ON THE CONSTRUCTION OF PERMUTATION COMPLEXES FOR PROFINITE GROUPS

PETER SYMONDS

Abstract. Goerss, Henn, Mahowald and Rezk construct a complex of permutation modules for the Morava stabilizer group 𝔾_{2} at the prime 3. We describe how this can be done using techniques from homological algebra.

1. Introduction

In [5], Goerss, Henn, Mahowald and Rezk consider the special extended Morava stabilizer group 𝔾_{2}^{1} = 𝕊_{2}^{1} ⋊ Gal at the prime 3 and construct an exact sequence of compact modules

0 → Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} → Ind_{SD_{16}}^{𝔾_{2}^{1}} ℤ̂_{3}(χ) → Ind_{SD_{16}}^{𝔾_{2}^{1}} ℤ̂_{3}(χ) → Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} → ℤ̂_{3} → 0,

where G_{24} is a subgroup of order 24 etc. and ℤ̂_{3}(χ) is a copy of ℤ̂_{3} on which SD_{16} acts via a character χ : SD_{16} → {±1}. They then use this to construct a certain tower of spectra.

The aim of this note is to show how methods from the homological algebra and representation theory of these groups can help in the algebraic part of this construction.

2. Background

Let G be a profinite group and let R be a complete noetherian local ring with finite residue class field k of characteristic p. For example, R could be the p-adic integers.

We work in the category of compact R[[G]]-modules, C_{R}(G), (see [10] for definitions, properties and more references).

The next result is basic, but does not seem to have appeared in the literature.

Proposition 2.1. If G is a virtual pro-p-group then the Krull-Schmidt property holds for (topologically) finitely generated modules in C_{R}(G), i.e. every such module can be expressed as a finite sum of indecomposable modules and this decomposition is essentially unique in the sense that the multiplicity of each isomorphism type is the same in any such decomposition.

Proof. The proof is just an adaptation of the one for finite groups (see [1]), but a little care is needed.

Let H ⊴_{o} G be an open normal pro-p subgroup. If M is a finitely generated R[[G]]- module then k ⊗_{R[[H]]} M is finite dimensional and we can decompose M as a finite sum of indecomposable modules using induction on dim_{k} k ⊗_{R[[H]]} M.

For this to work we need to know that our induction starts, that is that if M ≠ 0 then k ⊗_{R[[H]]} M ≠ 0. Let M′ be a finite quotient of M as an H-module; there is a surjection k ⊗_{R[[H]]} M → k ⊗_{R[[H]]} M′. The action of H on M′ factors through that of a finite p-group P, and in this case it is well known that k ⊗_{R[P]} M′ ≠ 0.

All we need to do now is to show that the endomorphism ring of a finitely generated indecomposable module is local, because then the uniqueness of decomposition follows formally (see e.g. [1] 1.4.3).

The proof is just a variant of the one for finite groups (see [1] 1.9). Let J be the Jacobson radical of R[[G]]. For any open normal subgroup N of G let I_{N} denote the augmentation ideal of R[[N]]. Given an endomorphism f of M ∈ C_{R}(G) we set Im(f^{∞}) = ∩_{n=1}^{∞} Im(f^{n}) and Ker(f^{∞}) = {x ∈ M|∀N ⊴_{o} G ∀n ≥ 0 ∃m ≥ 0 such that f^{m}(x) ∈ J^{n}M + I_{N}M}.

For each open normal subgroup N ⊴_{o} G define M_{N} = R ⊗_{R[[N]]} M ≅ M/I_{N}M ∈ C_{R}(G/N). Then M ≅ \underleftarrow{lim} M_{N}. Since M is finitely generated, M_{N} is too. Now f induces an endomorphism f_{N} of M_{N}. Define Im(f_{N}^{∞}) = ∩_{n=1}^{∞} Im(f_{N}^{n}) and Ker(f_{N}^{∞}) = {x ∈ M_{N}|∀n ≥ 0 ∃m ≥ 0 such that f^{m}(x) ∈ J^{n}M_{N}}. From the finite group case of Fitting’s Lemma we know that M_{N} = Im(f_{N}^{∞}) ⊕ Ker(f_{N}^{∞}).

But Im(f^{∞}) ≅ \underleftarrow{lim} Im(f_{N}^{∞}) and Ker(f^{∞}) ≅ \underleftarrow{lim} Ker(f_{N}^{∞}). Hence M = Im(f^{∞}) ⊕ Ker(f^{∞}).

Now suppose that M is indecomposable and let I be a maximal left ideal in End_{C_{R}(G)}(M) and let a be an endomorphism not in I. Then 1 = ba + f for some b ∈ End_{C_{R}(G)}(M) and f ∈ I. But f is not an isomorphism, so M = Ker(f^{∞}) and Im(f^{∞}) = 0.

Now (1 + f + ⋯ + f^{n−1})ba = 1 − f^{n}. Let N ⊴_{o} G be some arbitrary open normal pro-p subgroup. Since M is finitely generated, for sufficiently large n we have f^{n}(M) ⊆ JM + I_{N}M ⊆ JM. Thus 1 − f^{n} is onto, by the profinite version of Nakayama’s Lemma ([4] 1.4). Also if (1 − f^{n})(x) = 0 then x ∈ Im(f^{∞}) = 0, so 1 − f^{n} is injective. Thus 1 − f^{n} is an isomorphism and a has a left inverse, c say.

But c_{N} must also be a right inverse to a_{N} on each M_{N}, so c is also a right inverse and a is an isomorphism, as required. □

Projective covers exist in C_{R}(G) ([9]), thus so do minimal projective resolutions.

If S is a simple module, let P_{S} denote the projective cover of S. The P_{S} are precisely the indecomposable projective modules, and any other projective is a product of them.

If there is an open normal pro-p subgroup H ⊴_{o} G, then any simple module for R[[G]] is the inflation of one for k[G/H] so, in particular, there are only finitely many simple modules up to isomorphism.

The next result is well known for finite groups.

Proposition 2.2. Suppose that M ∈ C_{R}(G) is projective over R and let ⋯ → P_{r} → ⋯ → P_{1} → P_{0} → M be the minimal projective resolution of M. If S is a simple module then the multiplicity of P_{S} in P_{r} is dim_{End(S)} Ext_{R[[G]]}^{r}(M, S) = dim_{End(S)} H^{r}(G, (k ⊗_{R} M)^{∗} ⊗_{R} S).

Here S^{∗} denotes the dual over k, or rather the contragredient.

(If k is a splitting field for G/H, where H < G is open, normal and pro-p, then End(S) ≅ k.)

Proof. (cf. [11]) The multiplicity of P_{S} in P_{r} is dim_{End(S)} Hom_{R[[G]]}(P_{r}, S). The fact that the projective resolution is minimal implies that the differentials in the complex Hom_{R[[G]]}(P_{•}, S) are zero.

Combining these facts, we find that the multiplicity that we are calculating is equal to dim_{End(S)} Ext_{R[[G]]}^{r}(M, S). But Ext_{R[[G]]}^{r}(M, S) ≅ Ext_{R[[G]]}^{r}(R, Hom_{R}(M, S)) (see e.g. [1] 3.1.8) and Hom_{R}(M, S) ≅ (k ⊗_{R} M)^{∗} ⊗_{R} S. □

From now on we assume that G is of finite virtual cohomological dimension over R. The definition of Tate-Farrell cohomology appears in [8] for discrete coefficients and in [10] for compact ones, as does the next result. (See [3] for its basic properties in the case of an abstract group.)

Proposition 2.3. For M in C_{R}(G) or D_{R}(G), the Tate-Farrell cohomology Ĥ^{∗}(G, M) is isomorphic to the equivariant Tate-Farrell cohomology of the Quillen complex of G with coefficients in M.

Corollary 2.4. If G has p-rank 1 (i.e. no subgroups isomorphic to ℤ/p × ℤ/p) and only finitely many conjugacy classes of subgroups isomorphic to ℤ/p with representatives C_{1}, …, C_{n} then Ĥ^{∗}(G, M) ≅ ⊕_{i=1}^{n}Ĥ^{∗}(N_{G}(C_{i}), M) for any M in C_{R}(G) or D_{R}(G).

A similar result for M = k also appears in [7].

For M, N ∈ C_{R}(G) we can also define Tate-Farrell Ext groups Êxt_{G}^{∗}(M, N). This allows us to define the stable category St_{R}(G) to have the same objects as C_{R}(G) but morphism groups Êxt_{G}^{0}(M, N). We write ≃ for isomorphism in the stable category.

There is another description. We define the Heller translate Ω on C_{R}(G) by the short exact sequence ΩM → P_{M} → M, where P_{M} denotes the projective cover of M. We also define \underline{Hom}_{G}(M, N) to be the quotient of Hom_{C_{R}(G)}(M, N) by the submodule of all homomorphisms that factor through a projective module. Then Êxt_{G}^{r}(M, N) ≅ \underrightarrow{lim}_{i} \underline{Hom}_{G}(Ω^{r+i}M, Ω^{i}N). In fact we only need to take i ≥ vcd G.

For the basic properties of the stable category see [1] for finite groups and [2] for infinite abstract groups. In particular, it is a triangulated category with the inverse of Ω as translation and the exact triangles coming from short exact sequences in C_{R}(G).

The next statement is basic to our approach, although it is just a corollary of Yoneda’s Lemma.

Lemma 2.5. If f : A → B induces an isomorphism f^{∗} : Êxt_{G}^{0}(B, M) → Êxt_{G}^{0}(A, M) for all M ∈ C_{R}(G) then f is an isomorphism in the stable category.

Definition 2.6. A module M ∈ C_{R}(G) is cofibrant if it is projective on restriction to some open subgroup of G.

In fact, if M is cofibrant then it is projective on restriction to any p-torsion free subgroup.

Notice that Ω^{i}M is always cofibrant if i ≥ vcd G. If M and N are cofibrant then Êxt_{G}^{0}(M, N) ≅ \underline{Hom}_{G}(M, N).

The definition is taken from [2], as is the next lemma. As the terminology suggests, this is part of a the structure of a closed model category, but we do not need that here.

Lemma 2.7. If M ≃ N in St_{R}(G) and M and N are cofibrant then there exist projective modules P and Q such that M ⊕ P ≅ N ⊕ Q in C_{R}(G). If M and N are finitely generated then P and Q can be chosen to be finitely generated.

Proof. Let H ⊴_{o} G be open normal of finite cohomological dimension. The inclusion of the fixed points induces a map R → R[G/H], which is split over H. This induces a map M → R[G/H] ⊗ M ≅ Ind_{H}^{G} M, which is also split over H and where Q = Ind_{H}^{G} M is projective, and finitely generated if M is.

Consider the map M → Q ⊕ N, where the first component is the map constructed above and the second is a stable isomorphism. This map is split over H, so the cokernel, call it P, is cofibrant, and finitely generated if M and N are.

The long exact sequence for Êxt_{G}^{∗}(P, −) yields that 0 = Êxt_{G}^{0}(P, P) ≅ \underline{Hom}_{G}(P, P), so P is projective and the short exact sequence splits. □

3. The Calculation

We set R = ℤ̂_{3}, k = 𝔽_{3}. The Morava stabilizer group 𝕊_{2} at the prime 3 can be split as a product 𝕊_{2}^{1} × ℤ̂_{3}, where 𝕊_{2}^{1} is the kernel of the reduced norm. There is a natural action of the Galois group Gal = Gal(𝔽_{9}/𝔽_{3}), and we will consider the special extended Morava stabilizer group 𝔾_{2}^{1} = 𝕊_{2}^{1} ⋊ Gal.

Let S_{2}^{1} be the Sylow 3-subgroup of 𝕊_{2}^{1}. It is normal in 𝔾_{2}^{1} and 𝔾_{2}^{1} = S_{2}^{1} ⋊ SD_{16}, where SD_{16} is a subgroup isomorphic to the special dihedral group of order 16. In fact, if ϕ denotes the generator of Gal (of order 2) there is an element ω ∈ S_{2}^{1} of order 8 such that SD_{16} is generated by ϕ and ω. There is just one finite 3-subgroup, up to conjugation. It is cyclic of order 3 and we denote it by C_{3}. It is contained in a subgroup G_{24} of order 24, but there is no subgroup of order 48. We can, however, choose conjugacy class representatives so that SD_{16} ∩G_{24} = Q_{8}, a quaternion group of order 8 generated by ωϕ, which commutes with C_{3}, and ω^{2}, which does not. We refer to [5] for the details.

As a consequence, the simple modules in Cℤ̂_{3}(G) correspond to the simple modules for SD_{16} over 𝔽_{3}. In particular there is a character χ corresponding to the map SD_{16} → SD_{16}/Q_{8} ≅ {±1}, so χ(ϕ) = χ(ω) = −1. Define a module N_{1} by

0 → N_{1} → Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} → ℤ̂_{3} → 0,

where the right hand arrow is the natural augmentation.

Let S be a simple module and apply Ext_{𝔾_{2}^{1}}^{∗}(−, S). We obtain the long exact sequence

⋯ → Ext_{𝔾_{2}^{1}}^{∗}(ℤ̂_{3}, S) → Ext_{𝔾_{2}^{1}}^{∗}(Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3}, S) → Ext_{𝔾_{2}^{1}}^{∗}(N_{1}, S) → · · · .

The arrow on the left is just H^{∗}(𝔾_{2}^{1}, S) →^{res} H^{∗}(G_{24}, S), which is equivalent to H^{∗}(S_{2}^{1}, S)^{SD_{16}} →^{res}

H^{∗}(C_{3}, S)^{C_{8}} or (H^{∗}(S_{2}^{1})⊗S)^{SD_{16}} →^{res} (H^{∗}(C_{3})⊗S)^{C_{8}} or, more naturally, (H^{∗}(S_{2}^{1})⊗S)^{SD_{16}} →^{res}

((H^{∗}(C_{3})⊕H^{∗}(C′_{3}))⊗S)^{SD_{16}}, where C′_{3} is the conjugate of C_{3} by ω. (Where no coefficients for the cohomology are indicated they are just 𝔽_{3}.)

Now, for any finite 𝔽_{3}SD_{16}-module A, dim_{End(S)}(A⊗S)^{SD_{16}} ≅ dim_{End(S^{∗})} Hom_{SD_{16}}(S^{∗}, A) is just the multiplicity of the dual S^{∗} as a summand of A (A is completely reducible). So we are just decomposing the 𝔽_{3}SD_{16}-modules and identifying the map ρ : H^{∗}(S_{2}^{1}) →^{res} H^{∗}(C_{3}) ⊕ H^{∗}(C′_{3}). But this factors as

H^{∗}(S_{2}^{1}) →^{res} H^{∗}(C_{S_{2}^{1}}(C_{3})) ⊕ H^{∗}(C_{S_{2}^{1}}(C′_{3})) →^{res} H^{∗}(C_{3}) ⊕ H^{∗}(C′_{3}).

A standard calculation ([5, 7, 9]) shows that C_{S_{2}^{1}}(C_{3}) ≅ ℤ̂_{3} × C_{3}. Its cohomology is just H^{∗}(C_{S_{2}^{1}}(C_{3})) ≅ H^{∗}(ℤ_{3}) ⊗ H^{∗}(C_{3}) ≅ E(a_{1}) ⊗ (𝔽_{3}[y_{1}] ⊗ E(x_{1})) ≅ 𝔽_{3}[y_{1}] ⊗ E(x_{1}, a_{1}), where E denotes an exterior algebra, a_{1}, x_{1} are in degree 1 and y_{1} is in degree 2. The restriction to C_{3} just kills a_{1}. For C_{S_{2}^{1}}(C′_{3}) the result is similar, but we use the subscript 2 for the generators, which we take to be the images of those in the first case under conjugation by ω.

Henn [7] shows that the first of the maps above is injective. Its image is generated as an algebra by x_{1}, x_{2}, y_{1}, y_{2}, (x_{1}a_{1} − x_{2}a_{2}), y_{1}a_{1}, y_{2}a_{2}. The action of SD_{16} can now be calculated and is given in [5]:

ω_{∗}(x_{i}) = −(−1)^{i}x_{i+1}, ω_{∗}(y_{i}) = −(−1)^{i}y_{i+1}, ω_{∗}(a_{i}) = −(−1)^{i}a_{i+1},
ϕ_{∗}(x_{i}) = −x_{i+1}, ϕ_{∗}(y_{i}) = −y_{i+1}, ϕ_{∗}(a_{i}) = −a_{i+1},

(where the subscripts are taken modulo 2).

The map ρ is also explicitly calculated in [6].

From this we can read off that ρ is surjective, except in degree 0, where the cokernel is 𝔽_{3}(χ) as an SD_{16}-module. It is also injective in degrees 0 and 1. In degree 2 the kernel is generated by x_{1}a_{1} − x_{2}a_{2}, which gives a copy of 𝔽_{3}(χ) again. In degree 3 the kernel is generated by y_{1}a_{1} and y_{2}a_{2}, so consists of two simples: one trivial generated by y_{1}a_{1}+y_{2}a_{2} and a copy of 𝔽_{3}(χ) generated by y_{1}a_{1} − y_{2}a_{2}.

Thus the minimal projective resolution of N_{1} starts

⋯ → P_{𝔽_{3}} ⊕ P_{𝔽_{3}(χ)} → P_{𝔽_{3}(χ)} → P_{𝔽_{3}(χ)} → N_{1} → 0.

Now P_{𝔽_{3}(χ)} ≅ Ind_{SD_{16}}^{G} ℤ̂_{3}(χ), because the latter is projective and, for any simple S, Hom_{G}(Ind_{SD_{16}}^{G} ℤ̂_{3}(χ), S) ≅ Hom_{SD_{16}}(ℤ̂_{3}(χ), S), which is non-zero only for S ≅ 𝔽_{3}(χ) and then it has dimension 1. So if we define N_{3} = Ω^{2}N_{1} we have an exact sequence

0 → N_{3} → Ind_{SD_{16}}^{G} ℤ̂_{3}(χ) → Ind_{SD_{16}}^{G} ℤ̂_{3}(χ) → N_{1} → 0,

where N_{3} has projective cover P_{𝔽_{3}} ⊕ P_{𝔽_{3}(χ)}.

If we work stably we can obtain Ω^{2}N_{1} another way. Recall that C_{3} is the only cyclic subgroup of order 3 in 𝔾_{2}^{1} up to conjugacy. Write N = N_{𝔾_{2}^{1}}(C_{3}); because Q_{8} normalizes C_{3} it also normalizes C_{S_{2}^{1}}(C_{3}), and since the centralizer can be of index at most 2 in the normalizer we see that N ≅ C_{3} × ℤ̂_{3} ⋊ Q_{8}.

From 2.4 we see that res : Ĥ^{∗}(𝔾_{2}^{1}, M) → Ĥ^{∗}(N, M) is an isomorphism, or equivalently that the augmentation map ϵ : Ind_{N}^{𝔾_{2}^{1}} ℤ̂_{3} → ℤ̂_{3} induces an isomorphism Êxt_{𝔾_{2}^{1}}^{∗}(ℤ̂_{3}, M) →

Êxt_{𝔾_{2}^{1}}^{∗}(Ind_{N}^{𝔾_{2}^{1}} ℤ̂_{3}, M), for any M ∈ Cℤ̂_{3}(G). It follows from 2.5 that ϵ is a stable isomorphism.

So stably our complex starts Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} → Ind_{N}^{𝔾_{2}^{1}} ℤ̂_{3}, which is Ind_{N}^{𝔾_{2}^{1}} applied to the natural augmentation map Ind_{G_{24}}^{N} ℤ̂_{3} → ℤ̂_{3} over N.

But the subgroup D < G_{24} generated by C_{3} and ωϕ is normal in N, so N acts on Ind_{G_{24}}^{N} ℤ̂_{3} via its image N/D ≅ ℤ̂_{3} ⋊ C_{2}, the infinite virtually 3-adic dihedral group, so we can resolve to obtain

(†)  0 → Ind_{G_{24}}^{N} ℤ̂_{3}(θ) → Ind_{G_{24}}^{N} ℤ̂_{3} → ℤ̂_{3} → 0,

where θ : Q_{8} → {±1} is the character with θ(ωϕ) = 1 and θ(ω^{2}) = −1.

This can be seen systematically using cohomology, as before. More explicitly, the nonzero map on the left is determined by 1 ⊗ θ ↦ (g − g^{−1}) ⊗ 1, where g is a generator of the group ℤ̂_{3} and θ is considered as a basis element of ℤ̂_{3}(θ). The sequence is exact because on restriction to ℤ̂_{3} it is just a variation on the standard projective resolution for ℤ̂_{3}.

Similarly, since G_{24} has a quotient G_{24}/⟨ωϕ⟩ ≅ D_{6}, the dihedral group of order 6, we also have an exact sequence

0 → ℤ̂_{3} → Ind_{Q_{8}}^{G_{24}} ℤ̂_{3} → Ind_{Q_{8}}^{G_{24}} ℤ̂_{3}(θ) → ℤ̂_{3}(θ) → 0,

with middle map determined by 1 ⊗ 1 ↦ (c − c^{−1}) ⊗ θ where c is a generator of C_{3}. Inducing this to N gives

(‡)  0 → Ind_{G_{24}}^{N} ℤ̂_{3} → Ind_{Q_{8}}^{N} ℤ̂_{3} → Ind_{Q_{8}}^{N} ℤ̂_{3}(θ) → Ind_{G_{24}}^{N} ℤ̂_{3}(θ) → 0.

Now splice † and ‡ together at Ind_{G_{24}}^{N} ℤ̂_{3}(θ) and induce up to 𝔾_{2}^{1} to obtain

0 → Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} → Ind_{Q_{8}}^{𝔾_{2}^{1}} ℤ̂_{3} → Ind_{Q_{8}}^{𝔾_{2}^{1}} ℤ̂_{3}(θ) → Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} → Ind_{N}^{𝔾_{2}^{1}} ℤ̂_{3} → 0.

The second and third non-zero terms are projective, so stably Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} ≃ N_{3}. But Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} is cofibrant by construction and, on restriction to an open torsion free subgroup, N_{3} is a third syzygy hence also cofibrant, so, by 2.7, there are finitely generated projective modules P and Q such that Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} ⊕ P ≅ N_{3} ⊕ Q.

Let S be a simple ℤ̂_{3}[[𝔾_{2}^{1}]]-module (recall that these correspond to simple SD_{16}-modules). Then Hom_{𝔾_{2}^{1}}(Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3}, S) ≅ Hom_{G_{24}}(ℤ̂_{3}, S). For this to be non-zero we need Res_{G_{24}}^{𝔾_{2}^{1}} S ≅ 𝔽_{3}, so S must be either 𝔽_{3} or 𝔽_{3}(χ); in both cases the dimension of the Hom group is 1.

It follows that the projective cover of Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} is P_{𝔽_{3}} ⊕ P_{𝔽_{3}(χ)}. Now, taking projective covers in Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} ⊕ P ≅ N_{3} ⊕ Q, we obtain P_{𝔽_{3}} ⊕ P_{𝔽_{3}(χ)} ⊕ P ≅ P_{𝔽_{3}} ⊕ P_{𝔽_{3}(χ)} ⊕ Q, so P ≅ Q and thus Ind_{G_{24}}^{𝔾_{2}^{1}} ℤ̂_{3} ≅ N_{3}, by 2.1.

Remark. This construction generalizes to 𝔾_{p−1}^{1} for larger primes p. It is simpler to discuss if we restrict to the Sylow p subgroup. We now have N = C_{p} × ℤ̂_{p}^{p−2}. Since ℤ̂_{p}^{p−2} has cohomological dimension p − 2, we could take its projective resolution to the penultimate term and inflate to N. We then splice on a part induced from a partial projective resolution of ℤ̂_{3} over C_{p} that is long enough to make the last term cofibrant. It is not clear whether this has any significance in the homotopy theory.

Remark. The Tate-Farrell cohomology of 𝔾_{p−1}^{1} is easy to compute (see [9]). It is the lowdimensional cohomology that is difficult to calculate, but that is precisely what is needed to identify the projective modules in the complex. If we are satisfied with a complex with unknown projectives then the construction is much easier and only depends on the structure of N.

References

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[3] Brown, K.S., Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, Berlin 1982.

[4] Brumer, A., Pseudocompact algebras, profinite groups and class formations, Jour. Algebra 4 (1966), 442-470.

[5] Goerss, P., Henn, H.-W., Mahowald, M. and Rezk, C., A resolution of the K(2)-local sphere at the prime 3, preprint http://hopf.math.purdue.edu .

[6] Gorbounov, V., Siegel, S.F. and Symonds, P., The cohomology of the Morava stabilizer group S_{2} at the prime 3, Proc. Amer. Math. Soc. 126 (1998), 933-941.

[7] Henn, H.W., Centralizers of elementary abelian p-groups and mod-p cohomology of profinite groups, Duke Math. Jour. 91 (1998) 561-585.

[8] Scheiderer, C., Farrell cohomology and Brown theorems for profinite groups, Manuscripta Math. 91 (1996), 247-281.

[9] Symonds, P., The Tate-Farrell cohomology of the Morava stabilizer group S_{p−1} with coefficients in E_{p−1}, Contemporary Math. 346 (2004).

[10] Symonds, P., Permutation complexes for profinite groups, preprint, http://www.ma.umist.ac.uk/pas/preprints .

[11] Symonds, P. and Weigel, T., Cohomology of p-adic analytic groups, in New Horizons in pro-p groups, du Sautoy, Segal, Shalev, eds., Progress in Math. 184 (2000), Birkhäuser, Boston.

School of Mathematics, University of Manchester, P.O. Box 88, Manchester M60 1QD, England

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