G-algebras and Clifford theory [PhD thesis]

134 Pages·1997·1.304 MB·English

by  Hubert Fottner

G
{Algebras and Cli(cid:11)ord Theory
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
vorgelegt dem Rat der Fakulta(cid:127)t fu(cid:127)r Mathematik und Informatik
der Friedrich{Schiller{Universita(cid:127)t Jena.
von Dipl.-Math.
HUBERT FOTTNER
geboren am 4.5.1967
in Augsburg
Gutachter:
1. Prof. Dr. B. Ku(cid:127)lshammer
2. Prof. Dr. J. L. Alperin
3. Prof. Dr. G. R. Robinson
Tag des Rigorosums: 28.1.1997
Tag der (cid:127)o(cid:11)entlichen Verteidigung: 6.2.1997
Contents
Introduction ii
Preliminaries and notation v
Chapter I. The categories 1
1. The skew group algebra 1
2. S{, Q{ and SQ{idempotents 4
3. S{, Q{ and SQ{homomorphisms 12
4. Some functors and constructions 18
5. Morita theory for G{algebras 28
6. The category of SQ{embeddings 45
Chapter II. Induction and relative projectivity 61
1. Induction of SQ{embeddings 61
2. Higman’s criterion 77
3. Green’s indecomposability theorem revisited 85
4. Brauer characters of solvable groups 96
Chapter III. Representation rings and induction theorems 113
1. A module for the character ring 113
2. Lifting induction theorems 117
Bibliography 121
Index 122
i
Introduction
Originally, the notion of a G{algebra was introduced by Green to provide a uniform approach to
linear representations of a (cid:12)nite group, on the one hand, and blocks of the modular group algebra, on
the other hand. For instance, one wanted to relate vertex theory of modular representationswith defect
theory of blocks (or, more generally, Green theory with Brauer theory).
Let Gbe a(cid:12)nitegroup. A G{algebraisapair(A; (cid:1)) consistingofan algebraA(sayoversome(cid:12)eld
F) and a homomorphism (cid:1) from G into the automorphism group AutF(A) of A; i.e G acts on A by
automorphisms. AnimportantspecialcaseofthisnotionofaG{algebraistheoneofaninteriorG{algebra
due to Puig. An interior G{algebra is an F{algebra A together with a homomorphism from G into the
unitgroupUAofA. Clearly,anyinteriorG{algebraAmayberegardedasG{algebra,bycomposingthe
correspondinghomomorphismG!UAwiththecanonicalhomomorphismUA!AutF(A), whichmaps
a unit of A to the corresponding inner automorphism of A. Important examples of interior G{algebras
are linearrepresentationsG!GL(n,F) (n2N), where we regardthe generallinear groupGL(n,F) as
the unit group of the full matrix algebra Mat(n,F). Apart from that, any block B =FGe of the group
algebra FG (i.e. e is a primitive central idempotent in FG) is an interior G{algebra together with the
homomorphismG!UB, g 7!ge. In somerespect, interiorG{algebrashavemuch nicerpropertiesthan
arbitraryG{algebras. Forexample,thereisanotionofinductionofinteriorG{algebras,de(cid:12)nedbyPuig,
which is compatible with induction of linear representations. This induction process is closely related to
the defect theory of interior G{algebras; as one might expect having the de(cid:12)nition of a vertex in mind.
There is no complete analogue of this construction for arbitrary G{algebras. However, we will show
that { from a slightly di(cid:11)erent point of view { there is a similar construction for arbitrary G{algebras,
which, at least in some respect, behavesanalogously. For instance, one can describe the defect theory of
G{algebras in terms of this induction process. We will return to this point later.
In this thesis, we mainly deal with another aspect of G{algebra theory, namely Cli(cid:11)ord theory. In
some generalized sense, Cli(cid:11)ord theory is concerned with relating the representation theory of G with
the oneof G=N, where N is somenormalsubgroupof G. This obviousconcept form grouptheory is not
aseasytorealizein representationtheory. Forif M is an FG{moduleand N is anormalsubgroupofG,
N
the only way of associating an F[G=N]{module with M is to look at the N{(cid:12)xed points M of M (or
N N
theN{co(cid:12)xedpoints). However,ingeneral,M andM willhavenothingincommon(e.g.M mightbe
zero) unless N acts trivially on M. So this method won’t help, in general. What one does instead is to
lookat the actionof G onthe F{endomorphismsE :=EndF(M) of M, i.e. one considersE as(interior)
G{algebra together with the homomorphism G ! UE induced from the linear representation of M.
N
Then the N{(cid:12)xed points E of E (where G acts on E by conjugation; i.e. we regard E as G{algebra)
N
naturally carry the structure of a G=N{algebra. Though E is not interior, in general, it contains a
N
lot of information on M (or, vice versa, information on E provides information on M). This instance
suggests to investigate G{algebras.
If M isasemisimpleFG{module(e.g.if thecharacteristicofF doesnotdividetheorderofG), then
N N
E is a semisimple F{algebra. Hence E is the direct product of G{orbits of simple direct factors of
N N
E ; andtostudythe structureofE itsu(cid:14)cestoconsidertheactionofthe G=N{stabilizerU ofsucha
direct factor S on S (where S runs through a complete system of representatives of those G{orbits). If,
in addition, F is algebraically closed, then S is isomorphic to a full matrix algebra over F. In this case
S is, essentially, nothing but a projective representation (in the sense of Schur), by the Skolem{Noether
theorem. Thus S \corresponds" to a module of a twisted group algebra of U. So one can treat this
case without G{algebra theory; but, even in this case, the G{algebra approach could have advantages,
as section II.4 might show.
However, if F is of (cid:12)nite characteristic dividing the order of G and M is an arbitrary FG{module,
N
then E need not be semisimple; in fact, as far as we are aware, there is nothing known about the
N
structure of E , in general. So one has to deal with an arbitrary G=N{algebra. One method is then to
ii
INTRODUCTION iii
N N N N
try to reduce questions about E to questions about E =J E (where J E denotes the Jacobson
N
radicalofE ). If this is possible, one is in the situationdescribed above. Thismethod worksquite well,
(cid:0) (cid:1) (cid:0) (cid:1)
if M is relative projective to N. An important example of this concept is Th(cid:19)evenaz’ lifting theorem for
idempotents with transitive group action. We state an alternative (we believe simpli(cid:12)ed) proof of this
resultinsectionII.2 (cf. theoremII.2.16),since Th(cid:19)evenaz’theorem willserveasabasisfor severalofour
results. Some other examples of this method can be found in section II.3. Apart from that, there are
also other concepts (compare chapter III).
Our approach to G{algebra theory is rather module theoretic. Let A be a G{algebra. The most
important tool for this approach is the skew group algebra A(cid:3)G. The skew group algebra A(cid:3)G is,
on the one hand, an interior G{algebra, and, on the other hand, a crossed product. Besides, A may be
regardedas unitary G{subalgebraof A(cid:3)G. One wayof motivating this construction is the fact that the
skew group algebra provides a functor which is left adjoint to the canonical forgetful functor from the
category of interior G{algebras into the category of G{algebras (regardless whether one supplies these
categories with unitary or not necessarily unitary homomorphisms, respectively).
LetM beanA(cid:3)G{module. ThenthealgebraofA{endomorphismsEndA(M)ofM naturallycarries
the structure of a G{algebra, which we call the canonical G{algebra of M. This generalizes the notion
of a linear representation attached to an FG{module. However, the canonical G{algebra of M does not
determine the isomorphism type of M uniquely, in general. We raise the questions what information on
M iscoded in EndA(M) andwhat additionalinformation,besidesEndA(M), is neededto determinethe
isomorphism type of M.
G
Since (EndA(M)) =EndA(cid:3)G(M), anydirect summand of M is of the form e(M) for some idempo-
G
tente2(EndA(M)) uptoisomorphism. Moreover,givensuchanidempotente,thecanonicalG{algebra
ofe(M)isisomorphictoeEndA(M)e. Generalizingthisfact,wede(cid:12)nethenotionofasubquotientidem-
potent (SQI) in a G{algebra. We show that for any SQI e in EndA(M), e(M) is subquotient module of
M satisfying a certain property; and any subquotient module of M satisfying this property is ismorphic
tof(M),forsomeSQIf inEndA(M),uptoisomorphism. Roughlyspeaking,thispropertysaysthatthe
subquotient module isadirectsummand of M asanA{module. Moreover,for anySQIe ina G{algebra
B, one can de(cid:12)ne a G{algebra structure on eBe in such a way that fEndA(M)f becomes isomorphic to
the canonical G{algebra of f(M), for any SQI f in EndA(M).
Let e be an SQI in a G{algebra B. We de(cid:12)ne the notion of an SQ{embedding (or, more generally,
an SQ{homomorphism) of G{algebras such that the canonical direct embedding eBe ,! B of abstract
algebrasissuchan SQ{embedding; andanySQ{embeddingarisesas compositionof suchanSQ{embed-
ding with an isomorphism of G{algebras. We show that an SQ{homomorphism ’:A !B gives rise to
nicelybehaved\changeofringsfunctors"betweenthemodulecategoriesofA(cid:3)GandB(cid:3)G,respectively.
Moreover, any SQ{embedding :A !B induces an injective G{equivariant map P( ) from the G{set
of points of A into the G{set of points of B such that the cocycle attached to the multiplicity module of
a point (cid:11) of A coincides with the one of the multiplicity module of P( ) ((cid:11)) (cf. proposition I.3.9).
Furthermore, generalizing the notion of a bimodule, we de(cid:12)ne the notion of a twisted (A,G){B{bi-
(cid:2) (cid:3)
module. Forinstance,Beisatwisted(B,G){eBe{bimodule. Thesetwistedbimodulesgiverisetocertain
tensor product and hom functors on the module categories of A(cid:3)G and B(cid:3)G, respectively. (The above
\change of rings functors" are special instances of these functors.) Using these functors, we develop a
Morita theory for G{algebrasin section I.5, in complete analogy to ordinaryMorita theory. This theory
will be an important tool in this work. However, we believe it to be interesting in its own right.
Returning to our original question, we de(cid:12)ne a category G{Emb(B) in section I.6, the objects of
which are SQ{embeddings ’ : C ! Mat(n,F)(cid:10)F B, where C is some G{algebra and n 2 N. We show
thatonecanchooseBinsuchawaythatthereisanequivalenceofcategoriesFGfromG{Emb(B)intothe
categoryproA{A(cid:3)G of (right)A(cid:3)G{modules whichare(cid:12)nitely generatedand projectiveasA{modules.
Moreover,for any object ’:C !Mat(n,F)(cid:10)F B, the canonical G{algebraof FG’ is isomorphic to C.
Thus, in a sense, one can regardan object of G{Emb(B) as a generalizationof a linear representationof
G.
Let H be a subgroup of G. In chapter II we are concerned with the question of de(cid:12)ning a functor
G
H{Emb(B)!G{Emb(B)which\corresponds"tothecanonicalinduction functorIndH :proA{A(cid:3)H !
proA{A(cid:3)G under the above equivalence of categories. In particular, for an object ’ in H{Emb(B), we
G G
de(cid:12)neaG{algebraindH’whichisisomorphictothecanonicalG{algebraofIndHFH’. Thisde(cid:12)nition
isbasedonPuig’snotionofinductionofinteriorG{algebras,and,insomerespect,itbehavesanalogously.
We believe this de(cid:12)nition to be an adequate tool to deal with endomorphism rings of induced modules.
INTRODUCTION iv
To indicate this, we \compute" fairly well the endomorphism ring of a module induced from a subgroup
having a normal complement or supplement, respectively (cf. proposition II.3.3). In section II.3 we are
using this proposition to provegeneralizationsof Green’s indecomposability theorem (cf. theorem II.3.6,
corollary II.3.13 and theorem II.3.14). Moreover, in section II.4, this proposition will serve as a basis
for a result dealing with the question of extending modules (of twisted group algebras) from subgroups
having a normal complement (cf. proposition II.4.28, see also propositions II.4.27 and II.4.32). These
results will allow to prove that, under very speci(cid:12)c hypotheses, Green correspondents of simple modules
are simple (cf. corollariesII.4.37 and II.4.38).
Finally, in chapter III weare using the notion of a twisted bimodule to showthat K0(A(cid:3)G) maybe
regarded as (Green functor) module of the character ring. As a corollary, we prove that any induction
theorem for the character ring can be \lifted" to an induction theorem for the Green ring. This will
provideauniformproofofimportantinduction theorems,as,forexample,the onesofDressandConlon.
I wish to express my thanks to R. Boltje and G.-M. Cram for many helpful discussions. I was
lucky that these people shared their o(cid:14)ce with me during the time I was working on this thesis in
Augsburg. Inparticular,IthankR.Boltjeforfamiliarizingmewithhismethods,whichraisedmyinterest
in representation rings and Mackey functors. Moreover, I thank the Deutsche Forschungsgemeinschaft
(DFG)for(cid:12)nancialsupportwithinaGraduiertenkollegduringmytimeinAugsburgasdoctoralstudent.
Last but not least, I am indebted to Burkhard Ku(cid:127)lshammer for his support and his advice.
Preliminaries and notation
Numbers. DenotebyN :=f1,2, ::: gthesemigroupofnaturalnumbers. Let N0 :=N[f0g bethe
corresponding abelian monoid. Denote its Grothendieck group by Z,the ring of integers. The quotient
(cid:12)eld of Z,the (cid:12)eld of rational numbers, is denoted by Q.
Algebras. Throughout the whole work, we (cid:12)x a commutative (and associative) ring R with multi-
plicative identity 1=1R. Unless otherwise stated, any R{algebra A is assumed to be associative and to
possessamultiplicative identity 1A. Whereas, we donot requirehomomorphismsof algebrastopreserve
identity elements. Otherwise, we speak of unitary homomorphisms of R{algebras. Let A and B be
R{algebras. A direct embedding ’ : A !B of R{algebras is an injective homomorphism of R{algebras
such that the image of ’ is the whole of ’(1A)B’(1A). A standard example of a direct embedding is
the canonical inclusion eBe ,! B, where e is some idempotent in B. An arbitrary direct embedding of
R{algebrasarises as composition of such an embedding with an isomorphism of G{algebras.
Let A be an R{algebra. We denote by Z(A) = ZA the center of A and by U(A) = UA the group
of units of A. Whereas, for the ground ring R (and other ground rings), we usually prefer the notation
(cid:2)
R to denote the set of invertible elements in R. The abbreviation J(A) =JA stands for the Jacobson
radicalof A. We referto [Hu] forthe de(cid:12)nition andbasic propertiesof the Jacobsonradicalof arbitrary
rings and algebras. We denote the group of R{automorphisms of A by AutR(A). Moreover, for n 2 N,
Mat(n,R) denotes the full matrix algebraof n(cid:2)n{matrices with entries in R. The standard R{basis of
Mat(n,R) (n2N) is denoted by eij, i; j =1; ::: ; n. For a set X
(cid:14) :X(cid:2)X !f0; 1g
1 : x=y
(x; y)7!(cid:14)x;y :=
0 : x6=y
(cid:26)
denotes the Kroneckersymbol.
We(cid:12)xaprimenumberp2Z. Moreover,Odenotesacompletediscretevaluationringofcharacteristic
0withalgebraicallyclosedresidue(cid:12)eldF ofcharacteristicp. Weareusingthesymbolcharktodenotethe
characteristicofa(cid:12)eldk. LetpbetheuniquenonzeroprimeidealJOofO. Wheneverweneedaquotient
(cid:12)eld of O, it is denoted by K. However, in the absence of su(cid:14)ciently many letters, K will sometimes
denote a group, when no quotient (cid:12)eld of O is around. Essentially, we are interested in algebras over
O and F, respectively. However, within the (cid:12)rst chapter, there is no need to restrict ourselves to such
speci(cid:12)c ground rings; so, to begin with, we work with arbitrary R{algebras; and, when we talk about
algebras, this is supposed to mean R{algebras.
ByanR{orderwemean anR{algebrawhichis (cid:12)nitely generatedand projectiveasR{module. Thus
an O{order is (cid:12)nitely generated and free (or, equivalently, (cid:12)nitely generated and torsionfree) as O{
module, since O is a principal ideal domain. Instead of speaking of F{orders (which sounds silly, as far
as we are concerned), we implicitly assume F{algebras to be (cid:12)nite dimensional (as vector spaces over
F), unless otherwise stated. Most of the results we prove for O{orders are equally valid for F{algebras
(or, more generally, for artinian algebras), and vice versa. We feel free to apply these results this way,
without further comment.
LetAbeanO{order(F{algebra). Twoidempotentseandf inAarecalledassociated(inA),i(cid:11)there
areelementsa2eAf andb2fAesuchthate=abandf =ba. Idempotentseandf inAareassociated,
(cid:0)1
if andonly if they areconjugate, i.e. there is a unit u2UA such that e=ufu . (It is straightforward
to check that this lemma of [Ku] for F{algebras holds for O{orders, as well; alternatively, compare the
proof of lemma I.4.6.) This de(cid:12)nes an equivalence relation on the set of primitive idempotents of A.
The corresponding equivalence classes are called the points of A. We denote the set of points of A by
P(A) = PA. The points of A are in 1-1 correspondence with the points of A=JA, the maximal ideals
of A, the primitive central idempotents of A=JA and the isomorphism classes of simple A{modules,
v
PRELIMINARIES AND NOTATION vi
respectively (cf. [Ku] or[Th]). By Wedderburn, there is an isomorphism of F{algebras
A=JA(cid:24)= Mat(m(cid:11),F)
(cid:11)2PA
Y
for some m(cid:11) 2 N ((cid:11) 2 PA), since F is algebraically closed. (Note that pA (cid:18) JA, since pA is a (left
and right) quasi{regularideal in A (see [Th] and [Hu]). Hence A=JA is an algebraoverF =O=p.) The
uniquely determined natural numbers m(cid:11) are called the multiplicities of the points (cid:11) of A.
Groups. Throughout, group will always mean (cid:12)nite group. Of course, except for unit groups and
automorphismgroupsofalgebras. We(cid:12)xa(cid:12)nitegroupG. LetH (cid:20)Gstandfor\H isasubgroupof G";
and let H < G mean that H is a proper subgroup of G. Analogously, we write N (cid:2)G to indicate that
N is a normal subgroup of G, whereas N (cid:1)G means that N is a proper normal subgroup of G. Denote
by jGj the order of G (or, more generally, let jXj be the number of elements of a (cid:12)nite set X).
Suppose H and K are subgroups of G. Denote by G=H the set of cosets gH, g 2 G, let KnG be
the set of cosets Kg, g 2 G, and let KnG=H be the set of double cosets KgH, g 2 G. (We denote the
di(cid:11)erence set of two sets X and Y by X (cid:0)Y rather that X nY, for obvious reasons.) Furthermore, for
g (cid:0)1
g 2G, H denotes the conjugate subgroup fghg :h2Hg. Moreover, let NG(H) be the normalizer of
g
H in G, i.e. the subgroup fg 2G : H =Hg of G, and denote by CG(H):=fg 2G:gh=hg 8h2Hg
the centralizer of H in G.
A normal series N0(cid:2)N1(cid:2):::(cid:2)Nk of G is a set fN0; N1; ::: ; Nkg (k 2 N0) of normal subgroups
of G (!) such that Ni(cid:2)Ni+1 for i=0; ::: ; k(cid:0)1. (We do not require N0 =1 or Nk =G.)
For a homomorphism (cid:11) : G ! H between groups G and H, we denote by ker(cid:11) the kernel of (cid:11),
and let im(cid:11) stand for the image of (cid:11). Suppose a group G acts on a set (cid:10). Then the G{stabilizer of an
element ! of (cid:10) is denoted by StbG(!).
Let U and V be subgroups of G. Then [U; V] denotes the subgroup of G generated by elements of
(cid:0)1 (cid:0)1 0
the form [u; v] := uvu v , where u2 U and v 2 V. Moreover, G := [G; G] denotes the commutator
subgroup of G.
For a set (cid:25) (cid:18)Zof prime numbers, denote by O(cid:25)(G) the composite of all normal (cid:25){subgroups of G,
0
i.e. the unique maximal normal (cid:25){subgroupof G. Write Op(G) instead of Ofpg(G), and let p be the set
ofprimenumbersdi(cid:11)erentformp. Moreover,letF(G)betheFittinggroupofG,i.e.theuniquemaximal
normalnilpotentsubgroupofG. ThenF(G)isthecompositeofallOq(G),whereqrunsthroughallprime
numbersdividingtheorderofG. TheautomorphismgroupofG isdenoted byAut(G). Furthermore,let
Sylp(G) bethesetofSylow{p{subgroupsof G. Besides, Irr(G) denotesthesetof(absolutely)irreducible
Characters of G. Finally, the group algebraof G over R is denoted by RG or R[G].
Groups acting on algebras. A G{algebra(over R) is a pair (A; (cid:1)) consisting of an R{algebraA
and a homomorphism (cid:1):A!AutR(A) of groups. We call (cid:1) the structure map of (A; (cid:1)). We usually
g
omit the structure map (cid:1) and say that A is a G{algebra. We then write a instead of (cid:1)(g) (a) for
a2A and g2G. Then a G{algebra over R is, in particular, a left G{module (RG{module). Let A and
(cid:2) (cid:3)
B be G{algebras(over R). A homomorphism ’:A!B of R{algebrasis called a homomorphism of G{
algebras, i(cid:11) ’ is a homomorphism of G{modules. Moreover, a homomorphism ’:A!B of G{algebras
iscalled a direct embeddingof G{algebras,i(cid:11) ’is adirectembedding ofabstract algebras. A G{algebra
A over O is called a G{order, i(cid:11) A is an O{order as abstract algebra.
An interior G{algebra (over R) is a pair (A; (cid:1)) consisting of an R{algebraA and a homomorphism
G ! UA of groups. Again, (cid:1) is called the structure map of (A; (cid:1)). We usually omit the structure
map and write a(cid:1)g, g(cid:1)a and a(cid:1)g(cid:1)b instead of a(cid:1)(g), (cid:1)(g)a and a(cid:1)(g)b, respectively, for g 2 G and
a; b 2 A. Then an interior G{algebra is, in particular, an RG{RG{bimodule; and a homomorphism of
interior G{algebras is, by de(cid:12)nition, both, a homomorphism of abstract algebras and a homomorphism
of RG{RG{bimodules, as well. Any interior G{algebra may be regarded as G{algebra, when we de(cid:12)ne
g (cid:0)1
a:=g(cid:1)a(cid:1)g .
Let M be a left RG{module, let A be a G{algebra and let H be a subgroup of G. We denote
H
by M the H{(cid:12)xed points of M, i.e the R{submodule fm 2 M : hm = m 8h 2 Hg of M. Then
H h
A = fa 2 A : a = a 8h 2 Hg is even a unitary subalgebra of A. There is an R{linear map
G H G
TrH : M ! M , m 7! gH2G=Hgm, called the relative trace map. (Note that this de(cid:12)nition is
independent of the choice of representatives.) For G{algebras, relative trace maps are of particular
P
importance. We refer to [Ku] or [Th] for their properties.
PRELIMINARIES AND NOTATION vii
G
A G{algebra A over R is called primitive, i(cid:11) 1A is the unique nonzero idempotent in A . Then a
G G
G{orderAoverO (oraG{algebraoverF)isprimitive,ifandonlyifA islocal,i.e.J A istheunique
G G G
maximal ideal of A and any element in A is either a unit or contained in the Jacobson radical of A .
(cid:0) (cid:1) G
A may be regarded as H{algebra, by restricting the structure map. We indicate this by ResHA,
when we do so.
G G
Note that U A = UA\A , because of uniqueness of inverses. (Of course, for orders over O or
algebrasoverF, the analogousassertiondoesnotonly hold for(cid:12)xedpoint subalgebras,but for arbitrary
(cid:0) (cid:1)
unitary subalgebras, as well.) Let A be a G{order over O (or a G{algebra over F). A pointed group
H
on A is a pair (H; (cid:11)) consisiting of a subgroup H of G and a point (cid:11) of A . We write H(cid:11) instead of
(H; (cid:11)). The group G acts on the set of pointed groups on A, where, for g 2 G and any pointed group
g
g g g H g
H(cid:11) on A, H(cid:11) := H g(cid:11) and (cid:11) is the unique point of A containing e for some (all) e2(cid:11). The
G{stabilizer of a pointed group H(cid:11) on A is denoted by NG(H(cid:11)).
(cid:0) (cid:1) (cid:0) (cid:1)
Wereferto[Ku]or[Th]forthedefecttheoryofG{algebras,although,toacertainextent,wedevelop
this theory here from a module theoretic point of view; and, for the reader familiar with vertex theory,
this might su(cid:14)ce to understand the portion of the theory we need here. Note that, if A is primitive, we
speak of defect groups of A rather than defect groups of (the pointed group) Gf1Ag.
We will frequently use the following fact: Suppose A is a primitive G{order over O (or a G{algebra
over F) with defect group D and let N be a normal subgroup of G. Then DN=N is a defect group of
N N
the G=N{algebra A ; since, (cid:12)rstly, A is clearly DN=N{projective; secondly, suppose N (cid:20) Q (cid:20) DN
N g
such that A is Q=N{projective; then, clearly, A is Q{projective; thus D (cid:20) Q for some g 2 G; hence
g g
(DN)= DN (cid:20)Q(cid:20)DN, and thus Q=DN.
Modules. Let A be an R{algebra. In the following all A{modules will be unitary, i.e. 1A acts as
the identity endomorphism. Moreover, A{module means left A{module, although, to a great part, we
are considering right modules, as well. Denote by A{Mod (Mod{A) the category of (right) A{modules,
and let A{mod (mod{A) be the category of (cid:12)nitely generated (right) A{modules. Moreover, we denote
the category of (cid:12)nitely generated projective (right) A{modules by A{pro (pro{A). For (left or right) A{
modules M and N, HomA(M,N) denotes the R-module of A{homomorphisms M ! N and EndA(M)
denotes the R{algebraof A{endomorphisms of M.
Let (cid:3) be an R{order. A (cid:3){lattice is a (cid:3){module which is (cid:12)nitely generated and projective as
R{module. Denote by (cid:3){latt (latt{(cid:3)) the category of (right) (cid:3){lattices.
Suppose A is a G{algebra over R. Then, as abstract algebra, A is a unitary subalgebra of the
skew group algebra A(cid:3)G (cf. section I.1). We denote the category of (right) A(cid:3)G{modules which are
(cid:12)nitely generatedand projectiveas (right) A{modules by A(cid:3)G{proA (proA{A(cid:3)G). Let R be the trivial
G{algebra. Then R(cid:3)G=RG is the ordinary group algebra and R(cid:3)G{proR =RG{latt.
Categories. Our categorical terminology is a bit of a mixture of those of [HiSt] and [McL], and
what we (cid:12)nd convenient ourselves. However, we believe it should always be clear what is meant.
Let C be a category. We denote the identity morphism of an object C in C by 1C; and we keep
this notation for concrete categories, i.e. 1X : X ! X denotes the identity map of a set X. Moreover,
1C : C ! C denotes the identity functor, and 1F : F ! F denotes the identity natural transformation of
a functor F:C !D between categories C and D.
Suppose A; B and C are objects in C. The set of morphisms from A ! B in C is denoted by
C(A,B),unlessanothernotationismorecustomary(e.g.foramodulecategory). Moreover,weareusing
\mapping notation", i.e. composition in C is a map C(B,C)(cid:2)C(A,B) ! C(A,C), (g; f) 7! gf. To
denote composition of functors we are using the symbol (cid:14), i.e. if C; D; E are categories and F : C ! D,
G:D !E are functors, then G(cid:14)F:C !E denotes the composed functor.
Let C and D be categories. Suppose F : C ! D and G : D ! C are functors such that F(cid:14)G and
G(cid:14)F are naturally equivalent to 1D and 1C, respectively. Then F (and G) is called an equivalence of
categories. Moreover,wesaythatFandGaremutuallyinverseequivalencesofcategories. Alternatively,
we are using the phrase \F and G induce an equivalence of categories". Moreover, we write F a G to
indicate that F is left adjoint to G; and (cid:17) :FaG means that (cid:17) is an adjugant realizing this adjointness
relation.
Finally, we assume the existence of a universe containing all sets, groups, algebras, modules etc. we
are considering here; i.e. we don’t care about set theory.
CHAPTER I
The categories
In the following let G denote a (cid:12)nite group and let A be a G{algebra over R. It is well known that
there is a close connection between certain G{algebras and modules of the skew group algebra A(cid:3)G.
This relation can be seen as a generalization of the, as easy as important, fact of the equivalence of the
category of linear representations of G over F and the category of FG{modules, respectively. Although
this connection has become a standard tool in ring theory, s. [Mo] and [CoMo], rediscovered by some
representationtheorists,e.g.Dade(s. [Da])seealso[Pu]and[Ku1],ithasfoundapplicationsinmodular
representation theory, from our point of view, only to some degree (except for the special case of FG{
modules, ofcourse). Inparticular,aswebelieve,Puig’stheoryof G{algebrasdoesnot makemuchuse of
this relationship (at least not explicitly).
In this chapter we will de(cid:12)ne a category G{Emb(A), based on a G{algebra A, whose objects are
G{algebras together with some additional structure. This category will turn out to be equivalent to the
category of A(cid:3)G{modules which are (cid:12)nitely generated and projective as A{modules. We consider the
categoryG{Emb(A) asananalogueofthe categoryoflinear representationsof G. Usually, wewill useit
to compute things. Whereas the category of A(cid:3)G{modules will, (cid:12)rstly, serve as a tool for applying the
theory of rings and modules and will, secondly, allow us to de(cid:12)ne certain constructions, which are more
obvious from this point of view. In particular, we will de(cid:12)ne a notion of induction for G{algebras.
AnotheradvantageofthecategoryG{Emb(A)isthefactthatabigpartofPuig’stheorylivesinsideit
(e.g.defecttheory),thoughitis,incontrasttothe(resp.oneofthepossible)category(ies)ofG{algebras,
an R{additive category. Moreover, the category G{Emb(A) could be of interest for a computer{based
treatmentofA(cid:3)G{modules(whichareprojectiveasA{modules). Butwewon’tfollowthisthoughthere.
Finally, we hope this work to be a little indication of the usefulness of the above sight of things.
1. The skew group algebra
We recall the de(cid:12)nition of the skew group algebra A(cid:3)G. The algebra A(cid:3)G is, as an R{module,
isomorphic to A(cid:10)RRG. The images of the elements a(cid:10)g, a 2 A, g 2 G, under this isomorphism are
denoted by a(cid:3)g. Thus any element in A(cid:3)G can be written in the form g2Gag (cid:3)g, with uniquely
determined ag 2 A, g 2 G, since RG is R{free. Multiplication is de(cid:12)ned as follows. For a, b 2 A and
x P
x, y 2Gonesets(a(cid:3)x)(b(cid:3)y):=a b(cid:3)xyandextendsR-linearily. Onecheckseasilythat,togetherwiththe
abovemultiplication, A(cid:3)GbecomesanassociativeR{algebrawith multiplicativeidentity1A(cid:3)G =1A(cid:3)1.
Moreover A(cid:3)G is, on the one hand, together with the homomorphism G ! U A(cid:3)G , g 7! 1(cid:3)g,
an interior G{algebra. On the other hand A (cid:3) G represents a G{graded algebra with g{component
(cid:0) (cid:1)
A(cid:3)g :=fa(cid:3)g :a2Ag, g2G. Since 1(cid:3)g2A(cid:3)g, for g 2G, is a unit in A(cid:3)G, the skew group algebra
even is a crossed product, and is, therefore, sometimes called the trivial crossed product.
If AitselfisaninteriorG{algebra,thentheassignmenta(cid:3)g7!a(cid:1)g(cid:10)g,fora2Aandg2G,induces
an isomorphism ’:A(cid:3)G!A(cid:10)RRG of interior G{algebras;since, (cid:12)rst of all
x x
’((a(cid:3)x)(b(cid:3)y))=’(a b(cid:3)xy)=a b(cid:1)xy(cid:10)xy
(cid:0)1
=a(cid:1)x(cid:1)b(cid:1)x xy(cid:10)xy =(a(cid:1)x(cid:10)x)(b(cid:1)y(cid:10)y)
=’(a(cid:3)x)’(b(cid:3)y)
for a,b2A and x,y 2G. Furthermore ’(1A(cid:3)x)=x(cid:10)x for x2G. Finally, ’ is clearly bijective.
Let H be a subgroup of G. We may identify A (cid:3)H with a unitary subalgebra of A(cid:3)G in the
canonical way. In particular, A itself is a unitary subalgebra of A (cid:3) G, if we identify a with a (cid:3) 1,
G
a 2 A. Thus there are R{additive functors IndH := A (cid:3) G (cid:10)A(cid:3)H (cid:0) : A (cid:3) H{Mod ! A (cid:3) G{Mod
G
and ResH : A(cid:3)G{Mod ! A(cid:3)H{Mod, called the restriction and induction functor, respectively;
G
equivalentlyfor rightmodules. As usual, we omit the functor ResH when it’s clearfrom contextwhat is
g g (cid:0)1
meant. Supposeg 2G. Thencg,H :A(cid:3)H !A(cid:3) H,a(cid:3)h7! a(cid:3)ghg isanisomorphismofalgebras;since
1