2) as in the previous proposition.(iii) This follows from Poposition II.2.9 (ii).(iv) This follows from Diagram (2.2) noting tat πA, p, res, and res+ are multi-

plicative.(v) The first equation follows from Definition II.3.4 and the same remark on the

Mobius function on posets of cyclic subgroups as in the proof of Proposition 1.5 (vi).The second equation follows from the first one by applying bH .

(vi) This is proved in the same way as Proposition 1.2 (ix).(vii) The first equation follows from Equation (3.2) by counting degrees, and the

second equation follows from part (vi) by taking scalar products with the trivialcharacter.

(viii) This follows immediately form Lemma II.3.7.

1.9 Remark (i) The second equations in Propositions 1.5 (iv) and 1.8 (v)are exactly Brauer’s explicit version of Artin’s induction theorem from [Bra51], butthe two canonical induction formulae of Examples 1.4 and 1.7 are not equivalent(cf. Lemma II.3.9).

(ii) If one chooses a set T of subgroups of G which contains all cyclic subgroups,and which is closed under taking subgroups and conjugates, we may replace theset of cyclic subgrouups in Example 1.7 with T and obtain again a canonical in-duction formula for which Proposition 1.8 holds with the appropriate changes. In[Bo90, Cor. 3.13] it is shown that such a canonical induction formula is integralif and only if, whenever K H ≤ G with K ∈ T and H/K cyclic, then H ∈ T(cf. Remark 1.3 (ii)).


(iii) In general, the canonical induction formula a of Example 1.7 is not integral,is not a morphism of Mackey functors (by Lemma II.2.5 (ii)), and does not commutewith restrictions along arbitrary group homomorphisms.

The next example was suggested by H. Fottner.

1.10 Example Let N be a normal subgroup of a finite group G, and letϑ ∈ Irr(N). For a subgroup H ≤ G with N ≤ H we define Rϑ(H) as the span of

Irrϑ(H) := χ ∈ Irr(H) | ( gϑ, resHN (χ))N 6= 0 for some g ∈ G

and we define Rabϑ (H) as the span of

Irr∗ϑ(H) := χ ∈ Irr(H) | resHN (χ) = gϑ for some g ∈ G.

Then the definitions

M(H/N) := Rϑ(H), A(H/N) := Rabϑ (H), B(H/N) := Irr∗ϑ(H)

for N ≤ H ≤ G define a Z-Mackey functor M on G/N and a Z-restriction sub-functor A of M on G/N which has a stable basis B. For N ≤ K ≤ H ≤ Gand g ∈ G the conjugation, restriction and induction maps for M are defined by

cgN,H/N := cg,H , resH/NK/N := resHK , and ind

H/NK/N := indHK . Furthermore, we define a

morphism p : M → A of Z-conjugation functors for N ≤ H ≤ G and χ ∈ Irrϑ(H)by

pH/N (χ) =

χ, if χ ∈ Irr∗ϑ(H),

0, otherwise.

Note that mσ = 1 for all chains σ ∈ Γ(M(G/N)) and that C(M) is the set of allcyclic subgroups of G/N .


(i) The morphism a is an integral canonical induction formula.(ii) For N ≤ H ≤ G and χ ∈ Irr∗ϑ(H) we have aH/N (χ) = [H/N,χ]H/N .(iii) The morphism a is given explicitly by

aH/N (χ) =

=∑(

(H0/N,ϕ0)<...<(Hn/N,ϕn))∈H\Γ(M(H/N))

(−1)n(resHHn(χ), ϕn

)Hn

[H0/N,ϕ0]H/N

for N ≤ H ≤ G and χ ∈ Rϑ(H), and we have

V =∑(

(H0/N,ϕ0)<...<(Hn/N,ϕn))∈H\Γ(M(H/N))

(−1)nindHH0(V (Hn,ϕn))

in Rϑ(H) for a CH-module V whose character is in Rϑ(H), where V (Hn,ϕn) denotesthe sum of all CHn-submodules of resHHn(V ) which have character ϕn.

3.2. THE RING OF BRAUER CHARACTERS 73

Proof First we show that condition (∗π) in Theorem II.4.5 is satisfied forthe set π of all primes. Let N ≤ K H ≤ G with H/K cyclic and let ψ ∈Irr∗ϑ(K) be fixed under H. Let furthermore χ ∈ Irrϑ(H). Then resHK(χ) is a sumof H-conjugates of some ξ ∈ Irrϑ(K). If ξ 6= ψ, then the multiplicities of ψ inresHK(pH/N (χ)) and pK/N (resHK(χ)) are both zero. If ξ = ψ, then χ is a constituent

of indHK(ψ), which is the sum of the (H : K) different extensions of ψ, since H/K iscyclic and ψ is H-stable). Hence, χ is an extension of ψ, and therefore, χ ∈ Irr∗ϑ(H).This implies that resHK(pH/n(χ)) = ψ = pK/N (resHK(χ)). Now the integrality of aand part (iii) follow from Corollary II.4.8.

Next we show that a is a canonical induction formula using Proposition II.2.2.We have to show that

pH/N (χ)− χ ∈∑

N≤K<HindHK(Q⊗Rϑ(H))

for all N ≤ H ≤ G with H/K cyclic (cf. Proposition I.6.2), and for all χ ∈ Irrϑ(H).If χ ∈ Irr∗ϑ(H), then pH/N (χ) − χ = 0. If χ /∈ Irr∗ϑ(H), then the stabilizer S of ϑin H is strictly contained in H and χ is the induced character of an extension of aG-conjugate of ϑ to S by Clifford theory. This completes the proof of part (i).

Part (ii) follows from Lemma II.3.7.

1.12 Remark (i) If N is the trivial subgroup and ϑ the trivial character, thenExample 1.10 coincides with Example 1.1. Hence, Example 1.10 can be regardedas a generalization of Example 1.1.

(ii) Assume that ϑ is stable under G and that ϑ has an extension ϑ ∈ Irr(G).We define ϑH := resGH(ϑ) for N ≤ H ≤ G. Then it is well-known that

Irrϑ(H) = ϑH · infHH/N (χ) | χ ∈ Irr(H/N)

and this defines a bijection between Irrϑ(H) and Irr(H/N) which restricts to abijection between Irr∗ϑ(H) and Hom(H/N,C×). Moreover, this induces an isomor-phism of Mackey functors Rϑ(H) ∼= R(H/N), N ≤ H ≤ G, which identifies Rab

ϑ (H)and Rab(H/N) for N ≤ H ≤ G. Moreover pH/N is then identified with pH/N ofExample 1.1, and by Proposition II.3.11 we obtain a commutative digramm

Rϑ(H)aH/N−−−−→ Rab

ϑ +(H/N)

oy yo

R(H/N)aH/N−−−−→ Rab

+ (H/N),

where the lower map aH/N is the canonical induction formula of Example 1.1.

3.2 The ring of Brauer characters

In this section we will derive canonical induction formulae for the ring of Brauercharacters. Throughout this section let F denote an algebraically closed field ofcharacteristic p > 0. For a finite group G let RF (G) denote the Grothendieck ringof the category FG−mod of finite dimensional FG-modules. For V ∈ FG−mod,


the element in RF (G) associated to V will be denoted by [V ]. Arbitrary elementsin RF (G) will often be denoted by χ. The ring RF (G) is a free abelian group onthe elements [S], where S runs through a set of representatives for the isomorphismclasses of simple FG-modules. Whith the usual conjugation, restriction, inductionand multiplication, the rings RF (H), H ≤ G, form a Z-Green functor on G. ForH ≤ G let Rab

F (H) denote the span of those elements [S] ∈ RF (H) with S ∈FH−mod of dimension one. Then the rings Rab

F (H), H ≤ G, form a Z-algebrarestriction subfunctor Rab

F of RF on G. Obviously, for H ≤ G, the basis [S] | S ∈FH−mod,dimS = 1 of Rab

F (H) can be identified with H(F ) := Hom(H,F×).

For ϕ ∈ H(F ) we denote the corresponding FH-module by Fϕ. If ϕ = 1, then wesimply write F instead of F1.

Note that the set C(RF ) of coprimordial subgroups for RF is the set of cyclic p′-subgroups of G. This is clear from the well-known interpretation of RF (H), H ≤ G,as the ring of Brauer characters of H.

Similarly as in Section 1 the above construtions do not depend on the choice ofF , but only on p.

For a finite group G, any automorphism of the field F induces a ring automor-phism of RF (G) by using the induced automorphism on FG. These ring automor-phisms of RF (H), H ≤ G, form an isomorphism of Z-Green functors RF → RFon G. Similarly as in Section 1 we obtain actions of the group of automorphismsof F on RF , Rab

F , and RabF + by isomorphisms of Z-Green functors, resp. Z-algebra

restriction functors on G in the case of RabF , which will be called Galois actions.

If G′ is another finite group and f : G′ → G is a group homomorphism, we ob-tain, as in Section 1, ring homomorphisms resf : RF (G)→ RF (G′), resf : Rab

F (G)→RabF (G′), and res+f : Rab

F +(G) → RabF +(G′). The latter is given again by the for-

mula (3.3). The definition of res+f can be motivated by V.1.25. We also define

inflation maps on the level of RF , RabF , and Rab

F +, and note that formula (3.4) holds

also in this situation. With these constructions, RF , RabF , and Rab

F + can be consid-ered as contravariant functors from the category of finite groups to the category ofcommutative rings or to the category of abelian groups.

As in Section 1 we define a morphism ε : RF → RabF of Z-algebra restriction

functors by εH([V ]) := dimF (V ) · [F ] for H ≤ G and V ∈ FH−mod. Thisinduces a morphism ε+ : Rab

F + → RabF + of Z-Green functors on G which is given by

ε+H([K,ψ]H) = [K, 1]H for K ≤ H ≤ G, ψ ∈ K(F ).

Note that mσ = 1 for σ =((H0, ϕ0) < . . . < (Hn, ϕn)

)∈ Γ(M(G))) withM(G)

based on B(H) := H(F ) for H ≤ G.

2.1 Example Let G ba a finite group, M := RF ∈ Z−Mackalg(G), A :=

RabF ∈ Z−Resalg(G), B(H) := H(F ) for H ≤ G, and define p ∈ Z−Con(G)(M,A)

for H ≤ G and a simple FH-module S by

pH([S]) :=

[S], if dimF S = 1,

0, otherwise.

Note that mϕ([V ]) is the number of composition factors of V isomorphic to Fϕ forH ≤ G, V ∈ FH−mod, and ϕ ∈ H(F ), cf. Definition II.4.1.



(i) The morphism a is an integral canonical induction formula, i.e. it can beconsidered as a morphism a : M → A+.

(ii) For H ≤ G the homomorphism aH is A(H)-linear, i.e. aH(ϕ·[V ]) = [H,ϕ]H ·aH([V ]) for ϕ ∈ H(F ).

(iii) The morphism a respects the Galois action.(iv) For H ≤ G and ϕ ∈ H(F ) we have aH([Fϕ]) = [H,ϕ]H .(v) The morphism a is given explicitly by

aH([V ]) =∑(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)nmϕn(resHHn([V ]))[H0, ϕ0]H

for H ≤ G and V ∈ FH−mod, and we have

[V ] =∑(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)nmϕn(resHHn([V ]))indHH0([Fϕ0 ]).

(vi) For H ≤ G, V ∈ FH−mod and (K,ψ) ∈M(H) we have

αH(K,ψ)([V ]) =∑(

(H0,ϕ0)<...<(Hn,ϕn)

)∈H\Γ(M(H))

(H0,ϕ0)=H (K,ψ)

(−1)nmϕn(resHHn([V ]))

=|K|

|NH(K,ψ)|∑(

(H0,ϕ0)<...<(Hn,ϕn)

)∈Γ(M(H))

(H0,ϕ0)=(K,ψ)

(−1)nmϕn(resHHn([V ])).

(vii) For H ≤ G, S a simple FH-module, and (K,ψ) ∈M(H) we have

mψ

(resHK([S])

)= 0 =⇒ αH(K,ψ)([S]) = 0.

(viii) We have b ε+ a = ε, or more explicitly∑(K,ψ)∈H\M(H)

αH(K,ψ)([V ])indHK([F ]) = dimF (V ) · [F ]

for H ≤ G, V ∈ FH−mod.(ix) Let H ≤ G, S a simple FH-module, and let (Z(S), λ) ∈ M(H) be the

central pair of S, where Z(S) is defined as the maximal subgroup K of H whichacts on S by scalar multiplication via some element ψ ∈ K(F ), i.e. resHK(S) ∼=⊕dimS

i=1 Fψ, and where λ ∈ Hom(Z(S), F×) is such that resHZ(S)(S) ∼=⊕dimS

i=1 Fλ.

Then we have for (K,ψ) ∈M(H):

(Z(S), λ) 6≤ (K,ψ) =⇒ αH(K,ψ)([S]) = 0.

(x) For H ≤ G, V ∈ FH−mod and (K,ψ) ∈M(H) we have

Z(H)p′ 6≤ K =⇒ αH(K,ψ)([V ]) = 0,


where Z(H)p′ deonotes the p′-part of the centre Z(H) of H.(xi) Let f : G′ → G be a homomorphism of finite groups. Then the diagram

RF (G)aG−−−−→ Rab

F +(G)

resf

y yres+f

RF (G′)aG′−−−−→ Rab

F +(G′)

commutes. In particular, the canoncal induction formulae aG, where G runs overall finite groups, form a natural transformation between the contravariant functorsRF and Rab

F + with values in abelian groups.

Proof We only show that condition (∗π) in Theorem II.4.5 is satisfied for thesetπ of all primes. All other parts are proved in a similar way with the appropriatechanges as the corresponding parts in Proposition 1.2.

So let K H ≤ G with H/K cyclic, and let ψ ∈ K(F ) be fixed under H.Let furthermore S be a simple FH-module. Then resHK(S) is a direct sum of H-conjugates of some simple FK-module T . If T is not isomorphic to Fψ, thenthe multiplicity of ψ in pK(resHK([S])) is zero as well as the multiplicity of ψ inresHK(pH([S])). If T is isomorphic to Fψ, then

0 6= dimF HomFK(Fψ, resHK(S)) = dimF HomFH(indHK(Fψ), S),

and S is isomorphic to a simple quotient of indHK(Fψ). But indHK(Fψ) has only one-dimensional composition factors, since indHK(Fψ) comes via inflation from H/ kerψwhich is an abelian group (H fixes ψ). Therefore, dimS = 1 and resHK(S) ∼= Fψ.This implies that pK(resHK([S])) = ψ = resHK(pH([S])).

2.3 Remark (i) In general, the maps aH , H ≤ G, are not multiplicative, sincethe maps pH , H ≤ G, aren’t (cf. Proposition II.2.9 (i)). Neither is a : RF → Rab

F + amorphism of Mackey functors, since condition (ii) in Lemma II.2.5 fails in general.

(ii) Let T be a non-empty set of subgroups of G which is closed under takingsubgroups and conjugates such that T contains with a subgroup K ≤ G all sub-groups H of G with K ≤ H ≤ NG(K) and H/K cyclic. For example, T may be theset of solvable subgroups of G. If we define pH : M(H)→ A(H) as in Example 2.1for H ∈ T , and pH := 0 for H /∈ T , then we obtain again an integral canonicalinduction formula inducing only from the subgroups in T . In this situation parts(i), (ii), (iii) of Propostion 2.2 hold without changes, part (iv) holds for H ∈ T ,parts (v) and (vi) hold with the summations taken over chains with Hn ∈ T , andparts (vii) and (viii) hold without changes.

(iii) LetO be a complete discrete valuation ring with residue field F and quotientfield K of characteristic zero containing all |G|-th roots of unity. Then, for H ≤ G,there is the well-known decomposition map dH : RK(H)→ RF (H). The maps dH ,H ≤ G, form a morphism of Mackey functors and induce morphisms d : Rab

K → RabF

and d+ : RabK + → Rab

F + of restriction functors and Mackey functors respectively.One might expect that the diagram

RKa−−−−→ Rab

K +

d

y yd+

RFa−−−−→ Rab

F +


commutes, where the upper canonical induction formula a is the one of Example 1.1and the lower one comes from our last Example 2.1. But this is not true in generalby Proposition II.3.11, since dH(χ) may have one-dimensional constituents for χ ∈IrrK(G) with χ(1) > 1. Even worse, there are examples for groups G (e.g. G = S4)and virtual characters χ ∈ RK(G) (e.g. χ = 1 + ε−χ2, ε the sign character, χ2 theunique irreducible character of degree 2) with dG(χ) = 0, but d+G(aG(χ)) 6= 0. Thisshows that there can’t exist a map a : RF → Rab

F + rendering the above diagramcommutative.

2.4 Example Let G be a finite group, M := RF ∈ Z−Mackalg(G), A(H) :=

RabF (H) and B(H) := H(F ) for H ≤ G cyclic of p′-order, and A(H) := 0 andB(H) := ∅ otherwise. We define p ∈ Q−Con(G)(Q ⊗M,Q ⊗ A) by (cf. Proposi-tion I.1.4 and Remark I.3.3)

pH(χ) := e(H)H · χ =

1

|H|∑K≤H

|K|µ(K,H)indHK(resHK(χ)

)for H ≤ G, χ ∈ Q ⊗M(H), and note that e

(H)H · χ ∈ Q ⊗ A(H), since e

(H)H · χ = 0

for H /∈ C(RF ), and since M(H) = A(H) for H ∈ C(RF ). Note that automatically,αH(K,ψ)(χ) = 0 unless H is a cyclic subgroup of p′-order.


(i) The morphism a : Q⊗M → Q⊗A+ is a canonical induction formula.(ii) The morphism a : Q⊗M → Q⊗A+ is a morphism of Q-Green functors on

G.(iii) For H ≤ G, aH is a morphism of Q⊗A(H)-modules, cf. the last paragraph

in I.2.2.(iv) For H ≤ G and χ ∈ Q⊗RF (H) we have the explicit formulae

aH(χ) =1

|H|∑

L≤K≤HK cyclic, |K|p = 1

|L|µ(|K/L|)[K, indKL

(resHL (χ)

)]H

and

χ =1

|H|∑



),

where µ : N→ −1, 0, 1 denotes the Mobius function from number theory.(v) For H ≤ G and V ∈ FH−mod we have

αH(1,1)([V ]) =dimF V

|H|.

(vi) For H ≤ G and V ∈ FH−mod we have∑(K,ψ)∈H\M(H)

αH(K,ψ)([V ])indHK([F ]) =dimF V

|H|indH1 ([F ]).


(vii) For H ≤ G and V ∈ FH−mod we have∑(K,ψ)∈H\M(H)

1

|K|αH(K,ψ)([V ]) =

dimF V

|H|=

∑(K,ψ)∈H\M(H)

αH(K,ψ)([V ]).

(viii) The morphism a is Galois invariant.(ix) For H ≤ G we have

aH(ker(εH : Q⊗M(H)→ Q⊗A(H)

))⊆ ker

(ε+H : Q⊗A+(H)→ Q⊗A+(H)

),

i.e. virtual Brauer characters of degree zero are mapped to the span of [K,ψ]H −[K, 1]H , (K,ψ) ∈M(H) under aH .

Proof This follows by exactly the same arguments as in the proof of Propo-sition 1.5.

2.6 Remark (i) The canonical induction formula of Example 2.4 does notcommute with inflation or arbitrary group homomorphisms.

(ii) Let O be a complete discrete valuation ring with residue field F and quotientfield K of characteristic zero containing the |G|-th roots of unity. Then we have acommutative diagram

Q⊗RKa−−−−→ Q⊗Rab

K +

d

y yd+

Q⊗RFa−−−−→ Q⊗Rab

F +

with the upper and lower a being the canonical induction formulae of Example 1.4and Example 2.4 respectively, and with d being the decomposition map, cf. Re-mark 2.3 (iii). This follows from Proposition II.3.11, since

dH(e(H)H · χ) =

1

|H|∑K≤H

|K|µ(K,H)dH(indHK(resHK(χ))

)=

1

|H|∑K≤H

|K|µ(K,H)indHK(resHK(dH(χ))

)= e

(H)H · dH(χ),

for H ≤ G, χ ∈ Q⊗RK(H), since d is a morphism of Q-Green functors on G.

2.7 Example Let G be a finite group, M := RF ∈ Z−Mackalg(G), A(H) :=

RabF (H) and B(H) := H(F ) for H ≤ G a cyclic p′-group, and A(H) = 0 andB(H) = ∅ otherwise. Define p ∈ Z−Con(G)(M,A) for H ≤ G an χ ∈M(H) by

pH(χ) :=

χ, if H is a cyclic p′-group

0, otherwise.


(i) The morphism a is a canonical induction formula.


(ii) The morphism a is Galois invariant.(iii) For H ≤ G, aH is Q⊗A(H)-linear.(iv) For H ≤ G, aH is multiplicative.(v) For H ≤ G and χ ∈ Q⊗RF (H) we have

aH(χ) =1

|H|∑


|L|µ(|K/L|)[L, resHL (χ)]H

and

χ =1

|H|∑



).

(vi) For H ≤ G and V ∈ FH−mod we have∑(K,ψ)∈H\M(H)

αH(K,ψ)indHK([F ]) = dimF V · [F ].

(vii) For H ≤ G and V ∈ FH−mod we have∑(K,ψ)∈H\M(H)

(H : K)αH(K,ψ)([V ]) = dimF V =∑

(K,ψ)∈H\M(H)

αH(K,ψ)([V ]).

(viii) If H ≤ G is a cyclic p′-subgroup, then aH(ϕ) = [H,ϕ]H for ϕ ∈ H(F ).

Proof (i) This follows from Corollary II.2.4.(ii) This follows from Diagram (2.2).(iii) This follows from Proposition II.2.9 (ii).(iv) This follows from Diagram (2.2), since πA, p, res, and res+ are multiplica-

tive.(v) The first equation follows from Definition II.3.4, and the second equation

follows from applying bH to the first one.(vi) This is proved in the same way as Proposition 2.2 (viii).(vii) This follows from Equation (3.2) by counting dimensions and counting

multiplicites of [F ] on both sides of the equation of part (vi). Note that [F ] hasmultiplicity one in indHK([F ]), if H is a p′-group.

(viii) This follows from Lemma II.3.7.

2.9 Remark (i) In general, the canonical induction formula of Example 2.7is not a morphism of Mackey functors and does not commute with inflations orrestrictions along arbitrary group homomorphisms.

(ii) If T is a set of subgroups of G containing the cyclic p′-subgroups of G whichis closed under taking subgroups and conjugates, then there is a canonical inductionformula for RF from Rab

F on subgroups in T , which is defined in the same way asthe one in Example 2.7 by replacing the set of cyclic p′-subgroups with T . thiscanonical induction formula is integral if and only if T satisfies the condition ofRemark 2.3 (ii), since for H ≤ G, aH(1) is the sum of all primitive idempotents

e(H)K of Q ⊗ Ω(H) ⊆ Q ⊗ Rab

F + with K ∈ T , and it is well-known that this sum is


integral if and only if T has the property mentioned in Remark 2.3 (ii), cf. [Yo83a,Thm. 3.1].

If we choose T to be the set of all cyclic subgroups, then the resulting canonicalinduction formula and the one of Example 1.7 commute with the decompositionmap by Proposition II.3.11.

3.3 The Grothendieck group of (π-) projectivemodules

Throughout this section let F be an algebraically closed field of characteristic p > 0.For a finite group G let PF (G) denote the Grothendieck group of finite dimensionalprojective FG-modules. The abelian group PF (G) is free on the elements [P ] as-sociated to the projective indecomposable FG-modules P . By P ab

F (G) we denotethe span of the elements [P ], where P is a one-dimensional projective FG-module.Thus, P ab

F (G) = 0 unless G is a p′-group, and if G is a p′-group, then all FG-modules are projective, in particular PF (G) = RF (G) and P ab

F (G) = RabF (G). For a

p′-group G we can also identify the set of elements [P ], where P is an FG-module ofdimension one, with the multiplicative group G(F ) := Hom(G,F×), and note thatP abF (G) is free on G(F ). For ϕ ∈ H(F ) we denote the corresponding FH-module

by Fϕ, or just by F in case that ϕ = 1.For a finite group G, the groups PF (H), H ≤ G, form a Z-Mackey functor

on G with the usual conjugation, restriction and induction maps, and the groupsP abF (H), H ≤ G, form a Z-restriction subfunctor of PF . If we define for H ≤ G,

B(H) := H(F ), if H is a p′-group, and B(H) := ∅, if H is not a p′-group, thenPF , P ab

F , and B satisfy Hypothesis II.3.1. The set C(PF ) of coprimordial subgroupsfor PF is the set of cyclic p′-subgroups of G. In fact, since PF ⊆ RF , we haveC(PF ) ⊆ C(RF ), and one can verify easily that each cyclic p′-subgroup of G iscoprimordial.

As in the previous sections we have a Galois action of the automorphism groupof F on PF , P ab

F , and on P abF +, and the morphism bPF ,P

abF respects the Galois

action. Moreover, we have a morphism ε+ : P abF + → P ab

F + of Z-Mackey functors onG, given by ε+H([K,ψ]H) = [K, 1]H for (K,ψ) ∈M(H).

Note that PF , P abF , and P ab

F + cannot be considered as contravariant functors onthe categories of finite groups, since projectivity is not preserved under inflation.

For any chain σ ∈ Γ(M(G)) the multiplicity mσ is 1, cf. Definition II.4.1.

3.1 Example Let G be a finite group, M := PF ∈ Z−Mack(G), A := P abF ∈

Z−Res(G), B(H) := H(F ) for a cyclic p′-subgroup H ≤ G, B(H) := ∅ otherwise.We define p ∈ Z−Con(G)(M,A) by

pH([P ]) :=

[P ], if dimF P = 1,

0, otherwise,

for H ≤ G and a projective indecomposable FH-module P . Note that for a p′-subgroup H ≤ G, ϕ ∈ H(F ), and any (projective) FH-module P , the multiplicitymϕ([P ]) is the number of summands of P isomorphic to Fϕ in a direct decompositionof P .

3.3. PROJECTIVE CHARACTERS 81


(i) The morphism a is an integral canonical induction formula.

(ii) The morphism a respects the Galois action.

(iii) For a p′-subgroup H ≤ G and ϕ ∈ B(H) we have aH(ϕ) = [H,ϕ]H .

(iv) The morphism a is given explicitly by

aH([P ]) =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|NH(σ)p′ |mϕn(resHHn([P ]))

|NH(σ)|[H0, ϕ0]H

for H ≤ G and each projective P -module, with

|NH(σ)p′ |mϕn(resHHn([P ]))

|NH(σ)|∈ Z,

and we have

[P ] =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n|NH(σ)p′ ||NH(σ)|

indHH0

(P (Hn,ϕn)

),

where

P (Hn,ϕn) := x ∈ P | hx = ϕn(h)x for h ∈ Hn.

(v) For H ≤ G, P ∈ FH−mod projective indecomposable, and (K,ψ) ∈M(H)we have

mψ(resHK([P ])) = 0 =⇒ αH(K,ψ)([P ]) = 0.

(vi) For H ≤ G and P ∈ FH−mod projective, we have

∑(K,ψ)∈H\M(H)

αH(K,ψ)([P ]) =|Hp′ ||H|

dimF P.

Proof We first show that condition (∗π) in Theorem II.4.5 is satisfied for theset π of all primes different from p. Let KH ≤ G with H/K a cyclic p′-group, andlet ψ ∈ B(K) (in particular K, and hence also H, is a p′-group) be stable under H.Let furthermore P ∈ FH−mod be indecomposable (hence simple and projective).We have to show that the coefficients of ψ in resHK(pH([P ])) and pK(resHK([P ]))coincide. Since H is a p′-group, this can be proved exactly in the same way as inthe proof of Proposition 1.2.

(iv) The first equation follows from Theorem II.4.5, and the second by applyingbH . Note that |(NH(σ)/H0)p′ | = |NH(σ)p′ |/|H0|, since H0 is a p′-group. So itsuffices to prove that

|NH(σ)p′ ||NH(σ)|p′

andmϕn(resHHn([P ]))

|NH(σ)|p


are integers. In fact |NH(σ)|p′ divides |NH(σ)p′ | by [Hu67, V.19.14], and |NH(σ)|pdivides mϕ(resHHn([P ])) by the following argument:

Let Q be a Sylow p-subgroup of NH(σ), H ′ := QHn, and let V be an indecom-posable summand of resHH′(P ). It suffices to show that |Q| divides mϕ(resH

′Hn

([V ])).

We may assume that mϕ(resH′

Hn([V ])) 6= 0. Since V is projective, V is a direct

summand of indH′

1 (F ) ∼= indH′

Hn(indHn1 (F )). Since H ′/Hn∼= Q is a p-group, V is

isomorphic to indH′

Hn(W ) for some simple FHn-module W by Green’s indecompos-ability theorem. Mackey’s decomposition formula then yields

resH′

Hn(V ) ∼=⊕

s∈H′/Hn

indHnHn∩

sHn

(res

sHnHn∩

sHn

( sW )) ∼= ⊕

s∈H′/Hn

sW.

Since Fϕn is a summand of resH′

Hn(V ), and since ϕn is stable under H ′ ≤ NH(σ), we

have resH′

Hn(V ) ∼=

⊕|H′/Hn|i=1 Fϕn , hence mϕn(resH

′Hn

([V ]) = |H ′/Hn| = |Q|.(i) It follows from Corollary II.2.4 that a is a canonical induction formula. The

integrality of a follows from part (iv).

(ii) This follows from Diagram (2.2), since πA, p, res, and res+ respect the Galoisaction.

(iii) This follows from Lemma II.3.7.

(v) This follows immediately from part (iv).

(vi) This will be proved later in Remark IV.3.9.

3.3 Remark (i) The morphism a of the last proposition is in general not amorphism of Mackey functors, since condition (ii) in Lemma II.2.5 does not hold.

(ii) There are more canonical induction formulae for PF inducing from cyclicp′-subgroups. They are defined similar to Example 1.4 and Example 1.7 in the caseof the character ring.

(iii) Let O be a complete discrete valuation ring with algebraically residue fieldF and quotient field K of characteristic zero which contains all |G|-th roots of unity.Let H ≤ G. It is well-known that the functor F⊗O− induces a bijection between theset of isomorphism classes of projective modules in OH−mod and in FH−mod.If we define PO(H) as the Grothendieck group of the category of pojective OH-modules, we obtain a Z-Mackey functor PO with the usual conjugation, restrictionand induction maps. Tensoring with F over O induces an isomorphism PO → PFwhich identifies P ab

F (H) with

P abO (H) :=

ZHom(H,O×), if H is a p′-group,

0, otherwise.

The morphism p ∈ Z−Con(G)(PF , PabF ) of Example 3.1 is translated into a mor-

phism p ∈ Z−Con(G)(PO, PabO ), and we obtain a canonical induction formula

a ∈ Z−Res(G)(PO, PabO +).

Moreover, it is well-known that the functor K ⊗O − induces, for H ≤ G, anisomorphism between PO(H) and the subgroup RK,p(H) of RK(H) consisting ofall virtual characters vanishing on p-singular elements. These isomorphisms forman isomorphism PO → RK,p of Z-Mackey functors on G which identifies, for H ≤

3.3. PROJECTIVE CHARACTERS 83

G, P abO (H) with the subgroup Rab

K,p(H) of RK(H) spanned by one-dimensionalcharacters vanishing on p-singular elements, i.e.

RabK,p(H) =

RabK (H), if H is a p′-group,

0, otherwise.

Now the morphism pH : PO(H)→ P abO (H) translates into pH : RK,p(H)→ Rab

K,p(H)with

pH(χ) =

∑ϕ∈H(K)(χ, ϕ)H · ϕ, if H is a p′-group,

0, otherwise,

for χ ∈ RK,p(H). Again we obtain a canonical induction formula a : RK,p → RabK,p+

which corresponds to a : PO → P abO + under the canonical isomorphism. This leads

to the following generalization, where the group RK,p(G) of p-projective virtual

characters is replaced with the group RK,π(G) of π-projective virtual characters

for a set π of primes.

3.4 Example Let G be a finite group, π a set of primes, and K a field ofchracteristic zero which contains all |G|-th roots of unity. Let M ∈ Z−Mack(G)be defined by

M(H) := RK,π(H) := χ ∈ RK(H) | χ(h) = 0 for h ∈ H with hπ 6= 1

for H ≤ G, and let

B(H) := ϕ ∈ Hom(H,K×) | ϕ ∈ RK,π(H),

hence B(H) = H(K) for π′-subgroups H ≤ G and B(H) = ∅ for H ≤ G with|H|π 6= 1. Furthermore, for H ≤ G, let A(H) := Rab

K,π(H) denote the span of B(H).

Then RabK,π is a restriciton fubfunctor of the Mackey functor RK,π on G. We define

p ∈ Z−Con(G)(M,A) by

pH(χ) :=∑

ϕ∈B(H)

(χ, ϕ)H · ϕ

for H ≤ G and χ ∈ RK,π(H). In particular, pH = 0 for H ≤ G with |H|π 6= 1.


(i) The morphism a is an integral canonical induction formula.(ii) The morphism a respects the Galois action.(iii) For H ≤ G of π′-order and ϕ ∈ B(H) we have aH(ϕ) = [H,ϕ]H .(iv) The morphism a is given explicitly by

aH(χ) =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|NH(σ)π′ | · (ϕn, resHHn(χ))Hn

|NH(σ)|[H0, ϕ0]H


for H ≤ G and χ ∈ RK,π(H), with

|NH(σ)π′ | · (ϕn, resHHn(χ))Hn|NH(σ)|

∈ Z,

and we have

χ =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|NH(σ)π′ | · (ϕn, resHHn(χ))Hn

|NH(σ)|indHH0

(ϕ0).

(v) For H ≤ G and χ ∈ RK,π we have∑(U,µ)∈H\M(H)

αH(U,µ)(χ) =|Hπ′ ||H|

χ(1).

Proof We first show that (∗π′) in Theorem II.4.5 is satisfied. Let U H ≤ Gwith H/U a cyclic π′-group and let ψ ∈ B(U) (in particular U , and hence H, is a π′-group) be stable under H. Let furthermore χ ∈ RK,π(H). If π is empty, then we arein the situation of Example 1.1, and everything is proved there. If π is non-empty wechoose a prime p ∈ π. We may replace K without loss of generality by a completediscrete valuation field with algebraically closed residue field of characteristic p.Then we may interpret ψ and χ as an FU -module and an FH-module respectively,and we may proceed as in the proof of Proposition 1.2.

(iv) The first equation follows now from Theorem II.4.5, and the second byapplying bH . Note that |(NH(σ)/H0)π′ | = |NH(σ)π′ |/|H0|, since H0 is a π′-group.We will show that

|NH(σ)π′ ||NH(σ)|π′

and(ϕn, resHHn(χ))Hn|NH(σ)|π

are integers. In fact, |NH(σ)|π′ divides |NH(σ)π′ | by [Hu67, V.19.14], and it followsfrom the proof of Proposition 3.2 (iv) that for each p ∈ π the p-part |NH(σ)|pdivides (ϕn, resHHn(χ))Hn , since we may translate everything into Example 3.1 forthe prime p as already done above.

(i) This follows from part (iv).(ii) This follows from Diagram (2.2):(iii) This follows from Lemma II.3.7.(v) This will be proved in Remark IV.3.9.

3.6 Remark Remark 3.3 (i) and (ii) still hold in this situation with p replacedby π. Note that inRK,π(H) there is no distinguished basis having a module theoreticinterpretation as this was the case for π = p.

3.4 The trivial source ring and the linear source ring

Throughout this section G denotes a finite group and O is a complete discretevaluation ring with algebraically closed residue field F and with a quotient field ofcharacteristic zero which is a splitting field for all groups occurring in this section.

3.4. TRIVIAL AND LINEAR SOURCE RINGS 85

By OG−lin we denote the category of those O-free OG-modules of finite O-rankwhose indecomposable direct summands have vertices of O-rank one. The categoryOG−triv of direct sums of indecomposable O-free OG-modules of finite O-rankwith trivial source is a full subcategory of OG−lin. Considering linear source

modules, i.e. objects of OG−lin, does not add erious difficulties to the proofs inthis section compared with considering trivial source modules only, i.e. objects ofOG−triv.

The isomorphism classes of objects W ∈ OG−lin with rkOW = 1 are obviouslyparametrized by G(O) := Hom(G,O×), and those of objects W ∈ OG−triv withrkOW = 1 by the p′-part G(O)p′ = Hom(G,O×)p′ . In fact, if for ϕ ∈ G(O)the corresponding OG-module is denoted by Oϕ, the Sylow p-subgroups of G arevertices of Oϕ and a source of Oϕ is given by Oψ, where ψ is the restriction of ϕ toa Sylow p-subgroup. Obviously, ψ is trivial, if and only if ϕ is of p′-order.

An O-free OG-module of finite O-rank is called monomial, if it is isomorphicto a direct sum of modules of the type indGH(Oϕ) with H ≤ G, ϕ ∈ H(O). Thefollowing proposition, which is well-known for trivial source modules if ‘monomialmodule’ is replaced with ‘permutation module’, (cf. [Bro85, Props. (0.2), (0.4)] forexample), collects some basic properties of linear source modules. We will omit theproofs which are just modifications of the proofs in the trivial source case.

4.1 Proposition Let G be a finite group and let P denote a Sylow p-subgroupof G.

(a) For V ∈ OG−mod the following are equivalent:

(i) V ∈ OG−lin.

(ii) The module resGP (V ) is a monimial OP -module.

(iii) V is a direct summand of a monomial module.

(b) For H ≤ G, M,M ′ ∈ OG−lin, and N ∈ OH−lin we have:

(i) The modules M ⊕M ′ and M ⊗M ′ are again linear source modules.

(ii) The modules resGH(M) and indGH(N) are again linear source modules.

(iii) Any direct summand of M is a linear source module.

By LO(G) (resp. TO(G)) we denote the Grothendieck ring with respect to directsums of the category OG−lin (resp. OG−triv) and call it the linear source ring

(resp. trivial source ring). For V ∈ OG−lin (resp. V ∈ OG−triv) we denote by [V ]the associated element in LO(G) (resp. TO(G)). Since the Krull-Schmidt-Azumayatheorem holds for OG−lin and OG−triv (cf. [CR81, 6.12]), LO(G) and TO(G) arefree abelian groups on the finite set of isomorphism classes [V ] of indecomposableobjects V in OG−lin and OG−triv respectively. We view TO(G) in the naturalway as a subring of LO(G). We denote by Lab

O (G) the subring of LO(G) spanned

by the linear independent elements [Oϕ], ϕ ∈ G(O). Similarly, T abO (G) is defined

as the subring of TO(G) spanned by the elements [Oϕ], ϕ ∈ G(O)p′ , and we have

T abO (G) = TO(G)∩Lab

O (G). With the usual conjugation, restriction, induction, andmultiplication we obtain Z-Green functors TO ⊆ LO on G and Z-algebra restriction


subfunctors T abO ⊆ Lab

O of TO ⊆ LO on G. For ϕ ∈ G(O) we will abbreviate [Oϕ]

again by ϕ, and consider G(O) as a Z-basis of LabO (G).

By Conlon’s induction theorem [CR87, Thm. 80.51] and Proposition I.6.2, thesets C(TO) and C(LO) of coprimordial subgroups for TO and LO are contained inthe set of p-hypo-elementary subgroups of G, i.e. subgroups H such that H/Op(H)is a cyclic p′-group. Note that subgroups and factor groups of p-hypo-elementarygroups are again p-hypo-elementary. We will see in Proposition 4.15 that a p-hypo-elementary subgroup H ≤ G is in fact coprimordial for LO and TO.

Note that we may define Galois actions on TO, T abO , T ab

O +, LO, LabO , Lab

O + byautomorphisms of O.

If f : G′ → G is a homomorphism of finite groups, we obtain an inducedring homorphism resf : LO(G) → LO(G′) which restricts to ring homorphismsLabO (G) → Lab

O (G′), TO(G) → TO(G′), T abO (G) → T ab

O (G′), and induces ring ho-momorphisms res+f : Lab

O +(G) → LabO +(G′) and res+f : T ab

O +(G) → T abO +(G′) such

that the diagram

LabO +(G)

res+f−−−−→ LabO +(G′)x x

T abO +(G)

res+f−−−−→ T abO +(G′)

commutes, where the vertical maps are induced by the inclusion T abO ⊆ Lab

O . Notethat these maps form a morphism T ab

O + → LabO + of Z-Green functors on G which

is injective, since, for H ≤ G, the basis [K,ψ]H , K ≤ H, ψ ∈ K(O)p′ , of T abO +(H)

is mapped injectively into the basis [K,ψ]H , K ≤ H, ψ ∈ K(O) of LabO +(H). The

ring homomorphisms res+f are again described by formula (3.3). There are alsoinflation maps infGG/N and inf+

GG/N described by formula (3.4).

This allows to consider TO, T abO , T ab

O +, LO, LabO , Lab

O + as contravariant functorsfrom the category of finite groups to the category of commutative rings.

For two modules M and N over some ring we write M | N , and say that M isa summand of N , if M is isomorphic to a direct summand of N .

4.2 Example Let G be a finite group, M := LO ∈ Z−Mackalg(G), A :=

LabO ∈ Z−Resalg(G), B(H) := H(O) for H ≤ G, and let p ∈ Z−Con(G)(M,A) be

defined for H ≤ G, V ∈ OH−lin indecomposable, by

pH([V ]) :=

[V ], if rkOV = 1,

0, otherwise.

Note that mσ = 1 for all σ ∈ Γ(M(G)), and that mϕ([V ]) is the number of sum-mands isomorphic to Oϕ in a direct sum decomposition of V ∈ OH−lin for H ≤ G,ϕ ∈ B(H).


(i) The morphism a is an integral canonical induction formula.(ii) For H ≤ G the homomorphism aH is A(H)-linear.(iii) The morphism a respects the Galois action.


(iv) For H ≤ G and ϕ ∈ H(O) we have aH(ϕ) = [H,ϕ]H .

(v) The morphism a is given explicitly by

aH([V ]) =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|(NH(σ)/H0)p′ ||NH(σ)/H0|

mϕn

(resHHn([V ])

)[H0, ϕ0]H

for H ≤ G and V ∈ OH−lin, and we have

[V ] =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|(NH(σ)/H0)p′ ||NH(σ)/H0|

mϕn

(resHHn([V ])

)indHH0

([Oϕ0 ]).

(vi) For H ≤ G, V ∈ OH−lin, and (K,ψ) ∈M(H) we have

mψ

(resHK([V ])

)= 0 =⇒ αH(K,ψ)([V ]) = 0.

(vii) For H ≤ G and (K,ψ) ∈ M(H) we have αH(K,ψ)([V ]) = 0 unless a Sylowp-subgroup of K is contained in some vertex of V .

(viii) Let f : G′ → G be a homomorphism of finite groups. Then the diagram

LO(G)aG−−−−→ Lab

O +(G)

resf

y yres+f

LO(G′)aG′−−−−→ Lab

O +(G′)

is commutative. In particular, the maps aG, where G runs over all finite groups,form a natural transformation between the contravariant functors LO and Lab

O + withvalues in abelian groups.

Proof First we show that condition (∗p′) in Theorem II.4.5 is satisfied for LO,LabO , B, and p ∈ Z−Con(G)(LO, L

abO ). Let KH ≤ G with H/K a cyclic p′-group,

and let ψ ∈ Hom(K,O×) be stable under H. Let furthermore V ∈ OH−lin beindecomposable. If rkOV = 1, then

pK(resHK([V ])) = resHK([V ]) = resHK(pH([V ]));

and condition (∗p′) holds for V . If rkOV > 1, then pH([V ]) = 0, and we have toshow that the multiplicity of Oψ in resHK(V ) is zero. We assume the Oψ | resHK(V )and will obtain a contradiction.

Since (H : K) is a p′-number, V is K-projective, and there is some indecompos-able W ∈ OK−lin with V | indHK(W ). This implies

Oψ | resHK(V ) | resHK(indHK(W )) ∼=⊕

h∈H/K

indKK∩ hK

(reshK

K∩ hK( hW )) ∼=

⊕h∈H/K

hW.


Therefore, Oψ is isomorphic to hW for sume h ∈ H, and, since ψ is H-stable, thisimplies Oψ ∼= W . Hence,

V | indHK(Oψ) ∼=⊕

ϕ∈H(O)

resHK

(ϕ)=ψ

Oϕ,

and rkOV = 1 which is a contradiction.(v) This follows from Theorem II.4.5 with π the set of all primes different from p.(ii) This follows from Proposition II.2.9 (ii).(iii) This follows from Diagram (2.2).(iv) This follows from Lemma II.3.7.(vi) This follows from part (v).(vii) This follows from part (v). In fact, for (K,ψ) ∈M(H), mψ(resHK([V ])) 6= 0,

i.e. Oψ | resHK(V ), implies that a vertex of Oψ is contained in a vertex of V .(viii) This is proved exactly in the same way as Proposition 1.2 (viii). Here we

first prove that the Q-tensored version of the diagram commutes, since at this statewe do not know that a is integral at this state. The original version will then followfrom the integrality of a.

(i) Since condition (∗p′) in Theorem II.4.5 holds for LO, LabO , B, and p ∈

Z−Con(G)(LO, LabO ), Corollary II.4.7 implies that |H|p · aH is integral for H ≤ G.

We will define another induction formula a′ for Q⊗LO from Q⊗LabO in Example 4.5

such that |H|p′ · a′H is integral for H ≤ G, and a is a ‘homomorphic immage’ of a′

so that a is integral (cf. the paragraph after Proposition 4.8). Using the followinglemma we will be able to prove that b a = idQ⊗LO , i.e. that a is a canonicalinduction formula.

4.4 Lemma Let G be a p-hypo-elementary group with Sylow p-subgroup P .(i) Let V ∈ OG−lin be indecomposable with vertex P and source Oψ with ψ ∈

Hom(P,O×), and let H := NG(P,ψ). Then we have V ∼= indGH(Oϕ) for someextension ϕ ∈ Hom(H,O×) of ψ. Moreover, we have for H ≤ U ≤ G:

pU(resGU ([V ])

)=

∑g∈G/H

gϕ, if U = H,

0, if U > H.

(ii) The map

rG := ρG aG =(pH resGH)H≤G : LO(G)−→Lab

O+

(G)

is injective.(iii) We have p b a = p as morphisms in Q−Con(G)(Q⊗ LO,Q⊗ Lab

O ).

Proof (i) We have V | indGP (Oψ), and

indHP (Oψ) ∼=⊕

ϕ∈H(O)

resHP

(ϕ)=ψ

Oϕ,

hence V | indGH(Oϕ) for some extension ϕ ∈ H(O) of ψ. If K denotes the quotientfield of O, then K ⊗O indGH(Oϕ) is a simple KG-module, since H = stabG(ϕ).


Therefore indGH(Oϕ) is indecomposable. The last assertion follows immediatelyfrom Mackey’s decomposition theorem

resGU(indGH(Oϕ)

) ∼= ⊕g∈G/U

indUH(O( gϕ)),

and the fact, that indUH(O( gϕ)) is indecomposable, since indGH(O( gϕ)) ∼= indGH(Oϕ) ∼=V is.

(ii) We proceed by induction on |G|. If |G| = 1, the assertion is trivial. let|G| > 1. We show that for V,W ∈ OG−lin we have

pH(resGH([V ])

)= pH

(resGH([W ])

)for all H ≤ G =⇒ V ∼= W.

We may assume that V and W have no indecomposable summand in common.Under this condition we will show that V = W = 0.

First assume that V or W has an indecomposable summand with vertex P .Choose X among the indecomposable summands of V and W with vertex P withmaximal O-rank, and assume X | V . Let Oψ be a source of X with ψ ∈ P (O), let

H := NG(P,ψ), and let ϕ ∈ H(O) be an extension of ψ with X ∼= indGH(Oϕ) asgranted by part (i). From pH

(resGH([V ])

)= pH

(resGH([W ])

)and part (i) we can see

that Oϕ | resGH(W ). Hence W has an indecomposable summand Y with vertex Pand Oϕ | resGH(Y ). Since Oψ | resHP (Oϕ) | resGP (Y ), also Y has source Oψ, and by

part (i) we have Y ∼= indGH(Oϕ′) for some extension ϕ′ ∈ H(O) of ψ. But

Oϕ | resGH(Y ) ∼= resGH(indGH(Oϕ′)

) ∼= ⊕g∈G/H

O( gϕ′)

implies ϕ = gϕ′ for some g ∈ G, and

X ∼= indGH(Oϕ) ∼= indGH(O( gϕ′)) ∼= Y,

which contradicts our assumption that V an W have no indecomposable summandin common. Hence, all indecomposable summands of V and W have vertices strictlysmaller than P .

If V and W are not both trivial, we choose among the indecomposable directsummands of V and W one, say X, with maximal vertex Q. Assume that X | V .

If N := NG(Q) < G, then we have by our induction hypothesis that resGN (V ) ∼=resGN (W ). But then the Green correspondent X ′ ∈ ON−lin of X is also a summandof resGN (W ). Hence X is a summand of V by the Green correspondence. Thiscontradicts our assumption.

Hence, we are left with the case that our maximal vertex Q is normal in G. LetC ≤ G be a cyclic p′-subgroup with G = PC. Then by our induction hypothesiswe have resGQC(V ) ∼= resGQC(W ). Let Oψ be a source of X for some ψ ∈ Q(O).

Since X | indGQ(Oψ), there is an indecomposable module X ′ ∈ O[QC]−lin with X ′ |indQCQ (Oψ) andX | indGQC(X ′), henceX = indGQC(X ′) by Green’s indecomposabilitytheorem. By Mackey’s decomposition theorem we have

X ′ | resGQC(X) | resGQC(V ) ∼= resGQC(W ).


Therefore, W has an indecomposable summand Y with X ′ | resGQC(Y ). Since X ′

has vertex Q, and since Q is maximal among the vertices of summands of V and W ,also Y has vertex Q, and, similar as for X, Y ∼= indGQC(Y ′) for some indecomposablemodule Y ′ ∈ O[QC]−lin. Now we have

X ′ | resGQC(Y ) ∼= resGQC(indGQC(Y ′)

) ∼= ⊕g∈G/QC

gY ′,

and we obtain the contradiction

X ∼= indGQC(X ′) ∼= indGQC(Y ′) ∼= Y |W.

Hence, V = W = 0, and the proof of part (ii) is complete.(iii) It suffices to show pG bG aG = pG, since all subgroups of G are again

p-hypo-elementary. Let V ∈ OG−lin be indecomposable. If rkOV = 1, then

(pG bG aG)([V ]) = [V ] = pG([V ])

by Proposition 4.3 (iv). Now assume that rkOV > 1 and write

aG([V ]) =∑

(H,ϕ)∈G\M(G)

αG(H,ϕ)([V ])[H,ϕ]G.

If the vertices of V are strictly contained in P , then αG(H,ϕ)([V ]) = 0 for all (H,ϕ) ∈M(G) with P ≤ H by Proposition 4.3 (vii). For (H,ϕ) ∈ M(G) with P 6≤ H wehave always pG

(bG([H,ϕ]G)

)= pG

(indGH([Oϕ])

)= 0. Hence, (pGbGaG)([V ]) = 0.

If V has vertex P and source Oψ for some ψ ∈ P (O), then by the same argumentit suffices to show that

pG(αG(H,ϕ)([V ])indGH([Oϕ])

)= 0

for (H,ϕ) ∈ M(G) with P ≤ H. By Proposition 4.3 (vi), we may assume thatOϕ | resGH(V ), since otherwise αG(H,ϕ)([V ]) = 0. In this case we have

indGH(Oϕ) | indGH(resGH(V )

)| indGH

(resGH(indGP (Oψ))

)∼= indGH

( ⊕g∈G/H

indHP (O(gψ)))

∼=⊕

g∈G/H

indGP (O(gψ)).

But since ψ is not G-stable (by part (i)), the module indGP (O(gψ))∼= indGP (Oψ) has

no summands of O-rank one.

Now we are able to prove that b a = idQ⊗LO . It suffices to prove bH aH =idQ⊗LO(H) for all p-hypo-elementary subgroups H ≤ G, since a and b commute withrestrictions, and since elements of Q ⊗ LO(K), K ≤ G, are uniquely determinedby restrictions to all p-hypo-elementary subgroups of K. So let H ≤ G be p-hypo-elementary. Since ρAH aH = (pK resHK)

K≤H is injective by Lemma 4.4 (ii), it suffices


to show that pK resHK bH aH = pK resHK for K ≤ H. Since b and a commutewith restrictions, it suffices to prove pK bK aK = pK for all p-hypo-elementarysubgroups K of G. But this is granted by Lemma 4.4 (iii).

Next we turn to the integrality of a and introduce another canonical inductionformula a′ for Q⊗ LO from Q⊗ Lab

O .

4.5 Example Let G be a finite group, M := LO ∈ Z−Mackalg(G), and letfor H ≤ G, A′(H) ⊆ M(H) be defined as the Z-span of the set B′(H) of elements[V ] with V ∈ OH−lin indecomposable such that V/ ker(V ) is p-elementary , i.e. adirect product of a p-group and a cyclic p′-group. Then A′ ⊆ LO is a Z-algebrarestriction subfunctor of M . For H ≤ G and indecomposable V ∈ OH−lin wedefine

p′H([V ]) :=

[V ], if [V ] ∈ B′(H),

0, otherwise.

This definies a morphism p′ ∈ Z−Con(G)(LO, A′). We denote by M′, m′, α′, a′,

and b′ the features resulting from M , A′, p′, and B′.

4.6 Lemma (i) Let G be a p-elementary group, and let V ∈ OG−lin beindecomposable. Then there is a subgroup H ≤ G of p-power index and someϕ ∈ H(O) with V ∼= indGH(Oϕ). Moreover, the pair (H,ϕ) ∈ M(G) is uniquelydetermined up to G-conjugacy by the condition V ∼= indGH(Oϕ).

(ii) Let G be an arbitrary finite group, and let V ∈ OG−lin be indecomposablesuch that [V ] ∈ B′(G). Then there is a unique pair (H,ϕ) ∈ M(G) up to G-conjugacy such that V ∼= indGH(Oϕ). Moreover ker(V ) ≤ H and (G : H) is ap-power. In particular rkOV is a p-power.

Proof (i) Let G = P ×C with P = Op(G) and C = Op′(G). Let Q ≤ P be a

vertex of V , and let Oψ, ψ ∈ Q(O), be a source of V . Then

V | indGQ(Oψ) ∼= indGQC(indQCQ (Oψ)

) ∼= ⊕ϕ∈Hom(QC,O×)

resQCQ

(ϕ)=ψ

indGQC(Oϕ).

Since indGQC(Oϕ) is indecomposable by Green’s indecomposability theorem (CG of

p-power index), we have V ∼= indGH(Oϕ) with H = QC of p-power index in G. Nowlet V ∼= indGH′(Oϕ′) for an arbitrary subgroup H ′ ≤ G and some ϕ′ ∈ Hom(H ′,O×).From the first part of the proof we know that the O-rank of V is a p-power, henceC ≤ H ′. Now we have

Oϕ | resGH(indGH(Oϕ)

) ∼= resGH(V ) ∼= resGH(indGH′(Oϕ′)

) ∼=∼=

⊕g∈H\G/H′

indHH∩ gH′

(res

gH′

H∩ gH′(O( gϕ′))).

Since C ≤ H ∩ gH ′ for all g ∈ G, Green’s indecomposability theorem implies thateach of the summands in the last sum is indecomposable. Hence, there is someg ∈ G with (H,ϕ) ≤ ( gH ′, gϕ′). Now (G : H) = rkOV = (G : H ′) implies(H,ϕ) = ( gH, gϕ).


(ii) By part (i) there is a unique (H,ϕ) ∈ M(G) up to G-conjugacy withker(V ) ≤ H and indGH(Oϕ) ∼= V . Moreover (G : H) is a p-power. Let (H ′, ϕ′) be an-other element in M(G) with V ∼= indGH′(Oϕ′). We will show that ker(V ) ≤ H ′ andresH

′

ker(V )(ϕ′) = 1. Then part (i) implies that (H,ϕ) and (H ′, ϕ′) are G-conjugate.

With K := ker(V ) we have

O | resGK(V ) ∼= resGK(indGH′(Oϕ′)

) ∼= ⊕g∈K\G/H′

indKK∩ gH′

(res

gH′

K∩ gH′(O( gϕ′))).

Hence, there is some g ∈ G with O | indKK∩ gH′

(res

gH′

K∩ gH′(O( gϕ′))). Since the vertices

of O are the Sylow p-subgroups of K, the index (K : K ∩ gH ′) is a p′-number. Onthe other hand (K : K ∩ gH ′) is a p-power, since (G : H ′) = rkOV is a p-power

and (K : K ∩ gH ′) = (K · gH ′ : gH ′). Hence, K ∩ gH ′ = K and O | resgH′K (O( gϕ′)),

i.e. (K, 1) ≤ ( gH ′, gϕ′), and then also (K, 1) ≤ (H ′, ϕ′), since K is normal in G.

4.7 Lemma Let G be a finite group and H a normal subgroup of G of p-powerindex. Let V ∈ OG−lin and W ∈ OH−lin be indecomposable modules such thatW is stable under G and W | resGH(V ). Then, if H/ ker(W ) is p-elementary so isG/ ker(V ).

Proof Let K := ker(W ), then K is normal in G, since W is G-stable. Weassume that H/K is p-elementary. First note that G/K is again p-elementary. Infact, let K ≤ P,C ≤ H be intermediate groups with C/K = Op′(H/K) and P/K =Op(H/K), hence H/K = C/K ×P/K. Then W ∼= indHU (Oµ) for some C ≤ U ≤ Hand µ ∈ U(O) by Lemma 4.6 (i). Since K = ker(W ), K is also the kernel of resUC(µ).By the uniqueness part of Lemma 4.6 (ii), for each element g ∈ G there is someh ∈ H with g(U, µ) = h(U, µ). Therefore, gresUC(µ) = hresUC(µ) = resUC(µ), since Hcentralizes C/K. Since resUC(µ) has kernel K, also G centralizes C/K. Hence, G/Kis again p-elementary, and it suffices to prove that K ≤ ker(V ).

Let Q be a vertex of V and let ϕ ∈ Q(O) such that Oϕ is a source of V .

Then V | indGKQ(indKQQ (Oϕ)

). Hence there exists an indecomposable module V ′ ∈

O(KQ)−lin with V ′ | indKQQ (Oϕ) and V | indGKQ(V ′). We will show that V ′ ∼= Oϕ′for an extension ϕ′ ∈ Hom(KQ,O×) of ϕ with resKQK (ϕ′) = 1.

We have O | resKQK (V ′), since

O | resHK(W ) | resHK(resGH(V )

)∼= resGK(V ) | resGK

(indGKQ(V ′)

)∼=

⊕g∈K\G/KQ

indKK∩

g(KQ)

(res

g(KQ)

K∩g(KQ)

( gV ′))

∼=⊕

g∈G/KQ

gresKQK (V ′).


r

rrr

r

rr

HHHH

HH

HHHH

HH

HHHHHH

Oϕ Q

V ′ KQ

HQ

V G

K ∩Q

K

H W

From

O | resKQK (V ′) | resKQK(indKQQ (Oϕ)

)∼=

⊕g∈K\KQ/Q

indKK∩ gQ

(res

gQ

K∩ gQ( gOϕ)

)= indKK∩Q

(resQK∩Q(Oϕ)

)we see that K∩Q is a Sylow p-subgroup of K, since the vertices of O ∈ OK−lin arethe Sylow p-subgroups of K. Therefore, Q is a Sylow p-subgroup of KQ. Moreover,we obtain from O | indKK∩Q

(resQK∩Q(Oϕ)

)that resQK∩Q(ϕ) = 1, since O ∈ OK−lin

has trivial source. Hence we can extend ϕ ∈ Hom(Q,O×) to ϕ′ ∈ Hom(KQ,O×)by setting ϕ′(xy) := ϕ(y) for x ∈ K, y ∈ Q. Since the multiplicity of O in

resKQK(indKQQ (Oϕ)

) ∼= indKK∩Q(O)

equals 1, there is exactly one indecomposable summand X of indKQQ (Oϕ) with

O | resKQK (X). But both V ′ and Oϕ′ have this property. Hence, V ′ ∼= Oϕ′ , and

resGK(V ) | resGK(indGKQ(Oϕ′)

) ∼= ⊕g∈G/KQ

gresKQK (Oϕ′) ∼=⊕

g∈G/KQ

O.

Therefore, K ≤ ker(V ), and the result follows.

4.8 Proposition Let G be a finite group, let M , A′, B′, M′, p′, m′, α′, a′,and b′ be defined according to Example 4.5, and let Γ, A, B, M, p, m, α, a, and bbe defined according to Example 4.2.

(i) Condition (∗p) of Theorem II.4.5 holds for M , A′, B′, and p′.(ii) For H ≤ G we define ηH : A′+(H)→ A+(H) for K ≤ H and indecomposable

V ∈ OK−lin with [V ] ∈ B′(K) by

ηH([K, [V ]

]H

):= [L,ϕ]H ,


where [L,ϕ]H is uniquely determined by V ∼= indKL (Oϕ), cf. Lemma 4.7 (ii). Thenη ∈ Z−Res(G)(A′+, A+) and the diagram

Q⊗ LOa′−−−−→ Q⊗A′+

b′−−−−→ Q⊗ LOa

y η

y xbQ⊗A+ Q⊗A+ Q⊗A+

is commutative.(iii) The morphism a′ is a canonical induction formula for Q⊗LO from Q⊗A′.Proof (i) Let K H ≤ G with H/K a cyclic p-group, and let W ∈ OK−lin

be indecomposable and H-stable such that W/ ker(W ) is p-elementary (i.e. [W ] ∈B′(K)). Let furthermore V ∈ OH−lin be indecomposable. If [V ] ∈ B′(H), then

pK(resHK([V ])

)= resHK([V ]) = resHK

(pK([V ])

).

If V /∈ B′(H), then pH([V ]) = 0, and we have to prove that the multiplicity of Was a summand in resHK(V ) is zero. But this is exactly the statement of Lemma 4.7.In fact, if W | resHK(V ), then we obtain [V ] ∈ B′(H).

(ii) First we show that η is a morphism of Z-restriction functors on G. Obviouslyη commutes with conjugations. Now let U ≤ H ≤ G, and let (K, [V ]) ∈ M′(H),i.e. K ≤ H, V ∈ OK−lin indecomposable with [V ] ∈ B′(K). Let (L, µ) ∈ M(K)with V ∼= indKL (Oµ). Then we have

res+HU

(ηH([K, [V ]

]H

))= res+

HU ([L, µ]H) =

∑h∈U\H/L

[U ∩ hL, res

hL

U∩ hL( hµ)

]U,

and

ηU(res+

HU

([K, [V ]

]H

))=

∑x∈U\H/K

ηU([U ∩ xK, res

xKU∩ xK( x[V ])

]U

).

For x ∈ H we have

resxKU∩ xK( xV ) ∼= res

xKU∩ xK

(ind

xKxL

(O( xµ)))

∼=⊕

y∈U∩ xK\ xK/ xL

indU∩xK

U∩ xK∩ yxL

(res

yxL

U∩ xK∩ yxL(O( yxµ)))

∼=⊕

y∈U∩ xK\ xK/ xL

indU∩xK

U∩ yxK

(res

yxL

U∩ yxL(O( yxµ))).

Each of the summands in the last sum is indecomposable by Green’s indecompos-ability theorem. In fact, let C ≤ L ≤ K be such that C/ ker(V ) = Op′(K/ ker(V )).Then C is normal in K and (K : C) is a power of p. Hence, U ∩ yxC is normalin U ∩ yxK of p-power index, and U ∩ yxC ≤ U ∩ yxL ≤ U ∩ yxK, so that Green’sindecomposability theorem can be applied. Therefore, by the definition of η, weobtain

ηU

([U ∩ xK, res

xKU∩ xK( x[V ])

]U

)=

∑y∈U∩ xK\ xK/ xL

[U ∩ yxL, res

yxL

U∩ yxL(O( yxµ))]U,


and summing over x ∈ U\H/K we obtain exactly the the result of the abovetransformation of res+

HU

(ηH([K, [V ]

]H

)), since the elements yx run through a set

of representatives for U\H/L if x runs through a set of representatives for U\H/K,and for each x, y runs through a set of representatives for U ∩ xK\ xK/ xL. Hence,we have η ∈ Z−Res(G)(A′+, A+).

Next we show the commutativity of the diagram in the assertion. Let H ≤ Gand let K, V , L, and µ be as in the last paragraph, then we have

bH(ηH([K, [V ]

]H

))= bH([L, µ]H) = indHL ([Oµ]) =

= indHK(indKL (Oµ)) = indHK([V ]) = b′H([K, [V ]

]H

).

This shows the commutativity of the right square of the diagram. For the commu-ativity of left square of the diagram it suffices to show that ρAH ηH a′H = ρAH aHfor H ≤ G, since ρAH is injective. By the definition of ρAH and the commutativity ofa, a′, and η with restrictions it suffices to show that πAH ηH a′H = πAH aH forH ≤ G. Since πAH aH = pH , it suffices to show that

πAH ηH a′H = pH

for H ≤ G. Let V ∈ OH−lin be indecomposable. If [V ] ∈ B(H), i.e. V ∼= Oϕ forsome ϕ ∈ H(O), then

(πAH ηH a′H)([V ]) = (πAH ηH)([H, [V ]

]H

)= πAH([H,ϕ]H) = ϕ = pH(ϕ)

by Lemma II.3.7. If [V ] ∈ B′(H) \ B(H), and V ∼= indHK(Oϕ) with K < H,ϕ ∈ K(O), then

(πAH ηH a′H)([V ]) = (πAH ηH)([H, [V ]

]H

)= πAH([K,ϕ]H) = 0 = pH([V ])

again by Lemma II.3.7. If [V ] /∈ B′(H), then πA′

H (a′H([V ])) = p′H([V ]) = 0. Hencea′H([V ]) is a linear combination of basis elements

[K, [W ]

]H

, with K < H, [W ] ∈B′(K). This implies that ηH(a′H([V ])) is a linear combination of basis elements

[K,ψ]H with K < H, ψ ∈ K(O), and therefore we have (πAH ηH a′H)([V ]) = 0.On the other hand, also pH([V ]) = 0. This completes the proof of part (ii).

(iii) This follows from the commutative diagram in part (ii) and the fact thatb a = idQ⊗LO .

Now we can prove that a is integral. We already know that |H|p ·aH(LO(H)) ⊆LabO +(H) for H ≤ G. From Proposition 4.8 (i) and Corollary II.4.7 we obtain that|H|p′ · a′H(LO(H)) ⊆ A′+(H) for H ≤ G. Hence, Proposition 4.8 (ii) implies that

|H|p′ · aH(LO(H)) = |H|p′ · (ηH a′H)(LO(H)) ⊆ ηH(|H|p′ · a′H(LO(H))

)⊆ ηH(A′+(H)) ⊆ A+(H).

This implies aH(LO(H)) ⊆ A+(H) for H ≤ G, and completes the proof of (part (i)of) Proposition 4.3.

4.9 Example Let G be a finite group, M := TO ∈ Z−Mackalg(G), A :=

T abO ∈ Z−Resalg(G), B(H) := H(O)p′ for H ≤ G, and let p ∈ Z−Con(G)(M, A)


be defined for H ≤ G and for indecomposable V ∈ OH−triv by

pH([V ]) :=

[V ], if rkOV = 1,

0, otherwise.

4.10 Proposition Let G be a finite group, let M = LO, A = LabO , B, M, p,

m, α, a, and b be defined according to Example 4.2, and let M = TO, A = T abO , B,

M, p, m, α, a, and b be defined according to Example 4.9. Then the diagram

TOa−−−−→ T ab

O +b−−−−→ TO

i

y i+

y yiLO

a−−−−→ LabO +

b−−−−→ LO

commutes, where i : TO → LO and i : T abO → Lab

O denote the inclusion maps. Themorphism i+ is injective and αH(K,ψ)([V ]) = αH(K,ψ)([V ]) for H ≤ G, V ∈ OH−triv,

and (K,ψ) ∈ M(H). In particular, a is an integral canonical induction formula,and all the other parts of Proposition 4.3 hold with the appropriate changes in thesituation of Example 4.9.

Proof The morphism i+ is injective, as has already been observed at thebeginning of this section. The left part of the diagram commutes by Proposi-tion II.3.11. All the other assertions are easy consequences or can be reproved withsimilar arguments as used in the proof of Proposition 4.3.

In the next chapter we will need a stronger version of Lemma 4.4 for trivialsource modules.

4.11 Lemma Let G be a p-hypo-elementary group and let P denote the Sylowp-subgroup of G. Furthermore, let M = TO, A = T ab

O , ρ, B, M, p, m, α, a, and bbe defined according to Example 4.9.

(i) For indecomposable V ∈ OG−triv we have:

rkOV = 1 ⇐⇒ V has vertex P

(ii) The map

rG := ρG aG = (pH resGH)H≤G : TO(G)→ T ab

O+

(G)

is injective.(iii) For f ∈ Z−Res(G)(T ab

O , TabO ) we have

p b f+ a = f p, in particular, p b a = p.

(iv) pG is a ring homomomorphism.

Proof (i) Let rkOV = 1. Then V has vertex P . Conversely, if V has vertexP , then

V | indGP (O) =⊕

ϕ∈B(G)

Oϕ,


and rkOV = 1.(ii) This follows from Lemma 4.4 (ii).(iii) Let H ≤ G and let V ∈ OG−triv be indecomposable. If rkOV = 1, then

(pH bH f+H aH)([V ]) = fH([V ]) = (fH pH)([V ])

for H ≤ G by the analogue of Proposition 4.3 (iv). If rkOV > 1, then we havepH([V ]) = 0, and αH(K,ψ)([V ]) = 0 for all (K,ψ) ∈ M(H) with Op(H) ≤ K by

part (i) and the analogue Proposition 4.3 (vii). Hence, (bH f+H aH)([V ]) is alinear combination of elements indGK(ψ) with (K,ψ) ∈ M(H) such that the Sylowp-subgroup of K is strictly contained in Op(H). Therefore, (bH f+H aH)([V ])is a linear combination of elements [W ], where W ∈ OH−triv is indecomposablewith vertex strictly smaller than Op(H). Now part (i) implies that (pH bH f+H aH)([V ]) = 0.

(iv) Let H ≤ G and let V,W ∈ OH−triv be indecomposable. If rkOV =rkOW = 1, then obviously pH([V ] · [W ]) = [V ⊗W ] = [V ] · [W ] = pH [(V ]) · pH([W ]).Hence, we may assume that rkOV > 1. Then pH([V ]) = 0, and by part (i), V has avertex Q strictly contained in Op(H). Since V is Q-projective, so is V ⊗W by theFrobenius axiom (cf. Definition I.1.1 (vi)). Part (i) again shows that V ⊗W has nosummand of O-rank one. Hence pH([V ⊗W ]) = 0.

4.12 Remark (i) In Example 4.9 we may replace O by its residue field F . Itis well-known (see [Bro85, 3.5] for example) that reduction yields an isomorphismTO → TF of Z-Green functors identifying also T ab

O with T abF . Here TF is the Mackey

functor arising from the categories FH−triv defined in the same way as OH−trivfor H ≤ G, and T ab

F (H) is Z-free on the set H(F ) := Hom(H,F×) for H ≤ G. Themorphism p translates to a morphism TF → T ab

F annihilating the indecomposabletrivial source modules over F of dimension greater than 1, and being the identityon one-dimensional modules. The resulting canonical induction formula TF →T abF shares all properties of Proposition 4.10, or better of Proposition 4.3 with the

appropriate changes.Moreover, this canonical induction fromula commutes with the one of Exam-

ple 3.1 with respect to the embedding PF → TF . In contrast to the situation inExample 3.1, the coefficient

|(NH(σ)/H0)p′ | ·mϕ(resHHn([V ]))

|NH(σ)/H0|

in the explicit formula for a in Proposition 4.3 (v) is in general not an integer, asone can see easily for V = O and σ = ((1, 1)), where it reduces to |Hp′ |/|H|.

(ii) There are other canonical induction formulae for TO or LO in the spirit ofExamples 1.4, 1.7 and Remark 1.9 (ii).

(iii) In general, the canonical induction formulae a and a are not morphisms ofMackey functors, since condition (ii) in Lemma II.2.5 does not hold, and they arenot multiplicative, since p and p aren’t, cf. Proposition II.2.9 (i).

In the next chapter we will need the notion of a species (cf. [Be84b, 2.2 and2.13]) on TO(G), and some of its properties. For the rest of this section let K


denote the quotient field of O which is of characteristic zero and which is assumedto be a splitting field for all finite groups occurring in this section.

4.13 Definition Assume that notation of Example 4.9. (i) Let G be a finitegroup. A species of TO(G) is a ring homomorphism s : LO(G)→ K.

(ii) For a p-hypo-elementary group G, g ∈ G, and V ∈ OG−triv with [V ] ∈ T abO

we definesg : T ab

O (G)→ K, [V ] 7→∑

ϕ∈G(O)

mϕ([V ]) · ϕ(g).

In other words, sg([V ]) is the trace of the O-linear action of g on V . Note that sgis a ring homomorphism.

(iii) For a finite group G, a p-hypo-elementary subgroup H ≤ G, and h ∈ H wedefine a species (cf. Lemma 4.11 (iv))

s(H,h) := sh pH resGH : TO(G)→ K

of TO(G).

4.14 Proposition For a finite group G, the ring homomorphism

sG :=∏

(H,h)

s(H,h) : LO(G)→∏

(H,h)

K,

where (H,h) runs through all pairs with H ≤ G p-hypo-elementary and h ∈ H, isinjective. In particular, the K-algebra K ⊗ TO(G), is semisimple, i.e. isomorphicto a direct product of copies of K.

Proof This follows from Lemma 4.11 (ii) together with the fact that the setof p-hypo-elementary subgroups of G contains the set of coprimordial subgroups forTO.

4.15 Proposition Let G be a p-hypo-elementary group. Then G is coprimor-dial for TO and LO.

Proof Assume that G /∈ C(TO) = C(Q⊗ TO). Then by Proposition I.6.2, we

have e(G)G · 1TO(G) 6= 0, i.e

[O] =∑H<G

αH indGH([O])

in Q ⊗ TO(G) for certain rational coefficients αH ∈ Q, H < G, by the explicit

formula for e(G)G , cf. Remark I.3.3. We apply the species s(G,g) to both sides of this

equation, where g ∈ G is chosen in such a way that gOp(G) generates G/Op(G).Obviously,

s(G,g)([O]) = 1,

and we will show thats(G,g)(indGH([O]) = 0

for H < G, which yields the desired contradiction. In fact, if Op(G) ≤ H < G, thens(G,g)(indGH([O])) equals the K-character of indGH(1) evaluated at g, and this is zero.

3.5. THE GREEN RING 99

If Op(G) 6≤ H, then each indecomposable summand in indGH(O) has a vertex whichis strictly smaller than Op(G), and Lemma 4.11 (i) implies that pG(indGH([O])) = 0.This shows that G is coprimordial for TO. By Proposition I.6.2 the same is true forLO.

4.16 Remark It is surprising that the ring LO(G) is not treated in theliterature to the same extent as TO(G), since they are comparably easy or difficultto investigate. However, LO(G) has the advantage that it can serve as a link betweenRF (G) and RK(G), since we have surjections of Mackey functors

LO −−−−→ RFyRK

Moreover, one can show that K⊗LO(G) is semisimple, so that we have for all threerings a character theory. In fact, we may extend the species s(H,h) : TO(G) → K,for H ≤ G p-hypo-elementary and h ∈ H, to a ring homomorphism

s(H,h) : LO(G)−→K

by the following construction:For a p-hypo-elementary subgroup H of G with Sylow p-subgroup P and V ∈

OH−lin we definep∗H : LO(G)→ LO(G), [V ] 7→ [V ′],

where we use a decomposition V = V ′ ⊕ V ′′ with V ′ having only indecomposablesummands with vertex P , and V ′′ having only indecomposable summands withvertex strictly smaller than P . Note that p∗H is a ring homomorphism. Moreover,for h ∈ H and V ∈ OH−lin we denote by sh([V ]) the trace of the multiplicationby h on V . This defines a ring homomorphism

sh : LO(G)−→K.

Now, s(H,h) is defined as

s(H,h) := sh p∗H resGH .

Using arguments of Lemma 4.4 (ii) it is not difficult to see that the collection

sG :=∏

(H,h)

s(H,h) : LO(G)−→∏

(H,h)

K

is injective.

3.5 The Green ring

Throughout this section let O and F be defined as in the previous section.For a finite group G we denote by OG−lat the category of OG-lattices, i.e.

finitely generated O-free OG-modules. The Grothendieck ring of OG−lat withrespect to direct sums will be called the Green ring of G over O, and will be


denoted by GrO(G). The element in GrO(G) associated to V ∈ OG−lat willbe again denoted by [V ]. By the Krull-Schmidt-Azumaya theorem (cf. [CR81,6.12]) the ring GrO(G) is a free abelian group on the set of elements [V ], whereV ∈ OG−lat runs through a set of representatives of the isomorphism classes ofindecomposable OG-lattices.

With the usual conjugation, restriction and induction maps, the rings GrO(H),H ≤ G, form a Z-Green functor on G. Since Q⊗TO ⊆ Q⊗GrO is an inclusion of qu-Green functors on G, Proposition I.6.2 and Proposition 4.15 imply that C(GrO) =C(Q⊗GrO) is the set of p-hypo-elementary subgroups of G.

For a finite group G we define B as the set of elements [V ] ∈ GrO(G), whereV ∈ OG−lat is indecomposable and G/ ker(V ) is solvable with a normal Sylowp-subgroup. Further we define the subring Gr′O(G) of GrO(G) as the span of B(G).Obviously, Gr′O(G) is free on B(G) and the rings Gr′O(H), H ≤ G, form a Z-algebrarestriction subfunctor of GrO on G.

If f : G′ → G is a homomorphism of finite groups, there is an induced ringhomomorphism resf : GrO(G)→ GrO(G′) restricting along the obvious ring homo-morphism OG′ → OG. This ring homomorphism maps Gr′O(G) to Gr′O(G′), andwe can define a ring homomorphism

res+f : Gr′O+(G)→ Gr′O+(G′),[H, [V ]

]G7→

∑g∈f(G′)\G/H

[f−1( gH), resf : f−1(

gH)→ g

H( g[V ])]G′

as in Formula (3.3) which allows to consider GrO, Gr′O, and Gr′O+ as contravariantfunctors from the category of finite groups to the category of commutative rings.It is easily verified that bGrO,Gr

′O : Gr′O+ → GrO is a natural transformation. If f

is the inclusion of a subgroup H ≤ G, we write res+GH instead of res+f , which is

consistent with the definition of res+ in I.2.2. And if f is the natural epimorphismG→ G/N for N G we write inf+

GG/N instead of res+f .

As in the previous sections we have Galois actions of the automorphism groupof O on GrO, Gr′O, and Gr′O+.

5.1 Example Let G be a finite group, M := GrO ∈ Z−Mackalg(G), A :=Gr′O ∈ Z−Resalg(G), B(H) as defined at the beginning of this section, and letp ∈ Z−Con(G)(M,A) be defined for H ≤ G and indecomposable V ∈ OH−lat by

pH([V ]) :=

[V ], if [V ] ∈ B(H)

0, otherwise.

Note that, forH ≤ G and V,W ∈ OH−lat with [W ] ∈ B(H), the numberm[W ]([V ])is the multiplicity of W in a direct sum decomposition of V .


(i) The morphism a is a canonical induction formula and |H|p · aH is integralfor H ≤ G.

(ii) For H ≤ G the homomorphism aH is A(H)-linear.(iii) The morphism a respects the Galois action.

3.5. THE GREEN RING 101

(iv) For H ≤ G and indecomposable V ∈ OH−lat with [V ] ∈ B(H) we haveaH([V ]) = [H, [V ]]H .

(v) The morphism a is given explicitly by

aH([V ]) =∑

σ=(

(H0,[V0])<...<(Hn,[Vn]))∈G\Γ(M(H))

(−1)n×

×|(NH(σ)/H0)p′ ||NH(σ)/H0|

mσm[Vn]

(resHHn([V ])

)[H0, [V0]

]H

for H ≤ G and V ∈ OH−lat, and we have

[V ] =∑

σ=(

(H0,[V0])<...<(Hn,[Vn]))∈G\Γ(M(H))

(−1)n×

×|(NH(σ)/H0)p′ ||NH(σ)/H0|

mσm[Vn]

(resHHn([V ])

)indHH0

([V0])

in GrO(G).(vi) For H ≤ G, V ∈ OH−lat and (K, [W ]) ∈M(H) we have

m[W ]

(resHK([V ])

)= 0 =⇒ αH(K,[W ])([V ]) = 0.

(vii) Let f : G′ → G be a homomorphism of finite groups. Then the diagram

Q⊗GrO(G)aG−−−−→ Q⊗Gr′O+(G)

resf

y yres+f

Q⊗GrO(G′)aG′−−−−→ Q⊗Gr′O+(G′)

is commutative. In particular, the maps aG, where G runs over all finite groups,form a natural transformation between the contravariant functors Q ⊗ GrO andQ⊗Gr′O+ with values in Q-vector spaces.

Proof First we show that condition (∗p′) in Theorem II.4.5 is satisfied for M ,A, B, and p ∈ Z−Con(G)(M,A) for the set of primes different from p. Let KH ≤G with H/K acyclic p′-group, and let W ∈ OK−lat be indecomposable and stableunder H with [W ] ∈ B(H). Let furthermore V ∈ OH−lat be indecomposable. If[V ] ∈ B(H), then

pK(resHK([V ])

)= resHK([V ]) = resHK

(pH([V ])

)and the condition (∗p′) holds for V . Now assume that [V ] /∈ B(H). Then pH([V ]) =0, and we have to show that W is not a constituent of resHK(V ). We assume thatW | resHK(V ) and will deduce a contradiction. Since (H : K) is a p′-number, V isK-projective and there is some indecomposable V ′ ∈ OK−lat with V ′ | resHK(V )and V | indHK(V ′). This implies

W | resHK(V ) | resHK(indHK(V ′)

) ∼= ⊕h∈H/K

hV ′


by the Mackey decomposition formula. Hence W ∼= V ′, since W is H-stable.This implies that V is a summand of indHK(W ). But ker(W ) is normal in H andker(indHK(W )) ≥ ker(W ) by the Mackey decomposition formula. Hence H/ ker(V )is again solvable with normal p-Sylow subgroup, and therefore [V ] ∈ B(H) which isa contradiction.

(v) This follows from Theorem II.4.5.

(i) This follows from Corollary II.2.4, since Gr′O(H) = GrO(H) for p-hypo-elementary subgroups H ≤ G, and from Corollary II.4.7.

(ii) This follows from Proposition II.2.9 (ii).

(iii) This follows from Diagram (2.2).

(iv) This follows from Lemma II.3.7.

(vi) This follows from part (v).

(vii) This is proved by the appropriate modifications of the proof of Proposi-tion 1.2 (xiii).

5.3 Remark (i) We do not know whether a is integral or not. In particularwe do not know how to imitate the trick of the previous section, where a was provedto be a homomorphic image of a formula with p′-denominators.

(ii) It is not clear whether Gr′O is the right canididate for a canonical inductionformula for GrO. There are suggestions by Puig, to replace the condition ‘solvablewith normal Sylow p-subgroup’ with ‘nilpotent modulo p’ or ‘elementary modulop’. But then we are not even able to prove that |H|p · aH is integral. If we replaceour condition by ‘abelian mod p’ the resulting formula is definitely not integral asexamples show.

(iii) By a suggestion of Dipper we may replace the condition ‘solvable with nor-mal Sylow p-subgroup’ with ‘abelian modulo p’ and replace p ∈ Z−Con(G)(M,A)by taking fixed points under the smallest normal subgroup with ‘abelian modulop’-factor group. But again, it is definitely not integral as examples show. Onemight as well try a combination of the suggestions of Puig in (ii) and the modifieddefinition of p ∈ Z−Con(G)(M,A) using a kind of Harrish-Chandra restriction bytaking fixed points under suitable distinguished subgroups. We just have no answerto the question of integrality in all these examples.

(iv) We set M := GrO ∈ Z−Mackalg(G). Let A(H) := M(H) = GrO(H)for all solvable subgroups H ≤ G, and A(H) = 0, if H ≤ G is not solvable.Let B(H) consist of the isomorphism classes of indecomposable OH-lattices forsolvable H ≤ G, and let B(H) be the empty set for all other subgroups. Moreover,let pH : M(H) → A(H) be the identity for solvable H ≤ G, and the trivial mapotherwise. then it is very easy to see that condition (∗π) of Theorem II.4.5 issatisfied for the set π of all primes, and since each p-hypo-elementary subgroupis solvable, Corollary II.4.9 implies that the corresponding morphism aM,A,p is anintegral canonical induction formula for GrO from Gr′O. but it seems that weallow too many modules to be induced. We would rather like to induce modules ofrestricted type from arbitrary subgroups than inducing all modules from subgroupsof restricted type. At least this was the theme in all the previous sections.

(v) We may replace the ring O with its residue field F everywhere in this section,and obtain the corresponding results of Proposition 5.2 also in this situation. Thequestion of integrality is not easier over F than over O.

(vi) There are other non-integral canonical induction formulae for GrO inducing

3.6. SUMMARY 103

from p-hypo-elementary subgroups only. They are the precise analogues of Exam-ples 1.4 and 1.7.

3.6 Summary

Let G be a finite group and let O be a complete disctete valuation ring with alge-braically closed residue field F of characteristic p > 0, and with quotient field K ofcharacteristic zero containing all |G|-th roots of unity. Furthermore, let π be a setof primes containing p.

Recalling the definitions of the previous sections we have a commutative diagram

RK,π⋂RK,p

∼←− PO∼−→ PF⋂ ⋂ ⋂

RK ←− TO∼−→ TF

‖⋂

‖RK − LO − TF‖

⋂ ⋂RK − GrO −→ GrF‖

yRK

d−−−−−−−−−−→ RF

(3.5)

in the category Z−Mack(G), involving the different choices for M in the Examplesof the previous sections, where the horizontal left bound arrows are induced bythe functor K ⊗O −, and the horizontal right bound arrows are induced by thefunctor F ⊗O −. The vertical morphism GrF (H) → RF (H) maps [V ] to [V ] forV ∈ OH−lat and H ≤ G. The morphism d : RK → RF is uniquely determinedby GrO → RF , since GrO → RK is surjective. It is the well-known decompositionmap. The cde-triangle (cf. [Se78, §15]) is part of the diagram, with the Cartan mapc : PF → RF given by the vertical arrows, and e : PF ∼= PO → RK given by anypath in the diagram connecting the two Mackey functors. Moreover, we obtain asimilar commutative diagram

RabK,π⋂

RabK,p

∼←− P abO

∼−→ P abF⋂ ⋂ ⋂

RabK T ab

O∼−→ T ab

F

‖⋂

‖RabK

∼←− LabO − T ab

F

‖y

RabK

d−−−−−−−−−−→ RabF

(3.6)

in the category Z−Res(G), involving the different choices of A, and by functoriality


a commutative diagram

RabK,π+⋂

RabK,p+

∼←− P abO +

∼−→ P abF +⋂ ⋂ ⋂

RabK + T ab

O +∼−→ T ab

F +

‖⋂

‖RabK +

∼←− LabO + − T ab

F +

‖y

RabK +

d−−−−−−−−−−→ RabF +

(3.7)

in the category Z−Mack(G). The induction morphisms bM,A : A+ → M fromDiagram (3.7) to Diagram (3.5) yield a three-dimensional commutative diagram(which we do not write down). Since the maps p ∈ Z−Con(G)(M,A) from theappropriate part of Diagram (3.5) to Diagram (3.6) only commute on the skeleton

RK,p∼←− PO

∼−→ PF⋂ ⋂TO

∼−→ TF⋂LO

(3.8)

of Diagram (3.5), the canonical induction formulae commute only on Diagram (3.8),cf. Proposition II.3.11. In particular, we obtain a commutative diagram (cf. Propo-sition 4.10 and Remark 4.11 (i))

POa−−→ P ab

O +⋂ ⋂TO

a−−→ T abO +⋂ ⋂

LOa−−→ Lab

O +

(3.9)

Note moreover, that none of the three morphisms in the cde-triangle commutes withthe respective canonical induction formulae.

6.1 Remark If we consider the Q-tensored version of the Diagrams (3.5)–(3.7)together with the morphisms p ∈ Q−Con(G)(Q ⊗M,Q ⊗ A) which are defined

by multiplication with the primitive idempotent e(H)H ∈ Q ⊗ Ω(H) on M(H), for

H ≤ G, then all corresponding induction formulae commute with each other byProposition II.3.11, since for any morphism f ∈ Q−Mack(G)(M,N) between Q-Mackey functors on G we have

fH(e(H)H · x) = e

(H)H · x

for H ≤ G, x ∈M(H). See also Remark 2.6 (ii) for an example of this statement.

3.7. EXTRASPECIAL P -GROUPS 105

3.7 Extraspecial p-groups: an explicit example

In this section we will compute the map aG : R(G)→ Rab+ (G) in the case of Exam-

ple 1.1 for the extraspecial p-group G of exponent p and order p2n+1, where n ≥ 0and p is an odd prime. For details about extraspecial p-groups see for example[Hu67, III.13] and [Su86, IV.4, Exerc. 4], for symplectic forms see [Hu67, II.9], andfor the representation theory of G see [Hu67, V.16.14].

7.1 The centre Z of G has order p and G/Z is a symplectic Fp-vector spaceof dimension 2n, where the symplectic form is given by the commutator, if oneidentifies Z with Fp once and for all. Note that Z equals the Frattini subgroup of Gand also the commutator subgroup of G. A subgroup H ≥ Z of G is abelian if andonly ifH/Z is a totally isotropic subspace ofG/Z. The automorphism group Aut(G)has the normal subgroup Inn(G) of inner automorphisms which consists preciselyof those automorphisms which act trivially on Z and G/Z. Each automorphism ofG induces an automorphism of Z. This yields a surjective group homomorphismβ : Aut(G)→ Aut(Z) ∼= F×p , and we obtain a series

1 < Inn(G) < ker(β) < Aut(G)

of normal subgroups of Aut(G) with Aut(G)/ ker(β) ∼= F×p and ker(β)/Inn(G) ∼=Sp2n(Fp), the group of symplectic 2n by 2n matrices over Fp, since each element ofker(β) respects the symplectic structure of G/Z, and since each element in Sp2n(Fp)can be realized as an automorphism of G. Therefore we will denote ker(β) also

by Sp2n ≤ Aut(G). Note that, by a theorem of Witt, Sp2n(Fp) acts transitively

on the set of totally isotropic subspaces of G/Z of fixed dimension. Hence, Sp2n

acts transitively on the set of abelian subgroups H ≥ Z of G of fixed order. Themaximal abelian subgroups of G are all conjugate under Sp2n, since they all containZ. Moreover, a maximal abelian subgroup of G has order pn+1.

The set Irr(G) consists of p2n one-dimensional characters and p − 1 charactersof degree pn. Let λ be one of the p − 1 non-trivial irreducible characters of Z,let λ be an extension of λ to a maximal abelian subgroup H ≤ G, then indGH(λ) isirreducible and depends only on λ and not on H and λ. Since indGH(λ) is irreducible,the stabilizer of λ in G is H, and therefore, λ has exactly pn G-conjugates whichhave to be precisely the pn different extensions of λ to H, since λ is stable underG-conjugation.

7.2 Proposition Let χ ∈ Irr(G) be of degree pn and let λ ∈ Irr(Z) be suchthat resGZ (χ) = pn · λ. Then

aG(χ) =n∑d=0

(−1)dpd(d−1)∑

Z ≤ H abelian

|H/Z|=pn−d

[H,ϕ]G,

where ϕ ∈ H is an arbitrary extension of λ to H.

Proof We write

aG(χ) =∑

(H,ϕ)∈G\M(G)

αG(H,ϕ)(χ)[H,ϕ]G,


with αG(H,ϕ)(χ) ∈ Z as in Equation (3.1). By Proposition 1.2 (xi) we obtain

αG(H,ϕ)(χ) = 0 unless (Z, λ) ≤ (H,ϕ). But in this case H is abelian. In fact, if

the commutator group H ′ of H were non-trivial, then H ′ ≤ G′ = Z, hence H ′ = Z,and resHH′(ϕ) = 1, which is a contradiction to resHZ (ϕ) = λ. For given abelian sub-group H with Z ≤ H, the extensions ϕ of λ form a full G-orbit, since they do inthe case of a maximal abelian subgroup of G. Therefore, we obtain

aG(χ) =∑

Z ≤ H abelian

αH [H,ϕ]G,

where αH ∈ Z and ϕ is an extension of λ to H. Let X denote the poset of abeliansubgroups of G containing Z. For each d ∈ 0, . . . , n, the subgroup Sp2n ∈ Aut(G)(which fixes λ and hence χ) acts transitively on the set of elements H ∈ X with|H/Z| = pn−d. Hence, by Proposition 1.2 (xiii) we obtain αH = αH′ for H,H ′ ∈ Xwith |H| = |H ′|. Therefore we can write

aG(χ) =n∑d=0

αd∑H∈X

|H/Z|=pn−d

[H,ϕ]G,

where αd ∈ Z and ϕ is an extension of λ to H. Now it suffices to show

αd = (−1)dpd(d−1)

for d ∈ 0, . . . , n. Let H0 ∈ X be a maximal abelian subgroup of G, and ford ∈ 0, . . . , n let Hd ∈ X be a subgroup of H0 of index pd. We fix e ∈ 0, . . . , n.Then Diagram (2.2) implies

pHe(resGHe(χ) =

n∑d=0

αd∑H∈X

|H/Z|=pn−d

πHe(res+

GHe([H,ϕ]G)

)(3.10)

in Rab(He). Now we fix an extension λe ∈ Irr(He) of λ and compare the coefficientsof λe in both sides of the above equation. Since resGH0

(χ) is the sum of all different

extensions of λ to H0, we have resGHe(χ) = pe∑λ where the sum runs over all

extensions λ of λ to He. Hence the coefficient of λe in the left hand side of theabove equation equals pe. Recall that

res+GHe([H,ϕ]G) =

∑g∈He\G/H

[He ∩ gH, res

gHHe∩

gH

( gϕ)]He,

and that the coefficient of λe in (πHeres+GHe

)([H,ϕ]G) is the coefficient of [He, λe]Hein res+

GHe

([H,ϕ]G). If He 6≤ H, this coefficient is zero. If He ≤ H, then each of the|H/Z| different G-conjugates (extensions of λ) of ϕ occur (G : H)/(H : Z) times inthe above sum. Among these |H/Z| different G-conjugates of ϕ, precisely (H : He)ie above λe. Thus, the coefficient of λe in (πHe res+

GHe

)([H,ϕ]G) equals

|H||He|

· |G||H|· |Z||H|

= pe+d,


if |H/Z| = pn−d. Hence by comparing the λe-coefficients in both sides of Equa-tion (3.10) we obtain (after dividing by pe) for every e ∈ 0, . . . , n,

1 =

e∑d=0

αd · pd ·#H ∈ X | He ≤ H, |H/Z| = pn−d.

Since these equations determine αd inductively, it suffices to prove that for givenU ∈ X we have

1 =∑

U≤H∈X(−1)d(H)pd(H)2

,

where d(H) is defined by |H/Z| = pn−d(H). Proposition B.2 (ii) in Appendix Bshows that the above equations for U ∈ X are equivalent to the equations

(−1)d(U)pd(U)2=

∑U≤H∈X

µX(U,H),

for U ∈ X. Now we fix U ∈ X. Note that each H ∈ X with U ≤ H is containedin the centralizer CG(U), and CG(U)/Z = (U/Z)⊥ with respect to the symplecticform. The original symplectic form on G/Z induces a non-degenerate symplecticform on (CG(U)/Z)

/(U/Z) ∼= CG(U)/U , and the poset H ∈ X | U ≤ H is

isomorphic to the poset of totally isotropic subspaces in (CG(U)/Z)/

(U/Z). Nowthe above equation follows from the following proposition, noting that µX(U,H)is the number of chains σ connecting U and H counted with multiplicity (−1)|σ|,and that the above sum equals −µX .

∪1(U, 1), if we add an element 1 to X which isbigger than every element in X.

7.3 Lemma Let V be a non-degenerate symplectic Fp-vector space of dimen-sion 2d, and let XV denote the poset of all totally isotropic subspaces of V togetherwith a greatest element 1. Then we have

µXV (0, 1) = (−1)d+1pd2.

Proof We prove the above equation by induction on d. The cases d = 0 andd = 1 can be verified easily. Let d ≥ 2 and consider a decomposition

V = W ⊥ H

of V into a non-degenerate symplectic subspace W 6= 0 of dimension 2d − 2 and ahyperbolic plane H. We consider the two maps

f : XW → XopV , U 7→ U, 1 7→ 1,

g : XopV → XW , U 7→ U ∩W, 1 7→ 1,

where XopV denotes the opposite poset of XV , i.e. the elements are the same and

the relations are reversed. Obviously f and g are order reversing, and they form aGalois connection, i.e. f(g(v)) ≥ v and g(f(w)) ≥ w for v ∈ Xop

V and w ∈ XV . A


fundamental property of Galois connections (cf. [Walk81, Thm. 4.1] for example) isthe fact that, for each v ∈ Xop

V and w ∈ XW , we have∑t∈f−1(v)

µXW (w, t) =∑

s∈g−1(w)

µXopV

(v, s).

We apply this to v = 1 ∈ XopV and w = 0 ∈ XW , and obtain

µXW (0, 1) =∑

U∈XV \1U∩W=0

µXV (U, 1).

For arbitrary U,U ′ ∈ XV \1, the relation U ≤ U ′ implies U ′ ∈ U⊥. The factorspace U⊥/U is again a non-degenerate symplectic Fp-vector space of dimension2(dimV − dimU), and we have for given U ∈ XV \ 1 an isomorphism of posets

U ′ ∈ XV | U ≤ U ′ → XU⊥/U , U ′ 7→ U ′/U, 1 7→ 1.

If U ∩W = 0, we have dimU ≤ 2, and the last equation can be written as

ad−1 = n0ad + n1ad−1 + n2ad−2, (3.11)

where ai := µXV ′ (0, 1) for a non-degenerate symplectic Fp-vector space V ′ of di-mension 2i, i ∈ N, and where

ni := U ∈ XV \ 1 | dimU = i, U ∩W = 0

for i ∈ 0, 1, 2. Obviously, n0 = 1, and

n1 =p2d − p2d−2

p− 1= p2d−1 + p2d−2,

since we may generate U by a vector in V \W and have p− 1 choices.

The computation of n2 is more difficult. First note that there are

(p2d−1 − 1)(p2d−2 − 1)

(p2 − 1)(p− 1)

elements U ∈ XV \ 1 with dimU = 2. In fact, there are (p2d − 1)(p2d−1 − p)ordered pairs (v, v′) of linearly independent vectors v, v′ ∈ V with v′ ⊥ v. On theother hand each such U has (p2 − 1)(p2 − p) different ordered bases.

Among these U ∈ XV \ 1 with dimU = 2 there are by the same argument

(p2(d−1) − 1)(p2(d−1)−2 − 1)

(p2 − 1)(p− 1)

elements with dim(U ∩W ) = 2, and there are

(p2d−2 − 1)(p2d−2 − p2d−4)

(p− 1)2


elements with dim(U ∩W ) = 1, since there are (p2d−2 − 1)(p2d−1 − p2d−3) pairs(w, v) of linearly independent vectors w ∈ W and v ∈ W⊥ \ (W ∩W⊥), and eachsuch U has exactly (p− 1)(p2 − p) ordered bases (w, v) with w ∈W .

Hence we have

n2 =(p2d − 1)(p2d−2 − 1)− (p2d−2 − 1)(p2d−4 − 1)

(p2 − 1)(p− 1)−

− (p+ 1)(p2d−2 − 1)(p2d−2 − p2d−4)

(p2 − 1)(p− 1)

=p2d−2 − 1

(p2 − 1)(p− 1)[p2d − 1− p2d−4 + 1− p2d−1 + p2d−3 − p2d−2 + p2d−4]

=p2d−2 − 1

(p2 − 1)(p− 1)[p2d − p2d−2 − p2d−1 + p2d−3]

=p2d−2 − 1

(p2 − 1)(p− 1)(p2 − 1)(p2d−2 − p2d−3)

= p2d−3(p2d−2 − 1).

Using the induction hypothesis for ad−1 and ad−2, we obtain from Equation (3.11)

ad = (1− n1)ad−1 − n2ad−2

=(1− (p+ 1)p2d−2

)(−1)dp(d−1)2 − p2d−3(p2d−2 − 1)(−1)d−1p(d−2)2

= (−1)d+1(−pd2−2d+1 + pd2 + pd

2−1 − p4d−5+(d−2)2+ p2d−3+(d−2)2

)= (−1)d+1

(−pd2−2d+1 + pd

2+ pd

2−1 − pd2−1 + pd2−2d+1

)= (−1)d+1pd

2,

which completes the proof of the lemma.


Chapter 4

Applications and FurtherProperties of CanonicalInduction Formulae

In this chapter we give applications of the existence of the canonical inductionformulae constructed in Chapter III. In Section 1 we consider the question whethera morphism of restriction functors f : A → N between a restriction subfunctorA of a Mackey functor M and a Mackey functor N can be lifted to a morphismF : M → N of restriction functors. The answer is positive, if there is a canonicalinduction formula for M from A whose residue is the identity on A. Moreover, theextension is unique, if A and M coincide on the set of coprimordial subgroups ofN . This extension theorem is applied in Sections 2, 3, and 4 to the determinant,the Adams operations, and the Chern classes. By this procedure we are able togive explicit algebraic definitions for Chern classes of Brauer characters, trivialsource modules and linear source modules with all the natural properties that areknown for the classical Chern classes on the character ring. In Sections 5 and 6we consider the semidirect product GS of two finite groups G and S of coprimeorder (S acts on G). Using the canonical induction formulae of Chapter III we areable to define extension maps extG,S : M(G)S → M(GS) such that resGSG extG,S

is the identity, where M(G)S denotes the subgroup of S-fixed elements in M(G)with respect to the conjugation action. Moreover, we define a map glG,S : M(G)S →M(GS), where GS denotes the subgroup of S-fixed points of G. If M is the characterring Mackey functor and S is a p-group, then glG,S is related to the Glaubermancorrespondence. In Section 7 we construct canonical induction formulae for theprojectification of a complex character and of a Brauer character. Both formulaehave rational coefficients, and in the character ring case the denominators of thesecoefficients determine the defect of an irreducible character.

4.1 An extension theorem on morphisms of restrictionfunctors

Throughout this section we fix the following situation:Let k denote a commutative ring and G a finite group. We assume that we have a

111

112 CHAPTER 4. APPLICATIONS

k-restriction functor M on G (not necessarily a Mackey functor) and a k-restrictionsubfunctor A ⊆ M on G. Furthermore, let a ∈ k−Res(G)(M,A+) be such thataH(ϕ) = [H,ϕ]H for H ≤ G, ϕ ∈ A(H). Let p := πA a ∈ k−Con(G)(M,A)be the residue of a. Then we have pH(ϕ) = ϕ for H ≤ G, ϕ ∈ A(H). In fact,pH(ϕ) = πAH(aH(ϕ)) = πAH([H,ϕ]H) = ϕ.

1.1 Definition For M , A, a as above and N ∈ k−Mack(G) we define ak-linear map

ΦM,AN : k−Res(G)(M,N)→ k−Res(G)(A,N)

by restricting a morphism on M to A. Conversely we define a k-linear map

ΣM,A,aN : k−Res(G)(A,N)→ k−Res(G)(M,N), f 7→ bN,N f+ a,

and call ΣM,A,aN (f) the canonical extension of f ∈ k−Res(G)(A,N) with respect

to the morphism a.

The following theorem justifies this terminology.

1.2 Theorem Let M , A, and a be as above, and let N ∈ k−Mack(G). ThenΦM,AN ΣM,A,a

N = id. In particular, the map


is split surjective.

Proof We show that ΦM,AN ΣM,A,a

N = id, i.e. fH = bN,NH f+H aH on A(H)for H ≤ G. In fact, for ϕ ∈ A(H) we have by assumption aH(ϕ) = [H,ϕ]H , andfurther by the definitions of f+ and bN,N we obtain

(bN,NH f+H aH)(ϕ) = bN,NH (f+H([H,ϕ]H)) = bN,NH ([H, fH(ϕ)]H)

= indHH(fH(ϕ)) = fH(ϕ).

The following theorem describes a situation where ΦM,AN and ΣM,A,a

N are inverseisomorphisms.

1.3 Theorem Let M , A, and a be as above and let N ∈ k−Mack(G) suchthat A(H) = M(H) for H ∈ C(N). Then the restriction map


is an isomorphism with inverse ΣM,A,aN . In particular, ΣM,A,a

N is independent of a.

Proof It suffices to show that ΣM,A,aN ΦM,A

N = id, i.e. bN,NH f+H aH = fH forf ∈ k−Res(G)(M,N) and H ≤ G. Since b, f+, and a commute with restrictions,it suffices to prove this equation for H ∈ C(N). But for H ∈ C(N) we haveM(H) = A(H), and we obtain (bN,NH f+H aH)(ϕ) = fH(ϕ) for ϕ ∈M(H) = A(H)as in the proof of Theorem 1.2.

4.1. AN EXTENSION TEOREM 113

1.4 Definition Let M , A, and a be given as at the beginning of this section,and assume that M is a k-Mackey functor on G.

(i) We define the k-module k−Res(G)(A ⊆ M,A ⊆ M) as the set of elementsf ∈ k−Res(G)(M,M) satisfying f(A) ⊆ A. Similarly, if M ∈ k−Mackalg(G) andA ⊆ M is a k-algebra restriction subfunctor on G, we define k−Resalg(G)(A ⊆M,A ⊆M) as the set of elements f ∈ k−Resalg(G)(M,M) with f(A) ⊆ A.

(ii) Similar to Definition 1.1 we define the k-linear restriction function

ΦM,A : k−Res(G)(A ⊆M,A ⊆M)→ k−Res(G)(A,A)

and a k-linear map in the other direction,

ΣM,A,a : k−Res(G)(A,A)→ k−Res(G)(A ⊆M,A ⊆M), f 7→ bM,A f+ a.

We call ΣM,A,a(f) again the canonical extension of f ∈ k−Res(G)(A,A) withrespect to a.

If M ∈ k−Mackalg(G) and A ⊆ M is a k-algebra restriction subfunctor thek-linear map restricts to a function

ΦM,Aalg : k−Resalg(G)(A ⊆M,A ⊆M)→ k−Resalg(G)(A,A).

1.5 Theorem Let M , A, and a be defined as at the beginning of this section,assume that M ∈ k−Mack(G), and let N ∈ k−Mack(G). Then we have ΦM,A ΣM,A,a = id, in particular,


is split surjective.

Proof This is proved in the same way as Theorem 1.2.

1.6 Theorem Let M , A, and a be defined as at the beginning of this sectionand assume that M is a k-Mackey functor on G with A(H) = M(H) for H ∈ C(M).Then the restriction map


is an isomorphism with inverse ΣM,A,a.

Proof This is proved in the same way as Theorem 1.3.

We have the following multiplicative versions of Theorems 1.5 and 1.6.

1.7 Theorem Let M , A, a, and p be defined as at the beginning of thissection. Assume that M is a k-Green functor on G and that A ⊆M is a k-algebrarestriction subfunctor of M on G.

(i) If ρA : A+ → A+ is injective, and pH is a k-algebra homomorphism forH ∈ C(M), then

ΣM,A,a(k−Resalg(G)(A,A)

)⊆ k−Resalg(G)(A ⊆M,A ⊆M).


In particular,

ΦM,Aalg : k−Resalg(G)(A ⊆M,A ⊆M)→ k−Resalg(G)(A,A)

is surjective.(ii) If A(H) = M(H) for H ∈ C(M), then ΦM,A

alg is bijective with inverse ΣM,A,a.

Proof Using Theorem 1.6 it suffices for both parts to show that the H-component bM,A

H f+H aH : M(H)→M(H) of ΣM,A,a(f) is a k-algebra homomor-phism for f ∈ k−Resalg(G)(A,A). We have to proof that

(bM,AH f+H aH)(1M(H)) = 1M(H)

and

(bM,AH f+H aH)(x · y) = (bM,A

H f+H aH)(x) · (bM,AH f+H aH)(y)

for H ≤ G, x, y ∈ M(H). Since these equations of elements in M(H) can beverified by applying resHK for K ≤ H, K ∈ C(M), and since restriction maps arek-algebra homomorphisms and commute with b, f+, and a, it suffices to prove theabove equations for H ≤ G with H ∈ C(M).

In the situation of part (i), Corollary I.4.2 (iv) implies that aH is a k-algebrahomomorphism for H ∈ C(M), and the result follows, since bM,A

H and f+H arek-algebra homomorphisms.

In the situation of part (ii), the above equations hold for H ∈ C(M), since(bM,AH f+H aH)(x) = fH(x) for x ∈M(H) = A(H).

1.8 Theorem Let M , A, a, and p be defined as at the beginning of thissection with M ∈ k−Mack(G) and assume that a is injective (which is the case ifa is a canonical induction formula). Assume further that one of the following twoconditions holds:

(i) The mark morphism ρA : A+ → A+ is injective, and

pH bM,AH h+H aH = hH pH : M(H)→ A(H)

for h ∈ k−Res(G)(A,A) and H ∈ C(M).

(ii) For H ∈ C(M) we have M(H) = A(H).

Then we haveΣM,A,a(f g) = ΣM,A,a(f) ΣM,A,a(g)

for f, g ∈ k−Res(G)(A,A).

Proof We have to show that

bM,AH f+H aH bM,A

H g+H aH = bM,AH (fg)+H aH : M(H)→M(H)

for H ≤ G. Since all maps in the above equation commute with restrictions, itsuffices to prove the above equation for H ∈ C(M). In situation (ii) we have (bM,A

H h+H aH)(x) = hH(x) for all h ∈ k−Res(G)(A,A), H ≤ C(H), and x ∈ M(H) =A(H), and the result follows. In situation (i), we obtain from Diagram (2.2) that

4.1. AN EXTENSION TEOREM 115

the map (pK resHK)K≤H = ρAH aH is injective, since ρAH and aH are injective.

Therefore it suffices to show that the above equation holds after having composedboth sides with pK resHK , where K ≤ H and H ≤ C(M). Moreover we may alsoassume that K ∈ C(M). But everything follows from the hypothesis in (i):

pK resHK bM,AH f+H aH bM,A

H g+h aH= pK bM,A

K f+K aK bM,AK g+K aK resHK

= fK pK bM,AK g+K aK resHK = fK gK pK resHK

= (f g)K pK resHK = pK bM,AK (f g)+K aK resHK

= pK resHK bM,AH (f g)+H aH

1.9 Remark Recall from Chapter III the Examples 1.1 (RabK ⊆ RK), Exam-

ple 2.1 (RabF ⊆ RF ), Example 3.1 (P ab

F ⊆ PF or equivalently RabK,p ⊆ RK,p), Ex-

ample 3.4 (RabK,π ⊆ RK,π), Example 4.2 (Lab

O ⊆ LO) and Example 4.9 (T abO ⊆ TO).

In all these examples we have pH(x) = x and aH(x) = [H,x]H for H ≤ G andx ∈ A(H), so that our basic assumptions at the beginning of this section hold.

For the examples 1.1, 2.1, 3.1, and 3.4 of Chapter III we have A(H) = M(H)for H ∈ C(M) so that we have isomorphims

ΦM,A : k−Res(G)(A ⊆M,A ⊆M)∼−−→ k−Res(G)(A,A)⋃ ⋃

ΦM,Aalg : k−Resalg(G)(A ⊆M,A ⊆M)

∼−−→ k−Resalg(G)(A,A)

(4.1)

with inverse ΣM,A,a.

In the situation of Examples III.4.2 and III.4.9, the map ΦM,Aalg , and hence also

the upper one, is not injective in general, but of course surjective by Theorem 1.4.

In fact, consider Example II.4.2 or II.4.9 with G the cyclic subgroup of or-der p. Note that M(G) = A(G) ⊕ Z · [OG] and M(1) = A(1). The identity ink−Resalg(G)(A,A) has at least two extensions in k−Resalg(G)(A ⊆ M,A ⊆ M),namely the identity and the morphism f with fG([OG]) = p · [O].

1.10 Remark Let M , A, and a be given as at the beginning of this sectionwith M ∈ k−Mack(G). Assume that M and A can be considered as contravariantfunctors from the category of finite groups to the category of abelian groups withrespect to restrictions along arbitrary group homomorphisms. Then also A+ can beconsidered as such a functor, using the definition of res+f : A+(G)→ A+(G′), givenby Equation (3.3), for a group homomorphism f : G → G′. Assume further thata : M → A+ and bM,A : A+ → M are natural transformations. This is the case forthe Examples III.1.1 (character ring), II.2.1 (Brauer character ring), II.4.2 (linearsource ring), and III.4.9 (trivial source ring).

Let N be another k-Mackey functor on G which can also be regarded as a functorfrom the category of finite groups to the category of abelian groups by extendingthe definition of resGH (for H ≤ G) to resf (for f : G′ → G). If t : A→ N is a natural

transformation, so is t+, and it follows that also ΣM,A,aN (t) : M → N , defined by


ΣM,A,aN (t)G := gM,A

G t+G aG : M(G) → N(G) for any finite group, is a naturaltransformation. Hence, we obtain theorems which are similar to Theorems 1.1,1.2, and 1.4–1.7. So, for instance, the modified version of Theorem 1.5 states forM := R, A := Rab, using a as in Example III.1.1, that the restriction

Nat(Rab ⊆ R,Rab ⊆ R)→ Nat(Rab, Rab)

between the ‘sets’ of natural transformations is a bijection. The inverse is given byΣM,A,a as in the proof of Theorem 1.5. It seems to be within reach to determineNat(Rab, Rab) and thus obtain a nice description for Nat(Rab ⊆ R,Rab ⊆ R). TheAdams operations Ψn, n ∈ Z, cf. Section 3, form a set of Z-linearly independentelements in the abelian group Nat(Rab, Rab). But we won’t go into this at themoment.

4.2 Determinants

For a commuative ring k, a finite group G, and a k-free kG-module V of finitek-rank the determinant det(V ) ∈ Hom(G, k×) is defined as the homomorphism

Gρ−→GL(V )

det−→ k×,

where ρ is the representation of G on V induced by the kG-module structure. Firstwe show that determinants have values in a Mackey functor.

2.1 Example for a finite group G and a commutative ring k we define G(k)as the multiplicative abelian group Hom(G, k×). For a fixed group G, the abeliangroups H(k), H ≤ G, form a cohomological Z-Mackey functor on G with thefollowing structure maps:

( gϕ)(h) := ϕ(hg) for h ∈ H ≤ G, ϕ ∈ H(k), g ∈ G,(resHK(ϕ)

)(x) := ϕ(x) for x ∈ K ≤ H ≤ G, ϕ ∈ H(k),(

indHK(ϕ))(h) := ϕ trH/K νH for K ≤ H ≤ G, ϕ ∈ K(k), h ∈ H,

where νH : H → Hab is the canonical surjection, trH/K : Hab → Kab is the transfer

map (see [CR81, 13.10] for a definition) and ϕ : Kab → k× is induced by ϕ : K → k×.In fact, the Mackey functor axioms are verified by simple calculations, or by usingthe isomorphisms H(k) ∼= H1(H, k×), where k× is regarded as trivial H-module forH ≤ G. These isomorphisms translate the three structure maps defined above toconjugation, restriction and corestriction of cohomology, where the Mackey functoraxioms are well-known to hold. The set of coprimordial subgroups for H(k), H ≤ G,is certainly contained in the set of cyclic subgroups of prime power order of G, since(k× being abelian) a group homomorphism G → k× is uniquely determined byits values on elements of prime power order. If k× does not contain non-trivialp-elements for a prime p, then the set of coprimordial groups is even contained inthe set of cyclic p′-subgroups of prime power order.

The following proposition is well-known. A proof for k = C can be found in[CR81, 13.15]. This proof generalizes without changes to arbitrary commutativerings k.

4.2. DETERMINANTS 117

2.2 Proposition Let k be a commutative ring, G a finite group, H ≤ G, andV a k-free kH-module of finite k-rank. Then indGH(V ) is a k-free kG-module offinite k-rank and we have

det(indGH(V )

)= εrkkV

G/H ·(det(V ) trG/H νG

)∈ G(k),

where νG : G → Gab is the canonical surjection, trG/H : Gab → Hab the transfer

map, det(V ) : Hab → k× the map induced by det(V ), and εG/H ∈ G(k) is definedby mapping an element g ∈ G to the sign of the permutation by which g acts onG/H via left multiplication. (The dot in the above formula indicates multiplicationin G(k).)

2.3 Next we are going to apply Theorems 1.1 and 1.2 to determinants. Moreprecisely, we are going to define in different situations the following data for a finitegroup G: a Z-Mackey functor M , a Z-restriction subfunctor A ⊆ M , a canonicalinduction formula a ∈ Z−Res(G)(M,A+), a commutative ring k, a Z-Mackey func-tor N on G given by N(H) := H(k) for H ≤ G (cf. Example 2.1), and a morphismdet ∈ Z−Res(G)(A,N). Note that for H ≤ G, the abelian group M(H) is writ-ten additively and the abelian group N(H) multiplicatively. We distinguish thefollowing situations:

(i) Character ring: Let k := K be a field for G of characteristic zero whichcontains all |G|-th roots of unity, M := RK , A := Rab

K , a : RK → RabK + as in

Example III.1.1. Furthermore, for H ≤ G, let detH : Rab(H) = Z · H(K) →H(K) = N(H) be defined by detH(ϕ) = ϕ for ϕ ∈ H(K).

(ii) Brauer character ring: Let k := F be an algebraically closed field of charac-teristic p > 0, M := RF , A := Rab

F , a : RF → RabF + as in Example III.2.1.

Furthermore, for H ≤ G, let detH : A(H) = Z · H(F ) → H(F ) = N(H) bedefined by det(ϕ) = ϕ for ϕ ∈ H(F ). Note that C(N) is the set of cyclicp′-groups of prime power order, since F× contains no non-trivial p-elements.

(iii) p-projective character group: Let k := F be as in (ii), M := PF , A := P abF ,

a : PF → P abF + as in Example III.3.1. Furthermore, for a p′-group H ≤ G, let

detH : A(H) = Z · H(F ) → H(F ) = N(H) be defined as in (ii). Note thatC(N) is the set of cyclic p′-subgroups of prime power order.

(iv) π-projective character group: Let k := K be defined as for the character ringin (i), let M := RK,π, A := Rab

K,π, a : RK,π → RabK,π+

as in Example III.3.4.

Furthermore, for a π′-subgroup H ≤ G, let detH : A(H) = Z · H(K) →H(K) = N(H) be defined as in situation (i).

(v) Linear source ring: Let k := O be a complete discrete valuation ring withalgebraically closed residue field of characteristic p > 0, and a quotient fieldof characteristic zero which contains all |G|-th roots of unity. Let M := LO,A := Lab

O , and a : LO → LabO + as in Example III.4.2. Furthermore, for H ≤ G,

let detH : A(H) = Z · H(O)→ H(O) = N(H) be defined by detH(ϕ) = ϕ forϕ ∈ H(O).


(vi) Trivial source ring: Let k := O be as for the linear source ring, M := TO,A := T ab

O , a : TO → T abO + as in Example III.4.9. Furthermore, for H ≤ G, let

detH : A(H) = Z · H(O)p′ → H(O)p′ = N(H) be defined by detH(ϕ) = ϕ for

ϕ ∈ H(O)p′ .

Note that in each of these situations there is an extension det ∈ Z−Res(G)(M,N)of det ∈ Z−Res(G)(A,N), given by the usual determinant.

2.4 Proposition Assume that G, M , A, a : M → A+, k, N , and det ∈Z−Res(G)(A,N) are given as in one of the six prceeding situations.

(i) The morphism det ∈ Z−Res(G)(A,N) can be extended to a morphism det ∈Z−Res(G)(M,N) by defining det := ΣM,A,a

N (det) = bM,A det+ a.(ii) In the situations 2.3 (i) (character ring), 2.3 (ii) (Brauer character ring),

and 2.3 (iii) (p-projective character group), the extension is unique, and thereforecoincides with the usual morphism det ∈ Z−Res(G)(M,N).

(iii) In situations 2.3 (iv)–2.3 (vi) (π-projective characters, linear source ring,

trivial source ring), the morphisms det and det in Z−Res(G)(M,N) coincide, pro-vided that p 6= 2, resp. 2 /∈ π.

Proof (i) This is an immediate consequence of Theorem 1.2.(ii) This is an immediate consequence of Theorem 1.3, since A(H) = M(H) for

H ∈ C(N).(iii) Note that in all three situations, the coprimordial subgroups on N are

the set of cyclic subgroups of prime power order. Hence it suffices to show thatdetH = detH for a cyclic group of prime power order qn, q prime, n ∈ N. Ifq 6= p, resp. q /∈ π in situation (iv), then A(H) = M(H), and it follows easilythat detH = bN,NH det+H aH . Next we consider situation 2.3 (iv) (π-projectivecharacter group) and assume that q ∈ π, hence q 6= 2. Then M(H) = Z · indH1 (1),and we have aH(indH1 (1)) = [1, 1]H . Hence,

detH(indH1 (1)

)= (bN,NH det+H)([1, 1]H) = bN,NH

([1,det1(1)]H

)= indH1 (1) = 1.

On the other hand, detH(indH1 (1)) = εH/1 = 1, since q is odd.

Now we consider situation (2.3) (v) (linear source ring) and assume q = p. Then

LO(H) =⊕U≤Hϕ∈U(O)

Z · indHU ([Oϕ]).

It is easy to show that aH(indHU ([Oϕ])

)= [U,ϕ]H . Hence,

det(indHU ([Oϕ])

)=(bN,NH det+H aH

)(indHU ([Oϕ])

)= ϕ trH/U

On the other hand, by Proposition 2.2, detH(indHU ([Oϕ])

)= εH/U · (ϕ trH/U ) with

εH/U = 1, since p is odd.Finally we consider situation 2.3 (vi) (trivial source ring) and assume q = p.

Then

TO(H) =⊕U≤H

Z · indHU ([O]).

4.2. DETERMINANTS 119

It is easy to show that aH(indHU ([O])

)= [U, 1]H . Hence,

det(indHU ([O])

)= bN,NH ([U, 1]H) = 1 trH/U = 1.

On the other hand,

detH

(indHU ([O])

)= εH/U · (1 trH/U ) = εH/U = 1,

since p is odd.

2.5 Remark (i) First note that det 6= det in situation 2.3 (iv)–(vi), for Gthe cyclic group of order 2, and p = 2, resp. 2 ∈ π in situation 2.3 (iv). Considerthe KG-module KG with character indG1 (1) in situation 2.3 (iv) with 2 ∈ π. Thendet(KG) = det(indG1 (1)) = εG/1·(1trG/1) = εG/1 which is the non-trivial element in

the group G(K) of order 2. On the other hand it is easy to verify that aG(indG1 (1)) =

[1, 1]G, hence det(indG1 (1)) = 1 trG/1 = 1. Similar arguments shwo that in the

situatons 2.3 (v) and 2.3 (vi) we have det([OG]) = 1 and det([OG]) = εG/1 6= 1.

At a first glance, it is confusing that we obtain different results for PF and RK,p.The reason is that the Mackey functor N differs in the two cases. If we choose POinstead of PF in situation (2.3) (iii) and N = ?(O) instead of N = ?(F ), then weobtain the same ambiguities as in example (2.3) (iv), since C(N) changes from theset of cyclic p′-subgroups of prime power order to the set of all cyclic subgroupsof prime power order. But note that from an algebraic point of view none of theextensions det and det is distinguished. They can also both be considered as naturaltransformations in the sense of Remark 1.10.

(ii) The above proposition is certainly not surprising. The existence of an ex-tension of the determinant from A to M is clear, since we have the representationtheoretic interpretation of M which allows to define determinants without effort.And the uniqueness part is also clear, because detH(χ) is a function on H, andhence determined by its restrictions to cyclic subgroups (resp. cyclic p′-subgroups.But on these subgroups H, the value is already defined, since A(H) = M(H).

Nevertheless, we think that also this trivial application shows the power of thetheorems in Section 1. Assume that by what reasons ever, we knew the characterring only as a certain lattice in the C-vector space of class functions on a finitegroup, without having any module-theoretic interpretation. It would be very hardto extend the definition from Rab to R. Note that even without module theoreticinterpretation the map detH : Rab(H)→ H, ϕ 7→ ϕ, is very natural to be considered.The next section will provide an example, where because of the lack of a moduletheoretic interpretation of the Adams operations the definition in the literature isnot straight forward.

(iii) Although the statements about uniqueness and existence in the last propo-sition are not surprising, the explicity of the existence part has a surprising aspect.We write as in Equations (3.1) and (3.2), for H ≤ G and χ ∈M(H),

aH(χ) =∑

(K,ψ)∈H\M(H)

αH(K,ψ)(χ)[K,ψ]H ∈ A+(H)


with αH(K,ψ)(χ) ∈ Z and M(H) according to the situation. Then we have

χ =∑

(K,ψ)∈H\M(H)

αH(K,ψ)(χ)indHK(χ) ∈M(H) (4.2)

by applying bM,AH . Hence, applying detH and using Proposition 2.2 we obtain

detH(χ) =∏

(K,ψ)∈H\M(H)

(εH/K · (ψ trH/K νH)

)αH(K,ψ)

(χ)(4.3)

=∏

(K,ψ)∈H\M(H)

(εH/K)αH

(K,ψ)(χ) ·

∏(K,ψ)∈H\M(H)

(ψ trH/K νH)αH

(K,ψ)(χ).

On the other hand, we have detH(χ) = detH(χ) = (bM,AH det+H aH)(χ) in the

situations 2.3 (i)–(iii), which expands to

detH(χ) =∏

(K,ψ)∈H\M(H)

(ψ trH/K νH)αH

(K,ψ)(χ). (4.4)

This shows that we may forget about the extra factors εH/K if we use a canonical in-duction formula. This is obviously not true if we use an arbitrary induction formula.Another possible interpretation of Equations (4.3) and (4.4) is that starting withEquation 4.2 we may calculate detH(χ) by letting detH commute with induction ineach summand. Of course, detH does not commute with induction in general, butthe error terms, namely εH/K , just cancel over the whole sum (or product).

4.3 Adams operations

Throughout this section let G denote a finite group, and O a complete discretevaluation ring with algebraically closed residue field F of characteristic p > 0 andquotient field K of characteristic zero which contains the |G|-roots of unity.

We are going to apply the theorems of Section 1 to the morphisms Ψn, n ∈ Z,which are known as Adams operations. We first give a partial definition in eachsituation and extend the definition using Theorem 1.4.

3.1 In six situations we are going to specify the following data: A Z-Mackeyfunctor M on G, a Z-restriction subfunctor A ⊆ M which has a stable basisB(H), H ≤ G, carrying the structure of a multiplicative group in its own right(i.e. A(H) is a group ring) unless B(H) = ∅, and a canonical induction formulaa ∈ Z−Res(G)(M,A+). Moreover, we define in each of these situations the n-thAdams operation Ψn ∈ Z−Res(G)(A,A) for n ∈ Z by

ΨnH(ϕ) = ϕn

for H ≤ G, ϕ ∈ B(H). Here the six situations:

(i) Character ring: M := RK , A := RabK , B(H) := H(K) for H ≤ G, a : RK →

RabK + as in Example III.1.1.

4.3. ADAMS OPERATIONS 121

(ii) Brauer character ring: M := RF , A := RabF , B(H) := H(F ) for H ≤ G,

a : RF → RabF + as in Example III.2.1.

(iii) p-projective character group: M := PF , A := P abF , B(H) := H(F ) for H ≤ G

a p′-subgroup, B(H) = ∅ otherwise, a : PF → P abF + as in Example III.3.1.

(iv) π-projective character group: M := RK,π, A := RabKπ, B(H) := H(K) for

H ≤ G a π′-group, a : RK,π → RabKπ+ as in Example III.3.4.

(v) Linear source ring: M := LO, A := LabO , B(H) := H(O) for ≤ G, a : LO →

LabO + as in Example III.4.2.

(vi) Trivial source ring: M := TO, A := T abO , B(H) := H(O)p′ for H ≤ G,

a : TO → T abO + as in Example III.4.9.

Note that in situation (i), (ii), (v), and (vi), we have M ∈ Z−Mackalg(G), A ∈Z−Resalg(G), and Ψn ∈ Z−Resalg(G)(A,A). In situation (iii) and (iv), A(H) ⊆M(H) are still rings (but possibly without unity) and Ψn is still multiplicative.

3.2 Proposition Let M , A, a, Ψn for n ∈ Z, be defined as in one of the sixsituations in 3.1.

(i) The morphisms Ψn ∈ k−Res(G)(A,A), n ∈ Z, can be extended to morphismsΨn ∈ Z−Res(G)(A ⊆ M,A ⊆ M) by the definition Ψn := ΣM,A,a(Ψn) = bM,A Ψn

+ a.(ii) In situation 3.1 (i)–(iv) the extensions are unique.(iii) In situation 3.1 (i), (ii), and (vi), we have Ψn ∈ Z−Resalg(G)(A ⊆M,A ⊆

M) for n ∈ Z.(iv) In situation 3.1 (i)–(iv) and (vi) we have Ψn Ψm = Ψnm for m,n ∈ Z.(v) In situation 3.1 (i), (ii), (v), and (vi), the morphisms Ψn, n ∈ Z, can be

regarded as natural transformations from M to M in the sense of Remark 1.10,where M is considered as a contravariant functor from the category of finite groupsto the category of commutative rings.

Proof (i) This is immediate from Theorem 1.5.(ii) This follows from Theorem 1.6, since A(H) = M(H) for H ∈ C(M), in these

situations.(iii) This follows from Theorem 1.7 (i), since ρA : A+ → A+ is injective (cf. Prop-

ositions I.3.2 and I.2.4), and pH : M(H)→ A(H) is a k-algebra homomorphism forH ∈ C(M) in these situations. In fact, in situation 3.1 (i) and (ii), pH is the identity,and in situation 3.1 (vi) this is stated in Lemma III.4.11 (iv).

(iv) We apply Theorem 1.8 (i) in situation 3.1 (vi) (cf. Lemma III.4.11 (iii)) andTheorem 1.8 (ii) in situation 3.1 (i)–(iv), and observe that Ψn Ψm = Ψnm holdsobviously on A for m,n ∈ Z.

(v) This follows from Remark 1.10, since Ψ: A → A can be considered as anatural transformation.

3.3 Remark Note that in the situation 3.1 (i), (ii), (iii), and (vi) (with TFinstead of TO) Adams operations

ΨnG : M(G)→M(G)


are defined in the literature. In situation 3.1 (i) the definition is well-known:(ΨnG(χ)

)(g) := χ(gn)

for n ∈ Z, χ ∈ RK(G), g ∈ G (see [Se78, 9.1, Exercise 3] for example) Thesame definition holds for the ring of Brauer characters if they are interpretedas class functions on p-regular elements. But note that it is not at all trivialto show that Ψn

G(χ) ∈ RK(G), resp. ΨnG(χ) ∈ RF (G). Kervaire gives a treat-

ment of the situations 3.1 (i), (ii), and (iii) in [Ke75]. Note that his definitionof Ψn

G is complicated, and so are the proofs of the properties of ΨnG. Benson

defines ΨnG : GrF (G) → GrF (G) for the Green ring in [Be84a]. Again the defi-

nition seems complicated, using tensor induction, and also the proofs of the mainproperties are not trivial. But it is clear from the definition that the subgroupsPF (G) ≤ TF (G) ≤ GrF (G) are invariant under Ψn. It is also clear from the defini-tions in all four situations that the Adams operations are morphisms of Z-restrictionfunctors on G, and that they coincide with the definitions in 3.1 on A.

Next we show that our definitions coincide with the definitions in the literaturein situations 3.1 (i), (ii), (iii), and (vi).

3.4 Proposition Let M , A, a, Ψn ∈ Z−Res(G)(A,A), for n ∈ Z, be definedas in one of the situations 3.1 (i), (ii), (iii), or (vi), and let Ψn ∈ Z−Res(G)(A ⊆M,A ⊆M) be the canonical extension as described in Proposition 3.2. Then Ψn =Ψn.

Proof In situation 3.1 (i)–(iii) this follows from the uniqueness statement ofProposition 3.2 (ii). In situation 3.1 (vi) it suffices to show that

s(H,h)

(ΨnG(x)

)= s(H,h)

(ΨnG(x)

)for H ≤ G p-hypo-elementary, h ≤ G, n ∈ Z, x ∈ TO(G), cf. Proposition III.4.14.Benson showed in [Be84a, Lemma 2] that

s(H,h)

(ΨnG(x)

)= s(H[n],hn)(x),

where H [n] is the unique subgroup of H of index gcd(n, |H|p′). Hence, we obtainfrom the definition (see Definition III.4.13) of s(H,h) that

s(H,h)

(ΨnG(x)

)=(shn pH[n] resG

H[n]

)(x)

=(shn pH[n] resH

H[n]

)(resGH(x)

)On the other hand we have by Lemma III.4.11 (iii),

s(H,h)

(ΨnG(x)

)=(sh pH resGH b

M,AG Ψn

+G aG)(x)

=(sh pH bM,A

H Ψn+H aH

)(resGH(x)

)=(sh Ψn

H pH)(

resGH(x)),

and it suffices to show that(sh Ψn

H pH)([V ]) =

(shn pH[n] resH

H[n]

)([V ])

4.3. ADAMS OPERATIONS 123

for H ≤ G p-hypo-elementary, h ∈ H, V ∈ OH−triv indecomposable, n ∈ Z.If rkOV = 1, i.e. V ∼= Oϕ for some ϕ ∈ H(O)p′ , then both sides are equal toϕn(h) = ϕ(hn). If rkOV > 1, then pH([V ]) = 0, and V has vertex Q < Op(H) byLemma III.4.11 (i). Therefore,

resHH[n](V ) | resH

H[n]

(indHQ ([O])

) ∼= ⊕h∈H/H[n]

indH[n]

hQ

([O]),

and resHH[n](V ) has no indecomposable summand with vertex Op(H). Now part (i)

of Lemma III.4.11 implies

pH[n]

(resH

H[n]([V ]))

= 0.

Note that we can use the above proposition as proof that the Adams operationscarry the character ring RK(G) to RK(G). In fact, if we tensor the character ringMackey functor with K, we can be shure that this extended Mackey functor isstable under the Adams operations. But now the two constructions Ψn and Ψn areextensions of the same morphism on K ⊗ Rab. Hence they coincide, and with Ψn

also Ψ stabilizes RK .From now on we will denote Ψn by Ψn, since whenever both are defined, they

coincide.Recall that the Cartan map cG : PF (G) → RF (G) is defined by cG([V ]) = [V ]

for a projective FG-module V . This defines a morphism c : PF → RF of Z-Mackeyfunctors on G which restricts to a morphism c : P ab

F → RabF of Z-restriction functors

on G.

3.5 Proposition (cf. [Ke75, §5]) For n ∈ Z the diagram

PFΨn−−−−→ PF

c

y ycRF

Ψn−−−−→ RF

in Z−Res(G) is commutative.

Proof The two morphisms Ψn c and c Ψn in Z−Res(G)(PF , RF ) coincideon P ab

F . Hence they have to be equal by Theorem 1.3, since we have the canonicalinduction formula a from Example III.3.1.

3.6 Remark (i) Note that Ψn does not commute with the decomposition mor-phism d : RK → RF or the morphism e : PF → RK from the cde-triangle (cf. [Se78,§15]) which is given as the composition of the canonical morphisms PF ∼= PO ∼=RK,p ⊆ RK . It is easy to show that the Adams operations Ψn : PF → PF andΨn : RK,p → RK,p correspond to each other via the canonical isomorphism. But theAdams operations on RK do not restrict to the Adams operations on RK,p, sincethe power of a p-singular element in G is in general not p-singular.

(ii) For two sets π1 ⊆ π2 of primes we have an inclusion RK,π2 ⊆ RK,π1 of Z-Mackey functors on G which restricts to an inclusion Rab

K,π2⊆ Rab

K,π1of Z-restriction


functors on G. As in the above proposition one shows easily that the Adams op-erations on RK,π1 restrict to those of RK,π2. In particular, Adams operations onRK,p1 and RK,p2 coincide on the intersection RK,p1,p2.

3.7 Proposition The diagram

PO ⊆ TO ⊆ LO

Ψny Ψn

y yΨn

PO ⊆ TO ⊆ LO

commutes for n ∈ Z.

Proof The commutativity of the diagram

LOp−−→ Lab

OΨn−−→ Lab

O ⊆ LO⋃ ⋃ ⋃ ⋃TO

p−−→ T abO

Ψn−−→ T abO ⊆ TO⋃ ⋃ ⋃ ⋃

POp−−→ P ab

OΨn−−→ P ab

O ⊆ PO

implies by Proposition III.3.11 the commutativity of the diagram

LOa−−→ Lab

O +

Ψn+−−→ LabO +

bLO ,LabO−−→ LO⋃ ⋃ ⋃ ⋃

TOa−−→ T ab

O +

Ψn+−−→ T abO +

bTO ,TabO−−→ TO⋃ ⋃ ⋃ ⋃

POa−−→ P ab

O +

Ψn+−−→ P abO +

bPO ,PabO−−→ PO

3.8 Remark Let M , A, a, and Ψn ∈ Z−Res(G)(A ⊆ M,A ⊆ M) for n ∈ Zbe given in one of the situations 3.1 (i)–(vi), and write for χ ∈M(G):

aG(χ) =∑

(H,ϕ)∈G\M(G)

αG(H,ϕ)(χ)[H,ϕ]G

with αG(H,ϕ)(χ) ∈ Z andM as in Definition II.3.2. Then, applying bM,A one obtains

χ =∑

(H,ϕ)∈G\M(G)

αG(H,ϕ)(χ)indGH(ϕ). (4.5)

Now the definition of ΨnG as canonical extension is given by

ΨnG(χ) =

∑(H,ϕ)∈G\M(G)

αG(H,ϕ)(χ)indGH(ϕk). (4.6)

If we compare Equation (4.5) and (4.6) we see that we may commute inductionwith Ψn in each summand without changing the result. But in general, Ψn doesnot commute with induction, and there is an error term in each summand, if onecommutes them by force. However, the error terms cancel over the whole sum.

4.4. CHERN CLASSES 125

Hence, using the canonical induction formula (4.5) we may assume that Ψn com-mutes with induction, as long as we are only interested in the result of the wholesum.

Now we are able to prove Proposition III.3.2 (vi) and III.3.5 (v).

3.9 Remark Assume that we are in situation 3.1 (iv) (π-projective charactergroup). In order to prove Proposition III.3.5 (v) we have to show that∑

(H,ϕ)∈G\M(G)

αG(H,ϕ)(χ) =|Gπ′ ||G|

for χ ∈ RK,π(G). We consider Equation (4.6) with n = 0 and obtain

Ψ0G(χ) =

∑(H,ϕ)∈G\M(G)

αH(H,ϕ)(χ)indGH(1).

Note that by the compatibility with restrictions, Ψ0G(χ) is the class function which

is constantly χ(1) on π-regular elements, i.e. π′-elements, and zero on π-singularelements. Taking scalar products with the trivial character yields the result inProposition III.3.5 (v).

With π = p we obtain Proposition III.3.2 (vi) as a special case.

4.4 Chern classes

Let G, O, F , and K be given as at the beginning of the previous section.In this section we are going to apply Theorem 1.2 to construct Chern classes

for characters, Brauer characters, projective characters, trivial source modules, andlinear source modules. We start with the description of the Mackey functor N , theChern classes live in.

4.1 For a finite group G define an abelian group

H∗∗e (G,Z) :=∏n≥0

H2n(G,Z).

The upper index∗∗

indicates that we take the direct product (not the direct sumas for the cohomology ring), and the index ‘e’ indicates that we consider onlyeven degrees. It is easy to see that H∗∗e (G,Z) is a commutative ring with respectto the cup product. We will write the elements x of H∗∗e (G,Z) as infinite seriesx = x0 + x1 + x2 + . . . with xi ∈ H2i(G,Z). For a group homomorphism f : G′ → Gthere is an induced ring homomorphism

H∗∗e (f,Z) :=∏n≥0

H2n(f,Z) : H∗∗e (G,Z)→ H∗∗e (G′,Z).

With this definition we may consider H∗∗e (−,Z) as a contravariant functor fromthe category of finite groups to the category of commutative rings. Taking units,we obtain a contravariant functor H∗∗e (−,Z)× from the category of finite groups to


the category of abelian groups. In particular, for H ≤ G, g ∈ G, we have ringhomormophisms

cg,H : H∗∗e (H,Z)→ H∗∗e ( gH,Z),

resGH : H∗∗e (G,Z)→ H∗∗e (H,Z)

induced by the conjugation map gH → H and by the inclusion H ≤ G. For H ≤ G,Evens defined in [Ev63] a function

indGH : H∗∗e (H,Z)→ H∗∗e (G,Z)

which is multiplicative with respect to the cup product, but not additive. Hencewe also obtain a group homomorphism

indGH : H∗∗e (H,Z)× → H∗∗e (G,Z)×.

4.2 Proposition For a finite group G, the abelian groups H∗∗e (H,Z)×, H ≤G, together with the maps defined above, define a Z-Mackey functor on G.

Proof The Mackey functor axioms, where induction is not involved are ob-viously satisfied by functoriality. The Mackey functor axioms involving inductionmaps follow immediately from the properties of indGH proved in [Ev63].

4.3 For an integral domain k we may indentify the torsion subgroup of k× withthe group of roots of unity in the quotient field of k. Hence, it may be identified(non-canonically) with a subgroups of Q/Z. The exact sequence

0→ Z→ Q→ Q/Z→ 0

of G-modules with trivial action gives rise to a long exact cohomology sequence,which yields an isomorphism

H1(G,Q/Z)∼−→H2(G,Z).

Using the above identification k× ≤ Q/Z we may regard Hom(G, k×) as a subgrouupof Hom(G,Q/Z) = H1(G,Q/Z), and obtain an embedding

Hom(G, k×)→ H2(G,Z) (4.7)

which we keep fixed for k. Now we define a group homomorphism

chG : Hom(G, k×)→ H∗∗e (G,Z)×, ϕ 7→ 1 + ϕ,

where we consider 1 ∈ H0(G,Z) = Z and ϕ ∈ H2(G,Z) via the embedding in (4.7).The groups Hom(H, k×) and H∗∗e (H,Z)×, H ≤ G form Z-Mackey functors on Gwith the definitions in Example 2.1 and in 4.1. It is easy to check that chH , H ≤ Gform a morphism of Z-restriction functors on G. Now we extend chH for H ≤ G,linearly to

Z ·Hom(H, k×)→ H∗∗e (H,Z)×

4.4. CHERN CLASSES 127

and obtain, with suitable choices for k, morphisms

ch : RabK → H∗∗e (−,Z)×

ch : RabF → H∗∗e (−,Z)×

ch : RabK,π → H∗∗e (−,Z)×

ch : P abO → H∗∗e (−,Z)×

ch : T abO∼= T ab

F → H∗∗e (−,Z)×

ch : LabO → H∗∗e (−,Z)×

(4.8)

of Z-restriction functors on G with RabK , Rab

F , RabK,π, P ab

O , T abO , T ab

F , and LabO as

defined in Chapter III. Note that in all these cases, except for RabK,π and P ab

O , themorphism ch can be regarded as a natural transformation between contravariantfunctors from the category of finite groups to the category of abelian groups.

4.4 Proposition The morphisms in (4.8) can be extended to morphisms

ch : RK → H∗∗e (−,Z)×

ch : RF → H∗∗e (−,Z)×

ch : RK,π → H∗∗e (−,Z)×

ch : PO → H∗∗e (−,Z)×

ch : TO ∼= TF → H∗∗e (−,Z)×

ch : LO → H∗∗e (−,Z)×

of Z-restriction functors on G. These morphisms, except the ones defined on RK,πand PO, can be considered as natural transformations in the sense of Remark 1.10.

Proof This is immediate from Theorem 1.2.

We call chG(χ) ∈ H∗∗e (G,Z)× the total Chern class of χ ∈ RK(G) (resp. χ ∈RF , RK,π, PO, TO, TF , LO). We write

chG(χ) = ch0G(χ) + ch1

G(χ) + ch2G(χ) + . . .

with chiG(χ) ∈ H2i(G,Z) and ch0G(χ) = 1, and call chiG(χ) the i-th Chern class of

χ.

4.5 Remark (i) Classically, Chern classes were defined for ordinary complexcharacters, or on RK(G) (see [At61] for example), by using the theory of vectorbundles on the classifying space BG of G. On one-dimensional characters this defi-nition coincides with the one in (4.8). It was first observed by Symonds (cf. [Sy91])that the canonical induction formula a : RK → Rab

K + of Example III.1.1 can beused to give an algebraic definition of the Chern classes. Since the set of coprimor-dial subgroups of the cohomology Mackey functors is the set of p-subgroups for allprimes p, and since RK(H) 6= Rab

K (H) for arbitrary p-groups H, Theorem 1.3 cannot be applied to show that the two constructions coincide. Fortunately, there isan axiomatic description of Chern classes by Kroll in [Kr87] with states that Chernclasses are characterized by three properties, namely functoriality, additivity, and


the definition on one-dimensional characters. Obviously our construction has theseproperties, and therefore coincides with the original definition.

Using the explicit version (cf. Proposition III.1.2 (v)) of the canonical inductionformula, we obtain the explicit formula

chG(χ) =∏

(H0,ϕ0)<...<(Hn,ϕn)∈G\Γ(M(G))

indGH0(1 + ϕ0)(−1)n(resGHn (χ),ϕn)Hn

for the total Chern class of χ ∈ RK(G).

(ii) As far as the author knows, there are no definitions of Chern classes forRF (G), RK,π(G), PF (G) ∼= PO(G), TF (G) ∼= TO(G), or LO(G). Theorem 1.3 cannotguarantee the uniqueness of the extension, since we can’t say more about the set ofcoprimordial subgrops of H∗∗e (−,Z)× than that it is a subset of the set of subgroupsof prime power order. But it might be possible that there are characterizationsas the one of Kroll for the Chern classes on RF (G), TF (G) ∼= TO(G), and LO(G),which can be interpreted as contravariant functors in the sense of Remark 1.10.

(iii) Similar to the proof of Proposition 3.7 for Adams operations we can showthat the diagram

PO −−−−→ TO −−−−→ LO

ch

y ch

y ychH∗∗e (−,Z)× H∗∗e (−,Z)× H∗∗e (−,Z)×,

where the upper horizontal arrows denote the inclusions, is commutative.

(iv) In general, the diagram

POe−−−−→ RK

ch

y ychH∗∗e (−,Z)× H∗∗e (−,Z)×

is not commutative. Let G be the cyclic group of order 2 and consider OG ∈OG−mod. Then chG([OG]) = indG1 (ch1([O]) = indG1 (1) = 1, since aG([O]) =[1, 1]G. On the other hand chG(indG1 (1)) = chG(1 + τ), if τ denotes the non-trivial one-dimensional character of G. Hence, chG(indG1 (1)) = chG(1) · chG(τ) =1 · (1 + τ) = 1 + τ .

4.5 Virtual extensions of characters and modules

In this section we will use some of the canonical induction formulae from Chapter IIIto define virtual extensions of characters or modules.

Throughout this section let X = G o S be the semidirect product of a finitegroup G with a finite group S satisfying gcd(|G|, |S|) = 1. We consider both G andS as subgroups of G and denote by GS the fixed points of G under S. Some readersmay prefer the notation CG(S) for GS . Let π denote the set of prime divisors of|S|.

First we recall a Lemma on coprime action, cf. [Is76, 13.8 and 13.9].

4.5. VIRTUAL EXTENSIONS 129

5.1 Lemma Let D be a finite left X-set such that G acts transitively on D.Then we have:

(i) There exists an S-stable element in D, i.e. DS 6= ∅.(ii) The set DS is a transitive GS-set.

5.2 We adopt the notation of Section III.1. Our first aim is to define a mapextG,S : R(G)S → R(GS) with resGSG extG,S = idR(G)S , where R(G)S denotes thefixed points of R(G) under the conjugation action of S. We will do this by defining

extG,S : R(G)SaG−−→Rab

+ (G)SextG,S+−−→Rab

+ (GS)bR,R

ab

−−→R(GS),

with a map extG,S+ which will be defined in a moment. Note that aG maps S-fixedpoints to S-fixed points, since it commutes with conjugation.

Note that the basis elements [H,ϕ]G of Rab+ (G), which correspond to G\M(G),

are permuted by the S-conjugation maps. Therefore, the S-orbit sums∑s∈S/stabS([H,ϕ]G)

[ sH, sϕ]G, (H,ϕ) ∈ GS\M(G),

form a Z-basis of Rab+ (G)S . For (H,ϕ) ∈ M(G) let T := stabS([H,ϕ]G) be the

stabilizer of [H,ϕ]G in S. Then GT acts G-transitively on the G-orbit of (H,ϕ).Hence, by Lemma 5.1, there exists a T -fixed point (H ′, ϕ′) in the G-orbit of (H,ϕ),i.e. T = stabS([H ′, ϕ′]G) = stabS((H ′, ϕ′)), and the set of T -fixed points forms aGT -orbit. Now we define

extG,S+

( ∑s∈S/T

[ sH, sϕ]G)

:= [H ′ · T, ϕ′ · 1]GS

with ϕ′ · 1 ∈ Hom(H ′ · T,C×) defined by (ϕ′ · 1)(h′t) := ϕ′(h′) for h′ ∈ H ′, t ∈ T .This is a homomorphism, since ϕ′ is T -stable. It is easy to verify that [H ′ ·T, ϕ′ ·1]GSdoes not depend on the choice of [H,ϕ] in the S-orbit, and of (H ′, ϕ′) in the G-orbitof (H,ϕ). Here Lemma 5.1 (ii) is needed.

Note that the notation extG,S+ is misleading, since it is not the result of somemorphism under the functor −+. All the same we use this notation just to distin-guish extG,S : R(G)S → R(GS) and extG,S+ : Rab

+ (G)S → Rab+ (GS).

5.3 Proposition For U ≤ S the diagram

R(G)SextG,S−−→ R(GS)⋂ yresGSGU

R(G)UextG,U−−→ R(GU)

is commutative.

Proof It suffices to prove that the three squares in the diagram

R(G)SaG−−→ Rab

+ (G)SextG,S+−−→ Rab

+ (GS)bGS−−→ R(GS)⋂ ⋂ yres+

GSGU

yresGSGU

R(G)UaG−−→ Rab

+ (G)UextG,U+−−→ Rab

+ (GU)bGU−−→ R(GU)


commute. The left hand square commutes obviously, and the right hand squarealso, since b is a morphism of Mackey functors. We will show that also the middlesquare commutes.

Let (H,ϕ) ∈ M(G) such that T := stabS([H,ϕ]G) = stabS((H,ϕ)). Then wehave (

res+GSGU extG,S+

)( ∑s∈S/T

[ sH, sϕ]G)

=

= res+GSGU

([HT,ϕ · 1]GS

)=

∑x∈GU\GS/HT

[GU ∩ x(HT ), res

x(HT )

GU∩x(HT )

( x(ϕ · 1))]GU.

It is easy to verify that a set of representatives for the double cosets in U\S/T isalso a set of representatives for the double cosets in GU\GS/HT . For s ∈ S we

have GU∩ s(HT ) = GU∩( sH · sT ) = sH ·(U∩ sT ) and ress(HT )

GU∩s(HT )

( s(ϕ ·1)) = sϕ ·1.

Hence we obtain(res+

GSGU extG,S+

)( ∑s∈S/T

[ sH, sϕ]G)

=∑

s∈U\S/T

[sH · (U ∩ sT ), sϕ · 1

]GU.

On the other hand, if we want to apply extG,U+ , we first have to decompose theelement

∑s∈S/T [ sH, sϕ]G into U -orbit sums, namely∑

s∈S/T

[ sH, sϕ]G =∑

s∈U\S/T

∑u∈U/U∩ sT

[ usH, usϕ]G.

Note that stabU ( sH, sϕ) = U ∩ sT = stabU ([ sH, sϕ]G), and therefore we may use( sH, sϕ) as representative in each U -orbit sum in order to apply extG,U+ to theU -orbit sum

∑u∈U/U∩ sT [ usH, usϕ]G, and we obtain

extG,U+

( ∑s∈S/T

[ sH, sϕ]G)

=∑

s∈U\S/T

extG,U+

( ∑u∈U/U∩ sT

[ usH, usϕ]G)

=∑

s∈U\S/T

[sH · (U ∩ sT ), sϕ · 1]GU .

5.4 Corollary The composition

R(G)SextG,S−−→R(GS)

resGSG−−→R(G)S

is the identity map.

Proof This follows from Proposition 5.3 with U = 1. In fact, the mapextG,1+ : Rab

+ (G) → Rab+ (G) is the identity by definition, and b a is the identity,

since a is a canonical induction formula.

5.5 Proposition The map extG,S : R(G)S → R(GS) is Galois invariant, i.e.

σ(extG,S(χ)) = extG,S( σχ)

4.5. VIRTUAL EXTENSIONS 131

for χ ∈ R(G)S and σ a field automorphism of the algebraic closure of Q.

Proof Since aG and bG are Galois invariant, it suffices to show that extG,S+ isGalois invariant. This in turn follows easily from the fact that Galois action andconjugation action commute.

5.6 Remark (i) An analysis of the definition of extG,S+ : Rab+ (G)S → Rab

+ (GS)

shows that we may define extG,S+ : A+(G)S → A+(GS) for A ∈ Z−Res(GS) withstable basis B containing the unities 1A(H), H ≤ G, such that for HT ≤ GS withH ≤ G, T ≤ NS(H), and T -stable ϕ ∈ B(H), there exists a unique ϕ ∈ B(HT )with resHTH (ϕ) = ϕ and resHTT (ϕ) = 1A(T ) (extendibility property for B). Note that

RabK (cf. Section III.1), Rab

F (cf. III.2), T abO∼= T ab

F (cf. III.4), and LabO (cf. III.4) have

this property, and that P abO∼= P ab

F and RabK,π1

(cf. III.3) have this property whenchar(F ) /∈ π resp. π ∩ π1 = ∅. Moreover, if M ∈ Z−Mack(G) with A ⊆ M aZ-restriction subfunctor on G, and a : M → A+ is a canonical induction formula,then we can define extG,S : M(G)S → M(GS) by extG,S := bM,A

G extG,S+ aG.Proposition 5.3 and Corollary 5.4 hold in this general situation by the same proofs.If there is a Galois action which commutes with the conjugation action, then alsoProposition 5.5 holds. Note that all this applies to M = RK , RF , TO ∼= TF , LOalways, and to PO ∼= PF , resp. RK,π1 , when char(F ) - |S|, resp. |S|π1 = 1.

(ii) Note that this extendibility property is well-known for complex irreduciblecharacters χ ∈ Irr(G)S , cf. [Hu67, V.17.12] for example. There it is shown that onehas even an irreducible extension. More generally, Isaacs showed in [Is81] that eachirreducible S-stable LG-module is the restriction of an irreducible L[GS]-module,where L is an arbitrary field. Note that for χ ∈ Irr(G)S there is a canonical exten-sion χ ∈ Irr(GS) which is uniquely determined by the property detS(resGSS (χ)) = 1,cf. [Gla68, Thm. 1]. Since our map extG,S is defined in a natural way, one wouldassume that it produces the canonical extension for irreducible S-fixed characters.But this is not true in general. Even worse, the next example shows that extG,S

does not map Irr(G)S to Irr(GS), but may result in virtual characters. However, itseems that the prime 2 is responsible for this ‘bad’ behaviour in Example 5.7, a phe-nomenon encountered also in other situations, cf. Proposition 2.4 or Remark 2.5 (i).In fact, in all examples with odd |S| we know so far, the map extG,S produces thecanonical irreducible extension in the sense of [Gla68].

5.7 Example Let

G =< x, y, z | xp = yp = zp = 1, [x, y] = z, [x, z] = [y, z] = 1 >

be the extraspecial p-group of order p3 with exponent p for an odd prime p, andlet S = 1, s act on G by sx = x−1, sy = y−1, sz = z. G has p − 1 irreduciblecharacters of degree p. Let χ be one of them, and denote by χ ∈ Irr(GS) thecanonical extension and χ′ ∈ Irr(GS) the other extension of χ. Then

extG,S(χ) =

p+1

2 χ− p−12 χ′, p ≡ 1 (mod 4),

p+12 χ′ − p−1

2 χ, p ≡ 3 (mod 4).

This can be derived easily from the explicit formula for aG(χ) in Proposition III.7.2.


4.6 The Glauberman correspondence

Throughout this section we assume the notation of the previous section. In partic-ular, X is the semidirect product of a normal subgroup G and a subgroup S withrelatively prime orders.

Following a suggestion of Puig we will construct a map

glG,S : R(G)S → R(GS),

using the canonical induction formula aG : R(G) → Rab+ (G) from Example III.1.1.

This map is related to the Glauberman correspondence. Before we define glG,S , wewill recall the definition and some basic facts of the Glauberman correspondence,cf. [Gla68] or [Is76, §13].

First we state a lemma which is of fundamental importance for the Glaubermancorrespondence.

6.1 Lemma (cf. [Is76, 13.14]) Assume that S is a p-group and let χ ∈ Irr(G)S

be an S-invariant irreducible character of G. Then there is a unique constituentλ ∈ Irr(GS) of resG

GS(χ) whose multiplicity is not divisible by p. Moreover, this

multipicity is congruent to 1 or -1 modulo p.

6.2 Theorem (cf. [Is76, 13.1]) There is a unique bijection πG,S : Irr(G)S →Irr(GS), called the Glauberman correspondence, satisfying the following conditions

(i) If T S, then πG,T (Irr(G)T ) ⊆ Irr(GT )S.

(ii) In the situation of (i), πG,S = πGT ,S/T πG,T .

(iii) If S is a p-group and χ ∈ Irr(G)S, then πG,S(χ) is the unique irreducibleconstituent λ of resG

GS(χ) with p - (λ, resG

GS(χ))GS according to Lemma 6.1.

For χ ∈ Irr(G)S , the irreducible character λ := πG,S(χ) ∈ Irr(GS) is called theGlauberman correspondent of χ, and χ is called the Glauberman correspondentof λ. Note that it follows directly from Theorem 6.2 that πG,S(ϕ) = resG

GS(ϕ) for

ϕ ∈ Irr(G)S with ϕ(1) = 1.For χ ∈ Irr(G) we will denote by TrS(χ) ∈ R(G)S the S-orbit sum of χ, i.e. if

T := stabS(χ), then TrS(χ) =∑

s∈S/Tsχ. Note that the set TrS(χ), where χ runs

through a set of representatives of the S-orbits S\Irr(G) of Irr(G), is a Z-basis ofR(G)S . Moreover, we write R(G)S<S fo the Z-span of the elements TrS(χ) ∈ R(G)S

with stabS(χ) < S. then we have R(G)S = Z · Irr(G)S ⊕R(G)S<S .We will need the following consequence of Lemma 6.1 and Theorem 6.2 later.

6.3 Proposition Let S be a p-group.(i) For χ ∈ Irr(G)S with stabS(χ) < S we have

resGGS (TrS(χ)) ≡ 0 (mod pR(GS)).

(ii) For λ ∈ Irr(GS) we have indGGS (λ) ∈ R(G)S, and there is a unique χ ∈Irr(G)S with p - (χ, indGGS (λ))G; Moreover, (χ, indGGS (λ))G ≡ ±1 (mod p) for thisχ.

4.6. GLAUBERMAN CORRESPONDENCE 133

(iii) For λ ∈ R(GS) we have

resGGS (indGGS (λ)) ≡ λ (mod pR(GS)).

(iv) For χ ∈ R(G)S we have

indGGS (resGGS (χ)) ≡ χ (mod pR(G)S +R(G)S<S).

Proof (i) Let T := stabS(χ), then

resGGS (TrS(χ)) =∑s∈S/T

resGGS ( sχ) =∑s∈S/T

s(resGGS (χ)) = |S/T | · resGGS (χ).

(ii) Let s ∈ S, then s(indGGS (λ)) = indGGS ( sλ) = indGGS (λ). Furthermore, sinceπG,S : Irr(G)S → Irr(GS) is surjective, Theorem 6.2 implies that there is someχ ∈ Irr(G)S with p - (λ, resG

GS(χ))GS = (indGGS (λ), χ)G. Moreover, since πG,S is

injective, Theorem 6.2 (iii) implies that χ is unique.(iii) It suffices to prove the congruence for λ ∈ Irr(GS). If λ = πG,S(χ) for

χ ∈ Irr(G)S , then by part (ii) we have

indGGS (λ) ≡ ±χ (mod pR(G)S +R(G)S<S).

Hence resGGS

(indGGS (λ)) ≡ λ (mod pR(GS)) by part (i).(iv) It suffices to prove the congruence for TrS(χ), χ ∈ Irr(G). If stabS(χ) < S,

then (i) implies indGGS (resGGS

(TrS(χ))) ∈ pR(G)S + R(G)S<S , and we also haveTrS(χ) ∈ pR(G)S + R(G)S<S . If stabS(χ) = S and λ ∈ Irr(GS) is the Glauber-

man correspondent of χ, then resGGS

(χ) ≡ ±λ (mod pR(GS)) and indGGS (±λ) ≡ χ(mod pR(G)S +R(G)S<S) by part (ii). This yields the result for χ.

6.4 Definition We define the map

glG,S : R(G)SaG−→Rab

+ (G)SglG,S+−→ Rab

+ (GS)bGS−→R(GS),

where glG,S+ is defined for a basis element∑

s∈S/T [ sH, sϕ]G, (H,ϕ) ∈ M(G) withT := stabS([H,ϕ]G) = stabS(H,ϕ), by

glG,S+

( ∑s∈S/T

[ sH, sϕ]G)

:=

[GS ∩H, resH

GS∩H(ϕ)]GS, if T = S,

0, if T < S.

If S = T , we can write HS instead of GS ∩H, since S acts on H. Note that glG,S+

is well-defined by Lemma 6.1.

6.5 Proposition For ϕ ∈ Irr(G) with ϕ(1) = 1 we have

glG,S(TrS(ϕ)) =

resG

GS(ϕ), if stabS(ϕ) = S,

0, if stabS(ϕ) < S.


Proof This is immediate from Proposition III.1.2 (v) which states aG(ϕ) =[G,ϕ]G.

6.6 Proposition Assume that S is a p-group.(i) For χ ∈ R(G)S we have

glG,S(χ) ≡ resGGS (χ) (mod pR(GS)).

(ii) For χ ∈ R(G)S<S we have

glG,S(χ) ≡ 0 (mod pR(GS)).

Proof (i) Since glG,S = bGS glG,S+ aG and resG

GS= bGS res+

GGS aG, it

suffices to show thatbGS gl

G,S+ ≡ bGS res+

GGS .

Let (H,ϕ) ∈ M(G) with T := stabS([H,ϕ]G) = stabS(H,ϕ). If T < S, thenglG,S+ (

∑s∈S/T [ sH, sϕ]G) = 0 and

res+GGS

( ∑s∈S/T

[ sH, sϕ]G)

=∑s∈S/T

s(res+GGS ([H,ϕ]G)) = |S/T | · res+

GGS ([H,ϕ]G)

so that the congruence holds in this case. From now on we assume that (H,ϕ) isS-stable. Then we have by Proposition 6.3 (iii):

(bGS glG,S)([H,ϕ]G) ≡(resGGS indGGS bGS

)([HS , resHHS (ϕ)]GS

)(mod pR(GS))

=(resGGS indGGS indG

S

HS

)(resHHS (ϕ)

)=(resGGS indGH

)(indHHS (resHHS (ϕ))

).

By Proposition 6.3 (iv) we have

indHHS

(resHHS (ϕ) ≡ ϕ (mod pR(H) +R(H)S<S).

Since indGH(R(H)S<S) ⊆ pR(G)S +R(G)S<S , we obtain from Proposition 6.3 (i) that

(bGS glG,S)([H,ϕ]G) ≡ resGGS(indGH(ϕ)

)(mod pR(GS)).

On the other hand

(bGS res+GGS )([H,ϕ]G) = (resGGS bG)([H,ϕ]G) = resGGS

(indGH(ϕ)

).

(ii) This follows from part (i) and Proposition 6.3 (i).

6.7 Remark (i) Proposition 6.6 (i) together with Theorem 6.2 shows thatour map glG,S can be used as a definition for the Glauberman correspondence inthe case, where S is a p-group. But note, that glG,S is defined for arbitrary S and Gwith relatively prime order. It might be that the Glauberman correspondence canbe defined directly from glG,S without going step by step through a composition

4.6. GLAUBERMAN CORRESPONDENCE 135

series of S. But we have no good evidence for this at the moment. Interestingexamples tend to be too big even for computers. And without interesting exampleswe don’t even know what we would like to prove about glG,S in the general case. Thefew examples we know, still allow that glG,S(χ) is a multiple of a single λ ∈ Irr(GS)for χ ∈ Irr(G)S , but we don’t believe in this property in general.

(ii) Note that in the case, where S is not solvable, |S| is even by a theoremof Feit and Thompson. Hence G is odd. Generally, in the case, where G is odd(and hence solvable), Isaacs defined in ([Is73]) a bijection πG,S : Irr(G)S → Irr(GS)with coincides with the Glaubermann correspondence, whenever both are defined,which was proved by Wolf in [Wo78]. Isaacs’ correspondence is defined by totallydifferent methods as Glauberman’s. It would be very nice if the map glG,S alloweda unified construction of these two corrspondences. But again we don’t have anyserious results relating Isaacs’ correspondence to the map glG,S .

(iii) Note that glG,S can be defined similarly for the various other Grothendieckgoups we considered in Chapter III. But we didn’t examine their properties so far.

6.8 Example We adopt the notation of Section III.7. Let G be the extraspe-cial p-group of order p2n+1 and exponent p for an odd prime p. In Proposition III.7.2we have shown that, for χ ∈ Irr(G) with χ(1) = pn, we have

aG(χ) =n∑d=0

∑Z≤H abelian

|H/Z|=pn−d

(−1)dpd(d−1)[H,ϕ]G,

where ϕ is an arbitrary extension of λ ∈ Irr(Z), and λ is given by resGZ (χ) = pn · λ.

The automorphism group of G has a normal subgroup Sp2n which is an extension ofthe symplectic group Sp2n(Fp) by the normal subgroup Inn(G) ∼= G/Z ∼= F2n

p . The

group Sp2n acts trivially on Z and therefore stabilizes the irreducible characters

χ ∈ Irr(G) of degree pn. Let S be a p′-subgroup of Sp2n. A basis element [H,ϕ]Gis S-stable if an only if H is S-stable, since sϕ is again an extension of λ ∈ Irr(Z),and therefore G-conjugate to ϕ, cf. III.7.1. Since glG,S+ is trivial on S-orbit sums of[H,ϕ]G with stabS([H,ϕ]G) < S, we obtain

glG,S(χ) =

n∑d=0

∑Z≤HabelianH S-invariant|H/Z|=pn−d

(−1)dpd(d−1)indGS

HS (ϕ),

where ϕ ∈ Irr(HS) is an extension of λ ∈ Irr(Z), resGZ (χ) = pn · λ.In the special case where GS = Z, we obtain

glG,S(χ) =( n∑d=0

∑Z≤H abelianH S-invariant|H/Z|=pn−d

(−1)dpd(d−1))· λ.

If we specialize further, and assume that Z is the only S-stable abelian subgroupabove Z, then we obtain

glG,S(χ) = (−1)npn(n−1)λ.


If GS = Z and all abelian subgroups of G containing Z are S-invariant, we obtain

glG,S(χ) = pnλ.

In fact,

n∑d=0

(−1)dpd(d−1)#Z ≤ H abelian | |H/Z| = pn−d =

=∑

(H,ϕ)∈G\M(G)

αG(H,ϕ)(χ) = χ(1) = pn

by Proposition III.1.2 (x), where αG(H,ϕ)(χ) ∈ Z is given as in Equation (3.1).

4.7 Projectification

Throughout this section G denotes a finite group, K denotes a field of characteristiczero containing all |G|-th roots of unity, and F denotes an algebraically closed fieldof characteristic p > 0. We will define morphisms (see III.1, III.2 and III.3 for thenotation)

a : Q⊗RK −→Q⊗ (RabK,p)+, a : Q⊗RF −→Q⊗ P ab

F+,

such thatpr : Q⊗ a−−→Q⊗ (Rab

K,p)+bRK,p,R

abK,p

−−→ Q⊗RK,pis the multiplication with the characteristic function on p′-elements, and

pr : Q⊗RFa−−→Q⊗ P ab

F+bPF ,P

abF−−→ Q⊗ PF

is the inverse of the Cartan morphism c : PF → RF .

7.1 Recall from Section III.3 that

RabK,p(H) =

RabK (H), if H is a p′-group,

0, otherwise,

for H ≤ G. We can regard RabK,p ⊆ RK as a Z-restriction subfunctor on G, and

define p ∈ Z−Con(G)(RK , RabK,p) for H ≤ G and χ ∈ IrrK(H) by

pH(χ) :=

χ, if H is a p′-group and χ(1) = 1,

0, otherwise.

Note that RK , RabK,p, and B, where B(H) := Hom(H,K×) for a p′-subgroup H ≤ G

and B(H) := ∅ otherwise, satisfy Hypothesis II.3.1 so that we may define

a := aRK ,RabK,p,p ∈ Q−Res(G)(Q⊗RK ,Q⊗ (Rab

K,p)+).

Furthermore, let pr ∈ Q−Res(G)(Q⊗RK ,Q⊗RK,p) be defined as the composition

pr : Q⊗RKa−−→Q⊗ (Rab

K,p)+bRK,p,R

abK,p

−−→ Q⊗RK,p.

4.7. PROJECTIFICATION 137

We call pr the projectification morphism. Note that RK , RabK,p, B, and p satisfy

condition (∗p′) of Theorem II.4.5 by the same argument as in the proof of Theo-rem III.1.2.

7.2 Proposition Assume the notation of 7.1. The morphism pr : Q⊗RK →Q⊗RK,p is given by multiplication with the characteristic function on p′-elements,i.e. (

prH(χ))(h) :=

h, if h is a p′-element,

0, otherwise

for H ≤ G, χ ∈ RK(H). Moreover we have the explicit formula

prH(χ) =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|(NH(σ)/H0)p′ ||NH(σ)/H0|

(resHHn(χ), ϕn

)Hn

indHH0(ϕ0)

for H ≤ G and χ ∈ RK(H), where M(H) is the poset of monomial pairs associatedto B as in 7.1, cf. Definition II.3.2. Furthermore, |H|p ·aH(χ) is integral for H ≤ Gand χ ∈ RK(H).

Proof Multiplication by the characteristic function on p′-elements induces amorphism pr′ : K⊗RK → K⊗RK,p of K-restriction functors on G, and we may alsoextend pr to pr : K ⊗ RK → K ⊗ RK,p. In order to prove that pr′ and pr coincideit suffices to show pr′H = prH for cyclic p′-subgroups of G. But in this case it isobvious, since, for ϕ ∈ IrrK(H), we have pr′H(ϕ) = ϕ, and prH(ϕ) = bH([H,ϕ]G),because aH(ϕ) = [H,ϕ]G by Lemma II.3.7. The explicit formula for prH , H ≤ G,follows from Theorem II.4.5, since RK , Rab

K,p, B, and p satisfy condition (∗p′). The

integrality of |H|p · aH(χ) for H ≤ G and χ ∈ RK(H) follows from Corollary 4.7.

For χ ∈ IrrK(G) let the p-defect dp(χ) ∈ N0 be defined by χ(1)p · pdp(χ) = |G|p.In [BF59] Brauer and Feit prove the following result about the morphism pr.

7.3 Proposition Assume the notation of 7.1. If χ ∈ IrrK(G), then

pdp(χ) · prG(χ) ∈ RK,p(G) and pdp(χ)−1 · prG(χ) /∈ RK,p(G).

We obtain a similar result for aG.

7.4 Proposition Assume the notation of 7.1.

(i) For χ ∈ RK,p(G) we have aG(χ) ∈ (RabK,p)+(G), i.e. aG(χ) is integral.

(ii) For χ ∈ IrrK(G) we have

pdp(χ) · aG(χ) ∈ (RabK,p)+(G) and pdp(χ)−1 · aG(χ) /∈ (Rab

K,p)+(G).


Proof (i) Let p′ ∈ Z−Con(G)(RK,p, RabK,p) denote the morphism from Exam-

ple III.3.4 which led to an integral canonical induction formula

a′ ∈ Z−Res(G)(RK,p, (RabK,p)+).

Then we have a commutative diagram

RKp−−→ Rab

K,p⋃‖

RK,pp′−−→ Rab

K,p

Hence, we obtain a commutative diagram

Q⊗RKa−−→ Q⊗ (Rab

K,p)+⋃‖

Q⊗RK,pa′−−→ Q⊗ (Rab

K,p)+

by Proposition II.3.11. Since a′ is integral, the result follows.(ii) Let χ ∈ RK(G) with 0 = prG(χ) = (bG aG)(χ). Then aG(χ) = 0, since

resGH(χ) = 0 for all p′-subgroups H ≤ G, and since aG is determined by Dia-gram (2.2). Therefore, bG is injective on aG(Q⊗RK(G)). Together with prpr = prthis implies

aG bG aG = aG.

Now let χ ∈ IrrK(G). We already know from Proposition 7.2 that |G|p · aG(χ) ∈(Rab

K,p)+(G). For n ∈ N0 we have

pn · aG(χ) = aG(pn · prG(χ))

by the above equation. Hence, if pn · prG(χ) is integral, so is pn · aG(χ) by part (i).Conversely, if pn · aG(χ) is integral, so is bG(pn · aG(χ)) = pn · prG(χ). Therefore,Proposition 7.3 yields the result.

Next we turn to the case of positive characteristic p.

7.5 Recall from Section II.3 that

P abF (H) =

RabF (H), if H is a p′-group,

0, otherwise,

for H ≤ G. We can regard P abF ⊆ RF as a Z-restriction subfunctor on G, and define

p ∈ Z−Con(G)(RF , PabF ) for H ≤ G and irreducible S ∈ FH−mod by

pH([S]) :=

[S], if H is a p′-group and dimF S = 1,

0, otherwise.

Note that RF , P abF , and B, where B(H) := Hom(H,F×) for a p′-subgroup H ≤ G

and B(H) := ∅ otherwise, satisfy Hypothesis II.3.1 so that we can define

a := aRF ,PabF ,p ∈ Q−Res(G)

(Q⊗RF ,Q⊗ P ab

F+

).

4.7. PROJECTIFICATION 139

Furthermore, we define pr ∈ Q−Res(G)(Q⊗RF ,Q⊗ PF ) as the composition

pr : Q⊗RFa−−→Q⊗ P ab

F+bPF ,P

abF−−→ Q⊗ PF .

We call pr the projectification morphism. Note that RF , P abF , B, and p satisfy

condition (∗p′) of Theorem II.4.5 by the same argument as in the proof of Theo-rem III.3.2. Hence, we have |H|p ·aH([V ]) ∈ P ab

F+(H) for H ≤ G and S ∈ FH−modby Corollary II.4.7.

7.6 Proposition Assume the notation of 7.5. Let V ∈ FG−mod be projec-tive. Then aG([V ]) is integral, i.e. aG([V ]) ∈ P ab

F+(G). In particular, prG([V ]) isintegral, i.e. prG([V ]) ∈ PF (G).

Proof Let p′ ∈ Z−Con(G)(PF , PabF ) denote the morphism of Example III.3.1

which led to an integral canonical induction formula a′ ∈ Z−Res(G)(PF , PabF+).

Then we have a commutative diagram

PFc−−−−→ RF

p′y ypP abF P ab

F ,

where c denotes the Cartan morphism defined by cH([V ]) = [V ] for H ≤ G and aprojective FH-module V . By Proposition II.3.11 we obtain a commutative diagram

Q⊗ PFc−−−−→ Q⊗RF

a′y ya

Q⊗ P abF+ Q⊗ P ab

F+.

Since a′ is integral, the result follows.

7.7 Proposition Keep the notation of 7.5, and let c ∈ Z−Mack(G)(PF , RF )be the Cartan morphism, which is given by cH([V ]) = [V ] for H ≤ G and a projectiveFH-module V . Then we have pr c = idQ⊗PF and c pr = idQ⊗RF . Moreover, wehave the explicit formula

prH([V ]) =∑

σ=(

(H0,ϕ0)<...<(Hn,ϕn))∈H\Γ(M(H))

(−1)n×

×|(NH(σ)/H0)p′ ||NH(σ)/H0|

mϕn

(resHHn([V ])

)indHH0

([Fϕ0 ])

for H ≤ G and V ∈ FH−mod, where M(H) is the poset of monomial pairsassociated to B as in 7.5, cf. Definition II.3.2. Furthermore, |H|p ·aH(χ) ∈ P ab

F+(H)and |H|p · prH([V ]) ∈ PF (H) for H ≤ G and V ∈ FH−mod.

Proof Since the coprimordial subgroups for RF and PF are precisely the p′-subgroups of G, it suffices to show that prH(cH([Fϕ])) = [Fϕ] and cH(prH([Fϕ])) =[Fϕ] for cyclic p′-subgroups H and ϕ ∈ Hom(H,F×). From Lemma II.3.7 we


have aH([Fϕ]) = [H,ϕ]H , and that implies prH([Fϕ]) = [Fϕ]. Moreover, we havecH([Fϕ]) = [Fϕ], and the above equations follow.

The explicit formula follows from Theorem II.4.5, since condition (∗p′) is satisfiedfor RF , P ab

F , B, and p.For H ≤ G and V ∈ FH−mod, the element |H|p · aH([V ]) is integral by

Corollary II.4.7, as already stated in 7.5

7.8 Corollary The Cartan map cG : PF (G)→ RF (G) is injective and

RF (G)/cG(PF (G)) ∼= prG(RF (G))/PF (G).

In particular, RF (G)/cG(PF (G)) is a finite p-group.

Proof The isomorphism RF (G)/cG(PF (G)) ∼= prG(RF (G))/PF (G) is inducedby the map prG. Since |G|p · prG([V ]) ∈ PF (G) for an FG-module V , the lattergroup is a finite p-group.

7.9 Remark There are obvious generalizations of 7.1 and 7.5 to the case ofrelative projective modules instead of projective modules.

Chapter 5

Monomial Resolutions

In this chapter we are going to lift the canonical induction formula aG : R(G) →Rab

+ (G) to a categorical level, by interpreting R(G) as the Grothendieck group ofCG−mod, and Rab

+ (G) as the Grothendieck group of a category CG−mon, consist-ing of monomial CG-modules with an additional structure. The homomorphism aGis induced by a functor form CG−mod to the homotopy category of chain complexesin CG−mon, which we call the monomial resolution. The monomial resolution canbe defined more generally for suitable commutative rings k instead of C. Howeverthese general monomial resolutions are not in relation to other canonical induc-tion formulae from Chapter III. Following a suggestion of Rickard we interpret themonomial resolution in Section 3 as a projective resolution by embedding the wholesetup into a bigger category.

Throughout this chapter G denotes a finite group, E its trivial subgroup, and k anoetherian integral domain. Let G(k) = Hom(G, k×) be the multiplicative group ofgroup homomorphisms from G to k×. By kG−modpr (resp. k−modpr) we denotethe full subcategory of kG−mod (resp. k−mod) whose objects are projective ask-modules (resp. projective k-modules).

Furthermore, let Mk(G) denote the set of all pairs (H,ϕ) where H ≤ G andϕ ∈ H(k). Mk(G) is a poset by setting (K,ψ) ≤ (H,ϕ) if and only if K ≤ Hand ψ = ϕ|K . The group G acts from the left on Mk(G) by poset automorphisms:s(H,ϕ) := ( sH, sϕ) for s ∈ G, (H,ϕ) ∈Mk(G), where sH := sHs−1 and sϕ ∈ sH(k)is defined by sϕ(g) := ϕ(s−1gs) for g ∈ sH. We denote the G-orbit of (H,ϕ) by[H,ϕ]G and the set of G-orbits of Mk(G) by G\Mk(G). Rk(G) ⊆ Mk(G) alwaysdenotes a set of representatives of the G-orbits G\Mk(G). The set G\Mk(G) isagain a poset by the following definition: [K,ψ]G ≤ [H,ϕ]G if and only if there issome s ∈ G with (K,ψ) ≤ s(H,ϕ). Note that Rk(G) inherits the structure of aposet from Mk(G), but in general it is not clear that Rk(G) can be chosen suchthatRk(G) andG\Mk(G) are isomorphic posets. More precisely, the canonical mapRk(G) → G\Mk(G) is always a bijective map of posets, but not an isomorphismof posets, since there may be orbits [K,ψ]G ≤ [H,ϕ]G in G\Mk(G) such that theirrepresentatives in Rk(G) are not in relation.

We will omit the argument k in G(k), Mk(G), Rk(G), if k = C.

141

142 CHAPTER 5. MONOMIAL RESOLUTIONS

5.1 The category of kG-monomial modules

In this section we define the category kG−mon of kG-monomial modules. Westudy the morphisms in this category and determine its indecomposable objects.Moreover, we give various equivalent conditions for M,N ∈ kG−mon to be iso-morphic. At the end this results in the determination of the Grothendieck groupRabk+(G) of kG−mon, which coincides with the construction defined in I.2.2 applied

to a Z-restriction functor Rabk on G.

1.1 Definition Let V ∈ kG−mod and ϕ ∈ G(k). V is called ϕ-homogeneous

if gv = ϕ(g)v for all v ∈ V and all g ∈ G. V is called abelian if V is the sum ofϕ-homogenous kG-submodules, where ϕ runs through G(k).

For each pair (H,ϕ) ∈ Mk(G) we define the (H,ϕ)-homogeneous component

V (H,ϕ) of V by

V (H,ϕ) = v ∈ V | hv = ϕ(h)v for all h ∈ H.

Obviously V (H,ϕ) is the largest ϕ-homogeneous kH-submodule of the kH-moduleresGHV . If V ∈ kG−modpr, then the sum of the submodules V (G,ϕ), ϕ ∈ G(k),is always a direct sum, and V is abelian if and only if V is the direct sum of itsdifferent (G,ϕ)-homogeneous components V (G,ϕ).

We denote the full subcategory of kG−mod which consists of abelian kG-modules by kG−modab, and we define kG−modpr

ab := kG−modab ∩ kG−modpr.For each V ∈ kG−mod the abelian part P(V ) of V is defined to be the largestabelian kG-submodule of V , i.e. P(V ) =

∑ϕ∈G(k) V

(G,ϕ). This defines a functor

P : kG−mod → kG−modab, since for V,W ∈ kG−mod, each f ∈ HomkG(V,W )maps V (G,ϕ) to W (G,ϕ) for all ϕ ∈ G(k).

V is called a monomial kG-module, if V is isomorphic to a direct sum of kG-modules indGHi(Vi), where Hi ≤ G are arbitrary subgroups and Vi ∈ kHi−modpr

ab.This extends the definition preceeding Proposition III.4.1.

1.2 Remark The definition of V (H,ϕ) provides a G-equivariant Mk(G)-filtration on each V ∈ kG−mod, i.e. we have:

(i) If (K,ψ) ≤ (H,ϕ) ∈Mk(G), then V (H,ϕ) ⊆ V (K,ψ).

(ii) Vs(H,ϕ) = sV (H,ϕ) for all (H,ϕ) ∈Mk(G), s ∈ G.

(iii) V (E,1) = V .

In view of (ii), V (H,ϕ) is a kNG(H,ϕ)-module, where NG(H,ϕ) := s ∈ G |s(H,ϕ) = (H,ϕ) denotes the stabilizer of (H,ϕ) in G. The set V (H,ϕ) | (H,ϕ) ∈Mk(G) of submodules of V is again a G-poset (ordered by inclusion). This G-poset was used as a main ingredient in [Bo94, Thm. 3.2(c)]. In Section 2 it will playan important role in the definition of a monomial resolution and in upper boundsfor the monomial homological dimension of V (cf. Theorem 2.14).

1.3 Definition A kG-monomial module M is an object M ∈ kG−modpr

(hence finitely generated and projective over k) with the additional structure of a

5.1. KG-MONOMIAL MODULES 143

fixed decomposition M = M1 ⊕ . . . ⊕Mm into k-submodules M1, . . . ,Mm (whichare thus again finitely generated and projective over k), such that the two followingconditions are satisfied:

(i) The G-action on M makes M1, . . . ,Mm into a G-set, i.e. for each s ∈ G andfor each i ∈ 1, . . . ,m there is some j ∈ 1, . . . ,m such that sMi = Mj .

(ii) If for each i ∈ 1, . . . ,m we denote the stabilizer s ∈ G | sMi = Mi of Mi

by Hi, then the kHi-module Mi is ϕi-homogeneous for some ϕi ∈ Hi(k).

Note that ϕi ∈ Hi(k) is uniquely determined, since Mi is k-torsion free. We call(Hi, ϕi) the stabilizing pair of Mi.

Let M = M1 ⊕ . . .⊕Mm and N = N1 ⊕ . . .⊕Nn be kG-monomial modules. AkG-monomial homomorphism from M = M1 ⊕ . . .⊕Mm to N = N1 ⊕ . . .⊕Nn isa k-linear combination of proper kG-monomial homomorphisms which are definedto be kG-linear maps f : M → N respecting the decomposition of M and N , i.e. foreach i ∈ 1, . . . ,m there is some j ∈ 1, . . . , n such that f(Mi) ⊆ Nj .

The kG-monomial modules and their homomorphisms form an additive categorywhich we denote by kG−mon. For M,N ∈ kG−mon we denote the k-module ofkG-monomial homomorphisms fromM toN by monkG(M,N). Note that kG−monis not an abelian category. In general kernels and cokernels don’t carry the structureof a kG-monomial module.

An object M = M1 ⊕ . . .⊕Mm ∈ kG−mon is called ϕ-homogeneous for someϕ ∈ G(k), if all the stabilizing pairs (Hi, ϕi) are equal to (G,ϕ), i.e. the under-lying kG-module of M is ϕ-homogeneous. M is called abelian, if Hi = G for alli ∈ 1, . . . ,m (the ϕi need not necessarily to be the same). We denote the full sub-category of kG−mon consisting of abelian kG-monomial modules by kG−monab.

1.4 Remark (a) We will often write M ∈ kG−mon if there is no need forgiving the fixed decomposition which is part of an object in kG−mon.

(b) Note that the above definition is different from the definition given in [Bo89,I.1] and [Bo90, 1.4], where we worked over the field of complex numbers only andrequired that the componentsMi are one-dimensional, and where we only consideredproper monomial homomorphisms. With this new definition we have more freedomwith respect to the choice of the base ring k, kG−mon becomes an additive categoryby allowing k-linear combinations of proper kG-monomial homomorphisms, and theGrothendieck group remains the same for k = C (cf. Subsection 1.22 and [Bo90,1.4]). We also should mention that alternatively to the above definition one couldimpose on each Mi the additional restriction to be free over k of rank one. In thecase where all objects in k−modpr are even free (as for k local or a principal idealdomain), the embedding of this alternative category into kG−mon is actually anequivalence. In the meanwhile we learnt that W. F. Reynolds has introduced in[Re71, Sect. 3] the same category as we did in [Bo89] and [Bo90], but for differentreasons.

(c) By definition, each kG-monomial module is a kG-module and each kG-monomial homomorphism is kG-linear, which gives rise to the forgetful functor

V : kG−mon→ kG−modpr.


Note that V restricts to a functor on the abelian subcategories

V : kG−monab → kG−modprab.

(d) The two notions ‘monomial kG-module’ and ‘kG-monomial module’ shouldnot be confused. They are objects in different categories: The first one is an objectin kG−mon and the second one in kG−mod. Note also that non-isomorphicobjects in kG−mon may become isomorphic after applying V, cf. Example 1.5.The confusing similarity of the two notions has the following justification: Themonomial kG-modules in kG−mod form precisely the image of kG−mon underV. In fact, let M = M1 ⊕ . . . ⊕Mm ∈ kG−mon and let us fix some Mi, then thesum of all Mj in the G-orbit of Mi is a kG-submodule of M which is isomorphic toindGHi(Mi). Hence, we have

V(M) ∼=⊕

Mi∈M1,...,Mm/G

indGHi(Mi),

where the sum runs over a set of representatives of the G-orbits of M1, . . . ,Mm.Since Mi is a k-projective and an (even ϕi-homogeneous) abelian kHi-module bydefinition, V(M) is a monomial kG-module.

Conversely, let H ≤ G and let U ∈ kH−modpr be ϕ-homogeneous for someϕ ∈ H(k). Then indGH(U) = kG⊗kH U is the image under V of the same underlyingkG-module with the decomposition ⊕s∈G/H(s ⊗kH U), where s runs over a set ofrepresentatives of G/H. Note that G permutes the summands s⊗kH U in the sameway as it permutes the G-set G/H, that the stabilizer of s ⊗kH U is sH and thats ⊗kH U is an sϕ-homogeneous k sH-module, i.e. the stabilizing pair of s ⊗kH U iss(H,ϕ). This construction can be extended to direct sums. Hence, the notion of akG-monomial module can be seen as the notion of a monomial kG-module togetherwith a specified way of writing it as a sum of induced homogeneous modules, whereboth the subgroups H we induce from and the ϕ-homogeneous modules (ϕ ∈ H(k))which we induce are specified up to G-conjugacy.

1.5 Example Let G be the quaternion group of order 8 and let k = C. LetV be an irreducible kG-module of dimension 2. Then V is a monomial kG-module.In fact, if H1, H2, H3 denote the three cyclic subgroups of order 4, and if ϕi denotesa faithful one-dimensional character of Hi, i = 1, 2, 3, then the character χ of Vequals indGHi(ϕi) and also indGHi(ϕ

−1i ) for all i = 1, 2, 3. In order to provide V with

the structure of a kG-monomial module we have to choose a suitable decompositionof V .

Let us choose i ∈ 1, 2, 3, then resGHi(χ) = ϕi + ϕ−1i , and this gives rise to a

unique decomposition V = V ′i ⊕ V ′′i as kHi-modules, such that V ′i , resp. V ′′i , areone-dimensional ϕi–, resp. ϕ−1

i –homogeneous kHi-submodules. Together with thisdecomposition, V is a kG-monomial module which we denote by Vi. The stabilizingpair of V ′i , resp. V ′′i , is (Hi, ϕi), resp. (Hi, ϕ

−1i ).

It is easy to see that V1, V2, V3 are pairwise non-isomorphic as kG-monomialmodules. In Example 1.13 we will even see that monkG(Vi, Vj) = 0 for i 6= j.

1.6 Remark The following is a list of constructions in the category kG−monwhich can be lifted from well-known constructions in kG−mod, i.e. these con-


structions commute with the forgetful functor V. Let M = M1 ⊕ . . . ⊕Mm andN = N1 ⊕ . . .⊕Nn be in kG−mon.

(a) Direct sum: This is the kG-module M⊕N with the decomposition M1⊕ . . .⊕Mm ⊕ N1 ⊕ . . . ⊕ Nn. Together with the canonical inclusions M → M ⊕ Nand N →M ⊕N this is also a coproduct in the categorical sense.

(b) Tensor product: This is the kG-module M ⊗N (unadorned tensor productsare taken over k) with the decomposition

⊕i,j(Mi ⊗Nj).

(c) Homomorphisms: This is the kG-module Homk(M,N) with the decomposi-tion

⊕i,j Homk(Mi, Nj). Here the G-action is given by (sf)(m) := sf(s−1m),

for f ∈ Homk(M,N), s ∈ G, m ∈M .

(d) Dual: Let M∗ := Homk(M,k) be endowed with the induced decompositionHomk(G, k) =

⊕i Homk(Mi, k). Considering the trivial kG-module k with

one-term decomposition as an object in kG−mon, this is a special case of (c).

(e) Restriction: Let f : G′ → G be a homomorphism of finite groups. Then wehave a functor resf : kG−mon → kG′−mon given by resf (M) = M as ak-module and the old decomposition, where g′ ∈ G′ acts on resf (M) = Mvia f(g′). If f is the inclusion of a subgroup H ≤ G, then we write resGH(M)instead of resf (M).

(f) Induction: If H ≤ G, then we have a functor indGH : kH−mon → kG−mongiven by the following construction. Let L = L1 ⊕ . . .⊕Ll ∈ kH−mon, thenindGHL ∈ kG−mon is defined to be kG⊗kH L as a kG-module together withthe decomposition into k-submodules s⊗kH Li, where s runs through a set ofrepresentatives of G/H and i through 1, . . . , l.

(g) Note that the constructions (a)—(e) applied to abelian kG-monomial modulesresult again in abelian kG- (or kG′-) monomial modules.

(h) All the well-known canonical isomorphism in kG−mod, as for example M ⊕N ∼= N ⊕ M , M ⊗ N ∼= N ⊗ M , M ⊗ (N ⊕ P ) ∼= (M ⊗ N) ⊕ (M ⊗ P ),Homk(M,N) ∼= M∗ ⊗ N , etc., hold also in kG−mon, since the underlyingisomorphisms in kG−mod are proper kG-monomial homomorphisms. More-over we mention that the constructions (b)—(f) commute with taking directsums, and (d) and (e) commute with taking tensor products. The followingproposition lists three more canonical isomorphisms.

1.7 Proposition

(a) Frobenius reciprocity: Let H ≤ G, then the functor resGH : kG−mon →kH−mon is right adjoint to indGH : kH−mon→ kG−mon, i.e. for M ∈ kH−monand N ∈ kG−mon there is a k-linear isomorphism

monkG(indGH(M), N) ∼= monkH(M, resGH(N))

which is natural in M and N .


(b) Let H ≤ G, M ∈ kG−mon and N ∈ kH−mon. Then we have a canonicalisomorphism in kG−mon

M ⊗ indGH(N) ∼= indGH(resGH(M)⊗N

).

(c) Let U,H ≤ G and let M ∈ kH−mon. Then we have a canonical isomor-phism in kU−mon

resGU indGH(M) ∼=⊕

s∈U\G/H

indUU∩ sHres

sHU∩ sH( sM),

where sM denotes the k sH-monomial module resf (M) with f : sH → H, g 7→ s−1gs,and where s runs through a set of representatives of the double cosets U\G/H inG.

Proof (a) Let M ∈ kH−mon and N ∈ kG−mon, then it is easily checkedthat the well-known inverse k-linear isomorphisms

HomkG(indGH(M), N) ∼= HomkH(M, resGH(N)),

f 7→(m 7→ f(1⊗kH m)

),(

g ⊗kH m 7→ gf ′(m))←7 f ′,

map proper kG- (resp. kH-) monomial homomorphisms to proper ones. Moreover,these isomorphisms are natural in M and N .

(b) The well-known kG-linear inverse isomorphisms

M ⊗ (kG⊗kH N) ∼= kG⊗kH(resGH(M)⊗N

),

m⊗ (g ⊗kH n) 7→ g ⊗kH (g−1m⊗ n),

gm⊗ (g ⊗kH n)←7 g ⊗kH (m⊗ n),

are obviously proper kG-monomial homomorphisms.

(c) The well-known kU -linear inverse isomorphisms

kG⊗kH M ∼=⊕

s∈U\G/H

kU ⊗k(U∩ sH)sM

g ⊗kH mg=ush7→ u⊗k(U∩ sH) hm ∈ s-component,

us⊗kH m ←7 u⊗k(U∩ sH) m ∈ s-component,

are obviously proper kG-monomial homomorphisms.

Similar to the Mk(G)-filtration of any V ∈ kG−mod we a have an Mk(G)-filtration and also an Mk(G)-grading on any M ∈ kG−mon by the followingdefinition.

1.8 Definition For M = M1 ⊕ . . . ⊕Mm ∈ kG−mon and (H,ϕ) ∈ Mk(G)we define the (H,ϕ)-homogeneous component M(H,ϕ) of M to be the (direct)


sum of those Mi whose stabilizing pair (Hi, ϕi) equals (H,ϕ). More generally wedefine for M⊆Mk(G) the following k-submodule of M :

M(M) :=⊕

(H,ϕ)∈M

M(H,ϕ) =⊕

i∈1,...,m(Hi,ϕi)∈M

Mi.

If M = [H,ϕ]G is the G-orbit of some (H,ϕ) ∈Mk(G), then we call

M([H,ϕ]G

)=

⊕s∈G/NG(H,ϕ)

M( s(H,ϕ)

)=

⊕i∈1,...,m

(Hi,ϕi)∈[H,ϕ]G

Mi

the [H,ϕ]G-homogeneous component of M . The kG-module M([H,ϕ]G

)is a kG-

monomial module in its own right with the decomposition inherited from M .

For (H,ϕ) ∈Mk(G) we define

M≥(H,ϕ) := (H ′, ϕ′) ∈Mk(G) | (H,ϕ) ≤ (H ′, ϕ′)

and

M≥[H,ϕ]G := (H ′, ϕ′) ∈Mk(G) | [H,ϕ]G ≤ [H ′, ϕ′]G,

and we set

M (H,ϕ) := M(M≥(H,ϕ)) and M [H,ϕ]G := M(M≥[H,ϕ]G).

The following remarks are immediate consequences of the last definition.

1.9 Remark Let M ∈ kG−mon.

(a) M⊆M′ ⊆Mk(G) =⇒ M(M) ⊆M(M′).

(b) (K,ψ) ≤ (H,ϕ) ∈Mk(G) =⇒ M (H,ϕ) ⊆M (K,ψ).

(c) M =⊕

(H,ϕ)∈Mk(G)M(H,ϕ) = M (E,1).

(d) M( sM) = sM(M) for all M ⊆ Mk(G), s ∈ G. In particular, M(M) is akNG(M)-monomial module, where NG(M) := s ∈ G | sM =M.

(e) [K,ψ]G ≤ [H,ϕ]G ∈ G\Mk(G) =⇒ M [H,ϕ]G ⊆M [K,ψ]G .

(f) M =⊕

(H,ϕ)∈Rk(G)M([H,ϕ]G) = M [E,1]G as kG-monomial modules.

(g) For (H,ϕ) ∈Mk(G) we have M (H,ϕ) = (resGH(M))(H,ϕ) = (resGH(M))(H,ϕ).

(h) M (H,ϕ) ⊆ V(M)(H,ϕ) for all (H,ϕ) ∈Mk(G). But in general (cf. Remark 1.15)the other inclusion is false.

(i) Let M ∈ kG−monab, then M(G,ϕ) = M (G,ϕ) = V(M)(G,ϕ) for all ϕ ∈ G(k).


(j) Let M ∈ kG−mon and (H,ϕ) ∈ Mk(G). If we consider M(H,ϕ) as akNG(H,ϕ)-monomial module by part (d), then indGNG(H,ϕ)(M(H,ϕ)) andM([H,ϕ]G) are isomorphic as kG-monomial modules via the proper isomor-phism s⊗kNG(H,ϕ) m 7→ sm.

1.10 Remark By the decomposition in 1.9 (c) each M ∈ kG−mon is anMk(G)-graded k-module, and by 1.9 (f) each M ∈ kG−mon is a G\Mk(G)-graded kG-monomial module. Each kG-monomial module is the direct sum (inkG−mon) of its [H,ϕ]G-homogeneous components. Note that M ∈ kG−mon isϕ-homogeneous for ϕ ∈ G(k) if and only if M = M([G,ϕ]G) = M(G,ϕ) = M (G,ϕ),and note that M is abelian if and only if M =

⊕ϕ∈G(k)M(G,ϕ). Note also that

if M ∈ kH−mon is ϕ-homogeneous for some ϕ ∈ H(k), then indGHM is [H,ϕ]G-homogeneous, i.e. indGHM equals its [H,ϕ]G-homogeneous component.

In general, the Mk(G)– (resp. G\Mk(G)–) grading is not preserved underkG-monomial homomorphisms. However, as the following proposition shows, theMk(G)– (resp. G\Mk(G)–) filtration on M given by the k-submodules M (H,ϕ)

(resp. kG-monomial submodules M [H,ϕ]G) is compatible with the notion of kG-monomial homomorphisms.

1.11 Proposition Let M,N ∈ kG−mon and let f ∈ HomkG(M,N). Thenthe statements (a)—(c) are equivalent:

(a) f ∈ monkG(M,N).(b) f(M (H,ϕ)) ⊆ N (H,ϕ) for all (H,ϕ) ∈Mk(G).(c) f

(M(H,ϕ)

)⊆ N (H,ϕ) for all (H,ϕ) ∈Mk(G).

(d) If (a)—(c) hold, then we have f(M([H,ϕ]G)

)⊆ f

(M [H,ϕ]G

)⊆ N [H,ϕ]G for

all [H,ϕ]G ∈ G\Mk(G). In general (d) does not imply (a), (b) or (c).

Proof Let M = M1 ⊕ . . . ⊕ Mm and N = N1 ⊕ . . . ⊕ Nn denote the re-spective decompositions and let (Hi, ϕi) ∈ Mk(G) be the stabilizing pair of Mi,i ∈ 1, . . . ,m.

(a) =⇒ (b): Since N (H,ϕ) is a k-submodule of N , we may assume that f isproper. It suffices to show that for each i ∈ 1, . . . ,m with (H,ϕ) ≤ (Hi, ϕi) wehave f(Mi) ⊆ N (H,ϕ). Obviously we may assume that f(Mi) 6= 0. Since f isproper, there is some j ∈ 1, . . . , n such that f(Mi) ⊆ Nj . Let (U, µ) ∈Mk(G) bethe stabilizing pair of Nj . It suffices to show that (Hi, ϕi) ≤ (U, µ). So let h ∈ Hi

and v ∈Mi with f(v) 6= 0. Then

0 6= hf(v) = f(hv) ∈ hNj ∩Nj ,

which forces hNj = Nj , and hence h ∈ U , showing that Hi ≤ U . Now we have foreach h ∈ Hi:

µ(h)f(v) = hf(v) = f(hv) = f(ϕi(h)v) = ϕi(h)f(v),

and therefore µ(h) = ϕi(h), hence µ|Hi = ϕi.

(b) =⇒ (c): This is trivial, since M(H,ϕ) ⊆M (H,ϕ).(c) =⇒ (a): Since f is the sum of kG-linear maps which are zero on all but one of

the G-orbits of the G-set M1, . . . ,Mm, we may assume that G acts transitively on


M1, . . . ,Mm. Let J ⊆ 1, . . . , n be the subset of those j such that Nj ⊆ N (H1,ϕ1),and let f1 : M1 → N be the restriction of f to M1. Then f1 is kH1-linear and bythe hypothesis in (c), f1 =

∑j∈J ejf1, where ej : N → Nj → N is the projection

with respect to the decomposition N = N1⊕ . . .⊕Nn. Each ej , j ∈ J , is kH1-linear,hence each ejf1, j ∈ J , is kH1-linear. Since M ∼= indGH1

(M1) as kG-modules, eachejf1 extends uniquely to a kG-linear map fj : M → N . By the same uniquenessproperty we have f =

∑j∈J fj , since both of them extend f1. Finally, for any

j ∈ J , fj is a proper kG-monomial homomorphism, since for each i ∈ 1, . . . ,mthere is some s ∈ G with Mi = sM1, and hence fj(Mi) = fj(sM1) = sfj(M1) =sejf1(M1) ⊆ sNj .

(d) The first inclusion holds always, since M([H,ϕ]G) ⊆ M [H,ϕ]G . Using the

obvious equation M [H,ϕ]G =∑

s∈GMs(H,ϕ) we obtain

f(M [H,ϕ]G

)= f

(∑s∈G

Ms(H,ϕ)

)=∑s∈G

f(M

s(H,ϕ)

)⊆∑s∈G

Ns(H,ϕ) = N [H,ϕ]G

from the hypothesis in (b).

Finally we show the last assertion in (d) by giving an example. Let k = C forsimplicity, and let G be a finite group which has a non-normal subgroup H. LetM ∈ kG−mon be defined by M := indGH(k), where k ∈ kH−mon is the trivial kH-module with one-term decomposition. We claim that any f ∈ HomkG(M,M) satis-fies the condition in (d). In fact, let [K,ψ]G ∈ G\Mk(G). If [K,ψ]G 6≤ [H,ϕ]G, thenM [K,ψ]G = 0, and if [K,ψ]G ≤ [H,ϕ]G, then M [K,ψ]G = M . Applying Frobeniusreciprocity and the Mackey decomposition formula (Proposition 1.7 (c)) one seesimmediately that HomkG(M,M) has k-rank |H\G/H|. Whereas in Prop. 1.20 (b)we will see that monkG(M,M) has k-rank #s ∈ H\G/H | H ≤ sH which isdifferent from |H\G/H|, if H is not normal in G.

1.12 Corollary

(a) Let M ∈ kG−mon and N ∈ kG−monab, then

monkG(M,N) = HomkG(M,N).

(b) The forgetful functor V : kG−monab → kG−modprab is a category equiva-

lence with inverse functor given by

V 7→M =⊕

ϕ∈G(k)

V (G,ϕ).

Proof (a) This follows from Proposition 1.11 (b) =⇒ (a) and Remark 1.9 (g),(h), and (i): For f ∈ HomkG(M,N) and (H,ϕ) ∈Mk(G) we have

f(M (H,ϕ)

)= f

((resGHM)(H,ϕ)

)⊆ f

(V(resGHM)(H,ϕ)

)⊆ V(resGHN)(H,ϕ)

= (resGHN)(H,ϕ) = N (H,ϕ),

since resGH(N) is again abelian.

(b) This follows from part (a).


1.13 Example With the notation of Example 1.5 we have, for the quaterniongroup G of order 8, three kG-monomial modules V1, V2, V3 with V(V1) ∼= V(V2) ∼=V(V3), but monkG(Vi, Vj) = 0 for i 6= j. In fact, let f ∈ monkG(Vi, Vj), then

by Proposition 1.11 (d) we have f(Vi) = f(V[Hi,ϕi]Gi ) ⊆ V

[Hi,ϕi]Gj . But this last

submodule of Vj is 0, since the stabilizing pairs of the components V ′j and V ′′j of Vj

are (Hj , ϕj) and (Hj , ϕ−1j ) respectively, and since [Hi, ϕi]G 6≤ (Hj , ϕj)

G for i 6= j.

1.14 Definition For each M ∈ kG−mon the abelian part P(M) of M isdefined by

P(M) :=⊕

ϕ∈G(k)

M(G,ϕ) ⊆M.

Note that P(M) ∈ kG−monab with the above decomposition, and that M ∈kG−monab if and only if P(M) = M . For arbitrary M,N ∈ kG−mon, f ∈monkG(M,N) and ϕ ∈ G(k) we have by Proposition 1.11 f(M(G,ϕ)) ⊆ N (G,ϕ) =N(G,ϕ), showing that P defines a functor

P : kG−mon→ kG−monab.

1.15 Remark Although the concepts of taking abelian parts of objects inkG−mon and kG−mod seem to be very similar, they do not commute with theforgetful functor, i.e. in general,

VP 6∼= PV : kG−mon→ kG−modab

are not naturally isomorphic. Let for example G be non-trivial, M := indGE(k) ∈kG−mon, k considered as a kE-monomial module with one-term decomposition.Then M = M1 ⊕ . . . ⊕M|G| and G acts freely on M1, . . . ,M|G| showing that allthe stabilizing pairs are equal to (E, 1), and therefore P(M) = 0. On the otherhand V(M) ∼= kG has a non-trivial abelian part, since kG(G,1) 6= 0. This examplealso shows that in general M (H,ϕ) 6= V(M)(H,ϕ), cf. 1.9 (h).

1.16 Definition We call M ∈ kG−mon indecomposable, if M is not iso-morphic to a direct sum M ′ ⊕M ′′ of non-zero kG-monomial modules M ′ and M ′′.For each (H,ϕ) ∈Mk(G) we denote by kϕ the ϕ-homogeneous abelian kH-modulewith underlying k-module k. By Corollary 1.12 (b) we may consider kϕ as elementin kH−monab, namely with the one-term decomposition. We define

SG(H,ϕ) := indGH(kϕ) ∈ kG−mon.

By the definition of induction, the decomposition of SG(H,ϕ) is given by SG(H,ϕ) =⊕s∈G/H(s⊗kH kϕ). Each summand is isomorphic to k as k-module and G acts tran-

sitively on these summands . The stabilizing pair of s⊗kH kϕ is s(H,ϕ), s ∈ G/H.Hence, SG(H,ϕ) = SG(H,ϕ)

([H,ϕ]G

)is homogeneous in degree [H,ϕ]G ∈ G\Mk(G).

We will often need the following hypothesis for the base ring k, and will refer toit as condition (∗):

Each finitely generated projective k-module is free. (∗)


1.17 Proposition

(a) For (H,ϕ), (K,ψ) ∈ Mk(G) we have: SG(H,ϕ)∼= SG(K,ψ) ⇐⇒ [H,ϕ]G =

[K,ψ]G.

(b) SG(H,ϕ) is indecomposable for (H,ϕ) ∈Mk(G).

(c) Let M ∈ kG−mon be homogeneous in degree [H,ϕ]G ∈ G\Mk(G). Thereis some N ∈ kG−mon, homogeneous in degree [H,ϕ]G, and some n ∈ N, suchthat M ⊕ N is isomorphic to the direct sum of n copies of SG(H,ϕ). If k satisfies

condition (∗), then M is isomorphic to the direct sum of rkkM/(G : H) copies ofSG(H,ϕ), where rkk denotes the k-rank.

(d) Let M,N ∈ kG−mon. The following are equivalent:

(i) M and N are isomorphic as kG-monomial modules.

(ii) M(H,ϕ) and N(H,ϕ) are isomorphic as kNG(H,ϕ)-modules, for all (H,ϕ) ∈Mk(G).

(iii) M(H,ϕ) and N(H,ϕ) are isomorphic as kNG(H,ϕ)-monomial modules, forall (H,ϕ) ∈Mk(G).

(iv) M([H,ϕ]G) and N([H,ϕ]G) are isomorphic as kG-monomial modules, for all(H,ϕ) ∈Mk(G).

Proof (a) Let f : SG(H,ϕ) → SG(K,ψ) be an isomorphism of kG-monomial mod-

ules. Then by part (a) =⇒ (d) of Proposition 1.11 we have

SG(K,ψ) = f(SG(H,ϕ)) = f((SG(H,ϕ))

[H,ϕ]G)⊆ (SG(K,ψ))

[H,ϕ]G ⊆ SG(K,ψ).

Hence, equality must hold, and from (SG(K,ψ))[H,ϕ]G = SG(K,ψ) = SG(K,ψ)

([K,ψ]G

)we

can deduce [H,ϕ]G ≤ [K,ψ]G. By symmetry we also get the other relation.

Conversely, let (K,ψ) = s(H,ϕ) for some s ∈ G. Then the map

SG(H,ϕ) → SG(K,ψ), g ⊗kH α 7→ gs−1 ⊗kK α

for g ∈ G,α ∈ k is a kG-monomial isomorphism.

(b) Assume that SG(H,ϕ)∼= M ⊕ N with non-trivial M,N ∈ kG−mon. Then

M and N are homogeneous in degree [H,ϕ]G. It is clear that the k-rank of an[H,ϕ]G-homogeneous kG-monomial module is a multiple of (G : H). Since M andN are non-trivial, the k-rank of M ⊕N is at least 2(G : H) which contradicts thefact that the k-rank of SG(H,ϕ) is (G : H).

(c) If we know the assertion for two kG-monomial modules satisfying the hy-pothesis, we also know it for their direct sum. Hence, we may assume that G actstransitively on M1, . . . ,Mm, where M = M1⊕. . .⊕Mm is the decomposition of M .Certainly, one of the Mi has stabilizing pair (H,ϕ), and we may assume that thisis the case for M1. Then M ∼= indGH(M1) as kG-monomial modules, where M1 canbe viewed as a ϕ-homogeneous kH-monomial module with one-term decompositionby Corollary 1.12 (b). In fact,

kG⊗kH M1 →M, s⊗kH m 7→ sm


provides such an isomorphism.If M1 (which is k-projective in general) is k-free, then M1 is isomorphic to a

direct sum of rkkM1 copies of kϕ as kH-monomial module by Corollary 1.12 (b).Since induction commutes with taking direct sums, we obtain that M is isomorphicto the direct sum of rkkM1 copies of SG(H,ϕ). Finally, it is clear that (G : H)·rkkM1 =rkkM .

In general, there is N1 ∈ k−modpr such that M1⊕N1 is k-free. We may considerN1 as a ϕ-homogeneous kH-module and view it via Corollary 1.12 (b) as object inkH−monab. We define N := indGH(N1). Then M ⊕N ∼= indGH(M1⊕N1), and sinceM1⊕N1 is k-free and since its isomorphism type in kH−mon doesn’t change if weview it endowed with the one-term decomposition (cf. Corollary 1.12 (b)), we are inthe previous case which shows that M ⊕N is isomorphic to a direct sum of copiesof SG(H,ϕ).

(d) (i) =⇒ (ii): Let f ∈ monkG(M,N) be an isomorphism and (H,ϕ) ∈Mk(G). Proposition 1.11 (b) implies that f induces kNG(H,ϕ)-linear isomor-phisms M (H,ϕ) → N (H,ϕ) and

∑(H,ϕ)<(H′,ϕ′)M

(H′,ϕ′) →∑

(H,ϕ)<(H′,ϕ′)N(H′,ϕ′).

Therefore we get kNG(H,ϕ)-linear isomorphisms between the quotients which arekNG(H,ϕ)-isomorphic to M(H,ϕ) and N(H,ϕ).

(ii) =⇒ (iii): Let f(H,ϕ) : M(H,ϕ) → N(H,ϕ) be a kNG(H,ϕ)-linear isomor-phism for each (H,ϕ) ∈ Mk(G). We may consider M(H,ϕ) and N(H,ϕ) askNG(H,ϕ)-monomial modules with the original decomposition. Then each f(H,ϕ)

is a kNG(H,ϕ)-monomial isomorphism by Proposition 1.11 (c), since M(H,ϕ) andN(H,ϕ) are homogeneous in degree (H,ϕ) with respect to the Mk(NG(H,ϕ))-grading.

(iii) =⇒ (iv): By the hypothesis in (iii) we have kNG(H,ϕ)-monomial isomor-phisms f(H,ϕ) : M(H,ϕ)→ N(H,ϕ) for all (H,ϕ) ∈ Rk(G). From Remark 1.19 (j)

we know that M([H,ϕ]G) ∼= indGNG(H,ϕ)M(H,ϕ) as kG-monomial modules, and

the same holds for N . Hence, indGNG(H,ϕ)(f(H,ϕ)) is an isomorphism M([H,ϕ]G)→N([H,ϕ]G) for all (H,ϕ) ∈ Rk(G).

(iv) =⇒ (i): Each kG-monomial module is the direct sum of its [H,ϕ]G-homo-geneous components.

1.18 Corollary(a) Let k satisfy condition (∗). Each indecomposable kG-monomial module is

isomorphic to some SG(H,ϕ), (H,ϕ) ∈ Rk(G). Hence Rk(G) (or as well G\Mk(G))parametrizes the set of isomorphism classes of indecomposable kG-monomial mod-ules.

(b) Let k satisfy condition (∗) and let M ∈ kG−mon. Then

M ∼=⊕

(H,ϕ)∈Rk(G)

r[H,ϕ]G(M) · SG(H,ϕ),

where r[H,ϕ]G(M) := rkkM([H,ϕ]G)/(G : H), and n · SG(H,ϕ) denotes the direct sum

of n copies of SG(H,ϕ).

(c) Let k satisfy condition (∗). For M,N ∈ kG−mon the following statementsare equivalent:

(i) M ∼= N as kG-monomial modules.


(ii) r[H,ϕ]G(M) = r[H,ϕ]G(N) for all [H,ϕ]G ∈ G\Mk(G).

(iii) rkkM([H,ϕ]G

)= rkkN

([H,ϕ]G

)for all [H,ϕ]G ∈ G\Mk(G).

(iv) rkkM[H,ϕ]G = rkkN

[H,ϕ]G for all [H,ϕ]G ∈ G\Mk(G).

(v) rkkM(H,ϕ) = rkkN(H,ϕ) for all (H,ϕ) ∈Mk(G).

(vi) rkkM(H,ϕ) = rkkN

(H,ϕ) for all (H,ϕ) ∈Mk(G).

(d) For each M ∈ kG−mon we have

M ∼=⊕

(H,ϕ)∈Rk(G)

indGHL(H,ϕ),

where L(H,ϕ) ∈ kH−monab is a ϕ-homogeneous kH-monomial module with one-term decomposition. In particular the [H,ϕ]G-homogeneous component of M isisomorphic to indGH(L(H,ϕ)) by Proposition 1.17 (d). If k satisfies condition (∗),then L(H,ϕ) can be choosen k-free of k-rank r[H,ϕ]G(M).

Proof Part (a) and (b) follow immediately from Remark 1.9 (f) and fromProposition 1.17 (c).

(c) The statements (i), (ii), and (iii) are equivalent by part (b) and the definitionof r[H,ϕ]G(M).

(i) =⇒ (iv),(vi): This follows from Proposition 1.11 (a) =⇒ (d) resp. (a) =⇒ (b).(vi) =⇒ (v): Since M (H,ϕ) =

⊕(H,ϕ)≤(H′,ϕ′)M(H ′, ϕ′) for all (H,ϕ) ∈ Mk(G),

we get by Mobius inversion

rkkM(H,ϕ) =∑

(H,ϕ)≤(H′,ϕ′)

µMk(G)

((H,ϕ), (H ′, ϕ′)

)· rkkM (H′,ϕ′),

where µMk(G)is the Mobius function of the poset Mk(G), see Appendix B.

(iv) =⇒ (ii): Similarly we obtain

rkkM([H,ϕ]G) =∑

[H,ϕ]G≤[H′,ϕ′]G

µG\Mk(G)

([H,ϕ]G, [H

′, ϕ′]G)· rkkM [H′,ϕ′]G ,

where µG\Mk(G)

is the Mobius function of the poset G\Mk(G).

(v) =⇒ (ii): This follows immediately from the equation

rkkM([H,ϕ]G) = (G : NG(H,ϕ)) · rkkM(H,ϕ)

for all (H,ϕ) ∈Mk(G).(d) By Remark 1.9 (f) we may assume that M is [H,ϕ]G-homogeneous for some

(H,ϕ) ∈ Rk(G), and we have to show that M is isomorphic to indGH(L(H,ϕ)) forsome L(H,ϕ) as in the assertion. We also may assume that G acts transitively onM1, . . . ,Mm, if M = M1 ⊕ . . .⊕Mm is the decomposition of M , since inductioncommutes with taking direct sums, and since two ϕ-homogeneous kH-monomialmodules are isomorphic if and only if their underlying k-modules are isomorphic,cf. part (b) of Corollary 1.12. By renumeration we may assume that M1 has stabi-lizing pair (H,ϕ). Then we have M ∼= indGH(M1), as already observed in the proofof Proposition 1.17 (c). Hence we may choose L(H,ϕ) = M1.


If k satisfies condition (∗), then the special choice for L(H,ϕ) follows from part (b)

and the fact that SG(H,ϕ) is homogeneous in degree [H,ϕ]G.

1.19 If k satisfies condition (∗), there is a well-known pairing for V,W ∈kG−modpr defined by

(V,W )G := rkkHomkG(V,W ).

Similarly we define a pairing on kG−mon by

[M,N ]G := rkkmonkG(M,N),

for M,N ∈ kG−mon. Note that [−,−]G is bilinear with respect to direct sums, butin general [−,−]G is not symmetric, cf. Proposition 1.20 (b). However, Frobeniusreciprocity holds by Proposition 1.7 (a):

[indGH(M), N ]G = [M, resGH(N)]H

for all H ≤ G,M ∈ kH−mon, N ∈ kG−mon. We extend the definition of [M,N ]Gfor arbitrary k, provided that monkG(M,N) is a free k-module.

1.20 Proposition(a) For (K,ψ) ∈Mk(G) and N ∈ kG−mon we have a k-linear isomorphism

monkG(SG(K,ψ), N) ∼= N (K,ψ),

f 7→ f(1⊗kK 1),(s⊗kK α 7→ αsn

)←7 n,

where α ∈ k, s ∈ G. In particular,

[SG(K,ψ), N ]G = rkkN(K,ψ).

(b) Let (K,ψ), (H,ϕ) ∈ Mk(G). The k-module monkG(SG(K,ψ), SG(H,ϕ)) is free

with k-basis fs | s ∈ G/H, (K,ψ) ≤ s(H,ϕ) given by fs(g ⊗kK α) = gs⊗kH α forg ∈ G and α ∈ k. Moreover,

[SG(K,ψ), SG(H,ϕ)]G = #s ∈ K\G/H | (K,ψ) ≤ s(H,ϕ)

= #s ∈ G/H | (K,ψ) ≤ s(H,ϕ).

In particular, [SG(K,ψ), SG(H,ϕ)]G = 0, unless [K,ψ]G ≤ [H,ϕ]G, and

[SG(H,ϕ), SG(H,ϕ)]G = (NG(H,ϕ) : H).

(c) The k-module monkG(M,N) is finitely generated and projective for M,N ∈kG−mon.

Proof (a) By Frobenius reciprocity (Proposition 1.7 (a)) we have an isomor-phism

monkG(SG(K,ψ), N) ∼= monkK(kψ, resGKN).


By Proposition 1.11 (a) =⇒ (c), we have

monkK(kψ, resGKN) = HomkK(kψ, (resGKN)(K,ψ)),

since kψ is homogeneous in degree (K,ψ) with respect to theMK(k)-grading. Butobviously

HomkK

(kψ, (resGKN)(K,ψ)

)= Homk

(kψ, (resGKN)(K,ψ)

) ∼= (resGKN)(K,ψ)

and the latter k-module equals N (K,ψ) by Remark 1.9 (g). The composition of allthese identifications yields exactly the map described in the proposition.

(b) The isomorphism in part (a) applied to N = SG(H,ϕ) yields an isomorphism

monkG(SG(K,ψ), SG(H,ϕ))

∼=(SG(H,ϕ)

)(K,ψ).

The latter submodule of SG(H,ϕ) is the direct sum of those components s ⊗kH kϕ,

s ∈ G/H, with (K,ψ) ≤ s(H,ϕ). The k-basis s⊗kH1 | s ∈ G/H, (K,ψ) ≤ s(H,ϕ)of the right hand side of the above isomorphism then corresponds exactly to the setfs, described in the proposition.

Note that whenever s ∈ G satisfies (K,ψ) ≤ s(H,ϕ), then the coset sH =( sH)s = K( sH)s = KsH is equal to its double coset. This proves the equation ofthe cardinalities, and everything else is obvious.

(c) Since M and N are the direct sums of their [H,ϕ]G-homogeneous compo-nents, [H,ϕ]G ∈ G\Mk(G), we may assume that M and N are homogeneous withrespect to the G\Mk(G)-grading. Then, by Proposition 1.17 (c), we may assumethat M = SG(K,ψ) and N = SG(H,ϕ) for some (K,ψ), (H,ϕ) ∈ Mk(G). In this case

the assertion follows from part (b).

For (K,ψ), (H,ϕ) ∈ Mk(G) we define γG

(K,ψ),(H,ϕ):= [SG(K,ψ), S

G(H,ϕ)]G. This

definition coincides with the definition in [Bo90, 1.16]. Proposition 1.17 (a) shows

that γG

(K,ψ),(H,ϕ)depends only on the G-orbits of (K,ψ) and (H,ϕ), and therefore

we will also write γG

[K,ψ]G,[H,ϕ]G.

In addition to the characterizations of isomorphic objects of kG−mon in Corol-lary 1.18 (c) we have the following proposition:

1.21 Proposition Let k satisfy condition (∗). For M,N ∈ kG−mon thefollowing statements are equivalent:

(a) M ∼= N as kG-monomial modules.

(b) [L,M ]G = [L,N ]G for all L ∈ kG−mon.

(c) [M,L]G = [N,L]G for all L ∈ kG−mon.

Proof Obviously (a) implies (b) and (c) by the definition of [−,−]G.(b) =⇒ (a): Let (H,ϕ) ∈ Mk(G). Setting L := SG(H,ϕ) we obtain from the

hypothesis in (b) and from Proposition 1.20 (a) that rkkM(H,ϕ) = rkkN

(H,ϕ). Thisimplies M ∼= N by Corollary 1.18 (c).


(c) =⇒ (a): From Corollary 1.18 (b) we know

M ∼=⊕

(H,ϕ)∈Rk(G)

r[H,ϕ]G(M)SG(H,ϕ) and N ∼=⊕

(H,ϕ)∈Rk(G)

r[H,ϕ]G(N)SG(H,ϕ).

Let (U, µ) ∈ Mk(G), then setting L = SG(U,µ) we obtain from the hypothesis in (c)

and from Proposition 1.20 (b) that∑[H,ϕ]G∈G\Mk(G)

r[H,ϕ]G(M)γG

[H,ϕ]G,[U,µ]G=

∑[H,ϕ]G∈G\Mk(G)

r[H,ϕ]G(N)γG

[H,ϕ]G,[U,µ]G.

Using the last part of Proposition 1.20 (b) we can show by induction on |U | thatr[U,µ]G(M) = r[U,µ]G(N) for all [U, µ]G ∈ G\Mk(G). This implies M ∼= N byCorollary 1.18 (c).

1.22 Let Rabk+(G) be the Grothendieck group of the category kG−mon, i.e.

the free abelian group over the isomorphism classes M of kG-monomial modulesM , modulo the subgroup generated by the elements M+N−M ⊕N, whereM,N ∈ kG−mon. The class of M in the Grothendieck group will be denoted by[M ].

We will determine Rabk+(G) in the case where k satisfies condition (∗), and we

will keep this assumption for the rest of this section. If k = C we just write Rab+ (G)

instead of Rabk+(G). The following description of Rab

k+(G) justifies the coincidence ofnotation with Section III.1.

Since each M ∈ kG−mon is isomorphic to a direct sum of some SG(H,ϕ), (H,ϕ) ∈Rk(G), their images [SG(H,ϕ)], (H,ϕ) ∈ Rk(G), generate the Grothendieck group.Moreover, since kG−mon has the cancellation property, i.e. M ⊕ L ∼= N ⊕ L =⇒M ∼= N for L,M,N ∈ kG−mon (cf. Corollary 1.18 (b) and (c) (i) ⇐⇒ (ii)), theelements [SG(H,ϕ)], (H,ϕ) ∈ Rk(G), form a Z-basis of the Grothendieck group.

We will abbreviate the basis elements [SG(H,ϕ)] of Rabk+(G) by [H,ϕ]G for (H,ϕ) ∈

Mk(G). If M ∈ kG−mon, then the associated element [M ] ∈ Rabk+(G) is given by

[M ] =∑

[H,ϕ]G∈G\Mk(G)

r[H,ϕ]G(M) · [H,ϕ]G,

where r[H,ϕ]G(M) is defined as in Corollary 1.18 (b). If M = M1 ⊕ . . .⊕Mm is thedecomposition of M , we may describe [M ] also by

[M ] =∑

Mi∈M1,...,Mm/G

rkkMi · [Hi, ϕi]G,

where (Hi, ϕi) denotes the stabilizing pair of Mi. Corollary 1.18 (c) now impliesthat we have for arbitrary M,N ∈ kG−mon:

M ∼= N ⇐⇒ [M ] = [N ] ∈ Rabk+(G).

1.23 Remark Subsection 1.22 gives now a very concrete meaning to thegroups Rab

+ (G), RabF+(G), and Lab

O+(G) of Sections III.1, III.2, and III.4. They


are just the Grothendieck groups of CG−mon, FG−mon and OG−mon. If wedefine Rab

k (H) := ZH(k), for H ≤ G, then Rabk is a k-algebra restriction functor

on G with the usual conjugation and restriction maps. The discussion in 1.22shows that for H ≤ G, the construction Rab

k+(H) of I.2.2 and the definition of

Rabk+(H) as Grothendieck group yield isomorphic groups. Hence we may denote

both groups by the same symbol Rabk+(H). Note that Rab

k+ in the notation of I.2.2 isa Z-Green functor on G. In the sequel we will show that its structure maps comefrom the constructions on the category level. The constructions in Remark 1.6 forexample provide Rab

k+(G) with more structure than just a free abelian group. Inthe next subsections we will give translations of these constructions to structuremaps on Rab

k+(G). We will outline this translation in detail for the ring structure

on Rabk+(G) and leave the proofs for the other translations as an easy exercise to

the reader. Note that the verification of some properties of these structures, as forexample associativity and commutatity of multiplication, requires nasty calculationswith double coset representatives (cf. [Bo89, Sect. III.2]). However, if we interpretRabk+(G) as the Grothendieck group of kG−mon, this is all obvious, since tensor

products are commutative and associative. The same is true for restrictions alonggroup homomorphism.

1.24 Multiplication. The tensor product on kG−mon defines a multiplicationon Rab

k+(G). This provides Rabk+(G) with the structure of a commutative ring with

unity [G, 1]G, since the tensor product in kG−mon is commutative, associative andsince taking tensor product with SG(G,1) is up to a functorial isomorphism the identity

functor on kG−mon. In order to give an explicit multiplication formula in Rabk+(G)

we have to decompose SG(H,ϕ) ⊗ SG(K,ψ) as a sum of SG(U,µ), (U, µ) ∈ G\Mk(G):

SG(H,ϕ) ⊗ SG(K,ψ) = indGK(kψ)⊗ indGH(kϕ)

1.7(b)∼= indGK(kψ ⊗ resGK(indGH(kϕ))

)1.7(c)∼=

⊕s∈K\G/H

indGK(kψ ⊗ indK

K∩ sH(ressHK∩ sH(k sϕ))

)1.7(b)∼=

⊕s∈K\G/H

(indGK indKK∩ sH)

(resK

K∩ sH(kψ)⊗ ressHK∩ sH(k sϕ)

)∼=

⊕s∈K\G/H

SG(K∩ sH ,ψ|

K∩sH· sϕ|

K∩sH

).

Hence we obtain the following explicit multiplication rule in Rabk+(G):

[K,ψ]G · [H,ϕ]G =∑

s∈K\G/H

[K ∩ sH,ψ · sϕ]G,

where we consider ψ and sϕ as restricted to K ∩ sH. This coincides with themultiplication defined in I.2.2.

1.25 Restriction. Let f : G′ → G be a homomorphism of finite groups. Thefunctor resf : kG−mon→ kG′−mon which commutes with direct sums and tensor


products induces a unitary ring homomorphism

res+f : Rabk+(G)→ Rab

k+(G′), [H,ϕ]G 7→∑

s∈f(G′)\G/H

[f−1( sH), sϕ f

]G′.

For an inclusion K ≤ G of a subgroup we also write

res+GK : Rab

k+(G)→ Rabk+(K), [H,ϕ]G 7→

∑s∈K\G/H

[K ∩ sH, sϕ|K∩

sH

]G.

This coincides with the definition of res+ in I.2.2. Note that, if f ′ : G′′ → G′ isanother group homomorphism, then by the transitivity of the functors we have

res+f ′ res+f = res+ff ′ : Rabk+(G)→ Rab

k+(G′′).

1.26 Induction. Let K ≤ G, then the functor indGK : kK−mon → kG−moninduces a group homomorphism

ind+GK : Rab

k+(K)→ Rabk+(G), [H,ϕ]K 7→ [H,ϕ]G.

This coincides with the definition of ind+ in I.2.2. Note that induction is transitivewith respect to subgroup towers, since this is the case for the underlying functors.The canonical isomorphism in Proposition 1.7 (c) implies a Mackey formula forRabk+(G): Let U,H ≤ G and let x ∈ Rab

k+(H), then

res+GU (ind+

GH(x) =

∑U\G/H

ind+UU∩ sH

(res+

sHU∩ sH( sx)

),

where sx denotes the image of x under res+f for f : sH → H, shs−1 7→ h, h ∈ H.This defines for each s ∈ G and each subgroup H ≤ G a conjugation map

s− : Rabk+(H)→ Rab

k+( sH), [K,ψ]H 7→ [ sK, sψ] sH .

This coincides with the definition of c+ in I.2.2. Note that these conjugation mapsare transitive with respect to multiplication in G, and that they commute withrestriction and induction. Moreover, if s ∈ H, then conjugation with s on Rab

k+(H)

is the identity. This shows that Rabk+ is a Mackey functor for G.

The canonical isomorphism in Proposition 1.7 (b) implies for H ≤ G, x ∈Rabk+(G), y ∈ Rab

k+(H):

x · ind+GH(y) = ind+

GH

((res+

GH(x)) · y

).

This shows again that Rabk + is a Green functor.

1.27 Bilinear form. The bilinear form [−,−]G on kG−mon induces a bilinearform, which we still denote by [−,−]G, on Rab

k+(G) with values in Z. For the basis

elements of Rabk+(G) we have (cf. Proposition 1.20 (b))[

[K,ψ]G, [H,ϕ]G]G

= γG

[K,ψ]G,[H,ϕ]G= #s ∈ K\G/H | (K,ψ) ∈ s(H,ϕ).


Note that for k = C this definition of [−,−]G on Rab+ (G) coincides with the one

of Section III.1. Proposition 1.21 implies that [−,−]G is non-degenerate in botharguments, i.e. for all x ∈ Rab

k+(G) we have

[x, y]G = 0 for all y ∈ Rabk+(G) ⇐⇒ x = 0

and

[y, x]G = 0 for all y ∈ Rabk+(G) ⇐⇒ x = 0.

Moreover Proposition 1.7 (a) implies that res+GH is the right adjoint of ind+

GH with

respect to [−,−] for H ≤ G:

[x, res+GH(y)]H = [ind+

GH(x), y]G

for all x ∈ Rabk+(H), y ∈ Rab

k+(G).

1.28 Abelianization. The image of kG−monab in Rabk+(G) is given by the

elements∑

ϕ∈G(k) αϕ[G,ϕ]G with αϕ ∈ N0. The Grothendieck group Rabk (G) of

kG−monab can be considered as the subgroup of Rabk+(G) generated by the elements

[G,ϕ]G, ϕ ∈ G(k). This is a subring of Rabk+(G) which is isomorphic to the group

ring ZG(k), when we identify [G,ϕ]G ∈ Rabk+(G) with ϕ ∈ ZG(k). In this way we can

consider Rabk+(G) as a ZG(k)-algebra. Now the functor P : kG−mon→ kG−monab

induces the ring homomorphism

πG : Rabk+(G)→ ZG(k), (H,ϕ) 7→

ϕ, if H = G,

0, if H 6= G,

which coincides with the Brauer morphism πAG : A+(G)→ A(G) of Section I.3 withA(H) := Rab

k (H) for H ≤ G.

1.29 The forgetful functor. The forgetful functor V : kG−mon→ kG−modpr

induces the map

bG : Rabk+(G)→ Rk(G), [H,ϕ]G 7→ [indGH(kϕ)],

where Rk(G) is the Grothendieck ring of the category kG−modpr, and the brack-ets denote the image of an object in the Grothendieck group. If k is a field ofcharacteristic 0, then Rk(G) is the k-character ring (of generalized characters of k-representations) of G and [indGH(kϕ)] corresponds to the character indGH(ϕ). Since Vcommutes with⊕,⊗, resf and indGH , the map bG is a ring homomorphism commutingwith restriction and induction. Hence, the collection (bH)H≤G forms a morphismof Green functors. Moreover, bG is a ZG(k)-algebra homomorphism, if we viewRk(G) as a ZG(k)-algebra by the obvious identification of ZG(k) = Rab

k (G) (theGrothendieck ring of kG−modpr

ab) as subring of Rk(G) (the Grothendieck ring ofkG−modpr), cf. Corollary 1.12 (b). Note that (bH)

H≤G coincides with the induction

morphism bM,A of Definition II.1.1 with A(H) := Rabk (H) and M(H) := Rk(H) for

H ≤ G.


1.30 Remark Assume that k = C. Then the Grothendieck ring of CG−modis given by the ordinary character ring R(G). The Grothendieck ring of CG−modab

is the subring of R(G) generated by ϕ ∈ Irr(G) | ϕ ∈ G, and is isomorphic tothe group ring ZG, which we have already considered as the Grothendieck ring ofCG−monab. Therefore we denote the Grothendieck ring of CG−monab also byRab(G).

The abelianization functor P : CG−mod → CG−modab (which can also beconsidered as the functor of taking G′-fixed points of a CG-module, where G′ is thecommutator subgroup of G) induces the map

pG : R(G)→ ZG, Irr(G) 3 χ 7→

χ, if χ ∈ G,

0, if χ 6∈ G,

which we have already encountered in Example III.1.1. Note that pG in contrast toπG is not multiplicative.

Using the abelianization maps on CG−mon and CG−mod together with thefact that their abelian subcategories are equivalent, gives us a possibility to form alink between Rab

+ (G) and R(G) via Rab(G). More precisely, we can embed both ofthese two Grothendieck groups into the group

∏H≤GR

ab(H) by defining

rG = (pHresGH)H≤G : R(G)→

∏H≤G

Rab(H), χ 7→(pH(resGH(χ))

)H≤G

and

ρG = (πHres+GH)

H≤G : Rab+ (G)→

∏H≤G

Rab(H), x 7→(πH(res+

GH(x))

)H≤G

.

Note that ρG which equals the mark homomorphism of Section I.3 is a ring homo-morphism, since πH and res+

GH are ring homomorphisms for all H ≤ G. However

rG is not a ring homomorphism in general. Clearly rG is injective, since a characteris uniquely determined by restriction to all cyclic subgroups, and since pH is theidentity for H cyclic. Also ρG is injective as we have seen in Proposition I.3.2. Thiscan also be seen by the fact that ρG is induced from the functor

kG−mon→∏H≤G

kH−monab,M 7→(P(resGH(M))

)H≤G

.

However, P(resGH(M)) =⊕

ϕ∈HM(H,ϕ) by Remark 1.9 (g), and the isomorphism

type of M is uniquely determined by the numbers dimCM(H,ϕ), cf. part (vi) of

Corollary 1.18 (c). The canonical induction formula aG : R(G)→ Rab+ (G) of Exam-

ple III.1.1 makes the following diagram commutative (cf. Diagram (2.2)):


+ (G)

rG

y yρG∏H≤G ZH

∏H≤G ZH.

Now the fact that aG is integral is equaivalent to the fact, that the image of rG iscontained in the image of ρG.

5.2. THE MONOMIAL RESOLUTION 161

We have defined bG as induced from the forgetful functor V : CG−mon →CG−mod. Of course aG can not be induced by a functor CG−mod→ CG−mon,since in this case any CG-module would be isomorphic to an image under V, andhence monomial. Another reason for the non-existence of such a functor is, thataG(χ) may have negative coefficients (as examples show). Hence, the only way oflifting aG to a functor is to associate to each CG-module V a ‘virtual’ CG-monomialmodule M , with the property

dimCM(H,ϕ) = dimC V

(H,ϕ) for all (H,ϕ) ∈M(G),

In fact, the equation ρG aG = rG implies the above equation, since

pHresGH([V ]) =∑ϕ∈H

dimC V(H,ϕ)ϕ ∈ ZH for V ∈ CG−mod

and

πHres+GH([M ]) =

∑ϕ∈H

dimCM(H,ϕ)ϕ ∈ ZH for M ∈ CG−mon.

In the next section we will enlarge the category kG−mon to the homotopy categoryof chain complexes of kG-monomial modules in order to give the negative coefficientsin aG(χ) a meaning in terms of ‘real’ objects by considering Lefschetz invariants.

5.2 The monomial resolution

In this section we define the notion of a monomial resolution (Definition 2.1), provethe existence and uniqueness of monomial resolutions (Theorem 2.4), and showthat over the complex numbers there exist always finite monomial resolutions (The-orem 2.14). Moreover, we show in Corollary 2.16 that the Lefschetz element of afinite monomial resolution of V ∈ CG−mod equals the canonical induction formulaof the character of V , as defined in Example III.3.1.

We keep the notation of the previous section.

2.1 Definition Let V ∈ kG−mod. A kG-monomial resolution of V is a (notnecessarily finite) chain complex

M∗ = . . .∂2−−→ M2

∂1−−→ M1∂0−−→ M0

with Mi ∈ kG−mon and ∂i ∈ monkG(Mi+1,Mi) for all i ≥ 0, together with akG-linear map ε : M0 → V such that

. . .∂2−−→ M

(H,ϕ)2

∂1−−→ M(H,ϕ)1

∂0−−→ M(H,ϕ)0

ε−−→ V (H,ϕ) −→ 0 (5.1)

is an exact sequence of k-modules for all (H,ϕ) ∈ Mk(G). Note that the maps∂i in (5.1) are well defined by Proposition 1.11 (b) and also that ε in (5.1) is welldefined by Remark 1.9 (h). Note furthermore that (5.1) is an exact sequence ofkNG(H,ϕ)-modules. In particular we obtain for (H,ϕ) = (E, 1) an exact sequence

. . .∂2−−→ M2

∂1−−→ M1∂0−−→ M0

ε−−→ V −→ 0


of kG-modules which justifies the terminology.

2.2 Lemma LetPyf

Mg−−→ N

be a diagram, where P,M ∈ kG−mon, N ∈ kG−mod and f, g ∈ kG−mod.Assume furthermore that

f(P (H,ϕ)) ⊆ g(M (H,ϕ)) for all (H,ϕ) ∈Mk(G).

Then there exists h ∈ monkG(P,M) such that g h = f .

Proof By Corollary 1.18 (d) we may assume that P = indGHL(H,ϕ) for some(H,ϕ) ∈ Mk(G), where L(H,ϕ) is a ϕ-homogeneous kH-monomial module, andhence also a ϕ-homogenous kH-module. We consider the following diagram of kH-monomial modules and kH-modules:

1⊗kH L(H,ϕ) ⊆ P (H,ϕ)yfM (H,ϕ) g−−→ g(M (H,ϕ)) ⊆ N (H,ϕ),

where f is well defined by the hypothesis. Since 1 ⊗kH L(H,ϕ)∼= L(H,ϕ) is k-

projective, we obtain a k-linear map h : 1⊗kH L(H,ϕ) →M (H,ϕ) such that g h = fon 1 ⊗kH L(H,ϕ). We consider h as a kH-linear map L(H,ϕ) → M . Since L(H,ϕ) is

ϕ-homogeneous, we have h ∈ monkH(L(H,ϕ), resGH(M)) by Proposition 1.11 (c). By

Proposition 1.7 (a), h extends to a kG-monomial homomorphism h : indGHL(H,ϕ) →M . Since g h and f coincide on L(H,ϕ), they also coincide on indGHL(H,ϕ) bykG-linearity.

2.3 Proposition Let V,W ∈ kG−mod and f ∈ HomkG(V,W ). Assume that

M∗ε−−→ V and N∗

δ−−→ W

are kG-monomial resolutions of V and W . Then there exists a kG-monomial chainmap f∗ = (fi)i≥0 : M∗ → N∗ such that

M∗ε−−−−→ V

f∗

y f

yN∗

δ−−−−→ W

commutes. Moreover, f∗ is uniquely determined up to kG-monomial homotopyby the commutativity of the above diagram; i.e. if g∗ : M∗ → N∗ is another kG-monomial chain map which renders the above diagram commutative, then thereexist hi ∈ monkG(Mi, Ni+1), i ≥ 0, such that f0 − g0 = ∂0 h0 and fi − gi =hi−1 ∂i−1 + ∂i hi for i ≥ 1.


Proof First we prove the existence of (fi)i≥0 by induction on i. We considerthe following diagram

M0ε−−→ Vyf

N0δ−−→ W .

Since f(ε(M

(H,ϕ)0 )

)⊆ W (H,ϕ) for all (H,ϕ) ∈ Mk(G), and since δ : N

(H,ϕ)0 →

W (H,ϕ) is surjective, we can apply Lemma 2.2 and obtain f0 : M0 → N0 in kG−monsuch that δ f0 = f ε. Now we assume that f0, . . . , fn, n ≥ 0, are already definedin such a way that δ f0 = f ε, ∂i−1 fi = fi−1 ∂i−1 for 1 ≤ i ≤ n. Then weconsider the diagram

Mn+1∂n−−→ Mnyfn

Nn+1∂n−−→ Nn,

and observe that by Lemma 2.2 we obtain a morphism fn+1 ∈ monkG(Mn+1, Nn+1)

which makes the above diagram commutative, since (fn∂n)(M(H,ϕ)n+1 ) ⊆ ∂n(N

(H,ϕ)n+1 )

for all (H,ϕ) ∈ Mk(G). In fact, ∂n(N(H,ϕ)n+1 ) = ker

(∂n−1 : N

(H,ϕ)n → N

(H,ϕ)n−1

)and

∂n−1 fn ∂n = fn−1 ∂n−1 ∂n = 0. (If n = 0, we have to set ∂−1 := ε andN−1 := W .)

Now suppose that we have another chain map g∗ : M∗ → N∗ such that δ g0 =f ε. Then we define inductively kG-monomial homomorphisms hi : Mi → Ni−1,i ≥ 0, such that

fi − gi = ∂i hi + hi−1 ∂i−1, i ≥ 0. (5.2)

Here we set ∂−1 = 0 and h−1 = 0. For i = 0 consider the following diagram

M0yf0−g0

N1∂0−−→ N0.

We may apply Lemma 2.2, since (f0 − g0)(M(H,ϕ)0 ) ⊆ ∂0(N

(H,ϕ)1 ) for all (H,ϕ) ∈

Mk(G). In fact, ∂0(N(H,ϕ)1 ) = ker

(δ : N

(H,ϕ)0 → W (H,ϕ)

)and δ (f0 − g0) =

f ε− f ε = 0. Hence, there is h0 ∈ monkG(M0, N1) such that f0 − g0 = ∂0 h0.Now let hi ∈ monkG(Mi, Ni+1) be already constructed for 0 ≤ i ≤ n, such that(5.2) holds for all 0 ≤ i ≤ n. Then we consider the diagram

Mn+1∂n−−→ Mn

fn+1−gn+1

y hn

Nn+2∂n+1−−→ Nn+1

∂n−−→ Nn.

Here again we may apply Lemma 2.2 for the maps fn+1− gn+1− hn ∂n and ∂n+1.In fact, we have

∂n (fn+1 − gn+1 − hn ∂n) = (fn − gn) ∂n − (∂n hn) ∂n= (fn − gn) ∂n(fn − gn − hn−1 ∂n−1) ∂n= 0,


and hence

(fn+1 − gn+1 − hn ∂n)(M

(H,ϕ)n+1

)⊆ ker

(∂n : N

(H,ϕ)n+1 → N (H,ϕ)

n

)= ∂n+1(N

(H,ϕ)n+2 ),

for all (H,ϕ) ∈ Mk(G). Lemma 2.2 now implies the existence of a morphismhn+1 ∈ monkG(Mn+1, Nn+2) such that fn+1 − gn+1 = hn ∂n + ∂n+1 hn+1.

2.4 Theorem Each V ∈ kG−mod has a kG-monomial resolution, and anytwo kG-monomial resolutions M∗

ε−−→V and N∗δ−−→V of V are homotopy equiv-

alent as chain complexes in kG−mon, i.e. chain maps and homotopies are kG-monomial homomorphisms.

Proof First we prove the existence of a kG-monomial resolution M∗ε−−→V of

V . We define M0 by defining its homogeneous components M0

([H,ϕ]G

)in degree

[H,ϕ]G ∈ G\Mk(G) for all (H,ϕ) ∈ Rk(G) according to Remark 1.9 (f) (or seealso 1.10):

Let ε(H,ϕ) : P(H,ϕ)0 → V (H,ϕ) be k-linear and surjective with P

(H,ϕ)0 ∈ k−modpr

(here we use the assumption that k is noetherian). We can provide P(H,ϕ)0 with the

structure of a ϕ-homogeneous kH-module, and ε(H,ϕ) is automatically kH-linear.Now set

M0

([H,ϕ]G

):= indGH(P

(H,ϕ)0 ),

where we consider P(H,ϕ)0 ∈ kH−monab by Corollary 1.12 (b). We define ε : M0 →

V by defining its restrictions to the homogeneous components:

ε : M0

([H,ϕ]G

)→ V, g ⊗kH p 7→ gε(H,ϕ)(p),

for g ∈ G, p ∈ P(H,ϕ)0 , (H,ϕ) ∈ Rk(G). The map ε is obviously kG-linear and

induces surjective maps ε : M(H,ϕ)0 → V (H,ϕ) for all (H,ϕ) ∈ Rk(G) (we may choose

g = 1 in the above definition). Since ε is kG-linear, surjectivity holds also for all(H,ϕ) ∈Mk(G).

Now assume that we have already defined

Mn∂n−1−−→ Mn−1

∂n−2−−→ . . .∂0−−→ M0

ε−−→ V

such that

M (H,ϕ)n

∂n−1−−→ M(H,ϕ)n−1

∂n−2−−→ . . .∂0−−→ M

(H,ϕ)0

ε−−→ V (H,ϕ) −−→ 0

is exact for all (H,ϕ) ∈Mk(G). Then let Mn+1 ∈ kG−mon be defined by the fol-lowing construction for its homogeneous components in degree [H,ϕ]G ∈ G\Mk(G),

(H,ϕ) ∈ Rk(G). Let Ω(H,ϕ)n be the kernel of the kH-linear map ∂n−1 : M

(H,ϕ)n →

M(H,ϕ)n−1 (if n=0, ∂−1 is interpreted as ε), then Ω

(H,ϕ)n is a ϕ-homogeneous kH-module.

Let ∂(H,ϕ)n : P

(H,ϕ)n+1 → Ω

(H,ϕ)n be a surjective kH-linear map with a ϕ-homogeneous

module P(H,ϕ)n+1 ∈ kH−modpr (as for P

(H,ϕ)0 it is enough to choose a surjective k-

linear map from a finitely generated k-projective module to Ω(H,ϕ)n ). We consider

P(H,ϕ)n+1 ∈ kH−monab by Corollary 1.12 (b) and define

Mn+1

([H,ϕ]G

):= indGH(P

(H,ϕ)n+1 ).


Since P(H,ϕ)n+1 is ϕ-homogeneous, Mn+1

([H,ϕ]G

)is homogeneous in degree [H,ϕ]G ∈

G\Mk(G).Next we define for all (H,ϕ) ∈ Rk(G)

∂n : Mn+1

([H,ϕ]G

)→Mn, g ⊗kH p 7→ g∂(H,ϕ)

n (p),

for g ∈ G and p ∈ P (H,ϕ)n+1 . This defines a kG-linear map ∂n : Mn+1 →Mn. First note

that ∂n−1 ∂n = 0, since for g ∈ G and p ∈ P (H,ϕ)n+1 we have (∂n−1)∂n(g ⊗kH p) =

g(∂n−1∂n)(1⊗kH p) = g∂n−1

(∂

(H,ϕ)n (p)

)= 0, since ∂

(H,ϕ)n (p) ∈ ker ∂n−1. Next note

that ∂n satisfies the condition in Proposition 1.11 (c). In fact, if (H,ϕ) ∈ Rk(G),then

Mn+1(H,ϕ) =⊕

s∈NG(H,ϕ)/H

s⊗kH P(H,ϕ)n+1

and∂n(s⊗kH P

(H,ϕ)n+1 ) = s · Ω(H,ϕ)

n ⊆ sM (H,ϕ)n = M

s(H,ϕ)n = M (H,ϕ)

for s ∈ NG(H,ϕ). And if (H,ϕ) ∈ Mk(G) is arbitrary, then s(H,ϕ) ∈ Rk(G) forsome s ∈ G, and we have

∂n(Mn+1(H,ϕ)

)= s−1∂n

(Mn+1( s(H,ϕ))

)⊆ s−1M

s(H,ϕ)n = M (H,ϕ).

Hence ∂n is a kG-monomial homomorphism by Prop. 1.11. Finally,

M(H,ϕ)n+1

∂n−−→ M (H,ϕ)n

∂n−1−−→ . . .∂0−−→ M

(H,ϕ)0

ε−−→ V (H,ϕ) → 0

is exact for all (H,ϕ) ∈ Mk(G). In fact, we only have to show exactness at degreen. For (H,ϕ) ∈ Rk(G) we have

ker(M (H,ϕ)n

∂n−1−−→M (H,ϕ)n−1

)= Ω(H,ϕ)

n = ∂n(1⊗kH P(H,ϕ)n+1 ) ⊆ ∂n(M

(H,ϕ)n+1 ).

For arbitrary (H,ϕ) ∈ Mk(G) there is some s ∈ G such that s(H,ϕ) ∈ Rk(G).Then we have by kG-linearity and Remark 1.9 (d)

ker(M (H,ϕ)n

∂n−1−−→M (H,ϕ)n−1

)= s−1 ker

(M

s(H,ϕ)n

∂n−1−−→Ms(H,ϕ)n−1

)⊆ s−1∂n(M

s(H,ϕ)n+1 ) = ∂n(M

(H,ϕ)n+1 ).

Since ∂n−1 ∂n = 0, this completes the proof of existence.Now we prove the uniqueness part of the assertion. Let M∗

ε−−→V and N∗δ−−→V

be two kG-monomial resolutions of V , then by Proposition 2.3 we have chain mapsf∗ = (fi)i≥0 and g∗ = (gi)i≥0 in kG−mon from M∗ to N∗ such that they extend theidentity on V . By the uniqueness part of Proposition 2.3 we then obtain g∗f∗ ∼ 1M∗and f∗g∗ ∼ 1N∗ , where ‘∼’ means homotopic as chain maps in kG−mon.

2.5 Remark Proposition 2.3 and Theorem 2.4 can be interpreted as follows:The construction of kG-monomial resolutions provides a functor from kG−mod tothe homotopy category of chain complexes in kG−mon, i.e. the category of chaincomplexes in kG−mon with homotopy classes of kG-monomial chain maps as mor-phisms. This functor is a full embedding, since — in the situation of Proposition 2.3


— mapping f to the homotopy class of f∗ is an isomorphism between HomkG(V,W )and the k-module of homotopy classes of kG-monomial chain maps from M∗ to N∗.In fact, the inverse map is given by associating to a representative f∗ : M∗ → N∗ ofa homotopy class the unique map f : V →W such that

M∗ε−−−−→ V

f∗

y yfN∗

δ−−−−→ W

commutes. This argument also shows that the full embedding of kG−mod into thehomotopy category of kG−mon has the functor H0V (taking homology in degree0) as a left inverse.

2.6 Lemma Let f : G′ → G be a group homomorphism, V ∈ kG−modpr andM ∈ kG−mon. Moreover let (H ′, ϕ′) ∈Mk(G

′).

(a) If ker f ∩H ′ = ker(f |H′ ) 6≤ kerϕ′, then

(resfV )(H′,ϕ′) = 0 and (resfM)(H′,ϕ′) = 0.

(b) If ker f∩H ′ = ker(f |H′ ) ≤ kerϕ′, then there is a unique ϕ ∈ Hom(f(H ′), k×)

with ϕ f = ϕ′, and we have

(resfV )(H′,ϕ′) = V (f(H′),ϕ) and (resfM)(H′,ϕ′) = M (f(H′),ϕ).

Proof (a) Let h′ ∈ H ′ with f(h′) = 1 and ϕ′(h′) 6= 1. Then we have forall v ∈ V (resp. v ∈ M): h′v = f(h′)v = 1v = v. Hence, there is no 0 6= v ∈ V(resp. no 0 6= v ∈ M) with h′v = ϕ′(h′)v, since V and M are k-torsion free. This

implies that (resfV )(H′,ϕ′) = 0 and(V(resfM)

)(H′,ϕ′)= 0. But (resfM)(H′,ϕ′) ⊆(

V(resfM))(H′,ϕ′)

.

(b) The existence and uniqueness of ϕ ∈ Hom(f(H ′), k×) is clear. For v ∈ V wehave

v ∈ (resfV )(H′,ϕ′) ⇐⇒ h′v = ϕ′(h′)v for all h′ ∈ H ′

⇐⇒ f(h′)v = ϕ(f(h′))v for all h′ ∈ H ′

⇐⇒ hv = ϕ(h)v for all h ∈ f(H ′)

⇐⇒ v ∈ V (f(H′),ϕ).

And if M = M1 ⊕ . . .⊕Mm is the decomposition of M , then we have for each Mi,1 ≤ i ≤ m:

Mi ⊆ (resfM)(H′,ϕ′) ⇐⇒ h′v = ϕ′(h′)v for all h′ ∈ H ′, v ∈Mi

⇐⇒ f(h′)v = ϕ(f(h′))v for all h′ ∈ H ′, v ∈Mi

⇐⇒ hv = ϕ(h)v for all h ∈ f(H ′), v ∈Mi

⇐⇒ Mi ⊆M (f(H′),ϕ).


2.7 Proposition Let V ∈ kG−modpr and let f : G′ → G be a group homo-morphism. If M∗

ε→V is a kG-monomial resolution of V , then resf (M∗)ε→ resf (V )

is a kG′-monomial resolution of resf (V ).

Proof This is obvious with Lemma 2.6.

In the above proposition the hypothesis V ∈ kG−modpr cannot be weakenedto V ∈ kG−mod: Assume that G′ is non-trivial and f : G′ → G is the trivial map.Let 1 6= ϕ′ ∈ G′(k) such that 1− ϕ′(g′) | g′ ∈ G′ generates a proper ideal I in k.Set V := k/I with the trivial G-action. Then ker f ∩G′ = G′ 6≤ kerϕ′ and therefore(resfM∗)

(G′,ϕ′) = 0 by Lemma 2.6. But (resfV )(G′,ϕ′) = resfV = V .

2.8 Definition For a finite chain complex

M∗ = . . . 0 −→ Mn −→ Mn−1 −→ . . . −→ Mm −→ 0 · · ·

in kG−mon we define the Lefschetz invariant Λ(M∗) ∈ Rabk+(G) of M∗ by

Λ(M∗) :=

n∑i=m

(−1)i[Mi],

where [Mi] ∈ Rabk+(G) was defined in 1.22.

2.9 Lemma Let k satisfy condition (∗) and let

M∗ = . . . 0 −→ Mm −→ Mm−1 −→ . . . −→ Mr −→ 0 . . . ,

N∗ = . . . 0 −→ Nn −→ Nn−1 −→ . . . −→ Ns −→ 0 . . .

be finite homomtopy equivalent chain complexes in kG−mon. Then we have

Λ(M∗) = Λ(N∗).

Proof By the injectivity of ρRabk

G (cf. Proposition I.3.2), we only have to showthat

πH(res+

GH(Λ(M∗))

)= πH

(res+

GH(Λ(N∗))

)∈ ZH(k)

for all H ≤ G. If we define integers c(H,ϕ), d(H,ϕ), for (H,ϕ) ∈Mk(G), by setting

πH(res+

GH(Λ(M∗))

)=

∑ϕ∈H(k)

c(H,ϕ)ϕ and πH(res+

GH(Λ(N∗))

)=

∑ϕ∈H(k)

d(H,ϕ)ϕ,

it suffices to show c(H,ϕ) = d(H,ϕ) for all (H,ϕ) ∈Mk(G). However, by the definitionof Λ(M∗) we have

c(H,ϕ) =∑i∈Z

(−1)irkk(resGH(Mi))(H,ϕ)


and

d(H,ϕ) =∑i∈Z

(−1)irkk(resGH(Ni))(H,ϕ).

Now we know from Remark 1.9 (g) that (resGH(Mi))(H,ϕ) = M(H,ϕ)i and similarly

for Ni, so that it suffices to show∑i∈Z

(−1)irkkM(H,ϕ)i =

∑i∈Z

(−1)irkkN(H,ϕ)i

for all (H,ϕ) ∈ Mk(G). But this last equation holds, since with M∗ and N∗ also

the two chain complexes M(H,ϕ)∗ and N

(H,ϕ)∗ of k-modules are homotopy equiva-

lent. In fact, the original chain maps and homotopies for the homotopy equivalenceof M∗ and N∗ induce via the functor −(H,ϕ) again chain maps and homotopies(cf. Proposition 1.11 (b)).

2.10 Remark Let k be a field of characteristic p > 0 and let G be cyclicof order p. Then Mk(G) = (E, 1), (G, 1). Let Vi, 1 ≤ i ≤ p, denote the inde-composable kG-module of dimension i. Assume that there is a finite kG-monomialresolution

0 −→ Mn −→ . . . −→ M0 −→ V −→ 0.

Then∑n

i=0(−1)i dimkM(H,ϕ)i = dimk V

(H,ϕ)i for all (H,ϕ) ∈ Mk(G). Hence, by

Corollary 1.18 (b), the vector (1i

)=

(dimk V

(G,1)i

dimk V(E,1)i

)

is an integral linear combination of the vectors(11

)=

(dimk(S

G(G,1))

(G,1)

dimk(SG(G,1))

(E,1)

)and

(0p

)=

(dimk(S

G(E,1))

(G,1)

dimk(SG(E,1))

(E,1).

)

This implies that (i− 1)/p is an integer, which is only possible for i = 1.Hence we have seen that V2, . . . , Vp don’t have finite monomial resolutions. V1,

the trivial module has a monomial resolution, namely

0 −→ SG(G,1)ε−−→ V1 → 0,

where ε is the identity, if we identify both V1 and SG(G,1) with the trivial module k.

2.11 Definition Let V ∈ kG−mod. The monomial homological dimension

dimmh(V ) of V is defined to be ∞, if there is no finite kG-monomial resolution ofV , and it is defined to be n ∈ N0, if n is minimal with respect to the existence of akG-monomial resolution 0→Mn →Mn−1 → . . .→M0 → V → 0 of length n.

2.12 Remark If V ∈ kG−modprab, then we may consider M0 := V as a

kG-monomial module by Corollary 1.12 (b), and 0−→M0ε−−→V −→ 0 with ε the


identity becomes a kG-monomial resolution of V . Hence, all k-projective abeliankG-modules have monomial homological dimension zero.

For the rest of this section let k = C, the field of complex numbers. We omitthe index k in most of the notations. In this situation we are able to prove thefiniteness of the monomial homological dimension and to give upper bounds. Inorder to formulate these upper bounds we have to introduce the following notionsfrom [Bo89] or [Bo90]. Moreover we will prove in Corollary 2.16 that the CG-monomial resolution can be considered as a categorical version of the canonicalBrauer induction formula aG of Example III.1.1.

2.13 Definition (cf. [Bo90, 2.17–2.20]) Let V ∈ CG−mod. We call (H,ϕ) ∈M(G) admissible for V , if V (H,ϕ) 6= 0. Let S(V ) denote the set of non-zerosubspaces of V , and let A(V ) ⊆M(G) denote the set of admissible pairs for V . Wedefine

FV : A(V )→ S(V ), PV : S(V )→ A(V )

(H,ϕ) 7→ V (H,ϕ), W 7→ sup(H,ϕ) ∈M(G) |W ⊆ V (H,ϕ).

Note that PV is well-defined (although in general suprema don’t exist in M(G)),since W ⊆ V (E,1) = V , and since W ⊆ V (H,ϕ) and W ⊆ V (H′,ϕ′) implies that thesubgroup U := <H,H ′> generated by H and H ′ also acts by scalar multiplicationon W via a unique µ ∈ U which extends ϕ and ϕ′.

The sets A(V) and S(V) are G-posets (i.e. G acts by poset automorphisms) withthe poset structure inherited fromM(G) and the inclusion in S(V ). The maps FVand PV form a Galois connection in the sense of [Ro64] or [Walk81], i.e. FV and PVare order reversing and

W ⊆ (FV PV )(W ), (H,ϕ) ≤ (PV FV )(H,ϕ), (5.3)

for all W ∈ S(V ), (H,ϕ) ∈ A(V ). Moreover, FV and PV are maps of G-sets. Therelations in (5.3) imply

PV (W ) = (PV FV PV )(W ) and FV (H,ϕ) = (FV PV FV )(H,ϕ) (5.4)

for all W ∈ S(V ), (H,ϕ) ∈ A(V ). For (H,ϕ) ∈ A(V ) we define the V-closure

clV (H,ϕ) asclV (H,ϕ) := (PV FV )(H,ϕ),

in other words, clV (H,ϕ) is the biggest pair (H ′, ϕ′) in A(V ) such that V (H,ϕ) =V (H′,ϕ′). A pair (H,ϕ) ∈ A(V ) is called closed for V , if clV (H,ϕ) = (H,ϕ). Theclosure map is order preserving and has the following properties:

(H,ϕ) ≤ clV (H,ϕ), clV(clV (H,ϕ)

)= clV (H,ϕ), clV

( s(H,ϕ))

=s(

clV (H,ϕ))

(5.5)for all (H,ϕ) ∈ A(V ), s ∈ G. The subset Cl(V ) ⊆ A(V ) of closed pairs forms aG-subposet of M(G) which is isomorphic to the G-poset V (H,ϕ) | (H,ϕ) ∈ A(V )via FV and PV .

For (H,ϕ) ∈ Cl(V ) we define the V -rank rV (H,ϕ) by the maximal number nsuch that there is a strictly ascending chain (H,ϕ) = (H0, ϕ0) < . . . < (Hn, ϕn)


in Cl(V ). In particular the pairs of V -rank 0 are exactly the maximal elements inCl(V ). Finally we define the rank r(Cl(V )) of the poset Cl(V ) as the maximum ofall ranks rV (H,ϕ), (H,ϕ) ∈ Cl(V ).

2.14 Theorem Each V ∈ CG−mod has a CG-monomial resolution M∗ →V → 0 with the following properties:

(i) Mi(H,ϕ) = 0 for all (H,ϕ) /∈ Cl(V ), i ≥ 0.

(ii) Mi(H,ϕ) = 0 for all (H,ϕ) ∈ Cl(V ), i ≥ 0, with rV (H,ϕ) < i.

In particular, V has finite monomial homological dimension and

dimmhV ≤ r(Cl(V )).

Proof We define

Mn∂n−1−−→ Mn−1

∂n−2−−→ . . .∂0−−→ M0

ε−−→ V

by induction on n with the following properties:

(An) Mi(H,ϕ) = 0 for all (H,ϕ) /∈ Cl(V ), 0 ≤ i ≤ n.

(Bn) Mi(H,ϕ) = 0 for all (H,ϕ) ∈ Cl(V ), 0 ≤ i ≤ n, with rV (H,ϕ) < i.

(Cn) M(H,ϕ)n

∂n−1−−→ . . .∂0−−→ M

(H,ϕ)0

ε−−→ V (H,ϕ) −→ 0 is exact for all (H,ϕ) ∈M(G).

(Dn) 0 −→ M(H,ϕ)n

∂n−1−−→ . . .∂0−−→ M

(H,ϕ)0

ε−−→ V (H,ϕ) −→ 0 is exact for all(H,ϕ) ∈ Cl(V ) with rV (H,ϕ) ≤ n.

Note that for n = r(Cl(V ))) + 1 the properties (An) and (Bn) imply that Mn = 0,and property (Cn) shows that 0→Mn−1 → . . .→M0 → V is a finite kG-monomialresolution of V . Thus, part (i) and (ii) of the theorem hold

(a) First we note that it suffices to prove (Cn) only for closed pairs, providedthat (An) holds:

Let (H,ϕ) ∈M(G) be arbitrary. If (H,ϕ) 6∈ A(V ), then (H ′, ϕ′) /∈ A(V ) for all

(H ′, ϕ′) ≥ (H,ϕ). Hence, Mi(H,ϕ) = M(H,ϕ)i = 0 for all i = 0, . . . , n, by (An), and

also V (H,ϕ) = 0.If (H,ϕ) ∈ A(V ), then V (H,ϕ) = V cl(H,ϕ) by (5.4), and M

(H,ϕ)i = M

cl(H,ϕ)i for all

0 ≤ i ≤ n. In fact, Mcl(H,ϕ)i ⊆M (H,ϕ)

i , since (H,ϕ) ≤ cl(H,ϕ). On the other hand,for all (H ′, ϕ′) ≥ (H,ϕ) we have Mi(H

′, ϕ′) = 0 unless (H ′, ϕ′) is closed (by (An)),

but in this case (H ′, ϕ′) = cl(H ′, ϕ′) ≥ cl(H,ϕ) and thereforeMi(H′, ϕ′) ⊆M cl(H,ϕ)

i .Since (Cn) holds for cl(H,ϕ) by assumption, it also holds for (H,ϕ).

(b) We begin the induction on n by defining M0ε−−→V . Let (H,ϕ) ∈ R(G). If

(H,ϕ) ∈ Cl(V ), then we define the [H,ϕ]G-homogeneous component M0

([H,ϕ]G

)as indGH(V (H,ϕ)), where we consider V (H,ϕ) as object in CH−monab via Corol-lary 1.12 (b). The map ε is defined on indGH(V (H,ϕ)) by s ⊗CH v 7→ sv for s ∈ G,


v ∈ V (H,ϕ). (This is analogous to the proof of Theorem 2.4 with P(H,ϕ)0 := V (H,ϕ).)

If (H,ϕ) /∈ Cl(V ), then we define M0

([H,ϕ]G

)= 0 so that (A0) is satisfied.

Condition (B0) is empty. By part (a) it suffices to prove (C0) only for (H,ϕ) ∈Cl(V ), i.e. we have to show that ε : M

(H,ϕ)0 → V (H,ϕ) is surjective. But this is

clear from the definition of ε on 1 ⊗CH V (H,ϕ) ⊆ M0(H,ϕ) ⊆ M(H,ϕ)0 if (H,ϕ) ∈

R(G). If (H,ϕ) 6∈ R(G), then (C0) follows from the previous case by the kG-linearity of ε. In order to prove (D0) we have to assume that (H,ϕ) is a maximalelement in Cl(V ), hence it is also maximal in A(V ) (by the kG-linearity of ε we mayassume that (H,ϕ) ∈ R(G)). This implies that NG(H,ϕ) = H, since otherwise ϕcould be extended to a bigger subgroup H ′ . H, and V (H,ϕ) 6= 0 then implies byClifford theory that V (H′,ϕ′) 6= 0 for some (H ′, ϕ′) > (H,ϕ); this contradicts themaximality of (H,ϕ) in A(V ). Since (H,ϕ) is maximal in Cl(V ) we see, by (A0) and

NG(H,ϕ) = H, thatM(H,ϕ)0 = M0(H,ϕ) = 1⊗CHV

(H,ϕ). Thus ε : M(H,ϕ)0 → V (H,ϕ)

is an isomorphism and (D0) holds.

(c) Next we assume that we have already constructed

Mn∂n−1−−→ . . .

∂0−−→ M0ε−−→ V

such that (An) − (Dn) hold. If n = 0, we interpret ∂n−1 as ε and Mn−1 as V .We are going to define Mn+1

∂n−−→Mn by defining, for each (H,ϕ) ∈ R(G), the[H,ϕ]G-homogeneous component Mn+1

([H,ϕ]G

)and a CG-monomial homomor-

phism ∂n : Mn+1

([H,ϕ]G

)→Mn. So let (H,ϕ) ∈ R(G).

If (H,ϕ) /∈ Cl(V ) we define Mn+1

([H,ϕ]G

):= 0, hence (An+1) is satisfied.

If (H,ϕ) ∈ Cl(V ) and rV (H,ϕ) ≤ n, we also define Mn+1

([H,ϕ]G

):= 0. Then,

for this (H,ϕ), (Bn+1) is satisfied, and (Dn) implies (Cn+1) and (Dn+1) for this(H,ϕ). Obviously (Bn+1) − (Dn+1) are then also satisfied for all elements in theG-orbit [H,ϕ]G of (H,ϕ).

If (H,ϕ) ∈ Cl(V ) and rV (H,ϕ) > n + 1, then we define Mn+1

([H,ϕ]G

):=

indGH(Ω(H,ϕ)n ) with Ω

(H,ϕ)n := ker

(∂n−1 : M

(H,ϕ)n → M

(H,ϕ)n−1

), and as in the proof of

Theorem 2.4 we define ∂n : Mn+1

([H,ϕ]G

)→ Mn (we may take P

(H,ϕ)n+1 := Ω

(H,ϕ)n

with the identity map). Then (Cn+1) holds for this (H,ϕ) as is proved in The-orem 2.4, and the conditions (Bn+1) and (Dn+1) are empty for (H,ϕ). By thekG-linearity of ∂n, (Bn+1)− (Dn+1) are also satisfied for all elements in [H,ϕ]G.

If (H,ϕ) ∈ Cl(V ) and rV (H,ϕ) = n+ 1, we will show in Lemma 2.15 that

Ω(H,ϕ)n := ker

(M (H,ϕ)n

∂n−1−−→M (H,ϕ)n−1

) ∼= indNG(H,ϕ)H (L(H,ϕ)) (5.6)

as CNG(H,ϕ)-module for some ϕ-homogeneous CH-submodule L(H,ϕ) ⊆ Ω(H,ϕ)n .

Considering L(H,ϕ) as object in CH−monab by Corollary 1.12 (b), we can define

Mn+1

([H,ϕ]G

):= indGH(L(H,ϕ)),

and∂n : Mn+1

([H,ϕ]G

)→Mn, g ⊗CH v 7→ gv,

for g ∈ G, v ∈ L(H,ϕ). The map ∂n is a CG-monomial homomorphism by Proposi-

tion 1.7 (a), since ∂n(L(H,ϕ)) ⊆M(H,ϕ)n (use Proposition 1.11 (c) =⇒ (a)).


For this (H,ϕ), (Bn+1) is empty and (Dn+1) implies (Cn+1). We will show that(Dn+1) holds for (H,ϕ). Since (An+1) and (Bn+1) hold for all closed pairs of rankless than n+ 1 we have

M(H,ϕ)n+1 =

(indGH(L(H,ϕ))

)(H,ϕ)=

⊕s∈NG(H,ϕ)/H

s⊗CH L(H,ϕ) = indNG(H,ϕ)H (L(H,ϕ)),

and hence by (5.6), ∂n induces an isomorphism M(H,ϕ)n+1 → Ω

(H,ϕ)n . Again it is

obvious that (Bn+1)− (Dn+1) also hold for all G-conjugates of the pair (H,ϕ).

Therefore, the theorem is proved up to the following lemma.

2.15 Lemma Let Mn∂n−1−−→ . . .

∂0−−→M0ε−−→V be such that conditions (An)−

(Dn) in the proof of Theorem 2.14 are satisfied. Let (H,ϕ) ∈ Cl(V ) with rV (H,ϕ) =n+ 1, then the character θ of the CNG(H,ϕ)-module

Ω(H,ϕ)n := ker

(∂n−1 : M (H,ϕ)

n →M(H,ϕ)n−1

)is a multiple on indGH(ϕ), possibly zero.

Proof Let N := NG(H,ϕ), then we have the following exact sequence ofCN -modules

0 −→ Ω(H,ϕ)n −→M (H,ϕ)

n∂n−1−−→ . . .

∂0−−→ M(H,ϕ)0

ε−−→ V (H,ϕ) −→ 0.

As a consequence of the semisimplicity of CN we obtain the following equation inthe character ring R(N):

(−1)n+1θ +n∑i=0

(−1)iχi = ν,

where by χi ∈ R(N) we denote the character of M(H,ϕ)i , 0 ≤ i ≤ n, and by

ν ∈ R(N) we denote the character of V (H,ϕ). On the other hand we can consider

M(H,ϕ)i as kN -monomial module in its own right (cf. Remark 1.9 (d)), and since

χi = bN([M

(H,ϕ)i ]

)and ν = bNaN (ν), the last equation can be written as

(−1)n+1θ = bn

(aN (ν)−

n∑i=0

(−1)i[M(H,ϕ)i ]

). (5.7)

First note that all the stabilizing pairs of the components of M(H,ϕ)i ∈ CN−mon,

0 ≤ i ≤ n, are of the form (H ′ ∩ N,ϕ′|H′∩N ) for some (H ′, ϕ′) ≥ (H,ϕ) in M(G).

Hence, [M(H,ϕ)i ] ∈ Rab

+ (N) may have non-zero coefficients only at basis elements[K,ψ]N ∈ N\M(N) with (H,ϕ) ≤ (K,ψ), for all 0 ≤ i ≤ n. The same is true forthe element aN (ν) by Theorem III.1.2 (xi), since ν|H is a multiple of ϕ. This showsthat we can write

aN (ν)−n∑i=0

(−1)i[M(H,ϕ)i ] =

∑[H,ϕ]N≤[K,ψ]N∈N\M(N)

α[K,ψ]N [K,ψ]N (5.8)


with suitable α[K,ψ]N ∈ Z. We will show that

α[K,ψ]N = 0 for all [K,ψ]N > [H,ϕ]N . (5.9)

This implies that the element in (5.8) is a multiple of [H,ϕ]N , and substitution of(5.8) in (5.7) establishes the assertion of the lemma.

In order to show (5.9) let us assume that there is [K0, ψ0]N > [H,ϕ]N withα[K0,ψ0]N 6= 0 and which is maximal under this condition. This will lead us to acontradiction as follows. By the maximality of [K0, ψ0]N and by Proposition 1.20 (b)we have [

[K0, ψ0]N ,∑

[H,ϕ]N≤[K,ψ]N∈N\M(N)

α[K,ψ]N [K,ψ]N

]N

=∑

[H,ϕ]N≤[K,ψ]N∈N\M(N)

α[K,ψ]N

[[K0, ψ0]N , [K,ψ]N

]N

= α[K0,ψ0]N

[[K0, ψ0]N , [K0, ψ0]N

]N

= α[K0,ψ0]N (NN (K0, ψ0) : K0)

6= 0.

On the other hand we obtain[[K0, ψ0]N , aN (ν)−

n∑i=0

(−1)i[M(H,ϕ)i ]

]N

= [[K0, ψ0]N , aN (ν)]N −n∑i=0

(−1)i[[K0, ψ0]N , [M

(H,ϕ)i ]

]N

III.1.2 (iii)=

(indNK0

(ψ0), ν)N−

n∑i=0

(−1)i[SN(K0,ψ0),M

(H,ϕ)i

]N

1.20 (a)=

(ψ0, resNK0

(ν))K0−

n∑i=0

(−1)i dimC

(M

(H,ϕ)i

)(K0,ψ0)

(H,ϕ)≤(K0,ψ0)= dimC V

(K0,ψ0) −n∑i=0

(−1)i dimCM(K0,ψ0)i .

We will get the desired contradiction by showing that the last expression is zero. If(K0, ψ0) is not admissible for the CN -module V (H,ϕ), i.e. V (K0,ψ0) = 0, then also

M(K0,ψ0)i = 0 for all 0 ≤ i ≤ n by (An), and the above expression is zero. If (K0, ψ0)

is admissible for V (H,ϕ), then we have clV (H,ϕ) = (H,ϕ) < (K0, ψ0) ≤ clV (K0, ψ0)which implies rV

(cl(K0, ψ0)

)< rV (H,ϕ) = n + 1. Now (Dn) of the proof of

Theorem 2.14 implies that the sequence

0 −→ M cl(K0,ψ0)n −→ . . . −→ M

cl(K0,ψ0)0 −→ V cl(K0,ψ0) −→ 0

is exact. But V (K0,ψ0) = V cl(K0,ψ0) by (5.4), and by an argument we already used

in part (a) of the proof of Theorem 2.14 we know that M(K0,ψ0)i = M

cl(K0,ψ0)i for

all 0 ≤ i ≤ n. Hence, also in this case the above expression is zero.


2.16 Corollary Let V ∈ CG−mod with character χ, and let M∗ε−−→V be a

finite CG-monomial resolution of V . Then we have

aG(χ) = Λ(M∗).

Proof It suffices to show that ρG(aG(χ)) = ρG(Λ(M∗)

), since ρG is injective.

So let H ≤ G, then the commutativity of Diagram (2.2) implies

πH(res+

GH(aG(χ))

)= pH(resGH(χ)) =

∑ϕ∈H

dimC V(H,ϕ) · ϕ,

and by Remark 1.9 (g) and the exactness of the complex in (5.1) we obtain

πH(res+

GH

(Λ(M∗)

))=∑ϕ∈H

(∑i≥0

(−1)i dimCM(H,ϕ)i

)· ϕ

=∑ϕ∈H

dimC V(H,ϕ) · ϕ.

2.17 Remark The last theorem shows that the construction of CG-monomialresolutions provides a full embedding of CG−mod into the homotopy categoryof finite chain complexes of CG-monomial modules. The corollary to the the-orem has the following interpretation: The canonical Brauer induction formulaaG : R(G) → Rab

+ (G) is the shadow of the functor of taking monomial resolutionson the level of Grothendieck groups. More precisely, if Kb(CG−mon) denotes thehomotopy category of finite chain complexes in CG−mon, then we have the follow-ing commutative diagram

CG−mod −−−−→ Kb(CG−mon)y yΛ


+ (G),

where the vertical arrows stand for taking characters and Lefschetz invariants, andthe upper horizontal arrow is the functor of taking CG-monomial resolutions.

2.18 Lemma (Stopping criterion) Let n ≥ 0 and let

Mn∂n−1−−→ . . .

∂0−−→ M0ε−−→ V

be a chain complex with Mn, . . . ,M0 ∈ CG−mon, V ∈ CG−mod, ∂n−1, . . . , ∂0

CG-monomial homomorphisms and ε ∈ HomCG(M0, V ) such that

M (H,ϕ)n

∂n−1−−→ . . .∂0−−→ M

(H,ϕ)0

ε−−→ V (H,ϕ) −→ 0

is exact for all (H,ϕ) ∈M(G). Let

U := ker ∂n−1 and U (H,ϕ) := U ∩M (H,ϕ)n = ker

(M (H,ϕ)n

∂n−1−−→M (H,ϕ)n−1

)for all (H,ϕ) ∈ M(G) (for n = 0 we set ∂−1 = ε and M−1 = V ). Then thefollowing are equivalent:


(a) The above sequence can be completed to a CG-monomial resolution

0 −→ Mn+1∂n−−→ Mn

∂n−1−−→ . . .∂0−−→ M0

ε−−→ V −→ 0

of V .

(b) For each (H,ϕ) ∈M(G) the character of the CNG(H,ϕ)-module

U (H,ϕ)/ ∑

(H,ϕ)<(H′,ϕ′)

U (H′,ϕ′)

is a multiple of indNG(H,ϕ)H (ϕ).

Proof (a) =⇒ (b): Under the hypothesis (a), the map ∂n induces isomor-

phisms Mn+1∂n−−→U and M

(H,ϕ)n+1

∂n−−→U (H,ϕ) for all (H,ϕ) ∈ M(G). Hence, thereare resulting isomorphisms

∂n : M(H,ϕ)n+1

/ ∑(H,ϕ)<(H′,ϕ′)

M(H′,ϕ′)n+1 −−−−→ U (H,ϕ)

/ ∑(H,ϕ)<(H′,ϕ′)

U (H′,ϕ′)

of CNG(H,ϕ)-modules. But clearly

M(H,ϕ)n+1

/ ∑(H,ϕ)<(H′,ϕ′)

M(H′,ϕ′)n+1

∼= Mn+1(H,ϕ)

as CNG(H,ϕ)-modules, and the character of the latter module is a multiple of

indNG(H,ϕ)H (ϕ).

(b) =⇒ (a): For all (H,ϕ) ∈ R(G) let X(H,ϕ) ∈ CNG(H,ϕ)−mod be acomplement of

∑(H,ϕ)<(H′,ϕ′) U

(H′,ϕ′) in U (H,ϕ). Then (b) implies that there is aϕ-homogeneous CH-submodule L(H,ϕ) ⊆ X(H,ϕ) such that

X(H,ϕ) =⊕

s∈NG(H,ϕ)/H

sL(H,ϕ). (5.10)

Now we extend the definition of X(H,ϕ) and L(H,ϕ) to all (H,ϕ) ∈M(G) by setting

X( g(H,ϕ)

):= g ·X(H,ϕ) and L g

(H,ϕ):= g · L(H,ϕ) ⊆ X

( g(H,ϕ)).

Then (5.10) holds for all (H,ϕ) ∈M(G) and we have decompositions

U (H,ϕ) =⊕

(H,ϕ)≤(H′,ϕ′)

X(H ′, ϕ′)

for all (H,ϕ) ∈M(G) and in particular (with (H,ϕ) = (E, 1))

U =⊕

(H,ϕ)∈M(G)

X(H,ϕ) =⊕

(H,ϕ)∈M(G)s∈NG(H,ϕ)/H

sL(H,ϕ).


The last decomposition furnishes U with the structure of a CG-monomial module.We will denote this module by Mn+1. The (H,ϕ)-homogeneous component of Mn+1

is given by X(H,ϕ), so that

U (H,ϕ) =⊕

(H,ϕ)≤(H′,ϕ′)

X(H ′, ϕ′) = M(H,ϕ)n+1 .

Now we define ∂n : Mn+1 → Mn by the embedding U ⊆ Mn. Then ∂n is a CG-

monomial homomorphism, since M(H,ϕ)n+1 = U (H,ϕ) ⊆M (H,ϕ)

n for all (H,ϕ) ∈M(G).Finally,

0 −→ Mn+1∂n−−→ Mn

∂n+1−−→ . . .∂0−−→ M0

ε−−→ V −→ 0

is a CG-monomial resolution of V , since

∂n

(M

(H,ϕ)n+1

)= U (H,ϕ) = ker

(M (H,ϕ)n

∂n−1−−→M (H,ϕ)n−1

)for all (H,ϕ) ∈M(G).

We complete the stopping criterion by determining the CG-monomial modulesof monomial homological dimension zero.

2.19 Proposition Let V ∈ CG−mod, then we have: dimmhV = 0 ⇐⇒V ∈ CG−modab.

Proof That all abelian CG-modules have monomial homological dimensionzero has been proved in Remark 2.12 in a more general situation. Now assume that0−→M

ε−−→V −→ 0 is a CG-monomial resolution of V . Then Corollary 2.16 yieldsaG(χ) = [M ], where χ is the character of V . This implies that

aG(χ) =∑

[H,ϕ]G∈G\M(G)

α[H,ϕ]G(χ)[H,ϕ]G

with α[H,ϕ]G(χ) ∈ N0 for all [H,ϕ]G ∈ G\M(G). Now we have the equation∑(H,ϕ)∈R(G)

α[H,ϕ]G(χ)(G : H) =∑

(H,ϕ)∈R(G)

α[H,ϕ]G(χ)

by Theorem III.1.2 (x). Since all α[H,ϕ]G(χ) are nonnegative, these two equationsimply α[H,ϕ]G(χ) = 0 for H 6= G. Hence, χ = bGaG(χ) =

∑ϕ∈G α[H,ϕ]G(χ) · ϕ, and

V is abelian.

2.20 Propositon Let V,W ∈ CG−mod. Then

dimmh(V ⊕W ) = maxdimmhV,dimmhW.

Proof Taking the direct sum of CG-monomial resolutions of V and W resultsin a CG-monomial resolution of V ⊕W . Hence

dimmh(V ⊕W ) ≤ maxdimmhV,dimmhW.


By symmetry we only have to show dimmhV ≤ dimmh(V ⊕W ) in order to completethe proof. Let

0 −−→ Nn −−→ . . . −−→ N0 −−→ V ⊕W −−→ 0

be a CG-monomial resolution of V ⊕W with n = dimmh(V ⊕W ). And let

0 −−→ Mm∂m−1−−→ . . .

∂0−−→ M0ε−−→ V −−→ 0

be a CG-monomial resolution of V . It suffices to show that if m > n, then thisresolution of V can be shortened. By Proposition 2.19 we may assume that n ≥ 1,hencem ≥ 2. We apply Proposition 2.3 to the canonical inclusion ι : V → V⊕W andto the canonical projection π : V ⊕W → V . This yields the following commutativediagram with CG-monomial homomorphisms ιi, πi, 0 ≤ i ≤ n,

0 −−→ Mm∂m−1−−→ . . .

∂0−−→ M0ε−−→ V −−→ 0y y yι0 yι

0 −−→ 0 −−→ . . . −−→ N0 −−→ V ⊕W −−→ 0y y yπ0

yπ0 −−→ Mm

∂m−1−−→ . . .∂0−−→ M0

ε−−→ V −−→ 0.

Since πι = idV , the uniqueness part of Proposition 2.3 implies that π∗ι∗ : M∗ →M∗and idM∗ are homotopy equivalent. Hence there are CG-monomial homomorphismshi : Mi →Mi+1, i ≥ 0 (Mi = 0 for i > m), such that

idMi − πi ιi = hi−1 ∂i−1 + ∂i hi

for all i ≥ 0, where we interpret h−1 and ∂−1 as zero. In particular, for i = m weobtain

idMm = hm−1 ∂m−1. (5.11)

Now we apply the stopping criterion to the chain complex

Mm−2∂m−3−−→ . . .

∂0−−→ M0ε−−→ V −→ 0.

For each (H,ϕ) ∈M(G) let

U (H,ϕ) := ker(∂m−3 : M

(H,ϕ)m−2 −→M

(H,ϕ)m−3

)⊆Mm−2

as in Lemma 2.18. If m = 2, we interpret ∂m−3 as ε and Mm−3 as V . The stoppingcriterion says that it suffices to prove that the character of the CNG(H,ϕ)-module

U (H,ϕ)/ ∑

(H,ϕ)<(H′,ϕ′)

U (H′,ϕ′)

is a multiple of indNG(H,ϕ)H (ϕ), for all (H,ϕ) ∈M(G).

Now let T := ker(hm−1) ⊆Mm−1, and for all (H,ϕ) ∈M(G) let

T (H,ϕ) := ker(hm−1 : M

(H,ϕ)m−1 −→M (H,ϕ)

m

)⊆Mm−1.


Then (5.11) implies that Mm = ∂m−1(Mm)⊕ T and also that

M(H,ϕ)m−1 = ∂m−1(M (H,ϕ)

m )⊕ T (H,ϕ) (5.12)

for all (H,ϕ) ∈ M(G). Since M∗ → V → 0 is a CG-monomial resolution of V ,

we have U (H,ϕ) = ∂m−2

(M

(H,ϕ)m−1

)for all (H,ϕ) ∈ M(G). Together with (5.12) this

implies U (H,ϕ) = ∂m−2(T (H,ϕ)). But since

T (H,ϕ) ∩ ker ∂m−2 = T (H,ϕ) ∩ ∂m−1(Mm) ⊆ T ∩ ∂m−1(Mm) = 0,

the CNG(H,ϕ)-linear map

∂m−2 : T (H,ϕ) ∼−→U (H,ϕ)

is an isomorphisms for all (H,ϕ) ∈M(G). This induces a CNG(H,ϕ)-isomorphism

T (H,ϕ)/ ∑

(H,ϕ)<(H′,ϕ′)

T (H′,ϕ′) ∼= U (H,ϕ)/ ∑

(H,ϕ)<(H′,ϕ′)

U (H′,ϕ′)

for each (H,ϕ) ∈ M(G), and it suffices to show that the character of the left one

of these two CNG(H,ϕ)-modules is a multiple of indNG(H,ϕ)H (ϕ). In order to show

this we note that (5.12) induces a CNG(H,ϕ)-isomorphism

M(H,ϕ)m−1

/ ∑(H,ϕ)<(H′,ϕ′)

M(H′,ϕ′)m−1

∼=

∼= ∂m−1(M (H,ϕ)m )

/ ∑(H,ϕ)<(H′,ϕ′)

∂m−1(M (H′,ϕ′)m ) ⊕

⊕ T (H,ϕ)/ ∑

(H,ϕ)<(H′,ϕ′)

T (H′,ϕ′).

The first one of these isomorphic modules is CNG(H,ϕ)-isomorphic to Mm−1(H,ϕ)

whose character is a multiple of indNG(H,ϕ)H (ϕ). The same is true for the first

summand in the second module. In fact, since ∂m−1 is injective, we have

∂m−1(M (H,ϕ)m )

/ ∑(H,ϕ)<(H′,ϕ′)

∂m−1(M (H′,ϕ′)m ) ∼=

∼= M (H,ϕ)m

/ ∑(H,ϕ)<(H′,ϕ′)

M (H′,ϕ′)m

∼= Mm(H,ϕ)

which now completes the proof.

Repeated use of the above Proposition yields

2.21 Corollary Let V ∈ CG−mod, then the monomial homological dimen-sion of V is the maximum of the monomial homological dimensions of the irreducibleconstituents of V .

5.3. MONOMIAL RESOLUTIONS AS PROJECTIVE RESOLUTIONS 179

5.3 Monomial resolutions as projective resolutions

We keep the notation of the two previous sections. In this section we are going tointroduce an interpretation of the monomial resolution as a projective resolution byembedding both kG−mod and kG−mon into an abelian category C such that theobjects of kG−mon become projective objects in C. The construction of C fromkG−mon is a standard method, but for the reader’s convenience we will give it indetail. Recall that V : kG−mon→ kG−modpr denotes the forgetful functor.

3.1 Definition We define C as the category of contravariant k-functors fromkG−mon to k−mod, i.e. contravariant functors which are k-linear on morphisms.The morphisms in C are the natural transformations between the contravariantfunctors in C. We define k-functors

I : kG−mod→ C, V 7→ HomkG(V−, V )

and

J : kG−mon→ C, M 7→ monkG(−,M).

For V,W ∈ kG−mod and f ∈ HomkG(V,W ) we define

I(f) := HomkG(V−, f) : HomkG(V−, V )−→HomkG(V−,W ),

and for M,N ∈ kG−mon and g ∈ monkG(M,N) we define

J (g) := monkG(−, g) : monkG(−,M)−→monkG(−, N).

3.2 Proposition The category C is abelian, and I and J are full embeddings,i.e. injective on objects and bijective on morphisms.

Proof Clearly C is an abelian category by pointwise constructions in k−mod,since k−mod is abelian. For example, if ϕ : F → G is a morphism in C, thenker(ϕ) ∈ C is defined as the functor M 7→ ker(ϕM ) ⊆ F(M), M ∈ kG−mon.

Clearly J is a full embedding by Yoneda’s Lemmma. In order to see that I isa full embedding we have to work harder. First we define a k-functor

T : C −→ kG−mod

by the following construction. For F ∈ C set T (F) := F(SG(E,1)) which is a k-module

by definition. We endow V := F(SG(E,1)) with the structure of a kG-module: Letg ∈ G, then we define left multiplication on V by g as

F(g) : F(SG(E,1))−→F(SG(E,1)),

where g ∈ monkG(SG(E,1), SG(E,1)) is defined by

g : SG(E,1)−→SG(E,1), s⊗kE α 7→ sg ⊗kE α,


for s ∈ G and α ∈ k.

On morphisms we define T as follows. Let F ,G ∈ C and let ϕ : F → G be anatural transformation. Then

T (ϕ) := ϕSG(E,1)

: F(SG(E,1))−→G(SG(E,1)).

It is easy to check that T (ϕ) is kG-linear with respect to the G-action defined above.

The functor T I : kG−mod→ kG−mod is naturally equivalent to the identityfunctor via the following isomorphism

ζV : T I(V ) = HomkG(V(SG(E,1)), V )∼−→V, f 7→ f(1⊗kE 1),

for V ∈ kG−mod, since the following diagram

T(I(V)) = HomkG(V(SG(E,1)), V )−V(g)−−−−→ HomkG(V(SG(E,1)), V ) = T(I(V))

ζV

y yζVV

g·−−−−→ V

commutes. This shows that V can be recovered from I(V ).

It is easy to check that for V,W ∈ kG−mod and f ∈ HomkG(V,W ) the followingdiagram is commutative

T I(V )T I(f)−−−−→ T I(W )

ζV

y yζWV

f−−−−→ W .

This shows that I is injective on morphisms.

Finally we show that I is surjective on morphisms. Let V,W ∈ kG−mod andlet ϕ : I(V ) → I(W ) be a natural transformation. We define f : V → W by thecommutative diagram

T I(V )T (ϕ)−−−−→ T I(W )

ζV

y yζWV

f−−−−→ W .

Then f is kG-linear, since the other three maps in the above diagram are kG-linear.We will show that

ϕM = I(f)(M) : HomkG(V(M), V )−→HomkG(V(M),W )

for all M ∈ kG−mon. Let h ∈ HomkG(V(M), V ) and m ∈ M , then we have toshow that (

ϕM (h))(m) = (fh)(m).

There is a unique g ∈ monkG(SG(E,1),M) with g(1⊗kE1) = m, and the commutativity


of the following diagram

I(V )(M) = HomKG(V(M), V )φM−−−−→ HomkG(V(M),W ) = I(W )(M)

HomkG(V(g),V )

y yHomkG(V(g),W )

I(V )(SG(E,1)) = HomkG(V(SG(E,1)), V )

φSG

(E,1)−−−−→ HomkG(V(SG(E,1)),W ) = I(W )(SG(E,1))

ζV

y yζWV

f−−−−→ W

implies (ϕM (h)

)(m) =

(ϕM (h)

)(V(g)(1⊗kE 1)

)=(ϕM (h) V(g)

)(1⊗kE 1)

=(ζW HomkG(V(g),W ) ϕM

)(h)

=(f ζV HomkG(V(g), V )

)(h)

= (f ζV )(h V(g))

= f((hV(g))(1⊗kE 1)

)= f(h(m)).

3.3 Proposition The contravariant functor J (M) ∈ C is projective for M ∈kG−mon.

Proof Let F := monkG(−,M) ∈ C and let G,H ∈ C be arbitrary. Assumethat we have a diagram

FyϕG ψ−−→ H

(5.13)

in C, where ψ is an epimorphism, i.e. ψN : G(N) → H(N) is surjective for allN ∈ kG−mon.

We will define µ ∈ C(F ,G) with ψ µ = ϕ. First we evaluate the above diagramat M . Since F(M) = monkG(M,M) is k-projective (cf. Proposition 1.20 (c)), wecan complete diagram (5.13) evaluated atM by a k-linear map µM : F(M)→ G(M).Now let N ∈ kG−mon be arbitrary, and let f ∈ F(N) = kG−mon(N,M), thenwe define

µN (f) := G(f)(µM (idM )) ∈ G(N).

The map µN is k-linear, since G is a k-functor. The collection µ = (µN ), N ∈kG−mon, is a natural transformation from F to G, since the following diagram

F(N ′)F(g)−−−−→ F(N)

µN′

y yµNG(N ′)

G(g)−−−−→ G(N)


commutes for all N,N ′ ∈ kG−mon, g ∈ monkG(N,N ′). In fact, let h ∈ F(N ′) =monkG(N ′,M), then

µN(F(g)(h)) = µN (h g) = G(h g)(µM (idM ))

= G(g)(G(h)(µM (idM ))

)= G(g)

(µN ′(h)

).

3.4 Remark In general, C has more projective objects than J (M), M ∈kG−mon. In fact, let k = C, M := SG(H,ϕ), for some (H,ϕ) ∈ Mk(G), then

C(J (M),J (M)) ∼= monkG(M,M) by Yoneda’s Lemma, and the latter k-algebra isisomorphic to the group algebra k[NG(H,ϕ)/H] by the map

k[NG(H,ϕ)/H]∼−→monkG(SG(H,ϕ), S

G(H,ϕ))

n 7→(g ⊗kH 1 7→ gn⊗kH 1

).

Assume that H 6= NG(H,ϕ), then k[NG(H,ϕ)/H] may have non-trivial idempo-tents, and therefore J (M) is decomposable in C. But M is not decomposable inkG−mon. Hence, each proper summand of J (M) is projective in C and not of theform J (N) for some N ∈ kG−mon, since J is a full embedding.

3.5 Lemma For each M ∈ kG−mon and each V ∈ kG−mod there is ak-isomorphism

κM,V : HomkG(V(M), V )∼−→C(J (M), I(V ))

f 7→(

monkG(N,M) → HomkG(V(N), V )h 7→ f V(h)

)N∈kG−mon

whose inverse maps ϕ to ϕM (idM ). Moreover, κM,V is functorial in M and V ,i.e. K is a functorial equivalence

HomkG(V−,−)∼−→ C(J−, I−)

of functors from kG−monop × kG−mod to k−mod.

Proof The proof is an easy verification and is left to the reader.

3.6 Proposition Let

. . .∂1−−→ M1

∂0−−→ M0ε−−→ V −→ 0

be a chain complex with Mi ∈ kG−mon and ∂i ∈ monkG(Mi+1,Mi) for i ≥ 0,V ∈ kG−mod and ε ∈ HomkG(V(M0), V ). Then the following are equivalent:

(a) M∗ε−−→V −→ 0 is a monomial resolution of V , i.e.

. . .∂1−−→ M

(H,ϕ)1

∂0−−→ M(H,ϕ)0

ε−−→ V (H,ϕ) −→ 0

is exact for all (H,ϕ) ∈Mk(G).


(b) The sequence

. . .J (∂1)−−→ J (M1)

J (∂0)−−→ J (M0)κM0,V

(ε)−−→ I(V ) −→ 0

is exact in C.

Proof (a) =⇒ (b): We have to show that for all N ∈ kG−mon the followingsequence in k−mod is exact:

. . .∂1−−−→ monkG(N,M1)

∂0−−−→ monkG(N,M0)εV−−−→

HomkG(V(N),M) −−→ 0.

Clearly this sequence is a chain complex. First we prove exactness at monkG(N,Mi)for i > 0. Let f ∈ monkG(N,Mi) such that ∂i−1 f = 0. Consider the diagram

NyfMi+1

∂i−−→ Mi∂i−1−−→ Mi−1.

By the hypothesis in (a) we have

f(N (H,ϕ)) ⊆ ker(∂i−1 : M

(H,ϕ)i →M

(H,ϕ)i−1

)= ∂i(M

(H,ϕ)i+1 ).

Hence, we may apply Lemma 2.2, and we obtain g ∈ monkG(N,Mi+1) with f =∂i g.

Exactness at monkG(N,M0) and at HomkG(V(N), V ) is proved in a similar wayusing (a) and Lemma 2.2.

(b) =⇒ (a): Let (H,ϕ) ∈ Mk(G). We apply the exact sequence of functors in(b) to SG(H,ϕ). This yields an exact sequence of k-modules

. . .∂1−−−→ monkG

(SG(H,ϕ),M1

) ∂0−−−→ monkG(SG(H,ϕ),M0

) εV−−−→

HomkG

(V(SG(H,ϕ)), V

)−−→ 0.

By Frobenius reciprocity (Proposition 1.7 (a)) we have functorial isomorphisms(note that V(SG(H,ϕ)) = indGH(kϕ))

monkG(SG(H,ϕ),Mi) ∼= monkH(kϕ, resGH(Mi))

andHomkG(VSG(H,ϕ), V ) ∼= HomkH(kϕ, resGH(V )).

Hence we get an exact sequence of k-modules

. . .∂1−−−→ monkH(kϕ, resGH(M1))

∂0−−−→ monkH(kϕ, resGH(M0))εV−−−→

HomkH(kϕ, resGH(V )) −−→ 0.

Since kϕ ∈ kH−mon is homogeneous in degree (H,ϕ) ∈ Mk(H) with respect tothe Mk(H)-grading of Remark 1.9 (c), Proposition 1.11 (c) yields

monkH(kϕ, resGH(Mi)) = HomkH

(kϕ, (resGH(Mi))

(H,ϕ))

=

= Homk(k,M(H,ϕ)i ) ∼= M

(H,ϕ)i


by Remark 1.9 (g). Using these identifications one obtains exactly the exact se-quence of part (a).

3.7 Remark We may interpret the last proposition in the following way: letV ∈ kG−mod, then the map(

M∗ε−−→V

)7→

(J (M∗)

κM0,V(ε)

−−−−→I(V ))

is a bijection between the monomial resolutions of V , and the projective resolu-tions of I(V ) in C consisting of objects from the subcategory J (kG−mon). Infact Proposition 3.2 and Lemma 3.5 together with the last proposition imply thebijectivity of the above correspondence.

Finally we give alternative descriptions of the category C by introducing equiv-alences from C to two other categories without proving the details.

3.8 Let

S :=⊕

(H,ϕ)∈Rk(G)

SG(H,ϕ) ∈ kG−mon,

and let

E := monkG(S, S).

Then E is a k-algebra, free over k by Proposition 1.20 (a), and C is equivalentto mod−E, the category of finitely generated right E-modules. An equivalence isgiven by the following functor

Φ: C −→mod−E, F 7→ F(S).

If F ∈ C, then F(S) is a finitely generated k-module by definition, and the rightE-module structure is given by the canonical E-action

F(S)⊗ E−→F(S), v ⊗ f 7→ F(f)(v).

If ϕ : F → G is a morphism in C, i.e. a natural transformation from F to G, thenwe define

Φ(ϕ) := ϕS : F(S)−→G(S).

In the other direction we define the functor

Ψ: mod−E−→C, X 7→ HomE

(monkG(S,−), X

),

where monkG(S,M) is a right E-module for M ∈ kG−mon by composition ofkG-monomial homomorphisms. If f : X → Y is a morphism in mod−E, then wedefine

Ψ(f) : HomE

(monkG(S,−), X

)−→HomE

(monkG(S,−), Y

)by composition with f from the left.


Next we define natural transformations

η : Idmod−E −→Φ Ψ and ε : IdC −→Ψ Φ.

For X ∈ mod−E we have an obvious isomorphism of E-modules which is naturalin X:

ηX : X∼−→HomE(E,X) = Φ(Ψ(X)).

For F ∈ C we define

εF : F −→Hom(monkG(S,−),F(S)

)= Ψ(Φ(F))

(which is supposed to be a natural transformation) at M ∈ kG−mon by

εF ,M : F(M)−→HomE

(monkG(S,M),F(S)

),

v 7→(f 7→ F(f)(v)

).

(5.14)

We leave it to the reader to show that εF is well-defined and natural in F . Inorder to prove that (5.14) is an isomorphism we may assume that M = SG(H,ϕ)

for some (H,ϕ) ∈ Rk(G), since F is a k-functor and each M ∈ kG−mon is adirect summand of a direct sum of these objects (cf. Proposition 1.17 (c)). For(H,ϕ) ∈ Rk(G) let p(H,ϕ) : S → SG(H,ϕ) and i(H,ϕ) : SG(H,ϕ) → S be the canonicalprojection and inclusion. Then it is easy to verify that

F(SG(H,ϕ))∼←→HomE

(monkG(S, SG(H,ϕ)),F(S)

)v 7→

(f 7→ F(f)(v)

)F(i(H,ϕ))(g(p(H,ϕ))) ←7 g

defines inverse isomorphisms.We observe that SG(H,ϕ) is mapped under the embedding ΦJ to e(H,ϕ)E, where

e(H,ϕ) = i(H,ϕ) p(H,ϕ) is an idempotent in E. The idempotents e(H,ϕ), (H,ϕ) ∈Rk(G) are mutually orthogonal and their sum is 1.

Next we give an explicit description of E in terms of a k-basis and a multi-plication rule for this basis. By Proposition 1.20 (b), monkG(SG(K,ψ), S

G(H,ϕ)) has a

distinguished k-basis fs | s ∈ G/H with (K,ψ) ≤ s(H,ϕ). For each subgroupH ≤ G let TH ⊆ G be a fixed set of representatives for G/H. Then E has a basiswhich is described by triples

[(H,ϕ), s, (K,ψ)]

with (K,ψ), (H,ϕ) ∈ Rk(G), s ∈ TH , such that (K,ψ) ≤ s(H,ϕ). Such a triplecorresponds to the kG-monomial homomorphism fs : SG(K,ψ) → SG(H,ϕ), g ⊗kK α 7→gs⊗kH α for g ∈ G, α ∈ k, which was described in Proposition 1.20 (b). Hence, weobtain the following multiplication rule:

[(H,ϕ), s, (K,ψ)] · [(U, µ), t, (L, λ)] =

=

0, if (K,ψ) 6= (U, µ),

ϕ(u−1ts) · [(H,ϕ), u, (L, λ)], if (K,ψ) = (U, µ),


where u ∈ TH with tsH = uH.Since the chains inMk(G) have bounded length, this shows that E has a nilpo-

tent ideal generated by

[(H,ϕ), s, (K,ψ)] | (H,ϕ) < s(K,ψ)

whose quotient is isomorphic to∏(H,ϕ)∈Rk(G)

monkG(SG(H,ϕ), SG(H,ϕ))

∼=∏

(H,ϕ)∈Rk(G)

k[NG(H,ϕ)/H].

So, if k is a field of characteristic zero, then this ideal is exactly the radical of thek-algebra E.

3.9 Let D be the following subcategory of the category of G-equivariantsheaves on the G-poset Mk(G). The objects of D are triples (A, r, c), where

• A = A(H,ϕ) | (H,ϕ) ∈Mk(G) is a family of k-modules, called stalks,

• r = r(H,ϕ)(K,ψ) : A(H,ϕ) → A(K,ψ) | (K,ψ) ≤ (H,ϕ) in Mk(G) is a family of

k-linear maps, called restrictions, and

• c = cg(H,ϕ) : A(H,ϕ) → A g(H,ϕ)

| g ∈ G, (H,ϕ) ∈ Mk(G) is a family of

k-linear maps, called conjugations,

which are subject to the following conditions

(i) r(H,ϕ)(H,ϕ) = idA(H,ϕ)

and r(K,ψ)(U,µ) r

(H,ϕ)(K,ψ) = r

(H,ϕ)(U,µ) for all (U, µ) ≤ (K,ψ) ≤ (H,ϕ)

in Mk(G).

(ii) cg′g(H,ϕ)

cg(H,ϕ) = cg′g

(H,ϕ) for all g, g′ ∈ G, (H,ϕ) ∈ Mk(G), and ch(H,ϕ) is the

scalar multiplication by ϕ(h) for all (H,ϕ) ∈Mk(G), h ∈ H.

(iii) Restrictions and conjugations commute, i.e. cg(K,ψ) r(H,ϕ)(K,ψ) = r

g(H,ϕ)g(K,ψ)

cg(H,ϕ)

for all g ∈ G, (K,ψ) ≤ (H,ϕ) in Mk(G).

Let (A, r, c) and (B, s, d) be objects in D. A morphism f from (A, r, c) to (B, s, d)is a family

f(H,ϕ) : A(H,ϕ) → B(H,ϕ) | (H,ϕ) ∈Mk(G)

of k-linear maps such that

(iv) f(K,ψ) r(H,ϕ)(K,ψ) = s

(H,ϕ)(K,ψ) f(H,ϕ) for all (K,ψ) ≤ (H,ϕ) in Mk(G).

(v) f g(H,ϕ)

cg(H,ϕ) = dg(H,ϕ) f(H,ϕ) for all g ∈ G, (H,ϕ) ∈Mk(G).

We define a k-functor Σ: C → D by Σ(F) := (A, r, c), with

A(H,ϕ) := F(SG(H,ϕ)),

r(H,ϕ)(K,ψ) := F(p

(H,ϕ)(K,ψ)),

cg(H,ϕ) := F(g(H,ϕ)),


wherep

(H,ϕ)(K,ψ) : SG(K,ψ)−→SG(H,ϕ), s⊗kK α 7→ s⊗kH α,

andg(H,ϕ) : SGg

(H,ϕ)−→SG(H,ϕ), s⊗k gH α 7→ sg ⊗kH α,

for s ∈ G, α ∈ k.For a morphism µ : F → G in C we define Σ(µ) : Σ(F)→ Σ(G) by

Σ(µ)(H,ϕ) := µSG(H,ϕ)

: F(SG(H,ϕ))−→G(SG(H,ϕ)).

We also define a functor Π: D → C by

Π(A, r, c) := D(ΣσJ−, (A, r, c)

)on objects and the obvious definition on morphisms, and leave it to the reader toshow that Σ and Π are inverse equivalences.


Chapter 6

Class Group Relations

In this chapter we recall a theorem of Yoshida which gives an equivalence betweenthe category of cohomological Mackey functors and a functor category. The in-terpretation of a cohomological Mackey functor M as a functor in the usual senseallows to deduce from each relation between the permutation modules k[G/H],H ≤ G, the same relation for the k-modules M(H), H ≤ G. Since the class groupsof intermediate fields of a Galois extension of number fields form a cohomologicalMackey functor (we prove this very carefully, although it must be obvious for ev-eryone familiar with Dedekind domains), we obtain relations between these classgroups. Moreover, we determine explicitly all relations which can be obtained inthis way by using the idempotents in the Burnside ring. In can be deduced that theclass group of the top field is uniquely determined by all other fields, if the Galoisgroup is not hypo-elementary and its reduced Euler characteristic does not vanish.Interestingly the last condition is a topological property of the simplicial complexassociated to the poset of all non-trivial proper subgroups. We do not know anygroup theoretical translation of this property.

6.1 Cohomological Mackey functors

Throughout this section k denotes a commutative ring and G denotes a finitegroup. By kG−per we denote the full subcategory of kG−mod consisting of kG-permutation modules, i.e. finitely generated k-free kG-modules with a G-stablek-basis. In other words, each object in kG−per is isomorphic to a kG-modulekS, where S is a finite G-set and kS denotes the free k-module with basis S.Let Functk(kG−per, k−Mod) be the category whose objects are k-functors fromkG−per to k−Mod, i.e. functors which are k-linear on morphisms, and whosemorphisms are the natural transformations between such functors. The followingtheorem is due to Yoshida (cf. [Yo83b, Thm. 4.3] or [Ta89, Sect. 1]).

1.1 Theorem There is an equivalence of categories

Ψ: k−Mackc(G)→ Functk(kG−per, k−Mod)

with the property that for M ∈ k−Mackc(G) and H ≤ G we have(Ψ(M)

)(k[G/H]) ∼= M(H). (6.1)

189

190 CHAPTER 6. CLASS GROUP RELATIONS

In what follows we don’t need the fact that Ψ is an equivalence, but only thatthere exists a functor Ψ such that (6.1) holds.

1.2 Corollary Let H1, . . . ,Hm, U1, . . . , Un ≤ G be (not necessarily distinct)subgroups such that

m⊕i=1

k[G/Hi] ∼=n⊕j=1

k[G/Uj ]

in kG−per. Then we have

m⊕i=1

M(Hi) ∼=n⊕j=1

M(Uj)

in k−Mod for every cohomological k-Mackey functor M on G.

Proof This is obvious from Theorem 1.1, since the functor Ψ(M) maps iso-morphic objects to isomorphic objects.

Let Perk(G) denote the Grothendieck ring of the category kG−per with respectto direct sums. For V ∈ kG−per we denote by [V ] the associated element inPerk(G). With the usual conjugation, restriction, and induction maps, the ringsPerk(H), H ≤ G, form a Z-Green functor Perk on G. The functor

k− : G−set→ kG−per, S 7→ kS,

induces a surjective ring homomorphism

iG : Ω(G)→ Perk(G).

The ring homomorphisms iH , H ≤ G, form the unique morphism i : Ω → Perkof Z-Green functors on G (cf. Proposition I.1.4). We will determine the kernel ofiG : Q⊗ Ω→ Q⊗ Perk(G) for k = Zp, the p-adic completion of Z for a prime p.

1.3 Proposition Let p be a prime. The kernel of

iG : Q⊗ Ω(G)→ Q⊗ PerZp(G)

is free on the idempotents e(G)H , where H runs through a set of representatives for the

conjugacy classes of subgroups of G which are not p-hypo-elementary (i.e. H/Op(H)is not cyclic).

Proof First we observe that the set C(Q⊗ PerZp) of coprimordial subgroupsfor Q ⊗ PerZp is the set of p-hypo-elementary subgroups. In fact, by Conlon’sinduction theorem (cf. [CR87, 80.51]) and Proposition I.6.2 we know that C(Q ⊗PerZp) is contained in the set of p-hypo-elementary subgroups of G. Conversely,we can show that each p-hypo-elementary subgroup H ≤ G is coprimordial forQ ⊗ PerZp ⊆ Q ⊗ TZp by the same proof as in Proposition III.4.14 for the trivialsource ring. Note that in Section III.4 we assumed that the complete discrete

6.2. APPLICATION TO CLASS GROUPS 191

valuation ring O is big enough. But Proposition III.4.14 also holds for O = Zp.Now the proposition follows from Proposition I.6.4.

Recall the explicit formula

e(G)H =

1

|NG(H)|∑U≤H

|U |µ(U,H)[G/U ]

from Remark I.3.3 or (A.2) in Appendix A, where µ denotes the Mobius functionof the poset of subgroups of G.

Let K0(ab) denote the Grothendieck group of the category ab of finite abeliangroups with respect to direct sums. For A ∈ ab let [A] denote the associatedelement in K0(ab). Note that two finite abelian groups A and B are isomorphic ifand only if [A] = [B] in K0(ab), since the category ab has the cancellation property(A⊕ C ∼= B ⊕ C =⇒ A ∼= B for A,B,C ∈ ab). For A ∈ ab and a prime p let Apdenote the Sylow p-subgroup of A.

We call a k-Mackey functor M on G finite, if M(H) is finite for all H ≤ G.We call a finite group hypo-elementary, if it is p-hypo-elementary for some

prime p.

1.4 Corollary Let M be a finite cohomological Z-Mackey functor on G.(a) Let p be a prime and H ≤ G a subgroup which is not p-hypo-elmentary, then

we have the relation ∑U≤H

|U |µ(U,H)[M(U)p] = 0

in K0(ab).(b) Let H ≤ G be a subgroup which is not hypo-elementary, then we have the

relation ∑U≤H

|U |µ(U,H)[M(U)] = 0

in K0(ab).

Proof (a) Note that Mp(H) := M(H)p for H ≤ G defines a cohomologicalZp-Mackey functor on G. Hence, the result follows from Proposition 1.3 and Corol-lary 1.2, since two ZpG-permutation modules are isomorphic if and only if theirassociated elements in PerZp(G) are equal.

(b) This follows from (a), since A = ⊕pAp for any finite abelian group A, wherep runs over the set of all primes.

6.2 Application to class groups

In this section we are going to apply the results of the previous section to theparticular finite cohomological Z-Mackey functor formed by the class groups of theintermediate fields of a Galois extension of number fields.

For a number field K let OK denote its ring of integers and IK the multiplicativegroup of fractional ideals of K, i.e. finitely generated non-trivial OK-submodules ofK. The group IK is free on the set of non-trivial prime ideals of OK . Let PK ≤ IK


be the subgrop of principal fractional ideals, i.e. non-trivial OK-submodules of Kwhich are generated by one element. The factor group CL(K) := IK/PK is finiteand is called the ideal class group (or just class group) of K.

Now let L/K be a finite Galois extension of number fields, and denote by G theGalois group of L/K. For H ≤ G, let LH denote the fixed field of H in L/K. ForU ≤ H ≤ G and g ∈ G we have homomorphisms

cg,H : ILH −→ ILgH

= IgLH , ℘ 7→ g℘,

resHU : ILH −→ ILU , ℘ 7→ ℘ · OLU ,

where 0 6= ℘ is a prime ideal of OLH , and

indHU : ILU −→ ILH , P 7→ NLU/LH (P) = ℘fLU/LH

(P),

where 0 6= P is a prime ideal of OLU , ℘ := OLH ∩ P, and fLU/LH (P) denotes the

residue degree of P over LH . Note that these three maps take principal fractionalideals to principal fractional ideals. Hence, we also obtain group homomorphisms

cg,H : CL(LH)−→CL(LgH),

resHU : CL(LH)−→CL(LU ),

indHU : CL(LU )−→CL(LH),

for U ≤ H ≤ G and g ∈ G.

2.1 Lemma Keep the above notation.

(i) The map resHU : ILH → ILU is injective for U ≤ H ≤ G.

(ii) We have indG1(resG1 (℘)

)= ℘|G| for each non-zero prime ideal ℘ of OK .

(iii) We have resG1(indGH(P)) =

∏g∈G/H res

gH

1 ( gP) for H ≤ G and each non-zeroprime ideal P of OLH .

(iv) We have

resGU(indGH(P)

)=

∑g∈U\G/H

indUU∩ gH

(res

gHU∩ gH(gP)

)for U,H ≤ G and each non-zero prime ideal P of OLH .

Proof (i) The extensions of two different prime ideals have no prime ideal incommon.

(ii) If ℘OL = (P1 · · · · · Pr)e with prime ideals P1, . . . ,Pr of OL and e theramification index of each Pi over K, then NL/K(Pi) = ℘f , where f is the residuedegree of Pi over K, for i = 1, . . . , r. Now the result follows from the well-knownequation [L : K] = r · e · f .

(iii) We first prove the result in the special case where H = 1. Let ℘ := OK ∩P,and ℘ · OL = (P1 · · · · · Pr)e with prime ideals P1, . . . ,Pr of OL, where e is theramification index of Pi over K for i = 1, . . . , r. Then

resG1(indG1 (P)) = resG1 (℘f ) = (P1 · · · · · Pr)ef =

∏g∈G

gP,


where f denotes the residue degree of P over K. Now, if H is arbitrary, then

resG1(indGH(P)

)|H|=(resG1 indGH

)(indH1 (resH1 (P))

)=(resG1 indG1

)(resH1 (P)

)=∏g∈G

g(resH1 (P))

=( ∏g∈G/H

g(resH1 (P)))|H|

.

Since IL is torsion free, the result follows.(iv) Since resU1 is injective by (i), it suffices to show

(resG1 indGH)(P) =∑

g∈U\G/H

(resU1 indUU∩ gH)(res

gHU∩ gH(gP)).

By (iii) we have

(resG1 indGH)(P) =∑

g∈G/H

resgH

1 (gP)),

and for s ∈ G,

(resU1 indUU∩ sH)(res

sHU∩ sH(sP)) =

∑t∈U/U∩ sH

tresU∩sH

1 (ressHU∩ sH(sP))

=∑

t∈U/U∩ sH

restsH

1 (tsP).

Now the result follows, since if s runs through a set of representatives for the doublecosets U\G/H and for each s, t runs through a set of representatives for U/U ∩ sH,then ts runs through a set of representatives for G/H.

2.2 Corollary Keep the above notations. The groups ILH and CL(LH), H ≤G, form a finite cohomological Z-Mackey functor on G.

Proof The Mackey axiom and cohomologicality axiom for ILH , H ≤ G, followfrom Lemma 2.1 (iv) and (i). The other axioms in Definition I.1.1 are clearlysatisfied. Since the axioms hold for ILH , they also hold for CL(LH), H ≤ G.

2.3 Corollary We keep the above notations.(a) Let p be a prime. If H ≤ G is not p-hypo-elementary, then we have a relation∑

U≤H|U |µ(U,H)

[CL(LU )p

]= 0

in K0(ab). In particular, CL(LH)p is uniquely determined by the groups CL(LU )p,1 ≤ U < H, through this relation.

(b) If H ≤ G is not hypo-elementary, then we have a relation∑U≤H

|U |µ(U,H)[CL(LU )

]= 0


in K0(ab). In particular CL(LH) is uniquely determined by the group CL(LU ),1 ≤ U < H, through this relation.

(c) If H ≤ G is not hypo-elementary and µ(1, H) 6= 0, then the class groupCL(L) is uniquely determined by the class groups CL(LU ), 1 < U ≤ H, through therelation in (b).

Proof Part (a) and (b) follow immediately from Corollary 1.4, and part (c)follows from part (b).

2.4 Example We keep the above notation.

(a) Let G be isomorphic to C2×C2, where Cn denotes the cyclic group of ordern ∈ N. Let L1, L2, L3 denote the three intermediate fields different from L and K.Since G is not p-hypo-elementary for odd primes p, we obtain from Corollary 2.3 (b):

2[CL(L)p

]− 2[CL(L1)p

]− 2[CL(L2)p

]− 2[CL(L3)p

]+ 4[CL(K)p

]= 0

in K0(ab) for each odd prime p, or

CL(K)odd ⊕ CL(K)odd ⊕ CL(L)odd∼= CL(L1)odd ⊕ CL(L2)odd ⊕ CL(L3)odd

as abelian groups, where Aodd denotes the odd part of A for A ∈ ab.

(b) Let G be the symmetric group S3 on three letters. Let H ≤ G be the Sylow3-subgroup and let U be a Sylow 2-subgroup. G is not p-hypo-elementary for all

primes different from 3. The idempotent e(G)G is given by

e(G)G =

1

2

([G/1]− 2[G/U ]− [G/H] + 2[G/G]

).

Hence, we obtain an isomorphism

CL(K)3′ ⊕ CL(K)3′ ⊕ CL(L)3′∼= CL(LU )3′ ⊕ CL(LU )3′ ⊕ CL(LH)3′

of abelian groups.

(c) The smallest non hypo-elementary group is the dihedral group D12 of order12. Moreover µ(1, D12) = −6 6= 0 so that part (c) of Corollary 2.3 can be applied.The same holds for G = S4, the symmetric group on 4 letters, where we haveµ(1, S4) = −12. But there are also minimal non-hypo-elementary groups G withµ(1, G) = 0, so that part (c) of Corollary 2.3 cannot be applied as for example thegroup SL2(F3).

The results we obtained by using the Mackey functor structure should be com-pared with the following classical results. Let hK denote the class number of K,i.e. the order of CL(K).

2.5 Theorem (Dirichlet 1842, [Di42]) Let 1 < d ∈ N be square-free, and letL = Q(

√d,√−d). Then

2hL = Q · hQ(√d) · hQ(

√−d),

where Q ∈ 1, 2 depends on d.


2.6 Theorem (Brauer 1951, [Bra51]) Let G be a finite group and let∑H≤G

aH indGH(1) = 0

be a relation in the character ring R(G) with integers aH , H ≤ G. Then the product∏H≤G

haHLH

assumes only finitely many values, as L runs over all Galois extensions of Q withGalois group G.

2.7 Remark (a) Note that Theorem 2.6 can be considered as a generalizationof Theorem 2.5. Both theorems make use of number theoretic results. Brauer forinstance uses the theory of Artin L-functions and the analytic class number formula.In contrast to this we only used very little knowledge from number theory to verifythat there is a Mackey functor structure. The rest followed from the general theoryof Mackey functors. Note also that the character relations used in Theorem 2.6 are

given by the idempotents e(G)H ∈ Q ⊗ Ω(G) for non-cyclic subgroups H ≤ G. This

can be proved similar to Proposition 1.3. Hence Brauer obtains more relations thanwe do in Corollary 2.3. But the relations are only exact up to a factor which mayassume finitely many values. This finiteness property cannot be obtained by ourapproach. By this finiteness property Dirichlet obtains for the group G = C2 × C2

even for the prime 2 a relation up to the factor Q, whereas we can’t say anythingabout the 2-part of the class numbers involved, cf. Example 2.4 (a). But note thatin general, when the range of the factor in Brauer’s theorem is unknown we obtainfor almost all primes exact relations, even for the class groups (not only the classnumbers), and we are not forced to assume K = Q.

(b) There have been improvements of Brauer’s result. Walter proved in [Walt79,Thm. 2.1] that each relation∑

H≤GaH indGH(1) =

∑H≤G

bH indGH(1)

in the character ring R(G) with aH , bH ∈ N0, induces a relation⊕H≤G

CL(LH)aHπ∼=⊕H≤G

CL(LH)bHπ ,

where π is the set of all primes which do not divide |G|. This again is covered byour results, since each relation in the character ring comes from the idempotents

e(G)H , H ≤ G non-cyclic, and since each non-cyclic H ≤ G is non-p-hypo-elementary

if p does not divide |G|.(c) Roggenkamp and Scott have an approach in [RS82] which is very similar

to the one we outlined here. What they call ‘Hecke actions’ is basicly a functorin Funct(kG−per, k−Mod). They showed that Picard groups (in particular classgroups) give rise to a Hecke action. In particular they have a result which is analo-gous to Corollary 1.2. But they didn’t determine the possible relations in kG−perexplicitly.


(d) Assume that G is not hypo-elementary. Then the class group CL(K) ofthe bottom field is uniquely determined by the class groups of the bigger fields.In contrast to this, we need the additional information µ(1, G) 6= 0 for CL(L)being uniquely determined by the class groups of smaller fields. This rises thegeneral question: What group theoretic properties of a finite group G imply thatthe reduced Euler characteristic

χ(G) := −1 +∑

1<H0<···<Hn<G(−1)n = µ(1, G)

of the poset H | 1 < H < G does not vanish. We only have a negative state-ment: If G has a non-trivial Frattini subgroup, then µ(1, G) = 0, since the poset iscontractible with respect to the Frattini subgroup.

Appendix A

The Burnside Ring

For the reader’s convenience we fix the notation and recall some of the basic factsabout the Burnside ring; see [CR87, §80] for more details.

The Burnside ring Ω(G) of a finite group G is the Grothendieck ring of thecategoryG−set of finite leftG-sets. For any S ∈ G−set we denote by [S] ∈ Ω(G) itsimage in the Burnside ring. Each finite G-set S is a disjoint union S = S1

.∪ . . .

.∪Sr

of transitive G-sets S1, . . . , Sr which are uniquely determined (namely as the G-orbits of S), and each transitive G-set is isomorphic to the set of cosets G/Hfor some H ≤ G with the obvious left G-action. For K,H ≤ G one knows thatG/H ∼= G/K in G−set, if and only if H and K are G-conjugate. Therefore theelements [G/H] ∈ Ω(G), where H runs through a set RG of representatives of theconjugacy classes of subgroups of G, form a Z-basis of Ω(G). Two G-sets S and S′

are isomorphic if and only if [S] = [S′] ∈ Ω(G).The direct product S × S′ of two finite G-sets S and S′ is again a G-set by

the diagonal action g(s, s′) := ( gs, gs′) for g ∈ G, s ∈ S, s′ ∈ S′. This induces amultiplication [S] · [S′] := [S×S′] on Ω(G) which furnishes Ω(G) with the structureof a commutative ring with unity [G/G]. The multiplication is given explicitly by

[G/K] · [G/H] =∑

g∈K\G/H

[G/K ∩ gH]

for K,H ≤ G.The ring structure is most efficiently studied via the ring homomorphisms

φ(G)H : Ω(G)→ Z, [S] 7→ |SH |, H ≤ G,

where for S ∈ G−set, |SH | denotes the cardinality of the set SH of H-fixed points of

S. For G-conjugate subgroups K and H of G one has φ(G)K = φ

(G)H , and the collection

of these ring homorphisms forms a ring homomorphism, the mark homomorphism

ρG : Ω(G)→( ∏H≤G

Z)G ∼= ∏

H∈RG

Z,

whose image is contained in the set of G-fixed points of∏H≤G Z, where G acts by

permuting the components according to the conjugation action on the index set ofsubgroups of G. The mark homomorphism ρG is injective (i.e. two finite G-sets S

197

198 APPENDIX A. THE BURNSIDE RING

and S′ are isomorphic, if and only if |SH | = |S′H | for all H ≤ G). Therefore, bycounting ranks, the Q-algebra homomorphism

Q⊗ ρG : Q⊗ Ω(G)→ Q⊗( ∏H≤G

Z)G ∼= ( ∏

H≤GQ)G ∼= ∏

H∈RG

Q

is an isomorphism. This shows that Q ⊗ Ω(G) is semisimple with primitive idem-

potents e(G)H ∈ Q⊗ Ω(G) defined by

φ(G)K (e

(G)H ) =

1, if K and H are G-conjugate,

0, otherwise,(A.1)

for K,H ≤ G. Obviously e(G)H = e

(G)K for G-conjugate subgroups K and H of G,

and the set e(G)H | H ∈ RG is a set of primitive mutually orthogonal idempotents

which sum up to 1 ∈ Q⊗Ω(G). This set of idempotents forms a Q-basis of Q⊗Ω(G)

and there is an explicit formula (see [Glu81]) which expresses e(G)H for H ≤ G in

terms of the basis [G/K], K ∈ RG, namely

e(G)H =

1

|NG(H)|∑

K≤H≤G|K|µ(K,H) [G/K], (A.2)

where µ denotes the Mobius function on the poset of subgroups of G ordered byinclusion, c.f. Appendix B. From Equation (1) one can easily derive the explicitformula

(Q⊗ ρG)−1 :( ∏H≤G

Q)G

→ Q⊗ Ω(G),

(xH)H≤G 7→ 1

|G|∑

K≤H≤G|K|µ(K,H)xH [G/K].

(A.3)

For g ∈ G and H ≤ G we have conjugation, restriction and induction functorscg,H : H−set → gH−set, resGH : G−set → H−set, indGH : H−set → G−set whichinduce maps

cg,H : Ω(H)→ Ω( gH), [H/K] 7→ [ gH/ gK],

resGH : Ω(G)→ Ω(H), [G/U ] 7→∑

g∈H\G/U

[H/H ∩ gU ],

indGH : Ω(H)→ Ω(G), [H/K] 7→ [G/K],

for K ≤ H and U ≤ G, called conjugation, restriction, and induction maps.Conjugation and restriction maps are ring homomorphisms, whereas induction isjust a group homomorphism.

For K,H ≤ G, the element resGH(e(G)K ) is an idempotent in Q⊗Ω(H), since resGH

is a ring homomorphism. Since

φ(H)U

(resGH(x)

)= φ

(G)U (x)

199

for arbitrary U ≤ H ≤ G, x ∈ Ω(G), we have

resGH(e(G)K ) =

∑K′∈RHK′=GK

e(H)K′ (A.4)

where RH is a set of representatives for the H-conjugacy classes of subgroups of H.For a commutative ring k we can form the k-algebra k⊗Ω(G), whose elements we

write as k-linear combinations of the basis [G/H], H ∈ RG, i.e. we omit the tensor

product for elements in k⊗Ω(G) as already done in Equation (2). The maps φ(G)H ,

ρG, cg,H , resGH , indGH for g ∈ G, H ≤ G, have k-tensored versions which we denoteby the same symbols. If |G| is invertible in k, then ρG : k ⊗ Ω(G)→ (

∏H≤G k)G is

invertible and the formulae (A.2) and (A.3) remain valid.

200 APPENDIX A. THE BURNSIDE RING

Appendix B

Posets and Mobius Inversion

The explicit induction formulae in Chapter 2 involve alternating sums over sets ofchains in posets, which can be expressed in terms of the Mobius function of theconsidered poset. This appendix collects the relevant notions and results aboutMobius inversion and also fixes the notation. For further reference see for example[Ro64].

B.1 Definition A partially ordered set (or for short, a poset) (X,≤) is a setX together with a partial order ≤ on X, i.e. a relation ≤ on X satisfying thefollowing axioms:

(P1) (Reflexivity) x ≤ x for x ∈ X.

(P2) (Antisymmetry) If x, y ∈ X, x ≤ y, y ≤ x, then x = y.

(P3) (Transitivity) If x, y, z ∈ X, x ≤ y, y ≤ z, then x ≤ z.

For x, y ∈ X we will write x < y if x ≤ y and x 6= y. We will mostly write just Xinstead of (X,≤). The zeta function (or incidence function) on X, ζX : X ×X →Z, is defined by

ζX(x, y) :=

1, if x ≤ y,

0, otherwise,

for x, y ∈ X. If X is finite, and if we enumerate the elements of X = x1, . . . , xrin such a way that xi ≤ xj implies i ≤ j for i, j ∈ 1, . . . , r (this is always possibleby first enumerating the minimal elements of X, then the minimal elements of theremaining elements, and so on), then the incidence matrix of X,

ZX :=(ζX(xi, yj)

)i,j

=(ζX(x, y)

)x,y∈X ,

is an upper triangular matrix whose diagonal entries are all equal to 1. Hence, ZXis invertible over Z, and the entries of the Mobius matrix MX of X, where

MX := Z−1X =

(µX(x, y)

)x,y∈X ,

define the Mobius function µX : X ×X → Z.

A chain of length n ∈ N0 in X is a sequence (x0, . . . , xn) ∈ Xn+1 with xi−1 < xifor i = 1, . . . , n. We will use the notation (x0 < . . . < xn) for such a chain. The

201

202 APPENDIX B. POSETS AND MOBIUS INVERSION

length of the chain σ = (x0 < . . . < xn) will be denoted by |σ|. For x, y ∈ X andn ∈ N0 we denote by ΓnX(x, y) (resp. ΓnX , resp. Γ(X)) the set of all chains of lengthn in X with minimal element x and maximal element y (resp. the set of all chainsof length n, resp. the set of all chains in X). The cardinality of ΓnX(x, y) (resp. ΓnX)will be denoted by γnX(x, y) (resp. γnX).

Let I denote the identity matrix. An induction argument shows that, for x, y ∈X and n ∈ N0, the (x, y)-entry of (ZX − I)n is equal to γnX(x, y). Since ZX − I isnilpotent, we can form the matrix

∑n≥0(−1)n(ZX − I)n, which is equal to MX =

Z−1X by the usual property of geometric series. Hence we obtain an explicit formula

for the Mobius function:

µX(x, y) =∑n≥0

(−1)nγnX(x, y),

for x, y ∈ X. In particular, µX(x, y) = 0 unless x ≤ y, and µX(x, x) = 1 forx, y ∈ X.

B.2 Proposition Let A be an (additively written) abelian group, X a finiteposet and f, g : X → A arbitrary functions.

(i) g(y) =∑

x≤y f(x) for all y ∈ X, if and only if, f(y) =∑

x≤y µX(x, y)g(x)for all y ∈ X.

(ii) g(x) =∑

x≤y f(y) for all x ∈ X, if and only if, f(x) =∑

x≤y µX(x, y)g(y)for all x ∈ X.

Proof (i) We define row vectors(f(x)

)x∈X and

(g(x)

)x∈X . Then the first

equations can be written as one matrix equation,(g(y)

)y∈X =

(f(x)

)x∈X · ZX ,

and the second equations can be written as(f(y)

)y∈X =

(g(x)

)x∈X ·MX .

Since MX = Z−1X , the two equations are equivalent.

(ii) This follows similarly to part (i) by additionally transposing ZX and MX ,or what amounts to the same, by considering the ‘opposite’ poset of X, where allthe relations are reversed.

B.3 Corollary Let A be an abelian group, X a finite poset and f : X → A afunction.

(i) For z ∈ X we have

f(z) =∑x≤y≤z

µX(y, z)f(x)

andf(z) =

∑x≤y≤z

µX(x, y)f(x).

(ii) For x ∈ X we have

f(x) =∑x≤y≤z

µX(x, y)f(z)

203

andf(x) =

∑x≤y≤z

µX(y, z)f(z).

Proof (i) When we substitute the expression for g in Proposition B.2 (i) inthe expression for f , and the other way round, we obtain exactly the two equationsof part (i).

(ii) is proved as (i) using Proposition B.2 (ii).

B.3 Let G be a finite group. A (left) G-poset X is a poset (X,≤) togetherwith a left G-action on X such that the action of any fixed group element g ∈ G isa poset automorphism, i.e. if x, y ∈ X with x ≤ y then gx ≤ gy. The set G\X ofG-orbits is then again a poset, called the orbit poset, by the following definition:

[x]G ≤ [y]G :⇐⇒ x ≤ gy for some g ∈ G,

for x, y ∈ X, where [x]G denotes the G-orbit in X containing x.If X is a G-poset, then ΓX and ΓnX , for n ∈ N0, are G-sets by the obvious action

on chains. Note that there is a canonical map

G\ΓnX(x, y)→ ΓnG\X([x]G, [y]G

), [(x0 < . . . < xn)]G 7→

([x0]G < . . . < [xn]G

),

which is obviously surjective, but in general not injective.

204 APPENDIX B. POSETS AND MOBIUS INVERSION

Bibliography

[Al87] J. L. ALPERIN: Weights for finite groups. Proc. Sympos. Pure Math. 47(1987), 369–379.

[At61] M.F. ATIYAH: Characters and the cohomology of finte groups. Inst.Hautes Etudes Sci. Publ. Math. no. 9 (1961), 23–64.

[Be84a] D. BENSON: Lambda and psi operations on Green rings. J. Algebra 87(1984), 360–367.

[Be84b] D. BENSON: Modular representation theory: New trends and methods.Springer Lecture Notes 1081, Springer-Verlag 1984.

[Bo89] R. BOLTJE: Canonical and explicit Brauer induction in the characterring of a finite group and a generalization for Mackey functors. Thesis,Universitat Augsburg 1989.

[Bo90] R. BOLTJE: A canonical Brauer induction formula. Asterisque 181–182(1990), 31–59.

[Bo94] R. BOLTJE: Identities in representation theory via chain complexes.J. Algebra 167 (1994), 417–447.

[BCS93] R. BOLTJE, G.-M. CRAM, V. SNAITH: Conductors in the non-separableresidue field case. In: Algebraic K-Theory and Algebraic Topology,P.G. Goerrs and J.F. Jardine (editors), Kluwer 1993, 1–34.

[BSS92] R. BOLTJE, V. SNAITH, P. SYMONDS: Algebraicisation of explicitBrauer induction. J. Algebra 148 (1992), 504–527.

[Bra47] R. BRAUER: On Artin’s L-series with general group characters. Ann. ofMath. 48 (1947), 502–514.

[Bra51] R. BRAUER: Beziehungen zwischen Klassenzahlen von Teilkorpern einesGaloisschen Korpers. Math. Nachr. 4 (1951), 158–174.

[Bro85] M. BROUE: On Scott modules and p-permutation modules: An approachthrough the Brauer morphism. Proc. Amer. Math. Soc. 93 (1985), 401–408.

[BF59] R. BRAUER, W. FEIT: On the number of irreducible characters of finitegroups in a given block. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 361–36.

205

206 BIBLIOGRAPHY

[CR81] C.W. CURTIS, I. REINER: Methods of representation theory, Vol. 1.J. Wiley & Sons, New York 1981.

[CR87] C.W. CURTIS, I. REINER: Methods of representation theory, Vol. 2.J. Wiley & Sons, New York 1987.

[Di42] L. DIRICHLET: Recherches sur les formes quadratiques a coefficients eta indeterminees complexes. J. Reine Angew. Math. 24 (1842), 291–371.

[Dr71] A.W.M. DRESS: The ring of monomial representations, I. Structure the-ory. J. Algebra 18 (1971), 137–157.

[Dr73] A.W.M. DRESS: Contributions to the theory of induced representations.In: ed. H. Bass, Algebraic K-Theory II, Lecture Notes in Mathematics342, Springer-Verlag, Berlin Heidelberg New York 1973, 183–240.

[Ev63] L. EVENS: A generalization of the transfer map in the cohomology ofgroups. Trans. Amer. Math. Soc. 108 (1963), 54–65.

[Gla68] G. GLAUBERMAN: Correspondences of characters for relatively primeoperator groups. Canad. J. Math. 20 (1968), 1465–1488.

[Glu81] D. GLUCK: Idempotent formula for the Burnside algebra with applica-tions to the p-subgroup complex. Illinois J. Math. 25 (1981), 63–67.

[Gr71] J.A. GREEN: Axiomatic representation theory for finite groups J. PureAppl. Algebra 1 (1971), 41–77.

[Hu67] B. HUPPERT: Endliche Gruppen I. Springer-Verlag 1967.

[Is73] I.M. ISAACS: Characters of solvable and symplectic groups. Amer. J.Math. 95 (1973), 594–635.

[Is76] I.M. ISAACS: Character theory of finite groups. Academic Press 1976.

[Is81] I.M. ISAACS: Extensions of group representations over arbitrary fields.J. Algebra 68 (1981), 54–74.

[Ke75] M. KERVAIRE: Operations d’Adams en theorie des representationslineaires des groupes finis. Enseign. Math., t. XXII (1975), 1–28.

[KO74] M.-A. KNUS, M. OJANGUREN: Theorie de la descente et algebresd’Azumaya. Lecture Notes in Mathematics, Vol. 389, Springer-Verlag,Berlin, 1974.

[Kr87] O. KROLL: An algebraic characterisation of Chern classes of finite grouprepresentations. Bull. London Math. Soc. 19 (1978), 245–248.

[Ne92] J. NEUKIRCH: Algebraische Zahlentheorie. Springer Verlag 1992.

[Pu88] L. PUIG: Pointed groups and constructions of modules. J. Algebra 116(1988), 7–129.

BIBLIOGRAPHY 207

[Re71] W. F. REYNOLDS: Twisted group algebras over arbitrary fields. IllinoisJ. Math. 15 (1971), 91–103.

[RS82] K. ROGGENKAMP, L. SCOTT: Hecke actions on Pcard groups. J. PureAppl. Algebra 26 (1982), 85–100.

[Ro64] G. C. ROTA: On the foundations of combinatorial theory, I. Theory ofMobius functions. Z. Wahrsch. verw. Gebiete 2 (1964), 340–368.

[Se78] J.-P. SERRE: Representation lineaires des groupes finis. 3eme ed., Her-mann, Paris 1978.

[Sn88] V. SNAITH: Explicit Brauer induction. Invent. Math. 94 (1988), 455–478.

[Su86] M. SUZUKI: Group theory, Vol. 2. Springer Verlag 1986.

[Sy91] P. SYMONDS: A splitting principle for group representations. Comment.Math. Helvetici 66 (1991), 169–184.

[Ta89] D. TAMBARA: Homological properties of the endomorphism rings ofcertain permutation modules. Osaka J. Math. 26 (1989), 807–828.

[tD87] T. tom DIECK: Transformation groups. Walter de Gruyter, Berlin - NewYork 1987.

[Th88] J. THEVENAZ: Some remarks on G-functors and the Brauer morphism.J. Reine Angew. Math. 384 (1988), 24–56.

[TW90] J. THEVENAZ, P. WEBB: A Mackey functor version of a conjecture ofAlperin. Asterisque 181-182 (1990), 263–272.

[TW91] J. THEVENAZ, P. WEBB: The structure of Mackey functors. Preprint1991.

[Walk81] J.W. WALKER: Homotopy type and Euler characteristic of partiallyordered sets. Europ. J. Combin. 2 (1981), 373–384.

[Walt79] C.D. WALTER: Brauer’s class number relation. Acta Arith. 35 (1979),33–40.

[We91] P. WEBB: A split exact sequence of Mackey functors. Comment. Math.Helv. 66 (1991), 34–69.

[Wo78] T.R. WOLF: Character correspondences in solvable groups. Illinois J.Math. 22 (1978), 327–340.

[Yo80] T. YOSHIDA: On G-functors (I): Transfer theorems for cohomologicalG-functors. Hokkaido Math. J. 9 (1980), 222–257.

[Yo83a] T. YOSHIDA: Idempotents of Burnside rings and Dress induction theo-rem. J. Algebra 80 (1983), 90–105.

[Yo83b] T. YOSHIDA: On G-functors (II): Hecke Operations and G-functors.J. Math. Soc. Japan 35 (1983), 179–190.

208 BIBLIOGRAPHY

List of Symbols

The symbols are listed together with the pagenumber of first occurence or with thepagenumber where they are explained.

Chapter I

cg,H . . . . . . . . . . . . . . . . . . 18

resHK . . . . . . . . . . . . . . . . . . 19

indHK . . . . . . . . . . . . . . . . . .19

k−Mack(G) . . . . . . . . . .19

k−Mackc(G) . . . . . . . . . 19

k−Mackalg(G) . . . . . . . 20

k−Mackcalg(G) . . . . . . . 20

Ω(G) . . . . . . . . . . . . . . . . . 21

k−Res(G) . . . . . . . . . . . . 22

k−Resalg(G) . . . . . . . . . 23

Rab . . . . . . . . . . . . . . . . . . . 23

k−Con(G) . . . . . . . . . . . 24

k−Conalg(G) . . . . . . . . . 24

PX(H) . . . . . . . . . . . . . . . 24

X+(H) . . . . . . . . . . . . . . . 25

c+g,H . . . . . . . . . . . . . . . . . . 25

res+HK . . . . . . . . . . . . . . . . 25

ind+HK . . . . . . . . . . . . . . . . 25

f+H . . . . . . . . . . . . . . . . . . . .25

SA(H) . . . . . . . . . . . . . . . . 26

A+(H) . . . . . . . . . . . . . . . 26

I(kH) . . . . . . . . . . . . . . . . 26

[K, a]H . . . . . . . . . . . . . . . 26

c+g,H . . . . . . . . . . . . . . . . . 26

res+HK . . . . . . . . . . . . . . . . 27

ind+HK . . . . . . . . . . . . . . . . 27

f+H . . . . . . . . . . . . . . . . . . 27

Z . . . . . . . . . . . . . . . . . . . . . 27

ρH . . . . . . . . . . . . . . . . . . . .27

I(M) . . . . . . . . . . . . . . . . . 28

P(M) . . . . . . . . . . . . . . . . 29

K(A) . . . . . . . . . . . . . . . . . 29

C(A) . . . . . . . . . . . . . . . . . .29

M(H) . . . . . . . . . . . . . . . . 29

T (M) . . . . . . . . . . . . . . . . 30

πA . . . . . . . . . . . . . . . . . . . .31

ρA . . . . . . . . . . . . . . . . . . . . 31

σAH . . . . . . . . . . . . . . . . . . . 31

µ . . . . . . . . . . . . . . . . . . . . . 32

e(G)H . . . . . . . . . . . . . . . . . . .33

λA,X . . . . . . . . . . . . . . . . . .34

σA,B . . . . . . . . . . . . . . . . . . 34

ρA,B . . . . . . . . . . . . . . . . . . 34

θA,B . . . . . . . . . . . . . . . . . . 34

Chapter II

bM,A . . . . . . . . . . . . . . . . . . 44

B(H) . . . . . . . . . . . . . . . . . 49

(H,ϕ) . . . . . . . . . . . . . . . . 49

M(H) . . . . . . . . . . . . . . . . 49

m(H,ϕ)(K,ψ) . . . . . . . . . . . . . . . . 50

aM,A,p . . . . . . . . . . . . . . . . 51

mϕ(χ) . . . . . . . . . . . . . . . . 53

Γ(M(H)) . . . . . . . . 53, 202

mσ . . . . . . . . . . . . . . . . . . . 53

NπH(σ) . . . . . . . . . . . . . . . . 54

Γπ(M(H)) . . . . . . . . . . . .55

Γπi (M(H)) . . . . . . . . . . . .55

H(s) . . . . . . . . . . . . . . . . . .55

f . . . . . . . . . . . . . . . . . . . . . 55

g(χ, σ) . . . . . . . . . . . . . . . . 56

Chapter III

RK(G) . . . . . . . . . . . . . . . 62

RabK (G) . . . . . . . . . . . . . . . 62

G(K) . . . . . . . . . . . . . . . . . 62

IrrK(G) . . . . . . . . . . . . . . .62

209

210 LIST OF SYMBOLS

resf . . . . . . . . . . . . . . . . . . .62

res+f . . . . . . . . . . . . . . . . . 63

infGG/N . . . . . . . . . . . . . . . . 63

inf+GG/N . . . . . . . . . . . . . . 63

[−,−]G . . . . . . . . . . . . . . . 63

ε . . . . . . . . . . . . . . . . . . . . . 63

V (H,ϕ) . . . . . . . . . . . . . . . . 63

Z(χ) . . . . . . . . . . . . . . . . . .65

Rϑ(H) . . . . . . . . . . . . . . . . 71

Irrϑ(H) . . . . . . . . . . . . . . . 71

Rabϑ (H) . . . . . . . . . . . . . . . 71

Irr∗ϑ(H) . . . . . . . . . . . . . . . 71

RF (G) . . . . . . . . . . . . . . . . 73

RabF (G) . . . . . . . . . . . . . . . 73

[V ] . . . . . . . . . . . . . . . . . . . 73

Fϕ . . . . . . . . . . . . . . . . . . . . 73

ε . . . . . . . . . . . . . . . . . . . . . 74

dG . . . . . . . . . . . . . . . . . . . . 76

PF (G) . . . . . . . . . . . . . . . . 80

P abF (G) . . . . . . . . . . . . . . . 80

[P ] . . . . . . . . . . . . . . . . . . . 80

ε+ . . . . . . . . . . . . . . . . . . . . 80

PO . . . . . . . . . . . . . . . . . . . 82

P abO . . . . . . . . . . . . . . . . . . . 82

RK,p . . . . . . . . . . . . . . . . . . 82

RabK,p . . . . . . . . . . . . . . . . . . 82

RK,π . . . . . . . . . . . . . . . . . .83

RabK,π . . . . . . . . . . . . . . . . . .83

OG−lin . . . . . . . . . . . . . . 84

OG−triv . . . . . . . . . . . . . 85

G(O) . . . . . . . . . . . . . . . . . 85

Oϕ . . . . . . . . . . . . . . . . . . . 85

LO(G) . . . . . . . . . . . . . . . . 85

TO(G) . . . . . . . . . . . . . . . . 85

LabO (G) . . . . . . . . . . . . . . . 85

T abO (G) . . . . . . . . . . . . . . . 85

[Oϕ] . . . . . . . . . . . . . . . . . . 85

A′(H) . . . . . . . . . . . . . . . . 90

B′(H) . . . . . . . . . . . . . . . . 90

p′H . . . . . . . . . . . . . . . . . . . .91

a′H . . . . . . . . . . . . . . . . . . . .91

B(H) . . . . . . . . . . . . . . . . . 95

pH . . . . . . . . . . . . . . . . . . . .95

aH . . . . . . . . . . . . . . . . . . . .96

TF . . . . . . . . . . . . . . . . . . . .97

T abF . . . . . . . . . . . . . . . . . . . 97

sg . . . . . . . . . . . . . . . . . . . . 98

s(H,h) . . . . . . . . . . . . . . . . . 98

OG−lat . . . . . . . . . . . . . 100

GrO(G) . . . . . . . . . . . . . 100

Gr′O(G) . . . . . . . . . . . . . 100

GrF (G) . . . . . . . . . . . . . 102

Gr′F (G) . . . . . . . . . . . . . 102

Sp2n(Fp) . . . . . . . . . . . . .105

Sp2n . . . . . . . . . . . . . . . . . 105

Chapter IV

ΦM,AN . . . . . . . . . . . . . . . . 112

ΣM,A,aN . . . . . . . . . . . . . . .112

ΦM,A . . . . . . . . . . . . . . . . 113

ΣM,A,a . . . . . . . . . . . . . . .113

ΦM,Aalg . . . . . . . . . . . . . . . . 113

det(V ) . . . . . . . . . . . . . . .116

trH/K . . . . . . . . . . . . . . . 116

εG/H . . . . . . . . . . . . . . . . 117

det . . . . . . . . . . . . . . . . . . 118

Ψn . . . . . . . . . . . . . .120, 122

Ψn . . . . . . . . . . . . . . . . . . 121

cG . . . . . . . . . . . . . . . . . . . 123

H∗∗e (G,Z) . . . . . . . . . . . .125

chG . . . . . . . . . . . . . 126, 127

chiG . . . . . . . . . . . . . . . . . .127

GS . . . . . . . . . . . . . . . . . . 128

R(G)S . . . . . . . . . . . . . . . 129

Rab+ (G)S . . . . . . . . . . . . . 129

extG,S . . . . . . . . . . . . . . . 129

extG,S+ . . . . . . . . . . . . . . . 129

χ . . . . . . . . . . . . . . . . . . . . 131

πG,S . . . . . . . . . . . . . . . . . 132

Irr(G)S . . . . . . . . . . . . . . 132

TrS(χ) . . . . . . . . . . . . . . 132

R(G)S<S . . . . . . . . . . . . . 132

glG,S . . . . . . . . . . . . . . . . 133

glG,S+ . . . . . . . . . . . . . . . . 133

pr . . . . . . . . . . . . . . 136, 139

dp(χ) . . . . . . . . . . . . . . . . 137

Chapter V

kG−modpr . . . . . . . . . . 141

k−modpr . . . . . . . . . . . .141

Mk(G) . . . . . . . . . . . . . . 141

G\Mk(G) . . . . . . . . . . . 141

(H,ϕ) . . . . . . . . . . . . . . . 141

Rk(G) . . . . . . . . . . . . . . . 141

[H,ϕ]G . . . . . . . . . . . . . . 141

LIST OF SYMBOLS 211

V (H,ϕ) . . . . . . . . . . . . . . . 142

kG−modab . . . . . . . . . .142

kG−modprab . . . . . . . . . .142

P(V ) . . . . . . . . . . . . . . . . 142

kG−mon . . . . . . . . . . . .143

monkG(M,N) . . . . . . . 143

V . . . . . . . . . . . . . . . . . . . . 143

M(H,ϕ) . . . . . . . . . . . . .146

M(M) . . . . . . . . . . . . . . 146

M([H,ϕ]G) . . . . . . . . . . 147

M (H,ϕ) . . . . . . . . . . . . . . 147

M [H,ϕ]G . . . . . . . . . . . . . 147

P(M) . . . . . . . . . . . . . . . 150

SG(H,ϕ) . . . . . . . . . . . . . . . 150

r[H,ϕ]G(M) . . . . . . . . . . .152

[M,N ]G . . . . . . . . . . . . . 154

γG

(K,ψ),(H,ϕ). . . . . . . . . . . 155

γG

[K,ψ]G,[H,ϕ]G. . . . . . . . . .155

Rabk+(G) . . . . . . . . . . . . . .156

Rab+ (G) . . . . . . . . . . . . . . 156

[M ] . . . . . . . . . . . . . . . . . .156

Rabk . . . . . . . . . . . . . 156, 159

res+f . . . . . . . . . . . . . . . . 157

ind+GK . . . . . . . . . . . . . . . 158

[−,−]G . . . . . . . . . . . . . . 158

πG . . . . . . . . . . . . . . . . . . .159

bG . . . . . . . . . . . . . . . . . . . 159

Λ(M∗) . . . . . . . . . . . . . . . 167

dimmh(V ) . . . . . . . . . . . 168

M(G) . . . . . . . . . . . . . . . 169

S(V ) . . . . . . . . . . . . . . . . 169

A(V ) . . . . . . . . . . . . . . . . 169

FV . . . . . . . . . . . . . . . . . . 169

PV . . . . . . . . . . . . . . . . . . 169

clV (H,ϕ) . . . . . . . . . . . . 169

Cl(V ) . . . . . . . . . . . . . . . 169

rV (H,ϕ) . . . . . . . . . . . . .170

r(Cl(V )) . . . . . . . . . . . . .170

Kb(CG−mon) . . . . . . . 174

C . . . . . . . . . . . . . . . . . . . . 179

I . . . . . . . . . . . . . . . . . . . . 179

J . . . . . . . . . . . . . . . . . . . 179

T . . . . . . . . . . . . . . . . . . . 179

g . . . . . . . . . . . . . . . . . . . . 179

ζV . . . . . . . . . . . . . . . . . . . 180

κM,V . . . . . . . . . . . . . . . . 182

S . . . . . . . . . . . . . . . . . . . . 184

E . . . . . . . . . . . . . . . . . . . .184

mod−E . . . . . . . . . . . . . 184

Φ . . . . . . . . . . . . . . . . . . . . 184

Ψ . . . . . . . . . . . . . . . . . . . .184

η . . . . . . . . . . . . . . . . . . . . 184

ε . . . . . . . . . . . . . . . . . . . . 184

[(H,ϕ), s, (K,ψ)] . . . . 185

D . . . . . . . . . . . . . . . . . . . 186

A(H,ϕ) . . . . . . . . . . . . . . . 186

r(H,ϕ)(K,ψ) . . . . . . . . . . . . . . . .186

cg(H,ϕ) . . . . . . . . . . . . . . . . 186

Σ . . . . . . . . . . . . . . . . . . . . 186

Π . . . . . . . . . . . . . . . . . . . .187

Chapter VI

kG−per . . . . . . . . . . . . . 189

Funct . . . . . . . . . . . . . . . 189

Ψ . . . . . . . . . . . . . . . . . . . .189

Perk(G) . . . . . . . . . . . . . 190

[V ] . . . . . . . . . . . . . . . . . . 190

iG . . . . . . . . . . . . . . . . . . . 190

K0(ab) . . . . . . . . . . . . . . 191

OK . . . . . . . . . . . . . . . . . . 191

IK . . . . . . . . . . . . . . . . . . .191

PK . . . . . . . . . . . . . . . . . . 192

CL(K) . . . . . . . . . . . . . . .192

hK . . . . . . . . . . . . . . . . . . 194

χ(G) . . . . . . . . . . . . . . . . 196

Appendix A

Ω(G) . . . . . . . . . . . . . . . . 197

[G/H] . . . . . . . . . . . . . . . 197

φ(G)H . . . . . . . . . . . . . . . . . 197

ρG . . . . . . . . . . . . . . . . . . .197

e(G)H . . . . . . . . . . . . . . . . . 198

Appendix B

ζX(x, y) . . . . . . . . . . . . . 201

ZX . . . . . . . . . . . . . . . . . . 201

MX . . . . . . . . . . . . . . . . . 201

µX(x, y) . . . . . . . . . . . . . 201

|σ| . . . . . . . . . . . . . . . . . . . 202

ΓnX(x, y) . . . . . . . . . . . . . 202

ΓnX . . . . . . . . . . . . . . . . . . 202

Γ(X) . . . . . . . . . . . . . . . . 202

γnX(x, y) . . . . . . . . . . . . . 202

γnX . . . . . . . . . . . . . . . . . . 202

[x]G . . . . . . . . . . . . . . . . . 203

Documents

Mackey Functors and Related Structures in Representation ... · Mackey Functors and Related Structures in Representation Theory and Number Theory Robert Boltje Universit at Augsburg