HIGHER LIMITS VIA SUBGROUP COMPLEXES

JESPER GRODAL

1. Introduction

In this paper we give a new finite model for lim_{D}^{∗} F, the higher derived functors of the inverse limit functor, for a general functor F : D → Z_{(p)}- mod, when D is one of the three standard categories which occur in connection with homology decompositions the classifying space BG of a finite group G. More precisely we consider the (opposite) C-orbit category O_{C}^{op}, associated with the subgroup decomposition, the C-conjugacy category A_{C}, associated with the centralizer decomposition, and the (opposite) orbit simplex category ((sd C)/G)^{op} associated with the normalizer decomposition [27, 29]; we obtain similar results for all three categories, but will in this introduction concentrate on O_{C}^{op}, which we define shortly.

The interest in this is at least twofold. In homotopy theory the groups lim_{D}^{∗} F occur in the E_{2}-term of fundamental spectral sequences, such as the Bousfield-Kan homotopy (or cohomology) spectral sequence of a homotopy limit (or colimit)

E_{2}^{i,j} = lim_{D}^{−i}π_{j}(X(d)) ⇒ π_{i+j}(holim_{D} X(d))

where X is a functor D → (pointed spaces) [18] (or the corresponding spectral sequence for unpointed spaces [17]). It is hence important for the homotopy theory of classifying spaces to be able to calculate these groups, for example for finding homology decompositions [27, 29], calculating maps between classifying spaces [39], and settling uniqueness issues [47].

A second case of interest is in group theory where F typically is a Mackey functor such as group cohomology. Here one can often show that the higher limits vanish for ∗ > 0 for general reasons (cf. [38, 68, 29] and Section 8). In this seemingly trivial case the formulas for lim_{D}^{∗} F in fact specialize to give interesting expressions, in the form of finite exact sequences, for the ‘stable elements’ lim_{D}^{0} F.

By a collection we mean a set C of subgroups in G, closed under conjugation in G. We can view a collection as a poset under inclusion of subgroups, and the nerve |C| of this poset is what is called the subgroup complex associated to the collection C. Obviously it is a finite (ordered) G-simplicial complex. These subgroup complexes play an important role in finite group theory, where they are thought of as generalized buildings (or geometries) of G.

We show that for many collections C, such as the collection of all non-trivial p-subgroups, higher limits lim_{D}^{∗} F over any of the three categories can be interpreted as Bredon equivariant cohomology of |C| with values in a G-local coefficient system (or presheaf) constructed from F. Here a G-local coefficient system is understood in the sense of Ronan-Smith [52, 53, 10], which is close to the original definition of Bredon [19], but is more general than what is usually called a (Bredon) coefficient system, and this is a key generalization which the result depends upon. This result reveals a finite model for these higher limits, which is a large improvement over earlier Bredon cohomology models for higher limits over O_{C}^{op} and A_{C}, which have been infinite. It furthermore reveals a close link between higher limits and modern finite group theory, and gives an easy systematic way to see which subgroups can be omitted from C without changing the higher limits.

Date: November 12, 1999. Last minor revisions: February 5, 2000.

1991 Mathematics Subject Classification. Primary: 55R35; Secondary: 18G10, 55R37, 20J06, 55N91.

Key words and phrases. inverse limit, subgroup complex, Steinberg complex, orbit category, Quillen category.

We now proceed to give a more detailed description of our results in the case of O_{C}^{op}. In order to do so, we first need to introduce some notation. Let S_{p}(G) and B_{p}(G) denote the collection of non-trivial p-subgroups and p-radical subgroups respectively. Recall that a p-radical subgroup of G is a p-subgroup P with the property that O_{p}(NP/P) = e. (Here O_{p}(·) means largest normal p-subgroup.) We use the notation S_{p}^{e}(G) etc. to denote the same posets as above, but now not excluding the trivial subgroup e.

Let d̄ C denote the opposite category of the simplex category of |C|. The objects are the simplices of |C|, i.e., chains of subgroups (H_{0} ≤ ⋯ ≤ H_{n}), H_{i} ∈ C, and the morphisms are generated by the opposite of face and degeneracy maps. Let (d̄ C)_{G} be the category that has the same objects as d̄ C but where a morphism σ → σ′ is a pair (f : gσ → σ′, g), where f is a morphism in d̄ C and g ∈ G, and morphisms compose via (f, g) ∘ (f′, g′) = (f ∘ (gf′), gg′). (See Section 2.1.) By a G-local coefficient system (over Z_{(p)}) on |C| we mean a (covariant) functor F : (d̄ C)_{G} → Z_{(p)}- mod.

The orbit category O(G) is defined to be the category with objects G-spaces G/H, for all subgroups H of G, and morphisms are G-maps. The C-orbit category O_{C} is defined to be the full subcategory of O(G) with objects G/H for H ∈ C. For functors on O(G) and its subcategories we write F (H) for F evaluated on G/H.

Given a functor F : O_{C}^{op} → Z_{(p)}- mod we equip |C| with the G-local coefficient system F given by precomposing with the canonical map (d̄ C)_{G} → (C^{op})_{G} → O_{C}^{op} (see Remark 2.2). On objects we have F(H_{0} ≤ ⋯ ≤ H_{n}) = F (H_{0}).

Theorem 1.1. Let C be any collection of p-subgroups in a finite group G, closed under passage to p-radical p-supergroups. Let F : O_{C}^{op} → Z_{(p)}- mod be any functor, and let F be the G-local coefficient system on |C| given by precomposing F with (d̄ C)_{G} → (C^{op})_{G} → O_{C}^{op}. Then

lim_{O_{C}} ^{∗}F ≃ H_{G}^{∗}(|C|; F)

Furthermore, for any collection C′ satisfying C ∩ B_{p}^{e}(G) ⊆ C′ ⊆ C, the inclusion |C′| → |C| induces an isomorphism H_{G}^{∗}(|C|; F) → H_{G}^{∗}(|C′|; F), where F is a coefficient system on |C′| by restriction.

If F is a functor concentrated on conjugates of a single subgroup P, then

lim_{O_{C}} ^{i}F ≃ H^{i−1}(Hom_{NP/P} (St_{∗}(NP/P), F (P))).

Here St_{∗}(W) denotes the Steinberg complex of W, the reduced normalized chain complex C̃_{∗}(|S_{p}(W)|; Z_{(p)}). The result that one only needs the p-radical subgroups for calculating the higher limits is due to Jackowski-McClure-Oliver [39]. They used a Bredon cohomology model ([45],[40] see Prop. 2.10) but the space was infinite, and hence hard to use directly for calculations.

By the work of Webb ([67, 68], see Theorem 5.1) it is known that the Steinberg complex, over the p-adic integers Z_{p}, is G-chain homotopic to a (unique minimal) direct summand which is a chain complex of projective Z_{p}G-modules, and we denote this by 𝕊t_{∗}(G). This chain complex has been recognized by group theorist as a worthy generalization of the Steinberg module to arbitrary finite groups [5, 67, 68] and we study its properties in Section 5. (If G is a finite group of Lie type then 𝕊t_{∗}(G) is just the ordinary Steinberg module concentrated in degree one less than the semisimple rank of G.) Theorem 1.1 combined with (mostly straightforward) properties of the Steinberg complex give significant generalizations of the properties of higher limits over functors concentrated on one conjugacy class found in [39]. (Functors concentrated on one conjugacy class are called atomic functors.)

In homotopy theory, the functors which occur in practice often have the property that the centralizer CP of P acts trivially on F (P) (i.e., the action of NP/P on F (P) factors through NP/P C(P)). In this case one can remove additional subgroups from the collection. Let D_{p}(G) be the collection consisting of the subgroups which are p-centric (i.e., the center ZP of P is a Sylow p-subgroup in CP), and such that O_{p}(NP/P C(P)) = e. These subgroups are necessarily p-radical subgroups and they play a role in Alperin’s theory of weights for arbitrary finite groups, where they are the p-radical subgroups corresponding to the principal block [4, §3]. (We call them principal p-radical subgroups.)

Theorem 1.2. Let C be any collection of p-subgroups in a finite group G, closed under passage to p-radical p-supergroups. Assume that F is a functor O_{C}^{op} → Z_{(p)}- mod such that for all P ∈ C, CP acts trivially on F (P). Then

lim_{O_{C}} ^{∗}F ≃ lim_{O_{C′′}} ^{∗}F ≃ H_{G}^{∗}(|C′′|; F)

for any collection C′′ satisfying C ∩ D_{p}(G) ⊆ C′′ ⊆ C, where F is the G-local coefficient system on |C′′| induced via (d̄ C′′)_{G} → (d̄ C)_{G} → (C^{op})_{G} → O_{C}^{op}.

The more explicit results in Theorem 1.1 for atomic functors, provide a computation of the E_{1}-term of a spectral sequence converging to the higher limits of a general functor F, obtained by filtering F in such a way that the associated quotients are atomic functors. By a G-invariant height function ht on a collection C we mean a strictly increasing map of posets C/G → Z. An example of a G-invariant height function on a collection of p-subgroups is given by ht(P) = log_{p} |P |.

Theorem 1.3. Let C be any collection of p-subgroups in a finite group G, closed under passage to p-radical p-supergroups. For any functor F : O_{C}^{op} → Z_{(p)}- mod we have a spectral sequence

E_{1}^{i,j} = ⨁_{[P]∈C′/G,ht(P)=−i} H^{i+j−1}(Hom_{NP/P} (St_{∗}(NP/P), F (P))) ⇒ lim_{O_{C}} ^{i+j}F

where C′ = C ∩ B_{p}^{e}(G) and ht is some chosen G-invariant height function on C′. The differentials are Serre type cohomological differentials d_{n} : E_{n}^{i,j} → E_{n}^{i+n,j−(n−1)}.

If for all P ∈ C, CP acts trivially on F (P), then we may replace C′ by C ∩ D_{p}(G).

We can always normalize a height function such that for a Sylow p-subgroup P, ht(P) = 0, and then this spectral sequence occupies a triangle in the ‘eighth octant’. If G is a finite group of Lie type of characteristic p the poset B_{p}^{e}(G) has a relatively simple structure. (It is Cohen-Macaulay.) In this case, choosing the right height function, the E_{1}-term is concentrated on the horizontal axis, and hence yields a chain complex, with an explicitly described differential, for calculating the higher limits (Theorem 4.1).

Applications to group cohomology and other Mackey functors. In cases where lim^{i} vanishes for i > 0, the chain complex defining Bredon cohomology in Theorem 1.1 produce finite exact sequences starting with lim_{D}^{0} F. (We get similar exact sequences when treating higher limits over C-conjugacy categories and orbit simplex categories.) For group cohomology this gives exact sequences expressing H^{n}(G; M) in terms of H^{n}(H; M) for proper subgroups H of G. For instance we get the following result.

Theorem 1.4. Let G be a finite group, and let H^{n}(−) denote group cohomology with coefficients in Z_{(p)}. We have the following three exact sequences coming from acyclic complexes.

0 → H^{n}(G) → ⊕_{[P]∈D_{p}(G)/G}H^{n}(P)^{NP} → ⊕_{[P_{0}<P_{1}]∈|D_{p}(G)|_{1}/G}H^{n}(P_{0})^{NP_{0}∩NP_{1}} → ⋯ → 0

0 → H^{n}(G) → ⊕_{[V]∈A_{p}(G)/G}H^{n}(CV)^{NV} → ⊕_{[V_{0}<V_{1}]∈|A_{p}(G)|_{1}/G}H^{n}(CV_{1})^{NV_{0}∩NV_{1}} → ⋯ → 0

0 → H^{n}(G) → ⊕_{[P]∈D_{p}(G)/G}H^{n}(NP) → ⊕_{[P_{0}<P_{1}]∈|D_{p}(G)|_{1}/G}H^{n}(NP_{0} ∩ NP_{1}) → ⋯ → 0

Here A_{p}(G) denotes the collection of non-trivial elementary abelian p-subgroups, that is the subgroups of the form (Z/p)^{k}, for some k > 0. The last exact sequence is a version of the celebrated exact sequence of Webb, but obtained for the small collection D_{p}(G), whereas the rest appear to be new. We explain these results as part of a general framework in Section 8, and provide various variants.

The spectral sequence in Theorem 1.3 can also be used to give a useful description of lim_{O_{C}}^{0} F, in fact regardless of the vanishing of the higher limits, simply by looking at what happens at the (0, 0)-spot in the spectral sequence. In this way we recover expressions for the stable elements, usually proved using various versions of Alperin’s fusion theorem, via homotopy theoretic considerations. (See Section 10.)

Applications to homology decompositions. A homology decomposition is said to be sharp [29] if the Bousfield-Kan cohomology spectral sequence of the homotopy colimit collapses at the E_{2}-term onto the vertical axis. (This can also be formulated in other equivalent ways—see Section 9.) Our general formulas for higher limits can be used to obtain new sharpness results for homology decompositions. It is easy to see that the collection S_{p}^{e}(G) is sharp with respect to both the subgroup, centralizer and normalizer homology decompositions. Theorem 1.2, with F equal to mod p group cohomology, shows that the E_{2}-term of the cohomology spectral sequence associated to the subgroup decomposition using all psubgroups, remains unchanged when removing all subgroups not in D_{p}(G). This, together with the analogous result for the normalizer decomposition (Theorem 7.3), hence reveals

Theorem 1.5. The collection D_{p}(G) gives a subgroup and normalizer sharp mod p homology decomposition of BG.

(See Theorem 9.1 for a more elaborate version of this result.) The subgroup sharpness result improves a result of Dwyer [29], that the collection of p-radical and p-centric subgroups is subgroup sharp. The normalizer sharpness result improves a result of Webb [67] that the collection of p-radical subgroups is normalizer sharp and verifies a suspicion of Smith-Yoshiara [59] that the collection of p-radical and p-centric subgroups should be normalizer sharp.

Related work. This work grew out of a need to calculate higher limits (initially using [39]) in order to compute monoids of vector bundles over classifying spaces completed at a prime p, as a part of a larger program to understand the ‘homotopy representation theory’ of finite, compact Lie, or even p-compact groups [28], at the prime p. This motivating application of our results will be published separately in [35].

The relationship between groups, posets and spaces which underly everything in this paper dates back to the fundamental work of Brown [20] and Quillen [50]. We assume some familiarity with the methods of [50]. The approach taken in this paper can be seen as a continuation along the lines of Dwyer papers [27, 29], which we highly recommend to the reader, although this paper can be read independently of those.

The methods of this paper are also closely related to a ‘Lie theoretic’ approach to local representation theory of finite groups advocated by Alperin [4, 5] and Webb [67] amongst others, and likewise have ties to early ideas of Puig [49] who examines the basic but yet elusive question of what is a p-local group?

Results relating higher limits over various categories to higher limits over orbit simplex categories have been found independently by J. Słomińska [56]. I am grateful to S. Jackowski for informing me of her unpublished work and providing me with a copy.

Organization of the paper. In Section 2 we prove various facts about higher limits and Bredon cohomology, needed in the later sections. In Section 3 we prove our general results on higher limits over orbit categories, a couple of times referring to results from Section 5. Section 4 elaborates on these results in the special case where G is a finite group of Lie type. In Section 5 we examine the structure of the Steinberg complex St_{∗}(G) for an arbitrary finite group G. In Section 6 and 7 treat higher limits over conjugacy categories and orbit simplex categories respectively. In the last three sections we look at special cases of the general results, focusing on the case where the higher limits in fact vanish. In Section 8 explain how one can obtain exact sequences in the case where all higher limits vanish, and provide some examples when this is satisfied. In Section 9 we explain how the previous results lead to sharpness results on homology decompositions. Finally, in Section 10 we explain how our results relate to the calculation of stable elements and Alperin’s fusion theorem.

Notation and conventions. This paper is written simplicially so the word space, without any prefix, means a simplicial set. Hence the nerve construction | · | is a functor from categories to simplicial sets. We remark in this connection that that for G-categories, taking Gfixedpoints commutes with taking nerves. We will let R denote an arbitrary commutative unital ring. In most of this paper we however work over Z_{(p)}, and we adopt the convention that unless otherwise indicated, Z_{(p)} is taken as the ground ring. We denote the normalizer and centralizer of H in G by NH and CH respectively, and write W H for the Weyl group NH/H. The word ‘functor’ without any prefix means a covariant functor. The suspension ΣX of a G-space X should be understood as CX/X, where CX is the unreduced cone on X. This is a pointed G-space, with G-basepoint [X].

Acknowledgements. I would like to thank Bill Dwyer, Dan Kan, Bob Oliver, and Peter Symonds for helpful conversations. Exchange of calculations with Bob Oliver led to improvements in the statement of the results in Section 5. I thank Kasper Andersen and Peter Webb for making useful comments on a preliminary version of this manuscript, and Lars Hesselholt for providing helpful advice on the exposition. Finally, I would like to thank my advisor Haynes Miller for many helpful conversations and much encouragement.

2. Preliminaries

In this section we explain various facts about equivariant cohomology and higher limits. All of the results explained in this section are well known.

2.1. Equivariant cohomology and local coefficient systems. Let X be a space. Define the simplex category (or division [30]) d X of X to be the category with objects the simplices of X and morphisms generated by face and degeneracy maps. In other words the objects are maps σ : Δ[n] → X, and a morphism σ → σ′ is given by a commutative triangle

\xymatrix{Δ[n] \ar[rr] \ar[dr]_{σ′} &  & Δ[m] \ar[dl]^{σ} \\  & X}

Often we need the opposite category of d X, which we denote d̄ X. If X is the nerve of a small category D, then we abbreviate d |D| to d D. Define a local coefficient system on X (over R) to be a functor F : d̄ X → R- mod.

For any local coefficient system F we can define the simplicial cochain complex (C^{∗}(X; F), δ) of X with coefficients in F by setting C^{n}(X; F) = ∏_{σ∈X_{n}} F(σ) and (δf)(σ) = ∑_{i}(−1)^{i}F(d_{i}^{op})(f(d_{i}σ)). We define H^{∗}(X; F) to be the homology of this cochain complex (cf. also [32, p. 42ff]). As usual we may as well use the normalized simplicial cochain complex for calculating H^{∗}(X; F) (cf. [70, §8.3]).

Note that we have a natural transformation d̄ D → D on objects given by (x_{0} → ⋯ → x_{n}) ↦ x_{n}. We also have a natural transformation d̄ D → D^{op} on objects given by (x_{0} → ⋯ → x_{n}) ↦ x_{0}. If F : D → R- mod is a covariant (resp. contravariant) functor then F induces a local coefficient system F on |D| via d̄ D → D (resp. d̄ D → D^{op}). In both the co- and contravariant case we denote the cohomology groups by H^{∗}(D; F), i.e., we omit indicating taking nerves, and indicate that the local coefficient system comes from a functor on D by keeping the notation F instead of F.

We now want to define Bredon equivariant cohomology. For our purposes it is most natural to use a notion of a G-local coefficient system which is more general than the one given in e.g. [19], but which agrees with the usage in group theory (cf. [10, Ch. 7]). Roughly speaking, we want to allow different elements of G to induce different automorphisms of C^{∗}(X; F) even though they act the same way on X. For this purpose we introduce the following construction. Let D be small G-category, and let α be the functor α : G → Cat which defines this G-category. (G is viewed as a category with one object, and Cat denotes the category of small categories.) Note that if D is a G-category, then so is D^{op}, via op : Cat → Cat. Set D_{G} equal to the Grothendieck construction of α [63], i.e., the category which has the same objects as D but where the morphisms x → y in D_{G} are given by pairs (f : gx → y, g), where f is a morphism in D and g ∈ G, which compose via (f′, g′) ∘ (f, g) = (f′ ∘ (g′f), g′g). Note that this can be thought of as a sort of semi-direct product of D with G, and just amounts to enlarging the morphisms in D with the elements of G in a freest possible way.

Remark 2.2. Let D = C for some collection C. The categories C_{G}, O_{C}^{op} and A_{C} can all be viewed as having the same objects, namely the objects of C. The morphisms from H to H′ in C_{G} can be identified with the elements of g ∈ G such that gHg^{−1} ≤ H′. The morphisms from H′ to H in O_{C}^{op} may be identified with the elements [g] ∈ G/H′ such that gH′g^{−1} ≥ H and the morphisms from H to H′ in A_{C} may be identified with the elements [g] ∈ G/CH such that gHg^{−1} ≤ H′.

We have a canonical functors C_{G} → A_{C} and (C^{op})_{G} → O_{C}^{op}, which in the above picture takes g to [g]. Note furthermore that we have a functor α : A_{C} → O(G)^{op} on objects given by H ↦ G/CH. In practice, our functors on A_{C} will be constructed in this way from functors on O(G)^{op}.

Definition 2.3. Let X be a G-space. We define a G-local coefficient system on X to be a functor F : (d̄ X)_{G} → R- mod. Define a G-action on C^{∗}(X; F), by letting g act (gf)(σ) = F((1_{gσ}, g))(f(g^{−1})). Define the Bredon cohomology of X with coefficients in F as H_{G}^{∗}(X; F) = H(C_{G}^{∗}(X; F)), where C_{G}^{∗}(X; F) = C^{∗}(X; F)^{G}.

We might replace C^{∗}(X; F) by the corresponding normalized chain complex for the purpose of calculations. If M is an RG-module, then we denote cohomology with values in the ‘constant’ coefficient system σ ↦ M by H_{G}^{∗}(X; M).

Note also that to any G-local coefficient system F we may construct a new coefficient system F̃ given by σ ↦ F(σ)^{G_{σ}}, where G_{σ} denotes the isotropy subgroup of σ. We have a natural transformation F̃ → F which satisfies H_{G}^{∗}(X; F̃) →^{≃} H_{G}^{∗}(X; F), and ˜ F̃ →^{=} F̃. (E.g., the constant coefficient system M corresponds to the generic coefficient system H^{0}(−; M) in this manner.) Hence, for the purpose of Bredon cohomology, one could (as in [19]) without loss of generality have restricted attention to G-local coefficient systems which satisfied F = F̃. However, in practice making this restriction seems slightly unnatural and moreover the more general definition corresponds to the generality needed when studying homology representations (see [52],[10, Ch. 7]).

A generic coefficient system is a functor 𝔉 : O(G)^{op} → R- mod. Note that for any Gspace X we have a canonical (covariant) functor θ : (d̄ X)_{G} → O(G)^{op}. On objects this is given by σ ↦ G_{σ}, and on morphisms it is given by (f : gσ → σ′, g) ↦ (gG_{σ′}g^{−1} ≥ G_{σ}, g), in the picture of Remark 2.2. A generic coefficient system induces functorially a G-local coefficient system F on any G-space X by setting F = 𝔉θ. (In much of the literature generic coefficient systems are just called coefficient systems, or Bredon coefficient systems.) We call a G-local coefficient system which is induced by some generic coefficient system for an isotropy G-local coefficient system. (Bredon calls this a simple coefficient system on X.) (Note that any functor 𝔉 : O_{C}^{op} → R- mod can be extended (non-uniquely) to a functor 𝔉′ : O(G)^{op} → R- mod (see [42, X3.4]), so the exact domain of definition is not a major issue.) When there is no danger of confusion we will not distinguish between a generic coefficient system and the isotropy G-local coefficient system it induces. However, to avoid confusion we have chosen not to give meaning to the term coefficient system without any prefix.

In general, cohomology with values in a G-local coefficient system on X is of course not a G-homotopy invariant of X. However, G-equivariant Bredon cohomology with values in a generic coefficient system does define a G-homotopy invariant cohomology theory on G-spaces (see e.g., [19]).

Recall that a Mackey functor is a generic coefficient system, but defined on a category with more morphisms than the orbit category, where one also has a ‘transfer’ map (cf. e.g., [44],[69]). We actually only directly need the Mackey functor H^{n}(−; M) where the transfer map is the usual transfer map in group cohomology. This is a so-called cohomological Mackey functor, meaning that restriction from one subgroup to a smaller subgroup followed by the transfer back equals multiplication by the index.

Given a G-space X with a G-local coefficient system F, we define Borel equivariant cohomology as H_{hG}^{∗}(X; F) = H_{G}^{∗}(EG × X; F), where EG × X is given the diagonal Gaction and coefficient system pulled back via the projection map EG × X → X.

Given a G-category D and a functor F : D_{G} → R- mod (or F : (D^{op})_{G} → R- mod), we naturally get a G-local coefficient system on |D| via d̄ D → D (or d̄ D → D^{op}).

Remark 2.4. In most of the paper we work over Z_{(p)}, for the reason that we need multiplication by the index of a Sylow p-subgroup to be invertible in R. (Any result stated over Z_{(p)} holds verbatim over any Z_{(p)}-algebra.) As a technical point however, when dealing with the Steinberg complex, we need to work over Z_{p} instead, in order to appeal to Webb’s Theorem 5.1. One can translate results back to Z_{(p)} in the following way: Z_{p} is faithfully flat over Z_{(p)}, so a map of Z_{(p)}-modules is an isomorphism iff it is an isomorphism after tensoring with Z_{p}. Furthermore if X is of finite type, then one checks that H_{G}^{∗}(X; F) ⊗_{Z_{(p)}} Z_{p} = H_{G}^{∗}(X; F ⊗_{Z_{(p)}} Z_{p}), and the same statement is of course true non-equivariantly. Hence, when showing certain maps are isomorphisms, we can without problems work over Z_{p}.

2.5. Higher limits. Let RD- mod denote the category of functors D → R- mod. By lim_{D}^{∗} we mean the right derived functors of the inverse limit functor lim_{D} : RD- mod → R- mod. We have the following fundamental identification.

Proposition 2.6. (cf. [31, 3.3]) Let F : D → R- mod be any functor. Then

lim_{D}^{∗}F = Ext_{RD}^{∗}(R, F) = H^{∗}(D; F)

Proof. First note that we can write lim_{D}^{0} F = Hom_{RD}(R, F) so lim_{D}^{∗} F = Ext_{RD}^{∗}(R, F). Let D ↓ d denote the over-category of d ∈ D and consider the functor C_{∗}(|D ↓ −|; R) : D → R- (chain complexes), which to each d ∈ D associates the simplicial chain complex of |D ↓ d|. One easily checks that degree n of the cochain complex Hom_{RD}(C_{∗}(|D ↓ −|; R), F) equals ∏_{x_{0}→···→x_{n}} F (x_{n}) with boundary map coming from the simplicial structure on |D|. In particular we see that Hom_{RD}(C_{∗}(|D ↓ −|; R), −) preserves exactness, so C_{n}(|D ↓ −|; R) is projective in RD- mod. But since D ↓ d has an initial object, and therefore has contractible nerve, C_{∗}(|D ↓ −|; R) defines a projective resolution of the constant functor R in RD- mod. This shows lim_{D}^{∗} F equals H^{∗}(D; F).

Remark 2.7. The cohomology H^{∗}(D; F) can be viewed as sheaf cohomology of a certain sheaf Φ on Top(|D|), the topological space associated to |D|, constructed from F. In this remark we sketch this construction. Suppose for concreteness that F is contravariant. For each open sets U ⊆ Top(|D|) set

Φ(U) = {r ∈ ∏_{σ∈|D|,U∩σ≠∅} F (σ)|F (ϕ)(r_{σ′}) = r_{σ} for all ϕ : σ → σ′ in d D}

It is easy to see that this defines a sheaf on Top(|D|), and if y ∈ Top(|D|), then the stalk at y is equal to F (σ), where σ is the unique non-degenerate simplex such y lies in the open cell associated to σ. (An open zero cell is a point.)

We now sketch how to see that H^{∗}(D; F) = H^{∗}(Top(|D|); Φ). We will restrict ourself to the case where D is a poset—the general case can be obtained (rather messily) by passing to a double subdivision. If D is a poset Top(|D|) is an ordered simplicial complex. For σ ∈ |D|, let star(σ) ⊆ Top(|D|) be the open star of σ, i.e., the union of all open cells whose boundary contains σ. Note that star(σ) is open and contractible, with Φ(star(σ)) = F (σ).

Furthermore the Čech complex associated to the open covering given by star(σ), σ ∈ |D|, equals the cochain complex considered in the proof of Proposition 2.6. Since the Čech complex calculates sheaf cohomology [32], we are done.

2.8. Some auxiliary G-categories. Let ι : O_{C} → O_{C∪{e}} be the inclusion functor, and consider the undercategory EO_{C} = G/e ↓ ι (see [42, p. 46]). Note that EO_{C} has a Gaction in the obvious way by setting g · f = gf. (Alternatively, we may view EO_{C} as the category with objects pairs (H, x) where H ∈ C and x ∈ G/H, and with a unique morphism (H, x) → (H′, x′) if there exists a G-map G/H → G/H′ sending x to x′. The G-action in this picture is given by g · (H, x) = (H, gx).) Likewise, set EA_{C} = ι ↓ G, where ι : A_{C} → A_{C∪{G}} is the inclusion. We define a G action via g · i = c_{g}i, where (i : H → G) ∈ EA_{C} and c_{g} denotes conjugation by g. (Note that the category EO_{C} is isomorphic to the category denoted X_{C}^{β} in [29], and EA_{C} is equivariantly equivalent to the category denoted X_{C}^{α} in [29].)

Remark 2.9. For any collection C we have G-equivariant functors

EO_{C} → C ← EA_{C}

given by (f : G → G/H) ↦ G_{f} and (i : H → G) ↦ i(H) respectively. These are (in general non-equivariant) equivalences of categories with maps the other way being given by H ↦ (proj : G → G/H) and H ↦ (incl : H → G) respectively. (This was a key observation made by Dwyer [27, 29].)

Note that any functor F : O_{C} → R- mod induces a G-local coefficient system on |EO_{C}| via (d̄ EO_{C})_{G} → (d̄ C)_{G} → (C^{op})_{G} → O_{C}^{op}, and is hence a pullback of the coefficient system on |C|. This coincides with the isotropy G-local coefficient system induced by F viewed as

a generic coefficient system, and is on objects given by (G →^{f} G/H_{0} → ⋯ → G/H_{n}) ↦ F (G_{f}). We keep the notation F for this G-local coefficient system on |EO_{C}|. Likewise we get a G-local coefficient system on |EA_{C}| via (d̄ EA_{C})_{G} → (d̄ C)_{G} → C_{G} → A_{C}, on objects given by (H_{0} → ⋯ H_{n} →^{i} G) ↦ F (i(H_{n})), and we also keep the notation F for this coefficient system. (Note that if F is of the form F = 𝔉α, where α is the canonical functor α : A_{C} → O_{C}^{op} and 𝔉 : O(G)^{op} → R- mod is some functor, then this is the isotropy G-local coefficient system induced by 𝔉.)

Proposition 2.10. We have natural isomorphisms |EO_{C}|/G = |O_{C}| and |EA_{C}|/G = |A_{C}|. Furthermore H^{∗}(O_{C}; F) = H_{G}^{∗}(|EO_{C}|; F) and H^{∗}(A_{C}; F) = H_{G}^{∗}(|EA_{C}|; F).

Proof. The first two statements follows by inspection of the simplices. But by the definition of the coefficient systems and the first two statements, we now get that C^{∗}(|EO_{C}|; F)^{G} = C^{∗}(O_{C}; F) and C^{∗}(|EA_{C}|; F)^{G} = C^{∗}(A_{C}; F). Hence the last two statements follow by the definition of Bredon cohomology.

3. Higher limits over C-orbit categories

In this section we examine higher limits over C-orbit categories.

Theorem 3.1. Let H ∈ C and let F : O_{C}^{op} → Z_{(p)}- mod be a functor concentrated on conjugates of a subgroup H ∈ C. Then

1. H_{G}^{∗}(|EO_{C}|; F) ≃ H_{hW H}^{∗}(|C_{≥H}|, |C_{>H}|; F (H)) ≃ H̃_{hW H}^{∗}(Σ|C_{>H}|; F (H)).

2. If |C_{>H}|^{Q} is F_{p}-acyclic for all non-trivial p-subgroups Q in W H, then

H_{G}^{∗}(|EO_{C}|; F) ≃ H_{G}^{∗}(|C|; F) ≃ H_{W H}^{∗}(|C_{≥H}|, |C_{>H}|; F (H)) ≃ H̃_{W H}^{∗}(Σ|C_{>H}|; F (H)),

where F is the coefficient system obtained from F by precomposing with (d̄ C)_{G} → (C^{op})_{G} → O_{C}^{op}.

Proof. We first perform some preliminary rewritings. By definition

C_{G}^{n}(|C|; F) = ( ∏_{H_{0}<H_{1}<···<H_{n}} F (H_{0}))^{G} .

Since F is atomic this equals (∏_{H<H_{1}<···<H_{n}} F (H))^{NH} by Frobenius reciprocity, so

C_{G}^{n}(|C|; F) = C_{NH}^{n}(|C_{≥H}|, |C_{>H}|; F (H)).

We want to do a similar rewriting with EO_{C} in place of |C|. View for a moment EO_{C} as being replaced by its (G-equivalent) skeletal subcategory with objects G-maps G → G/H′ where H′ runs through a set of conjugacy classes of objects in C. To simplify the notation, set Y = ∪_{H′∈C_{>H}} |EO_{C}^{H′}|. As in the first rewriting we have

C_{G}^{n}(|EO_{C}|; F) = (  ∏_{G→^{f}G/H→G/H_{1}→···→G/H_{n}} F (G_{f}))^{G}

= (  ∏_{G→^{f}G/H→G/H_{1}→···→G/H_{n},f(e)∈W H} F (H))^{NH}

= C_{NH}^{n}(|EO_{C}^{H}|, Y; F (H)).

These rewritings are natural, and give us corresponding identifications:

[figure: H_{G}^{∗}(|C|; F); ≃; H_{W H}^{∗}(|C_{≥H}|, |C_{>H}|; F (H)); H_{G}^{∗}(|EO_{C}|; F); ≃; H_{W H}^{∗}(|EO_{C}^{H}|, Y; F (H))]

Note that on the right hand side the coefficient systems are constant coefficient systems, and hence come from the generic coefficient system H^{0}(−; F (H)).

We now prove 1. First note that the W H acts freely on the pair (|EO_{C}^{H}|, Y), or said differently the fixpoint pair under all non-trivial subgroups of W H becomes a trivial pair of the form (X, X). Hence

H_{W H}^{∗}(|EO_{C}^{H}|, Y; F (H)) = H_{hW H}^{∗}(|EO_{C}^{H}|, Y; F (H))

by the defining property of Borel cohomology. This shows the first isomorphism, since the W H-map (|EO_{C}^{H}|, Y) → (|C_{≥H}|, |C_{>H}|) is a homotopy equivalence of pairs. The second rewriting follows directly from our definition of the suspension.

We now proceed to prove 2. Consider again the homotopy equivalence

(|EO_{C}^{H}|, Y) → (|C_{≥H}|, |C_{>H}|).  (3.1)

Let P be a Sylow p-subgroup of W H. Since H^{0}(−; F (H)) is a cohomological Mackey functor we will be done, by an application of the transfer, if we can see that this is a P - Z_{(p)}-equivalence if and only if |C_{>H}|^{Q} is F_{p}-acyclic for all non-trivial p-subgroups Q in W H. To this end, we have already observed that the Q-fixpoint set of the left-hand side of 3.1 becomes the trivial pair. Also |C_{≥H}|^{Q} is contractible since C_{≥H}^{Q} has the minimal element H. Hence 3.1 is a P -Z_{(p)}-equivalence if and only if |C_{>H}|^{Q} is Z_{(p)}-acyclic for all non-trivial p-subgroups Q in W H. Since |C_{>H}|^{Q} is of finite type, we may replace Z_{(p)} by F_{p}.

Remark 3.2. By Smith theory (c.f. e.g. [44]) it is in Theorem 3.1 enough to verify that |C_{>H}|^{Q} is acyclic for all subgroups Q of order p.

Remark 3.3. Note that Theorem 3.1 shows that if F is concentrated on conjugates of a subgroup H, then the equality H_{G}^{∗}(|EO_{C}|; F) ≃ H_{G}^{∗}(|C|; F) is equivalent to the equality H̃_{W H}^{∗}(Σ|C_{>H}|; F (H)) ≃ H̃_{hW H}^{∗}(Σ|C_{>H}|; F (H)). The condition in part 2 of Theorem 3.1 just gives an obvious criterion for when relative Bredon and Borel cohomology with values in a cohomological Mackey functor over Z_{(p)} coincide, namely when the action of a Sylow p-subgroup is free (on the appropriate pair). Alternatively the condition can be viewed as ensuring the collapse of the E_{2}-term a certain (relative) isotropy spectral sequence onto the horizontal axis (see [29]). Conditions of this type feature prominently in the work of Webb [67, 66, 68], and were also exploited by Dwyer [29].

Remark 3.4. If H is a maximal element in C, and F is concentrated on conjugates of H, then the formula of Theorem 3.1 shows that

H_{G}^{∗}(|EO_{C}|; F) = H^{∗}(W H; F (H))

In particular if C contains a maximal element H which is a non-Sylow p-group, then p||W H|, by a standard property of p-groups. Thus, for example, when F (H) = Z_{(p)} with trivial W H-action, lim_{O_{C}}^{∗} F = H^{∗}(W H; Z_{(p)}) is non-zero in infinitely many degrees. Thus one does not in general get good vanishing results for lim_{O_{C}}^{∗} F, when C = A_{p}(G), the collection of non-trivial elementary abelian p-groups.

Corollary 3.5. Let F : O_{C}^{op} → Z_{(p)} be a functor, and assume that |C_{>H}|^{Q} is F_{p}-acyclic for all H ∈ C and for all non-trivial p-subgroups Q of W H. Then H_{G}^{∗}(|C|; F) →^{≃} H_{G}^{∗}(|EO_{C}|; F).

Proof. Theorem 3.1 establishes this corollary for atomic functors. We want to establish the corollary for an arbitrary functors F by induction. Choose a maximal subgroup H in C on which F is nonzero, and let F′ denote the subfunctor of F obtained by setting F′(H′) = 0 if H′ is conjugate to H and F′(H′) = F (H′) otherwise. We have the following diagram with exact rows:

\xymatrix{0 \ar[r] & C^{∗}(|C|; F′)^{G} \ar[r] \ar[d] & C^{∗}(|C|; F)^{G} \ar[r] \ar[d] & C^{∗}(|C|; F/F′)^{G} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & C^{∗}(|EO_{C}|; F′)^{G} \ar[r] & C^{∗}(|EO_{C}|; F)^{G} \ar[r] & C^{∗}(|EO_{C}|; F/F′)^{G} \ar[r] & 0} .

(The sequences remain exact after taking G-fixpoints, since they are split as modules.) By induction on the cardinality of the subgroups in C/G on which the functor is nonzero, we can assume that the statement is true for F′, and the statement is also true for F/F′, since it is an atomic functor. Hence the right and the left vertical arrow induce isomorphisms on homology, so by the five-lemma this is also true for the middle arrow.

Before the next proof, recall that for a poset P, and x ∈ P, star_{P} x denotes the subposet {y ∈ P|x ≤ y or x ≥ y} and likewise link_{P} x = {y ∈ P|x > y or y < x}.

Proof of Theorem 1.1. We first prove the first part of the theorem under the assumption that C is actually closed under passage to all p-supergroups, and will later show that nonp-radical subgroups do not play a role. We already know that lim_{O_{C}}^{∗} F = H_{G}^{∗}(|EO_{C}|; F) from Proposition 2.6 and 2.10. Hence to prove the first part of the theorem, we just have to show that the conditions of Corollary 3.5 apply. By the assumption that C is closed under passage to p-supergroups, we have that for all H ∈ C, C_{>P} = S_{p}(G)_{>P}. But for all nontrivial p-subgroups Q of W P, S_{p}(G)_{>P}^{Q} has contractible nerve. To see this note that for all H ∈ S_{p}(G)_{>P}^{Q}, identifying Q with its preimage in NH, QH ∈ S_{p}(G)_{>P}^{Q} so the inequalities H ≤ QH ≥ Q provide a contracting homotopy (cf. also [50, 1.5],[10, Lemma 6.4.5]). Hence Corollary 3.5 shows that H_{G}^{∗}(|EO_{C}|; F) ≃ H_{G}^{∗}(|C|; F).

Skipping slightly in the order in which the that theorem was stated, we now proceed to prove the atomic case. From Theorem 3.1 we know that lim_{O_{C}}^{∗} F = H̃_{NP}^{∗} (Σ|C_{>P} |; F (P)). By assumption C_{>P} = S_{p}(G)_{>P}. But the posets S_{p}(G)_{>P} and S_{p}(W P) have NP -homotopy equivalent nerves, induced by the maps H ↦ (H ∩ NP)/P and H/P ↦ HP (cf. [50, Prop. 6.1], [10]), so we can use the space |S_{p}(W P)| instead of |C_{>P} |. Writing out the definition of Bredon cohomology now gives the wanted formula.

The last step in the proof is to see that we can replace the collection C with the smaller collection C′, without changing higher limits or Bredon cohomology. By filtering the functor as in the proof of Corollary 3.5, it is enough to prove this when F is an atomic functor, concentrated on conjugates of a subgroup P ∈ C. If P ∈ C \ C′ then S_{p}(W P) is W P - contractible, a contracting homotopy being given by H ≤ HO_{p}(W P) ≥ O_{p}(W P). Hence H_{G}^{∗}(|C|; F) = 0, by the formula for atomic functors, which proves the wanted result, since F in this case is zero on C′. So, suppose that P ∈ C′. By Theorem 3.1 it is enough to prove that C_{>P} and C′_{>P} are W P -homotopy equivalent. We will do this by showing that we can successively, in order of increasing size, remove W P -conjugacy classes of subgroups in C_{>P} \ C′_{>P} from C_{>P}, without changing its W P -homotopy type. Let C̃ be a W P -subposet of C_{>P}, with C′_{>P} ⊆ C̃ ⊆ C_{>P}. Suppose that P′ ∈ C̃ \ C′_{>P} and that C̃_{>P′} = C_{>P′}. Let Ĉ be the poset obtained from C̃ by removing W P -conjugates of P′. It is easy to check that we have a pushout square, which is a homotopy pushout of NP -space.

\xymatrix{NP ×_{NP ∩NP′} | linkC̃ P′| \ar[r] \ar[d] & |Ĉ| \ar[d] \\ NP ×_{NP ∩NP′} | starC̃ P′| \ar[r] & |C̃|}

Obviously | starC̃ P′| is NP′-contractible. Note that linkC̃ P′ = C_{>P′} ⋆ C̃_{<P′}, where ⋆ denotes the join of posets (corresponding to the join of spaces; see [50]). By the previous manipulations of posets, |C_{>P′}| is NP′ contractible since P′ ∈ C \ C′. Hence | linkC̃ P′| is NP′-contractible as well, so the above pushout square shows that |Ĉ| → |C̃| is a NP - homotopy equivalence.

An induction now shows that |C_{>P} | and |C′_{>P} | are NP -homotopy equivalent as wanted. (Compare also e.g., [61, Prop 1.7],[27, Thm. 7.1].)

Remark 3.6. The p-radical subgroups were first introduced by Bouc [15], and play an important role in modern local representation theory [4, 6]. (They also played an indirect role e.g., in the work of Puig [49]—see Section 10.) Their pertinence to the homotopy theory of classifying spaces was (independently, and in the more general context of compact Lie groups) discovered by Jackowski-McClure-Oliver [39] who dubbed them p-stubborn subgroups, since they just would not go away.

Remark 3.7. Taking C = S_{p}^{e}(G) and P = e in the above proof, we get that S_{p}(G) and B_{p}(G) are G-homotopy equivalent. This is a well known result of Bouc [15] and Thevenaz-Webb [61].

Remark 3.8. Note that Theorem 1.1 tells us that we can remove a single conjugacy class [H] from a collection C∩B_{p}^{e}(G) as in Theorem 1.1 and still get the same result for all functors F only if St_{∗}(W H) is contractible. Hence finding an example where we can remove more than the non-p-radical groups for all F would amount to finding a group G where e is p-radical but H̃^{∗}(|S_{p}(G)|; Z_{(p)}) ≠ 0. A strong version of Quillen’s conjecture [50, Conj. 2.9] (which is known to be true in many cases [8]) asserts that H̃^{∗}(|S_{p}(G)|; Z) ≠ 0 if e is p-radical. We are not aware of a counterexample if Z is replaced by Z_{(p)} either. (See also [39, p. 206-207].)

Remark 3.9. The non-equivariant cohomology of subgroup complexes with values in a Glocal coefficient system has been studied previously by especially Ronan-Smith [52, 53, 51] (cf. also [10]), as a method for constructing modular representations. It would be interesting to try to calculationally exploit the interplay between the equivariant and non-equivariant cohomology.

Remark 3.10. The proof of Theorem 1.1 reveals that the conclusion of the theorem holds not only for collections of p-subgroups but also for all collections C with are closed under extensions by p-subgroups. More precisely we need that if H ∈ C and H′ ≤ G is the middle term in an exact sequence of groups 1 → H → H′ → P → 1 with P a p-group, then H′ ∈ C.

Proof of Theorem 1.2. The proof follows the pattern of the last part of the proof of Theorem 1.1, but uses the additional assumptions about F together with information about the structure of the Steinberg complex (for which we refer to Section 5) to remove the additional subgroups. As explained in Remark 2.4 it is enough to prove the statement with Z_{p} in place of Z_{(p)}. Hence, for the rest of the proof we adopt the convention that all coefficients are taken over Z_{p}.

By filtering the functor as in the proof of Corollary 3.5, it is enough to prove the statement when F is an atomic functor, concentrated on conjugates of a subgroup P ∈ C.

Suppose first that P ∈ C \ C′′. Since in this case H_{G}^{∗}(|C′′|; F) = 0 we need to see that H_{G}^{∗}(|C|; F) = 0, i.e., by Theorem 1.1. that Hom_{W P} (St_{∗}(W), F (P)), or equivalently Hom_{W P} (𝕊t_{∗}(W), F (P)) (see Section 5), is acyclic.

If P is non-p-centric, then the kernel of the action of W P on F (P) has order divisible by p so, by Theorem 5.3, Hom_{W P} (𝕊t_{m}(W P), F (P)) = 0 for all m and H_{G}^{∗}(|C|; F) = 0 as wanted.

If P is p-centric, then O_{p}(NP/P C(P)) ≠ e since P ∈ C \ C′. Because ZP is a Sylow p-subgroup in CP we can write CP = ZP × K where p |̸|K|. By Proposition 5.6 we have that St_{∗}(W P)_{K} = St_{∗}(NP/P C(P)), where (·)_{K} denotes coinvariants. But S_{p}(NP/P C(P)) is NP -contractible since O_{p}(NP/P C(P)) ≠ e, and hence Hom_{W P} (St_{∗}(W P), F (P)) = Hom_{W P} (St_{∗}(W P)_{K}, F (P)) is contractible as wanted.

Now suppose that P ∈ C′′. It is by Theorem 3.1 enough to prove that H̃_{W P}^{∗} (Σ|C_{>P} |; F (P)) = H̃_{W P}^{∗} (Σ|C′′_{>P} |; F (P)), and that the same statement holds for Borel cohomology instead of Bredon cohomology. As in the proof of Theorem 1.1, we do this by showing that we can successively, in order of increasing size, remove W P -conjugacy classes of subgroups in C_{>P} \ C′′_{>P} from C_{>P}, without changing the the Bredon or Borel equivariant cohomology with coefficients F (P).

Let the notation and assumptions be as in the last part of the proof of Theorem 1.1. The pushout square used there, together with the Mayer-Vietoris sequence in Bredon cohomology, shows that we need to see that H̃_{NP ∩NP′}^{∗}(Σ| linkC̃ P′|; F (P)) = 0, and that this also holds for Borel cohomology. Now linkC̃ P′ = C_{>P′} ⋆ C̃_{<P′}. In particular for all non-trivial psubgroups Q of NP′ we have that |(linkC̃ P′)^{Q}| is contractible, since |C_{>P′}^{Q}| is contractible. Hence reduced Bredon and Borel cohomology coincide, so we can restrict to considering Bredon cohomology (see Remark 3.3).

By the definition of the join

C̃_{∗}(| linkC̃ P′|) = C̃_{∗}(|C_{>P′}|) ⊗ ΣC̃_{∗}(|C̃_{<P′}|)

where Σ denotes the suspension (i.e., dimension shifting) of chain complexes. This chain complex is again NP′ chain homotopy equivalent to 𝕊t_{∗}(W P′) ⊗ ΣC̃_{∗}(|C̃_{<P′}|). Assume that P′ is non-p-centric. Then we have that

Hom_{NP ∩NP′}(𝕊t_{m}(W P′) ⊗ C̃_{n}(|C̃_{<P′}|), F (P)) = Hom_{NP ∩NP′}(𝕊t_{m}(W P′), C̃_{n}(|C̃_{<P′}|) ⊗ F (P))
= Hom_{NP′}(𝕊t_{m}(W P′), (C̃_{n}(|C̃_{<P′}|) ⊗ F (P)) ↑_{NP ∩NP′}^{NP′})
= 0

were the last equality follows from Proposition 5.3, using that CP′ acts trivially on (C̃_{n}(|C_{<P′}|)⊗ F (P)) ↑_{NP ∩NP′}^{NP′} and that P′ is non-p-centric. But this is true for all m, n, so we conclude that H̃_{NP ∩NP′}^{∗}(Σ| linkC̃ P′|; F (P)) = 0 as wanted.

Now assume that P′ is p-centric and write CP′ = ZP′ × K, where p |̸|K|. We have that | linkC̃ P′|/K = |C_{>P′}|/K⋆|C̃_{<P′}|. But |C_{>P′}|/K is NP′-equivalent to |S_{p}(NP′/P′C(P′))| by Proposition 5.6, which is NP′-contractible by the assumption that O_{p}(NP′/P′C(P′)) ≠ e. Therefore

H̃_{NP ∩NP′}^{∗}(Σ| linkC̃ P′|; F (P)) = H̃_{NP ∩NP′}^{∗}(Σ| linkC̃ P′|/K; F (P)) = 0,

also in this case, which finishes the proof of the theorem.

We note that the above proof can be simplified somewhat if one if one is willing to settle for removing all non-p-centric subgroups, using that the collection of p-centric subgroups is closed under passage to p-supergroups.

Example 3.11. The G-homotopy type of D_{p}(G) is in general different from that of S_{p}(G). For example if G = Σ_{3} × Z/2 then |S_{2}(G)| is G-contractible, whereas |D_{2}(G)| ≃ G/(Z/2 × Z/2).

Example 3.12. To see that the collection D_{p}(G) is in fact in general a proper subcollection of the collection of p-radical and p-centric subgroups, consider for example the group G = (Z/2 × Z/2) ⋊ Σ_{3}, where Σ_{3} acts on Z/2 × Z/2 via Σ_{3} → Z/2 →^{flip} Aut(Z/2 × Z/2). In this case the both the Sylow 2-subgroups and Z/2 × Z/2 are 2-radical and 2-centric, but only the Sylow 2-subgroups belong to D_{2}(G).

Proof of Theorem 1.3. Let F_{i} be the subfunctor of F obtained by setting F (P) = 0 for ht(P) > i, and let F_{i} be the corresponding G-local coefficient system. This induces an increasing filtration {C^{∗}(C′; F_{i})^{G}}_{i} on C^{∗}(C′; F)^{G}. The spectral sequence is now just the spectral sequence associated a filtered chain complex [70]. Since we will need it in a while, let us write up explicitly the differential d^{1} in the E^{1}-term. The differential goes:

C^{k}(|C′|; F_{i+1}/F_{i})^{G} →^{d^{1}} C^{k+1}(|C′|; F_{i}/F_{i−1})^{G}.

More precisely given

α ∈ C^{k}(|C′|; F_{i+1}/F_{i})^{G} = ⨁_{[P′]∈C′/G;ht(P′)=i+1} Hom_{NP′}(C_{k}(|C′_{≥P′}|, |C′_{>P′}|), F (P′))

we get by conjugation a unique element (also denoted α), which lives in the same sum, but now taken over all P′, and not just conjugacy classes. We can describe d^{1}(α) as the following composition:

d^{1}(α) : C_{k+1}(|C′_{≥P} |, |C′_{>P} |) →^{∂} ⨁_{P <P′;ht(P′)=ht(P)+1} C_{k}(|C′_{≥P′}|, |C′_{>P′}|) →^{α}

⨁_{P <P′;ht(P′)=ht(P)+1} F (P′) →^{F (P <P′)} F (P)

where ∂ is the boundary map associated to the triple (|C′_{≥P} |, |C′_{≥P,ht≥ht(P)+1}|, |C′_{≥P,ht≥ht(P)+2}|).

That we can replace C ∩ B_{p}^{e}(G) by C ∩ D_{p}(G) is CP acts trivially on F (P) for all P ∈ C, follows from Theorem 1.2.

4. The case of a finite group of Lie type

The purpose of this section is to examine the spectral sequence in Theorem 1.3 in the special case of a finite group of Lie type G of characteristic p, where it degenerates into a chain complex.

We first need to recall some standard Lie notation. To a finite group of Lie type G of characteristic p, we can associate a split BN-pair of characteristic p (cf. e.g. [24, p. 50],[26]), which we will always assume to be fixed once and for all. Let S be the set of simple reflections of the Weyl group of the BN-pair. (Beware that this Weyl group need not coincide with the Weyl group of the ambient algebraic group if the Frobenius map is twisted.) Any parabolic subgroup in G is conjugate to exactly one ‘standard’ parabolic subgroup, and the standard parabolic subgroups in G are in one-to-one correspondence with subsets J ⊆ S, and are denoted P_{J} correspondingly. One defines U_{J} as the largest normal p-subgroup in P_{J}. Recall that one has that N_{G}(U_{J}) = P_{J} and P_{J} = U_{J} ⋊ L_{J}, where L_{J} again is a finite group of Lie type, with a split BN-pair of rank |J|. To any finite group of Lie type G of characteristic p we can associate the Steinberg module St_{G} (see e.g. [37, 26]). One of the many definitions of St_{G} is St_{G} = H_{r−1}(Δ; Z_{(p)}) = H_{r−1}(Δ; Z) ⊗ Z_{(p)}, were Δ is the Tits building of the BN-pair of G and r is the rank of the BN pair. It is a projective indecomposable module of dimension |G|_{p}, whose reduction mod p is also simple.

A theorem of Borel and Tits [11, 21] says that any p-radical subgroup is conjugate to exactly one U_{J}, and it easily follows (cf. e.g. [61] or Lemma 4.2) that B_{p}^{e}(G) is equal to the opposite poset of the poset of parabolic subgroups in G. Hence the NP/P which occur in Theorem 1.3 will be of the form L_{J} and, e.g., by Webb’s Theorem 5.1, we may replace St_{∗}(L_{J}) with St_{L_{J}} in degree |J| − 1 (cf. Remark 2.4). Choosing as height function ht(U_{I}) = −|I|, Theorem 1.3 now takes the following simple form.

Theorem 4.1. Let G be a finite group of Lie type of characteristic p, C be a collection of p-subgroups closed under passage to p-supergroups, and F : O_{C}^{op} → Z_{(p)}- mod any functor. Then lim_{O_{C}}^{∗} F equals the homology of a cochain complex (C^{∗}(F), δ) with

C^{i}(F) = ⨁_{J⊆S;|J|=i;U_{J} ∈C} Hom_{L_{J}} (St_{L_{J}} , F (U_{J}))

We now proceede to examine the boundary map in Theorem 4.1, which will be the content of Theorem 4.6. It turns out that special properties of finite groups of Lie type makes this boundary map very tractable.

Let RO(G)- mod denote the category of functors O(G)^{op} → R- mod. For any subgroup H of G, we have a functor RO(G)- mod → RW H- mod given by evaluation F ↦ F (H). This functor has a right adjoint given by to an W H-module M associating a functor E_{H,M} given by

E_{H,M}(H′) = (⊕_{Mor_{G}(G/H,G/H′)}M)^{W H}.

Here W H acts on M in the prescribed way, and on Mor_{G}(G/H, G/H′) via g · α = α ∘ g^{−1}. (Mor_{G} denotes morphisms in the category of left G-sets.) In particular, for any functor F we get a natural transformation F → E_{H,F (H)} induced by the unit of this adjunction. (See [34] for a careful explanation of the role of such adjunctions.)

For finite groups of Lie type, this right adjoint has a particularly simple form.

Lemma 4.2. If G is a finite group of Lie type, and U_{I}, U_{J′} are the unipotent radicals of standard parabolic subgroups P_{I}, P_{I′}, then

E_{U_{I},M}(U_{I′}) = \{ \begin{matrix}0 & if I′ ⊈ I \\ M^{U_{I′}} & if I′ ⊆ I \end{matrix}

Proof. As usual we have

Mor_{G}(G/U_{I}, G/U_{I′}) = {g ∈ G|g^{−1}U_{I}g ≤ U_{I′}}/U_{I′}.

We claim that

\{g ∈ G|g^{−1}U_{I}g ≤ U_{I′}\}/U_{I′} = \{ \begin{matrix}P_{I}/U_{I′} & if I′ ⊆ I \\ ∅ & else \end{matrix} .

This statement easily seen to be equivalent to the fact that the poset of parabolic subgroups in G is equal to the opposite poset of the poset of unipotent radicals of parabolic subgroups in G, which is well known. However, since we have been unable to find a reference for an explicit proof of this fact, we will give a direct argument. We follow standard BN-pair notation as in [26, §64,65,69] (except using U_{I} for their V_{I}), and assume some familiarity with this material. If g ∈ P_{I}, it is clear that g^{−1}U_{I}g ⊆ U_{I′} iff I′ ⊆ I, so suppose g ∉ P_{I}. Using the Bruhat decomposition [26, Thm. 65.4] we can write g = bẇb′, where ẇ is a lift of a w ∈ W \ W_{I}, and b, b′ ∈ B. Hence we can without restriction assume g = ẇ, where w has an expression as a reduced word of simple reflections, starting with s_{α}, for some α ∈ S \ I. But then w^{−1}α is a negative root, by the theory of root systems [26, Thm 64.16(vii)]. This means that g^{−1}U_{I}g ⊈ U_{∅} [26, 69.5], and the claim is justified.

We now see that

E_{U_{I},M}(U_{I′}) = (⊕_{P_{I}/U_{I′}} M)^{P_{I}/U_{I}} = Hom_{P_{I}} (Z_{(p)}, M ↑_{U_{I′}}^{P_{I}}) = M^{U_{I′}} .

Remark 4.3. The proof of the above lemma easily adapts to give a proof of another related fact. For G a finite group of Lie type and C the collection of p-subgroups which contain a conjugate subgroup of U_{I}, then the category O_{C∩B_{p}^{e}(G)} is equivalent to O_{B_{p}^{e}(L_{I})}. Note that L_{I} has smaller semisimple rank than G if I ≠ S, which allows for proofs by induction. One can for instance use this to give an independent inductive proof of Theorem 4.1 and 4.6 which instead of subgroup complexes only relies on knowledge of the irreducible representations of G.

Lemma 4.4. We have that (St_{Z_{(p)}G})^{U_{J}} ≃ St_{Z_{(p)}L_{J}} as Z_{(p)}L_{J}-modules.

Proof. The corresponding statement over F_{p} instead of Z_{(p)} is a consequence of a general theorem of S. Smith about U_{J}-fixpoints of irreducible modules of finite groups of Lie type (cf. [58, 22]). We now show how to get the statement over Z_{(p)} from that over F_{p}, in a slightly indirect way. Note that Ext_{U_{J}}^{1} (St_{Z_{(p)}G}, Z_{(p)}) = 0 since St_{Z_{(p)}G} is projective. Hence we have an exact sequence

0 → Hom_{U_{J}} (St_{Z_{(p)}G}, Z_{(p)}) →^{p} Hom_{U_{J}} (St_{Z_{(p)}G}, Z_{(p)}) → Hom_{U_{J}} (St_{Z_{(p)}G}, F_{p}) → 0

which shows that (St_{Z_{(p)}G})^{U_{J}} ⊗_{Z_{(p)}} F_{p} ≃ (St_{F_{p}G})^{U_{J}}, using that St_{Z_{(p)}G} is self-dual.

Hence the two expressions in the lemma are isomorphic after reducing mod p. Since St_{Z_{(p)}L_{J}} is Z_{(p)}L_{J}-projective this isomorphism can be lifted over to over Z_{(p)}L_{J}, where it has to be an isomorphism by Nakayama’s lemma.

Lemma 4.5. Assume J′ ⊂ J, |J| = |J′| + 1, and let F = E_{U_{J} ,St_{L_{J}}}. Then the restriction and projection of the boundary map δ : C^{|J′|}(F) → C^{|J|}(F) onto the corresponding factors J′ and J,

δ : Hom_{L_{J′}} (St_{L_{J′}} , (St_{L_{J}})^{U_{J′}}) → Hom_{L_{J}} (St_{L_{J}} , St_{L_{J}})

is an isomorphism.

Proof. First note that by Lemma 4.4 we know that (St_{L_{J}})^{U_{J′}} ≃ St_{L_{J′}}. By Nakayama’s lemma it is enough to prove the stated isomorphism after reducing both sides modulo p. Furthermore, one immediately check as in Lemma 4.4, that the appropriate Ext^{1} vanishes so that we may commute − ⊗_{Z_{(p)}} F_{p} and Hom. We have hence reduced the question of showing the isomorphisms to a question over F_{p}, and we therefore make the notational convention that in the rest of this proof, everything will be taken over F_{p} instead of Z_{(p)} and we suppress this from the notation.

Since St_{L_{J′}} is absolutely irreducible over F_{p}, we have that Hom_{L_{J′}} (St_{L_{J′}} , St_{L_{J′}}) = F_{p} (cf. [16, §13.4]), and likewise for J. Hence to show the isomorphism, it is enough to show that the identity map of St_{L_{J′}} is carried to a non-zero map under δ.

The proof of Theorem 1.3 gives a concrete description of δ. Since the homology of the chain complexes appearing in the proof of Theorem 1.3 are projective, we can replace the chain complexes with their homology. Hence we get that the restriction and projection of the boundary map δ : C^{|J′|}(F) → C^{|J|}(F) onto the corresponding factors J′ and J is given as follows:

δ(1) : St_{L_{J}} → St_{L_{J′}} ↑_{P_{J′}}^{P_{J}} →^{1} St_{L_{J′}} ↑_{P_{J′}}^{P_{J}} → St_{L_{J}}

Here first map is injective, by the long exact sequence in homology inducing it. The last map is surjective, since it is seen to be a nontrivial map of P_{J}-modules by its definition. But it is well known that St_{L_{J′}} ↑_{P_{J′}}^{P_{J}} contains a unique summand of St_{L_{J}} (cf. [26, Thm. 67.10]), so we conclude that δ(1) is in fact non-trivial, and hence an isomorphism.

Theorem 4.6 (Addendum to Theorem 4.1). Assume J′ ⊂ J, |J| = |J′| + 1, and let e be a central idempotent in Z_{(p)}L_{J} corresponding to the Steinberg block [26]. The restriction and projection of the boundary map δ : C^{|J′|}(F) → C^{|J|}(F) onto the corresponding factors J′ and J is given as the top horizontal composite in the following diagram:

\xymatrix{δ : Hom_{L_{J′}} (St_{L_{J′}} , F (U_{J′})) \ar[r] & Hom_{L_{J′}} (St_{L_{J′}} , F (U_{J})^{U_{J′}}) \ar[r]^{δ′} \ar@{->>}[d] & Hom_{L_{J}} (St_{L_{J}} , F (U_{J})) \ar@{->>}[d]^{≃} \\  & Hom_{L_{J′}} (St_{L_{J′}} , (eF (U_{J}))^{U_{J′}}) \ar[r]^{δ′′}_{≃} & Hom_{L_{J}} (St_{L_{J}} , eF (U_{J}))}

Here the top left horizontal map is induced by the map in O_{C′} (landing in the invariants), and δ′ and δ′′ are the boundary maps in the chain complex for E_{U_{J} ,F (U_{J})} and E_{U_{J} ,eF (U_{J})} respectively.

Proof. The diagram is commutative, since it is induced by the natural transformations of functors F → E_{U_{J} ,F (U_{J})} → E_{U_{J} ,eF (U_{J})}. That δ′′ is an isomorphisms is the contents of Lemma 4.5.

5. The Steinberg complex

In this section we analyze the Steinberg complex St_{∗}(G) = C̃_{∗}(|S_{p}(G)|; Z_{(p)}). Many of the results in this section are known to group theorists, and mainly due to Webb [67, 66, 68] (see also [10, 54, 12, 13, 14]). However we find it worthwhile to include them, since they are important for calculations with the results of the previous section.

In this section we shall mainly work over the p-adic integers Z_{p}. The main reason for this is the following theorem.

Theorem 5.1 (Webb [68]). The Steinberg complex over Z_{p} has the following structure

St_{∗}(G; Z_{p}) = P_{∗} ⊕ D_{∗}

as Z_{p}G-modules, where P_{∗} is a complex of Z_{p}G-projective modules, and D_{∗} is a Z_{p}G-split acyclic complex.

(Unfortunately, the proof given in [68] is not very clear on how to pass from the corresponding statement over F_{p}, which is first proved, to the statement over Z_{p}, the problem being that although all morphisms lift, the lifting is not functorial—this was pointed out to me by P. Symonds, who also explained to me how to overcome this problem [60].) In the category of bounded chain complexes over Z_{p}G the Krull-Schmidt theorem holds, so there exists a unique minimal such P_{∗} (unique up to isomorphism of chain complexes) (see [10, 5.17.1]). We denote such a minimal P_{∗} by 𝕊t_{∗}(G)—it is argueably really 𝕊t_{∗}(G) which deserves the name Steinberg complex [5].

We now proceed to give restrictions on which projective modules that can occur in 𝕊t_{∗}(G). Since in any degree St_{∗}(G; Z_{p}) is a sum of permutation modules, we first need a method of seeing which projective modules can occur as a summand of a given permutation module. Recall that there is a bijection between projective modules over Z_{p} and F_{p} (cf. [55, 14.4]), so we can restrict our attention to fields. Our method is the following lemma (with T = k), which is hinted in [67, Prop. 5.3].

Lemma 5.2. Let k be a field of characteristic p, and assume that H is a subgroup of a finite group G. Let S be a simple kG-module and let T be a simple kH-module. Denote the projective covers P_{S} and P_{T}. Then [End_{G} S : k] times the multiplicity of P_{S} in T ↑_{H}^{G} equals [End_{H} T : k] times the multiplicity of P_{T} in S ↓_{H}^{G}.

Proof. Let Ω^{0}M denote the largest submodule of a kG-module M, which does not contain any projective summands (unique only up to isomorphism). We have a decomposition M = Ω^{0}M ⊕ P, where P is projective. Note that the multiplicity of P_{T} in S ↓_{H}^{G} is equal to the End_{H} T -dimension of coker(Hom_{H}(T, Ω^{0}(S ↓_{H}^{G})) ↪ Hom_{H}(T, S ↓_{H}^{G})). Likewise the multiplicity of P_{S} in T ↑_{H}^{G} equals the End_{G}(S)-dimension of coker(Hom_{G}(Ω^{0}(T ↑_{H}^{G} ), S) ↪ Hom_{G}(T ↑_{H}^{G}, S)). So in order to prove the lemma, we just need to justify that Hom_{H}(T, Ω^{0}(S ↓_{H}^{G})) ≃ Hom_{G}(Ω^{0}(T ↑_{H}^{G}), S). We will show this by showing that we may replace Hom in both expressions with \underline{Hom}, i.e., Hom in the stable module category, where the statement will follow by Frobenius reciprocity in the stable module category (cf. [3, p. 74]). Assume α : Ω^{0}(k ↑_{H}^{G}) → S is a kG-map which factors through a projective module, and hence through P_{S}.

\xymatrix{ & P_{S} \ar@{->>}[d] \\ Ω^{0}(T ↑_{H}^{G}) \ar@{-->}[ur]^{α̃} \ar[r]^{α} & S}

If α is non-trivial then Ω^{0}(T ↑_{H}^{G}) → P_{S} has to be surjective, since it is surjective on the head, contradicting that Ω^{0}(T ↑_{H}^{G}) does not contain any projective summands. Hence α is trivial and Hom_{G}(Ω^{0}(T ↑_{H}^{G}), S) ≃ \underline{Hom}_{G}(T ↑_{H}^{G}, S). The proof that Hom_{H}(T, Ω^{0}(S ↓_{H}^{G})) ≃ \underline{Hom}_{H}(T, S ↓_{H}^{G}) is dual.

By [68] the conclusion of Webb’s Theorem 5.1 does not change if we replace St_{∗}(G; Z_{p}) by a G-homotopy equivalent chain complex. Hence 𝕊t_{∗}(G) has to be a direct summand of the reduced normalized cellular chain complex of any space X G-equivalent to |S_{p}(G)|. Lemma 5.2 says that if the space X has ‘large’ isotropy groups, then the simple modules S, corresponding to projective modules P_{S} appearing in 𝕊t_{∗}(G), also have to be ‘large’. Note that a non-degenerate n-simplex σ = (P_{0} < ⋯ < P_{n}) ∈ |C|_{n} has stabilizer G_{σ} = ∩_{i=0}^{n}NP_{i}.

There are many spaces which are known to be G-equivalent to |S_{p}(G)|. This is true for both |B_{p}(G)| and |A_{p}(G)| ([15], [50], [61]; for proofs see also the proofs of Theorem 1.1 and 6.1). If G is a finite group of Lie type of characteristic p then |B_{p}(G)| is just the subdivision of the building Δ of G (by the theorem of Borel and Tits [11]), and especially they are G-homotopy equivalent. Furthermore |A_{p}(G)| is G-homotopy equivalent to the smaller complex given by the nerve of the collection of elementary abelian p-subgroups V satisfying that V is exactly the elements of order p in the center of CV [10, §6.6]. The so-called commuting complex K_{p}(G) of Alperin [5, 7], which has n-simplices (n + 1)-touples of pairwise commuting subgroups of order p is also G-equivalent to |S_{p}(G)| [62, p. 187]. (One way to see this is to note that the category with objects arbitrary touples of pairwise commuting subgroups of order p viewed as a poset under inclusion is equivariantly equivalent to A_{p}(G); functors back and forth are given by associating to an n-touple the elementary abelian p-subgroup it generates, and to an elementary abelian p-group associating the touple of all rank one p-subgroups.)

Fix a Sylow p-subgroup P of G and define the following subgroup

B = ∩_{Q∈B_{p}(G)_{≤P}} NQ.

By Sylow’s theorem it is uniquely defined up to conjugacy. It lies between CP and NP and so p||B| if p||G|. If G is a finite group of Lie type then B = NP and B is a Borel subgroup.

Proposition 5.3. Let P_{S} be the projective module over Z_{p}G corresponding to the projective cover over F_{p}G of a simple F_{p}G-module S. Assume P_{S} appears as a summand in 𝕊t_{m}(G). Then we can say the following about S:

1. No element of order p in G acts trivially on S.

2. The module S ↓_{B}^{G} contains the projective cover P_{F_{p}} of the trivial F_{p}B-module F_{p} as a direct summand. In particular dim_{F_{p}} S ≥ |B|_{p}.

3. There exists an elementary abelian p-subgroup V of G with rk V ≥ m + 1 such that S ↓_{CV}^{G} contains the projective cover P_{F_{p}} of the trivial CV -module F_{p} as a direct summand. In particular dim_{F_{p}} S ≥ |CV |_{p} ≥ p^{m+1}.

Proof. To see the first claim let K = ker(G → Aut(S)) and assume that P_{S} is a summand of Z_{p}[G/G_{σ}] for some σ ∈ |S_{p}(G)|. Pick a Sylow p-subgroup P containing the chain of p-subgroups defining σ. Since K is normal in G, P ∩ K is a Sylow p-subgroup of K. Now suppose there exists an element of order p which acts trivially on S, so P ∩ K ≠ e. Then, since P and P ∩ K are p-groups, we can find a non-trivial element g ∈ P ∩ K which is fixed under the conjugation action of P. This means that ⟨g⟩ ⊆ G_{σ}. By Lemma 5.2, S ↓_{G_{σ}}^{G} contains a summand of the projective F_{p}G_{σ}-module P_{F_{p}}. But then the further restriction S ↓_{⟨g⟩}^{G} has to contain a projective module as well. Hence it cannot be the trivial module, which is a contradiction since g ∈ K.

To see the second claim we choose to view the Steinberg complex as the reduced normalized chain complex of |B_{p}(G)|. If P_{S} appears as a direct summand in C̃_{∗}(|B_{p}(G)|; Z_{p}), then P_{S} appears as a direct summand of Z_{(p)}[G/G_{σ}], for some σ ∈ |B_{p}(G)|. Hence by Lemma 5.2, S ↓_{G_{σ}}^{G} contains the projective cover P_{F_{p}} of the trivial F_{p}G_{σ}-module F_{p} as a direct summand., so by Sylow’s theorem G_{σ} contains a conjugate of B, and S ↓_{B}^{G} has to contain a copy of the projective cover of the trivial F_{p}B-module as well. It is now obvious that dim_{F_{p}} S ≥ |B|_{p}, since a projective F_{p}B-modules has dimension divisible by |B|_{p}.

To see the last claim we choose to view the Steinberg complex as the reduced normalized chain complex of |A_{p}(G)|. Observe that if σ = (V_{0} < ⋯ < V_{m}) is a nondegenerate msimplex in |A_{p}(G)| then CV_{m} ≤ G_{σ} and V_{m} has at least rank m + 1. Hence the last claim follows from Lemma 5.2 as well.

Note that the above proposition, combined with Theorem 1.1, has e.g. [39, Prop. 5.5] as a special case (compare also [27, Thm. 6.3]).

Example 5.4. If G is finite group of Lie type, then the Steinberg module St_{G} is the only simple module S such that S ↓_{B}^{G} contains a summand of P_{F_{p}}, agreeing nicely with our knowledge that the Steinberg complex in this case is chain homotopy equivalent to the Steinberg module in degree equal to the semi-simple rank of G minus 1.

Example 5.5. The structure of the Steinberg complex for the symmetric group Σ_{n} is unknown for large n, and appears to be quite complicated. See [12, 13] for low dimensional calculations. Note for instance that the modules which occur do not seem to be limited to any particular set of blocks of F_{p}Σ_{n}. It would be nice to have a better understanding of this case. (See [6] for a list of the p-radical subgroups in Σ_{n}.)

The next observation (which has also been used by group theorists [8, p. 479]) gives, when combined with Theorem 1.1, a refinement of [39, Prop. 6.1(iii)].

Proposition 5.6. Let K be a normal subgroup in G of order prime to p. Then |S_{p}(G)|/K = |S_{p}(G/K)|, and in particular St_{∗}(G/K) = St_{∗}(G)_{K}, where (−)_{K} denotes coinvariants.

Proof. To see the first claim we have to see that if (P_{0} ≤ ⋯ ≤ P_{n}) and (P′_{0} ≤ ⋯ ≤ P′_{n}) are two chains of subgroups in G such that P_{i}K = P′_{i}K for all i, then the two chains are conjugate by a single k ∈ K. Since K is normal in G then by Sylow’s theorem, P K = P′K iff we can find k ∈ K such that P′ = kP k^{−1}. In our two chains above we can hence assume that P_{n} = P′_{n}. But then P_{i}, P′_{i} are subgroups of P_{n} conjugate by an element in K, so P_{i} = P′_{i} (since if g ∈ P_{n}, k ∈ K and kgk^{−1} ∈ P_{n} then kgk^{−1}g^{−1} ∈ K ∩ P_{n} = e). The second claim is a trivial consequence of the first.

Although the homology groups of St_{∗}(G) need not in general be projective [5, 54] this is however true for H_{0}, as shown by Quillen [50, Prop. 5.2] and Webb [66, pf. of Thm. E]. Fix a Sylow p-subgroup P of G and let

G̃ = subgroup of G generated by N_{G}(Q), Q ∈ S_{p}(G)_{≤P}

(Using that B_{p}(G) and S_{p}(G) are G-equivalent and arguing as in [50, Prop. 5.2], one sees that we may replace S_{p}(G) by B_{p}(G) in the above definition of G̃.)

Proposition 5.7. (Quillen [50], Webb [66]) We have that |S_{p}(G)| = G ×G̃ |S_{p}(G̃)|, and S_{p}(G̃) is connected. Furthermore H_{0}(St_{∗}(G)) = ker(Z_{(p)}[G/G̃] →^{aug} Z_{(p)}) is a projective Z_{(p)}G-module and St_{∗}(G) ≃ H_{0}(St_{∗}(G)) ⊕ St_{∗}(G̃) ↑G ; G̃. □

Remark 5.8. Groups G such that S_{p}(G) is disconnected are sometimes called p-isolated [33]. Groups such that S_{p}(G) is disconnected or empty ‘have a strongly p-embedded subgroup’. (See e.g., [50, Prop. 5.2].) Such groups have been ‘classified’—see [7, (6.2)].

Example 5.9. If G is a finite group of Lie type with G ≠ G̃ then G has semisimple rank 1. In this case there is, up to conjugacy, only one non-trivial p-radical subgroup namely the Sylow p-subgroup and G̃ = B, the Borel subgroup. The expression in Proposition 5.7 for H_{0}(St_{∗}(G)) is just the standard alternating sum formula (cf. [26, Thm. 66.35]) for the Steinberg module.

Corollary 5.10. Let C be a collection of p-subgroups closed under passage to p-supergroups and let F : O_{C}^{op} → Z_{(p)}- mod be a functor concentrated conjugates of a subgroup P. Then lim_{O_{C}}^{1} F = Hom_{W P} (H_{0}(St_{∗}(W P)), F (P)) and lim_{O_{C}}^{2} F = Hom_{W P} (H_{1}(St_{∗}(W P)), F (P))

Proof. The statement for lim^{1} follows directly from Theorem 1.1 combined with Proposition 5.7 above. The lim^{2} statement also follows from these two results, using that Hom_{G}(−, M) is left exact.

Note that this proposition has [39, Prop. 6.2] as a consequence.

Remark 5.11. If a projective module P_{S} corresponding to a simple F_{p}G-module S appears in 𝕊t_{0}(G) then by Lemma 5.2 S ↓G ; G̃ contains a summand of the projective F_{p}G̃ module P_{F_{p}}, and dim_{F_{p}} S ≥ |G|_{p}, a much stronger estimate than Proposition 5.3.

More precisely, using Frobenius reciprocity in the stable module category, as in the proof of Lemma 5.2, one sees that for any F_{p}G-module M

dim_{F_{p}} Hom_{G}(𝕊t_{0}(G; F_{p}), M) = (mult. of P_{F_{p},G̃} in M ↓G ; G̃) − (mult. of P_{F_{p},G} in M)

where P_{F_{p},G̃} and P_{F_{p},G} denotes the projective covers of F_{p} over F_{p}G̃ and F_{p}G respectively.

Example 5.12. Let G = Σ_{2p} and suppose that p ≥ 5. Note that G̃ = Σ_{p} ≀ Z/2 and that Σ̃_{p} = Z/p ⋊ Z/(p − 1). Let M = 𝕊t_{0}(Σ_{p}) = ker(Z_{p}[Σ_{p}/(Z/p ⋊ Z/(p − 1))] →^{aug} Z_{p}). Note also that S_{p}(G̃) = S_{p}(Σ_{p}) ⋆ S_{p}(Σ_{p}), so 𝕊t_{∗}(G̃) = 𝕊t_{1}(G̃) = M ⊗_{Z_{p}} M, with the obvious Σ_{p} ≀ Z/2 action. Combining this describes the Steinberg complex of Σ_{2p}. We have that 𝕊t_{0}(G) = ker(Z_{p}[Σ_{2p}/(Σ_{p} ≀ Z/2)] →^{aug} Z_{p}), 𝕊t_{1}(G) = (M ⊗_{Z_{p}} M) ↑_{Σ_{p}≀Z/2}^{Σ_{2p}}, and the differential is zero.

For later reference we also record the following useful fact, which follows directly from the definition of 𝕊t_{∗}(G) as a suitably minimal summand of a complex of permutation modules.

Proposition 5.13. The complex 𝕊t_{∗}(G) is dimensionwise self-dual i.e., 𝕊t_{n}(G) ≃ Hom_{Z_{p}}(𝕊t_{n}(G), Z_{p}) as Z_{p}G-modules. □

6. Higher limits over C-conjugacy categories

The C-conjugacy category A_{C} is the category with objects subgroups H ∈ C and morphisms homomorphisms φ : H → H′ for which there exists g ∈ G such that φ = g(·)g^{−1}. This category occurs e.g., when working with the centralizer homology decomposition (cf. [27, 29]).

Given a functor F : A_{C} → Z_{(p)}- mod we equip |C| with the G-local coefficient system F given by precomposing with the canonical map (d̄ C)_{G} → C_{G} → A_{C} (see Remark 2.2). On objects we have F(H_{0} ≤ ⋯ ≤ H_{n}) = F (H_{n}).

For higher limits of functors over A_{C} we obtain dual results to the ones for O_{C}.

Theorem 6.1. Let C be any collection of non-trivial p-groups closed under passage to nontrivial elementary abelian subgroups, and let F : A_{C} → Z_{(p)}- mod be any functor. Then

lim_{A_{C}} ^{∗}F ≃ H_{G}^{∗}(|C|; F)

where F is the G-local coefficient system on |C| induced via (d̄ C)_{G} → C_{G} → A_{C}.

Furthermore for any collection C′ satisfying C ∩ A_{p}(G) ⊆ C′ ⊆ C, the inclusion |C′| → |C| induces an isomorphism H_{G}^{∗}(|C|; F) → H_{G}^{∗}(|C′|; F).

If F is a functor concentrated on a single conjugacy class of subgroups with representative V, then

lim_{A_{C}} ^{i}F = \{ \begin{matrix}Hom_{NV/CV} (St_{GL V} , F (V)) & if V ∈ A_{p}(G) and i = rk V − 1 \\ 0 & otherwise. \end{matrix}

We prove this result later in this section. (To see the duality to the case of orbit categories we remark that C̃_{∗}(|S_{p}(G)_{<V} |) is chain homotopy equivalent to St_{GL V} in degree rk V − 2 if V is elementary abelian and is contractible otherwise.) The formula for the higher limits for an atomic functor is due to Oliver [48], and using the rank filtration on A_{p}(G) we recover his chain complex, argueing as in the proof of Theorem 1.3.

Theorem 6.2. [48] Under the assumptions and notation of Theorem 6.1 we have that lim_{A_{C}}^{∗} F equals the homology of a cochain complex (C^{∗}(F), δ) with

C^{i}(F) = ⨁_{[V]∈C∩A_{p}(G)/G;rk V =i+1} Hom_{NV/CV} (St_{GL(V)}, F (V))

The boundary map can be described as follows. Given α ∈ Hom_{NV/CV} (St_{GL(V)}, F (V)), then the projection of δ(α) onto Hom_{NU/U}(St_{GL(U)}, F (U)) where V ⊂ U and dim_{F_{p}} U = dim_{F_{p}} V + 1 is given by the composite

St_{GL(U)} → ⨁_{V′⊂U;[V′]=[V]∈A_{p}(G)/G} St_{GL(V′)} →^{α} ⨁_{V′⊂U;[V′]=[V]∈A_{p}(G)/G} F (V′) → F (U)
□

Before proving Theorem 6.1 we establish some results.

Theorem 6.3. Let F : A_{C} → Z_{(p)}- mod be an atomic functor concentrated on conjugates of H ∈ C. Then

1. H_{G}^{∗}(|EA_{C}|; F) ≃ H_{h(NH/CH)}^{∗}(|C_{≤H}|, |C_{<H}|; F (H)) ≃ H̃_{h(NH/CH)}^{∗}(Σ|C_{<H}|; F (H)).

2. If |C_{<H}|^{Q} is F_{p}-acyclic for all non-trivial p-subgroups in NH/CH, then

H_{G}^{∗}(|EA_{C}|; F) ≃ H_{G}^{∗}(|C|; F) ≃ H_{NH/CH}^{∗}(|C_{≤H}|, |C_{<H}|; F (H)) ≃ H̃_{NH/CH}^{∗}(Σ|C_{<H}|; F (H)).

Proof. The proof is dual to the proof of Theorem 3.1, so we will be brief. Writing out the definitions one sees that:

C_{G}^{n}(|EA_{C}|; F) = C_{NH}^{n}(|EA_{C}_{≤H}|, |EA_{C}_{<H}|; F (H))

where EA_{C}_{≤H} (resp. EA_{C}_{<H}) denotes the full subcategory of EA_{C} with objects the objects f : H′ → G in EA_{C} satisfying f(H′) ≤ H (resp. f(H′) < H). Hence we get corresponding identifications:

[figure: H_{G}^{∗}(|C|; F); ≃; H_{NH}^{∗}(|C_{≤H}|, |C_{<H}|; F (H)); H_{G}^{∗}(|EA_{C}|; F); ≃; H_{NH}^{∗}(|EA_{C}_{≤H}|, |EA_{C}_{<H}|; F (H))]

To show 1 we note that NH/CH acts freely on the pair (|EA_{C}_{≤H}|, |EA_{C}_{<H}|) and argue as before. Likewise 2 follows dually.

The same proof as Corollary 3.5 now reveals:

Corollary 6.4. Let F : A_{C} → Z_{(p)}- mod be a functor and assume that |C_{<H}|^{Q} is F_{p}- acyclic for all H ∈ C and all non-trivial p-subgroups Q of NH/CH. Then H_{G}^{∗}(|C|; F) →^{≃} H_{G}^{∗}(|EA_{C}|; F). □

Proof of Theorem 6.1. We already know that lim_{A_{C}}^{∗} F = H_{G}^{∗}(|EA_{C}|; F) from Proposition 2.6 and 2.10, so to prove the first part of theorem, we just have to show that the conditions of Corollary 6.4 apply.

We start by assuming that C is closed under passage to all non-trivial subgroups, and will later show that non-elementary abelian p-subgroups can be removed without changing either higher limits or Bredon cohomology. Hence, for all H ∈ C, C_{<P} = S_{p}(G)_{<P}. If Q is a non-trivial p-subgroup of W P, then |S_{p}(G)_{<P}^{Q} | is contractible by the inequalities H ≥ H^{Q} ≤ P^{Q}. (The fixpoint subgroup H^{Q} is not trivial since Q and H are p-groups.) Now Corollary 6.4 and Theorem 6.3 proves the statements of Theorem 6.1, except the statement about C′ and the last rewriting in the atomic case.

We address the last rewriting first. If P not elementary abelian, then the inequalities H ≤ HΦ(P) ≥ Φ(P) show that |S_{p}(G)_{<P} | is NP -equivariantly contractible (see also [61]). (Here Φ(P) denotes the Frattini subgroup of P; alternatively, use the inequalities H ≥ H ∩ [P, P] ≤ [P, P] if P is non-abelian, and the inequalities H ≥ _{p}H ≤ _{p}P if P is abelian, instead.) Hence C̃_{∗}(|S_{p}(G)_{<P} |) is NP -contractible if P is not elementary abelian. If P = V is elementary abelian then |S_{p}(G)_{<V} | is the space associated to the ordered simplicial complex of flags of non-trivial proper sub-vectorspaces of V, which again is equal to the barycentric subdivision of the Tits building of GL(V), by the definition of the Tits building [64]. But C̃_{∗}(|S_{p}(G)_{<V} |) is NV -chain homotopy equivalent to its homology, which is just the Steinberg representation of GL(V) in dimension rk V − 2. (One way to see this is to use Webb’s Theorem 5.1.) This shows the last rewriting in the atomic case.

To see that we can replace C by C′ we use the same argument as in the proof of Theorem 1.1. We can suppose that F is an atomic functor concentrated on conjugates of a subgroup P ∈ C. If P ∈ C \ C′ then we are done by the formula for an atomic functor just proved. If P ∈ C′ then we are done if we can show that C_{<P} and C′_{<P} are NP -homotopy equivalent. This is done by showing that we can remove subgroups in C_{<P} \ C′_{<P} from C in order of decreasing size, without changing the NP homotopy type, using the pushout technique of the proof of Theorem 1.1 together with the NP′-contractibility of S_{p}(G)_{<P′} when P′ is not elementary abelian, as shown above.

Remark 6.5. In applications of Theorem 6.1, the NV/CV -action on F (V) can often be viewed as being the restriction of a GL V -action. In this case, we have by Frobenius reciprocity that

Hom_{NV/CV} (St_{GL V} , F (V)) = Hom_{GL V} (St_{GL V} , F (V) ↑_{NV/CV}^{GL V}).

This is often a convenient rewriting in concrete calculations.

Remark 6.6. A natural question to consider is whether one can obtain a similar description of higher limits of functors over other quotient categories D of C_{G} or (C^{op})_{G} than just A_{C} and O_{C}^{op}. The proof of Theorem 3.1 and 6.3 show what has to be satisfied. For quotient categories of C_{G} one needs that |C_{<H}|^{Q} is F_{p} acyclic for all H ∈ C and all non-trivial psubgroups Q ≤ Aut_{D}(H) (and analogously for (C^{op})_{G}). We see that this property is rather special for A_{C} (and O_{C}^{op}), and generally does not hold for, for example, their opposite categories. For their opposite categories we however get formulas like the ones found in this paper for higher colimits.

7. Higher limits over orbit simplex categories

Let (sd C)/G be the orbit simplex category, i.e., the poset with objects G-conjugacy classes [σ] of proper chains of subgroups σ = (H_{0} < ⋯ < H_{n}) and with a morphism [σ] → [σ′] if we can find representatives σ, σ′, such that σ is a refinement of σ′. This category occurs when working with the normalizer homology decomposition (cf. [27, 29]). Higher limits of functors on ((sd C)/G)^{op} can likewise be expressed as equivariant cohomology of the subgroup complex |C|. Indeed, this follows by a formal argument, and does not require any assumptions on the collection C or the ground ring.

Since (sd C)/G is already a finite poset and hence |(sd C)/G| = | sd C|/G is a finite space (with associated topological space homeomorphic to |C|/G), the only improvement will be to remove the subdivision sd. This follows easily from an argument of Słomińska [57]. (Alternatively one can use Remark 2.7: The topological spaces associated to | sd C| and |C| are G-homeomorphic, and under the canonical homeomorphism the sheaves on the topological spaces associated to | sd C|/G and |C|/G coincide.) The following proposition is due to her.

Proposition 7.1. Let C be a any collection of subgroups of a finite group G, and let F : ((sd C)/G)^{op} → R- mod be an arbitrary functor. Then

lim_{(sd C)/G}^{∗}F = H_{G}^{∗}(|C|; F),

where F is the G-local coefficient system given via (d̄ C)_{G} → ((sd C)/G)^{op}.

Proof. Note that |C| is in fact an ordered simplicial complex, so that the subcomplex |σ| generated by an n-simplex σ is contractible. Hence the functor σ ↦ C_{∗}(|σ|; R), i.e., the functor which to σ associates the normalized chain complex of the n-simplex σ, gives a more economical projective resolution of R in R(sd C)/G- mod, than the one of Proposition 2.6. In this case we see that Hom_{R(sd C)/G}(C_{n}(| − |; R), F) = ∏_{[σ]} F (σ), where [σ] runs over the G-conjugacy classes of non-degenerate n-simplices in |C|. Hence lim_{(sd C)/G}^{∗} F = H^{∗}(|C|/G; F) = H_{G}^{∗}(|C|; F) as wanted.

In the topological applications occurring when using the normalizer decomposition, F will most often be induced by a generic coefficient system 𝔉 : O(G)^{op} → R- mod, i.e., F is the functor which on objects is given by [(P_{0} ≤ ⋯ ≤ P_{n})] ↦ 𝔉(NP_{0} ∩ ⋯ ∩ NP_{n}). In this case we have C_{G}^{∗}(|C|; F) = C_{G}^{∗}(|C|; 𝔉). (Note however that F is not exactly the Glocal coefficient system induced by 𝔉.) Hence in this case we may replace |C| with any G-homotopy equivalent space, for the purpose of calculating H_{G}^{∗}(|C|; F). More precisely we get:

Corollary 7.2. Let C be a collection of subgroups of a finite group G and let F : (sd C)/G → R- mod be a functor induced by a generic coefficient system 𝔉. Then

lim_{(sd C)/G}^{∗}F = H_{G}^{∗}(|C′|; 𝔉)

for any collection C′ such that |C′| is G-equivalent to |C|. □

We end this section by examining the special case were 𝔉 is the generic coefficient system 𝔉(−) = H^{n}(−; Z_{(p)}), n ≥ 0, which illustrates that for particular 𝔉 one can do better than Corollary 7.2.

Theorem 7.3. Let C be a collection of p-subgroups, closed under passage to p-supergroups. Let 𝔉(−) = H^{n}(−; Z_{(p)}), n ≥ 0, and let F be the associated functor on objects given by [(P_{0} ≤ ⋯ ≤ P_{n})] ↦ 𝔉(NP_{0} ∩ ⋯ ∩ NP_{n}).

lim_{(sd C)/G}^{∗}F ≃ lim_{(sd C′′)/G}^{∗}F ≃ H_{G}^{∗}(|C′′|; 𝔉),

for C′′ any collection C ∩ D_{p}(G) ⊆ C′′ ⊆ C.

Proof. As in the proof of Theorem 1.2 it is enough to prove the statement over Z_{p}, and we therefore adopt the convention that all coefficients are over Z_{p}. (In this case we even have 𝔉(−) = 𝔉(−)⊗_{Z_{(p)}} Z_{p}.) We want to show that we can successively remove conjugacy classes of subgroups in C \ C′′ in order of increasing size, without changing the Bredon cohomology. So suppose that C̃ is a subcollection obtained from C by removing G-conjugacy of subgroups in this way and let Ĉ denote the collection obtained from C̃ by removing the G-conjugacy class of a minimal subgroup P ∈ C̃\ C′′. As in the proof of Theorem 1.1 we have a pushout square, which is a homotopy pushout of G-spaces

\xymatrix{G ×_{NP} | linkC̃ P | \ar[r] \ar[d] & |Ĉ| \ar[d] \\ G ×_{NP} | starC̃ P | \ar[r] & |C̃|}.

As usual we need to see that H̃_{NP}^{∗} (| linkC̃ P |; 𝔉) = 0. In other words we must see that H^{n}(NP; C̃_{∗}(| linkC̃ P |)) is an acyclic chain complex. But as in the proof of Theorem 1.2, C̃_{∗}(| linkC̃ P |) is NP -chain homotopy equivalent to 𝕊t_{∗}(W P) ⊗ ΣC̃_{∗}(|C̃_{<P′}|).

Consider the Lyndon-Hochschild-Serre spectral sequence:

E_{2}^{i,j} = H^{i}(W P; H^{j}(P; C̃_{r}(|C̃_{<P} |)) ⊗ 𝕊t_{m}(W P)) ⇒ H^{i+j}(NP; 𝕊t_{m}(W P) ⊗ C̃_{r}(|C̃_{<P} |))

Since H^{j}(P; C̃_{r}(|C̃_{<P} |)) ⊗ 𝕊t_{m}(W P) is a projective W P -module, the spectral sequence collapses and yields that the chain complex H^{n}(NP; C̃_{∗}(| linkC̃ P |)) is chain homotopy equivalent to the complex Hom_{W P} (Z_{p}, H^{n}(P; C̃_{∗}(|C̃_{<P} |)) ⊗ 𝕊t_{∗}(W P)) But, using Proposition 5.13,

Hom_{W P} (Z_{p}, H^{n}(P; C̃_{r}(|C̃_{<P} |)) ⊗ 𝕊t_{m}(W P)) = Hom_{W P} (𝕊t_{m}(W P), H^{n}(P; C̃_{r}(|C̃_{<P} |)))

which is zero by Proposition 5.3 if P is non-p-centric. So in this case our complex is chain homotopy equivalent to the trivial complex, and hence acyclic. If P is p-centric write CP = ZP ×K, where p |̸|K|. Since for all subgroups H ≤ NP, H^{∗}(H; Z_{p}) = H^{∗}(HK; Z_{p}), we have that

H̃_{NP}^{∗} (Σ| linkC̃ P |; 𝔉) = H̃_{NP}^{∗} (Σ| linkC̃ P |/K; 𝔉).

But | linkC̃ P |/K is homotopy equivalent to |S_{p}(NP/P C(P))|⋆|C̃_{<P} |, which is NP -contractible since by assumption O_{p}(NP/P C(P)) ≠ e. Hence H̃_{NP}^{∗} (Σ| linkC̃ P |; 𝔉) = 0 also in this case, which finishes the proof of the theorem.

Remark 7.4. The key feature in the above argument was being able to replace St_{∗}(W P; Z_{p}) with the chain homotopic complex 𝕊t_{∗}(W P). This will more generally be possible if just F extends from a functor O(G)^{op} → Z_{(p)}- mod to a Z_{(p)}-linear functor Z_{(p)}G- mod → Z_{(p)}- mod. (Having an extension to a Z_{(p)}-linear functor Z_{(p)}G-(permutation modules) → Z_{(p)}- mod is equivalent to F being a cohomological Mackey functor by [71].) Note also that in concrete situations the method of the above proof can be used to determine if further subgroups can be removed.

Example 7.5. To give an example where one cannot get away with just considering D_{p}(G), consider the generic coefficient system F (H) = R(H) ⊗ Z_{(2)}, where R denotes the complex representation ring. (This is a non-cohomological Mackey functor.) If G = Σ_{3} × Z/2 then H_{G}^{∗}(|S_{2}(G)|; F) = R(G) ⊗ Z_{(2)} and H_{G}^{∗}(|D_{2}(G)|; F) = R(Z/2 × Z/2) ⊗ Z_{(2)}. These are obviously different, the first having rank 6 the latter 4. (See [62] for interesting work using this coefficient system.)

8. Generalized Webb exact sequences

The purpose of this section is to examine our results in the special case where F is a functor for which all the higher limits in fact vanish. In this case our results produce exact sequences, the one corresponding to the normalizer decomposition being the one found by Webb [68]. We note that we do not assume that our functors are Mackey functors, although a Mackey structure is often needed in order to show that the higher limits indeed vanish.

We expand on work of Dwyer [29] and Webb [68], the main new results being Example 8.8 and 8.9 and Corollary 8.13, 8.14, and 8.18.

Our general setup is the following. Suppose we have a space G-space X, a functor F : (S(G)^{op})_{G} → Z- mod, and a map (d̄ X)_{G} → (S(G)^{op})_{G} making F a G-local coefficient system on X. If H_{G}^{n}(X; F) = 0 for n > 0 and H_{G}^{0}(X; F) = F (G), then writing up the normalized cochain complex gives us an exact sequence

0 → F (G) → ∏_{σ_{0}∈X_{0}/G} F (σ_{0})^{G_{σ_{0}}} → ∏_{σ_{1}∈X_{1}/G} F (σ_{1})^{G_{σ_{1}}} → ⋯,

coaugmented by setting C^{−1}(X; F) = F (G). Here X_{i} denotes the set of non-degenerate i-simplices of X. This sequence expresses F (G) in terms of subgroups. In this section we will describe some instances of the above situation, as well as some related methods.

8.1. Webb’s exact sequence. We will first try to explain the work of Webb in the current context—we refer the reader to [68, 69] for much more information, as well as many beautiful applications.

We say that a functor 𝔉 : O(G)^{op} → R- mod is projective (as a generic coefficient system) relative to a collection C if the canonical map, given as the unit of an adjunction, ⊕_{H∈C}𝔉 ↓_{H}^{G}↑_{H}^{G}→ 𝔉 is split surjective in the category of functors O(G)^{op} → R- mod. (For example if C is a cohomological Mackey functor to Z_{(p)}- mod then a transfer argument shows that 𝔉 is projective relative to C = S_{p}(G).)

Theorem 8.2 (Webb [68]). Let 𝔉 : O(G)^{op} → R- mod be any functor. Let X be a G-space. Consider two collections C′ ⊆ C of subgroups of G both closed under passage to subgroups. Assume that 𝔉 is projective relative to C, that 𝔉(H) = 0 for all H ∈ C′, and that X^{H} is contractible for all H ∈ C \ C′.

Then X is acyclic with respect to 𝔉 and we have a split exact sequence.

0 → 𝔉(G) → ∏_{σ_{0}∈X_{0}/G} 𝔉(G_{σ_{0}}) → ∏_{σ_{1}∈X_{1}/G} 𝔉(G_{σ_{1}}) → ⋯

Proof. Since ⊕_{H∈C}𝔉 ↓_{H}^{G}↑_{H}^{G}→ 𝔉 is assumed to be split surjective it is enough to prove acyclicity for 𝔉 ↓_{H}^{G}↑_{H}^{G}. However, since for any H-generic coefficient system 𝔉 we have the adjunction H_{G}^{∗}(X; 𝔉 ↑_{H}^{G}) = H_{H}^{∗}(X; 𝔉) we can without restriction assume that C is the collection of all subgroups, when proving acyclicity. The general acyclicity result now follows from Lemma 8.3 below, considering the special case when Y = ∗. The sequence in the theorem is in fact split exact follows from the proof of Lemma 8.3, since we may replace X by the G-contractible subspace X′.

Lemma 8.3. Let f : X → Y be a map of G-spaces, C a collection of subgroups of G closed under passage to subgroups, and 𝔉 : O(G)^{op} → R- mod an arbitrary generic coefficient system. Assume that F (H) = 0 for all H ∈ C and that f^{H} : X^{H} → X^{H} is a homotopy equivalence for all H ∉ C. Then H_{G}^{∗}(Y; 𝔉) →^{≃} H_{G}^{∗}(X; 𝔉).

Proof. Define X′ = ∪_{K∉C}X^{K} and note that C^{∗}(X; 𝔉) = C^{∗}(X′; 𝔉) by the assumption on 𝔉. We define Y′ similarly. It hence suffices to show that the restriction f′ : X′ → Y′ is a G-equivalence. If H ∉ C then X′^{H} = X^{H} and Y′^{H} = Y^{H} so f′^{H} is a homotopy equivalence by our assumption. If H ∈ C we need to see that f′^{H} : X′^{H} = ∪_{H≤K,K∉C}X^{K} → ∪_{H≤K,K∉C}Y^{K} = Y′^{H} is a homotopy equivalence. But this is true since for each of the pieces, X^{K}, and their intersections which will again be of the same form, the map f′_{|X^{K}}^{H} : X^{K} → Y^{K} is homotopy equivalence. (The most appealing argument for why this implies that f′^{H} is a homotopy equivalence is perhaps to interpret the union as a homotopy colimit, and refer to the homotopy invariance property of homotopy colimits.) Hence f′ is a homotopy equivalence on all fixpoints, so we conclude that f′ is a G-equivalence.

It is interesting to compare the method of the above lemma to the ‘method of discarded orbits’ in [29, §6.7].

8.4. Exact sequences arising from an acyclic collection. Let C be a collection of subgroups in G and let 𝔉 : O(G)^{op} → Z_{(p)}- mod be any functor. To 𝔉 we can associate three other functors. Let F^{β} : O_{C}^{op} → Z_{(p)}- mod be the functor given by H ↦ 𝔉(H). Let F^{α} : A_{C} → Z_{(p)}- mod be the functor given by H ↦ 𝔉(CH). Finally let F^{δ} : ((sd C)/G)^{op} → Z_{(p)}- mod be the functor given by [H_{0} < ⋯ < H_{k}] ↦ 𝔉(NH_{0} ∩ ⋯ ∩ NH_{k}). Note that we have canonical maps 𝔉(G) → lim_{O_{C}}^{0} F^{β}, F (G) → lim_{A_{C}}^{0} F^{α} and 𝔉(G) → lim_{(sd C)/G}^{0} F^{δ} given by restriction.

Definition 8.5. We say that C is subgroup acyclic with respect to 𝔉 if lim_{O_{C}}^{i} F^{β} = 0 for i > 0 and 𝔉(G) →^{≃} lim_{O_{C}}^{0} F^{β}. Likewise we say that C is centralizer or normalizer acyclic with respect to 𝔉 if the analogous conditions holds for F^{α} or F^{δ}.

By the rewritings of Proposition 2.6, 2.10 and 7.1, this definition can be stated as saying that C is subgroup (resp. centralizer and normalizer) acyclic with respect to 𝔉 iff the Gspace |EO_{C}| (resp. |EA_{C}| and |C|) is acyclic with respect to the generic coefficient system 𝔉.

Example 8.6 (The collection C = S_{p}^{e}(G)). Consider the case C = S_{p}^{e}(G), and let M be an arbitrary Z_{(p)}G-module. By the analysis of Section 2, the spaces |EO_{C}|, |EA_{C}| and |C| are P -contractible, where P is a Sylow p-subgroup of G. Since H^{n}(−; M) is a cohomological Mackey functor a transfer argument shows that C = S_{p}^{e}(G) is both subgroup, centralizer and normalizer acyclic with respect to H^{n}(−; M), n ≥ 0.

Example 8.7 (The collection C = S_{p}(G)). Let 𝔉(−) = H^{n}(−; M) : O(G)^{op} → Z_{(p)}- mod, where M is some Z_{(p)}G-module. By our analysis in sections 3 and 6 the spaces |EO_{S_{p}(G)}|, |EA_{S_{p}(G)}| and |S_{p}(G)| are P -equivalent. Hence, by an application of the transfer, we see that

H_{G}^{∗}(|EO_{S_{p}(G)}|; 𝔉) ≃ H_{G}^{∗}(|S_{p}(G)|; 𝔉) ≃ H_{G}^{∗}(|EA_{S_{p}(G)}|; 𝔉).

For n > 0 the formula in Theorem 1.1 shows that H_{G}^{∗}(|EO_{S_{p}(G)}|; 𝔉) = H_{G}^{∗}(|EO_{S_{p}^{e}(G)}|; 𝔉), since H^{n}(e; M) = 0 for n > 0. So, by the previous example, we see that S_{p}(G) is both subgroup, centralizer and normalizer acyclic with respect to H^{n}(−; M), n > 0. (Alternatively, a proof can be obtained by appealing to Lemma 8.3.) For n = 0 we see by definition that |S_{p}(G)| is acyclic with respect to H^{0}(−; M) iff Hom_{G}(St_{∗}(G), M) is an acyclic cochain complex. Hence S_{p}(G) is subgroup acyclic with respect to H^{0}(−; M) iff it is centralizer acyclic iff it is normalizer acyclic, iff Hom_{G}(St_{∗}(G), M) is an acyclic cochain complex.

By removing subgroups we can now establish sharpness of smaller collections.

Example 8.8 (The collection D_{p}(G)). Example 8.6 combined with Theorem 1.2 and 7.3 and Corollary 7.2 show that D_{p}(G) is subgroup and normalizer acyclic with respect to 𝔉 = H^{n}(−; Z_{(p)}), n ≥ 0.

Example 8.9 (The collections D_{p,M}^{e}(G) and D_{p,M}(G)). The previous example can be generalized to deal with arbitrary coefficient modules. Let M be a Z_{(p)}G-module and let K be the kernel of the homomorphism G → Aut M. As in [27] a group is said to be M-centric if Z(P) ∩ K is a Sylow p-subgroup in C(P) ∩ K. (If P is M-centric then we can write C(P) ∩ K = (Z(P) ∩ K) × L for some subgroup L where p |̸|L|.) Let D_{p,M}^{e}(G) denote the collection of M-centric subgroups which satisfy O_{p}(NP/(P (C(P) ∩ K))) = e, and let D_{p,M}(G) = D_{p,M}^{e}(G) ∩ S_{p}(G). Note that D_{p}(G) ⊆ D_{p,M}^{e}(G) ⊆ B_{p}^{e}(G). In the extreme case where M is the trivial module and p||G| then D_{p,M}(G) = D_{p,M}^{e}(G) = D_{p}(G), since e is non-centric. On the other hand if G acts faithfully on M, then D_{p,M}^{e}(G) = B_{p}^{e}(G).

The proof of Theorem 1.2 and 7.3 reveal that the collection D_{p,M}^{e}(G) is subgroup and normalizer acyclic with respect to H^{n}(−; M), n ≥ 0. Likewise D_{p,M}(G) is subgroup and and normalizer acyclic with respect to H^{n}(−; M), n > 0 and has the same acyclicity properties as S_{p}(G) for n = 0.

Remark 8.10. It would be quite interesting to see if a smaller collection than D_{p,M}^{e}(G) can be used, using more elaborate information about M. This question is closely linked to getting a better understanding of the modules appearing in the Steinberg complex for various Weyl groups in G.

Remark 8.11. For the subgroup and normalizer acyclicity of H^{n}(−; M), n > 0 we can remove those subgroups P ∈ D_{p,M}^{e}(G) such that M ↓_{P}^{G} is projective. This is obvious for subgroup acyclicity, and for normalizer acyclicity follows from Lemma 8.3.

Example 8.12 (The collection A_{p}(G)). Theorem 6.1 and Corollary 7.2 shows that A_{p}(G) is centralizer and normalizer acyclic with respect to H^{n}(−; M) for n > 0 and has the same acyclicity properties as S_{p}(G) for n = 0.

Theorem 1.1, Theorem 6.1 and Proposition 7.1 immediately produce three exact sequences, where the last one is Webb’s result.

Corollary 8.13. Let C be a collection of p-subgroups closed under passage to p-radical psupergroups. Let 𝔉 : O(G)^{op} → Z_{(p)}- mod be any functor such that C is subgroup acyclic with respect to 𝔉. Then we have an exact sequence

0 → 𝔉(G) → ⊕_{[P]∈C′/G}𝔉(P)^{NP} → ⊕_{[P_{0}<P_{1}]∈|C′|_{1}/G}𝔉(P_{0})^{NP_{0}∩NP_{1}} → ⋯ → 0

where C′ = C ∩ B_{p}^{e}(G). The first boundary map is given by restriction and the latter comes from the boundary map in the in the cochain complex C_{G}^{∗}(|C′|; F^{β}).

If for all P ∈ C, CP acts trivially on 𝔉(P) then we may replace C′ by C ∩ D_{p}(G). □

Corollary 8.14. Let C be a collection of non-trivial p-subgroups in G closed under passage to non-trivial elementary abelian subgroups. Let 𝔉 : O(G)^{op} → Z_{(p)}- mod be any functor such that C is centralizer acyclic with respect to 𝔉.

Then we have an exact sequence

0 → 𝔉(G) → ⊕_{[V]∈C′/G}𝔉(CV)^{NV} → ⊕_{[V_{0}<V_{1}]∈|C′|_{1}/G}𝔉(CV_{1})^{NV_{0}∩NV_{1}} → ⋯ → 0

where C′ = C ∩ A_{p}(G). The first boundary map is given by restriction and the latter comes from the boundary map in the in the cochain complex C_{G}^{∗}(|C′|; F^{α}). □

Corollary 8.15. [68] Let C be any collection. Let 𝔉 : O(G)^{op} → R- mod be any functor such that C is normalizer acyclic with respect to 𝔉. Then we have an exact sequence.

0 → 𝔉(G) → ⊕_{[H]∈C/G}𝔉(NH) → ⊕_{[H_{0}<H_{1}]∈|C|_{1}/G}𝔉(NH_{0} ∩ NH_{1}) → ⋯ → 0

where the boundary map comes from the boundary map in the cochain complex C̃_{G}^{∗}(|C|; 𝔉), where 𝔉 is viewed as a generic coefficient system. (The sequence is split by Theorem 8.2.) □

Remark 8.16. A sequence of the form of Corollary 8.13 also shows up in the work of Villarroel-Flores [65], but his work did not reveal exactness, except in very special cases.

Example 8.17. Consider the sporadic group M_{11}. According to the Atlas [25] there are two conjugacy classes of elementary abelian 2-subgroups. One of rank one V_{1} with centralizer CV_{1} = GL_{2}(F_{3}) and one of rank two V_{2} which is self-centralizing CV_{2} = V_{2}. Furthermore NV_{1} = CV_{1}, NV_{2} = Σ_{4}, and NV_{1} ∩ NV_{2} = D_{8}. Example 8.12 and Corollary 8.14 hence produces an exact sequence for any n > 0 and any Z_{(2)}M_{11}-module N

0 → H^{n}(M_{11}; N) → H^{n}(GL_{2}(F_{3}); N) ⊕ H^{n}(V_{2}; N)^{Σ_{3}} → H^{n}(V_{2}; N)^{Z/2} → 0

The chain complexes of Theorem 4.1 and Theorem 6.2 likewise gives us exact sequences.

Corollary 8.18. Let G be a finite group of Lie type, and let C be a collection of p-subgroups closed under passage to p-supergroups. Let 𝔉 : O(G)^{op} → Z_{(p)}- mod be any functor such that C is subgroup acyclic with respect to 𝔉. Then we have an exact sequence

0 → 𝔉(G) → 𝔉(U_{∅})^{P_{∅}} → ⨁_{J⊆S;|J|=1;U_{J} ∈C} Hom_{L_{J}} (St_{L_{J}} , 𝔉(U_{J})) → ⋯ → 0

The first boundary map is restriction to the p-Sylow subgroup U_{∅} in G and the following ones are described in Theorem 4.6. □

Corollary 8.19. Let G be a finite group, and let C be a collection of non-trivial p-subgroups closed under passage to non-trivial elementary abelian p-subgroups. Let 𝔉 : O(G)^{op} → Z_{(p)}- mod be any functor such that C is centralizer acyclic with respect to 𝔉. Then we have an exact sequence

0 → 𝔉(G) → ⨁_{[V]∈C∩A_{p}(G)/G;rk V =1} 𝔉(CV)^{NV} → ⨁_{[V]∈C∩A_{p}(G)/G;rk V =2} Hom_{NV/CV} (St_{GL(V)}, 𝔉(CV)) → ⋯ → 0

The first boundary map is given by restriction and the following ones are described in Theorem 6.2. □

Theorem 1.3 can also be used to obtain information about 𝔉(G). However, in general, if G is not a finite group of Lie type, then we do not get an exact sequence, but we can get two other expressions. One is by looking at what happens at the (0, 0) spot in the spectral sequence, where we get a formula generalizing the Cartan-Eilenberg formula for stable elements in group cohomology [23]—we will explain this in Section 10. Another is by trading in the spectral sequence for an expression in the Grothendieck group of Z_{(p)}- modules, where we get a somewhat generalized version of a result of Webb, which we now explain.

Define the Lefschetz module L of a chain complex C_{∗} to be the virtual module L(C_{∗}) = ∑_{i}(−1)^{i}C_{i}, where the alternating sum is taken in the Grothendieck group of modules under direct sum. Recall that the Steinberg module St(G) of an arbitrary finite group G is defined to be the virtual Z_{(p)}G-module

St(G) = L(St_{∗}(G))

(This definition of course coincides with the preexisting definition for a finite group of Lie type up to a sign.) Note that this is a projective virtual Z_{(p)}G-module, since its reduction modulo p is projective.

Corollary 8.20 (cf. Webb [67, Thm. 6.8]). Let G be a finite group and let C be a collection of p-subgroups closed under passage to p-supergroups. Let 𝔉 : O(G)^{op} → (finitely generated Z_{(p)}- mod) be any functor. Suppose that C is subgroup acyclic with respect to 𝔉. Then

𝔉(G) = − ∑_{P ∈(C∩B_{p}^{e}(G))/G} Hom_{W P} (St(W P), 𝔉(P))

Proof. Let F be the G-local coefficient system on C given by precomposing with the composite C_{G} → O_{C} ↪ O(G). By Theorem 1.1 and our assumption we have that 𝔉(G) = H_{G}^{0}(|C|; F), and the higher cohomology is trivial. Hence 𝔉(G) = L(C_{G}^{∗}(|C|; F)). Let 𝔉_{P} denote the atomic functor associated to 𝔉, which is F (Q) on subgroups Q conjugate to P and zero otherwise, and let F_{P} denote the corresponding G-local coefficient system. By filtering F by the functors F_{P}, as in the proof of Corollary 3.5 and using that this filtration splits as Z_{(p)}-modules, we get that

L(C_{G}^{∗}(|C|; F)) = ∑_{[P]∈C/G} L(C_{G}^{∗}(|C|; F_{P}))

But the proof of Theorem 1.1 shows that C_{G}^{∗}(|C|; F_{P}) is Z_{(p)}-chain homotopy equivalent to Hom_{W P} (Σ St_{∗}(W Q), 𝔉(P)) where Σ here denotes suspension of chain complexes. Combining these statements and using the definition of St(G) proves the corollary.

Remark 8.21. The above techniques can also be used to obtain information even if acyclicity does not hold, although in that case the results often come as an equality between virtual F_{p}-modules, rather than as exact sequences. We will illustrate this with an example.

Suppose that G is a finite group and H is a normal subgroup. Then G acts on S_{p}(G/H) by conjugation. Consider the Bredon equivariant cohomology of |S_{p}(G/H)| with values in the generic coefficient system 𝔉(−) = H^{n}(−; M), where M is any F_{p}G-module. By a spectral sequence argument, as in the proof of Theorem 7.3, one sees that

H̃_{G}^{∗}(|S_{p}(G/H)|; H^{n}(−; M)) = H̃_{G}^{∗}(|S_{p}(G/H)|; H^{0}(−; H^{n}(H; M))),

which need not equal zero. However, writing out the defining chain complexes on the left and the right hand side and rearranging reveals

H^{n}(G; M) = ( ∑_{[σ]∈|S_{p}(G/H)|/G}(−1)^{|σ|}H^{n}(G_{σ}; M)) − Hom_{G/H}(St(G/H), H^{n}(H; M)).

This result is due to Adem-Milgram [1, Thm. 2.2].

Remark 8.22. Note that all our exact sequences exist if just lim_{D}^{∗} F for ∗ > 0, without making assumptions on lim_{D}^{0} F, if one is willing to replace 𝔉(G) with lim_{D}^{0} F. This weaker assumption is satified more frequently in nature. In fact, for any Mackey functor 𝔉 : O(G)^{op} → Z_{(p)}- mod we have lim_{O_{S^{e}_{p}(G)}}^{∗} F^{β} = 0 for ∗ > 0 (see [38, 41]). The condition that 𝔉(G) → lim_{O_{S^{e}_{p}(G)}}^{0} F^{β} is an isomorphism is in practice more restrictive, and requires 𝔉 to be projective relative to p-subgroups [69]. (Or, almost equivalently, satisfy an ‘induction theorem’ relative to p-subgroups [69] [9, 5.6].) For a Mackey functor 𝔉 which does not satisfy this, one can instead consider the smallest collection C which is closed under passage to subgroups, contains a collection relative to which 𝔉 is projective, and which satisfies the condition of Remark 3.10. C is now subgroup acyclic with respect to 𝔉 and Theorem 1.1 still applies. We hope to examine some interesting examples of this situation in detail in a future paper.

9. Sharpness

In this short section we explain how to obtain sharpness results from the acyclicity results of the previous section. We will be brief and refer the reader to [29] for a much fuller treatment.

Let M be some Z_{(p)}G-module. A collection C is called M-ample if H^{∗}(G; M) →^{≃} H_{hG}^{∗}(|C|; M). A collection C is called subgroup M-sharp if C is subgroup acyclic with respect to H^{n}(−; M) for all n ≥ 0. We define centralizer M-sharp and normalizer M-sharp similarly. (We abbreviate Z_{(p)}-sharp to just sharp, and similarly for ample.)

Recall that for any G-space X we have an isotropy spectral sequence

E_{2}^{i,j} = H_{G}^{i}(X; H^{j}(−; M)) ⇒ H_{hG}^{i+j}(X; M)

The rewritings of Proposition 2.6, 2.10 and 7.1 identify the E_{2} of the isotropy spectral sequence for |EO_{C}|, |EA_{C}|, and |C| with the higher limits over O_{C}^{op}, A_{C}, and ((sd C)/G)^{op} respectively. Therefore the definition of M-sharpness says that the E_{2}-term of the isotropy spectral sequence must be concentrated on the vertical axis, where we must have E^{0,∗} = H^{∗}(G; M). In particular, if C is either subgroup, centralizer or normalizer M-sharp then C is M-ample, since |EO_{C}|_{hG} ≃ |C|_{hG} ≃ |EA_{C}|_{hG}. (Compare the isotropy spectral sequence of |EO_{C}|, |EA_{C}|, or |C| to that of a point.)

We can now use the acyclicity results in the previous sections to obtain sharpness statements. We start by demonstrating that S_{p}(G) is normalizer M-sharp iff it is M-ample. We have seen that the isotropy spectral sequence for |S_{p}(G)| is concentrated on the two axes. On the horizontal axis we have E_{2}^{∗,0} = H_{G}^{∗}(S_{p}(G); M), whereas the vertical axis is E_{2}^{0,∗} = H^{∗}(G; M), for ∗ > 0. Furthermore, this spectral sequence has to collapse at E_{2}. To see this note the map S_{p}(G) → ∗ has a P -equivariant section given by ∗ ↦ ZP, so there can be no differentials for in the isotropy spectral sequence for the P -space |S_{p}(G)|. But by a transfer argument, the E_{2}-term for the G-space |S_{p}(G)| is a retract of the E_{2}-term for |S_{p}(G)| viewed as a P -space, so there can be no differential for the G-space either. Example 8.9 now gives us the following result.

Theorem 9.1. The collection D_{p,M}^{e}(G) of Example 8.9 is subgroup M-sharp and normalizer M-sharp (and is in particular M-ample). The collection D_{p,M}(G) of Example 8.9 is subgroup M-sharp iff it is normalizer M-sharp iff it is M-ample iff Hom_{G}(St_{∗}(G), M) is an acyclic chain complex. □

Note that in the extreme case where p |̸|G| we get that D_{p,M}(G) is M-ample iff M^{G} = 0. The above result generalizes the result of Dwyer [29] that the collection of p-centric and p-radical subgroups is subgroup sharp, and the result originally due to Webb [68] that the collection of p-radical subgroups is normalizer sharp. The consequence that the collection of p-radical and p-centric subgroups is normalizer sharp was suspected by Smith-Yoshiara [59]. We remark that the analysis shows that in the cases we have studied the collections lying in between our sharp collections and S_{p}^{e}(G) are also sharp.

10. Stable elements and Alperin’s fusion theorem

In this section we explain how our work relates to Cartan-Eilenberg’s formula for stable elements in group cohomology and Alperin’s fusion theorem.

Let C be a collection closed under passage to p-supergroups, and let F : O_{C}^{op} → Z_{(p)}- mod be any functor. Write lim_{O_{C}}^{0} F for the stable elements of F with respect to C. (These are the stable elements of Cartan-Eilenberg [23] if C = S_{p}^{e}(G) and F = H^{n}(−; M).)

More amenable formulas for the stable elements can be obtained from Alperin’s fusion theorem. This is mentioned e.g. in Holt [36], and a related approach to the fusion theorem and stable elements is presented in the pioneering work of Puig [49, Ch. III §2].

Our point is that looking at lim_{O_{C}}^{0} F in our model for lim_{O_{C}}^{∗}F gives the same kind of reduction in the formula for stable elements, and the method gives some additional insight as well. We start by explaining the results, and will afterwards relate them to Alperin’s fusion theorem.

Let H_{p}(G) denote the collection of p-subgroups Q in G such that S_{p}(NQ/Q) is either empty or disconnected (see Remark 5.8). In particular S_{p}(NQ/Q) will be non-contractible so H_{p}(G) ⊆ B_{p}^{e}(G). If G is a finite group of Lie type then one easily identifies H_{p}(G) inside B_{p}^{e}(G) as exactly the unipotent radicals of parabolic subgroups of rank less than or equal to one.

Remark 10.1. The role of the subgroups in H_{p}(G) for the p-local structure of G has been examined by especially Goldschmidt [33] and Puig [49], and a link to subgroup complexes was noted by Quillen [50, §6]. (Puig calls the non-Sylow subgroups in H_{p}(G) 1-essential (or just essential) [49, p. 25] and non-Sylow subgroups in H_{p}(G) ∩ D_{p}(G) C-essential [49, p. 38].)

Remark 10.2. Easy examples show that H_{p}(G) is in general not an ample collection. For example take G = GL_{4}(F_{2}). If H_{2}(G) was an ample collection it would be sharp and Corollary 8.18 should give an exact sequence e.g. for F = H^{1}(−; F_{2}). However this is not the case, as can perhaps easiest be seen by comparing it to the corresponding sequence for the collection B_{p}(G).

Examining what happens at the (0, 0)-spot of the spectral sequence of Theorem 1.3 one gets the following result, which says that one can determine the stable elements ‘locally’, i.e., via information coming from p-subgroups and their normalizers—one does not have to consider conjugation between different subgroups as it appears from the definition.

Theorem 10.3. Let C be a collection of p-subgroups in G closed under passage to psupergroups, and let F : O_{C}^{op} → Z_{(p)}- mod be any functor. Let P be a Sylow p-subgroup in G. Then

lim_{O_{C}} ^{0}F = {α ∈ F (P)^{W P} | res_{Q}^{P}(α) ∈ F (Q)^{W Q} for all [Q] ∈ C ∩ H_{p}(G)/G}

where Q is an arbitrary representative of [Q] such that Q ≤ P.

If CQ acts trivially on F (Q) for all subgroups Q ≤ P, then we may replace C ∩ H_{p}(G) by C ∩ H_{p}(G) ∩ B_{p}(G).

Proof. Choose a height function on C ∩ H_{p}(G) such that ht(P) = 0. Consider the spectral sequence of Theorem 1.3. We have that E_{1}^{i,−i} ≠ 0 only if i = 0, since St_{∗}(H) is (−1)- connected if p||H|. Hence, we can calculate lim_{O_{C}}^{0} F as the intersection of the kernels of the differentials originating at the (0, 0)-spot. By Proposition 5.7 we have

E_{1}^{i,−(i−1)} = ⨁_{[Q]∈(C∩B_{p}^{e}(G))/G;ht(Q)=−i} Hom_{W Q}(H̃_{0}(|S_{p}(W Q)|), F (Q)) = F (Q)W̃ Q/F (Q)^{W Q}

In particular, only differentials going to a subgroup Q ∈ C ∩H_{p}(G) can contribute anything.

Finally, one easily verifies from the construction of the spectral sequence (see the proof of Theorem 1.3) that kernel of the differential d_{n} : E_{n}^{0,0} → E_{n}^{n,−(n−1)} is just equal to the

kernel of F (P)^{NP} →^{res_{Q}^{P}} F (Q) → F (Q)/F (Q)^{W Q}. This proves the first part of the theorem.

The last part follows since if CQ acts trivially on F (Q) then Theorem 1.3 holds with C ∩ B_{p}^{e}(G) replaced by C ∩ D_{p}(G).

Specializing to the case of group cohomology, we find that the above result takes the following form, in view of the results of Section 8.

Corollary 10.4. Let M be any Z_{(p)}G-module, and let P be a Sylow p-subgroup in G. We have the following formula for the group cohomology of G with coefficients in M.

H^{∗}(G; M) = {α ∈ H^{∗}(P; M)^{W P} | res_{Q}^{P}(α) ∈ H^{∗}(Q; M)^{W Q} for all [Q] ∈ (H_{p}(G)∩D_{p,M}^{e}(G))/G}

where Q is an arbitrary representative of [Q] such that Q ≤ P. □

Example 10.5 (Swan groups). Recall that a p-group P is called a Swan group if for all finite groups G with P as Sylow p-subgroup we have that H^{∗}(G; F_{p}) →^{≃} H^{∗}(P; F_{p})^{W P}.

Suppose that P satisfies the condition that for all proper subgroups Q < P,

C_{P} (Q) = ZQ implies that (N_{P} (Q)/QC_{P} (Q)) ∩ O_{p}(Out(Q)) ≠ e.

Then for all G with P as Sylow p-subgroup we have that D_{p}(G)/G = [P]. Hence, by Corollary 10.4, H^{∗}(G; F_{p}) →^{≃} H^{∗}(P; F_{p})^{W P}, so P is a Swan group. This result was originally proved by Martino-Priddy [43, Thm. 2.3] using Goldschmidt’s version of Alperin’s fusion theorem (see below).

Remark 10.6. The proof of Theorem 10.3 in the case where CQ acts trivially on F (Q) allows for the seemingly stronger statement that we may restrict attention to p-centric subgroups Q ∈ C for which S_{p}(NQ/QC(Q)) is empty or disconnected. However, using some group theory one can (non-trivially) show that this is equivalent to assuming Q ∈ C ∩ H_{p}(G) ∩ D_{p}(G) (see [49, Ch. III Cor. 2]).

Remark 10.7. Note that the proof of Theorem 10.3 actually shows more than stated in the theorem. We already know that res_{Q}^{P}(α) ∈ F (Q)W̃ Q, so we only have to calculate invariants under a subgroup which together with W̃ Q generate W Q. Also, the proof shows that each Q ∈ C ∩ H_{p}(G) can only contribute if F (Q)^{W Q} ≠ F (Q)W̃ Q i.e., if Hom_{W Q}(H̃_{0}(S_{p}(W Q)), F (Q)) ≠ 0. Remark 5.11 gives us a good handle on when this happens, and can allow us to rule out more subgroups if we know something about F (Q). This can for instance be useful for low dimensional group cohomology calculations.

Remark 10.8. The conditions on subgroups are stated in a number of different ways in the literature. For the convenience of the reader, we state some relations between these.

1. A subgroup Q of P is a tame intersection in P if there exists a Sylow p-subgroup P′ of G such that Q = P ∩ P′ and that N_{P} (Q) and N_{P′}(Q) are both Sylow p-subgroups in NQ. Puig observed [49, p. 42] (see also [50, §6]) that if Q ∈ H_{p}(G) and if N_{P} (Q) is a Sylow p-subgroup in NQ, then such P′ always exists. Hence, by Sylow’s theorem, any Q ≤ P with Q ∈ H_{p}(G) is G-conjugate to a subgroup Q′ ≤ P with Q′ a tame intersection in P.

2. A subgroup Q ≤ P is p-centric in G iff for all subgroups Q′ ≤ P with Q′ G-conjugate to Q we have C_{P} (Q′) = ZQ′. (Use Sylow’s theorem.)

3. If Q is assumed to be a tame intersection in P and C_{P} (Q) = ZQ then Q is p-centric. To see this just note that since N_{P} (Q) is assumed to be a Sylow p-subgroup in NQ and CQ ⊴ NQ, C_{P} (Q) = CQ ∩ N_{P} (Q) is also a Sylow p-subgroup in CQ.

4. Note that we always have O_{p′}(NQ) = O_{p′}(CQ). Hence, if Q is p-centric then the condition that O_{p}(NQ/QC(Q)) = e is equivalent assuming that Q is a Sylow psubgroup of O_{p′,p}(NQ). If Q is assumed to be a tame intersection in P, it is again equivalent to assuming O_{p′,p}(NQ) ∩ P = Q.

By 2 we can view Dwyer’s p-centric condition as being a G-conjugacy invariant refinement of the condition C_{P} (Q) = ZQ, which occur more frequently in group theory. Also, 1 and 3 shows that the tame intersection condition picks out distinguished representatives from a G-conjugacy class of p-centric subgroups.

Definition 10.9. [2] Let G be a finite group and P a fixed Sylow p-subgroup in G. A set 𝕏 of pairs {(Q, T)}, where Q is a subgroup of P and T is a subset of NQ, is called a conjugation family provided the following condition is satisfied: For any subset A of P and any g ∈ G such that A^{g} ≤ P, then there exist pairs (Q_{1}, T_{1}), …, (Q_{n}, T_{n}) in 𝕏 and elements x_{1}, …, x_{n}, y of G such that g = x_{1} ⋯ x_{n}y, x_{i} ∈ T_{i} for i = 1, …, n, y ∈ NP, and A^{(x_{1}···x_{i})} ≤ Q_{i+1} for i = 0, …, n − 1. (A^{g} = g^{−1}Ag.)

Note that we for each conjugation family 𝕏 can form a new conjugation family replacing each (Q, T) with (Q, T̄), where T̄ is the subgroup generated by the union of the subsets {U|(Q, U) ∈ 𝕏}, which will be just as good for practical purposes. Hence we can when convenient assume that (Q, T) is uniquely determined by Q, and that T is a group. We say that a conjugation family is invariant if it, in addition to the just stated requirements, satisfies that if (Q, T) ∈ 𝕏 and Q^{g} ⊆ P then (Q^{g}, T^{g}) ∈ 𝕏.

It is non-obvious that conjugation families do indeed exist. However the following is immediate.

Proposition 10.10. Let C = S_{p}^{e}(G) and let F : O_{C}^{op} → Z_{(p)}- mod be any functor. Assume that 𝕏 is a conjugation family in G. Then

lim_{O_{C}} ^{0}F = {α ∈ F (P)^{W P} | res_{Q}^{P}(α) ∈ F (Q)^{T} for all (Q, T) ∈ 𝕏}

If 𝕏 is invariant then it is enough to check the restriction condition on one representative (Q, T) for each G-conjugacy class of pairs in 𝕏.

Proof. The first part of the statement follows by writing out the definition of lim^{0} (see e.g. [36]). The last part of the statement follows by a downward induction on the size of Q. (The statement is trivial for Q = P.)

Goldschmidt’s version of Alperin’s fusion theorem [2, 33] (stated in a stronger form due to Miyamoto [46, Cor. 1], and slightly reformulated) says that the set of pairs {(Q, T)} where Q is a tame intersection subgroup in P such that Q ∈ H_{p}(G) and T = NQ if Q ∈ D_{p}(G) and T = CQ otherwise, is a conjugation family. Hence, Proposition 10.10 combined with the fusion theorem produces a statement very close to Theorem 10.3. (Compare also [49, Ch. III §2].)

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

   E-mail address: jg@math.mit.edu WWW address: http://www-math.mit.edu/˜ jg