PERMUTATION COMPLEXES FOR PROFINITE GROUPS

PETER SYMONDS

Abstract. An important tool in the analysis of discrete groups of finite virtual cohomological dimension is the existence of a contractible CW-complex on which the group acts with finite stabilizers. We develop an analogue for profinite groups.

1. Introduction

The main aim of this paper is to develop the theory of profinite groups along the lines of that done for discrete groups, as in the book of Brown [6] for example. Some of this was addressed by the author and Weigel in [26], but here we are particularly interested in constructing an algebraic analogue of a contractible space on which the group acts with finite stabilizers and in developing its consequences. The torsion free case was considered by Serre [21].

We start with a review of the basic module theory behind the cohomology of profinite groups. This originated in the work of Brumer [7], and is well explained in the books of Ribes and Zaleskii [18] and of Wilson [28], but we collect together the main properties that we will use and describe them from our point of view. This part might be of independent interest. We devote relatively little space to basic homological algebra since this is a routine adaptation of the methods used in the discrete case once the module theory is understood.

We pay particular attention to permutation modules and the Brauer construction because this allows us to utilize some results of Swan on the cohomology of fixed point sets. This enables us to formulate precisely the complexes we want and their properties. We then define the Tate-Farrell cohomology for a profinite group of finite virtual cohomological dimension and use the complexes to develop its properties.

There are applications to questions about the finiteness of the number of conjugacy classes of finite subgroups and to the local cohomology of the cohomology of the group.

2. Background

2.1. Ambient ring. Let R be a commutative local ring with maximal ideal 𝔪 and residue class field k = R/𝔪. We assume that dim_{k} 𝔪/𝔪^{2} < ∞ and that R is complete, i.e. R ≅ \underleftarrow{lim} R/𝔪^{r}. We give the quotients R/𝔪^{r} the discrete topology and then let R have the inverse limit topology.

The obvious example is when R is the p-adic integers, ℤ̂_{p}, but we also allow such things as R = k or power series rings k[[x_{1}, …, x_{n}]] and we even allow p = 0. The reader who is only interested in profinite modules may assume that k is finite throughout, but there is little simplification in the proofs and it is sometimes natural to want k to be algebraically closed, for example.

2.2. Modules. We consider principally three categories of R-modules: the modules of finite length (i.e. artinian), F_{R}, the pseudocompact modules, C_{R}, which are inverse limits of modules in F_{R}, and the discrete torsion modules, D_{R}, which are direct limits of modules in F_{R}. Both are given their limit topologies: (if k is finite then the objects of C_{R} are compact and profinite; in any case the objects of D_{R} are discrete). All morphisms are supposed to be R-linear and continuous and the set of all morphisms is denoted by Hom_{R}. Both C_{R} and D_{R} are abelian categories, and C_{R} has exact inverse limits and D_{R} has exact direct limits. (All limits will be over directed systems, although we do not assume that all the maps are epimorphisms.) We will refer to modules in C_{R} and D_{R} as compact (rather than pseudo-compact) and discrete respectively.

Let T′ be the injective hull of k as a (non-topological) R-module and let T be its torsion submodule, i.e. the maximal submodule in D_{R}. For example, if R = k then T = k, if R = ℤ̂_{p} then T = ℤ/p^{∞} = p − torsion in ℚ/ℤ and if k is finite then T = Hom(R, ℚ/ℤ).

The categories C_{R} and D_{R} are dual by Hom_{R}(−, T); this is an instance of Pontryagin duality. We will denote Hom(A, T) by A^{∗}. Thus results for one category can be translated into results for the other. ([7] 2.3, [18], [12].)

It is sometimes also convenient to consider the category E_{R} of modules which are inverse limits of modules in D_{R}. This contains both C_{R} and D_{R} as full subcategories and in particular if R = ℤ̂_{p} then E_{R} contains ℚ̂_{p}, since it is the inverse limit of the system

⋯ →^{p} ℤ/p^{∞} →^{p} ℤ/p^{∞}.

We can, of course, consider continuous morphisms from an object of one category of R-modules to an object of another. If A ∈ C_{R} and B ∈ E_{R} then Hom_{R}(A, B) is naturally an object of E_{R}. It is in D_{R} if B ∈ D_{R} or if B ∈ C_{R} and B is finitely (topologically) generated.

In particular it makes sense to claim that Hom_{R}(A, B) ≅ Hom_{R}(B^{∗}, A^{∗}), A ∈ C_{R}, B ∈ D_{R}, and this is proved by taking the limit of the finite case.

2.3. Operations. On D_{R} we have the usual operations ⊕ and ⊗. The dual of ⊕ on C_{R} is just the product Π.

For A, A_{i}, C_{i} ∈ C_{R} and B, B_{i} ∈ D_{R} we have ([18] 5.1.4):

Hom_{R}(\underleftarrow{lim} A_{i}, B) ≅ \underrightarrow{lim} Hom_{R}(A_{i}, B),
Hom_{R}(A, \underrightarrow{lim} B_{i}) ≅ \underrightarrow{lim} Hom_{R}(A, B_{i}),
Hom_{R}(A, \underleftarrow{lim} C_{i}) ≅ \underleftarrow{lim} Hom_{R}(A, C_{i}),

and so

Hom_{R}(ΠA_{i}, B) ≅ ⊕ Hom_{R}(A_{i}, B),
Hom_{R}(A, ⊕B_{i}) ≅ ⊕ Hom_{R}(A, B_{i}),
Hom_{R}(A, ΠC_{i}) ≅ Π Hom_{R}(A, C_{i}).

A sort of tensor product ˆ⊗_{R} can be defined on C_{R} as follows. If A = \underleftarrow{lim} A_{i} and B = \underleftarrow{lim} B_{j} with A_{i}, B_{j} ∈ F_{R} then A ˆ⊗_{R}B = \underleftarrow{lim}(A_{i} ⊗_{R} B_{j}). ([16, 7])

The product ˆ⊗_{R} is symmetric, associative and commutes with inverse limits, in particular products. It is the ordinary tensor product if one of A or B is finitely generated. It is not the dual of ⊗, however, because ℤ/p^{∞} ⊗ℤ̂_{p} ℤ/p^{∞} = 0.

There is a natural identity

A ˆ⊗_{R}B ≅ Hom_{R}(A, B^{∗})^{∗},

for A, B ∈ C_{R}, by a ⊗ b ↦ [f ↦ (f(a))(b)]. This is well known in the artinian case and, since both sides commute with inverse limits, it is true in general. We could have used this identity as the definition of ˆ⊗_{R}, but it seems unnatural. It also motivates the next definition.

We define a product ⋆_{R} on D_{R} by A ⋆_{R} B = Hom_{R}(A^{∗}, B). This is symmetric by the properties of duality. It is also associative and commutes with direct limits. If R is a discrete valuation ring then A ⋆_{R} B ≅ Tor_{R}(A, B).

This allows us to define another product ˆ⋆_{R} on C_{R} by Aˆ⋆_{R}B = \underleftarrow{lim}(A_{i} ⋆_{R} B_{i}) for A = \underleftarrow{lim} A_{i} and B = \underleftarrow{lim} B_{j} with A_{i}, B_{j} ∈ F_{R}. We have:

(A ⋆_{R} B)^{∗} ≅ A^{∗} ˆ⊗_{R}B^{∗},
(A ⊗_{R} B)^{∗} ≅ A^{∗}ˆ⋆_{R}B^{∗}),

for A, B ∈ D_{R}.

C_{R} satisfies a form of Nakayama’s Lemma, that if U ≤ V is closed and U + rad V = V then U = V. The dual statement also holds. ([7] 1.4)

For any D ∈ D_{R}, soc D = ⊕_{I}k for some indexing set I and hence the injective hull of D is ⊕_{I}T. By duality, the projective cover of C ∈ C_{R} is the same as the projective cover of C/ rad C ≅ Π_{I}k, which is Π_{I}R.

As a consequence every projective in C_{R} is of the form Π_{I}R and there are enough projectives, and dually for D_{R}.

However Π_{I} ℤ̂_{p} is not projective in the category of ℤ̂_{p}-modules; see [14].

2.4. Modules for groups. If G is a profinite group then we can consider the categories of objects in one of the categories above endowed with a continuous action of G, and we denote these by C_{R}(G), D_{R}(G) and E_{R}(G). The morphisms will be denoted by Hom_{G}. All the constructions and identities above are natural, hence equivariant, and so yield versions for these new categories.

Every object in C_{R}(G) is, in fact, an inverse limit of objects in F_{R}(G), and similarly for D_{R}(G) and E_{R}(G) ([18] 5.3.3, [16] II 2.2.6). They both satisfy a weak version of the Krull-Schmidt property, that if M ∈ C_{R}(G) is a product of indecomposables (or N ∈ D_{R}(G) is a sum of indecomposables) with local endomorphism rings then this decomposition is essentially unique ([12]).

Suppose that G = \underleftarrow{lim} G/N_{i} for some system N_{i} of open subgroups. Then we define the complete group algebra R[[G]] ∈ C_{R} by R[[G]] = \underleftarrow{lim} R[G/N_{i}]. An object of E_{R} has a continuous action of G if and only if it has a continuous action of R[[G]]. This implies that, as a left module over itself, R[[G]] is projective in C_{R}(G). The dual of R[[G]] is F (G), the functions on G (see later).

We can define A ˆ⊗_{R[[G]]}B for A, B ∈ C_{R}(G) either as \underleftarrow{lim} A_{i} ⊗_{RG} B_{i}, where A = \underleftarrow{lim} A_{i} and B = \underleftarrow{lim} B_{j} with A_{i}, B_{j} ∈ F_{R}(G), or simply as the coinvariants of A ˆ⊗_{R}B. Similarly with ⋆.

Induction on C_{R}(G) and coinduction on D_{R}(G) are defined by Ind_{H}^{G} M = R[[G]] ˆ⊗_{R[[H]]}M and Coind_{H}^{G} N = F (G)⋆^{H} N. (The latter is called induced by Serre [22].) They are exact, dual to each other, and satisfy the customary adjointness properties

Hom_{G}(Ind_{H}^{G} M, N) ≅ Hom_{H}(M, N) ≅ Hom_{G}(M, Coind_{H}^{G} N).

As a consequence, restriction to a subgroup preserves projectives in C_{R}(G) and injectives in D_{R}(G). Also Ind preserves projectives and Coind preserves injectives. This makes it easy to show that C_{R}(G) has enough projectives and D_{R}(G) has enough injectives.

There is also the formula Ind_{H}^{G} Res_{H}^{G} M ≅ R[[G/H]] ⊗_{R} M.

2.5. Projectives. The following are equivalent for an object C of C_{R}(G) ([7] 2.2, 3.6):

• (i) C is projective in C_{R}(G),

• (ii) Hom_{G}(C, −) is exact on D_{R}(G),

• (iii) Hom_{G}(C, −) is exact on F_{R}(G).

The dual formulation is also true.

However Hom_{G}(C, −) is not necessarily exact on E_{R}(G) even when C is projective in C_{R}(G) . First note that, since C is a summand of a module induced from the trivial subgroup, it is enough to consider the case without group action. If A →^{i} B →^{j} C is a short exact sequence in E_{R} and we want it to remain exact after applying Hom_{R}(C, −) then we need i to induce a homeomorphism between A and its image under the subspace topology. This is sufficient if C is finitely generated. Otherwise we also need every compact submodule of C to be the image of a compact submodule in B, which is guaranteed under several general conditions, e.g. A compact, the sequence split over R, or R a discrete valuation ring and B a finite dimensional vector apace over the quotient field.

A useful result is that if A ∈ C_{R}(G) and A ≅ \underleftarrow{lim} A_{i} for A_{i} ∈ C_{R}(G) such that for each A_{i} there is an open normal subgroup N_{i} < G which acts trivially on A_{i} and A_{i} is projective over G/N_{i}, then A is projective. ([7] 3.3, but surjections are not necessary everywhere, compare A3 there with 2.3.)

Let S be a simple module in C_{R}(G), so necessarily a finite dimensional vector space over k. For each open normal subgroup N < G that acts trivially on S let A_{N} be the projective cover of S as an R(G/N)-module. For each M < N there is a unique map A_{M} → A_{N}. Now let P_{S} = \underleftarrow{lim} A_{N}. Then P_{S} is the projective cover of S and End_{G}(P_{S}) is local.

If M is any module in C_{R}(G) then M/ rad_{G}(M) = Π_{I}S_{i}, where the S_{i} are a collection of simple modules. The projective cover of M is then Π_{I}P_{S_{i}}. Thus C_{R}(G) not only has projective covers, but also every projective is of this form and the expression is unique by 2.4.

2.6. Cohomology. We define Ext_{G}^{∗}(A, B) on C_{R}(G) × D_{R}(G) in the usual way, using either a projective resolution of A or an injective resolution of B. It naturally takes values in D_{R}. This Ext_{G}^{∗} is continuous in the sense that ([7] 3.4, [18] 6.5.3)

Ext_{G}^{∗}(\underleftarrow{lim} A_{i}, B) ≅ \underrightarrow{lim} Ext_{G}^{∗}(A_{i}, B)
Ext_{G}^{∗}(A, \underrightarrow{lim} B_{i}) ≅ \underrightarrow{lim} Ext_{G}^{∗}(A, B_{i}).

Ext_{G}^{∗}(A, B) can even be defined on C_{R}(G) × E_{R}(G) by using a projective resolution of A. In this case it just takes values in abelian groups, although it will take values in C_{R} if A is of type FP_{∞} in C_{R}(G) and B ∈ C_{R}(G). If these conditions hold then we also have

Ext_{G}^{∗}(A, \underleftarrow{lim} C_{i}) ≅ \underleftarrow{lim} Ext_{G}^{∗}(A, C_{i}).

But we only obtain long exact sequences in the second variable if the conditions of 2.5 are satisfied, e.g. if A is of type FP_{∞} or the terms in the second variable are compact. We need to know that projectives are preserved under restriction (2.4) in order to define restriction maps.

There is also a derived functor Tor_{∗}^{G} defined on C_{R}(G) × C_{R}(G).

The cohomology of the group is H^{∗}(G, M) = Ext_{G}^{∗}(R, M). There are the usual standard and bar resolutions which can be used to show that if M ∈ D_{R}(G) then H^{∗}(G, M) ≅ \underrightarrow{lim} H^{∗}(G/N, M^{N}) as N runs through the open normal subgroups of G and the maps are restrictions.

Cup products can be defined in the usual way by taking the tensor product of the projective resolutions.

The usual Eckmann-Shapiro isomorphisms hold and there is a Lyndon-Hochschild-Serre spectral sequence H^{p}(G/H, H^{q}(H, M)) ⇒ H^{p+q}(G, M) for any (closed) subgroup H and M ∈ E_{R}(G).

Finally there is another continuity property for any K < G and A ∈ C_{R}(G), B ∈ D_{R}(G) that Ext_{K}^{n}(A, B) ≅ \underrightarrow{lim} Ext_{NK}^{n}(A, B), as N runs through the open normal subgroups of G.

This is because Ext_{K}^{n}(A, B) ≅ Ext_{G}^{n}(Ind_{K}^{G} A, B) ≅ Ext_{G}^{n}(R[[G/K]] ˆ⊗_{R}A, B) ≅ Ext_{G}^{n}(\underleftarrow{lim} R[[G/NK]] ˆ⊗_{R}A, B) ≅ \underrightarrow{lim} Ext_{G}^{n}(R[[G/NK]] ˆ⊗_{R}A, B) ≅ \underrightarrow{lim} Ext_{NK}^{n}(A, B).

3. Permutation Modules

Let X be a profinite space and M a discrete torsion R-module. Define F (X, M) to be the R-module of continuous functions X → M. We write F (X) for F (X, T).

If G is a profinite group that acts continuously on X and M then G acts continuously on the discrete torsion module F (X, M) by (gf)(x) = f(g^{−1}x), g ∈ G, f ∈ F (X, M), x ∈ X.

Definition 3.1. A discrete permutation G-module is an F (X) as above. An R-permutation module is a summand of a permutation module. A compact (R)-permutation module is the Pontryagin dual of a discrete one and we denote R[[X]] = F (X)^{∗}.

Lemma 3.2. Discrete permutation modules are closed under ⊕, hence compact ones are closed under Π. Similarly for R-permutation modules.

If H ≤ G also acts on the right of X, then F (X, M) is a G×H-module by ((g, h)f)(x) = hf(g^{−1}xh).

If we take X = G in the above, with the usual left and right actions by multiplication, then the H-fixed points F (G, M)^{1×H} form the coinduced G-module Coind_{H}^{G} M.

Note that if H ≤ G and X is an H-space then F (G ×_{H} X) ≅ Coind_{H}^{G} F (X).

Definition 3.3. (Brauer Construction) If M is a G-module (discrete or compact) and P ≤ G, define M^{[P]} = M^{P} /\overline{∑_{Q≨_{o}P} tr_{Q}^{P} M^{Q}}. This M^{[P]} is an N_{G}(P)-module (discrete or compact, according to M).

Lemma 3.4. If X is a profinite space on which G acts continuously and P is a pro-p group then F (X)^{[P]} ≅ F (X^{P}) as discrete N_{G}(P)-modules.

Proof. This is well known in the finite case (e.g. [5]). Let X = \underleftarrow{lim} X_{i}. Then F (X^{P}) ≅ F (\underleftarrow{lim} X_{i}^{P}) ≅ \underrightarrow{lim} F (X_{i}^{P}) ≅ \underrightarrow{lim}(F (X_{i})^{[P]}) ≅ (\underrightarrow{lim} F (X_{i}))^{[P]} ≅ F (X)^{[P]}. □

Lemma 3.5. Let G be a profinite group and H a closed subgroup: all limits are over G-equivariant directed systems.

(1) (\underrightarrow{lim} M_{i})^{[H]} ≅ \underrightarrow{lim}(M_{i}^{[H]}), where the M_{i} are discrete.

(2) (\underleftarrow{lim} M_{i})^{[H]} ≅ \underleftarrow{lim}(M_{i}^{[H]}), where the M_{i} are compact.

(3) The Brauer construction commutes with products Π for compact modules and with direct sums ⊕ for discrete modules.

(4) M^{∗}^{[H]} ≅ M^{[H]}^{∗}, where M is a permutation module, discrete or compact, and ∗ denotes the Pontryagin dual.

Proof. (1) and (2) follow easily from the definitions and (3) is a special case of them.

For (4) there is a natural map M^{∗}^{[H]} → M^{[H]}^{∗} obtained as follows: an element of M^{∗}^{[H]} can be lifted to an H-invariant function on M, which we then restrict to M^{H}, yielding an element of M^{[H]}^{∗}. For a finite permutation module this is easily checked to be an isomorphism, using the standard basis and 3.4.

In the general case we may suppose that M = F (X), where X = \underleftarrow{lim} X_{i} and the X_{i} are finite G-sets. Then F (X)^{∗}^{[H]} ≅ (\underleftarrow{lim} F (X_{i})^{∗})^{[H]} ≅ \underleftarrow{lim}(F (X_{i})^{∗}^{[H]}) and F (X)^{[H]}^{∗} ≅ (\underrightarrow{lim}(F (X_{i})^{[H]})^{∗} ≅ \underleftarrow{lim}((F (X_{i})^{[H]})^{∗}). □

Lemma 3.6. If H, K < G and H is finite then F (G/H)^{[K]} is isomorphic to some F (X) where X is a profinite N_{G}(K)-set with a finite number of orbits.

Proof. Notice that F (G/H)^{[K]} ≅ F ((G/H)^{K}), and N_{G}(K)\(G/H)^{K} is finite because it is in bijection with the conjugacy classes of subgroups of H whose members are conjugate to K in G, by the map N_{G}(K)gH ↦ K^{g}. □

Lemma 3.7. Let G be an (abstract) group and suppose that we have subgroups H ⊴G and S ≤ G such that S ∩ H = 1 and also P ≤ G/H. Let S_{H}(P) denote the set of subgroups Q ≤ G such that Q ∩ H = 1 and QH/H = P; G acts on this by conjugation. Then we have

(G/HS)^{P} ≅ ⨿_{Q∈S_{H}(P)/H}N_{H}(Q)\(G/S)^{Q}.

Proof. Let ρ : G → G/H and σ : G/S → G/HS be the quotient maps. Let Y = σ^{−1}((G/HS)^{P}), so that (G/HS)^{P} ≅ H\Y.

Notice that Y = {gS|P ≤ ρ(^{g}S)}. We claim that Y = ⨿_{S_{H}(Q)}(G/HS)^{Q}: the result follows.

First note that the (G/S)^{Q} are disjoint for different Q. If gS ∈ (G/S)^{Q_{1}} ∩ (G/S)^{G_{2}} then gS ∈ (G/S)^{⟨Q_{1},Q_{2}⟩} and so ⟨Q_{1}, Q_{2}⟩ ≤ ^{g}S. Now ⟨Q_{1}, Q_{2}⟩ ∩ H ≤ ^{g}S ∩ H = 1, so ρ restricted to ⟨Q_{1}, Q_{2}⟩ is injective. But ρ(⟨Q_{1}, Q_{2}⟩) = P, so Q_{1} = Q_{2}.

Now if gS ∈ (G/S)^{Q} then Q ≤ ^{g}S, so P ≤ ρ(^{g}S) and gS ∈ Y. Conversely, if gS ∈ Y then let Q = (ρ |_{^{g}S})^{−1}(P) and we find that gS ∈ (G/S)^{Q}. □

Corollary 3.8. In the circumstances of the previous lemma, but with G profinite and H open, we have

F (G/HS)^{[P]} ≅ ⊕_{Q∈S_{H}(P)/H}N_{H}(Q)\F (G/S)^{[Q]},

where the sum has only a finite number of non-zero terms.

Proof. This follows from 3.7 and 3.4. Note that, since H is open in G, both sides are finite-dimensional. □

Suppose that G acts continuously on X and that ∗ ∈ X is a fixed point. We say that G acts freely on (X, ∗) if it acts freely on X − {∗}.

If Y ⊆ X, we define F (X, Y) to be the kernel of restriction F (X) → F (Y), i.e, those functions that vanish on Y. Note that F (X, Y) ≅ F (X/Y, ∗) (where ∗ denotes the image of Y).

Lemma 3.9. If G acts freely on (X, ∗) then F (X, ∗) is injective as a discrete G-module, hence isomorphic to a summand of a sum of F (G)s.

Proof. (cf. [18] (5.6). This is well known in the case when X is finite.

First suppose that G is finite. Then, by [18] (5.6.4(c)), we know that (X, ∗) = \underleftarrow{lim}(X_{i}, ∗) for some system of finite G-sets X_{i} such that G acts freely on (X_{i}, ∗) . Then the result is true by 2.5, or rather its dual.

For the general case, for each N ◁_{o} G, G/H acts freely on (X/N, ∗) so, by the previous paragraph, F (G/N, ∗) is injective over G/N. Now apply 2.5 again. □

Define:

X_{≥H} = {x ∈ X| Stab_{G}(x) ≥ H}, X_{≥H} = {x ∈ X| Stab_{G}(x) ≥ H},
X_{(≥H)} = {x ∈ X| Stab_{G}(x) ≥_{G} H}, X_{(≥H)} = {x ∈ X| Stab_{G}(x) ≥_{G} H}.

Clearly X_{≥H} = X^{H} is closed. So is X_{(≥H)}, because it is the image of the map G×X_{≥H} → X, (g, x) ↦ gx.

We will always assume that X_{>H} (and hence X_{(>H)} ) is closed.

Lemma 3.10. R is a continuous ℤ̂_{p}-module if k has characteristic p and a ℚ-module if k has characteristic 0. In the first case F (X, T) ≅ F (X, ℤ/p^{∞}) ⋆ℤ̂_{p} T and in the second F (X, T) ≅ F (X, ℚ) ⋆_{ℚ} T.

Proof. The first part is clear and the second is because the image of X under any morphism is finite. □

Lemma 3.11. If Y ⊆ X is invariant as a set under G then F (X) ≅ F (X, Y) ⊕ F (Y) as G-modules.

Proof. There is a short exact sequence 0 → F (X, Y) → F (X) → F (Y) → 0. We need to show that it splits at the level of G-modules.

If X = \underleftarrow{lim} X_{i} and Y_{i} is the image of Y in X_{i} then Y = \underleftarrow{lim} Y_{i}. The short exact sequence 0 → F (X_{i}, Y_{i}) → F (X_{i}) → F (Y_{i}) → 0 is split by the finite case, which is well known and easy to prove.

To complete the proof, we can use 3.10 to assume that T is either ℤ̂_{p} or ℚ. We claim that the direct limit of split sequences splits. This is clear in the second case. Otherwise suppose that A_{i} → B_{i} → C_{i} be split for each i.

Let S_{i} denote the set of maps B_{i} → \underrightarrow{lim} A_{i} that, when composed with the map A_{i} → B_{i}, give the inclusion A_{i} → \underrightarrow{lim} A_{i}. The S_{i} form an inverse system of non-empty finite sets, so \underleftarrow{lim} S_{i} is non-empty, and any element gives a splitting \underrightarrow{lim} B_{i} → \underrightarrow{lim} A_{i}. □

Now we suppose that the action of G on X has only a finite number of conjugacy classes of stabilizers, S_{1}, …, S_{n}. Then each X_{>S_{i}} is closed, and an easy induction shows:

Lemma 3.12. F (X) ≅ ⨁_{i=1}^{n} F (X_{(≥S_{i})}, X_{(>S_{i})}).

Proposition 3.13. F (X_{(≥S_{i})}, X_{(>S_{i})}) is a summand of a sum of F (G/S_{i})s.

Proof. F (X_{(≥S_{i})}, X_{(>S_{i})}) ≅ F (G×_{N_{G}(H)}X_{≥S_{i}}, g×_{N_{G}(H)}X_{>S_{i}}) ≅ Coind_{N_{G}(S_{i})}^{G} F (X_{≥S_{i}}, X_{>S_{i}}) ≅ Coind_{N_{G}(S_{i})}^{G} F (X_{≥S_{i}}/X_{>S_{i}}, ∗).

We have already seen that F (X_{≥S_{i}}/X_{>S_{i}}, ∗) is a summand of a sum of F (N_{G}(S_{i})/S_{i})s. □

Definition 3.14. A strict discrete permutation module is a sum of discrete permutation modules of the form F (G/H). A strict compact permutation is a product of compact permutation modules of the form R[G/H]. As usual, a strict R-permutation module is a summand of a strict permutation module.

Notice that a strict permutation module is a permutation module by 3.2.

Just as in the finite case we may assume that an R-permutation module is a summand of permutation modules in which all the stabilizers are pro-p groups.

Corollary 3.15. If the action of G on X has only a finite number of conjugacy classes of stabilizers (e.g. if G is finite) then any R-permutation module is a strict R-permutation module.

Now we record some results of Bouc in the form in which we will need them. These appeared in [3] for finitely generated modules and [4] for infinitely generated modules.

Theorem 3.16. Let P be a finite p-group.

(a) Let C be a bounded complex of discrete (resp. compact) permutation kP -modules such that C^{[Q]} is exact for every non-trivial subgroup Q of P. Then C is homotopy equivalent to a bounded complex of injective (resp. projective) kP -modules.

(b) Let C be a bounded below complex of permutation R[[G]]-modules such that C^{[Q]} is exact for every subgroup G of P (including the trivial one). Then C is split.

Proof. In [4] 1.3 it is shown that if f : L → M is a morphism between two complexes of discrete permutation kP -modules, both bounded below or both bounded above, and such that for every Q ≤ P the induced map L^{[Q]} → M^{[Q]} is a quasi-isomorphism then f is a homotopy equivalence.

(a) Let f : C → I be an injective resolution of C. Then the theorem just mentioned shows that f is a homotopy equivalence. Now let n be a lower bound on C. Let U be the cokernel of I_{n} → I_{n−1}. Truncate I to form Ĩ according to the following diagram:

⋯ → I_{n+1} → I_{n} → U → 0 → · · · .

Then it easy to check that f̃: C → Ĩ is still a homotopy equivalence. Consider cone(f̃): it must be split exact, and it ends

⋯ C_{n} ⊕ I_{n+1} → I_{n} → U → 0.

Thus U is a summand of I_{0} and hence injective. So Ĩ is a bounded complex of injectives homotopy equivalent to C.

The compact case is just the dual.

(b) We obtain the result for k ⊗_{R} C over k instead of R by applying the theorem of Bouc to 0 → C. So we have a chain homotopy h : k ⊗ C → k ⊗ C such that Id_{k⊗C} = dh + hd.

By lemma 3.17 below we can lift h to h̃ : C → C and then Id_{C} = dh̃ + h̃d + m for some chain map m : C → 𝔪C. Let m^{r} denote the composition of m with itself m times and set ĥ = h̃ + h̃m + h̃m^{2} + · · ·. Then Id_{C} = dĥ + ĥd, as required. □

Lemma 3.17. Let U and V be compact strict permutation modules for a finite group G. Then the reduction map Hom_{R[[G]]}(U, V) → Hom_{k[[G]]}(k ⊗ U, k ⊗ V) is onto.

Proof. Because of the way that Hom commutes with products 2.3, we may assume that U = R[[G/H]] and V = R[[G/K]]. Now we are reduced to the well known finite case. □

4. Tate Hyper-Cohomology

Definition 4.1. If C is a complex of G-modules and P ≤ G is a finite p-subgroup then let q : C^{P} → C^{[P]} be the quotient map. A base map s : C^{[P]} → C^{P} is a map of complexes of vector spaces such that the composite qs : C^{[P]} → C^{[P]} is homotopic to the identity . We say that s is equivariant if it is a map of N_{G}(P)/P -modules.

Remark. If P is a finite p-group which acts admissibly on a finite dimensional CW-complex X and C(X) is the CW-chain complex of X then the inclusion X^{P} ⊆ X induces an isomorphism C(X)^{[P]} ≅ C(X^{P}) and hence a base map.

Lemma 4.2. Let P be a finite p-group and let C be a complex of modules in C_{k}(P), zero in negative degrees. Suppose that C is exact except, perhaps, in degree 0, that C → H_{0}(C) is split exact over kP and that q induces an isomorphism q_{0} : H_{0}(C)^{P} → H_{0}(C)^{[P]}. Then there is a base map s : C^{[P]} → C^{P}. If, in addition, C^{[P]} is a complex of projective compact N_{G}(P)/P -modules then the base map can be made equivariant.

Proof. The splitting implies that both C^{P} and C^{[P]} are resolutions of H_{0}(C)^{P}. The latter is clearly projective over k, and even over k[[N_{G}(P)/P]] in the presence of the extra hypothesis. The result is now an application of the basic comparison theorem ([27] 2.2.6,

[6] I 7.4). □

We define Tate hyper-cohomology groups over a finite group G in the usual way. If C is a bounded complex of compact G-modules then we take a complete resolution F_{C} of C over G by compact modules. We define 𝔼̂xt_{G}^{∗}(C, D) = H^{∗}(Hom_{G}(F_{C}, D)), whenever D is a bounded complex of G-modules, either compact or discrete. When C = R we write Ĥ_{P}^{∗} (D).

Remark. (a) The complete resolution of a bounded complex C can be formed in the same way as for a module (see [6] VI,X), and has similar uniqueness properties. Take a projective resolution P for C and choose a degree n higher than the upper bound on C. Truncate P by removing all terms in degree less than n. Now add projectives in all degrees less than n in such a way that the resulting complex is exact, just as in the case of a single module.

If C is projective over R then there is an alternative description of F_{C}. Let F_{R} be the complete resolution of the trivial module R. Then we can take F_{C} to be F_{R} ⊗ C.

In full generality the complete resolution should be defined as the total complex of the complete version of a Cartan-Eilenberg complex (see [27]), where all the projective resolutions of modules in the definition are replaced by complete resolutions.

(b) This definition is not quite what we would expect from [6] (VII), which uses the cohomology of Hom_{G}(F_{R}, Hom_{R}(C, D)). But this is isomorphic to Hom_{G}(F_{R} ⊗ C, D) and since in [6] C is always projective over R, we have that F_{R} ⊗ C is a complete resolution of C. For us such a definition would have the disadvantage that Hom_{R}(C, D) might only be in E_{R} if C is compact.

(c) The construction of complete resolutions via Cartan-Eilenberg resolutions leads to two spectral sequences:

IE_{1}^{p,q} = ⊕_{p′+p′′=p}Êxt_{G}^{q}(C_{p′}, D_{p′′}) ⇒ 𝔼̂xt_{G}^{p+q}(C, D),
IIE_{2}^{p,q} = ⊕_{q′+q′′=q}Êxt_{G}^{p}(H_{q′}(C), H_{q′′}(D)) ⇒ 𝔼̂xt_{G}^{p+q}(C, D).

The conditions on the complexes ensure that there are no convergence problems.

(d) The first spectral sequence shows that 𝔼̂xt^{∗}(C, D) vanishes if C is a complex of projectives. The second shows that it is invariant under a quasi-isomorphism of C or D.

(e) Since C is quasi-isomorphic to its projective resolution P, remark (d) shows that there would be problems associated with allowing unbounded complexes.

The next two lemmas are based on [6], VII 10, [23].

Lemma 4.3. Let P be a cyclic group of order p and let C be a bounded complex of discrete permutation kP -modules with base map s. Then Ĥ_{P}^{∗} (C) ≅ Ĥ_{P}^{∗} (C^{[P]}).

Proof. Since C consists of permutation modules it follows from 3.15 that, in each degree, C_{r} ≅ Πk ⊕ ΠkP.

Let cone(s) be the mapping cone of s : C^{[P]} → C. Consider what happens when we apply the Brauer construction. From the definition, cone(s)^{[P]} = cone(s′), where s′ is the induced map s′ : C^{[P]} → C^{[P]}. But s′ is a homotopy equivalence, by the definition of base map, so cone(s′) is exact.

Now 3.16 implies that cone(s) is homotopy equivalent to a bounded complex of injectives. Since H_{P}^{∗} vanishes on such a complex it follows that H_{P}^{∗} (s) induces an isomorphism. □

Proposition 4.4. Let P be a finite p-group and let C be a bounded complex of discrete permutation modules over k for P, with a base map for each subgroup of P. Then

∑_{r} dim H^{r}(C) ≥ ∑_{r} dim H^{r}(C^{[P]})

(i.e. if the left hand side is finite then so is the right hand side and the inequality holds).

Proof. Consider a composition series 1 = P_{0} < P_{1} < ⋯ < P_{n} = P, where |P_{i}| = p^{i}. The hypotheses imply that C^{[P_{i}]} → (C^{[P_{i}]})^{[P_{i+1}/P_{i}]} ≅ C^{[P_{i+1}]} has a base map, so we have a proof by induction on i provided that we can prove the case when |P | = p.

Note that, since P is cyclic, dim Ĥ_{P}^{r} (M) = dim H^{r}(P, M) ≤ dim M, for any discrete kP -module M.

Consider the spectral sequence

IIE_{2}^{p,q} = Ĥ_{P}^{p} (H_{q}(C)) ⇒ Ĥ_{P}^{p+q}(C).

We see that

∑_{r} dim H^{r}(C) ≥ ∑_{r} Ĥ_{P}^{−r}(H^{r}(C)) ≥ dim Ĥ_{P}^{0} (C).

Now, by Lemma 4.3, dim Ĥ_{P}^{0} (C) ≥ dim Ĥ_{P}^{0} (C^{[P]}). Also, by the Künneth theorem, Ĥ_{P}^{0} (C^{[P]}) ≅ ⊕_{r}H^{r}(C^{[P]}) ⊗ Ĥ^{−r}(P), so dim Ĥ_{P}^{0} (C^{[P]}) = ∑_{r} dim H^{r}(C^{[P]}) □

5. Weak R-Permutation Resolutions

Definition 5.1. A compact weak R-permutation resolution of the profinite group G is an exact complex of of finite length of discrete G-modules

0 → J_{n} → ⋯ → J_{1} → J_{0} → R → 0 (= J_{•}),

where each J_{i} is a summand of a permutation module R[[X_{i}]] for some profinite space X_{i} on which G acts with finite stabilizers.

The discrete version is just the Pontryagin dual.

A signed weak R-permutation resolution is similar except that we allow each J_{i} to be a summand of a product of modules of the form R[[X_{i,j}]] ⊗ R′, where X_{i,j} is as before and R′ is a copy of R on which G acts as ±1.

Remark. The signed version makes no difference unless p = 2 and 2R ≠ 0, but we want it for the next result.

The next theorem uses a method of Swan [24], following Serre. It was first proved in the profinite case by Scheiderer [20, 19] .

Theorem 5.2. If the profinite group G has finite virtual cohomological dimension over R then it has a signed weak R-permutation resolution.

Proof. For some H ≤_{o} G there is a finite projective resolution

0 → P_{n} → ⋯ → P_{1} → P_{0} → R → 0,

where each I_{i} is of the form R[[Y_{i}]] for some profinite space Y_{i} on which G acts freely (using the Eilenberg swindle).

Take J_{•} to be the tensor induced complex, as defined in [1] or [24]. Note that there is a rather subtle sign convention. The terms are all summands of R[[(⨿X_{i})^{⊗G}]] ≅ R[[(⨿_{π}Π_{i=1}^{s}g_{i}X_{π_{i}}]], where {g_{1}, …, g_{s}} are coset representatives of G/H and π : {1, …, s} → {1, …, m}, except that they might be twisted by a sign. □

Lemma 5.3. If H < G is closed and M is a discrete G-module and N a compact Gmodule then Ext_{H}^{∗}(N, M) = \underrightarrow{lim}_{H<K<_{o}G} Ext_{K}^{∗}(N, M) (the limit over restriction maps).

Proof.

Ext_{H}^{∗}(N, M) ≅ Ext_{G}^{∗}(N, Coind_{H}^{G} Res_{H}^{G} M)
≅ Ext_{G}^{∗}(N, F (G/H) ⋆ M)
≅ Ext_{G}^{∗}(N, (\underrightarrow{lim}_{H<K<_{o}G} F (G/K)) ⋆ M)
≅ Ext_{G}^{∗}(N, \underrightarrow{lim}_{H<K<_{o}G}(F (G/K) ⋆ M))
≅ \underrightarrow{lim}_{H<K<_{o}G} Ext_{G}^{∗}(N, F (G/K) ⋆ M)
≅ \underrightarrow{lim}_{H<K<_{o}G} Ext_{K}^{∗}(N, M)
□

Proposition 5.4. Let G act on the profinite space X and let M be a discrete G-module and N a compact G-module. Suppose that, for some n and any x ∈ X, Ext_{Stab_{G}(x)}^{n}(N, M) = 0. Then Ext_{G}^{n}(N ⊗ R[[X]], M) = 0.

Proof. For j ≥ i let ρ_{j,i} denote the map X_{j} → X_{i} and ρ_{j} the map X → X_{j}.

Ext_{G}^{n}(N ⊗ R[[X]], M) ≅ Ext_{G}^{n}(N ⊗ \underleftarrow{lim} R[[X_{i}]], M) ≅ \underrightarrow{lim} Ext_{G}^{n}(N ⊗ R[[X_{i}]], M) ≅ \underrightarrow{lim}(⊕_{x∈X_{i}} Ext_{Stab_{G}(x)}^{n}(N, M))^{G}.

So it suffices to show that, for any i, x ∈ X_{i}, h ∈ Ext_{Stab_{G}(x)}^{n}(N, M), there is a j ≥ i such that ρ_{j,i}^{n}(h) = 0 ∈ ⊕_{ρ_{j,i}(y)=x} Ext_{Stab_{G}(y)}^{n}(N, M).

Let z ∈ ρ^{−1}(x): then since Ext_{Stab_{G}(z)}^{n}(N, M) = 0, there is a subgroup H, Stab_{G}(z) ≤ H ≤_{o} G such that res_{H}^{G} H = 0 by 5.3. Now Stab_{G}(ρ_{j}z) ≤ H, which is what we want. □

Corollary 5.5. If G has a signed weak R-permutation resolution of finite length n, and M is a compact kG-module which is projective on restriction to each finite elementary abelian subgroup, then M has projective dimension less than or equal to n.

There is a dual result for discrete modules.

Proof. By Chouinard’s Theorem, M is projective on restriction to any finite subgroup. Tensor the resolution

0 → J_{n} → ⋯ → J_{1} → J_{0} → R → 0

with M to obtain

0 → J_{n} ⊗ M → J_{1} ⊗ M → J_{0} ⊗ M → ⋯ → M → 0.

By 5.4, each J_{i} ⊗ M satisfies Ext_{G}^{∗}(M ⊗ J_{i} ⊗ N) = 0 for all discrete modules N and so, by 2.5, is projective. □

The next useful lemma is taken from [2].

Lemma 5.6. Let H <_{o} G and suppose that M is a compact module for G of finite projective dimension whose restriction to H is projective. Then M is projective.

Proof. Let Y be the cokernel of the coaugmentation R → R[[G/H]]. We obtain a short exact sequence

0 → M → R[[G/H]] ⊗ M → Y ⊗ M → 0.

The middle term is projective, and if M has projective dimension r then Y ⊗ M has projective dimension less than or equal to r. It follows that either M has projective dimension less than of equal to r − 1, a contradiction, or that M is projective. □

Lemma 5.5 and the theory of profinite duality groups discussed in [26] are precisely what is needed to carry over the algebraic proof of the local cohomology theorem of Benson and Greenlees [2] from discrete virtual duality groups to profinite ones. In particular we obtain:

Theorem 5.7. If G is a virtual duality group of virtual dimension n and with dualizing module I, then for any compact G-module M there is a spectral sequence

E_{2}^{∗,∗} = H_{J}^{∗,∗}(H^{∗}(G, M)) ⇒ Σ^{n}H_{∗}(G, I ˆ⊗M),

where J is the ideal of positive degree elements in H^{∗}(G).

6. R-Permutation Resolutions

In this section we consider algebraic analogues of some geometric results of Kropholler and Mislin [15].

We count the trivial group as a finite p-subgroup, and the Brauer construction at the trivial group will be taken to mean reduction modulo p.

Definition 6.1. By an R-permutation resolution for G of length n we mean a complex of finite length of compact G-modules

0 → C_{n} → … → C_{1} → C_{0} → R (= C̃),

where each C_{i} is a strong R-permutation module.

In addition, if we apply the Brauer construction at any finite p-subgroup P < G, the resulting complex C̃^{[P]} is required to be exact.

We say that the permutation resolution is of finite type if each C_{i} is finitely generated. The smallest possible n is called the permutation dimension of G over R (possibly ∞).

We have given the definition in terms of the Brauer construction because this is what corresponds better with the geometric methods used for discrete groups. However, there is an alternative description, by a theorem of Bouc mentioned previously (3.16).

Proposition 6.2. The condition that the complex is exact at every Brauer construction at a finite p- subgroup is equivalent to the condition that it be split on restriction to every finite p-subgroup (or equivalently on restriction to every finite subgroup).

Proposition 6.3. If H <_{o} G and permdim G < ∞ then permdim H = permdim G.

Proof. Clearly permdim H ≤ permdim G.

This is really a result about relative cohomology Ext_{G/F}^{∗}(R, −), where F denotes the set of finite p-subgroups of G. All we need to know is that this is the homology of Hom_{G}(C, −), where C is an R-permutation resolution of R, and any R-permutation resolution gives the same answer.

If n = permdim G and C is an R-permutation of length n then Ext_{G/F}^{n}(R, C_{n}) ≠ 0, since otherwise C_{n} would split off C_{n−1} and we would obtain a resolution of smaller length.

The usual epimorphisms Ind_{F}^{G} Res_{F}^{G} → M, for F < G, F finite, can be combined to give a morphism ˆ⊕_{F ∈F/G} Ind_{F}^{G} Res_{F}^{G} M → M, which is split on restriction to every F ∈ F. It follows that Ext_{G,F}^{n}(R, ˆ⊕_{F ∈F/G} Ind_{F}^{G} Res_{F}^{G} M) ≠ 0 and thus that Ext_{G,F}^{n}(R, Ind_{F}^{G} Res_{F}^{G} M) ≠ 0 for some F ∈ F.

But Ext_{G,F}^{n}(R, Ind_{F}^{G} Res_{F}^{G} M) ≅ Ext_{G,F}^{n}(R, Coind_{H}^{G} Ind_{F}^{H} Res_{F}^{G} M) ≅ Ext_{H,F}^{n}(R, Ind_{F}^{H} Res_{F}^{G} M). Thus H cannot have an R-permutation resolution of dimension less than n. □

Definition 6.4. The relative dimension of G over R, reldim G is the least integer n such that every compact module for G that is projective on restriction to every finite p-subgroup has projective dimension less than or equal to n. If no such n exists then we set reldim G = ∞.

Lemma 6.5. reldim G ≤ permdim G.

Proof. If C̃ is a permutation resolution for G of minimum dimension n and M is a module that is projective on every finite p-subgroup then tensoring the two gives a projective resolution for M of dimension n by 2.4. □

Lemma 6.6. If H < G then reldim H ≤ reldim G.

Proof. Let M be a compact module for H which is projective on restriction to all finite p-subgroups. Then Ind_{H}^{G} M is projective on restriction to each finite p-subgroup of G too. Also Res_{H}^{G} Ind_{H}^{G} M ≅ M ⊕ X, for some X, so

reldim G ≥ projdim Ind_{H}^{G} M ≥ projdim Res_{H}^{G} Ind_{H}^{G} M ≥ projdim M.
□

Lemma 6.7. Let P < G be a finite normal p-subgroup. Then reldim G/P ≤ reldim H.

Proof. Let M be a compact module for G/P which is projective on restriction to any finite p-subgroup. Consider M as a module for G and let M̃ be its projective cover. Then M̃ is also projective on restriction to any finite p-subgroup and so if reldim G is finite then there is a projective resolution P_{∗} of M̃ of finite length. Taking coinvariants under P gives a projective resolution of M over G/P. □

Theorem 6.8. Suppose that there is a bound on the order of a finite p-subgroup P of G. Then the following are equivalent.

(1) G has a signed weak permutation complex,

(2) G has a permutation complex,

(3) reldim G is finite.

Proof. It is clear that (2) implies (1), and (1) implies (3) by lemma 5.5, so we assume that reldim G = n and construct a permutation complex.

By downwards induction on r, we construct a complex of compact R-permutation modules C_{n} → ⋯ → C_{0} → R which is exact after applying the Brauer construction at each P with |P | ≥ p^{r}.

For r large enough we can just take C to be 0. Suppose that we have this complex for r + 1. We claim, by induction on s, that we can add R-permutation modules to make the complex exact at the Brauer construction at P in degrees up to and including s at each P with |P | ≥ p^{r} and in degrees greater than s for P with |P | ≥ p^{r+1}.

Suppose that we have such a complex exact up to and including degree s − 1. For any finite p-subgroup P, let K_{P} = ker(C_{s}^{[P]} → C_{s−1}^{[P]}). Consider it as an N_{G}(P)/P -module and let ϕ_{P} : K̃_{P} → K_{P} be its projective cover. This lifts to ϕ̃_{P} : K̃_{P} → C_{s}^{P} ⊆ C_{s}. Consider this as a map of N_{G}(P)-modules and induce to θ_{P} : L_{P} = Ind_{N_{G}(P)}^{G} K̃_{P} → C_{s}.

If Q is a finite p-subgroup such that Q ≠ P and |P | ≥ p^{r} then L_{P}^{[T]} = 0. Also θ_{P}^{[P]} : L_{P}^{[P]} → C_{s}^{[P]} is just ϕ_{P} : K̃_{P} → C_{s}^{[P]}.

Thus we can replace C_{s} by C_{s} ⊕ Π̂_{P ∈P/G}L_{P} and the boundary map d_{s} by d_{s} + Πθ_{P}. Of course this might not yield a complex of finite dimension, so when we get to a degree

n for which C_{n} = 0 we do something different.

Let U be finite p-subgroup of N_{G}(P)/P, let V be any subgroup of U and let R̃V be its inverse image in N_{G}(P). Since C_{∗}^{[P]}[V] ≅ C_{∗}[Ṽ] it is exact, so by a result of [4] applied to C_{∗}^{[P]} as an augmented complex of U-modules, K_{P} is projective as a U-module.

By condition (3) and lemmas 6.6 and 6.7 we see that K_{P} has a projective resolution D_{∗} as an N_{G}(P)/P -module of length bounded by n. The map D_{0} → K_{P} lifts to η_{P} : D_{0} → C_{n}^{P}.

Now regard D_{∗} as a complex of N_{G}(P)-modules. We can finish our level r permutation resolution by taking ˆ⨁_{P ∈P/G} Ind_{N_{G}(P)}^{G} D_{∗} and splicing it on to C_{∗} with the map Π Ind_{N_{G}(P)}^{G} η_{P}. □

Corollary 6.9. If G has virtual cohomological dimension t over R then permdim G = reldim G = t.

Proof. By 5.2 G satisfies condition (1) of 6.3, so permdim G is finite. Thus permdim G = t by 6.3. Now reldim G ≤ t by 6.5, and, in fact, is equal by considering Ind_{H}^{G} R. □

Remark. In fact, when G has virtual cohomological dimension t, the construction in the proof above can be altered to yield a resolution of dimension t. First we add the condition that the complex should be of dimension less than or equal to t to the induction hypothesis on r. Now at degree t we can apply the dual of lemma 5.6 to K_{P} to see that it is projective and thus we can split it off.

Theorem 6.10. The following are equivalent:

(1) G has a permutation resolution of finite type and finite dimension.

(2) G has only finitely many conjugacy classes of finite p-subgroups P and each N_{G}(P)/P (including G) is virtually of type F P.

(3) reldim G is finite and in addition there are only finitely many conjugacy classes of finite p-subgroups P and each N_{G}(P)/P (including G) is of type F P_{∞}.

Proof. (1) ⇒ (2): Let P be a finite p-subgroup. Exactness after applying the Brauer construction at P implies that C_{0}^{[P]} ≠ 0. But if C_{0} is a summand of ⊕_{i=1}^{n}k[[G/S_{i}]] then P must be conjugate to a subgroup of some S_{i}. Thus there are only a finite number of conjugacy classes of finite p-subgroups.

Now we can find an open subgroup H ≤ G that avoids all the finite p-subgroups. The restriction of the complex to H is then a projective resolution, and still finitely generated, so G is virtually of type F P.

We claim that applying the Brauer construction at a finite p-subgroup P yields a similar complex for N_{G}(P)/P, so condition (2) follows. We need to check that the complex is still finitely generated. It is enough to check that k[[G/S]]^{[P]} is finitely generated over N_{G}(P). But this is dual to F (G/S)^{[P]} ≅ F ((G/S)^{P}), so the result follows by 3.6.

(2) ⇒ (3): The construction of theorem 6.8 yields a complex of finite type at each step in the induction. □

From now on we write H^{∗}(G) for H^{∗}(G, k).

Proposition 6.11. If G is of type vF P then there are only finitely many conjugacy classes of finite p-subgroups P, and each N_{G}(P) has an open subgroup with finite homology.

Proof. (cf. [6] IX 13.2) Let H ⊴_{o} G of type F P, hence p-torsion free. Let X be a compact permutation resolution of G of finite length, which exists by 5.2. The restriction of X to H is a projective resolution of k. Let Y = k ⊗_{H} X, so that H_{∗}(Y) ≅ H_{∗}(H), by the definition of homology, and so H_{∗}(Y) is finite.

Let P be a p-subgroup of G/H. We claim that Y^{P} → Y^{[P]} has a base map. To see this we use 3.8 to show that

Y^{[P]} ≅ k ˆ⊗_{H}Π_{Q∈S_{H}(P)}X^{[Q]}.

But each X^{Q} → X^{[Q]} has an equivariant base map by 4.2, and these piece together to give a map

Y^{[P]} ≅ k ˆ⊗_{H}Π_{Q∈S_{H}(P)}X^{[Q]} → k ˆ⊗_{H}Π_{Q∈S_{H}(P)}X^{Q}k ˆ⊗_{H}X = Y.

It follows that H_{∗}(Y^{[P]}) is finite, by 4.4.

Now

Y^{[P]} ≅ Π_{Q∈S_{H}(Q)/H}k ⊗_{N_{H}(Q)} X^{[Q]},

and X^{[Q]} is a projective resolution of N_{H}(Q)/Q by 5.2. Thus we have

H_{∗}(Y^{[P]}) ≅ Π_{Q∈S_{H}(Q)/H}H_{∗}(N_{H}(Q)).

In degree 0 this tells us that S_{H}(Q)/H is finite and hence, since there is only a finite number of possibilities for P, that there are only finitely many conjugacy classes of finite p-subgroups.

It also follows that each H_{∗}(N_{H}(Q)) is finite. □

Theorem 6.12. If G is a virtual pro-p group then the following conditions are equivalent: (1) G has a permutation resolution of finite type,

(2) G is virtually of type F P, (3) H^{∗}(G) is finitely generated as a ring,

(4) reldim G is finite and there is only a finite number of conjugacy classes of finite p-subgroups P and each H_{∗}(N_{G}(P)/P) is of finite type.

Proof. We claim that the conditions stated are equivalent to those of 6.10 under the hypothesis that G is a virtual pro-p-group, where finite homology implies type vF P.

Clearly (1) is the same in both. Also the present (2) is equivalent to 6.10 (2) by 6.11 and the present (4) is 6.10 (3).

Now (2) implies (3) by considering the Lyndon-Hochschild-Serre spectral sequence, and the converse is shown in [17]. □

Remark. In [17] it is shown that if H^{∗}(G) is finitely generated as a ring then there are only finitely many conjugacy classes of finite p-subgroups. The part of that proof used here is the fairly elementary result that if H^{∗}(G) is finitely generated then there is an open subgroup H <_{o} G such that res : H^{∗}(G) → H^{∗}(H) is zero in positive degrees (by an argument similar to that of 5.3 with H = 0 ). Since H^{∗}(H) is finitely generated over H^{∗}(G) it must be finite-dimensional.

The deeper part of that paper shows that the number of conjugacy classes of finite p-subgroups is finite. It uses results of Dwyer and Wilkerson [9], which in turn depend on unpublished work of Lannes on unstable algebras over the Steenrod algebra, to obtain a version of 6.14 below (see also [13]). The method of the present paper bypasses that.

Lemma 6.13. The number of conjugacy classes of chains of finite p-subgroups is finite if and only if the number of conjugacy classes of maximal finite p-subgroups is finite.

Proof. If the number of conjugacy classes of maximal finite p-subgroups is finite then any chain can be conjugated so that it lies within a finite collection of finite subgroups. But then the number of possibilities for such chains is finite.

The converse is trivial. □

Theorem 6.14. Let G be a virtual pro-p group with finitely generated cohomology. Then for any finite p-subgroup P, H^{∗}(N_{G}(P)/P) is finitely generated and H^{∗}(N_{G}(P)) is finitely generated as a module over H^{∗}(G).

Proof. For the first assertion we use 6.12 and and show that if condition (4) holds for G then it does so for N_{G}(P).

Certainly reldim H is finite, by 6.6.

Let K, L be non-trivial finite p-subgroups of N_{G}(P)/P. Let K′, L′ denote the inverse images in N_{G}(P). Then K and L are conjugate in N_{G}(P)/P if and only if the chains (P, K′) and (P, L′) are conjugate in G. Thus the number of conjugacy classes of finite p-subgroups of N_{G}(P)/P is finite, by 6.13.

Furthermore, N_{N_{G}(P)}(K)/K = N_{G}(P, K′)/K is of finite index in N_{G}(K)/K, so H_{∗}(N_{N_{G}(P)}(K)/K) is of finite type.

For the second claim, let U ◁_{o} G be pro-p of type FP (using 6.12 condition (2)). Now N_{G}(P)/P is of type FP, by the first claim and 6.12 again. It follows that the image of U ∩ N_{G}(P) in N_{G}(P)/P is of type FP, and this is isomorphic to U ∩ N_{G}(P).

Consider the diagram

U ∩ N_{G}(P) ⟶ N_{G}(P) ⟶ N_{G}(P)/(U ∩ N_{G}(P))
↓  ↓  ↓
U  ⟶ G ⟶  G/U

The Lyndon-Hochschild-Serre spectral sequence for the top row has E_{2} page H^{∗}(N_{G}(P)/(U∩ N_{G}(P)), H^{∗}(U ∩ N_{G}(P))). This is noetherian over H^{∗}(N_{G}(P)/(U ∩ N_{G}(P))) via inflation, hence so is H^{∗}(N_{G}(P)), by the theory of Evens [10, 11, 1].

By the same theory, H^{∗}(N_{G}(P)/(U ∩ N_{G}(P))) is noetherian over H^{∗}(G/U) via restriction.

Combining these we find that H^{∗}(N_{G}(P)) is noetherian over H^{∗}(G/U) via the natural map N_{G}(P) → G/U. But this map factors through G, so H^{∗}(N_{G}(P)) must be noetherian over H^{∗}(G). □

7. Tate-Farrell Cohomology

For any group of finite virtual dimension we can construct a complete resolution as in [6] (X) and so deal with the Tate-Farrell cohomology or hypercohomology, just as we did for the Tate cohomology of a finite group in §4. This was first considered for profinite groups by Scheiderer [20], but the approach given here is more elementary and allows compact coefficients, as required in certain applications, [25] for example.

This cohomology has the expected properties, as in [6]. In particular it can be calculated using the Brown complex, but in order to state this we need to define the chain complex of the Brown complex.

For any group U, let Δ(U) denote the usual Brown complex for U, with simplices corresponding to chains of elementary abelian p-subgroups of U (some fixed prime p is understood). If U is finite this is well understood. Let Q → D_{0} → ⋯ → D_{r} (= D̃(Δ(U))) denote the complex of cochains and C_{r} → ⋯ → C_{r} → R (= C̃(Δ(U))) the dual complex of chains. These are complexes of permutation modules; it is well known that Δ(U)^{P} is contractible for each non-trivial p-subgroup P, so D̃(Δ(U))^{[P]} and C̃(Δ(U))^{[P]} are exact.

If M, N ⊴_{o} G, M ≤ N, both p-torsion free, then there is a natural map Δ(G/M) → Δ(G/N). Define D̃(Δ(G)) to be the direct limit of the complexes D̃(Δ(G/N)) for open normal p-torsion free subgroups N. Similarly, define C̃(Δ(G)) to be the inverse limit of the complexes C̃(Δ(G/N)); this is dual to D̃(Δ(G)).

Lemma 7.1. For any non-trivial finite p subgroup P ≤ G both D̃(Δ(G))^{[P]} and C̃(Δ(G))^{[P]} are exact.

Proof. By 3.5 we see that D̃(Δ(G))^{[P]} ≅ (\underrightarrow{lim} D(Δ(G/N)))^{[P]} ≅ \underrightarrow{lim}(D(Δ(G/N))^{[P]}), and the direct limit of exact complexes is exact. □

Lemma 7.2. If the number of conjugacy classes of elementary abelian p-subgroups is finite then D_{i} ≅ ⊕_{σ∈Δ(G)_{i}/G}F (G/ Stab_{G}(σ)) and C_{i} ≅ ⊕_{σ∈Δ(G)_{i}/G}k[[G/ Stab_{G}(σ)]].

Proof. If we define ˜ D̃_{i}(Δ(G/N)) ⊆ D̃_{i}(Δ(G/N)) to be the sub-permutation module on the chains of subgroups that are images of a chain in G, then \underrightarrow{lim} ˜ D̃_{i}(Δ(G/N)) = \underrightarrow{lim} D̃_{i}(Δ(G/N)) = D̃_{i}(Δ(G)).

For a group with an open p-torsion free subgroup, F (G/S) = \underrightarrow{lim} F (G/SN) and this

extends to finite direct sums. Thus \underrightarrow{lim} ˜ D̃_{i}(Δ(G/N)) has the form claimed.

The case of C̃_{i} is dual. □

Remark. In [17] it is shown that a pro-p group has a finite number of conjugacy classes of elementary subgroups if and only if its cohomology is finitely generated as a ring modulo nilpotent elements.

Let C(Δ(G)) denote the complex C̃(Δ(G)), but without the augmentation. By analogy with the discrete case we expect the following result.

Theorem 7.3. Let M be a G-module, discrete or compact. Then 𝔼̂xt_{G}^{∗}(C(Δ(G)), M) ≅ Ĥ^{∗}(G, M).

Proof. From the long exact sequence associated to C → C̃ → 𝔽_{p} we see that it suffices to show that 𝔼̂xt_{G}^{∗}(C̃(Δ(G)), M) = 0.

Just as in the last stage (r = 0) of the proof of 6.8, we can add projectives to C̃(Δ(G)) to obtain a complex of finite length. This does not alter 𝔼̂xt_{G}^{∗}.

The spectral sequence ^{II}E_{2}^{p,q} = Êxt_{G}^{p}(H_{q}(C̃(Δ(G))), M) ⇒ 𝔼̂xt_{G}^{p+q}(C̃(Δ(G)), M) yields the required result. □

Remark. 𝔼̂xt_{G}^{∗}(C(Δ(G)), M) can be computed using the spectral sequence^{I}E_{1}^{p,q} = Êxt_{G}^{q}(C(Δ)_{p}, M) ⇒ 𝔼̂xt_{G}^{p+q}(C(Δ), M).

If there are only a finite number of conjugacy classes of elementary abelian p-subgroups then, by 7.2, ^{I}E_{1}^{p,q} ≅ ⊕_{σ∈Δ_{p}/G}Ĥ^{q}(Stab_{G}(σ̃), M), (where σ̃ denotes a chain in the class σ).

Remark. The case of p-rank 1 appears in the work of Henn [13].

Remark. We could easily define the complete cohomology of Kropholler [8] for profinite groups. But in the present framework, unless G is of finite virtual cohomological dimension so we revert to Tate-Farrell cohomology, we lack a compact version of the module B that he exploits so successfully in order to develop its properties.

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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