THE BREDON COHOMOLOGY OF SUBGROUP COMPLEXES

PETER SYMONDS

Abstract. 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

We examine various sequences for a coefficient system on a group G in terms of some class X of subgroups of G, in particular when these sequences are exact.

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 [21] 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 obscure. An infinite version of the third sequence can be found in Jackowski, McClure and Oliver [12]. Later Dwyer [10] had infinite versions of the second and third sequences, which arose from stable homotopy theory, 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 independently in work of Villarroel-Flores and Webb [20] and of Grodal [11].

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 homology 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 Slominska has recently appeared [13].

Our strategy is to show that all these sequences are representable as the modules of homomorphisms from some complex of coefficient systems C̃_{•} to L. It then turns out that C̃_{•} is equivalent to a complex of projectives, and this forces a spectral sequence to collapse.

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 the appendix.

We have been careful wherever possible to allow infinite groups, although this is not a direction that we pursue here, and it is abandoned in later sections, where we need a Sylow p-subgroup.

2. Coefficient Systems

We will always work over a fixed unital ring R and refer to a fixed prime p. 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. The superscript 1 will be used to denote the given class with the trivial subgroup removed.

A coefficient system C on a class W of subgroups of G assigns an R-module C(H) to each H ∈ W, along with conjugation maps for each g ∈ G and restriction maps for each inclusion. If h ∈ H then it must act trivially by conjugation on C(H). Another description is as contravariant functors on the orbit category. These form the objects of a category CS_{W}(G).

The morphisms from C to D in this category are the collections of R-linear maps from C(H) to D(H) for each H ∈ W which commute with the restriction and conjugation maps. They are denoted by hom_{CS_{W} (G)}(C, D). This can easily be extended to take values in the objects of CS_{W}(G) and is then denoted by Hom_{CS_{W} (G)}.

Natural examples are the character ring of a group, or its cohomology.

For a more formal description, see the appendix.

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, so we can do homological algebra. In particular, Ext is balanced [22]. Notice that, just as Hom_{CS_{W}} takes values in CS_{W}(G), or even CS_{S}(G), so does Ext_{CS_{W}}^{∗}.

Let R̄ denote the constant coefficient system R̄(H) = R for all H ∈ W and all the morphisms the identity map. If L ∈ CS_{W}(G), then \underleftarrow{lim}_{W} L is an element of CS_{W}(G) (or even CS_{S}(G)) defined on H ≤ G by letting (\underleftarrow{lim}_{W} L)(H) be the inverse limit of L over W ∩S(H) with respect to the restriction and conjugation maps. The restriction and conjugation maps on \underleftarrow{lim}_{W} are the natural ones. It follows easily from the definitions that Hom_{CS_{W}} (R̄, L) = \underleftarrow{lim}_{W} L for all L ∈ CS_{W}(G) (see 12.4), so Ext_{CS_{W}}^{n} (R̄, L) ≅ \underleftarrow{lim}_{W}^{n} L for all n ≥ 0 by the definition of the higher limits as the right derived functors of \underleftarrow{lim}.

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.

We say that H ≤ G is taut with respect to W if the functor M ↦ (\underleftarrow{lim}_{W} M)(H) from CS_{W}(G) to R − Mod is exact. This is certainly true if H ∈ W.

Lemma 3.1. For H ≤ H 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 W is closed under intersections with H (or just the intersections are taut with respect to W,

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

both in CS_{S}(G).

Proof. This follows from 12.10 and the fact that Ind preserves projectives and Coind preserves injectives and both are exact under the conditions stated (see [22]. □

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. [19], [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).

Proposition 3.2. 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 3.1.

But R̄ is projective in CS_{X} (H) by 12.8, so the higher Ext vanish, and for n = 0 we have the result required. □

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

Proposition 3.3. 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 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 12.16.

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) (see appendix) 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 12.18. □

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 3.4. 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 3.3 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}, it is enough to check 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}. □

For any poset X we define X_{≥H} = {K ∈ X |K ≥ H} and similarly X_{>H}.

Lemma 3.5. 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} R̄ is the free R-module on the components of X_{≥H}. □

Proposition 3.6. If each X_{≥H}, H ∈ W is connected then R^{i}(\underleftarrow{lim}_{X} Res_{X}^{W})L(G) = R^{i} \underleftarrow{lim}_{W} L(G), for L ∈ CS_{W}(G).

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

Proof. It suffices to prove the case i = 0 because the general case is obtained by applying this to an injective resolution of L.

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}), by 12.2. Now use the previous lemma. The second formula is proved similarly. □

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 [18].

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

Proposition 3.7. 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 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 (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 3.8. Suppose that there is a bound on the length of any chain in W. If Wess_{0}(W) ⊆ X ⊆ W then R^{i}(\underleftarrow{lim}_{X} Res_{X}^{W})L(G) = R^{i} \underleftarrow{lim}_{W} L(G), for L ∈ CS_{W}(G).

Proof. By the 3.6 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 3.7. □

The next result is an immediate consequence of Alperin’s Fusion Theorem, as stated in, for example, [18].

Proposition 3.9. Suppose that W ⊆ S_{p} is closed under supergroups in Ess_{0}(S_{p}), and that Ess_{0}(S_{p}) ∩ W ⊆ X ⊆ W. Then R^{i} \underleftarrow{lim}_{X} Res_{X}^{W} L(G) ≅ \underleftarrow{lim}_{W}^{i} L(G) for L ∈ GCS_{W}(G).

Proof. The Fusion Theorem gives the case i = 0. The general case follows from this by taking an injective resolution of L. □

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.

Let D_{p} = B_{p} ∩ C_{p}.

It is well known that Wess_{0}(S_{p}) ⊆ B_{p} and Ess_{0}(S_{p}) ⊆ D_{p} (see [18]). So, in particular, we have:

Corollary 3.10. If L ∈ CS_{S_{p}}(G) then R^{i}(\underleftarrow{lim}_{B_{p}} Res_{B_{p}}^{S_{p}})L(G) = R^{i} \underleftarrow{lim}_{S_{p}} L(G).

If L ∈ GCS_{S_{p}}(G) then R^{i}(\underleftarrow{lim}_{D_{p}} Res_{D_{p}}^{S_{p}})L(G) = R^{i}(\underleftarrow{lim}_{B_{p}} Res_{B_{p}}^{S_{p}})L(G) = R^{i} \underleftarrow{lim}_{S_{p}} L(G).

Remark. Later we will see that if R is p-local then in many cases we can replace R^{i}(\underleftarrow{lim} Res) by \underleftarrow{lim}^{i} Res. In this form the first part of the corollary appears in [12]. The second part is suggested there, but first appears explicitly in [11].

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). (We will often write 𝔼xt_{W} or just 𝔼xt.) These are the hyper-derived functors of Hom_{CS_{W}}, which takes its values in CS_{W}(G), so are themselves coefficient systems. Notice that the group G is implicitly understood in this notation, even when we are evaluating on a proper subgroup.

There are two spectral sequences (see e.g. [22]):

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

Note that when we apply plain Ext to a chain complex, we apply it term by term to obtain another chain complex.

The following is immediate:

Proposition 4.1. If

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

then 𝔼xt_{W}^{n}(C_{•}, L) ≅ (\underleftarrow{lim}_{W}^{n} L).

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

𝔼xt_{W}^{n}(C_{•}, L)(K) ≅ H^{n} Hom_{W}(C_{•}, L)(K)

.

Proposition 4.2. Suppose that C_{n} = 0 for n ≤ 0 and we have an augmentation C_{•} → R̄. Let X ⊆ W and suppose that for each H ∈ X we have that Res_{H}^{G} C_{•} is a complex of projectives, exact in positive degrees and also that L is injective relative to X. Then

L → Hom_{W}(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 supposed to be relatively injective as a coefficient system.

Proof. When we restrict to H ∈ X, R̄ becomes projective, by 12.8, so Res_{H}^{G} C_{•} → R̄ is an exact complex of projectives, and it splits. Thus Hom_{CS_{W} (H)}(R̄, L) → 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 12.10). We see that

\underleftarrow{lim}_{W} Coind_{H}^{G} Res_{H}^{G} L → Hom_{CS_{W} (G)}(C_{•}, Coind_{H}^{G} Res_{H}^{G} L)

is split exact. But \underleftarrow{lim}_{W} has no effect, since we remain within CS_{W}(G), so our sequence is a summand of a product of split sequences, so is itself split exact by [21].

The splitting as a Mackey functor comes from 12.15. □

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), then we can form a chain complex of coefficient systems C_{•}[Δ^{?}] by setting

C_{n}[Δ^{?}] = ⨁_{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 semi-simplicial G-set (in the language of [22]) 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. Notice that if H ∈ W then Hom_{W}(R[G/H^{?}], L) ≅ Coind_{H}^{G} Res_{H}^{G} L by 12.11, 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 [21], which we sometimes refer to as L_{Δ_{•}}. In particular:

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

L(G) → hom_{CS_{W} (G)}(C_{•}(|X |^{?}), L)

is the normaliser sequence of the introduction.

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_{W}(G).

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

Example. If W = {1}, the trivial group, then H_{CS_{{1}}}^{∗}(Δ, L)(G) is just the usual Gequivariant 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}^{n} L)(S) = 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}^{n} L)(G).

Proof. We check that the conditions of 4.1 are satisfied. The first is clear, and the second is because Ext_{CS_{W} (G)}^{n}(R[G/S^{?}], L) ≅ Ext_{CS_{W} (S)}^{n}(R̄, L) ≅ (\underleftarrow{lim}_{W}^{n} L)(S) by 3.1. □

Recall that H ≤ G is taut with respect to W if the functor M ↦ (\underleftarrow{lim}_{W} M)(H) from CS_{W}(G) to R − Mod is exact. This is certainly true if H ∈ W.

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

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

Then the chain complex

L → Hom_{CS_{W}} (C_{•}[Δ^{?}], L)

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

Proof. We see that the conditions of 4.2 are satisfied. □

Remark. The statement for Mackey functors is similar to the main theorem of Webb, [21]. He has W = S, 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.

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

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

Usually Δ is taken to be the Brown complex, i.e. the geometric realisation of the poset of chains of non-trivial p-subgroups of G, S_{p}^{1}(G), or something similar. 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 [17]: 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_{•}, L) ≅ L_{Δ_{•}}. But by Robinson’s reformulation of Alperin’s Conjecture [14], ∑(−1)^{i} dim L_{Δ_{i}} 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 terms of Steinberg complexes, in the manner of [12]. The result is the original formulation of Alperin’s conjecture, by essentially the same proof as in [14].

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:

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

Proof. Since \underrightarrow{lim} commute with Ind (their right adjoints commute), and R[G/H^{!}] ≅ Ind_{H}^{G} R̄, it suffices to prove the case G = H. But \underrightarrow{lim}^{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

Res_{\{1\}}^{W∪\{1\}} \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. If F ≤ G, of index invertible in R, and D_{•} is a complex of RGmodules, 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 RGmodules.

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.

□

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 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 12.8 and the definition of V, so must split. Thus \underrightarrow{lim}_{V∪\{1\}}^{V} Res_{V} Res_{F}^{G} C̃_{•}(Δ^{?}) is also a split exact complex of projectives, since \underrightarrow{lim} preserves projectives by 12.2 and the remark before 12.14.

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

\underrightarrow{lim}_{V∪\{1\}}^{V} 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 is projective and based at 1.

But in the homotopy category, a split exact complex is equal to 0, so we have that Res_{V∪{1}} Res_{F}^{G} C̃_{•}(Δ?) is homotopy equivalent to a complex of projectives, based at 1. Restricting to the subgroup 1 (so we are just dealing with RG-modules), we 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 with enough projectives, 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_{•} ⊕ SP_{•} ≅ P_{•} ⊕ E_{•},

where SP_{•} is an exact complex of projectives and E_{•} is an exact complex. If f is a homotopy equivalence, then E_{•} is split exact.

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. E_{•} is exact, bounded below and is split if f is a homotopy equivalence.

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 chain map 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}

An isomorphism Θ : C_{•} → E_{•} ⊕ P_{•} is now given by setting Θ(e, p) = (e + θp, p), e ∈ E, p ∈ P. □

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) ⊕ SP_{•} ≅ P_{•} ⊕ E_{•},

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

Proof. Apply 6.5 to the second statement of 6.4. □

Finally, we obtain Webb’s original result.

Corollary 6.7. In the circumstances of 6.4, 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. We can now apply the Krull-Schmidt Theorem to 6.6 to cancel SP_{•}. □

The standard example when the hypotheses of 6.4 are satisfied is 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 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 it 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. If G acts admissibly on a CW-complex Δ in such a way that Stab_{G}(Δ) ⊆ N_{p} and X is a class of subgroups containing S_{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, projective relative to 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):

a) Hom_{R?}(St_{•}(G), M(1)),

b) Hom_{W}(St_{•}^{0}(G), M),

c) Hom_{R?}(C̃_{•}(|S_{p}|), M(1)),

d) Hom_{W}(C̃_{•}(|S_{p}|^{?}), M(1)^{?}),

e) If W ⊆ N_{p}, L Hom_{W}(C̃_{•}(|S_{p}|^{?}), M). (Where L Hom is the complex defining 𝔼xt.)

Proof. Now (a) and (b) are homotopic because St_{•}^{0}(G) = \underrightarrow{lim}_{1}^{W} St_{•}(G), and \underrightarrow{lim} is the right adjoint of restriction. Also (a) and (c) are homotopic by the first part of 6.4. But (c) and (d) 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}|^{?}) by the last part of 6.4, so (e) is homotopic to (b) by definition. □

Lemma 7.2. If H ◁ G and p divides |H| then for any p-subgroup P of G: i) p divides |C_{G}(P) ∩ H|, ii) 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 (ii), let P be the largest subgroup in the chain and apply (i). □

Now we can state a fundamental result from [12], (although our proof is based on [11]).

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_{RG} → hom_{R}, (with the usual transfer on Hom). Using the complex (c) of 7.1, each term is a sum of pieces of the form hom_{RG}(R[G/S], M) ≅ hom_{RS}(R, M) ≅ M^{S}, (by tr_{S}^{G}), where S is the stabiliser of a chain σ.

By 7.2 with H as the kernel of M, we see that there S ∩H contains a p-subgroup Q, say, so Q is in both S and H. But tr_{1}^{S} factors through tr_{1}^{Q}, and this is 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 comples must split. □

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

Corollary 7.4. For simplicity we assume that P is a p-group and all coefficient systems are over S_{p}.

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

ii) hom_{CS_{Sp} (E)}(Ind_{N_{E}(P)}^{E} Inf_{N_{E}(P)/P}^{N_{E}(P)} St_{•}^{0}(N_{E}(P)/P), M) ≃ 0 for any M ∈ CS_{S_{p}}(E) 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_{E}(P)/P -module.

iii) G_{0} Ind_{N_{E}(P)}^{E} Inf_{N_{E}(P)/P}^{N_{E}(P)} St_{•}^{0}(N_{E}(P)/P) ≃ 0 if P is not a centric subgroup of G. (Where G_{0} is the left adjoint of the forgetful functor I : GCS → CS.)

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

For (ii), the adjoint properties show that we are just calculating hom_{CS_{Sp} (N_{E}(P)/P)}(St_{•}^{0}(N_{E}(P)/P), Q_{N_{E}(P)/P}^{N_{E}(P)} Res_{N_{E}(P)}^{E} M), and this is 0 if P is not centric in E by 7.1 and 7.3.

Now (iii) is a formal consequence of (ii). □

The next result is a version of one in [11].

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).

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, by downward 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 the result claimed. □

8. The Subgroup Sequence

There are other ways of obtaining a chain complex in CS_{W}(G) from a class of subgroups, which do not factor through the geometric realisation. We will always use a functor on the semisimplicial G-set of chains and inclusions. Previously our examples have been based on the Brown complex, or a variation of it, so the functor has always been F^{N} : (σ) ↦ R[G/N_{G}(σ)^{?}] ≅ Ind_{N_{G}(σ)}^{G} R̄, where we write (σ) for the orbit of σ.

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}.) For any chain σ in W, let σ_{b} be the smallest element and let σ_{t} be the largest. The functor which represents the subgroup sequence is F^{S}, which can be defined

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

Another description is F^{S}((σ)) = 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)}
□

Let C_{•}^{S}(X) be the complex obtained from X by using F^{S}

Corollary 8.2. If L ∈ CS_{W}(G) and X ⊆ W, then L(G) → hom_{CS_{W} (G)}(C_{•}^{S}(X), L) is the second sequence of the introduction, the subgroup sequence.

Lemma 8.3. If every X_{≥H}, H ∈ W, is R-acyclic (e.g. W ⊆ X) then 𝔼xt_{W}^{n}(C_{•}^{S}(X), L) ≅ (\underleftarrow{lim}_{X}^{W})^{n}L.

Proof. This is a consequence of 4.1, since C_{•}^{S}(X)(H) ≅ C_{•}(|X_{≥H}|). □

Lemma 8.4. In any Abelian category, let C_{•} be a complex and suppose that it has a finite filtration such that each of the composition factors is homotopy equivalent to a complex of projectives P_{•}^{i}. 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 triangle P_{•}^{1} → C_{•} → P_{•}^{2} ⇝ P_{•−1}^{1} in the homotopy category. It follows that C• is homotopy equivalent to the mapping cone of the map P_{•}^{2} ⇝ P_{•−1}^{1}, shifted in degree by one. This has the form required. □

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

If for every H ∈ X not Sylow, 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 less 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).

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 12.5.

Now we can apply 8.4. □

Corollary 8.6. Suppose that every prime dividing |G| except perhaps p is invertible in R, that X ⊆ Y ⊆ W ⊆ S_{p}(G) and that every maximal element of X is also maximal in W. Suppose also that, for every H ∈ W, X_{≥H} is R-acyclic and that for every non-maximal H ∈ X and K ∈ S_{p} with H ◁ K we have that (X_{>H})^{K} is R-acyclic.

Then, for L ∈ CS_{Y}(G) we have H^{n} Hom_{Y}(C_{•}^{S}(X), L) ≅ \underleftarrow{lim}_{Y}^{n} L.

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

Proof. By 8.5 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 Y by 12.8. Now we can apply 4.1 to see that H^{n} Hom_{Y}(C_{•}^{S}(X), L) ≅ 𝔼xt_{Y}^{n}(C_{•}^{S}(X), L). Now use 8.2 and 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 8.7. Suppose that W ⊆ S_{p} is closed under supergroups in Wess(S_{p}) and that Wess(S_{p}) ∩ W ⊆ X ⊆ Y ⊆ W.

Then, for any L ∈ CS_{Y}(G), we have that H^{n} Hom_{Y}(C_{•}^{S}(X), L) ≅ \underleftarrow{lim}_{Y}^{n} L.

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

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-contractibility instead of contractibility.

Proof. We need to check the conditions of 8.6.

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 this is clearly in Wess(S_{p})∩W, so by hypothesis is in X.

Thus we can apply 3.7 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. □

9. Change of Class of Groups

The results are based on those in [11].

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 8.7, \underleftarrow{lim}_{Y} L(G) ≅ \underleftarrow{lim}_{B_{p}∩W} Res_{B_{p}∩W}^{Y} L(G).

Proof. Use 8.7 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}_{B_{p}}^{n} Res_{B_{p}}^{S_{p}} L(G) = \underleftarrow{lim}_{S_{p}}^{n} L(G) = R^{n}(\underleftarrow{lim}_{B_{p}} Res_{B_{p}}^{S_{p}})L(G) for L ∈ CS_{S_{p}}(G) for L ∈ CS_{S_{p}}(G).

Proof. The first two are both equal by 9.1. They are equal to the third by 3.10. □

We extract for future use the main feature of the proof of 8.7.

Corollary 9.3. Suppose that W ⊂ S_{p} is closed under supergroups in B_{p}. Then there is a finite projective resolution of R̄ in CS_{W}(G) in which all the terms are based at B_{p} ∩ W.

Proof. We claim that C_{•}^{S}(B_{p} ∩ W) is such a resolution. it is complex of projectives by the previous proof and has just R̄ in degree 0 as its homology, by the proof of 8.3. □

Proposition 9.4. Suppose that W ⊆ S_{p} is closed under supergroups in B_{p}. Then there is a finite projective resolution of R̄ in GCS_{W}(G) in which all the terms are summmands of sums of terms of the form G_{0}R[G/H^{?}], for H ∈ D_{p} ∩ W.

Proof. 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.

We claim that G_{0} Res_{C_{p}∩W}^{S_{p}∩W} C_{•}^{S}(B_{p} ∩ W) is homotopy equivalent to the required resolution. It has the right homology, by the exactness property above. It is also equal to Res_{C_{p}∩W}^{S_{p}∩W} G_{0}C_{•}^{S}(B_{p} ∩ W) . Now C_{•}^{S}(B_{p} ∩ W) has a filtration with

composition factors of the form Ind_{N_{G}(P)}^{G} Inf^{N_{G}(P)} N_{G}(P)/P St_{•−1}^{0}(N_{G}(P)/P), for P ∈ B_{p} ∩ W . Thus G_{0}C_{•}^{S}(B_{p} ∩ W) has a filtration with composition factors of the form G_{0} Ind_{N_{G}(P)}^{G} Inf^{N_{G}(P)} N_{G}(P)/P St_{•−1}^{0}(N_{G}(P)/P) (because the terms are all projective). These are homotopic to 0 if P ∉ C_{p} by 7.4. Otherwise they are of the form claimed. □

Corollary 9.5. Suppose that W ⊆ S_{p} is closed under supergroups in B_{p} and that D_{p} ∩ W ⊆ X ⊆ Y ⊆ W.

Then, for any L ∈ GCS_{Y}(G), we have that H^{n} Hom_{Y}(C_{•}^{S}(X), L) ≅ \underleftarrow{lim}_{Y}^{n} IL.

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

Corollary 9.6. In the circumstances of 9.5, \underleftarrow{lim}_{Y} IL(G) ≅ \underleftarrow{lim}_{D_{p}∩W} Res_{D_{p}∩W}^{Y} IL(G).

Corollary 9.7. \underleftarrow{lim}_{D_{p}}^{n} Res_{D_{p}}^{S_{p}} IL(G) = \underleftarrow{lim}_{S_{p}}^{n} IL(G) for L ∈ GCS_{S_{p}}(G).

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

Proposition 9.8. Suppose that W ⊆ S_{p} is closed under supergroups in B_{p} and that D_{p} ∩ W ⊆ X ⊆ Y ⊆ W.

Then \underleftarrow{lim}_{X}^{n} L(G) = \underleftarrow{lim}_{X}^{n} IL(G) for L ∈ GCS_{X} (G).

Proof. By 3.9 and 9.6 we can reduce to the case X = D_{p} ∩ W. But since G_{0} is now exact, I must preserve injectives, and the result follows. □

Corollary 9.9. Suppose that every prime dividing |G| except perhaps for p is invertible. Then \underleftarrow{lim}_{S_{p}}^{n} L(G) = \underleftarrow{lim}_{S_{p}}^{n} IL(G) for L ∈ GCS_{S_{p}}(G).

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 taking F^{C}((σ))(P) to be the free R-module on the chains τ in the orbit of σ with C_{G}(τ_{t}) ≥ P. This is equivalent to 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 obtained from X using F^{S}.

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 𝔼xt_{W}^{n}(C_{•}^{C}(X), L) ≅ (\underleftarrow{lim}_{X}^{W})^{n}L.

Proof. This is a consequence of 4.1, 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 complex of projectives in CS_{W}(G).

Proof.

\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. 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}^{n} L)(G).

11. Projective Resolutions and the Steinberg Complex

Notice that 8.5 tells us how to find 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].

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).

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

For any RG-module V let V^{0} ∈ 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:

a) (\underleftarrow{lim}_{S_{p}}^{i} V^{0})(G) = 0, for all i ≥ 0 and all kG-modules V;

b) (\underleftarrow{lim}_{S_{p}}^{i}(kG)^{0})(G) = 0, for all i ≥ 0;

c) St_{•}(G) ≃ 0;

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

and all L ∈ CS_{S_{p}}(G); e) k ≅ (\underleftarrow{lim}_{S_{p}^{1}}(kG)^{?})(G).

Proof. It is clear that (a) ⇒ (b).

Now hom_{S_{p}}(−, V^{0}) vanishes on any projective based at a non-trivial p-subgroup. Thus 8.5 shows that if P_{•} → R̄ is a projective resolution then hom_{S_{p}}(P_{•}, V^{0}) is homotopy equivalent to hom_{S_{p}}(St_{•+1}^{0}(G), V^{0}), which, in turn, is isomorphic to Hom_{kG}(St_{•+1}(G), V). So (\underleftarrow{lim}_{S_{p}}^{i} V^{0})(G) is equal to the homology of Hom_{kG}(St_{•+1}(G), V).

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

For any L ∈ CS_{S_{p}}(G), let L_{0} denote the cokernel of the inclusion L(1)^{0} → L. We claim that (\underleftarrow{lim}_{S_{p}} L_{0})(G) ≅ (\underleftarrow{lim}_{S_{p}^{1}}^{i} Res_{S_{p}^{1}}^{S_{p}} L)(G). This is because L_{0} ≅

\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_{S_{p}}^{i}(R̄, L_{0})(G) ≅ Ext_{S_{p}^{1}}^{i}(R̄, Res_{S_{p}^{1}}^{S_{p}} L)(G).

The long exact Ext sequence for L(1)^{0} → L → L_{0} now shows that (a) ⇔ (d). Similarly (b) ⇔ (e), because the higher limits vanish since kG^{?} is injective. □

Remark. Quillen conjectured that |S_{p}^{1}(G)| is contractible if and only if G contains a non-trivial normal p-subgroup [16]. In fact, according to [11], no counterexample seems to be known if the contractibility condition is replaced by 𝔽_{p}-acyclicity.

In view of 11.1, the (stronger) conjecture is equivalent to the statement: k ≅ (\underleftarrow{lim}_{S_{p}^{1}}(kG)^{?})(G) if and only if G contains a non-trivial normal p-subgroup.

12. Appendix

Here we collect together some properties of coefficient systems and Mackey functors. The main ones 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 they always 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.

By a class of subgroups of a given group G, we mean a set of subgroups, closed under conjugation in G. Two obvious examples are the class of all subgroups S(G) (or just S), and the class of all p-subgroups S_{p}(G) (or just S_{p}, for some given prime p.

For a given class W 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(H, K).

The composition law is b_{h,K,L}b_{g,H,K} = b_{gh,H,L}, and we are identifying h ∈ H with b_{h,H,H}.

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} 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)}. Of course this is equivalent to defining CS_{W}(G) to be the category of covariant functors from G-sets with stabilisers in W to R − Mod.

We write c_{g,H,H^{g}}, or just c_{g}, for the map of R-modules 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, 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 (largest sub coefficient system) that is geometric.

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

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.

We can extend hom_{CS_{W} (G)} to a pairing Hom_{CS_{W}} : CS_{W}(G) × CS_{W}(G) → CS_{V}(G) for any class V, given by Hom_{CS_{W}} (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 V = W in most cases.

We will often abbreviate Hom_{CS_{W}} to Hom_{W}.

There is the usual adjunction:

Lemma 12.1. 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}. 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. Similarly \underrightarrow{lim}_{X}^{W} L(H) is the direct limit of the L(J) for all J ∈ X , J ≥ H where the limit is taken over inclusions only.

Proposition 12.2. \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}. □

From the definition of Hom we find that this extends to:

Lemma 12.3. If W ⊆ V and L, M ∈ CS_{W}(G) then Hom_{V}(L, M) ≅ \underleftarrow{lim}_{W}^{V} Hom_{W}(L, M).

Also, for N ∈ CS_{V}(G) we have Hom_{V}(N, \underleftarrow{lim}_{W}^{V} M) ≅ Hom_{W}(Res_{W}^{V} N, M) and Hom_{V}(\underrightarrow{lim}_{W}^{V} M, N) ≅ Hom_{W}(M, Res_{W}^{V} N) 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 12.4. Hom_{CS_{W}} (R̄, L) ≅ \underleftarrow{lim}_{W} L on any subgroup of G.

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

By a component of W we mean an equivalence class under the equivalence relation generated by inclusion.

Lemma 12.5. 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 invertible in R.

Proof. It follows from 12.4 that R̄ is projective if and only if \underleftarrow{lim}_{W} is exact.

The conditions imply that \underleftarrow{lim}_{W} L is the sum over the conjugacy classes of maximal elements M in W of L(M)^{N_{G}(M)}.

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 maximal elements of W, M and N say, such that M ∩ N contains an element of W. Let R̃ ∈ CS_{W}(G) take the value R on the conjugacy classes of M and N, and 0 elsewhere. There is an epimorphism \underrightarrow{lim}_{\{(M),(N)\}}^{W} Res_{\{(M),(N)\}} R → R̃. Now if

M and N are not conjugate, then \underleftarrow{lim}_{W} R̃ can have arbitrary values for its M and N coordinates, but in \underleftarrow{lim}_{W} \underrightarrow{lim}_{\{(M),(N)\}}^{W} Res_{\{(M),(N)\}} R these coordinates must be

equal. If M and N are conjugate, then \underleftarrow{lim}_{W} R̃ can take an arbitrary common values for its M and N coordinates, but in \underleftarrow{lim}_{W} \underrightarrow{lim}_{\{(M),(N)\}}^{W} Res_{\{(M),(N)\}} R these coordinates must be zero. □

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 inclusions, but the conjugation maps are less obvious. 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) for l ∈ L(J^{g}) ⊆ L̂_{G,H}(J). Now define conjugation by f ∈ G by c_{f}(g, l) = (fg, l). There is also an action of H on the right by (g, l)h = (gh, c_{h}^{−1}l). Then

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

is 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, 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 [19].

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 12.6. 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).

Notice that it follows from the definitions that:

Lemma 12.7. R[G/H^{?}] ≅ Ind_{H}^{G} R̄ in CS_{W}(G) for any W.

Corollary 12.8. R[G/H^{?}] is projective in CS_{W}(G) if H ∈ W.

Proof. Use 12.5, 12.6 and the fact that left adjoints of exact functors preserve projectives. □

Remark. Ind is always exact, but Coind only has to be exact in the classical case, when W is closed under intersections with H.

Lemma 12.9. 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_{G}(Ind_{H}^{G}(L ⊗ Res_{H}^{G} M), K) ≅ hom_{H}(L ⊗ Res_{H}^{G} M, Res_{H}^{G} K)
≅ hom_{H}(L, Hom(Res_{H}^{G} M, Res_{H}^{G} K))
≅ hom_{G}(Ind_{H}^{G} L, Hom(M, K))
≅ hom_{G}(Ind_{H}^{G} L ⊗ M, K).

The result now follows formally. □

The adjunction can be generalised:

Proposition 12.10. 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. The proof of the second is similar, but needs 12.9. □

Corollary 12.11. 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, 12.1 and 12.6. □

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 12.12. 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. 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_{W} 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,^{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_{J∩K^{g}}^{J} ^{g}x)^{K},

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

It is easy to check that SC and T C are Mackey functors on V.

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 12.13. 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 (cf. [19]):

Proposition 12.14. 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:

a) \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.

b) \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 12.15. 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 12.13.

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 12.16. 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 12.17. 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 [19] 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 12.11 is straightforward to check.

Lemma 12.18. 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 12.19. 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 [19].

Proof. Notice that S commutes with induction (in CS or MF, depending upon the side) because their right adjoints commute. Also it is clear from the definitions that ℤ[G/H^{?}] ≅ Ind_{H}^{G} ℤ, where ℤ denotes the constant coefficient system. Now we claim that Sℤ ≅ B^{G}, which again follows from the definitions. For the rest, see [19]. □

This is the starting point for the definition of cohomology with coefficients in a Mackey functor in [15]. If M is a Mackey functor over ℤ on S(G) then

M_{X}^{•} = Hom_{CS}(C_{•}(X^{!}), M) ≅ Hom_{MF}(SC_{•}(X^{!}), M)

is naturally a complex of Mackey functors.

Now

C_{n}(X^{!}) ≅ ⨁_{σ∈X_{n}/G} ℤ[G/ Stab_{G} σ^{!}].

Hence

M_{X_{n}} ≅ Hom_{MF}( ⨁_{σ∈X_{n}/G} B^{G}(!, Stab_{G} σ), M).

The right hand side is the chain complex in [15], so we see that the theory there is just Bredon cohomology. The transfers are the usual ones for the induced Mackey functors M_{X_{n}}. For a slightly different approach, see [9].

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

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