p-LOCAL FINITE GROUP COHOMOLOGY
RAN LEVI AND K'ARI RAGNARSSON
Abstract. We study cohomology for p-local finite groups with non-constan*
*t coef-
ficient systems. In particular we show that under certain restrictions t*
*here exists a
cohomology transfer map in this context, and deduce the standard consequ*
*ences.
Introduction
A p-local finite group is an algebraic object designed to encapsulate the in*
*formation
modeled on the p-completed classifying space of a finite group. More specifical*
*ly, it is a
triple (S, F, L), where S is a finite p-group, and F is a certain category whos*
*e objects
are the subgroups of S and whose morphisms are certain homomorphisms between
them, satisfying a list of axioms, which entitles it to be called a "saturated *
*fusion
system over S". The category F models conjugacy relations between subgroups of *
*S,
while L is again a category, whose objects are a certain subcollection of subgr*
*oups of
S, but whose morphism set contain enough structure as to allow one to associate*
* a
classifying space with (S, F, L) from which the full structure briefly describe*
*d above
can be retrieved. The classifying space of (S, F, L) is simply the p-completed*
* nerve
|L|^p.
The theory of p-local finite group was introduced in [BLO2 ] and has been s*
*tudied
quite extensively by various authors. In particular the mod p cohomology of the*
* clas-
sifying space of a p-local finite group satisfies a "stable elements theorem", *
*identical in
essence to the corresponding statement for the cohomology of finite groups with*
* con-
stant mod p coefficients [BLO2 , Thm B]. However, as we shall see below, the a*
*nalogy
does not carry forward when one considers a more general setup, i.e., the cohom*
*ology
of p-local finite groups in the context of functor cohomology. In particular w*
*e will
present counter examples to the statement that the cohomology of the p-local gr*
*oup,
in this sense, is given as the stable elements in the cohomology of its Sylow s*
*ubgroup
with restricted coefficient.
The purpose of this paper is to study the cohomology of p-local finite group*
*s with
arbitrary coefficients, and in particular to establish an algebraic definition *
*of a transfer
map for p-local finite groups in this setting. We deal with the cases where a g*
*eometric
transfer must exist, that is, when there is a finite index covering space assoc*
*iated with
the given setup. The necessary theory is provided by the papers [5A1 , 5A2 ], *
*where
"subgroups" of a p-local finite group of a given "index" are defined and studie*
*d. In
particular it is shown in [5A2 ] that with a given p-local finite group one can*
* associate
two finite groups - a p-group and a p0-group - and that, up to equivalence, p-l*
*ocal sub-
groups of p-power index and p0index are in one to one correspondence with subgr*
*oups
of these two groups. Subgroups of indices which are neither p-power nor p0can a*
*lso be
studied under restricted conditions, which essentially follow from the two basi*
*c cases.
In the current article we show the existence of a "transfer map" to the cohomol*
*ogy of
___________
Date: October 29, 2009.
This project was supported by EPSRC grant GR/S94667/01.
1
2 p-local finite group cohomology
a p-local finite group from the cohomology of any subgroup of index prime to p *
*with
respect to any system of coefficients, and from the cohomology of any subgroup *
*of
p-power index with respect to locally constant systems of coefficients. These t*
*ransfer
maps carry similar properties to the standard group cohomology transfer, and ca*
*n be
used to study cohomology of p-local finite groups with non-constant coefficient*
*s.
We now state our results. Our notion of a coefficient system on a p-local f*
*inite
group will be described in detail later, but for the purpose of presenting our *
*result, it
suffices to say that a coefficient system on a p-local finite group (S, F, L) i*
*s a functor
M :L ___! Ab , where Ab is the category of abelian groups (or sometimes Z(p)-m*
*odules,
in which case we say that the system is p-local). Unless otherwise specified, *
*we will
work with coefficient systems which are covariant functors.
As already indicated, we only define a transfer for arbitrary systems of coe*
*fficients
under certain, rather restrictive, conditions. We start our discussion with a p*
*roposition,
due to Bob Oliver, which shows that in general one cannot hope to do much bette*
*r.
Proposition A. There exist a p-local finite group (S, F, L) and a locally const*
*ant p-
local system of coefficients M on (S, F, L) such that eH*(L, M) 6= 0 but eH*(S,*
* '*M) = 0,
where '*M means the restriction of M to S.
Thus the proposition shows that unlike the situation in ordinary group cohom*
*ology,
the cohomology of a p-local finite group with p-local coefficients is not alway*
*s a retract
of the cohomology of its Sylow subgroup with the restricted coefficients.
If (S, F, L) is a p-local finite group, and (S0, F0, L0) is a subgroup of p-*
*power index
or index prime to p, then one has a an associated covering space up to homotopy
': |L0| ! |L|, and thus if M is a Z(p)[ss1(|L|)]-module, then there is a transf*
*er map
H*(|L0|, '*M) ! H*(|L|, M) associated with the covering. We present a categori*
*cal
analog of this setup, and use it to prove our first theorem.
Theorem B. Let (S, F, L) be a p-local finite group, and let (S0, F0, L0) be a s*
*ubgroup
of p-power index or index prime to p. Then for any locally constant coefficient*
* system
M on L, there exists a map Tr :H*(L0, M0) ___! H*(L, M), where M0 denotes the
restriction of M to L0, such that the composite
Res * Tr *
H*(L, M) ___! H (L0, M0) ___! H (L, M)
is multiplication by the index. Furthermore, the map Tr coincides with the tra*
*nsfer
map associated with the covering |L0| ! |L|.
Theorem B is proven below as Theorem 3.5. Next, we specialize to p-local fi*
*nite
subgroups of index prime to p. It is in this setting that we are able to obtain*
* our most
general result.
Theorem C. Let (S, F, L) be a p-local finite group, and let (S0, F0, L0) be a s*
*ubgroup
of index prime to p. Let M be an arbitrary p-local system of coefficients on L.*
* Then
there exists a map
Tr:H*(L0, M0) ___! H*(L, M),
where M0 denotes the restriction of M to L0, such that the composite TrO Res is*
* given
by multiplication by the index. Furthermore, the map Tr satisfies a double cose*
*t formula
and Frobenius reciprocity.
The construction of the transfer map is carried out in Section 4. The precis*
*e state-
ments and proof of its properties appear below as Propositions 5.1, 5.2 and 5.7.
R. Levi and K. Ragnarsson *
* 3
As a final standard application we conclude a stable elements theorem in our*
* con-
text. The statement of the theorem however requires some extra preparation, and
will therefore not be stated here. The reader who is familiar with [5A2 ] is re*
*ferred to
Theorem 5.6 for the precise statement.
The paper is organized as follows. In Section 1 we recall the definition of *
*a p-local
finite group, and revise some of the background material necessary for our disc*
*ussion.
Section 2 is devoted to some basic concepts of homological algebra of functors,*
* includ-
ing a discussion of cohomology with locally constant coefficients. We start Sec*
*tion 3 by
presenting a family of examples which show that in general one cannot expect a *
*stable
elements theorem to hold for p-local finite groups even with respect to locally*
* constant
coefficients. This is followed by a discussion of the transfer for locally cons*
*tant coeffi-
cient systems and subgroups of p-power index and index prime to p. Still in Sec*
*tion 3
we set up the background for the construction of a transfer map for subgroup of*
* index
prime to p. The construction itself is carried out in Section 4, where we also *
*present a
cochain level construction of the map, and a geometric interpretation of the co*
*nstruc-
tion for locally constant coefficients, where a transfer map exists for geometr*
*ic reasons.
Finally in Section 5 we derive the standard consequences of the existence of a *
*transfer
map to the cohomology theory of p-local finite groups with nontrivial coefficie*
*nts.
The authors would like to thank Bob Oliver for a number of useful conversati*
*ons
while we were working on this project, and particularly for his illuminating co*
*unter
example (Section 3). The second named author would like thank Lukas Vokrinek for
giving him a copy of his bachelor thesis, which contains useful background mate*
*rial on
functor cohomology.
1. p-local finite groups
In this section we recall some of the basic concepts of p-local finite group*
* theory.
We also recall the Stable Elements Theorem for the cohomology of p-local finite*
* groups
with p-local constant coefficients. The reader is referred to [BLO2 , 5A1 , 5A*
*2 ] for more
detail.
1.1. Saturated Fusion Systems. The fundamental object underlying a p-local fini*
*te
group is a finite p-group and a fusion system over it. The concept is originall*
*y due to
Puig [Pu ], but we will use the simpler, equivalent definition from [BLO2 ].
For a group G and subgroups P, Q G we denote by Hom G(P, Q) Hom (P, Q) t*
*he
set of all homomorphisms P ! Q obtained by restriction of inner automorphisms o*
*f G
to P . The set of all elements g 2 G such that gP g-1 Q is called the transpo*
*rter set
in G from P to Q, and is denoted NG (P, Q). Thus Hom G (P, Q) = NG (P, Q)=CG (P*
* ).
If g 2 G, then we denote by cg the conjugation x ! gxg-1.
Definition 1.1 ([Pu ] and [BLO2 , Definition 1.1]). A fusion system over a fin*
*ite p-
group S is a category F, where Ob (F) is the set of all subgroups of S, and whi*
*ch
satisfies the following two properties for all P, Q S:
(1)Hom S(P, Q) Hom F (P, Q) Inj(P, Q); and
(2)each ' 2 Hom F(P, Q) is the composite of an isomorphism in F followed by*
* an
inclusion.
If F is a fusion system over a finite p-group S, then two subgroups P, Q S*
* are
said to be F-conjugate if they are isomorphic as objects in F.
4 p-local finite group cohomology
Definition 1.2 ([Pu ], and [BLO2 , Definition 1.2]). Let F be a fusion system *
*over a
p-group S. A subgroup P S is said to be
(1)fully centralized in F if |CS(P )| |CS(P 0)| for all P 0 S which are*
* F-
conjugate to P ;
(2)fully normalized in F if |NS(P )| |NS(P 0)| for for all P 0 S which a*
*re
F-conjugate to P ;
(3)F-centric if CS(P 0) = Z(P 0) for all P 0 S which are F-conjugate to P .
(4)F-radical if OutF (P ) def=AutF(P )= Inn(P ) does not contain a nontrivi*
*al normal
p-subgroup.
A fusion system F is said to be saturated if the following two conditions hold:
(I)For every P S that is fully normalized in F, P is fully centralized in*
* F and
Aut S(P ) 2 Sylp(Aut F(P )).
(II)If P S and ' 2 Hom F (P, S)_are such that P 0= '(P ) is fully centrali*
*zed,
then ' extends to a morphism ' 2 Hom F(N', S), where
N' = {g 2 NS(P ) | 'cg'-1 2 AutS(P 0)}.
If G is a finite group and S 2 Sylp(G), then the category F = FS(G), whose o*
*bjects
are all subgroups P S, and whose morphisms are Hom F (P, Q) = Hom G (P, Q), i*
*s a
saturated fusion system ([BLO2 , Proposition 1.3]).
1.2. Centric Linking Systems. Let Fc F denote the full subcategory whose
objects are the F-centric subgroups of S.
Definition 1.3 ([BLO2 , Definition 1.7]). Let F be a fusion system over the p-*
*group
S. A centric linking system associated to F is a category L whose objects are *
*the
F-centric subgroups of S, together with a functor ss :L ! Fc, and "distinguishe*
*d"
ffiP
monomorphisms P --! Aut L(P ) for each F-centric subgroup P S, which satisfy
the following conditions.
(A) ss is the identity on objects and surjective on morphism sets. For each *
*pair of
objects P, Q 2 Ob (L), Z(P ) acts freely on Mor L(P, Q) by composition (*
*upon
identifying Z(P ) with ffiP(Z(P )) AutL(P )), and ss induces a bijecti*
*on
~=
MorL (P, Q)=Z(P ) ------! Hom F(P, Q).
(B) For each F-centric subgroup P S and each x 2 P , ss(ffiP(x)) = cx 2 Au*
*tF (P ).
(C) For each f 2 Mor L(P, Q) and each x 2 P , f OffiP(x) = ffiQ (ss(f)(x)) O*
*f.
A p-local finite group is a triple (S, F, L), where S is a finite p-group, F*
* is a saturated
fusion system over S, and L is a centric linking system associated to F. The cl*
*assifying
space of a p-local finite group (S, F, L) is the p-completed nerve |L|^p.
If F = FS(G) for some finite group G, then P S is F-centric if and only if*
* P
is p-centric in G; that is, if and only if Z(P ) 2 Sylp(CG (P )), or equivalent*
*ly if and
only if CG (P ) ~= Z(P ) x C0G(P ), where C0G(P ) is a group of order prime to *
*p. The
centric linking system of G is defined to be the category LcS(G), whose objects*
* are the
subgroups of S that are p-centric in G, and whose morphism sets are Mor L(P, Q)*
* =
NG (P, Q)=C0G(P ). The triple (S, FS(G), LcS(G)) then forms a p-local finite gr*
*oup whose
classifying space is equivalent to BG^p[BLO2 ].
R. Levi and K. Ragnarsson *
* 5
1.3. Compatible Systems of Inclusions. For a p-local finite group (S, F, L), a
morphism_': P ! Q in L can be thought of as a "lift" of the group homomorphism
' = ss(') in the fusion system F, and the number of such lifts is |Z(P )|. It w*
*ill often be
convenient to extend concepts associated to group homomorphisms_to morphisms in*
* L.
For instance, we define the image of ' to be the image of ', and denote it by '*
*(P ) or
P '. (Observe that since the morpism is written on the right, we have P _O'= (P*
* ')_.)
Just as in F, a morphism in L can be restricted to a subgroup, and also induces*
* an
isomorphism from its source to its image. To make sense of these restrictions, *
*we first
need a good notion of "inclusions" in L. This is developed in [5A2 , Definition*
* 1.11],
and we recall the definitions here.
Definition 1.4 ([5A2 , Def. 1.11(b)]). Let (S, F, L) be a p-local finite group.*
* A com-
patible set of inclusions for L is a choice of morphisms 'QP2 Mor L(P, Q), one *
*for each
pair of F-centric subgroups P Q, such that 'SS= IdS, and the following hold f*
*or all
P Q R,
(i)ss('QP) is the inclusion P ,! Q;
(ii)'RQO'QP= 'RP;
We often write 'P for 'SP. The existence of a compatible set of inclusions *
*for L is
proved in [5A2 , Proposition 1.13]. A compatible set of inclusions for L allows*
* us to talk
about restrictions of morphisms in L. That is, for a morphism ' 2 Mor L(P, Q) *
*and
F-centric subgroups P 0 P and Q0 Q such that ' O'PP0(P 0) Q0, there is a un*
*ique
0 0 0 Q Q0 P Q0
morphism 'QP02 Mor Lq(P , Q ) with 'Q0O'P0 = 'O'P0. We refer to 'P0 as the rest*
*riction
0
of ' to P 0. To simplify notation, we will often write '|P0instead of 'QP0when *
*there is
no danger of confusion. If P 0is clear from the context, we will sometimes omit*
* it from
the notation as well.
Fix a compatible system of inclusions. Let ss1(|L|, S) be the fundamental gr*
*oup of
|L| with basepoint at the vertex S, and let B(ss1(|L|, S)) be the category asso*
*ciated to
ss1(|L|, S), so B(ss1(|L|, S)) has exactly one object, and the endomorphism mon*
*oid of
that object is ss1(|L|, S). One obtains a functor
J :L ___! B(ss1(|L|, S))
that sends each object to the unique object in the target, and sends a morphism
f :P ! Q to the class of the loop 'Q * f * '-1P. Composing with the distinguis*
*hed
monomorphism ffiS :S ! AutL(S), we get a functor
B(ffiS) J
B(j): B(S) ___! B(Aut L(S)) L ___! B(ss1(|L|, S)),
and a corresponding homomorphism
j :S ___! ss1(|L|).
A construction of the functor J in a more general setting is described in Subse*
*ction
2.6.
1.4. A Stable Elements Theorem. The cohomology of a finite group G with coef-
ficients in a p-local module can be computed by a fundamental result due to Car*
*tan
and Eilenberg [CE ] (known as the stable elements theorem). Their original stat*
*ement
can be reinterpreted as follows. Let OS(G) denote the category whose objects a*
*re
the subgroups of S, and whose morphisms P ! Q are representations (Q-conjugacy
classes of homomorphisms) induced by conjugation in G. Given a Z(p)[G]-module M,
there is a functor on OS(G)op which sends P S to H*(P, M), where M becomes
6 p-local finite group cohomology
a Z(p)[P ]-module via restriction. The Cartan-Eilenberg stable elements theore*
*m can
then be restated as claiming that the following isomorphism holds
H*(G, M) ~= lim H*(-, M).
OS(G)op
Theorem 5.8 in [BLO2 ] is the analogous statement for p-local finite groups*
*, and
where the module of coefficients is the field Fp. Specifically, if (S, F, L) i*
*s a p-local
finite group, and O(F) is the orbit category of F, i.e., the category with the *
*same
objects, and where morphisms P ! Q are given by Hom F (P, Q)= Inn(Q), then
H*(|L|, Fp) ~= lim H*(-, Fp).
O(F)op
From this one can deduce that the same statement is true for any Z(p)-module of
coefficients with a trivial ss1(|L|)-action.
More generally, using a stable transfer construction it is shown in [Ra ] (s*
*ee also [CM ])
that the stable elements theorem holds for any (non-equivariant) stably represe*
*ntable
cohomology theory. However, as we will observe at the end of Section 2, this st*
*atement
is far from being true for cohomology with nontrivial coefficients.
1.5. Finite Index Subgroups in p-local Finite Groups. We now discuss the setup
in which we are able to define a transfer map on p-local group homology and coh*
*o-
mology. Given a saturated fusion system F over a p-group S, the paper [5A2 ] de*
*fines
what it means to be a subsystem of F of a finite index, which in this context i*
*s either
a power of p or prime to p. We refer to the latter case as a p0 index subsyste*
*m The
paper also gives a classification of all subsystems of F of p-power or p0index,*
* which we
recall here.
To any saturated fusion system F over a p-group S, one associates two finite*
* groups
p(F) and p0(F). If F admits an associated centric linking system L, then the*
*se
groups turn out to be the maximal p-power and p0 quotients of ss1(|L|), respect*
*ively.
Both groups depend only on F (and not on the existence or nature of an associat*
*ed
centric linking system L). The group p(F) is given by S=OpF(S), where the div*
*isor
is the hyperfocal subgroup of S with respect to F (see [5A1 , Sec. 2]). If F *
*admits
an associated centric linking system, then p(F) ~=ss1(|L|^p). The group p0(F)*
* has a
more complicated description in [5A2 ], but was observed by Aschbacher to be ss*
*1(|Fc|).
Subsystems of F of p-power or p0index p are in bijective correspondence with*
* sub-
groups of p(F) and p0(F), respectively. If (S, F, L) is a p-local finite grou*
*p, and we
let denote either p(F) or p0(F), then with each subgroup H one has an a*
*ssoci-
ated p-local finite group (SH , FH , LH ), and |LH | is homotopy equivalent to *
*a covering
space of |L| with fibre =H.
It is for this type of subgroups of a p-local finite group that we are able *
*to define
a transfer in cohomology (or homology) with coefficients M 2 L-mod . In the cas*
*e of
a subgroup of p0 index, we can do so in full generality, and we define a cochai*
*n-level
transfer in cohomology for any functor of coefficients. These general methods d*
*o not
carry over to the p-power index case, but we can still define a transfer map fo*
*r locally
constant coefficient systems.
We next state a summary of the main classification result from [5A2 ] for fu*
*sion
subsystem of p-power or p0index. Some minor modifications of the original state*
*ment
will be dealt with in the proof. Before stating the theorem, we set up some not*
*ation.
R. Levi and K. Ragnarsson *
* 7
Let (S, F, L) be a p-local finite group, and let Fq and Lq be the associated*
* quasi-
centric fusion and linking systems [5A1 , Sec. 3]. Let denote p(F) = ss1(|L|*
*^p, S) or
p0(F) = ss1(|Fc|, S), where in both cases S denotes the basepoint given by the*
* vertex
S. Fix a compatible choice of inclusions {'QP} for Lq, and let J :Lq ___! B(ss*
*1(|Lq|, S) be
the resulting functor as defined in Section 1.3. Composing with the obvious pro*
*jection
`b:ss1(|Lq|, S) = ss1(|L|, S) ! , one gets a functor
b: Lq ___! B( ).
We will denote the restriction of b to L by the same symbol. Notice that b s*
*ends
the chosen inclusions in Lq to the identity (since J does). Let ` :S ___! de*
*note the
restriction of b to S via the monomorphism ffiS :S ! AutLq(S) Lq. (Equivalent*
*ly,
` = b`Oj, where j :S ! ss1(|L|, S) is the homomorphism defined in Section 1.3.)*
* For
any subgroup H , let LoH Lq be the subcategory with the same objects and
with morphism set b -1(H), and let LqH LoH be the full subcategory obtained by
restricting to subgroups of SH def=`-1(H). Finally, let FH be the fusion syste*
*m over
SH generated by ss(LqH) Fq and restrictions of morphisms, and let LH LqHbe *
*the
full subcategory on those objects which are FH -centric.
Theorem 1.5 ([5A2 ]). Let (S, F, L) be a p-local finite group. Then, with the n*
*otation
above the following are satisfied.
(i)FH is a saturated fusion system over SH , and LH is an associated centri*
*c linking
system. Thus the triple (SH , FH , LH ) is a p-local finite group.
(ii)If = p0(F) then a subgroup P SH is FH -centric (fully FH -central*
*ized,
fully FH -normalized) if and only if it is F-centric (fully F-centralize*
*d, fully
F-normalized). If = p(F) then the same statements hold, replacing cen*
*tric
by quasicentric.
(iii)There is a 1-1 correspondence between subgroups H and p-local subgr*
*oups
of (S, F, L) with p-power or p0 index (as appropriate). The corresponden*
*ce is
given by H ! (SH , FH , LH ).
(iv)|LH | is homotopy equivalent to the covering space of |Lq| ' |L| with fi*
*bre =H.
(v)The homomorphism Aut L(S) ! induced by the restriction of b :Lq ! B( )
to Aut L(S) is surjective.
Proof. Parts (i), (ii), (iii) and (iv) are included in Proposition 3.8 and Theo*
*rem 3.9 of
[5A2 ]. Part (v) is clear in the case of p(F) = S=OpF(S), and follows from the*
* definition
of p0(F) given in [5A2 , Thm 5.4]).
2. Homological algebra of functors
In this section we develop the background we need from homological algebra. *
*Most
or all the results we present here are well known to the expert, but are includ*
*ed here
for the convenience of the reader.
Throughout this paper, let R denote a fixed commutative ring with a unit. L*
*et
R-mod denote the category of (left) R-modules, and let R-alg denote the catego*
*ry of
(left) R-algebras. All categories we consider in this article will be small. Fo*
*r a category
C, let C-mod and C-alg be the categories whose objects are functors C ___! R-*
*mod
or C ___! R-alg, respectively, and whose morphisms are natural transformations*
*. We
will refer to an object of C-mod as a C-module, and to an object of C-alg as a*
* C-
algebra. Depending on context we may sometimes refer to objects of C-mod as sy*
*stem
8 p-local finite group cohomology
of coefficients on C, or a C-diagram of R-modules, and similarly for objects of*
* C-alg.
Notice that we have not discussed variance of functors at all. By convention, a*
*ll functors
we deal with are covariant. If we need to discuss contravariant functors (and i*
*n certain
contexts we will), we shall consider them as objects in the module category ove*
*r the
opposite category.
2.1. Functor Cohomology. Given an R-module M, the constant C-diagram MC is
the functor which takes every object in C to M and every morphism to the identi*
*ty.
The inverse limit on a category C is a functor
lim:C-mod ___! R-mod .
C
i j ~
It comes with a morphism limU ___U! U, for any C-diagram U of R-modules, and*
* is
C
characterized by the universal property that if M is any R-module, and if ff: M*
*C ___! U
is any natural transformation, then there exists a unique R-module homomorphism
fbf:M ____! limCU such that ff = ~U OfbfC, where bffCis the functor induced on*
* the
respective constant diagrams by bff.
The universal property of the inverse limit functor implies an obvious ident*
*ification:
(1) lim U ~=Hom C-mod(RC, U).
C
This shows in particular that the inverse limit functor is left exact. Its righ*
*t derived
functors applied to U 2 C-mod are usually referred to as the higher limits of *
*U, or
as the cohomology of C with coefficients in U (being the right derived functors*
* of the
Hom functor). Thus if U ! Io is an injective resolution of U in C-mod , and Po *
*! RC
is a projective resolution of RC in C-mod , then
limiU = Hi(Hom C-mod(RC, Io)) = Hi(Hom C-mod(Po, U)).
C
Notation 2.1. Throughout this article we use the notation H*(C, U) to denote li*
*m*U.
C
This is standard notation in the subject, and is better suited for purposes.
Dual to the limit functor there is the colimit functor
colim:C-mod ___! R-mod .
C
One defines the homology of C with coefficients in U 2 C-mod as the higher col*
*imits
of U, i.e. the left derived functors of the colimit functor, applied to U. In t*
*his section,
and throughout the paper, we focus on cohomology and mostly leave the reader to
dualize the discussion to obtain analogous results in homology.
2.2. Cup Products in Functor Cohomology. Let C be a small category. For chain
complexes Co and C0oin C-mod , one obtains a chain complex (C C0)o with
X
(C C0)n def= Ck C0l
k+l=n
and differential ffi induced by the differentials of Co and C0o(which we also d*
*enote by
ffi) via the formula ffi(x y) = ffix y + (-1)kxforffiyx 2,Cky 2 C0l.
It is a standard result that if both Co and C0oare exact then (C C0)o is e*
*xact, and
likewise that (C C0)o is projective if both Co and C0oare projective. In part*
*icular,
given projective resolutions Po ! RC and Po0! RC of the constant functor on C*
*, one
obtains a projective resolution (P P 0)o ! RC RC ~=RC
R. Levi and K. Ragnarsson *
* 9
Let M, M0 2 C-mod . For oe 2 Hom C-mod(Pk, M)and oe02 Hom C-mod(Pl0, M0), l*
*et
oe x oe02 Hom C-mod((P P 0)k+l, M M0)be the natural transformation
proj 0 oe oe0 0
oe x oe0:(P P 0)k+l___! Pk Pl ___! M M .
This gives a bilinear pairing
Hom C-mod(Po, M) Hom C-mod(Po0, M0) ___! Hom C-mod((P P 0)o, M M0)
oe oe07- ! oe x oe0.
One can check that this is a map of cochain complexes, so after taking cohomolo*
*gy we
obtain a bilinear pairing on higher limits
H*(C, M) H*(C, N) -! H*(C, M N),
x y 7! x x y.
One can furthermore check that this is independent of the choice of projective *
*resolu-
tions. This pairing is called the cross product pairing.
For any A 2 C-alg one has a multiplication transformation ~: A A ! A which
induces a homomorphism on cohomology. Composing with the cross product pairing,
we obtain the cup product.
Definition 2.2. Let A 2 C-alg. For x 2 Hk(C, A) and y 2 Hl(C, A), the cup produ*
*ct
x [ y 2 Hk+l(C, A), (or xy,) is the image of x y under the homomorphism
x * ~* *
H*(C, A) H*(C, A) ___! H (C, A A) ___! H (C, A).
The cup product constructed here has the algebraic properties we expect, as *
*stated
in the next proposition. The proof is routine.
Proposition 2.3. For a functor A 2 C-alg, the cup product on H*(C, A) is associ*
*a-
tive, graded-commutative, and has a multiplicative unit. Thus H*(C, A) is a gr*
*aded-
commutative ring with a unit.
If F :C ___! D is any functor, one has an exact functor F *:D-mod ___! C-*
*mod
defined by F *(ff) def=ff OF . The next proposition says that the cup product i*
*s natural
in both C and A. Again, the proof is routine.
Proposition 2.4. Let F :C ! D be a functor, let A, B 2 D-alg, and let j :A ___!*
* B
be a natural transformation. Then the following hold for x, y 2 H*(D, A):
(a)The homomorphism F *:H*(D, A) ! H*(C, F *A) induced by F satisfies
F *(x [ y) = F *x [ F *y.
(b) The homomorphism j*: H*(D, A) ! H*(D, B) induced by j satisfies
j*(x [ y) = j*x [ j*y.
2.3. Kan Extensions and the Shapiro Lemma. Let ': C ___! Dbe any functor.
For each object d 2 D one defines the overcategory ' # d to be the category with
objects (c, ff), where c 2 C and ff: '(c) ! d is a morphism in D. Morphisms fr*
*om
(c, ff) to (c0, ff0) in ' # d are morphisms fl :c ! c0 in C such that ff0O '(fl*
*) = ff. The
undercategory d # ' is defined analogously. Both categories admit an obvious fo*
*rgetful
functor to C, and for M 2 C-mod we denote the composite of M with the forgetful
functor by M].
10 p-local finite group cohomology
The functor '*: D-mod ___! C-mod induced by ' has a right adjoint R' and a*
* left
adjoint L' called the right and left Kan extension along ', respectively. The *
*Kan
extensions of a given functor M 2 C-mod is determined objectwise by
R'(M)(d) = lim M# and L'(M)(d) = colimM# ,
d # ' ' # d
and a morphism d ___! d0in D induces homomorphisms
R'(M)(d) ___! R'(M)(d0) and L'(M)(d) ___! L'(M)(d0),
via the universal properties of the limit and colimit functors. From these desc*
*riptions
of Kan extensions one easily obtains the isomorphisms
limR'(M) = limM and colim L'(M) = colimM.
D C D C
For a functor ': C ____! D and a system of coefficients M 2 C-mod there i*
*s a
homomorphism called the Shapiro map
ShM :H*(D, R'(M)) ___! H*(C, M),
which is constructed as follows. Let M ___! Io be an injective resolution of M*
*. Since R'
is the right adjoint of the exact functor '* it preserves injectives, and we ha*
*ve a (possibly
non-exact) cochain complex R'M ___! R'I0 ___! R'I1 ___! . . .in which every *
*term
after R'M is injective. If R'M ___! I0ois an injective resolution of R'M, the*
*n the
identity transformation of R'M lifts (non-uniquely) to a cochain map O: I0o! R'*
*Io,
which induces a cochain map of limits
O* ae-1 def
limI0odef=HomD(RD , I0o) ___! Hom D (RD , R'Io) ___~! Hom C(RC, Io) = lim*
* Io,
D = C
and ShM is defined as the induced map in cohomology. Here,
ae: Hom C(RC, Io) = Hom C('*RD , Io) ___! Hom D (RD , R'Io)
is the adjunction isomorphism. The Shapiro map is independent of the choice of *
*the
injective resolutions Io and I0o, and the cochain map O, and is natural with re*
*spect to
morphisms M ___! M0 in C-mod . Hence it induces a natural transformation
Sh: H*(D, R'(-)) ___! H*(C, -)
which will be referred to as the Shapiro transformation. The important propert*
*y of
the Shapiro transformation is given by the following lemma.
Lemma 2.5 (The Shapiro Lemma). Let ': C ___! Dbe a functor. If the right Kan
extension R' is exact then the Shapiro transformation is a natural isomorphism *
*of
functors.
Proof. If R' is exact, then R'M ___! R'Io is an injective resolution, so we ca*
*n take
I0o= R'Io and O = Id in the construction of ShM for M 2 C-mod , and it follows*
* that
ShM is an isomorphism.
The right Kan extension functor is always left exact as it is a right adjoin*
*t. Similarly
the left Kan extension functor is right exact. In some instances the left and r*
*ight Kan
extension functors are naturally isomorphic. In this case both Kan extension fu*
*nctors
are exact, and in particular the Shapiro Lemma holds. The Shapiro lemma has an
analogous homological version, involving the derived functors of the colimit, a*
*nd the
left Kan extension.
R. Levi and K. Ragnarsson *
* 11
2.4. Deformation Retracts of Categories. We now study a condition on a functor
which ensures that it induces a natural isomorphism between the corresponding d*
*erived
functors of the limit and colimit.
Definition 2.6. Let f :C ___! D be a functor between small categories. We say *
*that
f is a left deformation retract of D if there exists a functor r :D ___! C suc*
*h that
r Of = idC, and a natural transformation j :f Or ___! 1D satisfying jf(c)= 1f(*
*c)for
c 2 C. When f is the inclusion of a subcategory, we say that C is a left deform*
*ation
retract of D. Right deformation retracts are defined dually.
Notice that this falls short of defining ' as left adjoint to r in the sense*
* that we do
not require that rj is the identity transformation on r. The definition should *
*remind
the reader of deformation retracts of spaces. Indeed, if C is a (left or right)*
* deformation
retract of D, then |C| is a deformation retract of |D|.
Lemma 2.7. Let f :C ___! Dbe a functor between small categories.
(a)If f is a left deformation retract then f preserves limits.
(b) If f is a right deformation retract then f preserves colimits.
Proof. It suffices to prove part (a), as part (b) follows by duality, so assume*
* that f
is a left deformation retract and let r :D ! C and j :f Or ! 1D be as in Defini*
*tion
2.6. Since r Of = 1, we can regard f as the inclusion of a subcategory. To show*
* that
f preserves limits, it therefore suffices ([MacL ]) to show that f is left cofi*
*nal in the
sense that for every object d in D, the overcategory f # d is connected (in par*
*ticular
nonempty).
First, the map jd: f(r(d)) ! d gives rise to an object (r(d), jd) in f # d. *
* Now, if
u
(c, u) is another object consisting of an object c in C and a morphism f(c) ___*
*! d in
D, then the natural transformation j gives rise to a commutative square
(fOr)(u)
(f Or)(f(c))__________//(f Or)(d)
jf(c)|||| |jd
|| u fflffl||
f(c)_________________//d .
Thus we have a morphism r(u): (c, u) ! (r(d), jd) in f # d, and in particular (*
*c, u) is
in the same connected component as (r(d), jd), proving that f # d.
Lemma 2.7 actually holds if f is a weak deformation retract, for which we re*
*quire
only that for c 2 C, we have jf(c)= f(h) for some morphism h in C. Using the f*
*ull
strength of deformation retracts, one can prove a stronger result, namely that *
*if f is
a left deformation retract then it induces an isomorphism on cohomology, and if*
* it is
a right deformation retract then it induces an isomorphism on homology (Lemma 2*
*.7
makes that claim only in dimension 0, which is all we need for our purposes).
2.5. Cohomology with Locally Constant Coefficients. We now discuss an impor-
tant special case of functor cohomology, namely the case where the coefficient *
*system
is locally constant.
Let C be a small connected category (meaning there exists a zigzag of morphi*
*sms
between any two objects in C). Let bCdenote the groupoid completion of C, i.e.*
*, the
category with the same objects as C, and where all morphisms are formally inver*
*ted.
Let ss :C ! bCbe the obvious projection functor. Choose an object c0 2 C. For*
* any
12 p-local finite group cohomology
other object c 2 C, choose a morphism OEc 2 Mor bC(c, c0), and take OEc0to be t*
*he identity.
Then one gets isomorphisms of sets
ja,b:Mor bC(a, b) ___! Aut bC(c0) by ' |__! OEbO ' OOE-1a.
Let def=AutbC(c0), and let ': B( ) ! Cbbe the inclusion functor. Assembling *
*the
isomorphisms ja,btogether for all pairs of objects in bC, one gets a functor
J :bC___! B( ),
which is an equivalence of categories (but depends on the choices made). In par*
*ticular,
J O' is the identity on B( ), while ' OJ is naturally isomorphic to the identit*
*y on bC.
Moreover, by [Qu , Prop. 1], the group is naturally isomorphic to ss1(|C|, c0*
*), and the
map induced on nerves by the composite J Oss induces an isomorphism on fundamen*
*tal
groups.
A system of coefficients M 2 C-mod is said to be locally constant if for *
*every
morphism ': a ! b in C, M(') is an isomorphism. Notice that M is locally consta*
*nt
if and only if M factors as M = cM Oss for a unique functor cM 2 bC-mod. Let R *
*be a
commutative ring with a unit. An R[ ]-module is a functor N :B( ) ! R-mod . The
next lemma shows that every locally constant system of coefficients on C is, up*
* to a
natural isomorphism, a R[ ]-module composed with the functor J constructed abov*
*e.
Lemma 2.8. Let C be a small connected category, and let M 2 C-mod be a locally
constant system of coefficients on C. Then M is naturally isomorphic to N OJ O*
*ss,
where N is the R[ ]-module cM O'. Moreover, this defines an equivalence of cate*
*gories
between the category of R[ ]-modules and the category of locally constant funct*
*ors on
C.
Proof. Since ' OJ is naturally isomorphic to the identity on bC, one has
M = cM Oss ~=cM O' OJ Oss = N OJ Oss,
as claimed.
For the second statement, notice that the correspondences which takes an R[ *
*]-
module N to N OJ Oss and a locally constant system of coefficients M 2 C-mod to
Mc O' are natural on N and M respectively, and using the first statement, defin*
*e the
equivalence of categories claimed.
Proposition 2.9. Let C be a small connected category, and let M 2 C-mod be a l*
*ocally
constant functor. Then, with the notation above, there is an isomorphism
H*(C, M) ~=H*(|C|, cMO ')
which is natural in M.
To prove the proposition some preparation is needed. Let fl :C ___! B( ) d*
*enote
the composite J Oss. Then one has the restriction functor fl*: R[ ]-mod ___! *
*C-mod
and its right adjoint, the right Kan extension Rfl:C-mod ___! R[ ]-mod . Sinc*
*e Rflis
right adjoint to an exact functor, it preserves injectives, and so there is a G*
*rothendieck
spectral sequence associated to the composition of functors
Rfl HomR[(]R,-)
C-mod _! R[ ]-mod ___________! Ab ,
with
Ep,q2= Hp( , Rq(Rfl)(M)) ) Hp+q(C, M).
R. Levi and K. Ragnarsson *
* 13
Let o denote the unique object in B( ). Then by definition
Rfl(M)(o) = limMo
o#fl
fo M
where Mo is the composite (o # fl) ____! C ____! Ab , and where fo is the obv*
*ious
forgetful functor.
Lemma 2.10. The functor f*o:C-mod ___! (o # fl)-mod preserves injectives.
Proof. It suffices to show that the left Kan extension Lfo:(o # fl)-mod ___! *
*C-mod
is exact. Exactness is determined objectwise, and for d 2 C the value of the le*
*ft Kan
extension at d is a colimit over the category fo # d. A typical object in this *
*category
has the form ((c, g), ff), where g 2 = Aut B( )(o), and ff: c ! d is a morphi*
*sm in
C. A morphism from ((c, g), ff) to ((c0, g0), ff0) is a morphism fi :c ! c0 in*
* C, such
that fl(fi)g = g0, and ff0fi = ff. Notice in particular that ff is the unique *
*morphism
from ((c, g), ff) to ((d, fl(ff)g), 1d), and thus ((d, fl(ff)g), 1d) is a termi*
*nal object in the
connected component of ((c, g), ff). Since every connected component has a term*
*inal
object, colimits are determined by their values on the terminal objects, and so*
* the left
Kan extension is exact, proving the lemma.
Let M ___! Io be an injective resolution of M in C-mod . Then
Mo = f*o(M) ___! f*o(Io)
is an injective resolution of Mo in (o # fl)-mod . Hence
Rq(Rfl)(M)(o) = Hq(Hom (o#fl)-mod(R, f*o(Io))) = Hq(o # fl, Mo).
Thus the Grothendieck spectral sequence takes the form
Ep,q2~=Hp( , Hq(o # fl, Mo)) ) Hp+q(C, M).
Lemma 2.11. Let M be a locally constant system of coefficients on a small conne*
*cted
category C, let denote Aut bC(c0) for some c0 2 C, and let fl :C ____! B( ) *
*be the
projection as before. Then the restriction Mo of M to (o # fl) is naturally iso*
*morphic
to the constant functor on (o # fl) with value M(c0).
Proof. Let ': B( ) ! bCbe the inclusion. For each c 2 C let OEc: c ! c0 be the *
*isomor-
phism in bCchosen in the construction of the functor J :bC! B( ) at the beginni*
*ng
of the section. Since M is locally constant, cM is well defined on any morphism*
* in bC.
Thus let j be the natural isomorphism which takes an object (c, g) in o # fl to*
* the
isomorphism cM(OE-1cO'(g)): M(c0) ! M(c). If ff: (c, g) ! (c0, g0) is a morphi*
*sm in
o # fl, then the following holds in AutCb(c0):
OEc0Off OOE-1cO'(g) def=fl(ff) O'(g) = '(g0).
Applying cM to this equation we see that j is a natural isomorphism from the co*
*nstant
functor on o # fl with value M(c0) to the functor Mo. This proves the claim.
We are now ready to prove Proposition 2.9. By Lemma 2.11 it follows that the*
* E2
page of the Grothendieck spectral sequence converging to H*(C, M) is
(2) Ep,q2~=Hp( , Hq(o # fl, M(c0))) ~=Hp( , Hq(|o # fl|, M(c0))),
where the second isomorphism holds because M(c0) is a trivial o # fl module.
Now consider the Serre spectral sequence associated to the fibration
f|C|__! |C| ___! B ,
14 p-local finite group cohomology
with coefficients in the R[ ]-module N = M(c0). For any morphism g 2 , the ind*
*uced
map
|g*|: |o # fl| ___! |o # fl|
is a self homotopy equivalence, since is a group. Hence the hypothesis of Qui*
*llen's
theorem B [Qu ] is satisfied, and |o # fl| is homotopy equivalent to the homoto*
*py fibre
|fC|of the map |fl|: |C| ! B over the object o. Thus the E2 term of the Serre *
*spectral
sequence for this fibration is
Ep,q2= Hp( , Hq(|o # fl|, N) ) Hp+q(|C|, N).
By construction it is now easy to check that this E2 page is naturally isomorph*
*ic to
the E2 page of the Grothendieck spectral sequence (2), and thus we conclude the*
* proof
of Proposition 2.9.
Corollary 2.12. Let o :C ___! D be a functor between small connected categorie*
*s, such
that |o| is a homotopy equivalence. Then for any locally constant system to coe*
*fficients
M 2 D-mod , o induces an isomorphism
~= *
H*(C, o*M) ___! H (D, M).
2.6. Covering Spaces, and Locally Constant Coefficients. Let C be a small
connected category, c0 2 C and = AutCb(c0) ~=ss1(|C|, c0). Let J :C ___! B( *
*) be as
before.
For each 0 , let E ( = 0) denote the category with = 0as objects, and wi*
*th a
unique morphism ^g:a 0 ! ga 0 for each g 2 and aH 2 = 0. Let C 0 denote the
category given by the pull back in the diagram
C 0_________! E ( = 0)
| |
ss| |ae 0
# #
J
C ____________! B
Thus, Obj(C 0) = Obj(C) x = 0, while morphisms (c, a 0) ! (c0, a0 0) are morph*
*isms
': c ! c0, such that a0 0= J(')a 0.
For each c 2 C the undercategory c # ss has objects ((d, a 0), ') where d 2 *
*C,
a 02 = 0, and ': c ! d is a morphism in C. A morphism ((d, a 0), ') ! ((d0, a0*
* 0), '0)
in c # ss is a map _ :d ! d0in C, such that _' = '0, and a0 0= J(_)a 0.
Let ((d, a 0), ') be an arbitrary object in c # ss. Then there is a unique m*
*orphism
((c, a 0), 1c) ! ((d, J(')a 0), ') which is induced by '. Thus every component *
*of c # ss
has an initial object and is therefore contractible. Notice that there is an ob*
*vious 1-1
correspondence between components of c # ss and = 0, and that every morphism c*
* ! e
in C induces a homotopy equivalence e # ss ! c # ss. Thus the hypothesis of Qui*
*llen's
theorem B are satisfied, and |C 0| ! |C| is a covering space up to homotopy, wi*
*th fibre
over c given by |c # ss| ' = 0.
Lemma 2.13. Let o :D ! C be a functor between small connected categories such
that the induced map |o|: |D| ! |C| is a covering space up to homotopy (i.e., h*
*as a
homotopically discrete homotopy fibre), and let 0= ss1(|D|, d0) for some d0 2 *
*D. Then
the following hold:
(i)There is a functor bo:D ! C 0, lifting o (i.e., ssbo= o), which induces *
*a homo-
topy equivalence on nerves.
R. Levi and K. Ragnarsson *
* 15
(ii)For any locally constant system of coefficients M 2 C-mod , there is a *
*natural
isomorphism Ro(o*M) ~=Rss(ss*M).
Proof. Choose some d0 2 D, and let c0 = o(d0). Let = ss1(|C|, c0), and 0 =
ss1(|D|, d0). Then, constructing a projection functor J :C ! B( ) as before, on*
*e gets
JOo :D ! B( ), whose image is 0, and thus a corresponding projection J0:D ! B(*
* 0).
The diagram
o ss
D ____________! C ____________C 0
| | |
J0| J | |
# # #
inc ae 0 0
B( 0)_________! B( ) ________E ( = )
commutes, where the right hand side square is a pull back square. The inclusion
of B( 0) in E ( = 0) as the full subcategory on the object 1 0 induces a homoto*
*py
equivalence on nerves, and a functor bo:D ! C 0. A simple diagram chase now sho*
*ws
that |bo| is a homotopy equivalence, and proves (i).
To prove (ii), notice that since ssbo= o, the functor boinduces a natural tr*
*ansformation
of functors C ! Cat
bo(-):(- # o) ___! (- # ss).
Thus for every c 2 C, and M 2 C-mod , one has a map induced by bo,
Rss(ss*M) def=lim(ss*M)] ___! lim(bo*ss*M)] = lim(o*M)] def=Ro(o*M).
c#ss c#o c#o
But since ss*M is also locally constant, and since boinduces a homotopy equival*
*ence on
nerves, it follows from Corollary 2.12 that this map is an isomorphism, thus pr*
*oving
(ii).
Lemma 2.13 allows us to compute the values of a right Kan extension of a sys*
*tem of
coefficients M 2 D-mod along a functor o :D ! C satisfying its hypothesis.
The following lemma allows a calculation of the right Kan extension explicit*
*ly in a
specialized case which will be of interest in our discussion.
Lemma 2.14. Let o :D ! C be a functor between small connected categories such t*
*hat
for each c 2 C, every connected component in the under category c # o has an in*
*itial
object. Then for any M 2 D-mod , the value of the right Kan extension Ro(M) 2
C-mod at c is the direct product of the values of M]on these initial objects. *
*In particular
Ro is an exact functor.
Proof. For each c 2 C one has
Y
Ro(M)(c) def=limM] ~= M(d0),
c#o
[(d,ff)]2[c#o]
where the product runs over all connected components of c # o, and (d0, ff0) is*
* initial
in the component of (d, ff). This follows at once from the fact that a limit o*
*ver a
category with an initial object is given by the value of the functor on that ob*
*ject.
This description also makes it clear that Ro is an exact functor, and so the pr*
*oof is
complete.
Proposition 2.15. Let o :D ! C be a functor between small connected categories *
*such
that the induced map |o|: |D| ! |C| is a covering space up to homotopy. Then fo*
*r any
16 p-local finite group cohomology
locally constant system of coefficients M 2 C-mod there is a functor M0 2 C-mo*
*d such
that
H*(C, M0) ~=H*(D, o*M).
Furthermore, if the index of the covering is finite, then there is a natural tr*
*ansformation
T :M0 ! M, such that the composite
o* * * * 0 T* *
H*(C, M) ___! H (D, o (M)) ~=H (C, M ) ___! H (C, M)
is multiplication by the index.
Proof. Let 0= ss1(|D|, d0) as before, let ss :C 0! C be the projection, and le*
*t bo:D !
C 0be the functor constructed in Lemma 2.13.
Since M is locally constant H*(D, o*M) ~= H*(C 0, ss*M) by Corollary 2.12. *
*By
Lemma 2.14 Rssis exact, and hence we have an isomorphism H*(C, Rss(ss*M)) ~=
H*(C 0, ss*M) by the Shapiro lemma. Set M0 = Rss(ss*M) 2 C-mod . Then
H*(D, o*M) ~=H*(C, M0).
This proves the first statement.
To prove the second statement, consider the functor M0. For each coset a 0 2
= 0, the object ((c, a 0), 1c) 2 c # ss is initial in its own path compone*
*nt, and
(ss*(M))]((c, a 0), 1c) = M(c). Thus, by Lemma 2.14, for each c 2 C,
Y
M0(c) def=limss*M] ~= M(c).
c#ss
= 0
Assuming = 0is a finite set, define a natural transformation of functors T :M0*
* ! M
in C-mod by
X
(3) Tc({xa 0}a 02 = 0) = xa 0.
a 02 = 0
Naturality of T is clear, since M is additive.
It remains to show that T* Oo* is multiplication by | : 0|. To do that it s*
*uffices to
show that the natural transformation induced by ss
ss* * * *
Hom C-mod(-, M) ___! Hom C 0(ss (-), ss M) ~=Hom C-mod(-, Rss(ss M)),
is the diagonal map. But for U 2 C-mod , a natural transformation j :U ! M takes
an object c 2 C to a morphism jc: U(c) ! M(c). Composing with ss we see that
the composition transformation takes each object (c, a 0) in C 0to the morphism*
* jc as
above. This shows that ss* is precisely the diagonal map and the proof is compl*
*ete.
The next corollary follows at once from the definitions.
Corollary 2.16. Let M 2 C-mod be a locally constant system of coefficients, an*
*d let
0 = ss1(|C|, c0) be a subgroup of finite index. Then the map
T* *
H*(C 0, ss*M) ~=H*(C, M0) ___! H (C, M)
coincides with the geometric transfer map associated to the covering space |C 0*
*| ! |C|.
R. Levi and K. Ragnarsson *
* 17
3. p-Local Finite Group Cohomology
The cohomology of a p-local finite group (S, F, L) with coefficients M 2 L-m*
*od
is the main object of study in this paper. In ordinary finite group cohomology*
* the
transfer map with respect to a subgroup is one of the most useful computational*
* and
theoretical tools available to us, and its properties are the key to proving th*
*e stable
element theorem in cohomology, among other things. We start our discussion of *
*the
p-local analog with an example showing that in p-local finite group cohomology *
*one
cannot associate a transfer map to an arbitrary p-local subgroup inclusion, eve*
*n if one
restricts to locally constant coefficients.
3.1. An Example. As already mentioned, in [BLO2 ] it is shown that if (S, F, L*
*) is a p-
local finite group then the mod p cohomology of |L| is given by the F-stable el*
*ements
in H*(BS, Fp). The next statement, due to Bob Oliver, provides an abundance of
examples that show that with our definition of local coefficients one cannot ex*
*pect a
stable elements theorem to hold in full generality.
Proposition 3.1. Let (S, F, L) be a p-local finite group, (with S nontrivial) a*
*nd let
= ss1(|L|). Fix a compatible system of inclusions, and assume that the compo*
*site
functor
' J
B(j): B(S) ___! L ___! B( )
B(ffiS)
is faithful, where ' is the inclusion B(S) ___! B(Aut L(S)) L, and J is the *
*functor
defined in Section 1.3. Assume further that the universal cover f|L|is not mod *
*p acyclic.
Let M be the group ring Fp[ ] regarded as a system of coefficients in L-mod vi*
*a J and
the obvious action. Then eH*(S, '*M) = 0, but eH*(L, M) 6= 0.
j *
Proof. Since S ___! is monic, ' M is a projective Fp[S] module, and since S *
*is finite,
it is also injective, and thus acyclic. On the other hand, since M is locally c*
*onstant,
H*(L, M) = H*(|L|, Fp[ ]) def=H*(Hom Fp[(]C*(|fL|), Fp[ ])) ~=
H*(Hom Fp(C*(|fL|), Fp)) ~=H*(|fL|, F*
*p),
where the first equality follows from Proposition 2.9, and the right hand side *
*is not
mod p acyclic by assumption.
One large family of examples which satisfy the conditions of Proposition 3.1*
* is the
general linear groups GLn(Fpk). The following argument is a sketch of a proof. *
*Since
the same argument applies for all k, we will replace Fpk by F, while p and k are
assumed fixed. The radical subgroups in these groups are all p-centric, and th*
*eir
centers are given by the diagonally embedded F* ~= Z=(pk - 1). Hence the centr*
*ic
radical linking systems of GLn(F) and the associated projective group P GLn(F) *
*have
the same objects, and |Lcr(GLn(F))| is a covering space of |Lcr(P GLn(F))| with*
* fibre
F*, and thus they share the same universal cover. But Lcr(P GLn(F)) coincides w*
*ith
the centric radical transporter system of P GLn(F) (the category with the same *
*objects,
and where morphism sets are the transporters NG (P, Q)), and thus admits an obv*
*ious
map
|Lcr(P GLn(F))| ___! BP GLn(F),
with fibre given by the nerve of the poset of centric radical subgroups in P GL*
*n(F)
[BLO1 ]. This nerve is known as the Tits building, and has the homotopy type *
*of a
wedge of spheres. Thus it is the universal cover of |Lcr(P GLn)|, while at the *
*same time
18 p-local finite group cohomology
it is not acyclic. This was first observed by Grodal. To finish the argument, w*
*e recall
from [5A1 ] that for any p-local finite group, the centric radical linking syst*
*em and the
centric linking system have homotopy equivalent nerves.
3.2. Transfer Maps for Locally Constant Coefficients. We now show that if
(S, F, L) is a p-local finite group and (S0, F0, L0) is a subgroup of p-power i*
*ndex or
index prime to p, then one can define an algebraic transfer map, which coincide*
*s with
the transfer map associated to the finite covering |L0| ! |L| (See Thm. 1.5(iv)*
*).
First we must discuss restrictions of coefficient systems. We defined a sys*
*tem of
coefficients on a p-local finite group to be a functor M 2 L-mod . When (S0, F0*
*, L0)
is a subgroup of (S, F, L) of p-power index, one does not have in general an in*
*clusion
functor L0 ! L, but rather an inclusion Lq0! Lq. Thus, in this case, a system *
*of
coefficients on (S, F, L) cannot be directly restricted to (S0, F0, L0). We wil*
*l show that
for locally constant coefficients, this is in fact an easily solvable problem. *
*We start by
observing that the restriction of a locally constant system of coefficients on *
*Lq to L
does not affect cohomology.
Lemma 3.2. Let (S, F, L) be a p-local finite group, and let Lq be the associate*
*d quasi-
centric linking system. For every locally constant system of coefficients N 2 L*
*q-mod ,
the restriction to L induces an isomorphism in cohomology H*(Lq, N) ~=H*(L, '*N*
*).
Proof. This follows directly from Corollary 2.12 since the inclusion Lq L ind*
*uces a
homotopy equivalence on nerves.
Below we will outline a functorial way to extend a locally constant coeffici*
*ent system
M on L to a locally constant coefficient system Mq on Lq. The extension Mq can
then be restricted to a locally constant coefficent system M0 on L0 via the inc*
*lusions
L0 Lq0 Lq. We will furthermore show that the extension Mq is unique up to un*
*ique
isomorphism extending the identity morphism of M, and thus it makes sense to th*
*ink
of M0 as the restriction of M to L0. Lemma 3.2 shows that this convention has *
*the
desired effect in cohomology.
Given a locally constant system of coefficients M on L, we construct an exte*
*nsion Mq
of M to Lq as follows. First recall that M extends uniquely to to a system of c*
*oefficients
Mc on the groupoid completion bL. Since the inclusion i: L ___! Lq induces a h*
*omotopy
equivalence on nerves |L| ' |Lq|, the induced inclusion of groupoid completions*
* bi:bL!
Lcqis an equivalence of categories. This implies that bLis a deformation retrac*
*t of cLq,
and we can choose an inverse r :cLq! bLsuch that r Oi = idLb. (One such r sends*
* a
' j(Q) j(P)-1
non-centric group P to S and a morphism P ___! Q to 'Q O' O('P ) , where '*
* is
a fixed compatible system of inclusions.) Now the composite
ss r cM
Mq :Lq ___! cLq___! bL___! R-mod ,
where ss is the groupoid completion map, is a locally constant system of coeffi*
*cients on
Lq that extends M.
Fixing a choice for the retraction r, the construction of Mq as the restrict*
*ion of cM
along r Oss clearly makes the assignment M 7! Mq functorial. However this exten*
*sion
functor is by no means unique or even canonical as a different choice of retrac*
*t would
give rise to a different extension functor. The next lemma shows that, while th*
*ere are
many such extension functors, the difference between them is inconsequential.
R. Levi and K. Ragnarsson *
* 19
Lemma 3.3. Let (S, F, L) be a p-local finite group, let Lq be the associated qu*
*asicentric
linking system, and let M be a locally constant system of coefficients on L. I*
*f M0 is
a locally constant system of coefficients on Lq extending M, then there is a un*
*ique
natural isomorphism j :Mq ) M0 that restricts to the identity transformation of*
* M.
Proof. The isomorphism j sends P to the composite
Mq('P) q 0 M0('P)-1 0
jP :Mq(P ) ____! M (S) = M(S) = M (S) ______! M (P ).
The verification that j is natural is routine and is left for the reader.
In light of this uniqueness result, we make the following definition.
Definition 3.4. Let (S, F, L) be a p-local finite group and let = p(F) or p*
*0(F).
Let H , let (SH , FH , LH ) be the corresponding p-local subgroup, and let '*
*q: LH ! L
be the inclusion. For a locally constant system of coefficients M on L, the res*
*triction
of M to LH is the composite
Mq
MH :LH LqH Lq ___! R-mod .
The restriction map ResH :H*(L, M) ! H*(LH , MH ) is the unique map that fits i*
*nto
the commutative diagram
* * q * q
H*(Lq, Mq) _____'____//H (LH , ' M )
|~=| |~=|
fflffl| ResH fflffl|
H*(L, M) __________//_H*(LH , MH ),
where the vertical maps are the isomorphisms from Lemma 3.2.
When = p0(F), there is a well-defined inclusion LH L, and we note that *
*MH
is just restriction along this inclusion, and ResH is the usual restriction map.
The following proposition holds for both centric and quasicentric linking sy*
*stems,
although we state it using the notation for centric linking systems.
Proposition 3.5. Let (S, F, L) be a p-local finite group and let = p(F) or *
*p0(F).
Let H , and let (SH , FH , LH ) be the corresponding p-local subgroup. For a*
*ny locally
constant system of coefficients M 2 L-mod there exists a map
T rH :H*(LH , MH ) ___! H*(L, M),
such that T rH ORes H is multiplication by | : H| on H*(L, M). Furthermore, the*
* map
thus defined coincides with the transfer map associated with the covering |LH |*
* ! |L|.
Proof. It suffices to prove this in the quasicentric case, as the centric case *
*is obtained
by restriction. Let ': LqH! Lq be the inclusion. By Theorem 1.5, the map |'|: |*
*LqH| !
|Lq| is a covering space up to homotopy, with homotopy fibre equivalent to =H.*
* Thus
Proposition 2.15 applies and one has a system of coefficients M0 2 L-mod , such*
* that
H*(Lq, M0) ~=H*(LqH, '*M),
and a natural transformation T :M0 ! M such that the composite
'* * q * * q 0 T* * q
H*(Lq, M) ___! H (LH , ' M) ~=H (L , M ) ___! H (L , M)
is multiplication by the index | : H|. Identifying the two middle terms in the *
*sequence
via the isomorphism between them, define T r to be the map T*.
The second statement follows at once from Corollary 2.16.
20 p-local finite group cohomology
3.3. Subsystems of Index Prime to p. We now restrict attention to subgroups of
p-local finite groups of index prime to p. It is in this context that we are a*
*ble to
obtain our most general results. However some of the discussion here has more g*
*eneral
implications as well.
Throughout this subsection, fix a p-local finite group (S, F, L) and let =*
* p0(F).
Fix a subgroup H and let ': LH ! L be the inclusion We assume throughout
that a compatible system of inclusions for L has been chosen, and by restrictio*
*n of a
morphism in L we always mean restriction with respect to the chosen inclusions *
*(see
Subsection 1.3). Let b :L ! B( ) denote the projection, as in Subsection 1.5. I*
*f ff is
a morphism in L, we denote by [ff] the class of ff in F.
Recall that a category is a discrete groupoid if each of its morphism sets e*
*ither
consists of a single isomorphism or is empty.
Lemma 3.6. For P 2 L, let GH (P ) P # ' and GH (P ) ' # P denote the full
subcategories, whose objects, respectively, are of the form (P ff, ff) and (P f*
*f, ff-1) for
some ff 2 Mor L(P, S). Then GH (P ) and GH (P ) are discrete groupoids. Further*
*more,
GH (P ) is a left deformation retract of P # ', and GH (P ) is a right deformat*
*ion retract
of ' # P .
Proof. Since LH and L have the same objects by Theorem 1.5, the category P # '
is nonempty for any P 2 L. Thus let ff, fi 2 Hom L (P, S) be any morphisms. T*
*hen
(P ff, ff) and (P fi, fi) are objects in GH (P ). A morphism between them is a *
*morphism
': P ff! P fiin LH , such that ' Off = fi. But ff: P ! P ffis an isomorphism in*
* L, and
hence invertible. Thus ' = fi Off-1, and so if a morphism between two given obj*
*ects
exists, then it is unique, and is obviously an isomorphism. This shows that GH*
* (P )
is a discrete groupoid. An analogous argument shows that GH (P ) is also a dis*
*crete
groupoid.
Let j :GH (P ) ____! P # ' denote the inclusion, and define r :P # ' ____*
*! GH (P )
as follows. On objects r(Q, ff) def=(P ff, ff), noting that P ff Q, while any *
*morphism
' 0 0 0 -1 *
* 0 -1
(Q, ff) ___! (Q , ff ) is taken by r to ff Off . For this to be well defined,*
* ff Off should
be a morphism in LH , and this is indeed the case since 'Off0= ff and b(') 2 H *
*implies
b(ff0O ff-1) 2 H. The composite r Oj is the identity functor on GH (P ), and t*
*here is
a natural transformation j :j Or ___! IdP#'that takes an object (Q, ff) in P #*
* ' to
the morphism (P ff, ff) ___! (Q, ff) induced by the inclusion. Clearly jj is t*
*he identity
transformation of j. This shows that GH (P ) is a left deformation retract of P*
* # ', as
claimed.
Next, let j denote the inclusion GH (P ) ___! ' # P . For an object (Q, ff*
*) in ' # P
we choose a morphism fl(Q,ff)2 Mor L(P, S) with b (fl(Q,ff)) = b(ff)-1 as follo*
*ws. If ff is
an isomorphism, then we just take fl(Q,ff)= ff-1. Otherwise, we choose, using T*
*heorem
1.5(v), a morphism ~fl(Q,ff)2 AutL (S) such that b (~fl(Q,ff)) = b (ff)-1, and *
*we let fl(Q,ff)
be the restriction of ~fl(Q,ff)to P . We now define the retraction r :' # P ___*
*! GH (P ) by
'
setting r(Q, ff) = (P fl(Q,ff), fl-1(Q,ff)), and sending a morphism (Q, ff) ___*
*! (R, fi) to the
unique morphism fl(R,fi)Ofl-1(Q,ff):(P fl(Q,ff), fl-1(Q,ff)) ___! (P fl(R,fi),*
* fl-1(R,fi)) in GH (P ). Observe
that this is well defined since
b (fl(R,fi)Ofl-1(Q,ff)) = b(fi)-1b (ff) = b(') 2 H,
and so fl(R,fi)Ofl-1(Q,ff)is in LH . With this definition we have r Oj = IdGH (*
*P). Furthermore,
if j :Id'#P ___! j Or is the natural transformation that sends an object (Q, f*
*f) to the
R. Levi and K. Ragnarsson *
* 21
morphism fl(Q,ff)Off, then jj is the identity transformation of j, and so GH (P*
* ) is a right
deformation retract of ' # P .
Recall that the homomorphism induced by the restriction of b :Lq ___! B( ) *
*to
Aut L(S) is surjective by Theorem 1.5. Thus the composite with the projection t*
*o the
right cosets
(4) Aut L(S) ___! ___! =H
is also surjective.
Lemma 3.7. Consider the set of left cosets =H as a category with objects =H a*
*nd
no nontrivial morphisms. Then for any P 2 L the functor
ffiP :GH (P ) ___! =H
taking an object (P ff, ff) to the coset b (ff)-1H is an equivalence of categor*
*ies.
Proof. Recall that an object in GH (P ) is of the form (P ff, ff) where ff: P !*
* P ff S
is an isomorphism in L. Surjectivity of ffiP on objects follows from the surjec*
*tivity of
b: AutL (S) ! .
Two objects (P ff, ff) and (P fi, fi) are in the same connected component of*
* GH (P ) if
and only if fi Off-1 :P ff__! P fiis a morphism in LH . This is the case if an*
*d only if
b(fi Off-1) 2 H, or equivalently if and only if ff-1 = fi-1H, or in other word*
*s if and
only if ffiP(P ff, ff) = ffiP(P fi, fi). This shows both that ffiP is well defi*
*ned and that it is
an equivalence of categories as claimed.
We next observe that calculating limits over a discrete groupoid is particul*
*arly easy.
Lemma 3.8. Let G be a groupoid, and let M :G ___! Ab be a functor. Then
Y Y
limM ~= lim M ~= M(xi)AutGi(xi),
G G Gi
i Gi
where the product runs over connected components Giof G, and xiis an arbitrary *
*object
in Gi. In particular if G is discrete, then
Y
limM ~= M(xi).
G G
i
Dually, M
colimM ~= M(xi)AutG(xi),
G G i
i
where the subscript means coinvariants. In particular if G is discrete, then
M
colim M ~= M(xi).
G G
i
Proof. The first isomorphism is clear. For the second, observe that each conne*
*cted
component of a groupoid is equivalent as a category to the group of automorphis*
*ms of
any object in it. Thus if xi is an object in Gi, then
limM ~= lim M = M(xi)AutGi(xi),
Gi B(AutGi(xi))
as claimed. The claim for colimits follows at once by duality. The conclusion*
*s for
discrete groupoids are clear.
22 p-local finite group cohomology
We are now ready to describe the right Kan Extension of a module along_the i*
*nclusion
': LH ___! L. To make notation less cumbersome, we will use the symbol g to de*
*note
a left coset gH 2 =H. Also for any morphism ' in L, let t' def=b(') 2 .
Lemma 3.9. Let M 2 LH -mod be any functor, and _fix a section
oe : =H ____! Aut L(S) of the projection. Let oeg denote oe(g). Then the rig*
*ht Kan
extension of M to L can be described as follows.
(a)For each P 2 LH , there is an isomorphism
~= Y oe-1
oeP:R'(M)(P ) ___! M(P g),
_g2 =H
given by the formula
( oeP({x(Q,fi)}(Q,fi)2P#')_g= x(Poe-1g,oe-1.
g )
In other words, oePtakes a compatible family
Y
{x(Q,fi)}(Q,fi)2P#'2 R'(M)(P ) def=limM] M(Q)
P#'
(Q,fi)2P#'
Q -1 _
to the element in _g2 =HM(P oeg) whose g-th coordinate is x(Poe-1g,oe-*
*1.
g*
* )
(b) If o : =H ___! Aut L(S) is another section, then one has a commutative *
*dia-
gram
Y -1
M(P oeg)
_g2 =H
ll66
~=llll |
lllloe |
llll P |
|QM(o-1
R'(M)(P ) ~=| g Ooeg)
RRR |
RRRRRo |
RPRRR |
~= R((RY fflffl|
-1
M(P og).
_g2 =H
(c)Let ': P ___! Q be a morphism in L. Then, under the identification in p*
*art
(a), the map
R'(M)('): R'M(P ) ___! R'M(Q)
sends Y -1
(xg)_g2 =H2 M(P oeg)
_g2 =H
to n o Y -1
M(oe-1gO' Ooeb(')-1g)(xb(')-1g)_ 2 M(Qoeg).
g2 =H _g2 =H
Proof. For each object P 2 L, the section oe defines an equivalence of categor*
*ies
-1 -1
toeP: =H ! GH (P ) which takes an object gH to (P oeg, oeg ), and is a right in*
*verse
for ffiP :GH (P ) ! =H of Lemma 3.7. Let jP :GH (P ) ! P # ' and rP :P # ' ! G*
*H (P )
denote the inclusion and retraction of Lemma 3.6 respectively.
The map oePis the map induced by the composite jP OtoeP: =H ! P # ',
j*P (toeP)* oe Y oe-1
R'(M)(P ) def=limM] ___! lim M]O jP ___! limM]O jP OtP = M(P g).
P#' GH(P) =H _
g2 =H
R. Levi and K. Ragnarsson *
* 23
-1
Notice that jp OtoePtakes a coset gH 2 =H to the object (P oeg, oe-1g) 2 P # '*
*. Thus
the formula of part (a) is clear. Furthermore, the first map in the sequence ab*
*ove is
an isomorphism since GH (P ) is a left deformation retract of P # ' by Lemma 3.*
*6, and
the second is an isomorphism since toePis an equivalence of groupoids.
For Part (b), notice that the morphisms o-1gOoeg define a natural isomorphis*
*m from
toePto toP, and hence a natural isomorphism jP OtoeP! jP OtoP. Thus oePand o*
*Pare
naturally isomorphic, and (b) is nothing but a diagramatic realization of this *
*natural
isomorphism.
To prove part (c), notice first that for each P 2 L the composite functor ff*
*iP OrP is
a left inverse to jP OtoeP, and since the latter induces the isomorphism oePof*
* part (a),
ffiP OrP induces ( oeP)-1. Thus to prove the statement weQneed to calculate the*
* effect of
-1
oeQOR'(M)(') O( oeP)-1 on a typical element {xg}_g2 =H2 _g2 =HM(P oeg). By d*
*irect
calculation one has for (R, fi) 2 P # ', (T, fl) 2 Q # ', and u 2 ,
o (( oeP)-1({xg}_g2 =H))(R,fi)= M(fioeb(fi)-1)(xb(fi)-1)
o (R'(M)(')({M(fioeb(fi)-1)(xb(fi)-1)}(R,fi)))(T,fl)= M(fl'oeb(fl')-1)(xb)*
*(fl')-1)
o ( oeQ({M(fl'oeb(fl')-1)(xb(fl')-1)}(T,fl)))_u= M(oe-1u'oeb(oe-1u')-1)(xb*
*(oe-1u')-1),
oe-1u oe-1 -1 -1 *
* -1
where oe-1urepresents the restricted map Q ___! Q u. Since b (oeu ') = b ('*
*) u,
we have
( oeQOR'(M)(') O( oeP)-1({xg}_g2 =H))_u= M(oe-1uO' Ooeb(')-1u)(xb(')-1u),
as claimed.
The following is an immediate corollary of Lemmas 3.9 and 2.5.
Corollary 3.10. The right Kan extension functor
R':LH -mod ! L-mod
is exact. Thus, the Shapiro map
ShM :H*(L, R'(M)) ! H*(LH , M)
is an isomorphism for every M 2 LH -mod .
There is a dual version of both Lemma 3.9, describing the left Kan extension*
* L'M,
and Corollary 3.10, showing that L' is exact and hence '* preserves injectives *
*and the
Shapiro lemma holds in homology. We will not state these results explicitly, b*
*ut we
draw the following corollary.
Corollary 3.11. The restriction functor '*: L-mod ____! LH -mod preserves bo*
*th
injective and projective resolutions.
4. Construction of the transfer for subsystems of index prime to p
Throughout this section let = p0(F), fix a subgroup H , and let ': LH *
*___! L
be the inclusion._Let oe : =H ___! Aut L(S) be a fixed section._As before, for*
* g 2
we denote by g the left coset gH 2 =H, and let oeg denote oe(g).
Proposition 4.1. Let M 2 L-mod . For each P 2 L , let
Pre-TrP:R'('*M)(P ) -! M(P )
24 p-local finite group cohomology
be the map that, under the identification of Lemma 3.9, is given by
X
(xg)_g2 =H7! M(oeg)(xg).
_g2 =H
Then, the maps Pre-TrP are independent of the choice of the section oe, and ass*
*emble
into a natural transformation
Pre-Tr: R'('*M) ) M
Proof. Let o be another section. Then by Part (b) of Lemma 3.9 (omitting '* fro*
*m the
notation),
iY j
oP= M(o-1_gOoe_g)O oeP.
Let x 2 R'('*M)(P ), and let (xg)_gand (x0g)_gdenote oeP(x) and oP(x) respect*
*ively.
Then
(x0g)_g= (M(o-1gOoeg)(xg))_g.
Thus using the section o to define Pre-Tr gives the map
X X X
(x0g)_g2 =H7! M(og)(x0g) = M(og)(M(o-1gOoeg)(xg)) = M(oeg)(xg),
_g2 =H _g2 =H _g2 =H
which agrees with the map defined using oe.
It remains to show that Pre-Tr is natural, meaning that for P, Q 2 L and ' 2
Mor L(P, Q), the square
Pre-TrP
R'('*M)(P ) __________//M(P )
| |
| |
| |
R'('*M)(')|| |M(')|
| |
| |
fflffl|Pre-TrQ fflffl|
R'('*M)(Q) __________//M(Q)
Y -1
commutes. Let x 2 R'('*M)(P ), and let (xg)_g= oeP(x) 2 M(P oeg), as in L*
*emma
_g2 =H
3.9. Then X
M(') OPre-TrP(x) = M(' Ooeg)(xg),
_g2 =H
while, by part (c) of Lemma 3.9,
X i j
Pre-TrQ OR'('*M)(')(x) = M(oeg) M(oe-1gO' Ooeb(')-1g)(xb(')-1g)
_g2 =H
X
= M(' Ooeb(')-1g)(xb(')-1g).
_g2 =H
Thus the two ways of composition in the square coincide.
The natural transformation Pre-Tr induces a map on higher limits, which we a*
*lso
denote by Pre-Tr. Composing this with the inverse of the Shapiro isomorphism we
obtain our transfer.
R. Levi and K. Ragnarsson *
* 25
Definition 4.2. For any subgroup H and a functor M 2 L-mod the associated
transfer, is the map
Sh-1M * * Pre-Tr *
Tr :H*(LH , '*M) ___~=!H (L, R'(' M)) ---! H (L, M).
Since the transfer is defined as a natural transformation of coefficient sys*
*tems it is
of course independent of the choice of resolutions used to compute cohomology.
4.1. Cochain-level Description. Just as in the classical group case, the transf*
*er map
constructed here can be described at the cochain level as a sum of conjugation *
*maps.
To this end we must first make sense of conjugation maps in our current context.
Definition 4.3. Let M and N be coefficient systems on L, and let K be a sub-
group of . Let 'K :LK ___! L denote the inclusion. For a natural transforma*
*tion
OE 2 Hom LK-mod('*KN, '*KM), and ff 2 AutL(S), let OEffbe the map which associa*
*tes with
an object P 2 L the homomorphism of R-modules OEffP:N(P ) ___! M(P ) given by
OEffP:= M(ff) OOEff-1(P)ON(ff-1) 2 Hom R(N(P ), M(P )).
We refer to this construction as conjugation. Notice that OEffneeds not be a*
* natural
transformation in general. From the definition we immediately get OE(ffOfi)= (O*
*Efi)ff, the
order being reversed since we write ff on the right. The following lemma invest*
*igates
other basic properties of OEff, and in particular provides a condition under wh*
*ich it is a
natural transformation.
Lemma 4.4. Let L, M and N be coefficient systems on L, let K, H , and let
ff 2 AutL(S).
(a)If
OE * j *
'*KN ___! 'K M ___! 'K L
is a composable sequence in LK -mod then
(j OOE)ff= jffOOEff.
(b) If OE 2 Hom LK-mod('*KN, '*KM), then for every ff 2 AutLK (S),
OEff= OE.
(c)If b (ff)-1 2 N (H, K)and OE 2 Hom LK-mod('*KN, '*KM), then OEffis a nat*
*ural
transformation '*HN ! '*HM , and the map which associates OEffwith OE is*
* a ho-
momorphism
c*ff:HomLK-mod ('*KN, '*KM) -! Hom LH-mod('*HN, '*HM).
Proof. For P 2 L and ff 2 AutL(S),
(j OOE)ffPdef=L(ff) O(j OOE)ff-1(P)ON(ff-1) =
L(ff) Ojff-1(P)OM(ff-1) OM(ff) OOEff-1(P)ON(ff-1) = jffPO*
*OEffP.
This proves part (a).
Part (b) is immediate, since by assumption OE is natural with respect to any*
* morphism
in LK , and in particular ff. Therefore, for any object Q,
OEffQ= M(ff) OOEff-1(Q)ON(ff-1) = M(ff) OM(ff-1) OOEQ = OEQ .
26 p-local finite group cohomology
It remains to prove part (c). Let ': P ! Q be a morphism in LH . Then b(') *
*2 H ,
so by assumption on ff we have
b(ff-1 O' Off) = b(ff)-1 Ob (') Ob (ff) 2 K,
so
ff-1 O' Off 2 Mor LK(ff-1(P ), ff-1(Q)).
Thus,
OEffQON(')= M(ff) OOEff-1(Q)ON(ff-1) ON(')
= M(ff) OOEff-1(Q)ON(ff-1 O' Off) ON(ff-1)
= M(ff) OM(ff-1 O' Off) OOEff-1(P)ON(ff-1)
= M(') OM(ff) OOEff-1(P)ON(ff-1)
= M(') OOEffP.
This shows that OEffis a natural transformation. That the map c*ffis a homomorp*
*hism
follows directly from the definition.
Remark 4.5. A consequence of part (b) of Lemma 4.4 is that if ff, ff0 2 Aut L(S)
with b (ff) = b(ff0)then c*ff= c*ff0, since ff-1 Off0 2 Aut LH(S) for any subgr*
*oup H
. Therefore we will often write c*xfor x 2 to denote the conjugation induc*
*ed
by any ff 2 AutL(S) with b(ff) = x, and write OEx for c*x(OE). When restrict*
*ed to
Hom LH-mod('*HN, '*HM), the conjugation_c*ffonly depends on the coset b (ff)H.*
* There-
fore_we will also write c*xand OEx for x 2 =H . Finally, we also write c*xand*
* OEx for
x 2 K\ =H when the maps involved are independent of the choice of representat*
*ive for
double cosets.
Corollary 4.6. Let M be a coefficient system on L with injective resolution M !*
* Io
and let K, H . For ff 2 AutL (S) such that b (ff)-1 2 N (H, K), the map indu*
*ced
by conjugation
c*ff:HomLK-mod ('*KR, '*KIo) -! Hom LH-mod('*HR, '*HIo)
is a map of cochain complexes, and the induced map on cohomology
c*ff:H*(LK , '*KM) -! H*(LH , '*HM).
is independent of the choice of the injective resolution Io.
Proof. First recall that '*H and '*K preserve injective resolutions by Corollar*
*y 3.11.
Now, denote the differential of Io by @. Then, by Lemma 4.4(b), @ff= @ for any
ff 2 Aut L(S). The differentials of the cochain complexes Hom LK-mod ('*KR, '*K*
*Io) and
Hom LH-mod('*HR, '*HIo) are both given by composition with @, and will be deno*
*ted ffiK
and ffiH respectively. For OE 2 Hom LK-mod('*KR, '*KIo)we therefore have
cffOffiK (OE) = cff(@ OOE) = @ffOOEff= @ OOEff= ffiH Ocff(OE).
This shows that cffOffiK = ffiH,Ocffso cffis a cochain map.
The proof of independence on choice of injective resolutions is routine.
Lemma 4.7. Let M and N be coefficent systems on L, and let H .*
* If
OE 2 Hom LH-mod('*HN, '*HM)then
X
OEg 2 Hom L-mod(N, M).
_g2 =H
R. Levi and K. Ragnarsson *
* 27
Proof. We need to show naturality with respect to morphisms in L. So let _ :P_!*
* Q
be a morphism in L._ Fix a section oe : =H ! Aut L(S), and let oeg denote oe(g)*
*, as
before. For each g 2 =H, let _g denote the composite
oeb(_)-1g _ oe-1g
oe-1b(_)-1g(P_)_! P ___! Q ___! oe-1g(Q),
where the appropriate restrictions on oeg and oe-1b(_)-1gare understood. Obser*
*ve that
b(_g) 2 H, so _g 2 Mor LH(oe-1b (P ), oe-1(Q)).
(_)-1g g
Now,
X X
OEgQON(_) = M(oeg) OOEoe-1g(Q)ON(oe-1g) ON(_)
g_2 =H _g2 =H
X
= M(oeg) OOEoe-1g(Q)ON(_g) ON(oe-1b -1)
_g2 =H (_) g
X
= M(oeg) OM(_g) OOEoe-1 (P)ON(oe-1b -1)
_g2 =H b(_)-1g (_) g
X
= M(_) OM(oeb(_)-1g) OOEoe-1 (P)ON(oe-1b -1)
_g2 =H b(_)-1g (_) g
X b(_)-1g
= M(_) OEP .
_g2 =H
X
= M(_) OEgP.
_g2 =H
The first and the fifth equalities follows from the definition of OEg (see Rema*
*rk 4.5), the
second and the fourth from the definition _g, the third from naturality of OE, *
*and the
sixth is clear. This shows naturality and proves the claim.
Definition 4.8. For a coefficient system M on L, a subgroup H and an inject*
*ive
resolution M ! Io, the associated cochain-level transfer is the cochain map
TrH : Hom LH('*HR, '*HIo) -! Hom L(R, Io)
given by X
TrH(OE) = OEg.
_g2 =H
Since T rH is a sum of conjugations, it follows by Corollary 4.6 that it is *
*indeed a
cochain map, and that the induced map on cohomology is independent of the choice
of injective resolution.
Proposition 4.9. For a coefficient system M on L and a subgroup H of , the map
H*(LH , '*M) -! H*(L, M)
induced by the cochain-level transfer T rH of Definition 4.8 coincides with the*
* transfer
associated to the same data, as defined in 4.2.
Proof. Let ': LH ___! L denote the inclusion. If M ___! Io is an injective re*
*solution,
then, by Corollary 3.11, '*M ___! '*Io is an injective resolution. Thus the S*
*hapiro
isomorphism ShM is given on the cochain-level by the ('*, R')-adjunction isomo*
*rphism
~= * *
ae: Hom L-mod(R, R'('*Io)) ___! Hom LH-mod (' R, ' Io).
28 p-local finite group cohomology
(cf. Subsection 2.3). Hence its inverse Sh-1Mis induced by ae-1, whose value*
* on
OE 2 Hom LH-mod('*R, '*Io)is the natural transformation which takes an object Q*
* 2 L
to the morphism ae-1(OE)Q given by the composite
Q -1 Q
_g2 =HR(oeg )Y -1 g_2 =HOEoe-1g(Q)Y -1
R = R(Q) --------! R(Qoeg) --------! Io(Qoeg) ~=R'('*Io)(Q).
_g2 =H _g2 =H
Therefore the composite
ae-1M * Pre-Tr
Hom LH-mod ('*R, '*Io) ___~=!Hom L-mod(R, R'(' Io)) ---! Hom L(R, Io),
which induces the transfer, is given by
X X
OE 7! Io(oeg) OOEoe-1g(Q)OR(oe-1g) = OEoeg= TrH(OE).
_g2 =H _g2 =H
4.2. Geometric Interpretation. We end this section by verifying that for a loca*
*lly
constant system of coefficients M 2 L-mod , and a subgroup LH L of index prime
to p, the transfer defined in this section coincides with the map defined in Se*
*ction 3.2.
Let ': LH ! L denote the inclusion. Let = ss1(|L|, S) and 0 = ss1(|LH |*
*, S).
Then, by Theorem 1.5(v), the map induced by ' on nerves is a covering space, up*
* to
homotopy, with homotopy fibre = 0~= p0(F)=H. Construct ss :L 0! L as in Secti*
*on
2.6. Then, by Lemma 2.13(i), there is a functor b':LH ! L 0, such that ss Ob'= *
*', and
by Part (ii) of the same lemma, for any locally constant M 2 L-mod , there is a*
* natural
isomorphism induced by b',
~= *
b'*:Rss(ss*M) ___! R'(' M).
Choose a section oe : =H ! Aut L(S) of the projection, and construct oeas in
Lemma 3.9. Then for any M 2 L-mod and P 2 L, we claim that the diagram
Pr Y
Rss(ss*M)(P )_______! M(P ) _________! M(Pw)
|| = 0 ww
| ww
| | w
| Q| ww
b'*| | M(oeg)-1 w
| | ww
| | w
| | w
# # w
oeP Y oe-1 oe
R'('*M)(P ) ______! M(P g) _______! M(P )
_g2 = 0
commutes. Here the map P r projects a compatible family of elements in the sou*
*rce
onto the coordinates of the respective initial objects, i.e., to the coordinate*
*s of the
objects of the form ((P, g 0), 1P). The map is the sum of coordinates, where *
* oeis
the twisting of the sum of coordinates map by the M(oeg), as in the constructio*
*n of
Pre-TrP in Proposition 4.1. The right square clearly commutes, while the verifi*
*cation
that the left square commutes only involves checking all the relevant definitio*
*ns.
R. Levi and K. Ragnarsson *
* 29
Since the map TP of Proposition 2.15 is the top composition, and the map Pre*
*-TrP is
the bottom composition, it follows that one has a commutative diagram in cohomo*
*logy
~= * * T* *
H*(L 0, ss*M)______! H (L, Rss(ss M))_____!H (L,wM)
| | ww
b'*~=| b'*~=| w
# ~ # w
= * * Pre-Tr**
H*(LH , '*M)______! H (L, R'(' M)) _____!H (L, M).
Here the top composition is the transfer associated to the covering |LH | ! |L|*
* by
Proposition 3.5, while the bottom composition is our algebraic transfer.
5. Standard Consequences
In this final section we show that the transfer constructed here for subsyst*
*ems of
index prime to p satisfies all the standard properties one expects to have in o*
*rdinary
group cohomology. Throughout this section = p0(F), and we let b :L ___! B( )
denote the projection functor, as before.
5.1. Transfer among Subgroups. We have defined a transfer associated to the in-
clusion LH L for a subgroup H . Just like in the group case, this construc*
*tion
can be generalized to obtain a transfer associated to the inclusion LH LK for*
* sub-
groups H K . Indeed, one can simply take LK in place of L in the construct*
*ion
of the transfer. The transfer associated to the inclusion 'KH:LH LK can th*
*en be
described on the cochain level as follows. Let M be a coefficient system on LK *
*, and
let M ! Io be an injective resolution of M. A cochain-level transfer
TrKH: Hom LH('*HR, ('KH)*Io) -! Hom LK ('*KR, Io)
is given by X
TrKH(OE) = OEx,
_x2K=H
as in Definition 4.8. This formula makes it clear that TrH = TrKHOTrK for subg*
*roups
H K .
5.2. Normalization. One of the most basic properties of the standard cohomology
transfer is that the restriction to the cohomology of a subgroup followed by th*
*e transfer
is given by multiplication by the index. This is exactly the case in our more g*
*eneral
context.
Proposition 5.1. Let (S, F, L) be a p-local finite group, and let H be a su*
*bgroup.
Let M be a system of coefficients for (S, F, L). Then the composite
Res * * Tr *
H*(L, M) ___! H (LH , ' M) ___! H (L, M)
is given by multiplication by the index | : H|.
Proof. Consider the diagram
TrH *
H*(L, M) ________Res______//H*(LH , '*M)________________//H (L, M)
PPPP || nnn66n
PPPP | nnnn
PPP(ffiMP)*P | -1 Pre-Trnnnn
PPP ~=ShM| nnnn
PPPP | nnnn
PPPP | nnnn
PPP(( fflffl|nnn
H*(L, R'('*M)),
30 p-local finite group cohomology
where the right triangle commutes by definition of the transfer, and where the *
*mor-
phism ffiM :M ! R'('*M) is the unit of the ('*, R')-adjunction evaluated at the*
* object
M. Notice that under the identification of Lemma 3.9(a), ffiM (x) = {M(oeg)-1(x*
*)}_g2 =H.
The left triangle is commutative by construction of the Shapiro isomorphism *
*(see
Subsection 2.3). To see this, notice that Sh-1Mis induced by the adjunction iso*
*morphism
ae: Hom LH-mod (RLH , '*Io) = Hom LH-mod('*RL, '*Io) ___! Hom L-mod(RL, R'('*
**Io)),
where Io is an injective resolution of M in L-mod . The adjunction, in turn, ap*
*plied
to a natural transformation _ :'*RL ! '*Io is the composite
ffiRL * R'_ *
RL ___! R'(' RL) ___! R'(' Io).
By naturality of ffi, for any natural transformation ': RL ! Io,
ffiIoO' = R'('*') OffiRL.
This proves commutativity of the left triangle, and therefore it suffices to sh*
*ow that
the composite Pre-TrO(ffiM )* is given by multiplication by | : H|. This map i*
*s induced
by a natural transformation of coefficients
ffiM * Pre-Tr
M -! R'(' M) ---! M,
which is given element-wise by
X
x 7! {M(oeg)-1(x)}_g2 =H7! M(oeg)(M(oe-1g)(x)) = | : H| . x
_g2 =H
for P 2 L and x 2 M(P ).
5.3. Double Coset Formula. We now show that the transfer map constructed here
satisfies a double coset formula which is essentially identical to that which h*
*olds for
ordinary group cohomology.
Proposition 5.2. Let (S, F, L) be a p-local finite group, and let H, K be s*
*ubgroups.
Then the composite
TrK * ResH * *
H*(LK , '*KM) ___! H (L, M) ___! H (LH , 'H M)
is given by X
ResH OTrK = TrHH\xKx-1Oc*xOResKx-1Hx\K
x2H\ =K
Proof. Using the cochain-level description of the transfer in Section 4.1, one *
*can easily
adapt most textbook proofs of the double coset formula for group cohomology to *
*prove
this proposition. The argument presented here follows the lines of [AM , Thm 6*
*.2], to
which the reader is referred for full detail.
`
Let = iHxiK be a double coset decomposition of with respect to K and *
*H.
For each i, put
Videf=x-1iHxi\ K and Widef=H \ xiKx-1i,
`
and let H = jzjiWi be a left coset decomposition of H with respect to Wi. The*
*n one
can rewrite the right K-coset decomposition of as
a
= zjixiK.
i,j
R. Levi and K. Ragnarsson *
* 31
Let M ! Io be an injective resolution of M in L-mod . Since x-1i2 N (Wi, Vi) *
*we
have induced chain maps
c*xi:Hom LVi-mod('*ViRL, '*ViIo) -! Hom LWi-mod('*WiRL, '*WiIo),
as in Corollary 4.6. Let OE 2 Hom LK-mod('*KRL, '*KIo). Then
0 1
X
ResH OTrK (OE)= ResH @ OEgA
_g2 =K
_ _ !!
X X
= ResH OE(zjixi) ,
i j
and
_ !
X X X zij
TrHWiOc*xiOResKVi(OE)= c*xi(Res KVi(OE))
i i j
_ !
X X (zjixi)
= Res KVi(OE) .
i j
As we can ignore restrictions when determining the effect of these natural t*
*rans-
formations, this calculation shows that the double coset formula holds already *
*at the
chain-level. Since the maps involved are maps of chain complexes, the result f*
*ol-
lows.
5.4. A Stable Elements Theorem. In standard group cohomology, the availability
of a transfer and a double coset formula allows one to prove that the cohomolog*
*y of
a finite group with a p-local module of coefficients is given by the so called *
*"stable
elements" in the cohomology of a Sylow p-subgroup with the restricted module of
coefficients. Using a "transfer-like" map, a similar theorem was proved in p-l*
*ocal
finite group cohomology with p-local constant coefficients in [BLO2 , Section *
*5]), and
generalized to any stably representable non-equivariant cohomology theory in [R*
*a ] (see
also [CM ]). We now show that the existence of a transfer and a double coset f*
*ormula
in our context implies a stable elements theorem in p-local finite group theory*
* for
arbitrary p-local modules of coefficients, and subgroups of index prime to p.
Fix a p-local finite group (S, F, L), and let = p0(F).
Definition 5.3. Let H K , and let M 2 L-mod be a system of coefficients. *
*An
element x 2 H*(LH , '*H(M))is K-stable if for every subgroup U H and every g*
* 2 K
with g-1 2 NK (U, H),
c*gOResHgUg-1(x) = ResHU(x) 2 H*(LU , '*U(M)).
Lemma 5.4. Let H K , and let M be a system of coefficients on L. The K-
stable elements in H*(LH , '*H(M)) form a submodule. Furthermore, if A is a sy*
*stem
of ring coefficients, then the submodule of K-stable elements in H*(LH , '*H(A)*
*) is a
subring.
Proof. This follows at once from the definition of stable elements, and the fac*
*t that the
maps induced by restriction and conjugation are module maps or, in the case of *
*ring
coefficients, ring maps.
32 p-local finite group cohomology
Lemma 5.5. Let H K , and let M 2 L-mod be a system of coefficients. If
x 2 H*(LK , '*K(M))then ResKH(x) is K-stable.
Proof. This follows from part (b) of Lemma 4.4.
We are now ready to state the Stable Elements Theorem for p-local finite gro*
*ups.
Theorem 5.6. Let = p0(F), let H K , and let M 2 L-mod be a p-local
system of coefficients. Then the restriction homomorphism
ResKH:H*(LK , '*K(M)) -! H*(LH , '*H(M))
is a split injection whose image is the submodule of K-stable elements.
Proof. By Proposition 5.1, the composite TrKHOResKH acts on H*(LK , '*K(*
*M)) as
multiplication by the index |K :H|. Since this index is prime to p,*
* the map
t def=|K :H|-1 . TrKHis a left inverse for ResKH, proving split injectivity. B*
*y Lemma
5.5 the image of Res KHlies in the submodule of stable elements. Conversely, *
* if
a 2 H*(LH , '*H(M))is K-stable, then the double coset formula gives us
X
ResKHOTrKH(a)= Tr HH\xHx-1Oc*xOResHx-1Hx\H(a)
x2H\K=H
X
= Tr HH\xHx-1OResHH\xHx-1(a)
x2H\K=H
X
= |H :H \ xHx-1| a
x2H\K=H
= |K :H| a,
where the equality
X
|H :H \ x-1Hx| = |K :H|
x2H\K=H
is obtained as in the first step of the proof of the double coset formula. It f*
*ollows that
K -1
x = ResKH TrH(|K :H| x)
is in the image of ResKH.
5.5. Frobenius Reciprocity. Here we show that the transfer and restriction maps
for p-local finite groups satisfy the standard Frobenius reciprocity formula.
Proposition 5.7. Let (S, F, L) be a p-local finite group, and let be p(F) or*
* p0(F).
Let H K , and let A be a system of ring coefficients on L. For x 2 H*(LK ,*
* '*KA)
and y 2 H*(LH , '*HA) we have
TrKH(Res KH(x) y) = x TrKH(y).
Proof. We prove this at the cochain level. Let Po ! R be a projective resoluti*
*on of
the constant functor on L, and let OE 2 Hom LK-mod ('*KPo, '*KA), (representing*
* x) and
R. Levi and K. Ragnarsson *
* 33
_ 2 Hom LH-mod('*HPo, '*HA)(representing y). Then
X X
TrKH(Res KH(OE)=_) (Res KH(OE) _)g = Res KH(OE)g_g
_g2K=H _g2K=H
X X
= Res KH(OE) _g = OE _g
_g2K=H _g2K=H
= OE TrKH(_),
where we have used the equality ResKH(OE)g = ResKH(OEg) = ResKH(OE) for g 2 K o*
*btained
in Lemma 4.4 (b). The result follows by passing to cohomology.
References
[AM] A. Adem, R.J. Milgram Cohomology of Finite Groups, Grundlehren der Mathe*
*matischen
Wissenschaften, 309. Springer-Verlag, Berlin, 1994.
[5A1] C. Broto, N. Castellana, J. Grodal, R. Levi, B. Oliver, Subgroup familie*
*s controlling p-local
finite groups, Proc. London Math. Soc. (3) 91 (2005), no. 2, 325-354.
[5A2] C. Broto, N. Castellana, J. Grodal, R. Levi, B. Oliver, Extensions of p-*
*local finite groups,
Trans. Amer. Math. Soc. 359 (2007), no. 8, 3791-3858.
[BLO1] C. Broto, R. Levi, B. Oliver, Homotopy equivalences of p-completed class*
*ifying spaces of finite
groups, Invent. Math. 151 (2003), no. 3, 611-664.
[BLO2] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. *
*Amer. Math. Soc. 16
(2003), no. 4, 779-856.
[CE] H. Cartan, S. Eilenberg, Homological algebra, Princeton University Press*
*, Princeton, N. J.,
1956.
[CM] N. Castellana, L. Morales, Vector bundles over classifying spaces of p-l*
*ocal finite groups, to
appear.
[MacL] S. MacLane, Categories for the working mathematicians, Graduate texts in*
* Mathematics, Vol.
5, Springer-Verlag, New York-Berlin, 1971.
[Pu] L. Puig Frobenius Categories., J Algebra 303(2006), 309-357
[Qu] D. Quillen, Higher algebraic K-theory. I.; Algebraic K-theory, I: Higher*
* K-theories (Proc.
Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147. Lectu*
*re Notes in Math.,
Vol. 341, Springer, Berlin 1973.
[Ra] K. Ragnarsson, Classifying spectra for saturated fusion systems, Algebr.*
* Geom. Topol. 6
(2006), 195-252.
Institute of Mathematics, University of Aberdeen, Fraser Noble Building 138,*
* Ab-
erdeen AB24 3UE, U.K.
E-mail address: r.levi@abdn.ac.uk
Department of Mathematical Sciences, Depaul University, 2320 N. Kenmore Aven*
*ue,
Chicago, IL 60614, USA
E-mail address: kragnars@math.depaul.edu