p-LOCAL FINITE GROUP COHOMOLOGY

RAN LEVI AND KÁRI RAGNARSSON

Abstract. We study cohomology for p-local finite groups with non-constant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences.

1. Introduction

A p-local finite group is an algebraic object designed to encapsulate the information modeled on the p-completed classifying space of a finite group. More specifically, it is a triple (S, F, L), where S is a finite p-group, and F is a certain category whose 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 subgroups 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 described 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 groups was introduced in [BLO2] and has been studied quite extensively by various authors. In particular the mod p cohomology of the classifying 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 constant mod p coefficients [BLO2, Thm B]. However, as we shall see below, the analogy does not carry forward when one considers a more general setup, i.e., the cohomology of p-local finite groups in the context of functor cohomology. In particular we will present counter examples to the statement that the cohomology of the p-local group, in this sense, is given as the stable elements in the cohomology of its Sylow subgroup with restricted coefficient.

The purpose of this paper is to study the cohomology of p-local finite groups 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 geometric transfer must exist, that is, when there is a finite index covering space associated 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 studied. 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 p′-group – and that, up to equivalence, p-local subgroups of p-power index and p′ index are in one to one correspondence with subgroups of these two groups. Subgroups of indices which are neither p-power nor p′ can also be studied under restricted conditions, which essentially follow from the two basic cases.

2010 Mathematics Subject Classification. 20J06, 20D20, 55R40.

Key words and phrases. fusion system, group cohomology, p-local finite group.

This project was supported by EPSRC grant GR/S94667/01.

In the current article we show the existence of a ”transfer map” to the cohomology of 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 transfer maps carry similar properties to the standard group cohomology transfer, and can be used to study cohomology of p-local finite groups with non-constant coefficients.

We now explain our setup and state our results. To a finite group G, one can associate a category B(G) with a single object whose endomorphism monoid is G. A G-module M can be regarded as a functor on B(G), and the cohomology of G with coefficients in M can be computed as the higher limits of that functor. By analogy, a coefficient system on a p-local finite group (S, F, L) is a functor M: L → Ab, where Ab is the category of abelian groups (or sometimes ℤ_{(p)}-modules, in which case we say that the system is p-local), and the cohomology groups H^{∗}(L, M) are defined as the higher limits of M over L. Unless otherwise specified, we will work with coefficient systems which are covariant functors. A coefficient system M is locally constant if it sends every morphism in L to an isomorphism of modules.

As already indicated, we only define a transfer for arbitrary systems of coefficients under certain, rather restrictive, conditions. We start our discussion with a proposition, due to Bob Oliver, which shows that in general one cannot hope to do much better.

Proposition 1.1. There exist a p-local finite group (S, F, L) and a locally constant plocal system of coefficients M on (S, F, L) such that H̃^{∗}(L, M) ≠ 0 but H̃^{∗}(S, ι^{∗}M) = 0, where ι^{∗}M means the restriction of M to S.

Thus the proposition shows that unlike the situation in ordinary group cohomology, the cohomology of a p-local finite group with p-local coefficients is not always a retract of the cohomology of its Sylow subgroup with the restricted coefficients.

If (S, F, L) is a p-local finite group, and (S_{0}, F_{0}, L_{0}) is a subgroup of p-power index or index prime to p, then one has a an associated covering space up to homotopy ι: |L_{0}| → |L|, and thus if M is a ℤ_{(p)}[π_{1}(|L|)]-module, then there is a transfer map H^{∗}(|L_{0}|, ι^{∗}M) → H^{∗}(|L|, M) associated with the covering. We present a categorical analog of this setup, and use it to prove our first theorem.

Theorem 1.2. Let (S, F, L) be a p-local finite group, and let (S_{0}, F_{0}, L_{0}) be a subgroup 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^{∗}(L_{0}, M_{0}) → H^{∗}(L, M), where M_{0} denotes the restriction of M to L_{0}, such that the composite

H^{∗}(L, M) ⟶^{Res} H^{∗}(L_{0}, M_{0}) ⟶^{Tr} H^{∗}(L, M)

is multiplication by the index. Furthermore, the map Tr coincides with the transfer map associated with the covering |L_{0}| → |L|.

Theorem 1.2 is proven below as Theorem 4.5. Next, we specialize to p-local finite subgroups of index prime to p. It is in this setting that we are able to obtain our most general result.

Theorem 1.3. Let (S, F, L) be a p-local finite group, and let (S_{0}, F_{0}, L_{0}) be a subgroup of index prime to p. Let M be an arbitrary system of coefficients on L. Then there exists a map

Tr: H^{∗}(L_{0}, M_{0}) → H^{∗}(L, M),

where M_{0} denotes the restriction of M to L_{0}, such that the composite Tr ∘ Res is given by multiplication by the index. Furthermore, the map Tr satisfies a double coset formula and Frobenius reciprocity.

The construction of the transfer map is carried out in Section 4. The precise statements and proof of its properties appear below as Propositions 6.1, 6.2 and 6.7.

As a final standard application we conclude a stable elements theorem in our context. 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 referred to Theorem 6.6 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 discussion. Section 2 is devoted to some basic concepts of homological algebra of functors, including a discussion of cohomology with locally constant coefficients. We start Section 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 constant coefficient systems and subgroups of p-power index and index prime to p. Still in Section 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 construction for locally constant coefficients, where a transfer map exists for geometric 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 coefficients.

Acknowledgements: The authors would like to thank Bob Oliver for a number of useful conversations while we were working on this project, and particularly for his illuminating counter 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 material on functor cohomology.

2. 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, 5A2] for more detail.

2.1. Saturated Fusion Systems. The fundamental object underlying a p-local finite group is a finite p-group and a fusion system over it. The concept is originally 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) the set of all homomorphisms P → Q obtained by restriction of inner automorphisms of G to P. The set of all elements g ∈ G such that gP g^{−1} ≤ Q is called the transporter set in G from P to Q, and is denoted N_{G}(P, Q). Thus Hom_{G}(P, Q) = N_{G}(P, Q)/C_{G}(P). If g ∈ G, then we denote by c_{g} the conjugation x → gxg^{−1}.

Definition 2.1 ([Pu] and [BLO2, Definition 1.1]). A fusion system over a finite pgroup S is a category F, where Ob(F) is the set of all subgroups of S, and which 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 φ ∈ 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.

Definition 2.2 ([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 |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ ≤ S which are F-conjugate to P;

(2) fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for for all P′ ≤ S which are Fconjugate to P;

(3) F-centric if C_{S}(P′) = Z(P′) for all P′ ≤ S which are F-conjugate to P.

(4) F-radical if Out_{F}(P) =^{def} Aut_{F}(P)/ Inn(P) does not contain a nontrivial 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) ∈ Syl_{p}(Aut_{F}(P)). (II) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that P′ = φ(P) is fully centralized,

then φ extends to a morphism φ ∈ Hom_{F}(N_{φ}, S), where

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(P′)}.

If G is a finite group and S ∈ Syl_{p}(G), then the category F = F_{S}(G), whose objects are all subgroups P ≤ S, and whose morphisms are Hom_{F}(P, Q) = Hom_{G}(P, Q), is a saturated fusion system ([BLO2, Proposition 1.3]).

2.2. Centric Linking Systems. Let F^{c} ⊆ F denote the full subcategory whose objects are the F-centric subgroups of S.

Definition 2.3 ([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 π: L → F^{c}, and “distinguished” monomorphisms P ⟶^{δ_{P}} Aut_{L}(P) for each F-centric subgroup P ≤ S, which satisfy the following conditions.

(A) π is the identity on objects and surjective on morphism sets. For each pair of objects P, Q ∈ Ob(L), Z(P) acts freely on Mor_{L}(P, Q) by composition (upon identifying Z(P) with δ_{P} (Z(P)) ≤ Aut_{L}(P)), and π induces a bijection

Mor_{L}(P, Q)/Z(P) ⟶^{≅} Hom_{F}(P, Q).

(B) For each F-centric subgroup P ≤ S and each x ∈ P, π(δ_{P} (x)) = c_{x} ∈ Aut_{F}(P). (C) For each f ∈ Mor_{L}(P, Q) and each x ∈ P, f ∘ δ_{P} (x) = δ_{Q}(π(f)(x)) ∘ 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 classifying space of a p-local finite group (S, F, L) is the p-completed nerve |L|_{p}^{∧}.

If F = F_{S}(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) ∈ Syl_{p}(C_{G}(P)), or equivalently if and only if C_{G}(P) ≅ Z(P) × C′_{G}(P), where C′_{G}(P) is a group of order prime to p. The centric linking system of G is defined to be the category L_{S}^{c}(G), whose objects are the subgroups of S that are p-centric in G, and whose morphism sets are Mor_{L}(P, Q) = N_{G}(P, Q)/C′_{G}(P). The triple (S, F_{S}(G), L_{S}^{c}(G)) then forms a p-local finite group whose classifying space is equivalent to BG_{p}^{∧} [BLO2].

2.3. Compatible Systems of Inclusions. For a p-local finite group (S, F, L), a morphism φ: P → Q in L can be thought of as a ”lift” of the group homomorphism φ = π(φ) in the fusion system F, and the number of such lifts is |Z(P)|. It will often be convenient to extend concepts associated to group homomorphisms to morphisms in L. For instance, we define the image of φ to be the image of φ, and denote it by φ(P) or P^{φ}. (Observe that since the morpism is written on the right, we have P^{ψ∘φ} = (P^{φ})^{ψ}.) 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 2.4 ([5A2, Def. 1.11(b)]). Let (S, F, L) be a p-local finite group. A compatible set of inclusions for L is a choice of morphisms ι_{P}^{Q} ∈ Mor_{L}(P, Q), one for each pair of F-centric subgroups P ≤ Q, such that ι_{S}^{S} = Id_{S}, and the following hold for all P ≤ Q ≤ R,

(i) π(ι_{P}^{Q}) is the inclusion P ↪ Q; (ii) ι_{Q}^{R} ∘ ι_{P}^{Q} = ι_{P}^{R};

We often write ι_{P} for ι_{P}^{S}. 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 φ ∈ Mor_{L}(P, Q) and F-centric subgroups P′ ≤ P and Q′ ≤ Q such that φ ∘ ι_{P′}^{P}(P′) ≤ Q′, there is a unique morphism φ_{P′}^{Q′} ∈ Mor_{L^{q}}(P′, Q′) with ι_{Q′}^{Q} ∘φ_{P′}^{Q′} = φ ∘ι_{P′}^{P}. We refer to φ_{P′}^{Q′} as the restriction of φ to P′. To simplify notation, we will often write φ_{|P′} instead of φ_{P′}^{Q′} when there is no danger of confusion. If P′ is clear from the context, we will sometimes omit it from the notation as well.

Fix a compatible system of inclusions. Let π_{1}(|L|, S) be the fundamental group of |L| with basepoint at the vertex S, and let B(π_{1}(|L|, S)) be the category associated to the fundamental group π_{1}(|L|, S). One obtains a functor

J: L → B(π_{1}(|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 ι_{Q} ∗ f ∗ ι_{P}^{−1}. Composing with the distinguished monomorphism δ_{S}: S → Aut_{L}(S), we get a functor

B(j): B(S) ⟶^{B(δ_{S})} B(Aut_{L}(S)) ⊆ L ⟶^{J} B(π_{1}(|L|, S)),

and a corresponding homomorphism

j: S → π_{1}(|L|).

A construction of the functor J in a more general setting is described in Subsection 3.5.

2.4. A Stable Elements Theorem. The cohomology of a finite group G with coefficients in a p-local module can be computed by a fundamental result due to Cartan and Eilenberg [CE] (known as the stable elements theorem). Their original statement can be reinterpreted as follows. Let O_{S}(G) denote the category whose objects are 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 O_{S}(G)^{op} which sends P ≤ S to H^{∗}(P, M), where M becomes a ℤ_{(p)}[P]-module via restriction. The Cartan-Eilenberg stable elements theorem can then be restated as claiming that the following isomorphism holds

H^{∗}(G, M) ≅ lim_{O_{S}(G)^{op}} H^{∗}(−, M).

Theorem 5.8 in [BLO2] is the analogous statement for p-local finite groups, and where the module of coefficients is the field 𝔽_{p}. Specifically, if (S, F, L) is 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|, 𝔽_{p}) ≅ lim_{O(F)^{op}} H^{∗}(−, 𝔽_{p}).

From this one can deduce that the same statement is true for any ℤ_{(p)}-module of coefficients with a trivial π_{1}(|L|)-action.

More generally, using a stable transfer construction it is shown in [Ra] (see also [CM]) that the stable elements theorem holds for any (non-equivariant) stably representable cohomology theory. However, as we will observe at the end of Section 3, this statement is far from being true for cohomology with nontrivial coefficients.

2.5. 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 cohomology. Given a saturated fusion system F over a p-group S, the paper [5A2] defines what it means to be a subsystem of F of a finite index, which in this context is either a power of p or prime to p. We refer to the latter case as a p′ index subsystem. The paper also gives a classification of all subsystems of F of p-power or p′ index, which we recall here. We point out that generically (i.e., without considering iterations) these are the only cases where it makes sense to talk about a subsystem where the notion of an index is well defined.

To any saturated fusion system F over a p-group S, one associates two finite groups Γ_{p}(F) and Γ_{p′}(F). If F admits an associated centric linking system L, then these groups turn out to be the maximal p-power and p′ quotients of π_{1}(|L|), respectively. Both groups depend only on F (and not on the existence or nature of an associated centric linking system L). The group Γ_{p}(F) is given by S/O_{F}^{p}(S), where the divisor is the hyperfocal subgroup of S with respect to F (see [5A1, Sec. 2]). If F admits an associated centric linking system, then Γ_{p}(F) ≅ π_{1}(|L|_{p}^{∧}). The group Γ_{p′}(F) has a more complicated description in [5A2], but was observed by Aschbacher to be π_{1}(|F^{c}|).

Subsystems of F of p-power or p′ index are in bijective correspondence with subgroups of Γ_{p}(F) and Γ_{p′}(F), respectively. If (S, F, L) is a p-local finite group, and we let Γ denote either Γ_{p}(F) or Γ_{p′}(F), then with each subgroup H ≤ Γ one has an associated p-local finite group (S_{H}, F_{H}, L_{H}), and |L_{H}| is homotopy equivalent to a covering space of |L| with fibre Γ/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 ∈ L-mod. In the case of a subgroup of p′ index, we can do so in full generality, and we define a cochain-level transfer in cohomology for any functor of coefficients. These general methods do not carry over to the p-power index case, but we can still define a transfer map for locally constant coefficient systems.

We next state a summary of the main classification result from [5A2] for fusion subsystem of p-power or p′ index. Some minor modifications of the original statement will be dealt with in the proof. Before stating the theorem, we set up some notation.

Let (S, F, L) be a p-local finite group, and let F^{q} and L^{q} be the associated quasicentric fusion and linking systems [5A1, Sec. 3]. Let Γ denote Γ_{p}(F) = π_{1}(|L|_{p}^{∧}, S) or Γ_{p′}(F) = π_{1}(|F^{c}|, S), where in both cases S denotes the basepoint given by the vertex S. Fix a compatible choice of inclusions {ι_{P}^{Q}} for L^{q}, and let J: L^{q} → B(π_{1}(|L^{q}|, S) be the resulting functor as defined in Section 2.3. Composing with the obvious projection θ̂: π_{1}(|L^{q}|, S) = π_{1}(|L|, S) → Γ, one gets a functor

Θ̂: L^{q} → B(Γ).

We will denote the restriction of Θ̂ to L by the same symbol. Notice that Θ̂ sends the chosen inclusions in L^{q} to the identity (since J does). Let θ: S → Γ denote the restriction of Θ̂ to S via the monomorphism δ_{S}: S → Aut_{L^{q}}(S) ⊆ L^{q}. (Equivalently, θ = θ̂ ∘ j, where j: S → π_{1}(|L|, S) is the homomorphism defined in Section 2.3.) For any subgroup H ≤ Γ, let L_{H}^{•} ⊆ L^{q} be the subcategory with the same objects and with morphism set Θ̂^{−1}(H), and let L_{H}^{q} ⊆ L_{H}^{•} be the full subcategory obtained by restricting to subgroups of S_{H} =^{def} θ^{−1}(H). Finally, let F_{H} be the fusion system over S_{H} generated by π(L_{H}^{q}) ⊆ F^{q} and restrictions of morphisms, and let L_{H} ⊆ L_{H}^{q} be the full subcategory on those objects which are F_{H}-centric.

Theorem 2.5 ([5A2]). Let (S, F, L) be a p-local finite group. Then, with the notation above the following are satisfied.

(i) F_{H} is a saturated fusion system over S_{H}, and L_{H} is an associated centric linking system. Thus the triple (S_{H}, F_{H}, L_{H}) is a p-local finite group.

(ii) If Γ = Γ_{p′}(F) then a subgroup P ≤ S_{H} is F_{H}-centric (fully F_{H}-centralized, fully F_{H}-normalized) if and only if it is F-centric (fully F-centralized, fully F-normalized). If Γ = Γ_{p}(F) then the same statements hold, replacing centric by quasicentric.

(iii) There is a 1–1 correspondence between subgroups H ≤ Γ and p-local subgroups of (S, F, L) with p-power or p′ index (as appropriate). The correspondence is given by H ⟷ (S_{H}, F_{H}, L_{H}).

(iv) |L_{H}| is homotopy equivalent to the covering space of |L^{q}| ≃ |L| with fibre Γ/H.

(v) The homomorphism Aut_{L}(S) → Γ induced by the restriction of Θ̂: L^{q} → B(Γ) to Aut_{L}(S) is surjective.

Proof. Parts (i), (ii), (iii) and (iv) are included in Proposition 3.8 and Theorem 3.9 of [5A2]. Part (v) is clear in the case of Γ_{p}(F) = S/O_{F}^{p}(S), and follows from the definition of Γ_{p′}(F) given in [5A2, Thm 5.4]). □

3. 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 included here for the convenience of the reader.

Throughout this paper, let R denote a fixed commutative ring with a unit. Let R-mod denote the category of (left) R-modules, and let R-alg denote the category of (left) R-algebras. All categories we consider in this article will be small. For 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 Calgebra. Depending on context we may sometimes refer to objects of C-mod as system 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, all functors we deal with are covariant. If we need to discuss contravariant functors (and in certain contexts we will), we shall consider them as objects in the module category over the opposite category.

3.1. Functor Cohomology. Given an R-module M, the constant C-diagram M_{C} is the functor which takes every object in C to M and every morphism to the identity. The inverse limit on a category C is a functor

lim_{C}: C-mod → R-mod.

It comes with a morphism (lim_{C} U) ⟶^{λ_{U}} U, for any C-diagram U of R-modules, and is characterized by the universal property that if M is any R-module, and if α: M_{C} → U is any natural transformation, then there exists a unique R-module homomorphism α̂: M → lim_{C} U such that α = λ_{U} ∘ α̂_{C}, where α̂_{C} is the functor induced on the respective constant diagrams by α̂.

The universal property of the inverse limit functor implies an obvious identification:

(1)  lim_{C} U ≅ Hom_{C-mod}(R_{C}, U).

This shows in particular that the inverse limit functor is left exact. Its right derived functors applied to U ∈ 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 → I_{•} is an injective resolution of U in C-mod, and P_{•} → R_{C} is a projective resolution of R_{C} in C-mod, then

lim^{i}_{C} U = H^{i}(Hom_{C-mod}(R_{C}, I_{•})) = H^{i}(Hom_{C-mod}(P_{•}, U)).

Notation 3.1. Throughout this article we use the notation H^{∗}(C, U) to denote lim_{C}^{∗}U. 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}: C-mod → R-mod.

One defines the homology of C with coefficients in U ∈ C-mod as the higher colimits of U, i.e. the left derived functors of the colimit functor, applied to U. In this section, and throughout the paper, we focus on cohomology and mostly leave the reader to dualize the discussion to obtain analogous results in homology.

3.2. Cup Products in Functor Cohomology. Let C be a small category. For chain complexes C_{•} and C′_{•} in C-mod, one obtains a chain complex (C ⊗ C′)_{•} with

(C ⊗ C′)_{n} =^{def} ∑_{k+l=n} C_{k} ⊗ C′_{l}

and differential δ induced by the differentials of C_{•} and C′_{•} (which we also denote by δ) via the formula δ(x ⊗ y) = δx ⊗ y + (−1)^{k}x ⊗ δy for x ∈ C_{k}, y ∈ C′_{l}.

It is a standard result that if both C_{•} and C′_{•} are exact then (C ⊗ C′)_{•} is exact, and likewise that (C ⊗ C′)_{•} is projective if both C_{•} and C′_{•} are projective. In particular, given projective resolutions P_{•} → R_{C} and P′_{•} → R_{C} of the constant functor on C, one obtains a projective resolution (P ⊗ P′)_{•} → R_{C} ⊗ R_{C} ≅ R_{C}

Let M, M′ ∈ C-mod. For σ ∈ Hom_{C-mod}(P_{k}, M) and σ′ ∈ Hom_{C-mod}(P′_{l}, M′), let σ × σ′ ∈ Hom_{C-mod}((P ⊗ P′)_{k+l}, M ⊗ M′) be the natural transformation

σ × σ′: (P ⊗ P′)_{k+l} ⟶^{proj} P_{k} ⊗ P′_{l} ⟶^{σ⊗σ′} M ⊗ M′.

This gives a bilinear pairing

Hom_{C-mod}(P_{•}, M) ⊗ Hom_{C-mod}(P′_{•}, M′) → Hom_{C-mod}((P ⊗ P′)_{•}, M ⊗ M′)
σ ⊗ σ′ ⟼ σ × σ′.

One can check that this is a map of cochain complexes, so after taking cohomology we obtain a bilinear pairing on higher limits

H^{∗}(C, M) ⊗ H^{∗}(C, N) ⟶ H^{∗}(C, M ⊗ N),
x ⊗ y ↦ x × y.

One can furthermore check that this is independent of the choice of projective resolutions. This pairing is called the cross product pairing.

For any A ∈ C-alg one has a multiplication transformation μ: A ⊗ A → A which induces a homomorphism on cohomology. Composing with the cross product pairing, we obtain the cup product.

Definition 3.2. Let A ∈ C-alg . For x ∈ H^{k}(C, A) and y ∈ H^{l}(C, A), the cup product x ∪ y ∈ H^{k+l}(C, A), (or xy,) is the image of x ⊗ y under the homomorphism

H^{∗}(C, A) ⊗ H^{∗}(C, A) ⟶^{×} H^{∗}(C, A ⊗ A) ⟶^{μ_{∗}} 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 3.3. For a functor A ∈ C-alg , the cup product on H^{∗}(C, A) is associative, graded-commutative, and has a multiplicative unit. Thus H^{∗}(C, A) is a gradedcommutative ring with a unit.

If F: C → D is any functor, one has an exact functor F^{∗}: D-mod → C-mod defined by F^{∗}(α) =^{def} α ∘ F. The next proposition says that the cup product is natural in both C and A. Again, the proof is routine.

Proposition 3.4. Let F: C → D be a functor, let A, B ∈ D-alg , and let η: A → B be a natural transformation. Then the following hold for x, y ∈ 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 η_{∗}: H^{∗}(D, A) → H^{∗}(D, B) induced by η satisfies

η_{∗}(x ∪ y) = η_{∗}x ∪ η_{∗}y.

3.3. Kan Extensions and the Shapiro Lemma. Let ι: C → D be any functor. For each object d ∈ D one defines the overcategory ι ↓ d to be the category with objects (c, α), where c ∈ C and α: ι(c) → d is a morphism in D. Morphisms from (c, α) to (c′, α′) in ι ↓ d are morphisms γ: c → c′ in C such that α′ ∘ι(γ) = α. The undercategory d ↓ ι is defined analogously. Both categories admit an obvious forgetful functor to C, and for M ∈ C-mod we denote the composite of M with the forgetful functor by M_{♯}.

The functor ι^{∗}: D-mod → C-mod induced by ι has a right adjoint R_{ι} and a left adjoint L_{ι} called the right and left Kan extension along ι, respectively. The Kan extensions of a given functor M ∈ C-mod is determined objectwise by

R_{ι}(M)(d) = {lim \atop d ↓ ι} M_{#} and L_{ι}(M)(d) = {colim \atop ι ↓ d} M_{#},

and a morphism d → d′ in D induces homomorphisms

R_{ι}(M)(d) → R_{ι}(M)(d′) and L_{ι}(M)(d) → L_{ι}(M)(d′),

via the universal properties of the limit and colimit functors. From these descriptions of Kan extensions one easily obtains the isomorphisms

lim_{D} R_{ι}(M) = lim_{C} M  and  colim_{D} L_{ι}(M) = colim_{C} M.

For a functor ι: C → D and a system of coefficients M ∈ C-mod there is a homomorphism called the Shapiro map

Sh_{M}: H^{∗}(D, R_{ι}(M)) → H^{∗}(C, M),

which is constructed as follows. Let M → I_{•} be an injective resolution of M. Since R_{ι} is the right adjoint of the exact functor ι^{∗} it preserves injectives, and we have a (possibly non-exact) cochain complex R_{ι}M → R_{ι}I_{0} → R_{ι}I_{1} → ⋯ in which every term after R_{ι}M is injective. If R_{ι}M → I′_{•} is an injective resolution of R_{ι}M, then the identity transformation of R_{ι}M lifts (non-uniquely) to a cochain map χ: I′_{•} → R_{ι}I_{•}, which induces a cochain map of limits

lim_{D} I′_{•} =^{def} Hom_{D}(R_{D}, I′_{•}) ⟶^{χ_{∗}} Hom_{D}(R_{D}, R_{ι}I_{•}) ⟶_{≅}^{ρ^{−1}} Hom_{C}(R_{C}, I_{•}) =^{def} lim_{C} I_{•},

and Sh_{M} is defined as the induced map in cohomology. Here,

ρ: Hom_{C}(R_{C}, I_{•}) = Hom_{C}(ι^{∗}R_{D}, I_{•}) → Hom_{D}(R_{D}, R_{ι}I_{•})

is the adjunction isomorphism. The Shapiro map is independent of the choice of the injective resolutions I_{•} and I′_{•}, and the cochain map χ, and is natural with respect to morphisms M → M′ in C-mod. Hence it induces a natural transformation

Sh: H^{∗}(D, R_{ι}(−)) → H^{∗}(C, −)

which will be referred to as the Shapiro transformation. The important property of the Shapiro transformation is given by the following lemma.

Lemma 3.5 (The Shapiro Lemma). Let ι: C → D be a functor. If the right Kan extension R_{ι} is exact then the Shapiro transformation is a natural isomorphism of functors.

Proof. If R_{ι} is exact, then R_{ι}M → R_{ι}I_{•} is an injective resolution, so we can take I′_{•} = R_{ι}I_{•} and χ = Id in the construction of Sh_{M} for M ∈ C-mod, and it follows that Sh_{M} is an isomorphism. □

The right Kan extension functor is always left exact as it is a right adjoint. Similarly the left Kan extension functor is right exact. In some instances the left and right Kan extension functors are naturally isomorphic. In this case both Kan extension functors are exact, and in particular the Shapiro Lemma holds. The Shapiro lemma has an analogous homological version, involving the derived functors of the colimit, and the left Kan extension.

3.4. Deformation Retracts of Categories. We now study a condition on a functor which ensures that it induces a natural isomorphism between the corresponding derived functors of the limit and colimit.

Definition 3.6. 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 such that r ∘f = 1_{C}, and a natural transformation η: f ∘ r → 1_{D} satisfying η_{f(c)} = 1_{f(c)} for c ∈ C. When f is the inclusion of a subcategory, we say that C is a left deformation retract of D. Right deformation retracts are defined dually.

Notice that this falls short of defining ι as left adjoint to r in the sense that we do not require that rη 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 3.7. Let f: C → D be 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 η: f ∘ r → 1_{D} be as in Definition 3.6. Since r ∘ f = 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 cofinal in the sense that for every object d in D, the overcategory f ↓ d is connected (meaning it is nonempty and there exists a zigzag of morphisms between any two objects).

First, the map η_{d}: f(r(d)) → d gives rise to an object (r(d), η_{d}) in f ↓ d. Now, if (c, u) is another object consisting of an object c in C and a morphism f(c) ⟶^{u} d in D, then the natural transformation η gives rise to the following commutative square.

[figure: (f ∘ r)(f(c)); (f ∘r)(u); η_{f(c)}; (f ∘ r)(d); η_{d}; f(c); u; d]

Thus we have a morphism r(u): (c, u) → (r(d), η_{d}) in f ↓ d, and in particular (c, u) is in the same connected component as (r(d), η_{d}), proving that f ↓ d. □

Lemma 3.7 actually holds if f is a weak deformation retract, for which we require only that for c ∈ C, we have η_{f(c)} = f(h) for some morphism h in C. Using the full 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 3.7 makes that claim only in dimension 0, which is all we need for our purposes).

3.5. Cohomology with Locally Constant Coefficients. We now discuss an important special case of functor cohomology, namely the case where the coefficient system is locally constant.

Let C be a small connected category, and let Ĉ denote the groupoid completion of C, i.e., the category with the same objects as C, and where all morphisms are formally inverted. Let π: C → Ĉ be the obvious functor. Choose an object c_{0} ∈ C. For any other object c ∈ C, choose a morphism ϕ_{c} ∈ MorĈ(c, c_{0}), and take ϕ_{c_{0}} to be the identity. Then one gets isomorphisms of sets

j_{a,b}: MorĈ(a, b) → AutĈ(c_{0}) by φ ↦ ϕ_{b} ∘ φ ∘ ϕ_{a}^{−1}.

Let Γ =^{def} AutĈ(c_{0}), and let ι: B(Γ) → Ĉ be the inclusion functor. Assembling the isomorphisms j_{a,b} together for all pairs of objects in Ĉ, one gets a functor

J: Ĉ → B(Γ),

which is an equivalence of categories (but depends on the choices made). In particular, J ∘ ι is the identity on B(Γ), while ι ∘ J is naturally isomorphic to the identity on Ĉ. Moreover, by [Qu, Prop. 1], the group Γ is naturally isomorphic to π_{1}(|C|, c_{0}), and the map induced on nerves by the composite J ∘ π induces an isomorphism on fundamental groups.

A system of coefficients M ∈ C-mod is said to be locally constant if for every morphism φ: a → b in C, M(φ) is an isomorphism. Notice that M is locally constant if and only if M factors as M = M̂ ∘ π for a unique functor M̂ ∈ Ĉ-mod. Let R be a commutative ring with a unit. An R[Γ]-module is a functor N: B(Γ) → R-mod. The next lemma shows that every locally constant system of coefficients on C is, up to a natural isomorphism, a R[Γ]-module composed with the functor J constructed above.

Lemma 3.8. Let C be a small connected category, and let M ∈ C-mod be a locally constant system of coefficients on C. Then M is naturally isomorphic to N ∘ J ∘ π, where N is the R[Γ]-module M̂ ∘ ι. Moreover, this defines an equivalence of categories between the category of R[Γ]-modules and the category of locally constant functors on C.

Proof. Since ι ∘ J is naturally isomorphic to the identity on Ĉ, one has

M = M̂ ∘ π ≅ M̂ ∘ ι ∘ J ∘ π = N ∘ J ∘ π,

as claimed.

For the second statement, notice that the correspondences which takes an R[Γ]- module N to N ∘ J ∘ π and a locally constant system of coefficients M ∈ C-mod to M̂ ∘ ι are natural on N and M respectively, and using the first statement, define the equivalence of categories claimed. □

Proposition 3.9. Let C be a small connected category, and let M ∈ C-mod be a locally constant functor. Then, with the notation above, there is an isomorphism

H^{∗}(C, M) ≅ H^{∗}(|C|, M̂ ∘ ι)

which is natural in M.

Proof. As a consequence of [Qu, Prop 1], Quillen shows that if C is a small connected category, and L is a ℤ[π_{1}(|C|)]-module, then there is a canonical isomorphism

H^{∗}(|C|, L) ≅ H^{∗}(C, L′),

where L′ ∈ C-mod is given by the composite

C ⟶^{π} Ĉ ⟶^{J} B(π_{1}(|C|)) ⟶^{L} Ab.

But by Lemma 3.8, every locally constant M ∈ C-mod is naturally isomorphic to a functor of the form M ∘ ι ∘ J ∘ π. Thus, if we set L = M̂ ∘ ι, then Quillen’s result reads

H^{∗}(|C|, M̂ ∘ ι) ≅ H^{∗}(C, M ∘ ι ∘ J ∘ π) ≅ H^{∗}(C, M).

This proves the claim. □

Corollary 3.10. Let τ: C → D be a functor between small connected categories, such that |τ| is a homotopy equivalence. Then for any locally constant system to coefficients M ∈ D-mod, τ induces a natural isomorphism

H^{∗}(C, τ^{∗}M) ⟶^{≅} H^{∗}(D, M).

3.6. Covering Spaces, and Locally Constant Coefficients. Let C be a small connected category, c_{0} ∈ C and Γ = AutĈ(c_{0}) ≅ π_{1}(|C|, c_{0}). Let J: C → B(Γ) be as before.

For each Γ′ ≤ Γ, let E_{Γ}(Γ/Γ′) denote the category with Γ/Γ′ as objects, and with a unique morphism ĝ: aΓ′ → gaΓ′ for each g ∈ Γ and aΓ′ ∈ Γ/Γ′. Let C_{Γ′} denote the category given by the pull back in the diagram

\xymatrix{C_{Γ′} \ar[r] \ar[d]_{π} & E_{Γ}(Γ/Γ′) \ar[d]^{ρ_{Γ′}} \\ C \ar[r]_{J} & BΓ}

Thus, Obj(C_{Γ′}) = Obj(C) × Γ/Γ′, while morphisms (c, aΓ′) → (c′, a′Γ′) are morphisms φ: c → c′, such that a′Γ′ = J(φ)aΓ′.

For each c ∈ C the undercategory c ↓ π has objects ((d, aΓ′), φ) where d ∈ C, aΓ′ ∈ Γ/Γ′, and φ: c → d is a morphism in C. A morphism ((d, aΓ′), φ) → ((d′, a′Γ′), φ′) in c ↓ π is a map ψ: d → d′ in C, such that ψφ = φ′, and a′Γ′ = J(ψ)aΓ′.

Let ((d, aΓ′), φ) be an arbitrary object in c ↓ π. Then there is a unique morphism ((c, aΓ′), 1_{c}) → ((d, J(φ)aΓ′), φ) which is induced by φ. Thus every component of c ↓ π has an initial object and is therefore contractible. Notice that there is an obvious 1–1 correspondence between components of c ↓ π and Γ/Γ′, and that every morphism c → e in C induces a homotopy equivalence e ↓ π → c ↓ π. Thus the hypothesis of Quillen’s theorem B are satisfied, and |C_{Γ′}| → |C| is a covering space up to homotopy, with fibre over c given by |c ↓ π| ≃ Γ/Γ′.

Lemma 3.11. Let τ: D → C be a functor between small connected categories such that the induced map |τ|: |D| → |C| is a covering space up to homotopy (i.e., has a homotopically discrete homotopy fibre), and let Γ′ = π_{1}(|D|, d_{0}) for some d_{0} ∈ D. Then the following hold:

(i) There is a functor τ̂: D → C_{Γ′}, lifting τ (i.e., πτ̂ = τ), which induces a homotopy equivalence on nerves.

(ii) For any locally constant system of coefficients M ∈ C-mod, there is a natural isomorphism R_{τ}(τ^{∗}M) ≅ R_{π}(π^{∗}M).

Proof. Choose some d_{0} ∈ D, and let c_{0} = τ(d_{0}). Let Γ = π_{1}(|C|, c_{0}), and Γ′ = π_{1}(|D|, d_{0}). Then, constructing a projection functor J: C → B(Γ) as before, one gets J ∘τ: D → B(Γ), whose image is Γ′, and thus a corresponding projection J′: D → B(Γ′). The diagram

\xymatrix{D \ar[r]^{τ} \ar[d]_{J′} & C \ar[d]^{J} & C_{Γ′} \ar[l]_{π} \ar[d] \\ B(Γ′) \ar[r]^{inc} & B(Γ) & E_{Γ}(Γ/Γ′) \ar[l]_{ρ_{Γ′}}}

commutes, where the right hand side square is a pull back square. The inclusion of B(Γ′) in E_{Γ}(Γ/Γ′) as the full subcategory on the object 1Γ′ induces a homotopy equivalence on nerves, and a functor τ̂: D → C_{Γ′}. A simple diagram chase now shows that |τ̂| is a homotopy equivalence, and proves (i).

To prove (ii), notice that since πτ̂ = τ, the functor τ̂ induces a natural transformation of functors C → Cat

τ̂_{(−)} : (− ↓ τ) → (− ↓ π).

Thus for every c ∈ C, and M ∈ C-mod, one has a map induced by τ̂,

R_{π}(π^{∗}M) =^{def} lim_{c↓π}(π^{∗}M)_{♯} → lim_{c↓τ}(τ̂^{∗}π^{∗}M)_{♯} = lim_{c↓τ}(τ^{∗}M)_{♯} =^{def} R_{τ}(τ^{∗}M).

But since π^{∗}M is also locally constant, and since τ̂ induces a homotopy equivalence on nerves, it follows from Corollary 3.10 that this map is an isomorphism, thus proving (ii). □

Lemma 3.11 allows us to compute the values of a right Kan extension of a system of coefficients M ∈ D-mod along a functor τ: D → C satisfying its hypothesis.

The following lemma allows a calculation of the right Kan extension explicitly in a specialized case which will be of interest in our discussion.

Lemma 3.12. Let τ: D → C be a functor between small connected categories such that for each c ∈ C, every connected component in the under category c ↓ τ has an initial object. Then for any M ∈ D-mod, the value of the right Kan extension R_{τ}(M) ∈ C-mod at c is the direct product of the values of M_{♯} on these initial objects. In particular R_{τ} is an exact functor.

Proof. For each c ∈ C one has

R_{τ}(M)(c) =^{def} lim_{c↓τ} M_{♯} ≅ ∏_{[(d,α)]∈[c↓τ]} M(d_{0}),

where the product runs over all connected components of c ↓ τ, and (d_{0}, α_{0}) is initial in the component of (d, α). This follows at once from the fact that a limit over a category with an initial object is given by the value of the functor on that object. This description also makes it clear that R_{τ} is an exact functor, and so the proof is complete. □

Proposition 3.13. Let τ: D → C be a functor between small connected categories such that the induced map |τ|: |D| → |C| is a covering space up to homotopy. Fix an object d_{0} in D, let Γ′ = π_{1}(|D|, d_{0}), and let π: C_{Γ′} → C be the projection.

For any locally constant system of coefficients M ∈ C-mod, there is a natural isomorphism

H^{∗}(C, R_{π}(π^{∗}M) ∈ C-mod) ≅ H^{∗}(D, τ^{∗}M).

Furthermore, if the index of the covering is finite, then there is a natural transformation T: R_{π}(π^{∗}M) ∈ C-mod → M, such that the composite

H^{∗}(C, M) ⟶^{τ^{∗}} H^{∗}(D, τ^{∗}(M)) ≅ H^{∗}(C, R_{π}(π^{∗}M) ∈ C-mod) ⟶^{T_{∗}} H^{∗}(C, M)

is multiplication by the index.

Proof. Since M is locally constant H^{∗}(D, τ^{∗}M) ≅ H^{∗}(C_{Γ′}, π^{∗}M) by Corollary 3.10. By Lemma 3.12 R_{π} is exact, and hence we have an isomorphism H^{∗}(C, R_{π}(π^{∗}M)) ≅ H^{∗}(C_{Γ′}, π^{∗}M) by the Shapiro lemma. This proves the first statement.

To prove the second statement, consider the functor R_{π}(π^{∗}M). For each coset aΓ′ ∈ Γ/Γ′, the object ((c, aΓ′), 1_{c}) ∈ c ↓ π is initial in its own path component, and (π^{∗}(M))_{♯}((c, aΓ′), 1_{c}) = M(c). Thus, by Lemma 3.12, for each c ∈ C,

R_{π}(π^{∗}M)(c) =^{def} lim_{c↓π} π^{∗}M_{♯} ≅ ∏_{Γ/Γ′} M(c).

Assuming Γ/Γ′ is a finite set, define a natural transformation of functors T: R_{π}(π^{∗}M) → M in C-mod by

(2)  T_{c}({x_{aΓ′}}_{aΓ′∈Γ/Γ′}) = ∑_{aΓ′∈Γ/Γ′} x_{aΓ′}.

Naturality of T is clear, since M is additive.

It remains to show that T_{∗} ∘ τ^{∗} is multiplication by |Γ: Γ′|. To do that it suffices to show that the natural transformation induced by π

Hom_{C-mod}(−, M) ⟶^{π^{∗}} Hom_{C_{Γ′}}(π^{∗}(−), π^{∗}M) ≅ Hom_{C-mod}(−, R_{π}(π^{∗}M)),

is the diagonal map. But for U ∈ C-mod, a natural transformation η: U → M takes an object c ∈ C to a morphism η_{c}: U(c) → M(c). Composing with π we see that the composition transformation takes each object (c, aΓ′) in C_{Γ′} to the morphism η_{c} as above. This shows that π^{∗} is precisely the diagonal map and the proof is complete. □

The next corollary follows at once from the definitions.

Corollary 3.14. Let M ∈ C-mod be a locally constant system of coefficients, and let Γ′ ≤ Γ = π_{1}(|C|, c_{0}) be a subgroup of finite index. Then the map

H^{∗}(C_{Γ′}, π^{∗}M) ≅ H^{∗}(C, R_{π}(π^{∗}M)) ⟶^{T_{∗}} H^{∗}(C, M)

coincides with the geometric transfer map associated to the covering space |C_{Γ′}| → |C|.

4. p-Local Finite Group Cohomology

The cohomology of a p-local finite group (S, F, L) with coefficients M ∈ L-mod 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 the 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, even if one restricts to locally constant coefficients.

4.1. An Example. As already mentioned, in [BLO2] it is shown that if (S, F, L) is a plocal finite group then the mod p cohomology of |L| is given by the F-stable elements in H^{∗}(BS, 𝔽_{p}). The next statement, due to Bob Oliver, provides an abundance of examples that show that with our definition of local coefficients one cannot expect a stable elements theorem to hold in full generality.

Proposition 4.1. Let (S, F, L) be a p-local finite group, (with S nontrivial) and let Γ = π_{1}(|L|). Fix a compatible system of inclusions, and assume that the composite functor

B(j): B(S) ⟶^{ι} L ⟶^{J} B(Γ)

is faithful, where ι is the inclusion B(S) ⟶^{B(δ_{S})} B(Aut_{L}(S)) ⊆ L, and J is the functor defined in Section 2.3. Assume further that the universal cover ˜|L| is not mod p acyclic. Let M be the group ring 𝔽_{p}[Γ] regarded as a system of coefficients in L-mod via J and the obvious Γ action. Then H̃^{∗}(S, ι^{∗}M) = 0, but H̃^{∗}(L, M) ≠ 0.

Proof. Since S ⟶^{j} Γ is monic, ι^{∗}M is a projective 𝔽_{p}[S] module, and since S is finite, it is also injective, and thus acyclic. On the other hand, since M is locally constant,

H^{∗}(L, M) = H^{∗}(|L|, 𝔽_{p}[Γ]) =^{def} H^{∗}(Hom_{𝔽_{p}[Γ]}(C_{∗}(˜|L|), 𝔽_{p}[Γ])) ≅

H^{∗}(Hom_{𝔽_{p}}(C_{∗}(˜|L|), 𝔽_{p})) ≅ H^{∗}(˜|L|, 𝔽_{p}),

where the first equality follows from Proposition 3.9, and the right hand side is not mod p acyclic by assumption. □

One large family of examples which satisfy the conditions of Proposition 4.1 is the general linear groups GL_{n}(𝔽_{p^{k}}). The following argument is a sketch of a proof. Since the same argument applies for all k, we will replace 𝔽_{p^{k}} by 𝔽, while p and k are assumed fixed. The radical subgroups in these groups are all p-centric, and their centers are given by the diagonally embedded 𝔽^{∗} ≅ ℤ/(p^{k} − 1). Hence the centric radical linking systems of GL_{n}(𝔽) and the associated projective group P GL_{n}(𝔽) have the same objects, and |L^{cr}(GL_{n}(𝔽))| is a covering space of |L^{cr}(P GL_{n}(𝔽))| with fibre 𝔽^{∗}, and thus they share the same universal cover. But L^{cr}(P GL_{n}(𝔽)) coincides with the centric radical transporter system of P GL_{n}(𝔽) (the category with the same objects, and where morphism sets are the transporters N_{G}(P, Q)), and thus admits an obvious map

|L^{cr}(P GL_{n}(𝔽))| → BP GL_{n}(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 |L^{cr}(P GL_{n})|, while at the same time it is not acyclic. This was first observed by Grodal. To finish the argument, we recall from [5A1] that for any p-local finite group, the centric radical linking system and the centric linking system have homotopy equivalent nerves.

4.2. Transfer Maps for Locally Constant Coefficients. We now show that if (S, F, L) is a p-local finite group and (S_{0}, F_{0}, L_{0}) is a subgroup of p-power index or index prime to p, then one can define an algebraic transfer map, which coincides with the transfer map associated to the finite covering |L_{0}| → |L| (See Thm. 2.5(iv)).

First we must discuss restrictions of coefficient systems. We defined a system of coefficients on a p-local finite group to be a functor M ∈ L-mod. When (S_{0}, F_{0}, L_{0}) is a subgroup of (S, F, L) of p-power index, one does not have in general an inclusion functor L_{0} → L, but rather an inclusion L_{0}^{q} → L^{q}. Thus, in this case, a system of coefficients on (S, F, L) cannot be directly restricted to (S_{0}, F_{0}, L_{0}). We will 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 L^{q} to L does not affect cohomology.

Lemma 4.2. Let (S, F, L) be a p-local finite group, and let L^{q} be the associated quasicentric linking system. For every locally constant system of coefficients N ∈ L^{q}-mod, the restriction to L induces an isomorphism in cohomology H^{∗}(L^{q}, N) ≅ H^{∗}(L, ι^{∗}N).

Proof. This follows directly from Corollary 3.10 since the inclusion L^{q} ⊆ L induces a homotopy equivalence on nerves. □

Below we will outline a functorial way to extend a locally constant coefficient system M on L to a locally constant coefficient system M^{q} on L^{q}. The extension M^{q} can then be restricted to a locally constant coefficent system M_{0} on L_{0} via the inclusions L_{0} ⊆ L_{0}^{q} ⊆ L^{q}. We will furthermore show that the extension M^{q} is unique up to unique isomorphism extending the identity morphism of M, and thus it makes sense to think of M_{0} as the restriction of M to L_{0}. Lemma 4.2 shows that this convention has the desired effect in cohomology.

Given a locally constant system of coefficients M on L, we construct an extension M^{q} of M to L^{q} as follows. First recall that M extends uniquely to a system of coefficients M̂ on the groupoid completion L̂. Since every morphism in L^{q} is both a monomorphism and an epimorphism in the categorical sense ([5A1, Corollary 3.10]), the inclusion i: L → L^{q} induces an inclusion of groupoid completions, î: L̂ → L̂^{q}. Since i induces a homotopy equivalence on nerves, and in particular an isomorphism of fundamental groups, î is an equivalence of categories. This implies that L̂ is a deformation retract of L̂^{q}, and we can choose an inverse r: L̂^{q} → L̂ such that r ∘ î = idL̂. Now the composite

M^{q}: L^{q} ⟶^{π} L̂^{q} ⟶^{r} L̂ \underrightarrow{M̂} R-mod,

where π is the groupoid completion map, is a locally constant system of coefficients on L^{q} that extends M.

Fixing a choice for the retraction r, the construction of M^{q} as the restriction of M̂ along r ∘ π clearly makes the assignment M ↦ M^{q} functorial. However this extension functor is by no means unique or even canonical as a different choice of retract would give rise to a different extension functor. The next lemma shows that, while there are many such extension functors, the difference between them is inconsequential.

Lemma 4.3. Let (S, F, L) be a p-local finite group, let L^{q} be the associated quasicentric linking system, and let M be a locally constant system of coefficients on L. If M′ is a locally constant system of coefficients on L^{q} extending M, then there is a unique natural isomorphism η: M^{q} ⇒ M′ that restricts to the identity transformation of M.

Proof. The isomorphism η sends P to the composite

η_{P} : M^{q}(P) ⟶^{M^{q}(ι_{P})} M^{q}(S) = M(S) = M′(S) ⟶^{M′(ι_{P})^{−1}} M′(P).

The verification of naturality and uniqueness of η is routine and is left for the reader.

□

In light of this uniqueness result, we make the following definition.

Definition 4.4. Let (S, F, L) be a p-local finite group and let Γ = Γ_{p}(F) or Γ_{p′}(F). Let H ≤ Γ, let (S_{H}, F_{H}, L_{H}) be the corresponding p-local subgroup, and let ι^{q}: L_{H}^{q} → L^{q} be the inclusion. For a locally constant system of coefficients M on L, the restriction of M to L_{H} is the composite

M_{H}: L_{H} ⊆ L_{H}^{q} ⊆ L^{q} ⟶^{M^{q}} R-mod.

The restriction map Res_{H}: H^{∗}(L, M) → H^{∗}(L_{H}, M_{H}) is the unique map that fits into the commutative diagram

\xymatrix{H^{∗}(L^{q}, M^{q}) \ar[d]^{≅} \ar[r]^{ι^{∗}} & H^{∗}(L_{H}^{q}, ι^{∗}M^{q}) \ar[d]^{≅} \\ H^{∗}(L, M) \ar[r]^{_{Res_{H}}} & H^{∗}(L_{H}, M_{H}),}

where the vertical maps are the isomorphisms from Lemma 4.2.

When Γ = Γ_{p′}(F), there is a well-defined inclusion L_{H} ⊂ L, and we note that M_{H} is just restriction along this inclusion, and Res_{H} is the usual restriction map.

The following proposition holds for both centric and quasicentric linking systems, although we state it using the notation for centric linking systems.

Proposition 4.5. Let (S, F, L) be a p-local finite group and let Γ = Γ_{p}(F) or Γ_{p′}(F). Let H ≤ Γ, and let (S_{H}, F_{H}, L_{H}) be the corresponding p-local subgroup. For any locally constant system of coefficients M ∈ L-mod there exists a map

T r_{H}: H^{∗}(L_{H}, M_{H}) → H^{∗}(L, M),

such that T r_{H} ∘ Res_{H} is multiplication by |Γ: H| on H^{∗}(L, M). Furthermore, the map thus defined coincides with the transfer map associated with the covering |L_{H}| → |L|.

Proof. It suffices to prove this in the quasicentric case, as the centric case is obtained by restriction. Let ι: L_{H}^{q} → L^{q} be the inclusion. By Theorem 2.5, the map |ι|: |L_{H}^{q}| → |L^{q}| is a covering space up to homotopy, with homotopy fibre equivalent to Γ/H. Thus Proposition 3.13 applies and one has a system of coefficients M′ ∈ L-mod, such that

H^{∗}(L^{q}, M′) ≅ H^{∗}(L_{H}^{q}, ι^{∗}M),

and a natural transformation T: M′ → M such that the composite

H^{∗}(L^{q}, M) ⟶^{ι^{∗}} H^{∗}(L_{H}^{q}, ι^{∗}M) ≅ H^{∗}(L^{q}, M′) ⟶^{T_{∗}} H^{∗}(L^{q}, M)

is multiplication by the index |Γ: 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 3.14. □

4.3. 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 able to obtain our most general results. However some of the discussion here has more general implications as well.

Throughout this subsection, fix a p-local finite group (S, F, L) and let Γ = Γ_{p′}(F). Fix a subgroup H ≤ Γ and let ι: L_{H} → L be the inclusion We assume throughout that a compatible system of inclusions for L has been chosen, and by restriction of a morphism in L we always mean restriction with respect to the chosen inclusions (see Subsection 2.3). Let Θ̂: L → B(Γ) denote the projection, as in Subsection 1.5. If α is a morphism in L, we denote by [α] the class of α in F.

Recall that a category is a discrete groupoid if each of its morphism sets either consists of a single isomorphism or is empty.

Lemma 4.6. For P ∈ L, let G_{H}(P) ⊆ P ↓ ι and G^{H}(P) ⊆ ι ↓ P denote the full subcategories, whose objects, respectively, are of the form (P^{α}, α) and (P^{α}, α^{−1}) for some α ∈ Mor_{L}(P, S). Then G_{H}(P) and G^{H}(P) are discrete groupoids. Furthermore, G_{H}(P) is a left deformation retract of P ↓ ι, and G^{H}(P) is a right deformation retract of ι ↓ P.

Proof. Since L_{H} and L have the same objects by Theorem 2.5, the category P ↓ ι is nonempty for any P ∈ L. Thus let α, β ∈ Hom_{L}(P, S) be any morphisms. Then (P^{α}, α) and (P^{β}, β) are objects in G_{H}(P). A morphism between them is a morphism φ: P^{α} → P^{β} in L_{H}, such that φ ∘ α = β. But α: P → P^{α} is an isomorphism in L, and hence invertible. Thus φ = β ∘ α^{−1}, and so if a morphism between two given objects exists, then it is unique, and is obviously an isomorphism. This shows that G_{H}(P) is a discrete groupoid. An analogous argument shows that G^{H}(P) is also a discrete groupoid.

Let j: G_{H}(P) → P ↓ ι denote the inclusion, and define r: P ↓ ι → G_{H}(P) as follows. On objects r(Q, α) =^{def} (P^{α}, α), noting that P^{α} ≤ Q, while any morphism (Q, α) ⟶^{φ} (Q′, α′) is taken by r to α′ ∘ α^{−1}. For this to be well defined, α′ ∘ α^{−1} should be a morphism in L_{H}, and this is indeed the case since φ ∘ α′ = α and Θ̂(φ) ∈ H implies Θ̂(α′ ∘ α^{−1}) ∈ H. The composite r ∘ j is the identity functor on G_{H}(P), and there is a natural transformation η: j ∘ r → Id_{P ↓ι} that takes an object (Q, α) in P ↓ ι to the morphism (P^{α}, α) → (Q, α) induced by the inclusion. Clearly ηj is the identity transformation of j. This shows that G_{H}(P) is a left deformation retract of P ↓ ι, as claimed.

Next, let j denote the inclusion G^{H}(P) → ι ↓ P. For an object (Q, α) in ι ↓ P we choose a morphism γ_{(Q,α)} ∈ Mor_{L}(P, S) with Θ̂(γ_{(Q,α)}) = Θ̂(α)^{−1} as follows. If α is an isomorphism, then we just take γ_{(Q,α)} = α^{−1}. Otherwise, we choose, using Theorem 2.5(v), a morphism γ̄_{(Q,α)} ∈ Aut_{L}(S) such that Θ̂(γ̄_{(Q,α)}) = Θ̂(α)^{−1}, and we let γ_{(Q,α)} be the restriction of γ̄_{(Q,α)} to P. We now define the retraction r: ι ↓ P → G^{H}(P) by setting r(Q, α) = (P^{γ_{(Q,α)}}, γ_{(Q,α)}^{−1}), and sending a morphism (Q, α) ⟶^{φ} (R, β) to the unique morphism γ_{(R,β)} ∘ γ_{(Q,α)}^{−1} : (P^{γ_{(Q,α)}}, γ_{(Q,α)}^{−1}) → (P^{γ_{(R,β)}}, γ_{(R,β)}^{−1}) in G^{H}(P). Observe that this is well defined since

Θ̂(γ_{(R,β)} ∘ γ_{(Q,α)}^{−1}) = Θ̂(β)^{−1}Θ̂(α) = Θ̂(φ) ∈ H,

and so γ_{(R,β)} ∘γ_{(Q,α)}^{−1} is in L_{H}. With this definition we have r ∘j = Id_{G^{H}(P)}. Furthermore, if η: Id_{ι↓P} → j ∘ r is the natural transformation that sends an object (Q, α) to the morphism γ_{(Q,α)} ∘α, then ηj is the identity transformation of j, and so G^{H}(P) is a right deformation retract of ι ↓ P. □

Recall that the homomorphism induced by the restriction of Θ̂: L^{q} → B(Γ) to Aut_{L}(S) is surjective by Theorem 2.5. Thus the composite with the projection to the right cosets

(3)  Aut_{L}(S) → Γ → Γ/H

is also surjective.

Lemma 4.7. Consider the set of left cosets Γ/H as a category with objects Γ/H and no nontrivial morphisms. Then for any P ∈ L the functor

δ_{P} : G_{H}(P) → Γ/H

taking an object (P^{α}, α) to the coset Θ̂(α)^{−1}H is an equivalence of categories.

Proof. Recall that an object in G_{H}(P) is of the form (P^{α}, α) where α: P → P^{α} ≤ S is an isomorphism in L. Surjectivity of δ_{P} on objects follows from the surjectivity of Θ̂: Aut_{L}(S) → Γ.

Two objects (P^{α}, α) and (P^{β}, β) are in the same connected component of G_{H}(P) if and only if β ∘ α^{−1} : P^{α} → P^{β} is a morphism in L_{H}. This is the case if and only if Θ̂(β ∘ α^{−1}) ∈ H, or equivalently if and only if Θ̂(α^{−1})H = Θ̂(β^{−1})H, or in other words if and only if δ_{P} (P^{α}, α) = δ_{P} (P^{β}, β). This shows both that δ_{P} is well defined and that it is an equivalence of categories as claimed. □

We next observe that calculating limits over a discrete groupoid is particularly easy.

Lemma 4.8. Let G be a groupoid, and let M: G → Ab be a functor. Then

lim_{G} M ≅ ∏_{G_{i}} lim_{G_{i}} M ≅ ∏_{G_{i}} M(x_{i})^{Aut_{G_{i}}(x_{i})},

where the product runs over connected components G_{i} of G, and x_{i} is an arbitrary object in G_{i}. In particular if G is discrete, then

lim_{G} M ≅ ∏_{G_{i}} M(x_{i}).

Dually,

colim_{G} M ≅ ⨁_{G_{i}} M(x_{i})_{Aut_{G_{i}}(x_{i})},

where the subscript means coinvariants. In particular if G is discrete, then

colim_{G} M ≅ ⨁_{G_{i}} M(x_{i}).

Proof. The first isomorphism is clear. For the second, observe that each connected component of a groupoid is equivalent as a category to the group of automorphisms of any object in it. Thus if x_{i} is an object in G_{i}, then

lim_{G_{i}} M ≅ lim_{B(Aut_{G_{i}}(x_{i}))} M = M(x_{i})^{Aut_{G_{i}}(x_{i})},

as claimed. The claim for colimits follows at once by duality. The conclusions for discrete groupoids are clear. □

We are now ready to describe the right Kan Extension of a module along the inclusion ι: L_{H} → L. To make notation less cumbersome, we will use the symbol g to denote a left coset gH ∈ Γ/H. Also for any morphism φ in L, let t_{φ} =^{def} Θ̂(φ) ∈ Γ.

Lemma 4.9. Let M ∈ L_{H}-mod be any functor, and fix a section σ: Γ/H → Aut_{L}(S) of the projection. Let σ_{g} denote σ(g). Then the right Kan extension of M to L can be described as follows.

(a) For each P ∈ L_{H}, there is an isomorphism

Φ_{P}^{σ} : R_{ι}(M)(P) ⟶^{≅} ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}}),

given by the formula

(Φ_{P}^{σ} (\{x_{(Q,β)}\}_{(Q,β)∈P ↓ι}))_{g} = x_{(P^{σ{−1 \atop g}},σ_{g}^{−1})}.

In other words, Φ_{P}^{σ} takes a compatible family

{x_{(Q,β)}}_{(Q,β)∈P ↓ι} ∈ R_{ι}(M)(P) =^{def} lim_{P ↓ι} M_{♯} ≤ ∏_{(Q,β)∈P ↓ι} M(Q)

to the element in ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}}) whose g-th coordinate is x_{(P^{σ{−1 \atop g}},σ_{g}^{−1})}.

(b) If τ: Γ/H → Aut_{L}(S) is another section, then one has a commutative diagram

\xymatrix{ & ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}}) \ar[dd]^{∏ M(τ_{g}^{−1}∘σ_{g})}_{≅} \\ R_{ι}(M)(P) \ar@{-->}[ur]^{≅}_{Φ_{P}^{σ}} \ar@{-->}[dr]_{≅}^{Φ_{P}^{τ}} \\  & ∏_{g∈Γ/H} M(P^{τ_{g}^{−1}}).}

(c) Let φ: P → Q be a morphism in L. Then, under the identification in part (a),

the map

R_{ι}(M)(φ): R_{ι}M(P) → R_{ι}M(Q)

sends

(x_{g})_{g∈Γ/H} ∈ ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}})

to

{M(σ_{g}^{−1} ∘ φ ∘ σ_{Θ̂(φ)^{−1}g})(x_{Θ̂(φ)^{−1}g})}_{g∈Γ/H} ∈ ∏_{g∈Γ/H} M(Q^{σ_{g}^{−1}}).

Proof. For each object P ∈ L, the section σ defines an equivalence of categories t_{P}^{σ} : Γ/H → G_{H}(P) which takes an object gH to (P^{σ_{g}^{−1}}, σ_{g}^{−1}), and is a right inverse for δ_{P} : G_{H}(P) → Γ/H of Lemma 4.7. Let j_{P} : G_{H}(P) → P ↓ ι and r_{P} : P ↓ ι → G_{H}(P) denote the inclusion and retraction of Lemma 4.6 respectively.

The map Φ_{P}^{σ} is the map induced by the composite j_{P} ∘ t_{P}^{σ} : Γ/H → P ↓ ι,

R_{ι}(M)(P) =^{def} lim_{P ↓ι} M_{♯} ⟶^{j_{P}^{∗}} lim_{G_{H}(P)} M_{♯} ∘ j_{P} ⟶^{(t_{P}^{σ})^{∗}} lim_{Γ/H} M_{♯} ∘ j_{P} ∘ t_{P}^{σ} = ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}}).

Notice that j_{p} ∘ t_{P}^{σ} takes a coset gH ∈ Γ/H to the object (P^{σ_{g}^{−1}}, σ_{g}^{−1}) ∈ P ↓ ι. Thus the formula of part (a) is clear. Furthermore, the first map in the sequence above is an isomorphism since G_{H}(P) is a left deformation retract of P ↓ ι by Lemma 4.6, and the second is an isomorphism since t_{P}^{σ} is an equivalence of groupoids.

For Part (b), notice that the morphisms τ_{g}^{−1} ∘ σ_{g} define a natural isomorphism from t_{P}^{σ} to t_{P}^{τ}, and hence a natural isomorphism j_{P} ∘ t_{P}^{σ} → j_{P} ∘ t_{P}^{τ}. Thus Φ_{P}^{σ} and Φ_{P}^{τ} are naturally isomorphic, and (b) is nothing but a diagramatic realization of this natural isomorphism.

To prove part (c), notice first that for each P ∈ L the composite functor δ_{P} ∘ r_{P} is a left inverse to j_{P} ∘ t_{P}^{σ}, and since the latter induces the isomorphism Φ_{P}^{σ} of part (a), δ_{P} ∘ r_{P} induces (Φ_{P}^{σ})^{−1}. Thus to prove the statement we need to calculate the effect of Φ_{Q}^{σ} ∘ R_{ι}(M)(φ) ∘ (Φ_{P}^{σ})^{−1} on a typical element {x_{g}}_{g∈Γ/H} ∈ ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}}). By direct calculation one has for (R, β) ∈ P ↓ ι, (T, γ) ∈ Q ↓ ι, and u ∈ Γ,

• ((Φ_{P}^{σ})^{−1}({x_{g}}_{g∈Γ/H}))_{(R,β)} = M(βσ_{Θ̂(β)^{−1}})(x_{Θ̂(β)^{−1}})
• (R_{ι}(M)(φ)({M(βσ_{Θ̂(β)^{−1}})(x_{Θ̂(β)^{−1}})}_{(R,β)}))_{(T,γ)} = M(γφσ_{Θ̂(γφ)^{−1}})(x_{Θ̂)(γφ)^{−1}})
• (Φ_{Q}^{σ}({M(γφσ_{Θ̂(γφ)^{−1}})(x_{Θ̂(γφ)^{−1}})}_{(T,γ)}))_{u} = M(σ_{u}^{−1}φσ_{Θ̂(σ_{u}^{−1}φ)^{−1}})(x_{Θ̂(σ_{u}^{−1}φ)^{−1}}),

where σ_{u}^{−1} represents the restricted map Q ⟶^{σ_{u}^{−1}} Q^{σ_{u}^{−1}}. Since Θ̂(σ_{u}^{−1}φ)^{−1} = Θ̂(φ)^{−1}u, we have

(Φ_{Q}^{σ} ∘ R_{ι}(M)(φ) ∘ (Φ_{P}^{σ})^{−1}({x_{g}}_{g∈Γ/H}))_{u} = M(σ_{u}^{−1} ∘ φ ∘ σ_{Θ̂(φ)^{−1}u})(x_{Θ̂(φ)^{−1}u}),

as claimed. □

The following is an immediate corollary of Lemmas 4.9 and 3.5.

Corollary 4.10. The right Kan extension functor

R_{ι}: L_{H}-mod → L-mod

is exact. Thus, the Shapiro map

Sh_{M}: H^{∗}(L, R_{ι}(M)) → H^{∗}(L_{H}, M)

is an isomorphism for every M ∈ L_{H}-mod.

There is a dual version of both Lemma 4.9, describing the left Kan extension L_{ι}M, and Corollary 4.10, showing that L_{ι} is exact and hence ι^{∗} preserves injectives and the Shapiro lemma holds in homology. We will not state these results explicitly, but we draw the following corollary.

Corollary 4.11. The restriction functor ι^{∗}: L-mod → L_{H}-mod preserves both injective and projective resolutions.

5. Construction of the transfer for subsystems of index prime to p

Throughout this section let Γ = Γ_{p′}(F), fix a subgroup H ≤ Γ, and let ι: L_{H} → L be the inclusion. Let σ: Γ/H → Aut_{L}(S) be a fixed section. As before, for g ∈ Γ we denote by g the left coset gH ∈ Γ/H, and let σ_{g} denote σ(g).

Proposition 5.1. Let M ∈ L-mod. For each P ∈ L, let

Pre-Tr_{P} : R_{ι}(ι^{∗}M)(P) ⟶ M(P)

be the map that, under the identification of Lemma 4.9, is given by

(x_{g})_{g∈Γ/H} ↦ ∑_{g∈Γ/H} M(σ_{g})(x_{g}).

Then, the maps Pre-Tr_{P} are independent of the choice of the section σ, and assemble into a natural transformation

Pre-Tr: R_{ι}(ι^{∗}M) ⇒ M

Proof. Let τ be another section. Then by Part (b) of Lemma 4.9 (omitting ι^{∗} from the notation),

Φ_{P}^{τ} = (∏ M(τ_{g}^{−1} ∘ σ_{g})) ∘ Φ_{P}^{σ} .

Let x ∈ R_{ι}(ι^{∗}M)(P), and let (x_{g})_{g} and (x′_{g})_{g} denote Φ_{P}^{σ} (x) and Φ_{P}^{τ} (x) respectively. Then

(x′_{g})_{g} = (M(τ_{g}^{−1} ∘ σ_{g})(x_{g}))_{g}.

Thus using the section τ to define Pre-Tr gives the map

(x′_{g})_{g∈Γ/H} ↦ ∑_{g∈Γ/H} M(τ_{g})(x′_{g}) = ∑_{g∈Γ/H} M(τ_{g})(M(τ_{g}^{−1} ∘ σ_{g})(x_{g})) = ∑_{g∈Γ/H} M(σ_{g})(x_{g}),

which agrees with the map defined using σ.

It remains to show that Pre-Tr is natural, meaning that for P, Q ∈ L and φ ∈ Mor_{L}(P, Q), the square

\xymatrix{R_{ι}(ι^{∗}M)(P) \ar[r]^{^{Pre-Tr^{P}}} \ar[d]_{R_{ι}(ι^{∗}M)(φ)} & M(P) \ar[d]^{M(φ)} \\ R_{ι}(ι^{∗}M)(Q) \ar[r]^{_{Pre-Tr_{Q}}} & M(Q)}

commutes. Let x ∈ R_{ι}(ι^{∗}M)(P), and let (x_{g})_{g} = Φ_{P}^{σ} (x) ∈ ∏_{g∈Γ/H} M(P^{σ_{g}^{−1}}), as in Lemma

4.9. Then

M(φ) ∘ Pre-Tr_{P} (x) = ∑_{g∈Γ/H} M(φ ∘ σ_{g})(x_{g}),

while, by part (c) of Lemma 4.9,

Pre-Tr_{Q} ∘R_{ι}(ι^{∗}M)(φ)(x) = ∑_{g∈Γ/H} M(σ_{g}) (M(σ_{g}^{−1} ∘ φ ∘ σ_{Θ̂(φ)^{−1}g})(x_{Θ̂(φ)^{−1}g}))

= ∑_{g∈Γ/H} M(φ ∘ σ_{Θ̂(φ)^{−1}g})(x_{Θ̂(φ)^{−1}g}).

Thus the two ways of composition in the square coincide. □

The natural transformation Pre-Tr induces a map on higher limits, which we also denote by Pre-Tr. Composing this with the inverse of the Shapiro isomorphism we obtain our transfer.

Definition 5.2. For any subgroup H ≤ Γ and a functor M ∈ L-mod the associated transfer, is the map

Tr: H^{∗}(L_{H}, ι^{∗}M) ⟶^{Sh_{M}^{−1}}≅ H^{∗}(L, R_{ι}(ι^{∗}M)) ⟶^{Pre-Tr} H^{∗}(L, M).

Since the transfer is defined as a natural transformation of coefficient systems it is of course independent of the choice of resolutions used to compute cohomology.

5.1. Cochain-level Description. Just as in the classical group case, the transfer 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 5.3. Let M and N be coefficient systems on L, and let K be a subgroup of Γ. Let ι_{K}: L_{K} → L denote the inclusion. For a natural transformation ϕ ∈ Hom_{L_{K}-mod}(ι_{K}^{∗}N, ι_{K}^{∗}M), and α ∈ Aut_{L}(S), let ϕ^{α} be the map which associates with an object P ∈ L the homomorphism of R-modules ϕ_{P}^{α}: N(P) → M(P) given by

ϕ_{P}^{α} := M(α) ∘ ϕ_{α^{−1}(P)} ∘ N(α^{−1}) ∈ Hom_{R}(N(P), M(P)).

We refer to this construction as conjugation. Notice that ϕ^{α} needs not be a natural transformation in general. In the case where (ϕ^{α}) is a natural transformation, so (ϕ^{α})^{β} is defined, it follows immediately from the definition that (ϕ^{α})^{β} = ϕ^{(β∘α)}, the order being reversed since we write α on the right. The following lemma investigates other basic properties of ϕ^{α}, and in particular provides a condition under which it is a natural transformation.

Lemma 5.4. Let L, M and N be coefficient systems on L, let K, H ≤ Γ, and let α ∈ Aut_{L}(S).

(a) If

ι_{K}^{∗}N ⟶^{ϕ} ι_{K}^{∗}M ⟶^{η} ι_{K}^{∗}L

is a composable sequence in L_{K}-mod then

(η ∘ ϕ)^{α} = η^{α} ∘ ϕ^{α}.

(b) If ϕ ∈ Hom_{L_{K}-mod}(ι_{K}^{∗}N, ι_{K}^{∗}M), then for every α ∈ Aut_{L_{K}}(S),

ϕ^{α} = ϕ.

(c) If Θ̂(α)^{−1} ∈ N_{Γ}(H, K) and ϕ ∈ Hom_{L_{K}-mod}(ι_{K}^{∗}N, ι_{K}^{∗}M), then ϕ^{α} is a natural transformation ι_{H}^{∗}N → ι_{H}^{∗}M, and the map which associates ϕ^{α} with ϕ is a homomorphism

c_{α}^{∗}: Hom_{L_{K}-mod}(ι_{K}^{∗}N, ι_{K}^{∗}M) ⟶ Hom_{L_{H}-mod}(ι_{H}^{∗}N, ι_{H}^{∗}M).

Proof. For P ∈ L and α ∈ Aut_{L}(S),

(η ∘ ϕ)_{P}^{α} =^{def} L(α) ∘ (η ∘ ϕ)_{α^{−1}(P)} ∘ N(α^{−1}) =

L(α) ∘ η_{α^{−1}(P)} ∘ M(α^{−1}) ∘ M(α) ∘ ϕ_{α^{−1}(P)} ∘ N(α^{−1}) = η_{P}^{α} ∘ ϕ_{P}^{α} .

This proves part (a).

Part (b) is immediate, since by assumption ϕ is natural with respect to any morphism in L_{K}, and in particular α. Therefore, for any object Q,

ϕ_{Q}^{α} = M(α) ∘ ϕ_{α^{−1}(Q)} ∘ N(α^{−1}) = M(α) ∘ M(α^{−1}) ∘ ϕ_{Q} = ϕ_{Q}.

It remains to prove part (c). Let φ: P → Q be a morphism in L_{H}. Then Θ̂(φ) ∈ H, so by assumption on α we have

Θ̂(α^{−1} ∘ φ ∘ α) = Θ̂(α)^{−1} ∘ Θ̂(φ) ∘ Θ̂(α) ∈ K,

so

α^{−1} ∘ φ ∘ α ∈ Mor_{L_{K}}(α^{−1}(P), α^{−1}(Q)).

Thus,

ϕ_{Q}^{α} ∘ N(φ) = M(α) ∘ ϕ_{α^{−1}(Q)} ∘ N(α^{−1}) ∘ N(φ)
= M(α) ∘ ϕ_{α^{−1}(Q)} ∘ N(α^{−1} ∘ φ ∘ α) ∘ N(α^{−1})
= M(α) ∘ M(α^{−1} ∘ φ ∘ α) ∘ ϕ_{α^{−1}(P)} ∘ N(α^{−1})
= M(φ) ∘ M(α) ∘ ϕ_{α^{−1}(P)} ∘ N(α^{−1})
= M(φ) ∘ ϕ_{P}^{α} .

This shows that ϕ^{α} is a natural transformation. That the map c_{α}^{∗} is a homomorphism follows directly from the definition. □

Remark 5.5. A consequence of part (b) of Lemma 5.4 is that if α, α′ ∈ Aut_{L}(S) with Θ̂(α) = Θ̂(α′) then c_{α}^{∗} = c_{α′}^{∗}, since α^{−1} ∘ α′ ∈ Aut_{L_{H}}(S) for any subgroup H ≤ Γ. Therefore we will often write c_{x}^{∗} for x ∈ Γ to denote the conjugation induced by any α ∈ Aut_{L}(S) with Θ̂(α) = x, and write ϕ^{x} for c_{x}^{∗}(ϕ). When restricted to Hom_{L_{H}-mod}(ι_{H}^{∗}N, ι_{H}^{∗}M), the conjugation c_{α}^{∗} only depends on the coset Θ̂(α)H. Therefore we will also write c_{x}^{∗} and ϕ^{x} for x ∈ Γ/H. Finally, we also write c_{x}^{∗} and ϕ^{x} for x ∈ K\Γ/H when the maps involved are independent of the choice of representative for double cosets.

Corollary 5.6. Let M be a coefficient system on L with injective resolution M → I_{•} and let K, H ≤ Γ. For α ∈ Aut_{L}(S) such that Θ̂(α)^{−1} ∈ N_{Γ}(H, K), the map induced by conjugation

c_{α}^{∗}: Hom_{L_{K}-mod}(ι_{K}^{∗}R, ι_{K}^{∗}I_{•}) ⟶ Hom_{L_{H}-mod}(ι_{H}^{∗}R, ι_{H}^{∗}I_{•})

is a map of cochain complexes, and the induced map on cohomology

c_{α}^{∗}: H^{∗}(L_{K}, ι_{K}^{∗}M) ⟶ H^{∗}(L_{H}, ι_{H}^{∗}M).

is independent of the choice of the injective resolution I_{•}.

Proof. First recall that ι_{H}^{∗} and ι_{K}^{∗} preserve injective resolutions by Corollary 4.11. Now, denote the differential of I_{•} by ∂. Then, by Lemma 5.4(b), ∂^{α} = ∂ for any α ∈ Aut_{L}(S). The differentials of the cochain complexes Hom_{L_{K}-mod}(ι_{K}^{∗}R, ι_{K}^{∗}I_{•}) and Hom_{L_{H}-mod}(ι_{H}^{∗}R, ι_{H}^{∗}I_{•}) are both given by composition with ∂, and will be denoted δ_{K} and δ_{H} respectively. For ϕ ∈ Hom_{L_{K}-mod}(ι_{K}^{∗}R, ι_{K}^{∗}I_{•}) we therefore have

c_{α} ∘ δ_{K}(ϕ) = c_{α}(∂ ∘ ϕ) = ∂^{α} ∘ ϕ^{α} = ∂ ∘ ϕ^{α} = δ_{H} ∘ c_{α}(ϕ).

This shows that c_{α} ∘ δ_{K} = δ_{H} ∘ c_{α}, so c_{α} is a cochain map.

The proof of independence on choice of injective resolutions is routine. □

Lemma 5.7. Let M and N be coefficent systems on L, and let H ≤ Γ. If ϕ ∈ Hom_{L_{H}-mod}(ι_{H}^{∗}N, ι_{H}^{∗}M) then

∑_{g∈Γ/H} ϕ^{g} ∈ Hom_{L-mod}(N, M).

Proof. We need to show naturality with respect to morphisms in L. So let ψ: P → Q be a morphism in L. Fix a section σ: Γ/H → Aut_{L}(S), and let σ_{g} denote σ(g), as before. For each g ∈ Γ/H, let ψ_{g} denote the composite

σ_{Θ̂(ψ)^{−1}g}^{−1}(P) ⟶^{σ_{Θ̂(ψ)^{−1}g}} P ⟶^{ψ} Q ⟶^{σ_{g}^{−1}} σ_{g}^{−1}(Q),

where the appropriate restrictions on σ_{g} and σ_{Θ̂(ψ)^{−1}g}^{−1} are understood. Observe that

Θ̂(ψ_{g}) ∈ H, so ψ_{g} ∈ Mor_{L_{H}}(σ_{Θ̂(ψ)^{−1}g}^{−1}(P), σ_{g}^{−1}(Q)).

Now,

∑_{g∈Γ/H} ϕ_{Q}^{g} ∘ N(ψ) = ∑_{g∈Γ/H} M(σ_{g}) ∘ ϕ_{σ_{g}^{−1}(Q)} ∘ N(σ_{g}^{−1}) ∘ N(ψ)

= ∑_{g∈Γ/H} M(σ_{g}) ∘ ϕ_{σ_{g}^{−1}(Q)} ∘ N(ψ_{g}) ∘ N(σ_{Θ̂(ψ)^{−1}g}^{−1})

= ∑_{g∈Γ/H} M(σ_{g}) ∘ M(ψ_{g}) ∘ ϕ_{σ_{Θ̂(ψ)^{−1}g}^{−1}(P)} ∘ N(σ_{Θ̂(ψ)^{−1}g}^{−1})

= ∑_{g∈Γ/H} M(ψ) ∘ M(σ_{Θ̂(ψ)^{−1}g}) ∘ ϕ_{σ_{Θ̂(ψ)^{−1}g}^{−1}(P)} ∘ N(σ_{Θ̂(ψ)^{−1}g}^{−1})

= M(ψ) ∑_{g∈Γ/H} ϕ_{P}^{Θ̂(ψ)^{−1}g}.

= M(ψ) ∑_{g∈Γ/H} ϕ_{P}^{g} .

The first and the fifth equalities follows from the definition of ϕ^{g} (see Remark 5.5), the second and the fourth from the definition ψ_{g}, the third from naturality of ϕ, and the sixth is clear. This shows naturality and proves the claim. □

Definition 5.8. For a coefficient system M on L, a subgroup H ≤ Γ and an injective resolution M → I_{•}, the associated cochain-level transfer is the cochain map

Tr_{H}: Hom_{L_{H}}(ι_{H}^{∗}R, ι_{H}^{∗}I_{•}) ⟶ Hom_{L}(R, I_{•})

given by

Tr_{H}(ϕ) = ∑_{g∈Γ/H} ϕ^{g}.

Since T r_{H} is a sum of conjugations, it follows by Corollary 5.6 that it is indeed a cochain map, and that the induced map on cohomology is independent of the choice of injective resolution.

Proposition 5.9. For a coefficient system M on L and a subgroup H of Γ, the map

H^{∗}(L_{H}, ι^{∗}M) ⟶ H^{∗}(L, M)

induced by the cochain-level transfer T r_{H} of Definition 5.8 coincides with the transfer associated to the same data, as defined in 5.2.

Proof. Let ι: L_{H} → L denote the inclusion. If M → I_{•} is an injective resolution, then, by Corollary 4.11, ι^{∗}M → ι^{∗}I_{•} is an injective resolution. Thus the Shapiro isomorphism Sh_{M} is given on the cochain-level by the (ι^{∗}, R_{ι})-adjunction isomorphism

ρ: Hom_{L-mod}(R, R_{ι}(ι^{∗}I_{•})) ⟶^{≅} Hom_{L_{H}-mod}(ι^{∗}R, ι^{∗}I_{•}).

(cf. Subsection 3.3). Hence its inverse Sh_{M}^{−1} is induced by ρ^{−1}, whose value on ϕ ∈ Hom_{L_{H}-mod}(ι^{∗}R, ι^{∗}I_{•}) is the natural transformation which takes an object Q ∈ L to the morphism ρ^{−1}(ϕ)_{Q} given by the composite

R = R(Q) ⟶^{∏_{g∈Γ/H} R(σ_{g}^{−1})} ∏_{g∈Γ/H} R(Q^{σ_{g}^{−1}}) ⟶^{∏_{g∈Γ/H} ϕ_{σ{−1 \atop g}(Q)}} ∏_{g∈Γ/H} I_{•}(Q^{σ_{g}^{−1}}) ≅ R_{ι}(ι^{∗}I_{•})(Q).

Therefore the composite

Hom_{L_{H}-mod}(ι^{∗}R, ι^{∗}I_{•}) ⟶^{ρ_{M}^{−1}}≅ Hom_{L-mod}(R, R_{ι}(ι^{∗}I_{•})) ⟶^{Pre-Tr} Hom_{L}(R, I_{•}),

which induces the transfer, is given by

ϕ ↦ ∑_{g∈Γ/H} I_{•}(σ_{g}) ∘ ϕ_{σ_{g}^{−1}(Q)} ∘ R(σ_{g}^{−1}) = ∑_{g∈Γ/H} ϕ^{σ_{g}} = Tr_{H}(ϕ).

□

5.2. Geometric Interpretation. We end this section by verifying that for a locally constant system of coefficients M ∈ L-mod, and a subgroup L_{H} ≤ L of index prime to p, the transfer defined in this section coincides with the map defined in Section 4.2.

Let ι: L_{H} → L denote the inclusion. Let Γ = π_{1}(|L|, S) and Γ′ = π_{1}(|L_{H}|, S). Then, by Theorem 2.5(v), the map induced by ι on nerves is a covering space, up to homotopy, with homotopy fibre Γ/Γ′ ≅ Γ_{p′}(F)/H. Construct π: L_{Γ′} → L as in Section 3.6. Then, by Lemma 3.11(i), there is a functor ι̂: L_{H} → L_{Γ′}, such that π ∘ ι̂= ι, and by Part (ii) of the same lemma, for any locally constant M ∈ L-mod, there is a natural isomorphism induced by ι̂,

ι̂_{∗}: R_{π}(π^{∗}M) ⟶^{≅} R_{ι}(ι^{∗}M).

Choose a section σ: Γ/H → Aut_{L}(S) of the projection, and construct Φ^{σ} as in Lemma 4.9. Then for any M ∈ L-mod and P ∈ L, we claim that the diagram

[figure: R_{π}(π^{∗}M)(P); P r; ι̂_{∗}; ∏_{Γ/Γ′} M(P); Σ; ∏ M(σ_{g})^{−1}; M(P); R_{ι}(ι^{∗}M)(P); Φ_{P}^{σ}; ∏_{g∈Γ/Γ′} M(P^{σ_{g}^{−1}}); Σ^{σ}; M(P)]

commutes. Here the map P r projects a compatible family of elements in the source onto the coordinates of the respective initial objects, i.e., to the coordinates of the objects of the form ((P, gΓ′), 1_{P}). The map Σ is the sum of coordinates, where Σ^{σ} is the twisting of the sum of coordinates map by the M(σ_{g}), as in the construction of Pre-Tr_{P} in Proposition 5.1. The right square clearly commutes, while the verification that the left square commutes only involves checking all the relevant definitions.

Since the map T_{P} of Proposition 3.13 is the top composition, and the map Pre-Tr_{P} is the bottom composition, it follows that one has a commutative diagram in cohomology

[figure: H^{∗}(L_{Γ′}, π^{∗}M); ≅; ι^{∗}; ≅; H^{∗}(L, R_{π}(π^{∗}M)); T_{∗}; ι̂_{∗}; ≅; H^{∗}(L, M); H^{∗}(L_{H}, ι^{∗}M); ≅; H^{∗}(L, R_{ι}(ι^{∗}M)); _{Pre-Tr_{∗}}; H^{∗}(L, M).]

Here the top composition is the transfer associated to the covering |L_{H}| → |L| by Proposition 4.5, while the bottom composition is our algebraic transfer.

6. Standard Consequences

In this final section we show that the transfer constructed here for subsystems of index prime to p satisfies all the standard properties one expects to have in ordinary group cohomology. Throughout this section Γ = Γ_{p′}(F), and we let Θ̂: L → B(Γ) denote the projection functor, as before.

6.1. Transfer among Subgroups. We have defined a transfer associated to the inclusion L_{H} ⊆ L for a subgroup H ≤ Γ. Just like in the group case, this construction can be generalized to obtain a transfer associated to the inclusion L_{H} ⊆ L_{K} for subgroups H ≤ K ≤ Γ. Indeed, one can simply take L_{K} in place of L in the construction of the transfer. The transfer associated to the inclusion ι_{H}^{K}: L_{H} ⊆ L_{K} can then be described on the cochain level as follows. Let M be a coefficient system on L_{K}, and let M → I_{•} be an injective resolution of M. A cochain-level transfer

Tr_{H}^{K}: Hom_{L_{H}}(ι_{H}^{∗}R, (ι_{H}^{K})^{∗}I_{•}) ⟶ Hom_{L_{K}}(ι_{K}^{∗}R, I_{•})

is given by

Tr_{H}^{K}(ϕ) = ∑_{x∈K/H} ϕ^{x},

as in Definition 5.8. This formula makes it clear that Tr_{H} = Tr_{H}^{K} ∘ Tr_{K} for subgroups H ≤ K ≤ Γ.

6.2. Normalization. One of the most basic properties of the standard cohomology transfer is that the restriction to the cohomology of a subgroup followed by the transfer is given by multiplication by the index. This is exactly the case in our more general context.

Proposition 6.1. Let (S, F, L) be a p-local finite group, and let H be a subgroup of Γ = Γ_{p′}(F). Let M be a system of coefficients for (S, F, L). Then the composite

H^{∗}(L, M) ⟶^{Res} H^{∗}(L_{H}, ι^{∗}M) ⟶^{T r} H^{∗}(L, M)

is given by multiplication by the index |Γ: H|.

Proof. Consider the diagram

\xymatrix{H^{∗}(L, M) \ar[r]^{Res} \ar[dr]^{(δ_{M})_{∗}} & H^{∗}(L_{H}, ι^{∗}M) \ar[d]^{Sh_{M}^{−1}}_{≅} \ar[r]^{T r_{H}} & H^{∗}(L, M) \\  & H^{∗}(L, R_{ι}(ι^{∗}M)), \ar[ur]^{Pre-Tr}}

where the right triangle commutes by definition of the transfer, and where the morphism δ_{M}: M → R_{ι}(ι^{∗}M) is the unit of the (ι^{∗}, R_{ι})-adjunction evaluated at the object M. Notice that under the identification of Lemma 4.9(a), δ_{M}(x) = {M(σ_{g})^{−1}(x)}_{g∈Γ/H}.

The left triangle is commutative by construction of the Shapiro isomorphism (see Subsection 3.3). To see this, notice that Sh_{M}^{−1} is induced by the adjunction isomorphism

ρ: Hom_{L_{H}-mod}(R_{L_{H}}, ι^{∗}I_{•}) = Hom_{L_{H}-mod}(ι^{∗}R_{L}, ι^{∗}I_{•}) → Hom_{L-mod}(R_{L}, R_{ι}(ι^{∗}I_{•})), where I_{•} is an injective resolution of M in L-mod. The adjunction, in turn, applied to a natural transformation ψ: ι^{∗}R_{L} → ι^{∗}I_{•} is the composite

R_{L} ⟶^{δ_{R_{L}}} R_{ι}(ι^{∗}R_{L}) ⟶^{R_{ι}ψ} R_{ι}(ι^{∗}I_{•}).

By naturality of δ, for any natural transformation φ: R_{L} → I_{•},

δ_{I_{•}} ∘ φ = R_{ι}(ι^{∗}φ) ∘ δ_{R_{L}}.

This proves commutativity of the left triangle, and therefore it suffices to show that the composite Pre-Tr ∘(δ_{M})_{∗} is given by multiplication by |Γ : H|. This map is induced by a natural transformation of coefficients

M ⟶^{δ_{M}} R_{ι}(ι^{∗}M) ⟶^{Pre-Tr} M,

which is given element-wise by

x ↦ {M(σ_{g})^{−1}(x)}_{g∈Γ/H} ↦ ∑_{g∈Γ/H} M(σ_{g})(M(σ_{g}^{−1})(x)) = |Γ : H| · x

for P ∈ L and x ∈ M(P). □

6.3. Double Coset Formula. We now show that the transfer map constructed here satisfies a double coset formula which is essentially identical to that which holds for ordinary group cohomology.

Proposition 6.2. Let (S, F, L) be a p-local finite group, and let H and K be subgroups of Γ = Γ_{p′}(F). Then the composite

H^{∗}(L_{K}, ι_{K}^{∗}M) ⟶^{T r_{K}} H^{∗}(L, M) ⟶^{Res_{H}} H^{∗}(L_{H}, ι_{H}^{∗}M)

is given by

Res_{H} ∘Tr_{K} = ∑_{x∈H\Γ/K} Tr_{H∩xKx^{−1}}^{H} ∘ c_{x}^{∗} ∘ Res_{x^{−1}Hx∩K}^{K} .

Proof. Using the cochain-level description of the transfer in Section 5.1, 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 Γ = ∐_{i} Hx_{i}K be a double coset decomposition of Γ with respect to K and H. For each i, put

V_{i} =^{def} x_{i}^{−1}Hx_{i} ∩ K and W_{i} =^{def} H ∩ x_{i}Kx_{i}^{−1},

and let H = ∐_{j} z_{ji}W_{i} be a left coset decomposition of H with respect to W_{i}. Then one can rewrite the right K-coset decomposition of Γ as

Γ = ∐_{i,j} z_{ji}x_{i}K.

Let M → I_{•} be an injective resolution of M in L-mod. Since x_{i}^{−1} ∈ N_{Γ}(W_{i}, V_{i}) we have induced chain maps

c_{x_{i}}^{∗}: Hom_{L_{V_{i}}-mod}(ι_{V_{i}}^{∗}R_{L}, ι_{V_{i}}^{∗}I_{•}) ⟶ Hom_{L_{W_{i}}-mod}(ι_{W_{i}}^{∗}R_{L}, ι_{W_{i}}^{∗}I_{•}),

as in Corollary 5.6. Let ϕ ∈ Hom_{L_{K}-mod}(ι_{K}^{∗}R_{L}, ι_{K}^{∗}I_{•}). Then

Res_{H} ∘Tr_{K}(ϕ) = Res_{H} ( ∑_{g∈Γ/K} ϕ^{g})

= Res_{H} (∑_{i} (∑_{j} ϕ^{(z_{ji}x_{i})})) ,

and

∑_{i} Tr_{W_{i}}^{H} ∘ c_{x_{i}}^{∗} ∘ Res_{V_{i}}^{K}(ϕ) = ∑_{i} (∑_{j} (c_{x_{i}}^{∗}(Res_{V_{i}}^{K}(ϕ)))^{z_{ij}})

= ∑_{i} (∑_{j} (Res_{V_{i}}^{K}(ϕ))^{(z_{ji}x_{i})}) .

As we can ignore restrictions when determining the effect of these natural transformations, 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 follows. □

6.4. A Stable Elements Theorem. In standard group cohomology, the availability of a transfer and a double coset formula allows one to prove that the cohomology 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-local finite group cohomology with p-local constant coefficients in [BLO2, Section 5]), and generalized to any stably representable non-equivariant cohomology theory in [Ra] (see also [CM]). We now show that the existence of a transfer and a double coset formula 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 Γ = Γ_{p′}(F).

Definition 6.3. Let H ≤ K ≤ Γ, and let M ∈ L-mod be a system of coefficients. An element x ∈ H^{∗}(L_{H}, ι_{H}^{∗}(M)) is K-stable if for every subgroup U ≤ H and every g ∈ K with g^{−1} ∈ N_{K}(U, H),

c_{g}^{∗} ∘ Res_{gUg^{−1}}^{H}(x) = Res_{U}^{H}(x) ∈ H^{∗}(L_{U}, ι_{U}^{∗}(M)).

Lemma 6.4. Let H ≤ K ≤ Γ, and let M be a system of coefficients on L. The Kstable elements in H^{∗}(L_{H}, ι_{H}^{∗}(M)) form a submodule. Furthermore, if A is a system of ring coefficients, then the submodule of K-stable elements in H^{∗}(L_{H}, ι_{H}^{∗}(A)) is a subring.

Proof. This follows at once from the definition of stable elements, and the fact that the maps induced by restriction and conjugation are module maps or, in the case of ring coefficients, ring maps. □

Lemma 6.5. Let H ≤ K ≤ Γ, and let M ∈ L-mod be a system of coefficients. If x ∈ H^{∗}(L_{K}, ι_{K}^{∗}(M)) then Res_{H}^{K}(x) is K-stable.

Proof. This follows from part (b) of Lemma 5.4. □

We are now ready to state the Stable Elements Theorem for p-local finite groups.

Theorem 6.6. Let Γ = Γ_{p′}(F), let H ≤ K ≤ Γ, and let M ∈ L-mod be a p-local system of coefficients. Then the restriction homomorphism

Res_{H}^{K}: H^{∗}(L_{K}, ι_{K}^{∗}(M)) ⟶ H^{∗}(L_{H}, ι_{H}^{∗}(M))

is a split injection whose image is the submodule of K-stable elements.

Proof. By Proposition 6.1, the composite Tr_{H}^{K} ∘ Res_{H}^{K} acts on H^{∗}(L_{K}, ι_{K}^{∗}(M)) as multiplication by the index |K: H|. Since this index is prime to p, the map t =^{def} |K: H|^{−1} · Tr_{H}^{K} is a left inverse for Res_{H}^{K}, proving split injectivity. By Lemma 6.5 the image of Res_{H}^{K} lies in the submodule of stable elements. Conversely, if a ∈ H^{∗}(L_{H}, ι_{H}^{∗}(M)) is K-stable, then the double coset formula gives us

Res_{H}^{K} ∘Tr_{H}^{K}(a) = ∑_{x∈H\K/H} Tr_{H∩xHx^{−1}}^{H} ∘ c_{x}^{∗} ∘ Res_{x^{−1}Hx∩H}^{H}(a)

= ∑_{x∈H\K/H} Tr_{H∩xHx^{−1}}^{H} ∘ Res_{H∩xHx^{−1}}^{H}(a)

= ∑_{x∈H\K/H} |H: H ∩ xHx^{−1}| a

= |K: H| a,

where the equality

∑_{x∈H\K/H} |H: H ∩ x^{−1}Hx| = |K: H|

is obtained as in the first step of the proof of the double coset formula. It follows that

x = Res_{H}^{K} (Tr_{H}^{K}(|K: H|^{−1} x))

is in the image of Res_{H}^{K}. □

6.5. Frobenius Reciprocity. Here we show that the transfer and restriction maps for p-local finite groups satisfy the standard Frobenius reciprocity formula.

Proposition 6.7. Let (S, F, L) be a p-local finite group, and let Γ be Γ_{p}(F) or Γ_{p′}(F). Let H ≤ K ≤ Γ, and let A be a system of ring coefficients on L. For x ∈ H^{∗}(L_{K}, ι_{K}^{∗}A) and y ∈ H^{∗}(L_{H}, ι_{H}^{∗}A) we have

Tr_{H}^{K}(Res_{H}^{K}(x) y) = x Tr_{H}^{K}(y).

Proof. We prove this at the cochain level. Let P_{•} → R be a projective resolution of the constant functor on L, and let ϕ ∈ Hom_{L_{K}-mod}(ι_{K}^{∗}P_{•}, ι_{K}^{∗}A), (representing x) and ψ ∈ Hom_{L_{H}-mod}(ι_{H}^{∗}P_{•}, ι_{H}^{∗}A) (representing y). Then

Tr_{H}^{K}(Res_{H}^{K}(ϕ) ψ) = ∑_{g∈K/H}(Res_{H}^{K}(ϕ) ψ)^{g} = ∑_{g∈K/H} Res_{H}^{K}(ϕ)^{g} ψ^{g}

= ∑_{g∈K/H} Res_{H}^{K}(ϕ) ψ^{g} = ϕ ∑_{g∈K/H} ψ^{g}

= ϕ Tr_{H}^{K}(ψ),

where we have used the equality Res_{H}^{K}(ϕ)^{g} = Res_{H}^{K}(ϕ^{g}) = Res_{H}^{K}(ϕ) for g ∈ K obtained in Lemma 5.4 (b). The result follows by passing to cohomology. □

References

[AM] A. Adem and R.J. Milgram Cohomology of Finite Groups, Grundlehren der Mathematischen Wissenschaften, 309. Springer-Verlag, Berlin, 1994.

[5A1] C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver, Subgroup families 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 and B. Oliver, Extensions of p-local finite groups, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3791–3858.

[BLO1] C. Broto, R. Levi and B. Oliver, Homotopy equivalences of p-completed classifying spaces of finite groups, Invent. Math. 151 (2003), no. 3, 611–664.

[BLO2] C. Broto, R. Levi and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779-856.

[CE] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956.

[CM] N. Castellana and L. Morales, Vector bundles over classifying spaces of p-local finite groups, preprint.

[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. Lecture 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, Aberdeen AB24 3UE, U.K.

  E-mail address: r.levi@abdn.ac.uk

Department of Mathematical Sciences, Depaul University, 2320 N. Kenmore Avenue, Chicago, IL 60614, USA

  E-mail address: kragnars@math.depaul.edu