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ON LOCAL CATEGORIES OF FINITE GROUPS
FEI XU
Abstract. Let G be a finite group. Over any finite G-poset P we
may define a transportercategory as the corresponding Grothendieck
construction. Transporter categories are general-izations of
subgroups of G, and we shall demonstrate the finite generation of
their cohomology.We record a generalized Frobenius reciprocity and
use it to examine some important quotientcategories of transporter
categories, customarily called local categories.
1. Introduction
Let C be a finite category in the sense that Mor C is finite.
Then we can define a finite-dimensional algebra kC, over a chosen
field k. This algebra has a distinguished module, the trivialmodule
k, which is the tensor identity of the closed symmetric monoidal
category kC-mod [18].The tensor structure allows us to define a cup
product on Ext∗kC(k, k) = ⊕i≥0Ext
ikC(k, k) so that
there is a ring isomorphismExt∗kC(k, k)
∼= H∗(BC, k),
in which BC is the classifying space of C. This ring is called
the ordinary cohomology ring ofkC. In general such a ring is not
finitely generated, but for various reasons we would like tohave a
good class of finite categories with their ordinary cohomology
rings finitely generated. In[18] we discussed the so-called
transporter categories G ∝ P, defined over a poset P with theaction
by a group G, and proved that both the ordinary cohomology ring and
the Hochschildcohomology ring of k(G ∝ P) are finitely generated.
In this article, we continue to show that forany M,N ∈ k(G ∝
P)-mod, Ext∗k(G∝P)(M,N) is finitely generated over Ext
∗k(G∝P)(k, k), which
generalizes the well known theorem of Evens and Venkov on group
cohomology [2, 14]. Our mainresult is the following.
Theorem 1.1. Let G be a finite group and P a finite G-poset.
Suppose k(G ∝ P) is the transportercategory algebra. Then for any
M,N ∈ k(G ∝ P)-mod, Ext∗k(G∝P)(M,N) is finitely generated overthe
Noetherian ring Ext∗k(G∝P)(k, k).
The proof is different from that for group algebras, or more
generally for finite-dimensionalcocommutative Hopf algebras [6],
where the internal hom plays a significant role. Here the proofis
based on functor cohomology and is very similar to Venkov’s proof
for finite groups. Howeverinterestingly it depends on Hochschild
cohomology.
In this article, we shall first recall the definition of a
transporter category as well as basicfacts about representations
and cohomology of category algebras. We explain why
transportercategories are interesting by examples in group
representations and cohomology. These will be inSections 2 and 3.
Particularly we provide a generalized Frobenius Reciprocity in
Section 3. Thefinite generation theorem is proved in Section 4,
which itself is an evidence of the interests ontransporter
categories. The last section looks into various cases of the
Frobenius Reciprocity.
I would like to thank the anonymous referee for helpful
suggestions and for pointing out amistake in an earlier
version.
2010 Mathematics Subject Classification. Primary 20C05;
Secondary 20J99.The author (徐斐) was supported in part by a Beatriu
de Pinós research fellowship from the government
of Catalonia of Spain, and a Grant MTM2010-20692 “Analisis local
en grupos y espacios topologicos" from theMinistry of Science and
Innovation of Spain.
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2 FEI XU
2. preliminaries
In this section, we recall the definition of a transporter
category and some background incategory algebras. Throughout this
article we will only consider finite categories, in the sense
thatthey have finitely many morphisms. Thus a group G, or a G-poset
P, is always finite.
2.1. Transporter categories as Grothendieck constructions. In
the literature the trans-porter categories are mostly considered as
auxiliary constructions before passing to various quo-tient
categories of them. Here we want to stress on the perhaps unique
property, among variouscategories constructed from a group, that
transporter categories admit natural functors to thegroup itself.
It singles out this particular type of categories and is the
starting point of this article.Here in order to emphasize the
similarities and connections between transporter categories
andsubgroups, we follow a definition which is well known to some
algebraic topologists. It is intendedto be phrased explicitly in a
way such that an algebraist, say a representation theorist, may
readilyaccept as an entirely algebraic or categorical
construction.
We deem a group as a category with one object, usually denoted
by •. The identity of a groupG is always named e. We say a poset P
is a G-poset if there exists a functor F from G to sCats,the
category of small categories, such that F (•) = P. It is equivalent
to say that we have a grouphomomorphism G → Aut(P). The
Grothendieck construction on F will be called a
transportercategory. In the following explicit definition, the
morphisms in a poset are customarily denotedby ≤.
Definition 2.1. Let G be a group and P a G-poset. The
transporter category G ∝ P has thesame objects as P, that is, Ob(G
∝ P) = ObP. For x, y ∈ Ob(G ∝ P), a morphism from x to yis a pair
(g, gx ≤ y) for some g ∈ G.
If (g, gx ≤ y) and (h, hy ≤ z) are two morphisms in G ∝ P, then
their composite is (hg, (hg)x ≤z). One can check directly that if
HomG∝P(x, y) 6= ∅ then both AutG∝P(x) and AutG∝P(y) actfreely on
HomG∝P(x, y).
The symbol G ∝ P is used because this particular Grothendieck
construction resembles asemidirect product, yet is different. From
the definition one can easily see that there is a naturalembedding
ιP : P ↪→ G ∝ P via (x ≤ y) 7→ (e, x ≤ y). On the other hand, the
transportercategory admits a natural functor πP : G ∝ P → G, given
by x 7→ • and (g, gx ≤ y) 7→ g. Thuswe always have a sequence of
functors
P ιP↪→G ∝ P πP−→G
such that πP ◦ ιP(P) is the trivial subgroup or subcategory of
G. For convenience, in the rest ofthis article we often neglect the
subscript P and write ι = ιP , π = πP . Topologically it is
wellknown that B(G ∝ P) ' EG×GBP. Passing to classifying spaces, we
obtain a fibration sequence
BP Bι−→EG×G BPBπ−→BG.
Forming the transporter category over a G-poset eliminates the
G-action, and thus is the algebraicanalogy of introducing a Borel
construction over a G-space.
This neat but seemingly abstract definition can be easily seen
to give the usual transportercategories. For example, when P = Sp
is the poset of non-trivial p-subgroups, we get G ∝ Sp =Trp(G), the
p-transporter category of G. The advantage of taking our approach
is that we maygive group-theoretic interpretations to transporter
categories, which is shown by the upcomingexamples, where each
subgroup of G is identified as a transporter category, up to a
categoryequivalence.
Example 2.2. If G acts trivially on P, then G ∝ P = G× P.
Example 2.3. Let G be a finite group and H a subgroup. We
consider the set of left cosets G/Hwhich can be regarded as a
G-poset: G acts via left multiplication. The transporter categoryG
∝ (G/H) is a connected groupoid whose skeleton is isomorphic to H.
In this way one canrecover all subgroups of G, up to category
equivalences.
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ON LOCAL CATEGORIES OF FINITE GROUPS 3
A category equivalence D → C induces a Morita equivalence
between their category algebras-tobe recalled in Section 2.2, kD '
kC as well as a homotopy equivalence BD ' BC (see [15]).It means
there is no essential difference between kH and k(G ∝ (G/H)) (and
their modulecategories), or between BH and B(G ∝ (G/H)). Hence it
makes sense if we deem transportercategories as generalized
subgroups for a fixed finite group.
As two other motivating examples, transporter categories were
implicitly considered by MarkRonan and Steve Smith [11] in the
1980s for constructing group modules, and later on played a keyrole
in Bill Dwyer’s work [5] on homology decomposition of classifying
spaces. Roughly speaking,one often finds in various situations a
diagram of categories and functors
G ∝ Pπ
{{
ρ
##G C
for C a quotient category of G ∝ P and π the canonical functor.
(The quotient categories we havein mind are orbit categories,
Brauer categories, Puig categories or even transporter
categoriesthemselves.) Dwyer used this diagram to establish
connections among various homotopy colimits(e.g. classifying
spaces), while Ronan and Smith constructed kG-modules via
representations ofG ∝ P (using the language of G-presheaves on
P).
2.2. Category algebras, representations and cohomology. We
recall some facts about cat-egory algebras. The reader is referred
to [15, 17] for further details. Let C be a finite category andk a
field. One can define the category algebra kC, which, as a vector
space, has a basis the set ofall morphisms in C, and in which
multiplication is determined by compositions of base elements.When
C is a group, kC is the group algebra. As a convention, throughout
this article, kG-modulesare usually written as M,N etc, while the
modules of a (non-group) category algebra kC aredenoted by M,N
etc., except those special modules, namely k and κM , which are
restrictions ofkG-modules (to be defined shortly).
A k-representation of C is a covariant functor from C to V ectk,
the category of finite dimensionalk-vector spaces. All
representations of C form the functor category V ectCk . By a
theorem of B.Mitchell, the finitely generated left kC-modules are
the same as the k-representations of C, in thesense that there
exists a natural equivalence
V ectCk ' kC-mod.
In the module category, there is a distinguished module k,
sometimes called the trivial module,which can be defined as a
constant functor taking k as its value at every object of C. Since
V ectk isa symmetric monoidal category, V ectCk inherits this
structure. It means there exists an (internal)tensor product, or
the pointwise tensor product, written as ⊗̂, such that for any two
kC-modulesM,N, (M⊗̂N)(x) := M(x) ⊗k N(x). Let α ∈ Mor C be a base
element of kC. Then α acts onM⊗̂N via α⊗ α. Obviously k is the
identity with respect to ⊗̂ and M⊗̂N ∼= N⊗̂M. We can alsoconstruct
a function object, the internal hom, Hom(M,N) ∈ kC-mod such
that
HomkC(L⊗̂M,N) ∼= HomkC(L,Hom(M,N)),
for any L ∈ kC-mod, and the isomorphism is natural in L,M and N,
see [12, 18]. When C = G isa group, we recover the usual
construction −⊗̂− = −⊗k − and Hom(−,−) = Homk(−,−).
For any two kC-modules it makes sense to consider the Ext groups
Ext∗kC(M,N). The tensorproduct ⊗̂ induces a cup product as
follows
∪ : ExtikC(M,N)⊗ ExtjkC(M
′,N′)→ Exti+jkC (M⊗̂M′,N⊗̂N′).
In particular Ext∗kC(k, k) is a graded commutative ring and we
have an isomorphism Ext∗kC(k, k)
∼=H∗(BC, k). This ring is called the ordinary cohomology ring of
kC and it acts on Ext∗kC(M,N).The ordinary cohomology ring of a
category algebra is usually far from finitely generated, butit is
so when C = G ∝ P is finite (implicitly in Venkov’s proof of the
finite generation of groupcohomology).
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4 FEI XU
One concept that we will refer to is the bar resolution, a
combinatorally constructed projectiveresolution BC∗ , of k ∈
kC-mod.
Finally we record the basic tools for comparing category
algebras and their modules. Whenτ : D → C is a functor between two
finite categories. There are functors for comparing
thererepresentations. The functor τ induces a restriction
Resτ : kC-mod→ kD-mod.
If we regard a kC-module as a functor, then its restriction is
the precomposition of it with τ . Ifwe consider the functor π : G ∝
P → G, then any kG-module M restricts to a k(G ∝ P)-module,written
as κM = ResπM , with only one exception k = κk = Resπ k. The
functor Resτ is equippedwith two adjoints: the left and right Kan
extensions along τ
LKτ , RKτ : kD-mod→ kC-mod.
The definition of the left and right Kan extensions [7] depend
on the so-called over-categories andunder-categories, respectively.
For each x ∈ Ob C, one can define an over-category τ/x and
anunder-category x\τ . These concepts are frequently used in the
representations and cohomologyof category algebras. For example the
bar resolution is constructed by using a certain collectionof
overcategories. The analysis of over- and undercategories
associated with π leads to veryinteresting results, as they reveal
the structures of various prominent kG-modules and complexesof
kG-modules. The reader is referred to [19] for the work based on
studying these categories,where we build the Becker-Gottlieb
transfer map.
3. The role of transporter categories
From this section to the end of the article, we illustrate the
role of transporter categories ingroup representations and
cohomology. The purpose is to show that transporter categories
shouldbe further investigated, for better understanding of groups,
or as prototypes for research in otherlocal categories.
3.1. A diagram of categories. We have seen in the introduction
that given a G-poset P thereexists a diagram of categories and
functors
G ∝ Pπ
{{
ρ
##G C
for any given local category C.
3.2. Frobenius reciprocity. Applying the adjunctions, described
in Section 2.2, to the firstdiagram, we obtain a diagram of module
categories
k(G ∝ P)-modLKπ,RKπ
{{
LKρ,RKρ
##kG-mod
Resπ
;;
kC-mod.Resρ
cc
The adjunctions between the restrictions and Kan extensions have
the following consequences.
Proposition 3.1 (Frobenius Reciprocity). Suppose P is a G-poset
and C is a quotient categoryof G ∝ P as in the preceding diagrams.
Let M,N ∈ kG-mod and M,N ∈ kC-mod. Then
(1) HomkG(M,RKπ ResρN) ∼= HomkC(LKρ ResπM,N);(2) HomkG(LKπ
ResρM, N) ∼= HomkC(M, RKρ Resπ N).
By direct calculations, these particular Kan extensions in
Proposition 3.1 are simplified:• LKπ ∼= lim−→P , RKπ
∼= lim←−P (used by Ronan-Smith. See also [18]);
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ON LOCAL CATEGORIES OF FINITE GROUPS 5
• LKρ ∼=↑kCk(G∝P) (the induction), RKρ ∼=⇑kCk(G∝P) (the
co-induction), since ρ induces an
algebra homomorphism k(G ∝ P) → kC when C is a quotient
category. (One can easilyprove that a functor τ : D → C induces a
k-linear map τ : kD → kC, and moreover thismap becomes an algebra
homomorphism if and only if the functor is injective on
objects.)
Remark 3.2. For any N ∈ kC-mod, we shall write the k(G ∝
P)-module ResρN as N becausethey share the same underlying vector
space. Recall that we have set κM = ResπM . Then theFrobenius
reciprocity can be rewritten as
(1’) HomkG(M, lim←−P N)∼= HomkC(κM ↑kCk(G∝P),N);
(2’) HomkG(lim−→PM, N)∼= HomkC(M, κN ⇑kCk(G∝P)).
When P = G/H for some subgroup H, we have natural equivalences
lim←−G/H∼= lim−→G/H
∼=↑GH .Then the above isomorphisms certainly become the usual
adjunctions between ↑GH and ↓GH (theusual Frobenius Reciprocity)
with C = G ∝ (G/H) and ρ = Id, in light of the Morita
equivalencebetween kC and kH, see Example 2.3.
Remark 3.3. Our Frobenius reciprocity is different from a
similar result of Ronan-Smith, see [2,7.2.4], where they
(implicitly) had a diagram of the same shape. However their C = G ∝
Q,not necessarily a quotient of G ∝ P, is another transporter
category and ρ is induced by a G-map P → Q. This prohibits us from
considering various quotients of transporter categories.Moreover
since a G-map P → Q usually is not injective on objects, it does
not induce an algebrahomomorphism from k(G ∝ P) to k(G ∝ Q). Hence
their Kan extensions along ρ cannot beinterpreted as induction and
coinduction.
The functors ↑kCk(G∝P) and ⇑kCk(G∝P) admit interesting
interpretations when G ∝ P → C is part
of an extension (or an opposite extension) sequence of
categories. Under the circumstance ↑kCk(G∝P)and ⇑kCk(G∝P) on
certain k(G ∝ P)-modules can be very well understood. We shall
discuss it inSection 5.
4. Finite generation of cohomology
The functor π is very useful to study cohomology of transporter
categories. The reader can findmore applications in [19] where we
construct the Becker-Gottlieb transfer map. We do not putthe
construction here because it requires substantial analysis of
various over- and under-categoriesassociated with π and related
functors, and accompanying permutation G-modules.
4.1. First attempt with internal hom. Suppose P is a G-poset and
π is the natural functorfrom the transporter category G ∝ P to G,
regarded as a category with one object •. The nextresult is a
direct generalization of the fact that the two obvious kG-module
structures on P⊗M areisomorphic, for P,M ∈ kG-mod with P
projective. It reveals a connection between representationsof
groups and of transporter categories.
Recall that for any finite category C, the regular module kC
decomposes into a direct sum⊕x∈Ob C kC ·1x because 1 =
∑x∈Ob C 1x is an orthogonal decomposition by idempotents. A
direct
summand kC · 1x is often written as kHomC(x,−) [15].
Theorem 4.1. Let P ∈ k(G ∝ P)-mod be a projective module and κM
= ResπM for someM ∈ kG-mod. Then P⊗̂κM is a projective k(G ∝
P)-module. Consequently BG∝P∗ ⊗̂κM →k⊗̂κM = κM → 0 is a projective
resolution.
Proof. We will prove P⊗̂κM ∼= P⊗M , with k(G ∝ P) acting on the
latter via left multiplication.To this end, we assume P =
kHomG∝P(x,−). The proof is entirely analogues to the case whenP =
•, i.e. when G ∝ • = G.
We define a k-linear map ϕ : kHomG∝P(x,−) ⊗M → kHomG∝P(x,−)⊗̂κM
as follows. Onbase elements ϕ((g, gx ≤ y) ⊗m) = (g, gx ≤ y) ⊗ (g,
gx ≤ y)m, where the latter m is consideredas an element in κM (x).
For any (h, hy ≤ z), we readily verify
(h, hy ≤ z)ϕ[(g, gx ≤ y)⊗m] = ϕ[(h, hy ≤ z)((g, gx ≤ y)⊗m)].
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6 FEI XU
Thus ϕ is a homomorphism of k(G ∝ P)-modules.We remind the
reader that, following definition, given any pair of x, y ∈ Ob(G ∝
P) with
HomG∝P(x, y) non-empty, both AutG∝P(x) and AutG∝P(y) act freely
on HomG∝P(x, y). Thisimplies that kHomG∝P(x, y) is a free
kAutG∝P(x)- or kAutG∝P(y)-module. Consequently ϕrestricts on each y
to the classical kAutG∝P(y)-isomorphism (see [1, 3.1.5])
kHomG∝P(x, y)⊗M → (kHomG∝P(x,−)⊗̂κM )(y) = kHomG∝P(x, y)⊗M.
Furthermore because as vector spaces
kHomG∝P(x,−)⊗̂κM =⊕
y∈Ob(G∝P)
kHomG∝P(x, y)⊗M = kHomG∝P(x,−)⊗M,
the linear map ϕ is actually one-to-one and hence an isomorphism
of k(G ∝ P)-modules. �
The following result from [18] is needed.
Proposition 4.2. Let G be a finite group and P a finite G-poset.
Then Ext∗k(G∝P)(k, k) is finitelygenerated, and for an arbitrary N
∈ k(G ∝ P)-mod, Ext∗k(G∝P)(k,N) is finitely generated over
theExt∗k(G∝P)(k, k).
The proof is entirely analogous to Venkov’s proof for group
cohomology [2, 14]. The maindifference here is the use of the
Grothendieck spectral sequence, instead of the Leray-Serre
spectralsequence, in order to allow arbitrary functors as
coefficients.
Proposition 4.3. For any M ∈ kG-mod and N ∈ k(G ∝ P)-mod,
Ext∗k(G∝P)(κM ,N) is finitelygenerated over Ext∗k(G∝P)(k, k).
Proof. One can easily deduce from Theorem 4.1, together with the
internal hom in Section 2.2,an Eckmann-Sharpiro type
isomorphism
Ext∗k(G∝P)(κM ,N)∼= Ext∗k(G∝P)(k,Hom(κM ,N)).
Then we apply the finite generation result that we just quoted.
�
However in general Ext∗k(G∝P)(M,N) 6∼= Ext∗k(G∝P)(k,Hom(M,N)),
showing the limit of inter-
nal hom. It is striking that Hochschild cohomology occurs when
we solve the finite generationproblem in the next section.
4.2. Finite generation through Hochschild cohomology. In [17],
we proved that the ordi-nary cohomology ring Ext∗kC(k, k) is
closely related to the Hochschild cohomology ring Ext
∗kCe(kC, kC).
In order to see the connection we have to rely on F (C) (another
finite category), the category offactorizations in C. There are
natural functors and a commutative diagram
F (C) ∇ //
t!!
Ce = C × Cop
p
yyC .This diagram induces various functors among the module
categories of the category algebras ofthese three categories. We
shall not recall the definitions of these categories and functors
as we donot need them here. For applications in the present
article, in that paper we demonstrated that
(1) Ext∗kCe(kC,M) ∼= Ext∗kF (C)(k,Res∇M).
for any M ∈ kCe-mod [17]. Thus our result is a generalization of
a well know theorem in groupcohomology. In fact when C = G is a
finite group, Ge ∼= G × G and F (G) ' ∆G. We commentthat although F
(C) and C usually are not equivalent as categories, their
classifying spaces arehomotopy equivalent. Consequently
(2) Ext∗kF (C)(k, k)∼= Ext∗kC(k, k).
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ON LOCAL CATEGORIES OF FINITE GROUPS 7
Before moving to transporter categories, we remind the reader
that in order to prove Evens-Venkov theorem, we depend on the
following isomorphism
Ext∗kG(M,N)∼= Ext∗kG(k,Homk(M,N)),
where Homk(M,N) is the internal hom. This is not the case for
category cohomology as wementioned earlier. Fortunately we find a
replacement for the above isomorphism. The trick is thatwe must
replace C with F (C). To this end we recall a theorem of
Cartan-Eilenberg [4, ChapterIX, Corollary 4.4]. Let A be a
finite-dimensional algebra over a field k and M,N ∈ A-mod. Then
(3) Ext∗A(M,N)∼= Ext∗Ae(A,Homk(M,N)).
Proposition 4.4. Suppose M,N are two kC-modules. Then we
have
Ext∗kC(M,N)∼= Ext∗kF (C)(k,Res∇Homk(M,N)).
Proof. By (3) and (1)
Ext∗kC(M,N)∼= Ext∗kCe(kC,Homk(M,N)) ∼= Ext
∗kF (C)(k,Res∇Homk(M,N)).
�
Thus in order to prove the finite generation of Ext∗kC(M,N), we
only need a result like Propo-sition 4.2, but for F (G ∝ P). This
is also done in [18].
Proposition 4.5. Let G be a finite group and P a finite G-poset.
Then Ext∗kF (G∝P)(k, k) is finitelygenerated, and for any N ∈ kF (G
∝ P)-mod, Ext∗kF (G∝P)(k,N) becomes a finitely generatedExt∗kF
(G∝P)(k, k)-module.
The proof is similar to that of Proposition 4.2, using the
Grothendieck spectral sequence.However it requires the
understanding of undercategories associated to the composite of
functorsF (G ∝ P) → G ∝ P → G, which allows us to show that the
spectral sequence has only finitelymany rows.
Theorem 4.6. Suppose M,N are two k(G ∝ P)-modules. Then the
module Ext∗k(G∝P)(M,N) isfinitely generated over Ext∗k(G∝P)(k,
k).
Proof. By Proposition 4.4
Ext∗k(G∝P)(M,N)∼= Ext∗kF (G∝P)(k,Res∇Homk(M,N)).
Hence by Proposition 4.5 and (2), Ext∗kF (G∝P)(k,Res∇Homk(M,N))
is finitely generated over thering Ext∗kF (G∝P)(k, k) ∼= Ext
∗k(G∝P)(k, k). We are done. �
Because of the above theorem, we can define the variety for a
k(G ∝ P)-module M to be themaximal ideal spectrum of Ext∗k(G∝P)(k,
k)/IM, where IM is the annihilator of the following map
−⊗̂M : Ext∗k(G∝P)(k, k)→ Ext∗k(G∝P)(M,M).
It implies that there exists a support variety theory [2].
However we shall investigate it in anotherarticle [20].
5. The functor ρ: invariants and coinvariants
Here we return to the Frobenius Reciprocity and take the
opportunity to write out explicitlythe formulas in some interesting
cases. In practice one often finds that the functor G ∝ P → Cis
part of an extension (or an opposite extension) sequence of
categories. (Among examples arevarious orbit categories, Brauer
categories and Puig categories.) It means that for such a
quotient
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8 FEI XU
category C there exists a category K which is a disjoint union
of subgroups Kx ⊂ AutG∝P(x), xrunning over ObP = Ob(G ∝ P), such
that we can add K into the picture
Kι
##G ∝ P
π
{{
ρ
##G C
and moreover K ↪→ G ∝ P � C satisfies some natural conditions.
The explicit definition of acategory extension, due to G. Hoff,
will be provided in Section 5. It will helps us to
understandrelationship between kG-mod and kC-mod.
To this end we need to understand the functor ρ : G ∝ P → C,
where C is a quotient categoryof G ∝ P. In general situation it
seems hard to make group theoretic interpretation of ↑kCk(G∝P)and
⇑kCk(G∝P). However we can do so when we have certain quotient
categories, which are part ofsome category extension sequences in
the sense of G. Hoff [8].
An extension E of a category C via a category K is a sequence of
functors
K ι−→E ρ−→C,
which has the following properties:(1) ObK = Ob E = Ob C, ι is
injective and ρ is surjective on morphisms;(2) if ρ(α) = ρ(β), for
two morphisms α, β ∈ Mor(E), if and only if there is a unique g
∈
Mor(K) such that β = ι(g)α.The following properties can be
deduced from the definition.(3) if αι(h) exists for α ∈ Mor(E) and
h ∈ Mor(K), then there exists a unique h′ ∈ Mor(K)
such that ι(h′)α = αι(h);(4) for any α ∈ HomC(x, y), K(y) acts
regularly on ρ−1(α).It is known to Hoff that K is a disjoint union
of the groups ρ−1(1x) for all 1x ∈ Mor(C) (regarded
as categories), and can be identified with a functor K : E →
Groups. Usually from the context,one can easily see when we take K
to be a category and when it is regarded as a functor.
A sequence K ↪→ E � C is called an opposite extension if Kop ↪→
Eop � Cop is an extension.For a fuller discussion of category
extensions, see [16, Section 4].
The advantage of considering π : G ∝ P → C which is part of an
extension (or oppositeextension) is that it enables us to provide a
good characterization of the left (or right) Kanextension. Indeed
it is the case for many familiar category constructions in
representation theoryand homotopy theory.
The following statement is a special case of [16, Lemma
4.2.1]
Lemma 5.1. Let K → E ρ→C a sequence of three EI-categories and M
∈ kE-mod.(1) Suppose K → E → C is an extension. Then LKρM ∼= MK,
where MK as a functor overC is given by MK(x) = M(x)K(x)
(K(x)-coinvariants of the kAutE(x)-module M(x)), forany x ∈ Ob C =
Ob E = ObK.
(2) Suppose K → E → C is an opposite extension. Then RKρM ∼= MK,
where MK as afunctor over C is given by MK(x) = M(x)K(x)
(K(x)-invariants of the kAutE(x)-moduleM(x)), for any x ∈ Ob C = Ob
E = ObK.
In the above lemma, there is another way to express the Kan
extensions. Under the sameassumptions, they are MK = H0(K;M) and MK
= H0(K;M) respectively.
In what follows, we shall apply the above statements to various
local categories of G, in com-bination with the Frobenius
reciprocity (see Proposition 3.1 and Remark 3.2)
(1’) HomkG(M, lim←−P N)∼= HomkC(LKρκM ,N);
(2’) HomkG(lim−→PM, N)∼= HomkC(M, RKρκN ).
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ON LOCAL CATEGORIES OF FINITE GROUPS 9
In what follows, we shall bring up several local categories
considered in modular representationtheory and homotopy theory of
classifying spaces. Here the information coming from their
cate-gorical structures is our chief concern. To this end, the
reader does not have to understand thatmuch of these categories and
their functions.
5.1. Orbit categories. Suppose P is a collection of subgroups of
G, which is closed under conju-gation, and on which G acts by
conjugation. Then it forms a G-poset and we can define an
orbitcategory OP as the quotient category of G ∝ P by asking
HomOP (P,Q) = Q\NG(P,Q).Then we have an extension sequence
S ↪→ G ∝ P � OP ,where S is the disjoint union of all objects in
P, regarded as a subcategory of G ∝ P.
When M = κM for someM ∈ kG-mod, (LKρκM )(P ) = MP for any P ∈
ObP. We denote sucha kOP -module by MS := (κM )S = LKρκM . Since
giving a morphism (g, gP ≤ Q) is the same asgiving a group
homomorphism P → Q, the conjugation induced by g, there is a
natural way toconstruct a map MP → MQ, identical to the natural map
H0(P ;M) = k ⊗kP M → k ⊗kQ M =H0(Q;M). Hence we know how kOP acts
on MS .
Combining the Frobenius Reciprocity (1’) and Lemma 5.1 (1), we
obtain the following isomor-phism.
Proposition 5.2. Let M ∈ kG-mod and N ∈ kOP -mod. ThenHomkG(M,
lim←−PN)
∼= HomkOP (MS ,N),
where MS is as above.
Corollary 5.3. With the same notations as above, we have
lim←−PN∼= HomkOP (kGS ,N) and
HomkG(k, lim←−PN)∼= HomkOP (k,N).
As an example we let H be a subgroup of G and P the subgroups
that are conjugate to H. Thesize of the discrete poset P is
G/NG(H). Note that both G ∝ P and OP are connected groupoids,the
former equivalent to NG(H) and the latter NG(H)/H. Thus the
isomorphism in Proposition5.2 can be interpreted as
HomkG(M,HomkNG(H)(kG,N))∼= HomkNG(H)(M,N)∼=
Homk(NG(H)/H)(k(NG(H)/H)⊗kNG(H) M,N)∼= Homk(NG(H)/H)(MH , N),
where M ∈ kG-mod and N ∈ k(NG(H)/H)-mod.
5.2. Brauer categories, fusion and linking systems. Suppose b is
a p-block of the groupalgebra kG and Pb is the poset of b-Brauer
pairs [13, Section 47]. Then for any G-subposetP ⊂ Pb we can
introduce the Brauer category BP as the quotient category of G ∝ P
such that
HomBP (P,Q) = HomG∝P(P,Q)/CG(P ).
This gives us an opposite extension, which means that the
following sequence
CG ↪→ (G ∝ P)op → BopPis an extension sequence, given that CG is
the disjoint union of all CG(P ), P ∈ ObP. Dual to theextension
situation we examined before, now we are able to describe the right
Kan extension ofmodules.
If M = κM for some M ∈ kG-mod, we denote by MCG the kFP -module
RKρκM . Since amorphism (g, gP ≤ Q) provides a group homomorphism P
→ Q and thus induces an injectioncg−1 : CG(Q)→ CG(P ), we obtain an
injection MCG(P ) →MCG(Q). This leads to the kBP -actionon MCG
.
Combining the Frobenius Reciprocity (2’) and Lemma 5.1 (2), we
obtain the following isomor-phism.
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10 FEI XU
Proposition 5.4. Let M ∈ kBP -mod and N ∈ kG-mod. Then
HomkG(lim−→PM, N)∼= HomkBP (M, NCG),
where NCG is as above.
As an example we assume b is the principal block b0 and P is the
conjugacy class of a fixedp-subgroup H. Then the discrete poset P
has |G/NG(H)| objects. Both G ∝ P and BP areconnected groupoids,
the former equivalent to NG(H) and the latter NG(H)/CG(H). Thus
theabove isomorphism can be interpreted as
HomkG(kG⊗kNG(H) M,N) ∼= HomkNG(H)(M,N)∼=
Homk(NG(H)/CG(H))(M,HomkNG(H)(k(NG(H)/CG(H)), N))∼=
Homk(NG(H)/CG(H))(M,NCG(H)),
where N ∈ kG-mod and M ∈ k(NG(H)/CG(H))-mod.
Corollary 5.5. We have HomkG(lim−→PM, k)∼= HomkBP (M, k) for any
M ∈ kBP -mod.
Let Bb = BPb . If we fix a maximal object (S, eS) and take all
objects (Q, eQ) with Q ⊂ S, thenthe full subcategory of Bb,
consisting of all these objects, is called a fusion system, usually
writtenas Fb or FS . The inclusion Fb ⊂ Bb is an equivalence. There
is a general theory of (abstract)fusion systems and p-local finite
groups introduced by Broto, Levi and Oliver[3]. We recall
onlynecessary ingredients for our case. Let us take the full
subcategory Fcb ⊂ Fb whose objects areself-centralizing b-Brauer
pairs (Q, eQ) [13]. According to a newly established difficult
theorem ofChermak [10] (true in general for all fusion systems),
there exists a unique category, the centriclinking system Lc,
fitting into the middle of a sequence
Z ↪→ Lc � Fc
which is an opposite extension. Here Z is the disjoint union of
Z(Q) for all objects (Q, eQ) of Fcb .From Lemma 5.1 (2) we obtain
the following isomorphism.
Proposition 5.6. Let N ∈ kLc-mod and M ∈ kFc-mod. Then
HomkLc(ResρM,N) ∼= HomkFc(M,NZ),
where NZ is defined by NZ(P ) = N(P )Z(P ).
Let Bcb be the full subcategory of Bb for a block b, consisting
of self-centralizing b-Brauer pairs.Since Fcb naturally embeds into
Bcb and it induces an equivalence, we similarly can consider
anopposite extension
Z ↪→ L̃cb � Bcb .There exists a natural embedding Lcb → L̃cb
inducing an category equivalence. By taking the largercategory
L̃cb, we can write down
CG/Z ↪→ G ∝ Pcb � L̃cb,another opposite extension. Here CG/Z is
the disjoint union of CG(P )/Z(P ) in which P runsover all
F-centric subgroups. For the sake of convenience, we introduce a
notation C ′G = CG/Zso that C ′G(P ) = CG(P )/Z(P ) ∼= PCG(P )/P
for each P .
When M = κM for some M ∈ kG-mod, we write the kL̃cb-module RKρκM
as MC′G . Given
a morphism (g, gP ≤ Q) it induces an injection C ′G(Q) → C ′G(P
) thus a morphism MC′G(P ) →
MC′G(Q). Hence we get the kL̃cb-action on MC
′G .
Combining the Frobenius Reciprocity (2’) and Lemma 5.1 (2), we
obtain the following isomor-phism.
Proposition 5.7. Let M ∈ k(G ∝ Pcb )-mod and N ∈ kG-mod.
Then
HomkG(lim−→PM, N)∼= HomkL̃cb(M, N
C′G),
where NC′G is defined as above.
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ON LOCAL CATEGORIES OF FINITE GROUPS 11
In particular if M = ResρM′ for some M′ ∈ kBcb-mod then
HomkG(lim−→PM′, N) ∼= HomkBcb (M
′, NCG),
a special situation of Proposition 5.2.
Proof. The first part is a direct consequence of Corollary 5.3.
As for the special case We need tonotice that lim−→P ResρM
′ ∼= lim−→PM′ as kG-modules. Then
HomkG(lim−→P ResρM′, N) ∼= HomkL̃cb(ResρM
′, NC′G)
∼= HomkBcb (M′, (NC
′G)Z)
∼= HomkBcb (M′, NCG).
�
5.3. Puig categories. If we take PA to be the poset of pointed
subgroups on an interior G-algebra A, then analogues to the Brauer
category for any G-subposet P ⊂ PA we can introducethe Puig
category [13, Section 47] LP as a quotient category of G ∝ P such
that
HomLP (Pγ , Qδ) = HomG∝P(Pγ , Qδ)/CG(P ).
Then some results in last section can be obtained accordingly.If
M = κM for some M ∈ kG-mod, we denote by MCG the kLP -module RKρκM
. Since a
morphism (g, gP ≤ Q) provides a group homomorphism P → Q and
thus induces an injectioncg−1 : CG(Q)→ CG(P ), we obtain an
injection MCG(P ) →MCG(Q). This leads to the kLP -actionon MCG
.
Combining the Frobenius Reciprocity (2’) and Lemma 5.1 (2), we
obtain the following isomor-phism.
Proposition 5.8. Let M ∈ kLP -mod and N ∈ kG-mod.
ThenHomkG(lim−→PM, N)
∼= HomkLP (M, NCG),
where NCG is as above.
5.4. Orbit categories of fusion systems. This method also works
for the orbit category of afusion system [9]. Since we are not
going to recall the definition of an abstract fusion system, herewe
only deal with a special case. Suppose Bb is a Brauer category and
Fb is a fusion system asintroduced after Corollary 5.5. One may
continue to define the orbit category OFb as a quotientcategory
with
HomOFb (P,Q) = Q\HomFb(P,Q).Then we obtain an extension
sequence
S ↪→ Fb � OFb ,where S is given by S(Q, eQ) = Q for any object
(Q, eQ) ∈ ObFb. The above constructions stillwork if we replace Fb
by Fcb . Lemma 5.1 (1) leads to the following statements.
Proposition 5.9. If M ∈ kFb-mod and N ∈ kOFb-mod,
thenHomkFb(M,ResρN)
∼= HomkOFb (MS ,N).
Given M ∈ kFcb -mod, we get HomkG(lim−→PM, k)∼= HomkOFc
b(MS , k).
Proof. The first isomorphism comes directly from Lemma 5.1 (1).
Replacing Fb and OFb byFcb and OFcb , respectively, we obtain
HomkFcb (M,ResρN) ∼= HomkOFcb (MS ,N). Then we applyCorollary 5.5
to it and get
HomkG(lim−→PM, k)∼= HomkBcb (M, k)∼= HomkFcb (M, k)∼=
HomkOFc
b(MS , k).
�
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12 FEI XU
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Departament de Matemàtiques, Universitat Autònoma de Barcelona,
08193 Bellaterra, Spain.E-mail address: [email protected]