THE BREDON COHOMOLOGY OF SUBGROUP COMPLEXES

PETER SYMONDS

Abstract. We develop the homological algebra of coefficient systems on a group, in particular from the point of view of calculating higher limits. We show how various sequences of modules associated to a class of subgroups of a given group can be analysed by methods from homological algebra. We are particularly interested in when these sequences are exact, or if not, when their homology is equal to the higher limits of the coefficient system.

1. Introduction

This paper is concerned with the homological algebra of coefficient systems on a class of subgroups of a group G . It is partly structured around the investigation of three sequences associated to some class X of subgroups of G, in particular their cohomology.

The three sequences of particular interest are:

L(G) → ∏_{σ∈ch_{0}(X)/G} L(N_{G}(σ)) → ∏_{σ∈ch_{1}(X)/G} L(N_{G}(σ)) → …

L(G) → ∏_{σ∈ch_{0}(X)/G} L(σ_{b})^{N_{G}(σ)} → ∏_{σ∈ch_{1}(X)/G} L(σ_{b})^{N_{G}(σ)} → …

L(G) → ∏_{σ∈ch_{0}(X)/G} L(C_{G}(σ_{t}))^{N_{G}(σ)} → ∏_{σ∈ch_{1}(X)/G} L(C_{G}(σ_{t}))^{N_{G}(σ)} → …

Here ch_{n}(X) denotes the set of chains in X (without repetition) of length n + 1. The smallest element of a chain σ is denoted σ_{b} and the largest by σ_{t}.

The first of these sequences was investigated by Webb [24], when L is a Mackey functor, and is by now well known. The second sequence first appeared in work of Bouc [5], again for Mackey functors. The preprint dates from 1991, but remained unpublished until 1998. An infinite version of the third sequence is implied by results of Jackowski and McClure [13]. Later Dwyer [10] had infinite versions of the second and third sequences, which arose from topology, and he investigated the properties of all three. Following him, we will sometimes refer to these as the normaliser, subgroup and centraliser sequences. Finally the version of the third sequence given above (and also the second) appeared in work of Grodal [12] and of Villarroel-Flores and Webb [23].

The original interest was in conditions on L and X which forced these sequences to be exact. But when they are not exact, it turns out that their cohomology can often be described as the higher limits of L, and these are often of interest in their own right.

Our aim is to present a unified treatment of all the results entirely within the homological algebra of coefficient systems. The only geometric results used are some standard ones about the fixed point sets of a group acting on some subgroup complex, and then only for examples.

A non-geometric but more category-theoretic treatment of some of these results by Jackowski and Słomińska has recently appeared [15].

Our strategy is to show that each these sequences is representable as the complex of homomorphisms from some complex of coefficient systems C̃_{•} to L, in other words that the sequence is of the form Hom(C̃_{•}, L). It then turns out that C̃_{•} is homotopy equivalent to a projective resolution of the trivial coefficient system R̄. Thus the cohomology of our sequence is Ext^{∗}(R̄, L), which is the definition of the higher limits of L.

When these sequences are not exact then their cohomology is usually equal to the higher limits of L. This insight was developed by Grodal in [12], and the later sections of the present paper were inspired by his work, being essentially an attempt to formulate the geometric proofs given there in algebraic terms.

Many statements in group theory can be phrased succinctly in terms of higher limits. For example Robinson’s reformulation of Alperin’s Weight Conjecture (§5) and Quillen’s conjecture on the contractibility of subgroup complexes (§11).

We use a lot of basic results about coefficient systems, in particular we make great use of the adjoint functors of several forgetful functors and their properties, and these are collected together in §2.

2. Coefficient Systems

Here we collect together some constructions on coefficient systems and Mackey functors and record their properties.

We will always work over a fixed unital ring R and refer to a fixed prime p. We have been careful wherever possible to allow infinite groups in the basic

definitions, although this is not a direction that we pursue here, and it is abandoned later, when we need a Sylow p-subgroup.

For a given group G, we will consider various classes of subgroups, assumed to be closed under conjugation. For example S(G), the class of all subgroups; S_{p}(G), the class of finite p-subgroups; or A_{p}(G), the class of finite elementary abelian p subgroups. We will often omit G from the notation. The superscript 1 will be used to denote the given class with the trivial subgroup removed.

Many of our results are phrased in terms of adjoint properties. Recall that if A and B are two categories then two functors L : B → A and R : A → B are adjoint if and only if there exist two natural transformations, the unit η : I_{B} → RL and the counit ϵ : LR → I_{A}, such that the compositions (Rϵ)(ηR) : R → R and (ϵL)(Lη) : L → L are both the identity. Our proofs will usually consist of giving explicit formulas for η and ϵ and leaving to the reader the straightforward task of checking these identities. We will also omit many sub- and superscripts where this simplifies the formulas.

Note that these adjoint functors are known to exist for abstract reasons and can be defined in much greater generality, but we want explicit formulas so that we can investigate their properties.

When we refer to results or proofs in the literature the authors of these results usually assume that W and any other class of groups are either S or S_{p}, but the change to general W does not present any difficulties. They also tend to assume that G is finite and here again the generalisation is straightforward in the cases mentioned, except that we have to be careful to distinguish ⊕ and Π.

For a given class W of subgroups of a group G we construct two categories S_{W} and T_{W}. They both have the elements of W as objects. The morphisms are given as follows:

S_{W}(H, K) = {b_{g,H,K}|g ∈ G, H^{g} ≤ K},
T_{W}(H, K) = H\S_{W}(H, K).

The composition law is b_{h,K,L}b_{g,H,K} = b_{gh,H,L}.

A weak coefficient system is an object of the category WCS_{W}(G) of contravariant functors from S_{W} to R − Mod, and a coefficient system is an object of the category CS_{W}(G) of contravariant functors from T_{W} to R − Mod. The morphisms are the natural transformations of functors and are denoted by hom_{CS_{W} (G)}. This is equivalent to defining CS_{W}(G) to be the category of contravariant functors from G-sets with stabilisers in W to R − Mod, by setting the value of C ∈ CS_{W}(G) on the G-set G/H to be C(H) and in general taking the value of C on an arbitrary G-set to be the direct sum of its values on the orbits. (See [7], and also [25] for the similar case of Mackey functors.)

We write c_{g,H,H^{g}}, or just c_{g}, for the map of R-modules C(H^{g}) → C(H) induced by b_{g,H,H^{g}}, and call these the conjugations. Then c_{g}c_{h} = c_{gh}, and for any weak coefficient system C, this makes C(H) into a left RN_{G}(H)-module, and if C is a coefficient system then H acts trivially, so C(H) is naturally an RN_{G}(H)/H-module. We also write res_{H}^{K} for the map induced by b_{e,K,H}, and call these restrictions. Since b_{g,H,K} = b_{e,H^{g},K}b_{g,H,H^{g}}, we see that it is enough to check identities for conjugations and restrictions only.

The forgetful functor from coefficient systems to weak coefficient systems has a right adjoint given by taking invariants under H at each evaluation C(H), and a left adjoint given by taking coinvariants.

A coefficient system C ∈ CS_{W}(G) is called geometric if, for all H ∈ W, C_{G}(H) acts trivially on C(H). This occurs, for instance, if C is the restriction to G of a global coefficient system or Mackey functor. The full subcategory of geometric objects in CS_{W}(G) will be denoted by GCS_{W}(G). The inclusion I : GCS_{W}(G) → CS_{W}(G) has both left and right adjoints, G_{0} and G^{0} respectively. They are formed by taking the largest quotient (respectively largest sub coefficient system) that is geometric. G_{0} has the explicit description (G_{0}L)(P) = H_{0}(C_{G}(P)/Z(P), L(P)).

We will also occasionally mention primitive coefficients systems (PCS), which have no conjugations at all, only restrictions.

For a finite group, each of these categories of coefficient systems is equivalent to the category of modules for some finite rank R-algebra. In this way we can import various results from the theory of representations of algebras: for example if R is a complete local ring then we have the Krull-Schmidt property.

If H ≤ G there is a forgetful map Res_{H}^{G} : CS_{W}(G) → CS_{W}(H) (we should really write CS_{W∩S(H)}(H)).

There is an obvious concept of tensor product ⊗ : CS_{W}(G) × CS_{W}(G) → CS_{W}(G), defined groupwise by (M ⊗ N)(J) = M(J) ⊗ N(J) and the obvious restriction and conjugation maps.

Given L, M ∈ CS_{W}(G), an element of hom_{CS_{W} (G)} can be considered as a collection of maps of R-modules L(H) → M(H) for each H ∈ W that commute with the restriction and conjugation maps. Notice that hom_{CS_{W} (G)}(L, M) is naturally an R-module.

We can extend hom_{CS_{W} (G)} to a pairing Hom_{CS_{W} (G)} : CS_{W}(G) × CS_{W}(G) → CS_{T} (G) for any class T , given by Hom_{CS_{W} (G)}(M, N)(J) = hom_{CS_{W} (J)}(Res_{J}^{G} M, Res_{J}^{G} M). The restrictions are just the forgetful maps, and the conjugations are the usual ones, c_{g}(f) = c_{g}fc_{g}^{−1}.

Thus Hom takes its values in coefficient systems. We will assume that T = W unless otherwise indicated. Notice that if we take T = {G} then we recover hom.

We will often abbreviate Hom_{CS_{W} (G)} to Hom_{CS_{W}}.

There is the usual adjunction:

Lemma 2.1. For L, M, N ∈ CS_{W}(G) we have Hom_{CS_{W}} (L ⊗ M, N) ≅ Hom_{CS_{W}} (L, Hom_{CS_{W}} (M, N)).

Proof. The isomorphism Φ is given by ((Φ_{H}f)_{J}(l))_{I}(m) = f_{I}(res_{I}^{J} l ⊗ m), for H ≥ J ≥ I, f ∈ Hom_{CS_{W}} (L ⊗ M, N)(H), m ∈ M(I). □

When X ⊆ W then there is the forgetful functor Res_{X}^{W} : CS_{W}(G) → CS_{X} (G). There are functors \underleftarrow{lim}_{X}^{W} and \underrightarrow{lim}_{X}^{W} in the opposite direction. Now \underleftarrow{lim}_{X}^{W} L is obtained on H ∈ W by taking the inverse limit of L(I) for I ∈ X ∩ S(H): the limit is over all inclusions and conjugations in H. We define \underrightarrow{lim}_{X}^{W} L(H) to be the direct limit of the L(J) for all J ∈ X , J ≥ H where the limit is taken over inclusions only.

Proposition 2.2. \underrightarrow{lim}_{X}^{W} is the left adjoint and \underleftarrow{lim}_{X}^{W} is the right adjoint of Res_{X}^{W} : CS_{W}(G) → CS_{X} (G).

Proof. For \underleftarrow{lim}, we have η_{M} : M → \underleftarrow{lim} Res M is the identity on subgroups in X, and this extends uniquely by the definition of \underleftarrow{lim}. Then ϵ_{N} : Res \underleftarrow{lim} N → N is the identity.

For \underrightarrow{lim}, we have η_{M} : M → Res \underrightarrow{lim} M is the identity, and ϵ_{N} : \underrightarrow{lim} Res → N follows from the definition of \underrightarrow{lim}. □

Clearly \underleftarrow{lim} and \underrightarrow{lim} are transitive.

Notice that from the definition of hom we find that hom_{CS_{W} (G)}(L, M) = (\underleftarrow{lim}_{W}^{S} Hom_{CS_{W} (G)}(L, M))(G), where we regard Hom_{CS_{W} (G)}(L, M)) ∈ CS_{W}(G). This observation and the previous proposition now yield:

Lemma 2.3. If W ⊆ V and L, M ∈ CS_{W}(G) then Hom_{CS_{W}} (L, M) ≅ \underleftarrow{lim}_{W}^{V} Hom_{CS_{W}} (L, M) in CS_{V}(G) (but we regard Hom_{CS_{W}} (L, M) ∈ CS_{W}(G)) .

Also, for N ∈ CS_{V}(G) we have Hom_{CS_{V}} (N, \underleftarrow{lim}_{W}^{V} M) ≅ Hom_{CS_{W}} (Res_{W}^{V} N, M) and Hom_{CS_{V}} (\underrightarrow{lim}_{W}^{V} M, N) ≅ Hom_{CS_{W}} (M, Res_{W}^{V} N), both in CS_{S}(G).

If we denote the constant coefficient system by R̄, then the following is a consequence of the definition of \underleftarrow{lim}.

Lemma 2.4. Hom_{CS_{W}} (R̄, L) ≅ \underleftarrow{lim}_{W}^{S} L in CS_{S}(G).

Proof. Hom_{CS_{W}} (R̄, L) ≅ Hom_{CS_{S}} (R̄, \underleftarrow{lim}_{W}^{S} L) ≅ \underleftarrow{lim}_{W}^{S} L, by taking the image of 1 ∈ R̄(H), where H is the group that we are evaluating on. □

For convenience we denote \underleftarrow{lim}_{W}^{G} L = (\underleftarrow{lim}_{W}^{S} L)(G), i.e. the usual inverse limit of L. By a component of W we mean an equivalence class under the equivalence relation generated by inclusion.

Lemma 2.5. In CS_{W}(G), R̄ is projective if and only if each component of W has a unique maximal element M, say, and |N_{G}(M) : M| is finite and invertible in R.

When the conditions of this lemma are satisfied we say that G is tight with respect to W.

Proof. It follows from 2.4 that R̄ is projective if and only if the functor L ↦ \underleftarrow{lim}_{W}^{G} L is exact on CS_{W}(G).

If the conditions involving M are satisfied then we claim that \underleftarrow{lim}_{W}^{G} L is the sum over the conjugacy classes of maximal elements M in W of L(M)^{N_{G}(M)/M}. From this it will follow that \underleftarrow{lim}_{W}^{G} is exact, since for any group A the functor X ↦ X^{A} is exact on RA-modules if and only if |A| is finite and invertible in R.

This claim is equivalent to the one that \underleftarrow{lim}_{W}^{G} L ≅ (∏_{M} L(M))^{G}, where M runs over all maximal elements of W. We will denote the right hand side by X and indicate the components of x ∈ X by x = (x_{M}). There is a family of maps ϕ_{H} : X → L(H) for H ∈ W defined by ϕ_{H}(x) = res_{H}^{M} x_{M}, where M is the unique maximal element of W containing H (the uniqueness is one of the conditions). These maps are compatible with restriction and conjugation, and it is easy to see that any other such compatible family of maps must factor through X, proving the claim.

Conversely, if R̄ is projective then, by considering coefficient systems that are non-zero only on a maximal element M and its conjugates, we see that taking fixed points in R[N_{G}(M)/M]-modules must be exact, so the condition on the index must hold. If the other condition is not met then it easy to see that we can find two different maximal elements of W, M and N say, such that M ∩ N contains an element U of W.

Let R̃ ∈ CS_{W}(G) take the value R on the conjugacy classes of M and N (which we denote by ⟨M⟩ and ⟨N⟩), and 0 elsewhere. There is a surjection \underrightarrow{lim}_{\{⟨M⟩,⟨N⟩\}}^{W} R̄ →

R̃. Now if we apply \underleftarrow{lim}_{W}^{G} then, by the observation at the beginning of this proof, we obtain another surjection \underleftarrow{lim}_{W}^{G} \underrightarrow{lim}_{\{⟨M⟩,⟨N⟩\}}^{W} R̄ → \underleftarrow{lim}_{W}^{G} R̃.

The domain of this surjection is 0, because the images of the restrictions from M and N to (\underrightarrow{lim}_{\{⟨M⟩,⟨N⟩\}}^{W} R̄)(U) are linearly independent.

Now if M and N are not conjugate then \underleftarrow{lim}_{W}^{G} R̃ ≅ R^{2}. If M and N are conjugate then \underleftarrow{lim}_{W}^{G} R̃ ≅ R. In either case we have a contradiction. □

Next, if H ≤ G, consider the restriction functor Res_{H}^{G} : CS_{W}(G) → CS_{W}(H). One functor in the other direction is Ind_{H}^{G}. This is defined on subgroups by

(Ind_{H}^{G} L)(J) = ⨁_{g∈^{J} (G/H)} L(J^{g}).

The restrictions are the obvious compositions of restriction in L and inclusion, but the conjugation maps are less clear. For this purpose it is better to use a representative-free description.

First set

L̂_{G,H}(J) = ⨁_{g∈G, J^{g}≤H} L(J^{g}).

This is a PCS on W. In fact it is a WCS as follows. Write (g, l)^{J} for l ∈ L(J^{g}) ⊆ L̂_{G,H}(J) and define res_{K}^{J}(g, l)^{J} = (g, res_{K}^{J} l)^{K}. In this way L̂_{G,H} becomes a PCS. Now define conjugation by f ∈ G by c_{f}(g, l)^{J^{f}} = (fg, l)^{J}. This makes L̂_{G,H} in to a WCS.

There is also an action of H on the right by (g, l)^{J}h = (gh, c_{h}^{−1}l)^{J}. These two actions commute and we set

(Ind_{H}^{G} L)(J) = H_{0}(H, L̂_{G,H}(J)).

The conjugation and restriction maps are the induced ones, and it is routine to check that if f ∈ J then c_{f} acts trivially on the evaluation at J, so we have a coefficient system.

In fact Ind is the left adjoint of Res. The unit and counit of the adjunction are: η_{M} : M → Res Ind M is the inclusion of M(J) as M(J^{e}), and ϵ_{N} : Ind Res N → N is ⊕c_{g}.

For the right adjoint we have coinduction, which, at least if Stab_{W} (G/H) ∈ W for each W ∈ W, can be defined in terms of G-sets just as induction is for Mackey functors, by (Coind_{H}^{G} L)(G/J) = L(Res_{H}^{G}(G/J)).

In explicit form, this is

(Coind_{H}^{G} L)(J) = ∏_{g∈J\G/H} L(H ∩ J^{g}).

For a representative-free form, set

Ľ_{G,H}(J) = ∏_{g∈G} L(H ∩ J^{g}).

Just as before, there is a left action of G and a right action of H. There are restriction maps given for I ≤ J by specifying that res_{I}^{J}(g, l) = (g, res_{H∩^{g}J}^{H∩^{g}I} l). We can now set (Coind_{H}^{G} L)(J) = H^{0}(J × H, Ľ_{G,H}(J)) (where J acts on the left).

The adjunction is given as follows: η_{M} : M → Coind Res M is (res c_{g}^{−1}), and ϵ_{N} : Res Coind N → N is projection on to N(H ∩^{e}J) = N(J). The identities can be checked just as in [21].

If W is not closed under intersections with H then it is not clear what to use for L(H ∩^{g}J) in the formulas above. In fact we fill in these gaps using \underleftarrow{lim}, i.e. we define Coind_{H,W}^{G} L = Res_{W}^{V} Coind_{H,V}^{G} \underleftarrow{lim}_{W}^{V} L, for some V ⊇ W that is closed under intersections with H, e.g. V = S.

The unit and counit extend in the obvious way, but in case this seems too much like sleight of hand, and since the matter is important for this paper, we will give a proof of the adjunction using only the properties of the functors already defined.

Working always within CS, we have:

hom_{CS_{W} (G)}(L, Coind_{H}^{G} M) = hom_{CS_{W} (G)}(L, Res_{W}^{V} Coind_{H}^{G} \underleftarrow{lim}_{W}^{V} M)
≅ hom_{CS_{V} (G)}(\underrightarrow{lim}_{W}^{V} L, Coind_{H}^{G} \underleftarrow{lim}_{W}^{V} M)
≅ hom_{CS_{V} (H)}(Res_{H}^{G} \underrightarrow{lim}_{W}^{V} L, \underleftarrow{lim}_{W}^{V} M)
≅ hom_{CS_{V} (H)}(\underrightarrow{lim}_{W}^{V} Res_{H}^{G} L, \underleftarrow{lim}_{W}^{V} M)
≅ hom_{CS_{W} (H)}(Res_{H}^{G} L, Res_{W}^{V} \underleftarrow{lim}_{W}^{V} M)
≅ hom_{CS_{W} (H)}(Res_{H}^{G} L, M)

Summing up we have shown:

Proposition 2.6. Ind_{H}^{G} is the left adjoint and Coind_{H}^{G} is the right adjoint of Res_{H}^{G} : CS_{W}(G) → CS_{W}(H).

If X is a left G-set we define R[X^{?}] ∈ CS_{W}(G) by letting its value on H ∈ W be the free R-module on the points of X fixed under H, that is on^{H}X. We will usually denote this by R[X^{H}]. Writing ? and H on the right is confusing but traditional.

The restrictions are induced by the inclusions of subsets and conjugation c_{g} is induced by left multiplication by g.

Notice that it follows from the definitions that:

Lemma 2.7. R[G/H^{?}] ≅ Ind_{H}^{G} R̄ in CS_{W}(G) for any W.

Corollary 2.8. R[G/H^{?}] is projective in CS_{W}(G) if H ∈ W. The R[G/H^{?}] for H ∈ W satisfy hom_{CS_{W} (G)}(R[G/H^{?}], L) ≅ L(H) for L ∈ CS_{W}(G) and they provide enough projectives in CS_{W}(G)

Proof. ([7]) R[G/H^{?}] is projective by 2.5, 2.6 and the fact that left adjoints of exact functors preserve projectives.

Now hom_{CS_{W} (G)}(R[G/H^{?}], L) ≅ hom_{CS_{W} (H)}(R̄, L) ≅ L(H) for H ∈ W. It is easy to see that this isomorphism is given by evaluating the homomorphism at H ∈ ^{H}(G/H).

Finally we need to show that any L is the surjective image of a projective. But, using the previous isomorphism, we can construct a map from a sum of copies of R[G/H^{?}] to L that is surjective on evaluation at H. Now we take the sum of these over the H ∈ W. □

If W is a class of subgroups of G and H ≤ G we say that H is taut with respect to W if, for each W ∈ W, H ∩ W is tight with respect to W ∩ S(H ∩ W) in the sense defined after 2.5.

Lemma 2.9. Working in CS_{W}, (1) Ind_{H}^{G} is always exact and preserves projectives,

(2) Coind_{H}^{G} always preserves injectives,

(3) Res_{H}^{G} is always exact and preserves injectives.

(4) Coind_{H}^{G} is exact and Res_{H}^{G} preserves projectives if and only if H is taut with respect to W (so in particular if H ∈ W).

Proof. Res_{H}^{G} and Ind_{H}^{G} are exact by construction. Therefore the left adjoint of Res_{H}^{G}, which is Ind_{H}^{G}, preserves projectives; its right adjoint, which is Coind_{H}^{G}, preserves injectives and the right adjoint of Ind_{H}^{G}, which is Res_{H}^{G}, preserves injectives.

Since the R[G/W^{?}] with W ∈ W provide enough projectives, Res_{H}^{G} preserves projectives if and only if each Res_{H}^{G} R[G/W^{?}] is projective. But this is R[(Res_{H}^{G} G/W)^{?}], and by the double coset formula is a sum of pieces of the form R[H/(H ∩^{g}W)^{?}], which are all projective if and only if H is taut with respect to W, by 2.5. □

Lemma 2.10. If H ≤ G, L ∈ CS_{W}(G), M ∈ CS_{W}(H) then

Ind_{H}^{G}(L ⊗ Res_{H}^{G} M) ≅ (Ind_{H}^{G} L) ⊗ M,

in CS_{W}(G).

Proof. For any K ∈ CS_{W}(G),

hom_{CS_{W} (G)}(Ind_{H}^{G}(L ⊗ Res_{H}^{G} M), K) ≅ hom_{CS_{W} (H)}(L ⊗ Res_{H}^{G} M, Res_{H}^{G} K)
≅ hom_{CS_{W} (H)}(L, Hom(Res_{H}^{G} M, Res_{H}^{G} K))
≅ hom_{CS_{W} (G)}(Ind_{H}^{G} L, Hom(M, K))
≅ hom_{CS_{W} (G)}(Ind_{H}^{G} L ⊗ M, K).

This is natural in K and the result now follows formally. (hom(A, −) ≅ hom(B, −) ⇒ A ≅ B.) □

The adjunction can be generalised:

Proposition 2.11. For H ≤ G, L ∈ CS_{W}(G), M ∈ CS_{W}(H) we have

Hom_{CS_{W} (G)}(L, Coind_{H}^{G} M) ≅ Coind_{H}^{G} Hom_{CS_{W} (H)}(Res_{H}^{G} L, M),
Hom_{CS_{W} (G)}(Ind_{H}^{G} M, L) ≅ Coind_{H}^{G} Hom_{CS_{W} (H)}(M, Res_{H}^{G} L),

in CS_{W}(G).

Proof. For any K ∈ CS_{W}(G),

hom_{CS_{W} (G)}(K, Hom_{CS_{W} (G)}(L, Coind_{H}^{G} M)) ≅ hom_{CS_{W} (G)}(K ⊗ L, Coind_{H}^{G} M)
≅ hom_{CS_{W} (H)}(Res_{H}^{G} K ⊗ Res_{H}^{G} L, M)
≅ hom_{CS_{W} (H)}(Res_{H}^{G} K, Hom_{CS_{W} (H)}(Res_{H}^{G} L, M))
≅ hom_{CS_{W} (G)}(K, Coind_{H}^{G} Hom_{CS_{W} (H)}(Res_{H}^{G} L, M)).

Now the first formula follows formally as in 2.10. The proof of the second is similar, but needs 2.10. □

Corollary 2.12. If L ∈ CS_{W}(G) and H ≤ G, then Ind_{H}^{G} Res_{H}^{G} L ≅ R[G/H^{?}] ⊗ L and Coind_{H}^{G} Res_{H}^{G} L ≅ Hom_{CS_{W}} (R[G/H^{?}], L) in CS_{W}(G).

Proof. There is a map Θ : Ind_{H}^{G} Res_{H}^{G} L → R[G/H^{?}]⊗L given by Θ(g, l) = gH⊗c_{g}l. Its inverse is given by Φ(gH ⊗ l) = (g, c_{g}^{−1}l).

For the second part note that both sides have the same left adjoint, by the first part, 2.1 and 2.6. □

We will occasionally need to deal with quotient groups, so suppose that H ◁ G and let ρ : G → G/H be the quotient map. Let W be a class of subgroups of G/H and let V be a class of subgroups of G such that V ⊇ ρ^{−1}(W) and ρ(V) ⊆ W.

Given L ∈ CS_{W}(G) define Q_{G/H}^{G}L ∈ CS_{V}(G/H) by Q_{G/H}^{G}L(W) = L(ρ^{−1}W), for W ∈ W.

In the other direction we have two functors, defined on M ∈ CS_{W}(G/H) by: (Inf_{G/H}^{G} M)(V) = M(ρ(V)), for V ∈ V, and (Coinf_{G/H}^{G} M)(V) is M(ρ(V)) if H ≤ V ∈ V and 0 otherwise. The next result is left as an easy exercise for the reader.

Proposition 2.13. Inf_{G/H}^{G} is the left adjoint and Coinf_{G/H}^{G} is the right adjoint of Q_{G/H}^{G} : CS_{V}(G) → CS_{W}(G/H).

Now we consider Mackey functors, so for simplicity assume that G is finite. These have been described in many other places, e.g. [25]. The only difference in our treatment is that we only evaluate the functor on a class W of subgroups of G and we assume that this class is closed under intersections. We require the double coset formula for res_{W}^{V} tr_{U}^{V} whenever U, V, W ∈ W. Notice that its terms are all defined because of the condition on intersections.

Let V and W be two classes of subgroups of G, such that V is closed under intersections and W is closed under intersections with V (that is, if W ∈ W and V ∈ V, then V ∩ W ∈ W).

Given C ∈ CS_{W}, define Ĉ ∈ WCS_{V} by:

Ĉ(J) = ⨁_{I∈W, I≤J} C(I).

An element x of C(I) ⊆ Ĉ(J) will be denoted (I, x)^{J}.

For g ∈ G define ^{g}(J, x)^{H} = (^{g}J, c_{g}x)^{^{g}H}. The conjugation maps in C combine to yield a map ĉ_{g} : Ĉ(H) → Ĉ(^{g}H), where ĉ_{g}(J, x)^{H} = ^{g}(J, x)^{H}.

Whenever K ≤ L ≤ G there are restriction morphisms between the values of Ĉ given by r_{K}^{L}(J, x)^{L} = (J ∩ K, res_{J∩K}^{J} x)^{K}. These make Ĉ into a weak coefficient system. There are also inclusion morphisms given by i_{K}^{L}(J, x)^{K} = (J, x)^{L}, which give Ĉ the dual structure i.e. make it into a covariant functor on the category of conjugation and inclusion morphisms.

Define:

T C(H) = H^{0}(H; Ĉ(H))

and

SC(H) = H_{0}(H; Ĉ(H)).

We now define restriction and transfer maps, denoted by R and I, on these groups. We denote the restriction and transfer in cohomology by res and tr.

On T C, R_{K}^{L} = (r_{K}^{L})_{∗} res_{K}^{L} and I_{K}^{L} = tr_{K}^{L}(i_{K}^{L})_{∗}.

On SC,

R_{K}^{L}(J, x)^{L} = (r_{K}^{L})_{∗} ∑_{g∈K\L/J} ^{g}(J, x)^{L} = ∑_{g∈K\L/J}(^{g}J ∩ K, res_{^{g}J∩K^{g}}^{^{g}J} c_{g}x)^{K},

and I_{K}^{L} = res_{K}^{L}(i_{K}^{L})_{∗}.

One can verify that SC and T C are Mackey functors on V. The case of S goes back at least to [9] (see also [5]) and T appears in [20].

There is a forgetful functor F : MF_{W}(G) → CS_{W}(G) and another functor G : MF_{W}(G) → CS_{W}(G) given by

GM(J) = M(J)/ ∑_{I≨J} Im tr_{I}^{J},

with the zero restriction maps.

Proposition 2.14. S is the left adjoint of F and T is the right adjoint of G.

Proof. For T, η_{J} : M → T GM is ∏ res_{I}^{J}, and ϵ_{J} : GT N → N is projection on to L(J).

For S, η_{J} : M → F SM takes m to (J, m)^{J}, and ϵ_{J} : SF N → N takes (I, n)^{J} to tr_{I}^{J} n. □

Remark. Some authors use a slightly different definition of T. Instead of Ĉ, they use a WCS Č, which differs from Ĉ only in that the restriction maps are given by

ř_{K}^{L}(J, x)^{L} = \begin{cases}(J, x)^{K} & if J ≤ K, \\ 0 & otherwise. \end{cases}

However there is a map ϕ : Ĉ → Č given by

ϕ_{L}(J, x)^{L} = ∑_{I∈W, I≤J}(I, res_{I}^{J} x)^{L}.

This ϕ is compatible with the maps r and ř, and also with the i and the c_{g}. It is an isomorphism because it is the identity on the factors if we filter according to the order of the group.

Thus the two definitions are equivalent.

This offers a good way of constructing projective and injective coefficient systems or Mackey functors. Note that if H is normal in G then CS_{{H}}(G) is naturally equivalent to the category of R(G/H)-modules. Since left adjoints of exact functors preserve projectives and right adjoints of exact functors preserve injectives, we have:

Proposition 2.15. Suppose that H ∈ W, and H ≤ N ≤ N_{G}(H), (and W is closed under intersections when we refer to MF_{W}(G)). Let P be a projective RN/Hmodule, I an injective RN/H-module (both regarded as elements of CS_{{H}}(N) as above) and let ⟨H⟩ denote the set of conjugates of H in G. Then:

(1) \underrightarrow{lim}_{⟨H⟩}^{W} Ind_{N}^{G} P ∈ CS_{W}(G) and S \underrightarrow{lim}_{⟨H⟩}^{W} Ind_{N}^{G} P ∈ MF_{W}(G) are projective.

(2) \underleftarrow{lim}_{⟨H⟩}^{W} Coind_{N}^{G} I ∈ CS_{W}(G) is injective.

The following result is key to many applications, including obtaining a splitting of Mackey functors in 5.3.

Proposition 2.16. If L ∈ CS_{W}(G) and M, N ∈ MF_{W}(G) then Hom_{CS_{W}} (L, M) and Hom_{MF_{W}} (N, M) are naturally Mackey functors in MF_{W}(G). These structures are consistent in the sense that they are compatible with the isomorphism Hom_{CS_{W}} (L, M) ≅ Hom_{MF_{W}} (SL, M) given on each subgroup in W by 2.14.

Proof. Hom_{MF_{W}} (N, M) is defined, at least as a coefficient system, in a similar way to Hom_{CS_{W}} (N, M).

The transfer on Hom_{CS_{W}} (L, M) is defined as follows: If K ≤ H ≤ G and f ∈ Hom_{CS_{W} (K)}(L, M) then tr_{K}^{H}(f) is defined on J ≤ H, J ∈ W as

∑_{g∈J\H/K}(L(J) →^{res} L(J ∩^{g} K) →^{c_{g}} L(J^{g} ∩K) →^{f} M(J^{g} ∩K) →^{c_{g^{−1}}} M(J ∩^{g} K) →^{tr} M(J)).

A similar definition works for Hom_{MF_{W}} (N, M). For full details see [6]. □

Proposition 2.17. Suppose that X ⊂ W, both closed under pairwise intersection, and M ∈ MF_{X} (G). Then \underleftarrow{lim}_{X}^{W} M is naturally a Mackey functor. It is the right adjoint of Res_{X}^{W} : MF_{W}(G) → MF_{X} (G).

Proof. To determine tr_{H}^{K} we only need to specify res_{I}^{K} tr_{H}^{K} x for each I ∈ X in a consistent way. This can be done by setting

res_{I}^{K} tr_{H}^{K} x = ∑_{g∈I\K/H} tr_{^{g}H∩I}^{I} c_{g} res_{H∩I^{g}}^{H} x.

□

Corollary 2.18. If M ∈ MF_{W}(G) and M ≅ \underleftarrow{lim}_{X}^{W} Res_{X}^{W} M as a coefficient system then M ≅ \underleftarrow{lim}_{X}^{W} Res_{X}^{W} M as a Mackey functor.

Proof. The formula used to define the transfer on \underleftarrow{lim} is clearly necessary, so it must agree with the transfers on L. □

In [21] there is constructed a functor Ind_{H}^{G} : MF_{S}(H) → MF_{S}(G), which is both right and left adjoint to restriction. The same recipe will work if we replace S by X, provided that X is closed under intersections with H, and, if we ignore transfers then we see that it agrees with our construction of Coind for coefficient systems. For this reason we prefer to denote it by Coind. If X is not closed under intersections with H then we use \underleftarrow{lim} as before. The following version of 2.12 is straightforward to check.

Lemma 2.19. If M ∈ MF_{W}(G) then Coind_{H}^{G} Res_{H}^{G} M ≅ Hom_{CS_{W}} (R[G/H^{?}], M) in MF_{W}(G).

There is an important property of the functor S above.

Lemma 2.20. Sℤ[G/H^{?}] ≅ B^{G}(?, H), where B^{G}(?, H) is the functor induced up to G from the Burnside ring Mackey functor B^{H} on H. In fact B^{G}(?, H) ≅ Hom_{Ω_{R}(G)}(G/H, −) in the notation of [21].

Proof. Notice that S commutes with induction (in CS or MF, depending upon the side) because their right adjoints commute. Also Sℤ[G/H^{?}] ≅ S Ind_{H}^{G} ℤ̄. Now we claim that Sℤ ≅ B^{G}, which can be checked from the definitions. For the rest, see [21]. □

3. Higher Limits

We work in the category CS_{W}(G) of coefficient systems on a class W of a group G over some fixed unital ring R. This is an abelian category with enough projectives and injectives (a consequence of 2.15), so we can use homological algebra. We could just consider the derived functors of hom, but instead we look at Hom_{CS_{W} (G)}, considered as taking values in CS_{T} (G) for some class T . This class T should, perhaps, be indicated in the notation but, instead, we will regard it as implicitly understood or mention it in the text. We take the right derived functors as a functor in the second variable, obtaining Ext_{CS_{W} (G)}^{∗} ∈ CS_{T} (G). This can be confusing, but at least some potential sources of confusion do not arise:

Lemma 3.1. (1) Ext_{CS_{W} (G)}^{∗}(M, N)(J) does not depend on T , as long as J ∈

T .

(2) If J ≤ H ≤ G, M, N ∈ CS_{W}(G) and J ∈ T then Ext_{CS_{W} (G)}^{∗}(M, N)(J) ≅

Ext_{CS_{W} (H)}^{∗}(Res_{H}^{G} M, Res_{H}^{G} N)(J).

Proof. Part (1) is clear from the definitions. Part (2) follows from the definition of the right derived functors and the fact that Res_{H}^{G} is exact and preserves injectives (2.9). □

A problem that does arise is that Hom is not always right balanced in the sense of, for example, [26] 2.2.7.

Lemma 3.2. If T is a class of subgroups of G that are taut (defined just before 2.9) with respect to W then Hom_{CS_{W} (G)}(−, −) is right balanced as a functor taking values in CS_{W}(G).

The advantage of having a balanced functor is that its derived functors in the first and in the second variable coincide.

Proof. We need to check that if the second variable N is injective then Hom(−, N) is exact as a functor of the first variable. We can do this by evaluating on each J ∈ T , so we are just looking at hom_{CS_{W} (J)}(−, Res_{J}^{G} N). But Res_{J}^{G} N is also injective, by 2.9.

We must also check that if the first variable M is projective then Hom(M, −) is exact. The argument is dual to the previous one, except that for Res_{J}^{G} M to be projective we need J to be taut with respect to W. □

Remark. A possible choice for T that satisfies the conditions of 3.2 is {G}, and this amounts to considering the derived functors of hom.

The higher limit coefficient systems are, by definition, the right derived functors of \underleftarrow{lim}_{X}^{W} : CS_{X} (G) → CS_{W}(G) and we will write (\underleftarrow{lim}_{X}^{W})^{i} for R^{i} \underleftarrow{lim}_{X}^{W}

What are normally thought of as the higher limits of L ∈ CS_{X} (G) are the Rmodules ((\underleftarrow{lim}_{X}^{S})^{i}L)(G).

Strictly speaking, we only defined \underleftarrow{lim}_{W}^{T} when W ⊆ T , but the definition without this restriction is clear. It comes to the same as Res_{T}^{S} \underleftarrow{lim}_{W}^{S}.

Lemma 3.3. For all n ≥ 0, Ext_{CS_{W}}^{n} (R̄, L) ≅ (\underleftarrow{lim}_{W}^{S})^{n}L in CS_{S}(G).

Proof. The case n = 0 is just 2.4. Both sides are, by definition, the derived functors in L of the n = 0 case. □

Remark. We can not calculate Ext_{CS_{W}}^{n} (R̄, L) above by taking a projective resolution of R̄ unless we are able to invoke 3.2.

Notice that if W consists of just the trivial group 1, then an object of CS_{{1}}(G) is just an RG-module and the higher limits are just the usual cohomology groups.

We see that the higher inverse limits are natural and unavoidable objects to consider. However if W is large enough they often vanish.

Lemma 3.4. Consider Ext_{CS_{W} (G)}^{∗} to take values in CS_{S}(G). For H ≤ G and L ∈ CS_{W}(H), M ∈ CS_{W}(G), we have

Ext_{CS_{W} (G)}^{∗}(Ind_{H}^{G} L, M) ≅ Coind_{H}^{G} Ext_{CS_{W} (H)}^{∗}(L, Res_{H}^{G} M),

and if H is closed under intersections with W (or just H is taut with respect to W) then

Ext_{CS_{W} (G)}^{∗}(M, Coind_{H}^{G} L) ≅ Coind_{H}^{G} Ext_{CS_{W} (H)}^{∗}(Res_{H}^{G} M, L).

Proof. The zeroth terms are isomorphic by 2.11. We need to check that both sides calculate the right derived functors in the second variable of this common functor. Notice that the first and third occurrences of Coind_{H}^{G} are applied to CS_{W}(H), so are exact by 2.9. The second occurrence is exact, by 2.9, because of the restrictions imposed on the intersections.

For the first formula, notice that an injective resolution of M in CS_{W}(G) becomes an injective resolution of Res_{H}^{G} M in CS_{W}(G) on applying Res_{H}^{G} since Res_{H}^{G} preserves injectives by 2.9 . For the second formula, notice that an injective resolution of L in CS_{W}(H) becomes an injective resolution of Coind_{H}^{G} L after applying Coind_{H}^{G} since Coind_{H}^{G} is exact and it preserves injectives by 2.9. □

We say that a coefficient system L is injective relative to a set of subgroups X ⊆ S(G) if L is a direct summand of ∏_{H∈X} Coind_{H}^{G} Res_{H}^{G} L. This has many equivalent formulations along the lines of Higman’s criterion (cf. [21], [1]). There is also an analogous concept for Mackey functors (where it is customarily referred to as projective relative to since the right and left adjoints of restriction are then isomorphic).

Remark. Since the forgetful functor F from Mackey functors to coefficient systems commutes with Coind_{H}^{G} (their left adjoints commute), a Mackey functor that is injective relative to X as a Mackey functor is also injective relative to X as a coefficient system.

Proposition 3.5. For L ∈ CS_{W}(G) and X ⊆ W with W closed under intersections with X , if L is injective relative to X then \underleftarrow{lim}_{X}^{W} Res_{X}^{W} L ≅ L and (\underleftarrow{lim}_{X}^{W})^{n} Res_{X}^{W} L = 0 for n > 0 in CS_{S}(G).

Proof. It is enough to prove this for Coind_{H}^{G} Res_{H}^{G} L, H ∈ X. But Ext_{CS_{X} (G)}^{n}(R̄, Coind_{H}^{G} Res_{H}^{G} L) ≅ Coind_{H}^{G} Ext_{CS_{X} (H)}^{n}(R̄, Res_{H}^{G} L) by 3.4.

But R̄ is projective in CS_{X} (H) by 2.8, so the higher Ext vanish, and for n = 0 we have the result required. □

The following vanishing result is a version of one in [14].

Proposition 3.6. Let X be a class of p-subgroups of G which is closed under intersections, and such that X contains a Sylow p-subgroup P, and assume that all positive numbers of the form |G/P | − np, n ∈ ℕ_{0} are invertible in R (e.g. R is p-local). Let M be a Mackey functor on X. Then \underleftarrow{lim}_{X}^{S} M is injective relative to X in MF_{S}(G). In particular M is injective relative to X in CS_{X} (G), and so (\underleftarrow{lim}_{X}^{S})^{n}M = 0 for n ≥ 1 in CS_{S}(G).

Proof. Notice that \underleftarrow{lim}_{X}^{S} M is naturally a Mackey functor by 2.17.

The natural augmentation yields a map π : R[G/P^{?}] → R̄, which is onto in CS_{X} (G).

Now the functor S : CS_{X} (G) → MF_{X} (G) is right exact, since its construction involves coinvariants, so Sπ : SR[G/P^{?}] → SR̄ is onto. We claim that Sπ splits. To see this, use the second (Č) model for S. The splitting is induced by sending

(J, 1)^{H} ↦ \frac{1}{|^{J}(G/P)|}(J, ∑_{g∈G/P, J≤^{g}P} gP)^{H}.

Note that, when J acts on G/P, the orbit of every non-fixed point has size divisible by p, so the denominators are indeed invertible, by hypothesis.

Now \underleftarrow{lim}_{X}^{S} M ≅ Hom_{MF_{X} (G)}(SR̄, M) is a summand of Hom_{MF_{X} (G)}(SR[G/P^{?}], M) ≅ Hom_{CS_{X} (G)}(R[G/P^{?}], M) ≅ Coind_{P}^{G} Res_{P}^{G} M by 2.19. □

Let MF_{S}(G, 1) denote the full subcategory of Mackey functors which are projective relative to S_{p} (often denoted Mack(G, 1) by other authors). We can deduce a result of Bouc [5].

Proposition 3.7. If R is p-local then \underleftarrow{lim}_{S_{p}}^{S} and Res_{S_{p}}^{S} provide an equivalence of categories between MF_{S_{p}}(G) and MF_{S}(G, 1).

Proof. From 3.6 we see that \underleftarrow{lim}_{S_{p}}^{S} takes values in MF_{S}(G, 1).

Clearly Res_{S_{p}}^{S} \underleftarrow{lim}_{S_{p}}^{S} = Id. As for \underleftarrow{lim}_{S_{p}}^{S} Res_{S_{p}}^{S} = Id, it is enough to check this

on a functor of the form Coind_{P}^{G} Res_{P}^{G} M for P ∈ S_{p}. But the left adjoint of \underleftarrow{lim}_{S_{p}}^{S} Res_{S_{p}}^{S} Coind_{P}^{G} Res_{P}^{G} is Ind_{P}^{G} Res_{P}^{G} \underrightarrow{lim}_{S_{p}}^{S} Res_{S_{p}}^{S} ≅ Ind_{P}^{G} Res_{P}^{G} , which in turn

has right adjoint Coind_{P}^{G} Res_{P}^{G}, so \underleftarrow{lim}_{S_{p}}^{S} Res_{S_{p}}^{S} Coind_{P}^{G} Res_{P}^{G} ≅ Coind_{P}^{G} Res_{P}^{G} as required. □

For any poset X we define X_{≥H} = {K ∈ X |K ≥ H} and similarly X_{>H}. Recall that G_{0} is the left adjoint of the inclusion GCS → CS

Lemma 3.8. If X ⊆ W and for each H ∈ W, the poset X_{≥H} is connected, then \underrightarrow{lim}_{X}^{W} R̄ = R̄.

If for each H ∈ W, C_{G}(H) acts transitively on the components of X_{≥H}, then G_{0} \underrightarrow{lim}_{X}^{W} R̄ = R̄.

Proof. From the definition, (\underrightarrow{lim}_{X}^{W} R̄)(H) is the free R-module on the components of X_{≥H}. □

Recall that we made the abbreviation \underleftarrow{lim}_{W}^{G} L = (\underleftarrow{lim}_{W}^{S} L)(G).

Proposition 3.9. If each X_{≥H}, H ∈ W is connected then \underleftarrow{lim}_{X}^{G} Res_{X}^{W} ≅ \underleftarrow{lim}_{W}^{G} on CS_{W}(G).

If C_{G}(H) acts transitively on the components of each X_{≥H}, H ∈ W then \underleftarrow{lim}_{X}^{G} Res_{X}^{W} ≅ \underleftarrow{lim}_{W}^{G} on GCS_{W}(G).

Proof. For the first formula we need to show that hom_{CS_{W} (G)}(R̄, L) ≅ hom_{CS_{X} (G)}(R̄, Res_{X}^{W} L). But hom_{CS_{W} (G)}(\underrightarrow{lim}_{X}^{W} R̄, L) ≅ hom_{CS_{X} (G)}(R̄, Res_{X}^{W} L), by 2.2. Now use the previous lemma. The second formula is proved similarly. □

For any poset W we denote the geometric realisation by |W |. This is the simplicial complex where the simplices correspong to chains in W

For any poset W we define the weakly essential elements to be

Wess_{0}(W) = {H ∈ W | |W_{>H}|is empty or has more than one component},

and also

Wess(W) = {H ∈ W ||W_{>H}|is not contractible}.

Also, if W is a G-poset, we define the essential elements to be

Ess_{0}(W) = {H ∈ W ||W_{>H}|/C_{G}(H)is empty or has more than one component}.

Notice that Ess_{0}(W) ⊆ Wess_{0}(W) ⊆ Wess(W).

Remark. We allow maximal elements of W to be essential, in contrast to [20].

The next proposition will be very useful for changing classes of groups. It is based on §6.6 of [1] vol. II, attributed to Bouc.

Proposition 3.10. Let W be a poset such that there is a bound on the length of any chain in W, and let X be a subposet.

If X contains Wess W then the inclusions of the geometric realisations |WessW | ⊆ |X| ⊆ |W | are homotopy equivalences.

If X contains Wess_{0} W then the inclusions of the geometric realisations |Wess_{0}W | ⊆ |X| ⊆ |W | induce a bijection on the connected components.

Proof. The first part is due to Bouc [2, 3, 4], see also II.6.6.5 of [1]. The second part is proved in the same way, replacing homotopy equivalence by induces a bijection on the connected components. The same must be done for Quillen’s Lemma (II.6.6.2 in [1]), either by considering only the E_{0,0}-term of the spectral sequence in the proof, or just by elementary means. □

Proposition 3.11. Suppose that there is a bound on the length of any chain in W. If Wess_{0}(W) ⊆ X ⊆ W then \underleftarrow{lim}_{X}^{G} Res_{X}^{W} ≅ \underleftarrow{lim}_{W}^{G} on CS_{W}(G).

If, in addition, X is closed under supergroups in W then (\underleftarrow{lim}_{X}^{G})^{n} Res_{X}^{W} ≅ (\underleftarrow{lim}_{W}^{G})^{n}.

Proof. By the 3.9 we need to show that X_{≥H} is connected for each H ∈ W. But Wess_{0}(W_{≥H}) ⊆ X_{≥H} ⊆ W_{≥H}, so we can apply 3.10.

For the higher limits the result will follow if we know that Res_{X}^{W} is exact (which it clearly is) and it preserves projectives. The latter is equivalent to the right adjoint \underleftarrow{lim}_{X}^{W} being exact, which it clearly is under the condition on supergroups. □

The next result is an immediate consequence of Alperin’s Fusion Theorem, as stated in, for example, [11], [20]. For the rest of this section we suppose that G is finite.

Proposition 3.12. Suppose that W ⊆ S_{p} is closed under supergroups in Ess_{0}(S_{p}), and that Ess_{0}(S_{p}) ∩ W ⊆ X ⊆ W. Then \underleftarrow{lim}_{X}^{G} Res_{X}^{W} ≅ \underleftarrow{lim}_{W}^{G} on GCS_{W}(G).

If, in addition, X is closed under supergroups in W then (\underleftarrow{lim}_{X}^{G})^{n} Res_{X}^{W} ≅ (\underleftarrow{lim}_{W}^{G})^{n}.

Proof. Pick a Sylow p-subgroup P of G and use the method of stable elements to realise \underleftarrow{lim}_{X}^{G} L as the set of elements x ∈ L(P) such that, whenever H ∈ W, H ≤ P, g ∈ G and ^{g}H ≤ P, then x satisfies res_{^{g}H}^{P} x = c_{g} res_{H}^{P} x.

The Fusion Theorem states that the group homomorphism c_{g} : H → ^{g}H given by conjugation by g is equal to the composition of a sequence of conjugations c_{v} : U → ^{v}U for v ∈ G, U,^{v}U ≤ E ≤ P for some essential subgroup E ∈ Ess_{0}(S_{p}) such that v normalises E. Since U ∈ W and W is closed under supergroups in Ess_{0}(S_{p}) we see that E ∈ Ess_{0}(W) and hence is in X.

But (res_{^{v}U}^{P} −c_{v} res_{U}^{P})x = res_{^{v}U}^{E}(1 − c_{v}) res_{E}^{P} x. It follows that all the conditions that we want to impose on x ∈ L(P) are already imposed when we just consider subgroups in X.

Note that the factorisation of c_{g} given is only as a group homomorphism, so ignores C_{G}(H). This is why we need to work in GCS not CS.

The claim about the higher limits follows as in the previous proof. Note that the right adjoint of Res_{X}^{W} in GCS is G^{0} \underleftarrow{lim}_{X}^{W} I, so is still exact. □

Let B_{p} denote the class of subgroups P of S_{p} satisfying P = O_{p}N_{G}(P), often known as the radical subgroups, and let C_{p} denote the class of subgroups P in S_{p} for which the centre of P is the Sylow p-subgroup of C_{G}(P), sometimes known as the centric or self-centralising subgroups.

It is well known that Wess_{0}(S_{p}) ⊆ B_{p} and Ess_{0}(S_{p}) ⊆ C_{p} ∩ B_{p} (see [20]).

The next lemma is well known and easy to prove.

Lemma 3.13. If N ◁ G and N is of order coprime to p then |S_{p}^{1}(G/N)| ≅ |S_{p}^{1}(G)/N|.

Following Grodal [12], let D_{p} be the set of centric subgroups P of G for which N_{G}(P)/P C_{G}(P) has no non-trivial normal p-subgroup.

Lemma 3.14. Ess_{0}(S_{p}) ⊆ D_{p} ⊆ C_{p} ∩ B_{p}.

Proof. D_{p} ⊆ C_{p} by definition. If P ∈ C_{p} ∖ B_{p} then there is a subgroup Q such that P ≨ Q ◁ N_{G}(P) and Q ≰ C_{G}(P). The image of Q in N_{G}(P)/P C_{G}(P) is non-trivial, so P ∉ D_{p}.

Now if P is centric then |S_{p}^{1}(G)^{>P} |/C_{G}(P) ≅ |S_{p}^{1}(N_{G}(P)/P)|/(C_{G}(P)/Z(P)) ≅ |S_{p}^{1}(N_{G}(P)/P C_{G}(P))|, by the lemma above. So if P ∉ D_{p} then these spaces are contractible and P ∉ Ess_{0}(S_{p}). □

Corollary 3.15. L ∈ CS_{S_{p}}(G) then \underleftarrow{lim}_{B_{p}}^{G} Res_{B_{p}}^{S_{p}} ≅ \underleftarrow{lim}_{S_{p}}^{G} on CS_{S_{p}}(G).

\underleftarrow{lim}_{D_{p}}^{G} Res_{D_{p}}^{S_{p}} ≅ \underleftarrow{lim}_{B_{p}}^{G} Res_{B_{p}}^{S_{p}} ≅ \underleftarrow{lim}_{S_{p}}^{G} on GCS_{S_{p}}(G).

Remark. In section 9 we will see that if R is p-local then we can extend these results to higher limits. In this form the first part of the corollary appears in [14] and the second part is in [12].

4. Hyper Cohomology

Given a chain complex C_{•} in CS_{W}(G) which is bounded below, and L ∈ CS_{W}(G) we can consider the hyper-Ext groups 𝔼xt_{CS_{W}}^{n} (C_{•}, L). These are the hyper-derived functors of Hom_{CS_{W}}, which takes its values in CS_{T} (G) (where T is any class of subgroups), so are themselves coefficient systems. This is not consistent with our previous definition of Ext as the derived functor on the second variable, unless we are in the circumstances of 3.2. But confusion will rarely arise, and when it does we will write R^{i} Hom(A, −)(B), for example.

Lemma 4.1. 𝔼xt_{CS_{W} (G)}^{n}(C_{•}, L)(J) does not depend on T , provided J ∈ T .

Proof. When we apply Hom_{CS_{W} (G)}(−, L) we do so groupwise. □

There are two spectral sequences (see e.g. [26] 5.7.9):

(1)  ^{II}E_{2}^{p,q} = Ext_{CS_{W} (G)}^{p}(H_{q}(C_{•}), L) ⇒ 𝔼xt_{CS_{W} (G)}^{n}(C_{•}, L)
(2)  ^{I}E_{2}^{p,q} = H^{p}(Ext_{CS_{W} (G)}^{q}(C_{•}, L)) ⇒ 𝔼xt_{CS_{W} (G)}^{n}(C_{•}, L)

We adopt the convention that when we apply plain Ext to a chain complex, we apply it term by term to obtain another chain complex.

The following proposition is the basic result that we will use to obtain the sequences of the introduction and to identify their cohomology.

We say that a class of subgroups T is taut with respect to another class W if each T ∈ T is taut with respect to W as defined after 2.8 .

Proposition 4.2. (1) If, in CS_{W}(G),

H_{n}(C_{•}) = \begin{cases}R̄ & if n = 0, \\ 0 & otherwise, \end{cases}

then 𝔼xt_{CS_{W} (G)}^{n}(C_{•}, L) ≅ Ext_{CS_{W} (G)}^{n}(R̄, L). If T is taut with respect to W then this is also equal to (\underleftarrow{lim}_{W}^{T})^{n}L.

(2) If K ≤ G and H^{∗} Ext_{CS_{W} (G)}^{n}(C_{•}, L)(K) = 0 for n ≥ 1 then
𝔼xt_{CS_{W} (G)}^{n}(C_{•}, L)(K) ≅ H^{n} Hom_{W}(C_{•}, L)(K)

.

Remark. We can always take T = {G} and then T is taut with respect to W. In this way we can always obtain (\underleftarrow{lim}_{W}^{G})^{n}L in part (1).

Proof. For (1) we apply the^{II}E spectral sequence and see that it collapses. The second part follows using 3.2 to see that Ext_{CS_{W} (G)}^{n}(R̄, L) ≅ R^{n} Hom_{CS_{W} (G)}(R̄, −)(L) and then 3.3 to identify this with (\underleftarrow{lim}_{W}^{T})^{n}L. □

Proposition 4.3. Let C_{•} be a complex in CS_{W}(G). Let X ⊆ W and suppose that L ∈ CS_{W}(G) is injective relative to X and also that for each H ∈ X we have that Res_{H}^{G} C_{•} is split exact. Then Hom_{CS_{W} (G)}(C_{•}, L) is split exact. If L is also a Mackey functor then this is split exact as a complex of Mackey functors.

Note that L being a Mackey functor entails W being closed under intersections. Also L is only required to be relatively injective as a coefficient system.

Proof. When we restrict to H ∈ X, Res_{H}^{G} C_{•} becomes split exact. Thus Hom_{CS_{W} (H)}(Res_{H}^{G} C_{•}, L) is split exact. Now apply Coind_{H}^{G} and use

Coind_{H}^{G} Hom_{CS_{W} (H)}(Res_{H}^{G} −, L) ≅ Hom_{CS_{W} (G)}(−, Coind_{H}^{G} Res_{H}^{G} L)

(by 2.11). We see that

Hom_{CS_{W} (G)}(C_{•}, Coind_{H}^{G} Res_{H}^{G} L)

is split exact. Thus our sequence is a summand of a product of split exact sequences, so is itself split exact by [24].

The splitting as a Mackey functor comes from 2.16. □

5. Bredon Cohomology

If Δ is a G-CW-complex on which G acts admissibly (i.e. the stabiliser of each cell fixes it pointwise), let Δ_{n} denote the G-set of n-cells. We can form a chain complex of coefficient systems C_{•}[Δ^{?}] in CS_{S}(G) by setting

C_{n}[Δ^{?}] = R[Δ_{n}^{?}] ≅ ⨁_{n-cells σup to G-conjugacy} R[G/ Stab_{G} σ^{?}],

with the natural boundary morphisms, as described in [7].

More succinctly, we regard Δ as a simplicial G-set (in the language of [26]) and apply the functor G/H ↦ R[G/H^{?}] to obtain a semi-simplicial coefficient system, and then take C_{•}[Δ^{?}] to be the associated chain complex of coefficient systems.

We often restrict this chain complex to some class W where it is better behaved.

We will also use the augmented complex, C̃_{•}[Δ^{?}], where we add the term R̄ in degree −1 and the map R[Δ_{0}^{?}] → R̄ takes each 0-cell to 1.

The definition of the cohomology of Δ with coefficients in a Mackey functor M in

[17] amounts to saying that it is the cohomology of the complex Hom_{MF S(G)}(SC_{•}[Δ^{?}], M). But this is isomorphic to Hom_{CS S(G)}(C_{•}[Δ^{?}], M). For a slightly different approach, see [22].

Notice that if H ∈ W then Hom_{CS_{W} (G)}(R[G/H^{?}], L) ≅ Coind_{H}^{G} Res_{H}^{G} L by 2.12, and so if the stabiliser of every cell in Δ is contained in W then Hom_{W}(C_{•}[Δ^{?}], L) takes the form described explicitly in [7] and [24], which we sometimes refer to as L_{Δ_{•}}.

If X is a class of subgroups of G then we can regard X as a G-poset and form the geometric realisation |X |.

Lemma 5.1. If the normaliser of each chain in X is in W then

hom_{CS_{W} (G)}(C̃_{•}(|X |^{?}), L)

is the normaliser sequence of the introduction, except that we have \underleftarrow{lim}_{W}^{G} L in the first place instead of L(G).

We define the Bredon cohomology of Δ with coefficients in L to be

H_{CS_{W}}^{∗} (Δ, L) = 𝔼xt_{CS_{W}}^{∗} (C_{•}[Δ^{?}], L).

This is again an element of CS_{T} (G).

Remark. For Bredon in [7], W = S always, so C_{•}[Δ^{?}] is a complex of projectives and H_{CS_{W}}^{∗} (Δ, L) = H^{∗} Hom_{CS_{W}} (C_{•}[Δ^{?}], L), but this is not always true for general W.

Remark. In view of 2.20 we can see that the definition of cohomology with coefficients in a Mackey functor M given in [17] is equivalent to H^{∗} Hom_{MF_{S}} (SC_{•}[Δ^{?}], M) ≅ H^{∗} Hom_{CS_{S}} (C_{•}[Δ^{?}], M), so is just Bredon cohomology, with the transfer given as in 2.16.

Example. If W = {1}, the trivial group, and T = {G} then H_{CS_{{1}}}^{∗}(Δ, L)(G) is just the usual G-equivariant cohomology of Δ as in [8], i.e. the cohomology of the Borel construction.

Theorem 5.2. If L ∈ CS_{W}(G) and H̃_{∗}(Δ^{H}, R) = 0 (i.e. R-acyclic) for every H ∈ W and (\underleftarrow{lim}_{W}^{S})^{n}L = 0 for every n ≥ 1 and every S which is the stabiliser of a cell in Δ (e.g. S ∈ W), then

H^{n}(hom_{CS_{W} (G)}(C_{•}[Δ^{?}], L)) ≅ (\underleftarrow{lim}_{W}^{G})^{n}L.

Proof. We take T = {G} and check that the conditions of 4.2 are satisfied (with K = G). This is clear for the first part. For the second we calculate instead with T = S, knowing that this will not matter by 4.1. Now Ext_{CS_{W} (G)}^{n}(R[G/S^{?}], L)(G) ≅ Coind_{S}^{G} Ext_{CS_{W} (S)}^{n}(R̄, L)(G) ≅ Ext_{CS_{W} (S)}^{n}(R̄, L)(S) by 3.4. Next we work in CS_{W}(S) with T = {S} (invoking 4.1 again). But now Hom is balanced so, by 3.3, we have Ext_{CS_{W} (S)}^{n}(R̄, L)(S) ≅ (\underleftarrow{lim}_{W}^{S})^{n}L, which is 0 by hypothesis. □

Theorem 5.3. Let L ∈ CS_{W}(G) and X ⊆ S. Suppose that L is injective relative to X and that Δ^{K} is R-acyclic for every K ∈ W, K ⊆ H ∈ X.

Suppose also that for each subgroup H ∈ X we know that H is taut with respect to W and also for each cell σ of Δ, Stab_{H}(σ) is taut with respect to W.

Then the chain complex

Hom_{CS_{W}} (C̃_{•}[Δ^{?}], L)

is split exact in CS_{S}(G). If L is a Mackey functor then the complex is split as a complex of Mackey functors.

Proof. Let H ∈ X and consider Res_{H}^{G} C̃[Δ^{?}]. It is exact, by the condition on the Δ^{K}, and a complex of projectives, by the tautness conditions. The conditions of 4.3 are now satisfied and the result follows . □

Remark. (1) The statement for Mackey functors is similar to the main theorem of Webb, [24]. He has W = S \Y, and our X is his X \Y. Notice firstly that relative injectivity is the same as relative projectivity for Mackey functors, and secondly that if a coefficient system L is injective relative to X and L vanishes on Y, then L is injective relative to X \ Y, at least if Y is closed under subgroups in X.

(2) This proof of Webb’s theorem, shorn of the general notation, is in fact very short. The relative injectivity condition allows us to reduce to the the case of a group in X, and then the complex C̃_{•}[Δ^{?}] is an exact complex of projectives, so splits.

(3) If R is p-local and Δ^{H} is R-acyclic for every H ≤ G of order p, then Δ^{H} is R-acyclic for every non-trivial p-subgroup H by Smith theory, or by using equivariant cohomology as in [8].

(4) Another proof of Webb’s Theorem has been given by Bouc [5].

Usually Δ is taken to be the Quillen complex, i.e. |S_{p}^{1}(G)|, or some variant. Webb gives many examples of 5.3, but 5.2 is also useful. It can be used to give a simpler proof of the main results in [19]: here is another application.

Example. Fix a prime p and let B denote the ring of Brauer characters, considered as a coefficient system over ℂ on some finite group G. Let N_{p} be the class of subgroups which contain a non-trivial normal p-subgroup, and let Δ denote the usual Brown complex.

Since each stabiliser of a cell in Δ is in N_{p}, 5.2 applies to B ∈ CS_{N_{p}}(G) on Δ and also Hom_{N_{p}}(C_{•}, B) ≅ B_{Δ_{•}}. But by Robinson’s reformulation of Alperin’s Conjecture [16], ∑(−1)^{i} dim B_{Δ_{i}}(G) should be equal to the number of non-projective simple modules for G, denoted f_{o}(G).

It follows that (the non-blockwise version of) Alperin’s weight conjecture is true for all finite groups if and only if for all finite groups G

f_{o}(G) = ∑_{i}(−1)^{i} dim(\underleftarrow{lim}_{N_{p}}^{i} B)(G).

It is interesting to try and understand this by filtering B by functors which are non-zero on only one conjugacy class and then calculating the higher limits of these in the manner of [14]. The result is the original formulation of Alperin’s conjecture, by essentially the same proof as in [16].

6. The structure of C_{•}([Δ^{?}])

First we need some lemmas.

Lemma 6.1. For any H ≤ G and any class W of subgroups of G, let π_{0}(|W∩S(H)|) denote the H-set of components in W ∩ S(H). Then, as RG-modules,

Res_{\{1\}}^{W∪\{1\}} \underrightarrow{lim}_{W}^{W∪\{1\}} R[G/H^{!}] ≅ R[G ×_{H} π_{0}(|W ∩ S(H)|)].

Proof. Since \underrightarrow{lim} commutes with Ind (their right adjoints commute) and R[G/H^{!}] ≅ Ind_{H}^{G} R̄ it suffices to prove the case G = H, observing that both sides are induced modules. But \underrightarrow{lim}_{W}^{W∪\{1\}} R̄ is formed by taking one basis element for each element of W ∩ S(H) and then identifying two basis elements if there is an inclusion between the corresponding subgroups. This yields R[π_{0}(|W ∩ S(H)|)]. □

It will be convenient to define a coefficient system to be based at H ≤ G if it is a summand of a sum of R[G/H^{?}]s.

Corollary 6.2. Consider the canonical map

\underrightarrow{lim}_{W}^{W∪\{1\}} R[G/H^{!}] → R[G/H^{!}]

in CS_{W∪{1}}(G).

It is onto if H contains some element of W and an isomorphism if W ∩ S(H) is connected, in particular if H ∈ W.

If H does not contain any subgroup in W then the left hand side is 0 and the right hand side is based at H.

Lemma 6.3. Let F ≤ G be of index invertible in R, and assume all complexes to be bounded below. If D_{•} is a complex of RG-modules, then Res_{F}^{G} D_{•} is homotopy equivalent to a complex of projective RF -modules if and only if D_{•} is homotopy equivalent to a complex of projective RG-modules.

Proof. If Res_{F}^{G} D_{•} ≃ P_{•} then Ind_{F}^{G} Res_{F}^{G} D_{•} ≃ Ind_{F}^{G} P_{•}, which is a complex of projectives. But D_{•} is a summand of Ind_{F}^{G} Res_{F}^{G} D_{•} by the maps

d ↦ ∑_{g∈G/F} g ⊗ g^{−1}d, h ⊗ d ↦ |G : F |^{−1}hd, d ∈ D_{•}, h ∈ G.

In order to simplify the notation we write D_{•} ⊕ X_{•} ≃ Q_{•}, where Q_{•} is a complex of projectives. Let P_{•}^{D} →^{ρ_{D}} D_{•} and P_{•}^{X} →^{ρ_{X}} X_{•} be projective resolutions. The composition P_{•}^{D} ⊕ P_{•}^{X} ⟶^{ρ_{D}⊕ρ_{X}} D_{•} ⊕ X_{•} →^{≃} Q_{•} is a quasi-isomorphism of bounded below complexes of projectives. Thus it is a homotopy equivalence and hence ρ_{D} ⊕ ρ_{X} is also a homotopy equivalence. It follows that ρ_{D} must be a homotopy equivalence. □

If G acts admissibly on a CW-complex Δ we define Stab_{G}(Δ) to be the set of subgroups Stab_{G}(δ), where δ is a cell of Δ.

Theorem 6.4. Suppose that G acts admissibly on a CW-complex Δ. Suppose also that there is a subgroup F ≤ G with |G : F | invertible in R such that Δ^{F} is R-acyclic and also for every K ∈ Stab_{F} (Δ), K ≠ 1, we have that Δ^{K} is R-acyclic.

Then C̃_{•}(Δ) is homotopy equivalent as a complex in CS_{{1}}(G) ≅ RG-Mod to a bounded complex of projectives.

If W is a class of subgroups of G such that for each K ∈ W, K ≠ 1 we have that Δ^{K} is R-acyclic, then C̃_{•}(Δ^{?}) is equal in the derived category of CS_{W}(G) to a complex of projective coefficient systems which are projective relative to 1.

Proof. Let V = {Stab_{F} (Δ) \ {1}} ∪ {F }, and work in CS_{V}(F). Res_{V} Res_{F}^{G} C̃_{•}(Δ^{?}) is an exact complex of projectives, by 2.8 and the definition of V, so must split. Thus \underrightarrow{lim}_{V}^{V∪\{1\}} Res_{V} Res_{F}^{G} C̃_{•}(Δ^{?}) is also a split exact complex of projectives, since \underrightarrow{lim} preserves projectives by 2.2 and the remark before 2.15.

Now, in CS_{V∪{1}}(F), the natural map

\underrightarrow{lim}_{V}^{V∪\{1\}} Res_{V} Res_{F}^{G} C̃_{•}(Δ^{?}) → Res_{V∪\{1\}} Res_{F}^{G} C̃_{•}(Δ^{?})

is a sum of those in 6.2. Each stabiliser is either in V or is 1, so the map is injective and the cokernel P_{•} is a complex of projectives based at 1.

Now Res_{V∪{1}} Res_{F}^{G} C̃_{•}(Δ^{?}) → P_{•} is a quasi-isomorphism of bounded below complexes of projectives, so must be a homotopy equivalence. Restricting to the subgroup 1 (so we are just dealing with RF -modules), we see that C̃_{•}(Δ) is homotopy equivalent to P_{•}(1), which is a complex of projective RF -modules.

Now apply 6.3 to obtain the first claim.

For the second claim, we work in CS_{W}(G), and notice that C̃_{•}(Δ^{?}) is equal in the derived category to the complex obtained from it by changing the evaluations to 0 on every subgroup not equal to 1. Now we use the first claim. □

Lemma 6.5. In any Abelian category, if there is a map of chain complexes f : C_{•} → P_{•}, which is a quasi-isomorphism, and where P_{•} and C_{•} are bounded below and P_{•} is a complex of projectives, then

C_{•} ⊕ S_{•} ≅ P_{•} ⊕ E_{•},

where S_{•} is a split exact complex of projectives and E_{•} is an exact complex. If f is a homotopy equivalence, then E_{•} is split exact. If C_{•} and P_{•} are bounded or of finite type (when this makes sense) then so are S_{•} and E_{•}.

Proof. By adding a split exact complex of projectives bounded below to C_{•}, we can assume that f is an epimorphism. Let E_{•} be the kernel, so E_{•} is exact and bounded below.

Now, since P_{•} is a complex of projectives, we can take a splitting in each degree, and in this way identify C_{•} with E_{•} ⊕ P_{•}, but with boundary map d_{E⊕P} + ϕ, where ϕ is a collection of maps ϕ_{i} : P_{i} → E_{i−1}. If we set ϕ′_{i} = (−1)^{i}ϕ_{i} then these combine to give a chain map ϕ′ : P_{•} → E_{•} of degree −1. But ϕ′ must factor through a projective resolution of E_{•}, which must be split, so ϕ′ is nullhomotopic. Thus we have maps θ_{i} : P_{i} → E_{i} such that ϕ′ = d_{E}θ + θd_{P}

Set θ′_{i} = (−1)^{i}θ_{i}. An isomorphism Θ : C_{•} → E_{•} ⊕ P_{•} is now given by setting Θ(e, p) = (e + θ′p, p), e ∈ E, p ∈ P.

If f was a homotopy equivalence then E_{•} must be split. □

Corollary 6.6. In the circumstances of 6.4, set C̃_{•}(Δ) = Res_{{1}} C̃_{•}(Δ^{!}), and regard it as a complex of RG-modules. Then

C̃_{•}(Δ) ⊕ S_{•} ≅ P_{•} ⊕ E_{•},

as bounded complexes of RG-modules, where S_{•} is a split exact complex of projectives, P_{•} is a complex of projectives and E_{•} is a split exact complex.

Proof. Apply 6.5 to the first claim of 6.4. □

Finally, we obtain Webb’s original result.

Corollary 6.7. In the circumstances of 6.4, but with Δ finite and R a complete local ring, set C̃_{•}(Δ) = Res_{{1}} C̃_{•}(Δ^{!}), and regard it as a complex of RG-modules. Then

C̃_{•}(Δ) ≅ P_{•} ⊕ E_{•},

as complexes of RG-modules, where P_{•} is a complex of projectives and E_{•} is a split exact complex.

Proof. The complexes in the isomorphism of 6.6 are of finite type so we can apply the Krull-Schmidt Theorem to cancel S_{•}. □

Example. The standard examples where the hypotheses of 6.4 are satisfied are when R is p-local, F is the Sylow p-subgroup of a finite group G and either:

(i) Δ is the geometric realisation of S_{p}^{1}(G), or

(ii) P is a p-subgroup of some group E, G = N_{E}(P)/P and Δ is the geometric realisation of S_{p}(E)_{>P}.

In case (i) we call the homotopy class of complexes of projective modules homotopy equivalent to C̃_{•}(Δ) the Steinberg complex of G, and denote it by St_{•}(G). Whenever we mention this complex it will be implicit that every prime dividing |G|, except perhaps for p, is invertible in R. If p does not divide |G| then St_{•}(G) consists just of the trivial coefficient system R̄ in degree −1.

In both cases the stabiliser of any cell contains a non-trivial normal p-subgroup, and conversely for any K ∈ (N_{p})_{>P}, Δ^{K} is R-acyclic.

In these circumstances we have a uniqueness result.

Proposition 6.8. Suppose that G acts admissibly on a CW-complex Δ in such a way that Stab_{G}(Δ) ⊆ N_{p}, and that X is a class of subgroups containing N_{p} such that, for each 1 ≠ K ∈ X, Δ^{K} is R-acyclic. Then C̃_{•}(Δ^{!}) ≃ C̃_{•}(|S_{p}^{1}(G)|^{!}) in CS_{X} (G) and, in particular, C̃_{•}(Δ) ≃ St_{•}(G) in RG-Mod.

Proof. Both C̃_{•}(Δ^{!}) and C̃_{•}(|S_{p}^{1}|^{!}) are projective resolutions of R̄ in CS_{X \{1}}(G), so are homotopy equivalent by maps which are the identity on R̄ and where the homotopies take the value 0 on R̄.

We can now apply \underrightarrow{lim}_{X \\{1\}}^{X} to recover C_{•}(Δ) and C_{•}(|S_{p}^{1}|), by 6.2. We also

obtain maps between them and the necessary homotopies, which we extend to R̄ by the identity and 0 respectively. □

For example this shows that in case (ii) above the complex obtained is in fact St_{•}(G). Of course in this case the defining simplicial complexes are known to be equivariantly homotopy equivalent anyway.

7. Properties of the Steinberg Complex

It is often convenient to consider St_{•}(G) as a coefficient system in CS_{W}(G) by giving it the value 0 on all non-trivial subgroups. This is in fact formally \underrightarrow{lim}_{1}^{W} St_{•}(G), and is still a complex of projectives, based at 1. We will denote it by St_{•}^{0}(G).

There is an alternative description St_{•}^{0}(G) ≃ C̃_{•}(|S_{p}^{1}|^{?}) ⊗ R̄_{1}, where R̄_{1} is the coefficient system which takes the value R on 1 and 0 elsewhere.

Proposition 7.1. For any class of subgroups W of G containing 1 and any M ∈ CS_{W}(G), the following are all homotopy equivalent as complexes in CS_{W}(G):

(1) Hom_{R?}(St_{•}(G), M(1)),

(2) Hom_{W}(St_{•}^{0}(G), M),
(3) Hom_{R?}(C̃_{•}(|S_{p}^{1}|), M(1)),
(4) Hom_{W}(C̃_{•}(|S_{p}^{1}|^{?}), M(1)^{?}),

(5) If W ⊆ N_{p}, R Hom_{W}(C̃_{•}(|S_{p}^{1}|^{?}), M). (Where R Hom is the complex used to define 𝔼xt.)

Proof. Now (1) and (2) are homotopy equivalent because St_{•}^{0}(G) = \underrightarrow{lim}_{1}^{W} St_{•}(G), and \underrightarrow{lim} is the left adjoint of restriction. Also (1) and (3) are homotopy equivalent by the first part of 6.4. But (3) and (4) are homotopic because M(1)^{?} ≅ \underleftarrow{lim}_{1}^{W} M(1) and \underleftarrow{lim} is the right adjoint of restriction. Finally St_{•}^{0}(G) is a projective resolution of C̃_{•}(|S_{p}^{1}|^{?}) by the last part of 6.4, so (5) is homotopic to (2) by definition. □

Lemma 7.2. If H ◁ G and p divides |H| then for any p-subgroup P of G: (1) p divides |C_{G}(P) ∩ H|,

(2) If σ is any chain of p-subgroups of G then p divides |N_{H}(σ)|.

Proof. P permutes the Sylow p-subgroups of H by conjugation. The number of these is coprime to p, so at least one of them is fixed: call it S.

Now P permutes the non-trivial elements of S. Again, the number of these is coprime to p, so one is fixed, say s. Now s ∈ C_{G}(P) ∩ H.

For (2), let P be the largest subgroup in the chain and apply (1). □

Now we can state a fundamental result from [14], (although our proof is based on a preliminary version of [12]).

Theorem 7.3. Suppose that R is a p-local discrete valuation ring, and that M is an RG-module such that either M is finitely generated or M is projective over R . If the order of the kernel (i.e. the subgroup of elements of G which act trivially on M) is divisible by p, then the complex of RG-modules hom_{RG}(St_{•}(G), M) is split exact.

Proof. First we assume that pM = 0.

Define ĥom_{RG} to be hom_{RG} modulo the image of tr_{1}^{G} : hom_{R} → hom_{RG}, (with the usual transfer on Hom). Using the complex (3) of 7.1, each term is a sum of pieces of the form hom_{RG}(R[G/S], M), where S is the stabiliser of a chain σ.

By 7.2 with H as the kernel of M, we see that S ∩ H contains a non-trivial psubgroup Q, say, so Q is in both S and H. Under the isomorphism hom_{RG}(R[G/S], M) ≅ hom_{RS}(R, M) the image of tr_{1}^{G} on the left corresponds to the image of tr_{1}^{S} on the right. But tr_{1}^{S} factors through tr_{1}^{Q}, and this is multiplication by |Q|, which is equal to 0. Thus hom_{RG}(St_{•}(G), M) = ĥom_{RG}(St_{•}(G), M); but ĥom vanishes on projectives.

In general, since St_{•}(G) is homotopy equivalent to a complex of projectives, hom_{RG}(St_{•}(G), M)/p ≅ hom_{RG}(St_{•}, M/p), and M/p also satisfies the conditions of the theorem, so is split by the proof above.

If M is finitely generated over R, then so is hom_{RG}(St_{•}(G), M), and so split modulo p implies split, (by an obvious generalisation of Nakayama’s Lemma).

If M is not finitely generated then it is a direct limit of finitely generated submodules, and the homology of hom_{RG}(St_{•}(G), −) commutes with direct limits, so the latter is exact. It is also a complex of projective R-modules if M is, because its terms are summands of sums of terms hom_{RG}(RG, M) ≅ M. Thus the complex must split. □

We can generalise this slightly in a way that will be useful in section 9. Example (ii) in §6 suggests that we consider the complex Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•}^{0}(N_{G}(P)/P).

Corollary 7.4. For simplicity we assume that P is a p-subgroup of G, that all coefficient systems are over S_{p} and that R is p-local.

(1) Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•}^{0}(N_{G}(P)/P) is a complex of projectives, based at P.

(2) hom_{CS_{Sp} (G)}(Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•}^{0}(N_{G}(P)/P), M) ≃ 0 for any M ∈ CS_{S_{p}}(G) with M(P) either finitely generated or projective over R and on which p divides the order of the kernel of M(P) as an N_{G}(P)/P -module.

(3) G_{0} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•}^{0}(N_{G}(P)/P) ≃ 0 if P is not in D_{p}(G). (Where G_{0} is the left adjoint of the forgetful functor I : GCS → CS.)

Proof. For (1), observe that the functors preserve projectives because their right adjoints are exact.

For (2), the adjoint properties show that we are just calculating hom_{CS_{Sp} (N_{G}(P)/P)}(St_{•}^{0}(N_{G}(P)/P), Q_{N_{G}(P)/P}^{N_{G}(P)} Res_{N_{G}(P)}^{G} M), and this is 0 by 7.3.

Now (3) is a formal consequence of (2) if P is not centric, since any geometric M will have P C_{G}(P) in the kernel of M(P), and p divides the order of P C_{G}(P)/P, so p will divide the order of the kernel of M(P) as an N_{G}(P)/P -module. If P is centric but not in D_{p} then P is not in B_{p}, by 3.14, so the complex is acyclic. □

The next result is a version of one in [12].

Theorem 7.5. If L ∈ GCS_{S_{p}}(G) then H_{CS_{Sp} (G)}^{∗}(|S_{p}|, L)(G) ≅ H_{CS_{Sp} (G)}^{∗}(|C_{p}|, L)(G), (as R-modules).

Proof. We work in CS_{S_{p}}(G). Define S_{p}^{≥n} to be the subclass of S_{p} of elements of order greater than or equal to n. Then C_{•}(|S_{p}|^{?}) is filtered by the complexes C_{•}(|S_{p}^{≥n}|^{?}). Now the terms of the factor C_{•}(|S_{p}^{≥n}|^{?})/C_{•}(|S_{p}^{≥n+1}|^{?}) correspond to chains with bottom element of order n. There are no inclusions between such chains with the same bottom element, so this factor splits as a direct sum of pieces indexed by the conjugacy class of the bottom element P of the chain, and this piece is induced from N_{G}(P).

C_{•}(|S_{p}^{≥n}|^{?})/C_{•}(|S_{p}^{≥n+1}|^{?}) ≅ ⨁_{P ∈S_{p}/G, |P |=n} Ind_{N_{G}(P)}^{G} C̃_{•+1}(|(S_{p})_{>P} |^{?})

≅ ⨁_{P ∈S_{p}/G, |P |=n} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} C̃_{•+1}(|(S_{p})_{>P} |^{?})

≃ ⨁_{P ∈S_{p}/G, |P |=n} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•+1}^{0}(N_{G}(P)/P).

The last equivalence is a consequence of the well-known fact that |(S_{p}(G))_{>P} | is equivariantly homotopy equivalent to |(S_{p}(N_{G}(P)))_{>P} |, (by the assignment Q ≥ N_{Q}(P)), and the latter is clearly isomorphic to |S_{p}^{1}(N_{G}(P)/P)|.

If we apply G_{0} then the summands with P not centric will vanish by 7.4. Note also that (S_{p})_{>P} = (C_{p})_{>P} if P is centric.

Since the last line of the formulas above is a complex of projectives, it follows that the inclusion of C_{p} into S_{p} induces an isomorphism

𝔼xt^{∗}(C_{•}(|(C_{p})^{≥n}|^{?})/C_{•}(|(C_{p})^{≥n+1}|^{?}), L) ≅ 𝔼xt^{∗}(C_{•}(|(S_{p})^{≥n}|^{?})/C_{•}(|(S_{p})^{≥n+1}|^{?}), L).

We can now show, induction on n, the long exact sequence for 𝔼xt and the five lemma, that there is an isomorphism

𝔼xt^{∗}(C_{•}(|C_{p}|^{?})/C_{•}(|C_{p}^{≥n}|^{?}), L) ≅ 𝔼xt^{∗}(C_{•}(|S_{p}|^{?})/C_{•}(|S_{p}^{≥n}|^{?}), L).

The case n = 0 is trivial and the case n large is the result claimed. □

8. The Subgroup Sequence

There are other ways of obtaining a chain complex in CS_{W}(G) from a class X of subgroups, which do not factor through the geometric realisation. Although the examples that we will consider are simple and concrete it seems helpful to mention the general context.

We consider the category ch(X) of chains in X and inclusions. We need a contravariant functor F : ch(X) → PCS_{W}(G) together with a collection of conjugation maps c_{g} : F (σ)(H) → F (^{g}σ)(^{g}H) for each g ∈ G, σ ∈ ch(X), H ∈ W. These conjugation maps must satisfy the usual properties c_{1} = Id, c_{g_{1}}c_{g_{2}} = c_{g_{1}g_{2}} and they must commute with restriction in PCS_{W}(G) and also with restriction in ch(X) (induced by inclusion). In addition we require that F (σ)(H) = 0 if H ≰ N_{G}(σ) and, for h ∈ H ≤ N_{G}(σ), we need c_{h} = Id : F (σ)(H) → F (σ)(H).

This naturally makes F (σ) into an element F̃(σ) ∈ CS_{W}(N_{G}(σ)), extended to H ≰ N_{G}(σ) by 0. These F̃(σ) and the restriction maps F (σ) → F (τ) uniquely determine the structure defined above.

For a G-subset X of ch(X) we define F (X) = ⊕_{σ∈X}F (σ). With the natural action of G via the c_{g} we have F (X) ∈ CS_{W}(G). In particular, writing (σ) for the orbit of σ, we have F ((σ)) ≅ Ind_{N_{G}(σ)}^{G} F̃(σ).

Up until now we have always used the functor F^{N}, where F̃^{N}(σ) = R̄ ∈ CS_{W}(N_{G}(σ)), so F^{N}((σ)) ≅ Ind_{N_{G}(σ)}^{G} R̄ ≅ R[G/N_{G}(σ)^{?}].

Recall that for any chain σ in X we denote by σ_{b} the smallest element and by σ_{t} the largest.

The functor which represents the subgroup sequence is F^{S}, defined by

F^{S}(σ)(P) = \begin{cases}R & if σ_{b} ≥ P , \\ 0 & otherwise. \end{cases}

Thus F^{S}((σ))(P) is the free R-module on the chains τ in the orbit of σ with τ_{b} ≥ P. The restrictions are the canonical inclusion maps.

For P ≤ G, define R̄_{P} ∈ CS_{W}(N_{G}(P)) by R̄_{P} = \underrightarrow{lim}_{W(P)}^{W(N_{G}(P))} R̄, so that for

H ∈ W, R̄_{P} (H) is R if H ≤ P and 0 otherwise. (So, in fact, R̄_{P} = Inf_{N_{G}(P)/P}^{N_{G}(P)} R̄_{1}.)

Then we have F^{S}((σ)) = Ind_{N_{G}(σ)}^{G} Res_{N_{G}(σ)}^{N_{G}(σ_{b})} R̄_{σ_{b}}, which we will abbreviate to Ind_{N_{G}(σ)}^{G} R̄_{σ_{b}}.

Lemma 8.1. hom_{CS_{W} (G)}(Ind_{N_{G}(σ)}^{G} R̄_{σ_{b}}, L) ≅ ((\underleftarrow{lim}_{W(σ_{b})} L)(σ_{b}))^{N_{G}(σ)}, and if σ_{b} ∈

W, then this is also isomorphic to L(σ_{b})^{N_{G}(σ)}.

Proof.

hom_{CS_{W} (G)}(Ind_{N_{G}(σ)}^{G} R̄_{σ_{b}}, L) ≅ hom_{CS_{W} (N_{G}(σ))}(R̄_{σ_{b}}, L)
≅ hom_{CS_{W} (N_{G}(σ))}(\underrightarrow{lim}_{W(σ_{b})}^{W(N_{G}(σ_{b}))} R̄, L)
≅ hom_{CS_{W(σ_{b})}(N_{G}(σ))}(R̄, L)
≅ (\underleftarrow{lim}_{W(σ_{b})} L)(σ_{b})^{N_{G}(σ)}
□

Now C_{•}^{S}(X) is defined to be the complex in CS_{W}(G) with terms F^{S}(ch_{n}(X)) and the usual boundary maps arising from the semisimplicial structure of ch(X). There is also an augmented version C̃_{•}^{S}(X).

The next two results follow directly from the definitions and 8.1 respectively.

Lemma 8.2. C_{•}^{S}(X)(H) ≅ C_{•}(|X_{≥H}|).

Corollary 8.3. If L ∈ CS_{W}(G) and X ⊆ W then hom_{CS_{W} (G)}(C̃_{•}^{S}(X), L) is the second sequence of the introduction, the subgroup sequence, except that L(G) is replaced by \underleftarrow{lim}_{W}^{G} L.

Lemma 8.4. If every X_{≥H}, H ∈ W, is R-acyclic (e.g. W ⊆ X) then 𝔼xt_{CS_{W} (G)}^{n}(C_{•}^{S}(X), L) ≅ R^{n} Hom_{CS_{W} (G)}(−, L)(R̄) in CS_{W}(G). If T is taut with respect to W then this is also equal to (\underleftarrow{lim}_{W}^{T})^{n}L.

Proof. This is a consequence of 4.2, in view of 8.2. □

Lemma 8.5. In any Abelian category, let C_{•} be a chain complex and suppose that it has a finite filtration such that each of the factors is homotopy equivalent to a complex of projectives P_{•}^{i} and such that in each degree the filtration splits to give a direct sum decomposition. Then C_{•} is homotopy equivalent to a complex of projectives: ignoring the boundary maps, this complex can be taken to be ⊕_{i}P_{•}^{i}.

Proof. By induction, we may reduce to the case where there are only two composition factors, so we have a short exact sequence X_{•}^{1} →^{a} C_{•} → X_{•}^{2}, which is split in each degree and where X_{•}^{i} ≃ P_{•}^{i}, i = 1, 2. Since a is split in each degree we know that X_{•}^{2} ≃ cone(a), so X_{•}^{1} →^{a} C_{•} → X_{•}^{2} extends to a triangle in the homotopy category.

Thus we have a triangle P_{•+1}^{2} →^{b} P_{•}^{1} → C_{•} → P_{•}^{2}, so C_{•} ≃ cone(b), which is a complex of projectives. □

We say that a coefficient system is based at X if it is a summand of a sum of terms of the form R[G/X^{?}] with X ∈ X.

Proposition 8.6. Suppose that every prime dividing |G| except perhaps p is invertible in R, that X ⊆ S_{p}(G) and that X contains the Sylow p-subgroups of G.

If for every H ∈ X that is not a Sylow p-subgroup of G and every p-subgroup K with H < K ≤ N_{G}(H) we have (X_{>H})^{K} is R-acyclic, then C_{•}^{S}(X) is, as a complex in CS_{S_{p}}(G), homotopy equivalent to a complex of projectives, based at X.

If we ignore the boundary maps, then this complex can be taken to be

⨁_{P ∈X /G} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•−1}^{0}(N_{G}(P)/P).

Proof. Define X^{≥n} to be the subclass of X of elements of order greater than or equal to n. Then C_{•}^{S}(X) is filtered by the complexes C_{•}^{S}(X^{≥n}). Now the terms of the factor C_{•}^{S}(X^{≥n})/C_{•}^{S}(X^{≥n+1}) correspond to chains with bottom element of order n. There are no inclusions between such chains with the same bottom element, so this factor splits as a direct sum of pieces indexed by the conjugacy class of the bottom element P of the chain, and this piece is induced from N_{G}(P).

C_{•}^{S}(X^{≥n})/C_{•}^{S}(X^{≥n+1}) ≅ ⨁_{P ∈X /G, |P |=n} Ind_{N_{G}(P)}^{G} C̃_{•−1}(|X_{>P} |^{?}) ⊗ R̄_{P}

≅ ⨁_{P ∈X /G, |P |=n} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} C̃_{•−1}(|X_{>P} |^{?}) ⊗ R̄_{1}

≃ ⨁_{P ∈X /G, |P |=n} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•−1}^{0}(N_{G}(P)/P),

by 6.8. These terms are projective, based at X, if P is not maximal in X. If P is Sylow then we just have Ind_{N_{G}(P)}^{G} R̄_{P} in degree 0. But this is projective in CS_{W}(N_{G}(P)) by 2.5.

Now we can apply 8.5. □

Corollary 8.7. ([12]) Suppose that every prime dividing |G| except perhaps p is invertible in R, that X ⊆ W ⊆ S_{p}(G) and that X contains the Sylow p-subgroups of G. Suppose also that, for every H ∈ W, X_{≥H} is R-acyclic and that for every non-Sylow H ∈ X and K ∈ S_{p} with H ◁ K we have that (X_{>H})^{K} is R-acyclic.

Then C_{•}^{S}(X) is homotopy equivalent to a projective resolution of R̄ in CS_{W}(G) by projectives based at X.

So, for L ∈ CS_{W}(G), we have H^{n} Hom_{CS_{W} (G)}(C_{•}^{S}(X), L) ≅ R^{n} Hom_{CS_{W} (G)}(−, L)(R̄) in CS_{T} (G).

In particular, the homology of the subgroup sequence of the introduction (after removing the first term) is (\underleftarrow{lim}_{W}^{G})^{n}L.

Proof. By 8.6 we see that C_{•}^{S}(X) is homotopy equivalent to a complex of projectives in CS_{S_{p}}(G) based at X. This remains projective on restriction to W by 2.8. Its cohomology is R̄, by 8.2. This proves the claim about the projective resolution; for the rest use 8.3. □

For the rest of this section we continue to suppose that every prime dividing |G| except perhaps p is invertible in R.

Corollary 8.8. Suppose that W ⊆ S_{p} is non-empty and closed under supergroups in Wess(S_{p}) and that Wess(S_{p}) ∩ W ⊆ X ⊆ W. Then the hypotheses of 8.7 are satisfied.

Note that we could replace Wess(S_{p}) by B_{p} to obtain a simpler statement. (That the two are the same is a conjecture of Quillen.) We could also use a weaker definition of Wess, in terms of R-acyclicity instead of contractibility.

Proof. We need to check the conditions of 8.7.

Since W is closed under supergroups in Wess(S_{p}), it contains the Sylow psubgroups of G. Thus these are in Wess(S_{p}) ∩ W and so in X. So the Sylow p-subgroups are the only maximal elements of X and certainly remain maximal in W.

Notice that, for any class of subgroups V and any H, K ∈ S_{p}, we have that Wess(V_{>H}^{K}) = Wess(V^{K}) ∩ V_{>H}.

Also, if K normalises J, then (S_{p}^{K})_{>J} contracts to JK unless K ≤ J. It follows that Wess(S_{p}^{K}) = Wess(S_{p}) ∩ (S_{p})_{≥K}.

Now, if H, K ∈ S_{p} and H ◁ K, H ≠ K then Wess((S_{p}^{K})_{>H}) = Wess(S_{p}) ∩ (S_{p})_{≥K}. If also H ∈ W then, since W is closed under supergroups in Wess(S_{p}), we have Wess((S_{p}^{K})_{>H}) ⊆ W_{>H}^{K}. But the left hand side is clearly in Wess(S_{p}) ∩ W, so by hypothesis is in X, and thus Wess((S_{p}^{K})_{>H}) ⊆ X_{>H}^{K}.

Thus we can apply 3.10 to the inclusion X_{>H}^{K} ⊆ (S_{p}^{K})_{>H} to see that we have a homotopy equivalence on the geometric realisations. But we have just seen that the right hand side is contractible, so X_{>H}^{K} is contractible.

A similar proof shows that X_{≥H} is contractible. □

Corollary 8.9. If W is closed under intersections, M ∈ MF_{W}(G) is injective relative to W and also the hypotheses of 8.7 are satisfied, then the subgroup sequence is split as a complex of Mackey functors.

Proof. Just as in the proof of 4.3, the condition of relative injectivity allows us to reduce to the case of a group H ∈ W. But R̄ is projective in CS_{W}(G), so C̃_{•}^{S}(X) splits. □

9. Change of Class of Groups

The results are based on those in [12].

As before, we continue to suppose that every prime dividing the order of |G|, except perhaps p, is invertible in R.

Corollary 9.1. In the circumstances of 8.8, (\underleftarrow{lim}_{W}^{G})^{n} ≅ (\underleftarrow{lim}_{B_{p}∩W}^{G})^{n} Res_{B_{p}∩W}^{W} on CS_{W}(G).

Proof. Use 8.7 and 8.8 with X = B_{p} ∩ W and notice that the terms of the subgroup sequence only evaluate L on groups in X. □

Corollary 9.2. (\underleftarrow{lim}_{S_{p}}^{G})^{n} ≅ (\underleftarrow{lim}_{B_{p}}^{G})^{n} Res_{B_{p}}^{S_{p}} on CS_{S_{p}}(G).

We extract for future use the main feature of the proof of 8.6.

Corollary 9.3. Suppose that W ⊆ S_{p} is closed under supergroups in B_{p}. Then C_{•}^{S}(W) and C_{•}^{S}(B_{p} ∩ W) are both homotopy equivalent to a projective resolution of R̄ in which all the terms are based at B_{p} ∩ W.

Proof. Both complexes have cohomology R̄ in degree 0 and 0 elsewhere, by 8.2. They are homotopy equivalent to a complex of projectives, by 8.6, and this also shows that C_{•}^{S}(B_{p} ∩ W) is homotopy equivalent to a complex of projectives based at B_{p} ∩ W. □

Proposition 9.4. Suppose that D_{p} ∩ W ⊆ X ⊆ W and that X , W ⊆ S_{p} are closed under supergroups in B_{p}. Then G_{0}C_{•}^{S}(X) ≃ G_{0}C_{•}^{S}(W): on restriction to C_{p} ∩ W these are homotopy equivalent to a projective resolution of R̄ in CS_{C_{p}∩W}(G) in which all the terms are summands of sums of terms of the form G_{0}R[G/H^{?}], for H ∈ D_{p} ∩ W.

Proof. Consider the inclusion map i : G_{0}C_{•}^{S}(X) → G_{0}C_{•}^{S}(W). We claim that both sides are filtered by complexes which are homotopy equivalent to sums of complexes of the form G_{0} Ind_{N_{G}(P)}^{G} Inf_{N_{G}(P)/P}^{N_{G}(P)} St_{•−1}^{0}(N_{G}(P)/P), where P appears (once for its conjugacy class) if and only if it is in X (respectively W). This is true before applying G_{0} from the proof of 8.6 and 8.8, and remains true afterwards by 8.5 since all the complexes in the filtration are homotopy equivalent to complexes of projectives in CS_{W}(G).

Now if P ∈ X then the P terms are the same in both C_{•}^{S}(X) and C_{•}^{S}(W). If P ∉ X then P ∉ D_{p} ∩ W, so the P terms in C_{•}^{S}(X) are 0 and those in C_{•}^{S}(W)are homotopy equivalent to 0 by 7.4. It follows that i is a homotopy equivalence.

Notice that G_{0} and Res_{C_{p}∩W}^{W} commute. Also Res_{C_{p}∩W}^{W} is clearly exact and G_{0} : CS_{C_{p}∩W}(G) → GCS_{C_{p}∩W}(G) is exact because it has the explicit description (G_{0}L)(P) = H_{0}(C_{G}(P)/Z(P), L(P)) on P ∈ C_{p}, and C_{G}(P)/Z(P) has order coprime to p.

Finally, Res_{W}^{W} G_{0}C_{•}^{S}(W) ≅ G_{0} Res_{C_{p}∩W}^{W} C_{•}^{S}(B_{p} ∩ W) has homology just R̄ in degree 0, by 9.3 and the exactness property mentioned above. Also the previous part of the proof shows that C_{•}^{S}(W) is homotopy equivalent to a complex of projectives of the form claimed, and these remain projective on restriction. □

Recall that I denotes the inclusion functor GCS → CS.

Corollary 9.5. In the circumstances of 9.4, for any L ∈ GCS_{W}(G) we have that H^{n} Hom_{CS_{W} (G)}(C_{•}^{S}(X), IL) ≅ R^{n} Hom_{CS_{W} (G)}(−, IL)(R̄).

In particular, the homology of the subgroup sequence of the introduction (after removing the first term) is (\underleftarrow{lim}_{W}^{G})^{n}IL.

Proof. We know that H^{n} Hom_{CS_{W} (G)}(C_{•}^{S}(W), IL) ≅ R^{n} Hom_{CS_{W} (G)}(−, IL)(R̄) from 8.6. Now Hom_{CS_{W} (G)}(C_{•}^{S}(W), IL) ≅ Hom_{GCS_{W} (G)}(G_{0}C_{•}^{S}(W), L) ≅ Hom_{GCS_{W} (G)}(G_{0}C_{•}^{S}(X), L) ≅ Hom_{CS_{W} (G)}(C_{•}^{S}(X), IL) by 9.4. □

Corollary 9.6. In the circumstances of 9.4, (\underleftarrow{lim}_{W}^{G})^{n}I ≅ (\underleftarrow{lim}_{C_{p}∩W}^{G})^{n} Res_{C_{p}∩W}^{W} I ≅

(\underleftarrow{lim}_{D_{p}∩W}^{G})^{n} Res_{D_{p}∩W}^{W} I on GCS_{W}(G).

Proof. By 9.5 and 8.1 we see that (\underleftarrow{lim}_{W}^{G})^{n}L ≅ hom_{CS_{W} (G)}(C_{•}^{S}(W), L) ≅ hom_{CS_{X} (G)}(C_{•}^{S}(W), L) ≅ (\underleftarrow{lim}_{X}^{G})^{n}L. □

Corollary 9.7. (\underleftarrow{lim}_{S_{p}}^{G})^{n}I ≅ (\underleftarrow{lim}_{C_{p}}^{G})^{n} Res_{C_{p}}^{S_{p}} I ≅ (\underleftarrow{lim}_{D_{p}}^{G})^{n} Res_{D_{p}}^{S_{p}} I on GCS_{S_{p}}(G).

Finally, we relate higher limits in CS and in GCS in certain circumstances.

Lemma 9.8. Suppose that X ⊆ C_{p} and M, N ∈ GCS_{X} (G). Then R^{n} hom_{GCS_{X} (G)}(−, M) ≅ R^{n}(hom_{CS_{X} (G)}(−, IM)) ∘ I on GCS_{X} (G).

Proof. Because hom_{GCS_{W} (G)}(−, M) ≅ hom_{CS_{W} (G)}(−, IM)∘I we can deduce R^{n} hom_{GCS_{W} (G)}(−, M) ≅ R^{n} hom_{CS_{W} (G)}(−, IM)∘I provided that I is exact (which it clearly is) and preserves injectives.

But the right adjoint of I is G_{0}, which is exact on CS_{X} (G) (see proof of 9.4), so I does preserve injectives.

Now observe that hom is balanced. □

Corollary 9.9. Suppose that every prime dividing |G| except perhaps for p is invertible and that W is closed under supergroups in B_{p}. Then (\underleftarrow{lim}_{W}^{G})^{n} ≅ (\underleftarrow{lim}_{W}^{G})^{n}I on GCS_{C_{p}}(G) (where the first higher limit is in GCS_{W}(G) and the second is in CS_{W}(G)).

Proof. Using 2.3 and 9.6 and the fact that hom_{GCS}(M, N) ≅ hom_{CS}(IM, IN) for M, N ∈ GCS, we find that, for N ∈ GCS_{W}(G), hom_{GCS_{W} (G)}(−, N) ≅ \underleftarrow{lim}_{W}^{G} Hom_{CS_{W} (G)}(−, N) ≅ \underleftarrow{lim}_{C_{p}∩W}^{G} Hom_{GCS_{Cp∩W} (G)}(Res_{C_{p}∩W}^{W} −, Res_{C_{p}∩W}^{W} N) ≅ hom_{GCS_{Cp∩W} (G)}(Res_{C_{p}∩W}^{W} −, Res_{C_{p}∩W}^{W} N) ≅

hom_{GCS_{Cp∩W} (G)}(−, Res_{C_{p}∩W}^{W} N) ∘ Res_{C_{p}∩W}^{W} on GCS_{W}(G).

But Res_{C_{p}∩W}^{W} is exact and preserves projectives in GCS, just as in the proof of 3.12. From this we obtain that R^{n} hom_{GCS_{W} (G)}(−, N) ≅ R^{n}(hom_{GCS_{Cp∩W} (G)}(−, Res_{C_{p}∩W}^{W} N))∘ Res_{C_{p}∩W}^{W} on GCS_{W}(G).

Similarly, since Hom_{CS}(M, IN) is naturally in GCS for M ∈ CS, N ∈ GCS, we obtain that R^{n} hom_{CS_{W} (G)}(−, IN) ≅ R^{n}(hom_{CS_{Cp∩W} (G)}(−, Res_{C_{p}∩W}^{W} IN))∘I Res_{C_{p}∩W}^{W} on CS_{W}(G).

Now R^{n} hom_{GCS_{Cp∩W} (G)}(−, Res_{C_{p}∩W}^{W} N) ≅ R^{n}(hom_{CS_{Cp∩W} (G)}(−, Res_{C_{p}∩W}^{W} IN))∘ I on GCS_{C_{p}∩W}(G), by 9.8. We deduce that R^{n} hom_{GCS_{W} (G)}(−, N) ≅ R^{n} hom_{CS_{W} (G)}(−, IN) on GCS_{W}(G). □

Remark. In general the higher limits in CS and GCS are not the same. For example if W consists only of the trivial group then the higher limits in CS_{{1}}(G) are the cohomology groups H^{∗}(G, −), whilst the higher limits in GCS_{{1}}(G) vanish.

10. The Centraliser Sequence

The construction is analogous to that of the subgroup sequence and we will be brief.

We use the functor F^{C} defined on a chain σ by

F^{C}(σ)(P) = \begin{cases}R & if C_{G}(σ_{t}) ≥ P , \\ 0 & otherwise. \end{cases}

this implies that F^{C}((σ))(P) is the free R-module on the chains τ in the orbit of σ with C_{G}(τ_{t}) ≥ P or, equivalently, F^{C}((σ)) ≅ Ind_{N_{G}(σ)}^{G} R̄_{C_{G}(σ_{t})}.

Lemma 10.1. hom_{CS_{W} (G)}(Ind_{N_{G}(σ)}^{G} R̄_{C_{G}(σ_{t})}, L) ≅ ((\underleftarrow{lim}_{W(C_{G}(σ_{t}))} L)(C_{G}(σ_{t})))^{N_{G}(σ)},

and if C_{G}(σ_{t}) ∈ W, then this is also isomorphic to L(C_{G}(σ_{t}))^{N_{G}(σ)}.

Let C_{•}^{C}(X) be the complex of coefficient systems obtained from the class of subgroups X using F^{C}, and C̃_{•}^{C}(X) the augmented version.

Corollary 10.2. If L ∈ CS_{W}(G) and for each X ∈ X we have C_{G}(X) ∈ W, then L(G) → hom_{CS_{W} (G)}(C_{•}^{C}(X), L) is the third sequence of the introduction, the centraliser sequence.

Lemma 10.3. If every X (C_{G}(H)), H ∈ W, is R-acyclic (e.g. X = A_{p}^{1} or S_{p}^{1} and for any H ∈ W, Z_{p}(H) ≠ 1) then H^{∗}(C_{•}^{C}(X)) ≅ R̄ and 𝔼xt_{W}^{n}(C_{•}^{C}(X), L) ≅ (\underleftarrow{lim}_{X}^{W})^{n}L in CS_{W}(G).

Proof. This is a consequence of 4.2, since C_{•}^{C}(X)(H) ≅ C_{•}(|X (C_{G}(H))|). □

Proposition 10.4. If A_{p}^{1} ⊆ X ⊆ S_{p} and W contains the centraliser of every element of X then C_{•}^{C}(X) is homotopy equivalent to a projective resolution of R̄ in CS_{W}(G).

Proof. The complex has the correct homology, by 10.3.

\begin{aligned}C_{•}^{C}(X^{≤n})/C_{•}^{C}(X^{≤n−1}) & ≅ ⨁_{P ∈X /G,|P |=n} Ind_{N_{G}(P)}^{G} C̃_{•}(|X_{<P} |) ⊗ R̄_{C_{G}(P)} \\ & ≅ ⨁_{P ∈X /G,|P |=n} Inf_{N_{G}(P)/C_{G}(P)}^{N_{G}(P)} C̃_{•+1}(|X_{<P} |)^{0}. \end{aligned}

So we just have to show that C̃_{•+1}(|X_{<P} |)^{0} is homotopy equivalent to a complex of projectives, or equivalently that C̃_{•}(|X_{<P} |) is homotopy equivalent to a complex of projective N_{G}(P)/C_{G}(P)-modules.

If P is trivial or cyclic of order p then everything is projective, since N_{G}(P)/C_{G}(P) is trivial.

For the other cases we check the conditions of 6.4.

If P is elementary abelian of rank greater than 1, then, for any p-subgroup H of N_{G}(P)/C_{G}(P), let E ≤ P be the subgroup of elements centralised by H. Let F be a subgroup of P of index p, containing E and normalised by H. Now if X ∈ X_{<P} is normalised by H then X ∩ F ≠ 1: this is seen by considering the codimensions of vector spaces over 𝔽_{p} if the rank of X is at least 2, and if X has rank 1 then it must be in E. We see that |X_{<P} |^{H} contracts to F by X → X ∩ F → F.

Otherwise, for any p-subgroup H of N_{G}(P)/C_{G}(P), let 1 ≠ E ≤ Φ(P)∩Z_{p}(P) be centralised by H. Then we see that |X_{<P} |^{H} contracts to E by X → X.E → E. □

Corollary 10.5. ([12]) Suppose that A_{p}^{1} ⊆ X = S_{p}^{1} and that X is closed under products with elementary abelian groups (i.e. if X ∈ X and Y < G, Y = X × E, E ∈ A_{p}, then Y ∈ X). Suppose also that W contains the centraliser of every element of X and also that for any H ∈ W, Z_{p}(H) ≠ 1. Then the homology of the centraliser sequence of the introduction (after removing the first term) is (\underleftarrow{lim}_{W}^{G})^{n}L.

11. Projective Resolutions and the Steinberg Complex

Notice that 8.6 describes a projective resolution of R̄ in CS_{S_{p}}(G) in terms of Steinberg complexes. In particular R̄ has finite projective dimension, originally a result of Bouc [5], (it also appears without proof in [14]).

Conversely we can calculate Steinberg complexes from a projective resolution L_{•} → R̄ by

St_{•}(N_{G}(P)/P) ≃ (\underrightarrow{lim}_{P <Q≤N_{G}(P)} Res_{N_{G}(P)}^{G} L)(P).

For each complex in the homotopy class of St_{•}(G) consider the highest degree in which the complex is non-zero, and define σ(G) to be the minimum of these.

If R is p-complete then we have the Krull-Schmidt property, so there is actually a smallest representative of St_{•}(G).

Similarly, define ρ(G) to be the shortest possible length of a projective resolution of R̄ in CS_{S_{p}}(G).

So ρ(G) = max_{P ∈S_{p}} σ(N_{G}(P)/P) + 1 = max_{P ∈B_{p}} σ(N_{G}(P)/P) + 1.

Also σ(G) ≤ p-rank(G) − 1

and σ(G) ≤maximum length of a chain in B_{p}(G) − 1.

Thus ρ(G) ≤ max_{P ∈B_{p}(G)} p-rank(N_{G}(P))

and ρ(G) ≤maximum length of a chain in B_{p}(G).

In fact the Steinberg complex also controls the difference between higher limits over S_{p} and S_{p}^{1}.

To see this, for any RG-module V let V_{1} ∈ CS_{S_{p}}(G) denote the coefficient system which takes the value V on 1 and 0 elsewhere.

Proposition 11.1. Assume that R is a field k, of characteristic p. For a fixed group G, the following are equivalent:

(1) (\underleftarrow{lim}_{S_{p}}^{G})^{i}V_{1} = 0, for all i ≥ 0 and all kG-modules V;
(2) (\underleftarrow{lim}_{S_{p}}^{G})^{i}(kG)_{1} = 0, for all i ≥ 0;
(3) St_{•}(G) ≃ 0;

(4) The canonical map yields (\underleftarrow{lim}_{S_{p}}^{G})^{i} ≅ (\underleftarrow{lim}_{S_{p}^{1}}^{G})^{i} Res_{S_{p}^{1}}^{S_{p}}, for all i ≥ 0, on

CS_{S_{p}}(G);
(5) \underleftarrow{lim}_{S_{p}^{1}}^{G}(kG)^{?} ≅ k and (\underleftarrow{lim}_{S_{p}^{1}}^{G})^{i}(kG)^{?} = 0 for i ≥ 1.

Proof. It is clear that (1) ⇒ (2).

Now hom_{CS_{Sp} (G)}(−, V_{1}) vanishes on any projective based at a non-trivial psubgroup, by 2.8. Thus 8.6 shows that if P_{•} → R̄ is a projective resolution then hom_{CS_{Sp} (G)}(P_{•}, V_{1}) is homotopy equivalent to hom_{CS_{Sp} (G)}(St_{•+1}^{0}(G), V_{1}), which, in turn, is isomorphic to Hom_{kG}(St_{•+1}(G), V). So (\underleftarrow{lim}_{S_{p}}^{G})^{i}V_{1} is equal to the cohomology of Hom_{kG}(St_{•+1}(G), V).

Clearly now (3) ⇒ (1). Also Hom_{kG}(St_{•+1}(G), kG) will detect any non-exactness in St_{•}(G), so (2) ⇒ (3).

For any L ∈ CS_{S_{p}}(G), let L^{1} denote the cokernel of the inclusion L(1)_{1} → L. We claim that (\underleftarrow{lim}_{S_{p}}^{G})^{i}L^{1} ≅ (\underleftarrow{lim}_{S_{p}^{1}}^{G})^{i} Res_{S_{p}^{1}}^{S_{p}} L. This is because L^{1} ≅ \underleftarrow{lim}_{S_{p}^{1}}^{S_{p}} Res_{S_{p}^{1}}^{S_{p}} L.

But \underleftarrow{lim} preserves injectives (because its left adjoint is exact) and is itself exact, so Ext_{CS_{Sp} (G)}^{i}(R̄, L^{1}) ≅ Ext_{CS_{Sp} (G)}^{i}(R̄, \underleftarrow{lim}_{S_{p}^{1}}^{S_{p}} Res_{S_{p}^{1}}^{S_{p}} L) ≅ Ext_{CS_{S^{1}_{p}} (G)}^{i}(R̄, Res_{S_{p}^{1}}^{S_{p}} L).

The long exact Ext sequence for L(1)_{1} → L → L^{1} now shows that (1) ⇔ (4). Similarly (2) ⇔ (5), because kG^{?} is injective on S_{p} so the higher limits vanish. □

Remark. Quillen conjectured that |S_{p}^{1}(G)| is contractible if and only if G contains a non-trivial normal p-subgroup [18]. In fact, according to [12], no counterexample seems to be known if the contractibility condition is replaced by 𝔽_{p}-acyclicity.

Notice that 𝔽_{p}-acyclicity is equivalent to condition (3) of 11.1, so we see that the (stronger) conjecture is equivalent to the statement: G contains a non-trivial normal p-subgroup if and only if the conditions of 11.1 are satisfied.

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THE BREDON COHOMOLOGY OF SUBGROUP COMPLEXES 31

Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

E-mail address: Peter.Symonds@umist.ac.uk