CENTRALIZERS OF ELEMENTARY ABELIAN p - SUBGROUPS AND MOD - p - COHOMOLOGY OF PROFINITE GROUPS

by

Hans–Werner Henn

1. Introduction

\underline{1.1} Let G be a profinite group and p be a fixed prime. In this paper we will be concerned with H_{c}^{∗}(G; 𝔽_{p}), the continuous cohomology of G with coefficients in the trivial module 𝔽_{p}. We will abbreviate H_{c}^{∗}(G; 𝔽_{p}) by H^{∗}(G; 𝔽_{p}), or simply by H^{∗}G if p is understood from the context. We recall that if G is the (inverse) limit of finite groups G_{i} then H^{∗}G = colim H^{∗}G_{i}. Throughout this paper we will assume that H^{∗}G is finitely generated as 𝔽_{p} - algebra. By work of Lazard [La] it is known that this holds for many interesting groups, for example for profinite p - analytic groups like GL(n, ℤ_{p}), the general linear groups over the p - adic integers. In case H^{∗}G is finitely generated as 𝔽_{p} - algebra Quillen has shown [Q1] that there are only finitely many conjugacy classes of elementary abelian p - subgroups of G (i.e. groups isomorphic to (ℤ/p)^{n} for some natural number n). In other words, the following category A(G) is equivalent to a finite category: objects of A(G) are all elementary abelian p - subgroups of G, and if E_{1} and E_{2} are elementary abelian p - subgroups of G, then the set of morphisms from E_{1} to E_{2} in A(G) consists precisely of those homomorphisms α : E_{1} ⟶ E_{2} of abelian groups for which there exists an element g ∈ G with α(e) = geg^{−1} ∀e ∈ E_{1}. The category A(G) plays an important role both in Quillen’s results and in the work presented here. This category entered into Quillen’s work as follows. The assignment E ↦ H^{∗}E extends to a functor from the opposite category A(G)^{op} to graded 𝔽_{p} - algebras and the restriction homomorphisms H^{∗}G ⟶ H^{∗}E (for E running through the elementary abelian p - subgroups of G) induce a canonical map of algebras q : H^{∗}G ⟶ lim_{A(G)^{op}} H^{∗}E.

THEOREM 1.2 [Q1]. Let G be a profinite group and assume H^{∗}G is a finitely generated 𝔽_{p} - algebra. Then the canonical map q : H^{∗}G ⟶ lim_{A(G)^{op}} H^{∗}E is an F - isomorphism, in other words q has the following properties.

• If x ∈ Ker q, then x is nilpotent.

• If y ∈ lim_{A(G)^{op}} H^{∗}E then there exists an integer n with y^{p^{n}} ∈ Im q.

\underline{1.3} In our main result we use the full subcategory A_{∗}(G) of A(G) whose objects are all elementary abelian p - subgroups except the trivial subgroup. The centralizer C_{G}(E) of an elementary abelian p - subgroup E is a closed subgroup and hence inherits a natural profinite structure from G. The assignment E ↦ H^{∗}C_{G}(E) extends to a functor from A_{∗}(G) to graded 𝔽_{p} - algebras and the restriction homomorphisms H^{∗}G ⟶ H^{∗}C_{G}(E) (for E running through the non-trivial elementary abelian p - subgroups of G) induce a canonical map ρ : H^{∗}G ⟶ lim_{A_{∗}(G)} H^{∗}C_{G}(E). Our main result reads as follows.

THEOREM 1.4. Let G be a profinite group and assume H^{∗}G is a finitely generated 𝔽_{p} - algebra. Then the canonical map ρ : H^{∗}G ⟶ lim_{A_{∗}(G)} H^{∗}C_{G}(E) has finite kernel and cokernel.

1.5. Remarks/Questions:

a) The map ρ of Theorem 1.4 is an actual isomorphism if G is a finite group or a compact Lie group (in the latter case H^{∗}G has to be interpreted as the mod p cohomology of the classifying space BG) provided G contains any elements of order p. This was first proved by Jackowski and McClure [JM] and then reproved and extended to certain classes of unstable algebras over the Steenrod algebra by Dwyer and Wilkerson [DW]. One’s first reaction might be that because the continuous cohomology of a profinite group is the colimit of the cohomology of finite groups, the profinite case should be a direct consequence of the finite case by passing to appropriate (co-)limits. However, one gets confronted with a subtle problem of interchanging limits and colimits, and this has the effect that ρ need not be an isomorphism for a profinite group. In fact, our proof requires a different approach: in [He] we investigated an appropriately defined map ρ for any unstable algebra K over the Steenrod algebra which is finitely generated as 𝔽_{p} - algebra and showed that this map has always finite kernel and cokernel (see Theorem 2.5 below). In section 2 we explain this algebraic result and show how Theorem 1.4 can be deduced from it. We emphasize that we do not know any proof of Theorem 1.4 which does not use the Steenrod algebra, in particular Lannes’ T - functor in a crucial way.

b) Obviously Theorem 1.4 gives more precise information than Theorem 1.2, but on the other hand its applicability is more limited. For example, the major reason for working with A_{∗}(G) instead of A(G) was to avoid the appearance of H^{∗}G in the limit. However, if G contains central elements of order p, then H^{∗}G does appear in the limit anyway, and Theorem 1.4 is not very useful. In other cases the functor E ↦ H^{∗}C_{G}(E) may be too complicated to be evaluated. However, in these cases Theorem 1.4 may still be of some theoretical interest (see 2.10 and 2.11 below for examples).

c) Theorem 1.4 says in particular that there is a least integer d such that ρ is an isomorphism in cohomological degrees > d. It would be interesting to have effective upper bounds for d in group theoretical terms. For profinite p - analytic groups the work of Lazard [La] suggests the dimension of such a group as a candidate for an upper bound for the number d. In fact, if such a group does not contain any elements of order p, as well as in the examples discussed in section 4 and 5 below, this gives actually a correct upper bound.

\underline{1.6} If G does not contain elements of order p then the target of ρ is the trivial algebra and Theorem 1.4 says that H^{∗}G is a finite algebra. Of course, this could have also been directly deduced from Theorem 1.2.

However, Theorem 1.4 may already be quite interesting in case the p - rank of G is equal to one. We recall that the p - rank r_{p}(G) of G is defined as the supremum of all natural numbers n such that G contains an elementary abelian p - subgroup E of rank n, i.e. E ≅ (ℤ/p)^{n}. In case r_{p}(G) = 1 the inverse limit simplifies substantially because A_{∗}(G) is equivalent to the following discrete category: objects are in one to one correspondence with conjugacy classes of subgroups E ≅ ℤ/p; the only morphisms of this category are automorphisms and the automorphism group of an object E identifies with N_{G}(E)/C_{G}(E) with N_{G}(E) denoting the normalizer of E in G. In particular we get the following corollary.

COROLLARY 1.7. Let G be a profinite group and assume H^{∗}G is a finitely generated 𝔽_{p} - algebra and r_{p}(G) = 1. Then the restriction maps induce a map

ρ : H^{∗}G ⟶ ∏_{(E)}(H^{∗}C_{G}(E))^{N_{G}(E)}

with finite kernel and cokernel. (Here the product is taken over conjugacy classes of elementary abelian p - subgroups of rank 1.)

Note that the group N_{G}(E)/C_{G}(E) is of order prime to p if E is of rank 1. Therefore the invariants (H^{∗}C_{G}(E))^{N_{G}(E)} are isomorphic to H^{∗}N_{G}(E) and Corollary 1.7 can be considered as a “profinite analogue” of a result of Ken Brown on the Farell cohomology of discrete groups of p - rank 1 [B]. In this case the number d introduced in 1.5c) above corresponds to the virtual cohomological dimension of G.

\underline{1.8.} As mentioned above section 2 will be concerned with the proof of Theorem 1.4 and related results. In sections 3 and 4 we will study certain subgroups of the group of units in p - adic division algebras and in section 5 we will touch upon the general linear group over the p - adic integers.

To get more explicit, we fix a prime p and a natural number n. Consider 𝔻_{n}, the division algebra with invariant \frac{1}{n} over the field of p - adic numbers ℚ_{p} and O_{n}, the maximal compact subring of 𝔻_{n}. O_{n} is a local ring and reducing modulo its maximal ideal gives a homomorphism from O_{n} to the finite field 𝔽_{q} with q = p^{n}. Let O_{n}^{×} denote the units of O_{n}, and S_{n} denote the kernel of the map O_{n}^{×} ⟶ 𝔽_{q}^{×}. In stable homotopy theory these groups are known as Morava stabilizer groups and their cohomology is known to play a central role in the chromatic theory of stable homotopy (see [M], [Ra 2,3], [D], [HG] for example). It is well known that r_{p}(S_{n}) = r_{p}(O_{n}^{×}) ≤ 1 and equality holds iff n ≡ 0 mod(p − 1). Our results give new insight if n ≡ 0 mod(p − 1). Using standard facts about division algebras we determine the categories A_{∗}(O_{n}^{×}) and A_{∗}(S_{n}) for n ≡ 0 mod(p−1) and describe the structure of the centralizers of the elementary abelian p - subgroups (Theorem 3.2.2). Furthermore for n = p−1 the centralizers turn out to be abelian and we can compute the target of ρ explicitly, hence H^{∗}S_{n} up to finite ambiguity. In particular we obtain in 3.3 the following result in which E(−) denotes an exterior algebra over 𝔽_{p} on the specified elements. The elements y_{i} are of degree 2 and the elements x_{i} and a_{i,j} are of degree 1. For more details about the definition of these elements the reader is referred to section 3.3 and 3.4.

THEOREM 1.9. Let p be an odd prime, n = p − 1 and s = \frac{p^{n}−1}{(p−1)^{2}} . Then there is a homomorphism

ρ : H^{∗}S_{n} ⟶ ∏_{i=1}^{s} 𝔽_{p}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i,1}, ⋯, a_{i,n})

with finite kernel and cokernel.

We remark that previously the mod - p - cohomology of S_{n} was computed by Ravenel [Ra 1] in the following cases: H^{1}S_{n} and H^{2}S_{n} for all n and p, all of H^{∗}S_{n} if either n ≤ 2 and p arbitrary or if n = 3 and p ≥ 5. So the only overlap between Ravenel’s computation and Theorem 1.9 occurs for p = 3 and n = 2 [Ra 1, Theorem 3.3]. In section 4 we use Theorem 1.9 together with some more detailed group theoretical analysis of S_{2} to give an independent computation of H^{∗}S_{n} if p = 3 and n = 2. In this case we find that ρ is injective and we use this to describe H^{∗}S_{2} as an explicit subalgebra of ∏_{i=1}^{2} 𝔽_{3}[y_{i}]⊗E(x_{i})⊗E(a_{i,1}, a_{i,2}) (Theorem 4.2). The multiplicative structure of the result derived here differs from that of Ravenel although additively the two results agree. Ravenel has informed me that he now believes that there is a mistake in his calculation. Finally we remark that Gorbounov, Siegel and Symonds [GSS] have independently and with very different methods confirmed the calculation in Theorem 4.2. The calculations of H^{∗}S_{2} for primes p > 3 have been used by Shimomura and Yabe [SY] to determine the stable homotopy groups of L_{2}S_{p}^{0}, the second stage in the chromatic tower of the p - local sphere. The computation of H^{∗}S_{2} at the prime 3 will be relevant for understanding L_{2}S_{p}^{0} if p = 3 and for this reason we have decided to give a rather detailed presentation in section 4. In fact, Shimomura [S] has already used the corrected computation of H^{∗}S_{2} to compute the homotopy groups of the L_{2} - localization of the Toda - Smith complex V (1) at the prime 3 up to a certain ambiguity; this ambiguity will be settled in joint work with Mahowald [HM] using the approach towards H^{∗}S_{2} via centralizers of elementary abelian p - subgroups that we introduce in this paper.

\underline{1.10.} Our second application concerns the mod - p - cohomology of the general linear groups GL(n, ℤ_{p}). The following result should be compared with Ash’s computations [A] of the Farrell cohomology of GL(n, ℤ). As above the element y has degree 2 while all other elements are of degree 1.

THEOREM 1.10. Let p be an odd prime and n = p − 1. Then there is a homomorphism

ρ : H^{∗}GL(n, ℤ_{p}) ⟶ (𝔽_{p}[y] ⊗ E(x) ⊗ E(a_{1}, ⋯, a_{n}))^{ℤ/n}

with finite kernel and cokernel. (Here (−)^{ℤ/n} denotes the invariants with respect to the following action of ℤ/n by algebra homomorphisms. After choosing a suitable isomorphism τ : ℤ/n ⟶ 𝔽_{p}^{×} this action is given by gy = τ(g)y, gx = τ(g)x and ga_{j} = τ(g)^{j}a_{j} for j = 1, …, n if g ∈ ℤ/n.)

This result looks very similar to 1.9. In fact, the similarity becomes even stronger if one compares (for n = p − 1) the groups O_{n}^{×} and GL(n, ℤ_{p}). In this case the targets of the two maps ρ agree (cf. Theorem 3.2.2 and Theorem 5.2). As in the case of S_{n} we give the complete computation for p = 3 and n = 2 (Proposition 5.5).

\underline{Acknowledgements:} The author was partially supported by a Heisenberg fellowship of the Deutsche Forschungsgemeinschaft. The origin of this paper can be traced back to a one month visit to Aarhus in the fall of 1990. This work was then pushed further during a three months visit at the Mittag Leffler institute in the fall of 1993. I would like to thank the staff at both institutions for providing such pleasant and stimulating working conditions. It is a pleasure to express special thanks to Bob Oliver on both occassions.

2. The proof of Theorem 1.4 and related results

As already indicated in the introduction, Theorem 1.4 is deduced from a general result about certain unstable algebras over the Steenrod algebra A [He]. We will begin by recalling some facts about Lannes’ T functor which are necessary to explain the main algebraic result of [He] and to deduce Theorem 1.4 from it.

\underline{2.1.} Let U resp. K denote the category of unstable modules resp. unstable algebras over the mod p Steenrod algebra A (see [L1]). The cohomology of a space is an unstable algebra, in particular the cohomology of any finite group and then also the cohomology of any profinite group is such an algebra. The Steenrod algebra is actually a Hopf algebra and its diagonal gives rise to a tensor product on the categories U resp. K. Now let V be an elementary abelian p - group with mod p cohomology H^{∗}V. Lannes [L1] has introduced the functor T_{V} : U ⟶ U. It is left adjoint to tensoring with H^{∗}V, i.e Hom_{U}(T_{V} M, N) ≅ Hom_{U}(M, H^{∗}V ⊗ N) for all unstable modules M and N. T_{V} has a number of remarkable properties. In particular, T_{V} lifts to a functor from K to itself and the adjunction relation continues to hold in K: Hom_{K}(T_{V} K, L) ≅ Hom_{K}(K, H^{∗}V ⊗ L) for all unstable algebras K and L.

\underline{2.2.} Now let G be a finite group. The following computation of T_{V} H^{∗}G in [L2] (for a more accessible reference see also [L1,3.4]) is quite crucial for the proof of Theorem 1.4. Denote by Rep(V, G) the set of G - conjugacy classes of homomorphisms from V to G. For each conjugacy class choose a representative φ and denote the centralizer of Im φ in G by C_{G}(φ). The homomorphism c_{φ} : V × C_{G}(φ) → G, (v, g) ↦ vφ(g) induces a map of unstable algebras c_{φ}^{∗} : H^{∗}G ⟶ H^{∗}V ⊗ H^{∗}C_{G}(φ) which is adjoint to a map of unstable algebras ad(c_{φ}^{∗}) : T_{V} H^{∗}G ⟶ H^{∗}C_{G}(φ).

THEOREM 2.2 [L2]. The homomorphism of unstable algebras

T_{V} H^{∗}G ⟶ ∏_{φ∈Rep(V,G)} H^{∗}C_{G}(φ)

whose components are the maps ad(c_{φ}^{∗}) is an isomorphism for each finite group G.

Note that the theorem shows in particular that the natural map from Rep(V, G) to Hom_{K}(H^{∗}G, H^{∗}V), which sends φ to its induced map φ^{∗}, is a bijection.

We also see that the terms in the inverse limit occuring in Theorem 1.4 appear in the computation of T_{V} H^{∗}G. Dwyer and Wilkerson [DW] noticed that this allows a purely algebraic approach to the map ρ of Theorem 1.4 which makes sense for a much larger class of unstable algebras. In order to explain this we need some more preparations (see [DW] and [HLS, I.4 and I.5]).

\underline{2.3.} To an unstable algebra K we associate a category R(K) as follows. Its objects are the morphisms of unstable algebras φ : K → H^{∗}V, V an elementary abelian p - group, for which H^{∗}V becomes a finitely generated K - module via φ; sometimes it will be convenient to denote such an object by the pair (V, φ). Then the set of morphisms from (V_{1}, φ_{1}) to (V_{2}, φ_{2}) are all homomorphisms V_{1} ⟶^{α} V_{2} of abelian groups such that φ_{1} = α^{∗}φ_{2}. If K is noetherian the opposite of this category was first investigated by Rector [Rc]. The full subcategory of R(K) having as objects all (V, φ) with V non-trivial will be denoted by R_{∗}(K) If K = H^{∗}BG then R(K) is equivalent to Quillen’s category A(G) and R_{∗}(K) is equivalent to A_{∗}(G). In fact, in this case the computation of Hom_{K}(H^{∗}G, H^{∗}V) (see 2.2 above) can be used to identify the objects of R(K) with the monomorphic representations of elementary abelian p - groups in G and an equivalence between R(K) and a skeleton of A(G) is induced by associating to a homomorphism φ : V ⟶ G the unique object in the skeleton of A(G) which is isomorphic to the image of φ (see [HLS, I.5.3]).

\underline{2.4.} Now consider the unstable algebra T_{V} K. For a morphisn φ : K ⟶ H^{∗}V we obtain a connected component T_{V} (K; φ) of T_{V} K: it is defined as T_{V} (K; φ) := 𝔽_{p}(φ) ⊗_{T_{V}^{0} K} T_{V} K where 𝔽_{p}(φ) denotes 𝔽_{p} considered as a module over T_{V}^{0} K via the adjoint of φ. More generally, we can consider the category K − U whose objects are unstable A - modules M with A - linear K - module structure maps K ⊗ M ⟶ M, and whose morphisms are all A - linear maps which are also K - linear. The full subcategory of K − U consisting of those objects which are finitely generated as K - modules is denoted by K_{fg} − U. If M is in K − U then T_{V} M is in T_{V} K − U and one can define components T_{V} (M; φ) := 𝔽_{p}(φ) ⊗_{T_{V}^{0} K} T_{V} M which are modules over the corresponding components T_{V} (K; φ). From the adjoint of the identity of T_{V} K we obtain canonical algebra morphisms ρ_{K,(V,φ)} from K to T_{V} (K; φ), hence T_{V} (M; φ) can be considered as a K - module and the assignment (V, φ) ↦ T_{V} (M; φ) gives rise to a functor from R(K) to K − U. Furthermore the adjoint of the identity of T_{V} M gives rise to maps ρ_{M,(V,φ)} from M to T_{V} (M; φ) which are all K - linear, and we obtain a natural transformation between the constant functor from R(K) to K − U with value M and the functor (V, φ) ↦ T_{V} (M; φ). If G is a finite group and K = M = H^{∗}G then Theorem 2.2 implies that the functor (V, φ) ↦ T_{V} (M; φ) corresponds via the equivalence between R(K) and A(G) to the functor E ↦ H^{∗}C_{G}(E) that appears in Theorem 1.4. Furthermore, the maps ρ_{M,(V,φ)} correspond to the restriction maps H^{∗}G ⟶ H^{∗}C_{G}(φ).

Now we are ready to formulate the general algebraic theorem from which Theorem 1.4 is deduced.

THEOREM 2.5 [He, Cor. 3.10]. Let K be a noetherian unstable algebra and let M be an object in K_{fg} − U. Then the maps ρ_{M,(V,φ)} induce a map

ρ : M ⟶ lim_{R_{∗}(K)} T_{V} (M; φ)

which has finite kernel and cokernel.

In fact, it is shown in [He] that this map is localization away from finite objects in K_{fg} − U. Theorem 2.5 together with Theorem 2.2 yield Theorem 1.4 in the case of a finite group (in which case ρ is even an isomorphism by [JM] and [DW]). To prove Theorem 1.4 for profinite groups G for which H^{∗}G is a finitely generated 𝔽_{p} - algebra, it suffices to extend Theorem 2.2 to this setting. We recall that a profinite group G is given as the (inverse) limit lim_{i} G_{i} of finite groups G_{i} along a directed partially ordered set (I, ≤) which we think of as a category (denoted I for simplicity) in the usual way. Then H^{∗}G can be identified with colim_{i} H^{∗}G_{i}. Here is the extension of Theorem 2.2.

THEOREM 2.6. Assume G is a profinite group such that H^{∗}G is a finitely generated 𝔽_{p} - algebra. Then the homomorphism of unstable algebras

T_{V} H^{∗}G ⟶ ∏_{φ∈Rep(V,G)} H^{∗}C_{G}(φ)

whose components are the maps ad(c_{φ}^{∗}) is an isomorphism.

In this result the set Rep(V, G) of representations is defined as before, i.e. the topology on G does not play any role. However, the centralizer of an elementary abelian p - subgroup of a profinite group inherits a natural structure of a profinite group and this structure is used in 2.6.

\underline{Proof of 2.6:} We deduce Theorem 2.6 from Theorem 2.2. For this we note that T_{V} commutes with arbitrary colimits, i.e. we get

T_{V} H^{∗}G = colim_{i} T_{V} H^{∗}G_{i} ≅ colim_{i} ∏_{φ∈Rep(V,G_{i})} H^{∗}C_{G_{i}}(φ) .

Let us look more carefully at the maps in the inverse system. For a morphism λ : i → j in I we use the same letter for the associated maps from G_{j} to G_{i} and from C_{G_{j}}(φ) to C_{G_{i}}(λφ) (with φ ∈ Rep(V, G_{j})). If we identify T_{V} H^{∗}G_{i} as in 2.2 then the map T_{V} λ^{∗} : T_{V} H^{∗}G_{i} ⟶ T_{V} H^{∗}G_{j} is given as follows: the φ - th component of it (φ ∈ Rep(V, G_{j})) sends the family {x_{φ′}} ∈ ∏_{φ′∈Rep(V,G_{i})} H^{∗}C_{G_{i}}(φ′) to the element λ^{∗}(x_{λφ}) ∈ H^{∗}C_{G_{j}}(φ). Here λ^{∗} is the induced map from H^{∗}C_{G_{i}}(λφ) to H^{∗}C_{G_{j}}(φ). In fact, in the same way we get maps T_{V} π_{i}^{∗} from ∏_{φ′∈Rep(V,G_{i})} H^{∗}C_{G_{i}}(φ′) to ∏_{φ∈Rep(V,G)} H^{∗}C_{G}(φ) if π_{i} denotes the canonical map from G to G_{i}. These maps fit together and thus give a map

colim_{i} ∏_{φ′∈Rep(V,G_{i})} H^{∗}C_{G_{i}}(φ′) ⟶ ∏_{φ∈Rep(V,G)} H^{∗}C_{G}(φ)

which we denote T_{V} π^{∗} by abuse of notation and which we claim to be an isomorphism. In order to show this we need the following lemmas.

LEMMA 2.7. Let G = lim G_{i} be any profinite group.

a) Then the maps π_{i} : G ⟶ G_{i} induce a bijection Rep(V, G) ⟶ lim_{i} Rep(V, G_{i}) of profinite sets for each elementary abelian p - group V.

b) For any homomorphism φ from an elementary abelian p - group V into G the maps π_{i} : G ⟶ G_{i} induce an isomorphism C_{G}(φ) ⟶ lim_{i} C_{G_{i}}(π_{i}φ) of profinite groups.

LEMMA 2.8. Let G be any profinite group for which H^{∗}G is a finitely generated 𝔽_{p} - algebra. Then the set Rep(V, G) is finite for each elementary abelian p - group V.

We postpone the proofs of 2.7 and 2.8 and continue with the proof of 2.6. First we show that T_{V} π^{∗} is onto. So let x = {x_{φ}} ∈ ∏_{φ∈Rep(V,G)} H^{∗}C_{G}(φ) be given. For each φ ∈ Rep(V, G) we find by 2.7.b) an object i = i(φ) of I and an element y_{i} ∈ H^{∗}C_{G_{i}}(π_{i}φ) such that x_{φ} = (π_{i})^{∗}y_{i}. By 2.8 there are only finitely many φ and therefore we can assume that i is independent of φ. Furthermore, by 2.7.a) and 2.8 we can choose i such that in addition the natural map Rep(V, G) → Rep(V, G_{i}) is injective. For such an i and any φ′ ∈ Rep(V, G_{i}) let z_{φ′} ∈ H^{∗}C_{G_{i}}(φ′) be equal to y_{i(φ)} if φ′ = π_{i}φ for some (necessarily unique) φ, and choose arbitrarily elements z_{φ′} if there is no φ such that φ′ = π_{i}φ. Then the family z = {z_{φ′}} ∈ ∏_{φ′∈Rep(V,G_{i})} H^{∗}C_{G_{i}}(φ′) satisfies T_{V} π_{i}^{∗}(z) = x, and hence T_{V} π^{∗} is onto. To see that T_{V} π^{∗} is mono we show that for any element z = {z_{φ′}} ∈ ∏_{φ′∈Rep(V,G_{i})} H^{∗}C_{G_{i}}(φ′) with T_{V} π^{∗}(z) = 0 there is μ : i ⟶ j in I such that T_{V} μ^{∗}(z) = 0. Now T_{V} π^{∗}(z) = 0 implies by 2.7.b) that for each φ′ there is λ_{φ′} : i ⟶ j(φ′) such that for each lift φ of φ′ to Rep(V, G_{j(φ′)}) which further lifts to Rep(V, G) we get (λ_{φ′})^{∗}(z_{φ′}) = 0 in H^{∗}C_{G_{j(φ′)}}(φ). As there are only finitely many φ′ we can choose a common λ : i ⟶ j such that λ^{∗}(z_{φ′}) = 0 in H^{∗}C_{G_{j}}(φ) for each φ′ and each lift φ of φ′ to Rep(V, G_{j}) which further lifts to Rep(V, G). Furthermore by 2.7.a) we can find λ′ : j ⟶ j′ such that each element φ ∈ Rep(V, G_{j}) which does not lift to Rep(V, G) does not lift to Rep(V, G_{j′}) either. If μ = λ′λ : i ⟶ j′ then it is clear that T_{V} μ^{∗}(z) = 0.

\underline{Proof of 2.7:} The proof is an exercise in elementary point set topology. We sketch part (a) and leave part (b) to the reader. Let us first show surjectivity. So assume we have a compatible family of elements φ_{i} ∈ Rep(V, G_{i}). For a subset J of I let Rep_{J} be the set of families of homomorphisms {ϕ_{i}} ∈ ∏_{i∈I} Hom(V, G_{i}) such that the conjugacy class of ϕ_{i} is equal to φ_{i} whenever i ∈ J and such that the ϕ_{i} are compatible as long as i ∈ J. We have to show that Rep_{I} is non-empty. Because any two elements of I have an upper bound it is clear that Rep_{J} is non-empty whenever J is finite. Furthermore Rep_{J} is easily seen to be closed for any J. Now we use that the intersection of closed sets in a compact space is non-empty if every finite intersection is non-empty and we are done. Next assume that we have two elements φ and φ′ in Rep(V, G) represented by homomorphisms ϕ and ϕ′ such that π_{i}ϕ and π_{i}ϕ′ are conjugate for each i, i.e. there is an element g_{i} ∈ G_{i} with π_{i}ϕ(v) = g_{i}π_{i}ϕ′(v)g_{i}^{−1} for all v ∈ V. For injectivity in (a) it is enough to show that the family g_{i} can be chosen to be compatible. Again this is easy for any finite subset of I, and the general case follows again from the fact that an intersection of closed sets in a compact space is non-empty if every finite intersection is non-empty.

\underline{Proof of 2.8:} For any profinite G we have by 2.2 and 2.7.a)

Rep(V, G) ≅ lim_{i} Rep(V, G_{i}) ≅ lim_{i} Hom_{K}(H^{∗}G_{i}, H^{∗}V) .

Furthermore we can identify lim_{i} Hom_{K}(H^{∗}G_{i}, H^{∗}V) with Hom_{K}(H^{∗}G, H^{∗}V). Finally, if H^{∗}G is a finitely generated 𝔽_{p} - algebra, then the set Hom_{K}(H^{∗}G, H^{∗}V) is clearly finite and hence we are done.

\underline{2.9.} We would like to point out that Theorem 2.2 can be used to derive numerous qualitative results on cohomology of finite groups such as detection results, information on nilpotent elements in H^{∗}G or characterizations of the “support of elements” in H^{∗}G (cf. [HLS, I.5], [CH]). Because of Theorem 2.6 all these results have anologues for continuous cohomology of profinite groups G as long as H^{∗}G is a finitely generated 𝔽_{p} - algebra. We list here only the following result which is a special case of the profinite analogue of Theorem 2 of [CH].

PROPOSITION 2.10. Let G be profinite with p - rank one and assume H^{∗}G is finitely generated as 𝔽_{p} - algebra. Then the map

ρ : H^{∗}G ⟶ ∏_{(E)}(H^{∗}C_{G}(E))^{N_{G}(E)}

(cf. Corollary 1.7) is a monomorphism if and only if H^{∗}G is free over a polynomial subalgebra of H^{∗}G with one generator.

\underline{Proof:} Assume that H^{∗}G is free over a polynomial subalgebra on one generator. Then the same is true for any non-trivial ideal in H^{∗}G, in particular for the kernel of the map ρ. However, by Corollary 1.7 this ideal is finite, hence it must be trivial.

Conversely, assume that ρ is a monomorphism. Take any element x in H^{∗}G which restricts to a non-nilpotent element on all non-trivial elementary abelian p subgroups of rank 1. Such an x exists by Theorem 1.2. We can consider the cohomology of H^{∗}C_{G}(E) as a module over the polynomial subalgebra of H^{∗}G generated by x (again via a restriction homomorphism). Now the proof of Theorem 1.1 in [BC] (see also Remark 2.3 in the same paper) shows that H^{∗}C_{G}(E) is free over this polynomial subalgebra. By assumption H^{∗}G is a submodule of the free module ∏_{(E)} H^{∗}C_{G}(E) and hence it is also free.

\underline{2.11.} Finally we want to point out that Theorem 1.4 can also be used to get information on cohomology with non-trivial coefficients. For example, assume M is a finite continuous G - module with a composition series for which all successive subquotients are trivial modules. (Such composition series exist always if G is a pro - p - group.) Furthermore assume that the p - rank of G is one and ρ : H^{∗}G ⟶ ∏_{(E)}(H^{∗}C_{G}(E))^{N_{G}(E)} is an isomorphism in degrees > d. Then playing with the long exact sequences in cohomology associated to short exact sequences of coefficient modules shows that the map ρ : H^{∗}(G; M) ⟶ ∏_{(E)}(H^{∗}(C_{G}(E); M))^{N_{G}(E)} is an isomorphism in all degrees > d + 1.

3. The case of the stabilizer groups

\underline{3.1.} In this section we will apply our general results to certain subgroups of p - adic division algebras which play an important role in stable homotopy theory. We begin by recalling the definition and basic properties of these groups. The reader is referred to [Rn, Chap. 3,7] and [Ha, 20.2.16, 23.1.4] for background information on division algebras.

\underline{3.1.1.} Let p be a prime. For each integer n let 𝕎_{n} be the ring of Witt vectors of the finite field 𝔽_{q} with q = p^{n} elements and let σ : 𝕎_{n} → 𝕎_{n}, w ↦ w^{σ} be the lift of the Frobenius automorphism x ↦ x^{p} on 𝔽_{q}. Adjoin an element S to 𝕎_{n} subject to the relations S^{n} = p, Sw = w^{σ}S for each w ∈ 𝕎_{n}. The resulting non-commutative ring will be denoted by O_{n}. It is the maximal order in the central division algebra 𝔻_{n} over the field ℚ_{p} of rational numbers with invariant 1/n and is a free module over 𝕎_{n} of rank n with generators the elements S^{i}, 0 ≤ i < n. An important property of 𝔻_{n} that we will use below is that the degree (over ℚ_{p}) of each commutative subfield of 𝔻_{n} divides n, and each extension of ℚ_{p} whose degree divides n can be embedded as commutative subfield of 𝔻_{n}. We recall that O_{n} can be identified with the endomorphism ring of a certain formal group law over 𝔽_{q} of height n [Ha, Theorem 20.2.13]. Its group of units O_{n}^{×} is often called the n - th (full) Morava stabilizer group. The element S generates a two sided maximal ideal 𝔪 in O_{n} with quotient O_{n}/𝔪 = 𝔽_{q}. The kernel of the resulting epimorphism of groups O_{n}^{×} ⟶ (𝔽_{q})^{×} will be denoted by S_{n} and is also called the (strict) Morava stabilizer group; it can be identified with the group of strict automorphisms of the same height n formal group law over 𝔽_{q}.

\underline{3.1.2.} The groups O_{n}^{×} and S_{n} have natural profinite structures which can be described as follows. The valuation v on ℚ_{p} (normalized such that v(p) = 1) extends uniquely to a valuation on 𝔻_{n} such that v(S) = \frac{1}{n} and O_{n} = \{x ∈ 𝔻_{n}|v(x) ≥ 0\}. The two sided maximal ideal 𝔪 is given by 𝔪 = {x ∈ 𝔻_{n}|v(x) > 0}. The valuation gives subgroups

F_{i}S_{n} := {x ∈ S_{n}|v(1 − x) ≥ i} = {x ∈ S_{n}|x ≡ 1 mod S^{ni}}

for positive multiples i of \frac{1}{n} with

S_{n} = F_{\frac{1}{n}} S_{n} ⊃ F_{\frac{2}{n}} S_{n} ….

The intersection of all these subgroups is empty and S_{n} is complete with respect to this filtration, i.e. we have S_{n} = lim_{i} S_{n}/F_{i}S_{n}. Furthermore, we have canonical isomorphisms

F_{i}S_{n}/F_{i+ \frac{1}{n}} S_{n} ≅ 𝔽_{q}

induced by

x = 1 + aS^{i} ↦ ā .

Here a is an element in O_{n}, i.e. x ∈ F_{i}S_{n} and ā is the residue class of a in O_{n}/𝔪 = 𝔽_{q} In particular, all the quotients S_{n}/F_{i}S_{n} are finite p - groups and hence S_{n} is a profinite p - group which is the p - Sylow subgroup of the profinite group O_{n}^{×}.

\underline{3.1.3.} The associated graded object grS_{n} with gr_{i}S_{n} = F_{i}S_{n}/F_{i+ \frac{1}{n}} S_{n} becomes a graded Lie algebra with Lie bracket [ā, b̄] induced by the commutator xyx^{−1}y^{−1} in S_{n}. Furthermore, if we define a function φ from the positive real numbers to itself by φ(i) := min{i + 1, pi} then the p - th power map on S_{n} induces maps P : gr_{i}S_{n} ⟶ gr_{φ(i)}S_{n} which define on grS_{n} the structure of a mixed Lie algebra in the sense of Lazard [La, Chap. II.1]. If we identify the filtration quotients with 𝔽_{q} as in 3.1.2 above then the Lie bracket and the map P are explicitly given as follows.

LEMMA 3.1.4.

a) Let ā ∈ gr_{i}S_{n}, b̄ ∈ gr_{j}S_{n}. Then

[ā, b̄] = āb̄^{p^{ni}} − b̄ā^{p^{nj}} ∈ gr_{i+j}S_{n} .

b) Let ā ∈ gr_{i}S_{n}. Then

P ā = \{ \begin{matrix}ā^{1+p^{ni}+...+p^{(p−1)ni}} & if i < (p − 1)^{−1} \\ ā + ā^{1+p^{ni}+...+p^{(p−1)ni}} & if i = (p − 1)^{−1} \\ ā & if i > (p − 1)^{−1} . \end{matrix}

\underline{Proof:} a) Write i = \frac{k}{n}, j = \frac{l}{n} and choose representatives x = 1 + aS^{k} ∈ F_{i}S_{n}, y = 1 + bS^{l} ∈ F_{j}S_{n}. Then x^{−1} = 1 − aS^{k} mod S^{k+1}, y^{−1} = 1 − bS^{l} mod S^{l+1} and the formula

xyx^{−1}y^{−1} = 1 + ((x − 1)(y − 1) − (y − 1)(x − 1))x^{−1}y^{−1}

shows

xyx^{−1}y^{−1} = 1 + (aS^{k}bS^{l} − bS^{l}aS^{k}) mod S^{k+l+1} .

Because O_{n}/𝔪 ≅ 𝕎_{n}/(p) we can choose a and b from 𝕎_{n}. Then Sw = w^{σ}S and w^{σ} ≡ w^{p} mod(p) give the stated formula.

b) Again we write i = \frac{k}{n} and choose a representative x = 1 + aS^{k} with a ∈ 𝕎_{n}. Consider the expression x^{p} = ∑_{r} \binom{p}{r}(aS^{k})^{r}. Because \binom{p}{r} is divisible by p for 0 < r < p and because S^{n} = p we get

x^{p} ≡ 1 + aS^{n+k} + … + (aS^{k})^{p} mod S^{2k+n} .

Furthermore

(aS^{k})^{p} = aa^{σ^{k}} … a^{σ^{(p−1)k}}S^{pk} ≡ aa^{p^{k}} … a^{p^{(p−1)k}}S^{pk} ≡ a^{1+p^{k}+...+p^{(p−1)k}}S^{pk} mod S^{pk+1} .

Now we only have to determine whether pk is smaller resp. equal resp. larger than n + k. i.e. whether pi is smaller resp. equal resp. larger than 1 + i. These cases are equivalent to i < (p − 1)^{−1} resp. i = (p − 1)^{−1} resp. i > (p − 1)^{−1} and hence we are done.

\underline{Remark:} One can use 3.1.4 to compute H^{1}(F_{i}S_{n}) resp. H_{1}(F_{i}S_{n}) (coefficients are, as always, in 𝔽_{p}). For example, by using [Lazard, III.2.1, in particular III.2.1.8] we can derive the following. If i > (p − 1)^{−1} then the quotient map F_{i}S_{n} ⟶ F_{i}S_{n}/F_{i+1}S_{n} induces an isomorphism on H_{1}. Furthermore, if i ≥ 1 then F_{i}S_{n}/F_{i+1}S_{n} is elementary abelian of rank n^{2} and, if p is odd and i ≥ 1, then H^{∗}F_{i}S_{n} is an exterior algebra on H^{1}F_{i}S_{n} (see [Lazard, V.2.2.7]). Ravenel claims in [Ra 2, Theorem 6.3.7] that H^{∗}F_{i}S_{n} is exterior on n^{2} generators in dimension 1 as soon as i > \frac{p}{2(p−1)}. (Note that Ravenel’s i corresponds to \frac{i}{n} in our notation!) However, if p = 5, n = 4 and i = \frac{3}{4} it is not hard to show (using 3.1.4) that the abelianization of F_{i}S_{n}/F_{i+1}S_{n} is F_{i}S_{n}/F_{i+ \frac{n−1}{n}} S_{n} which is elementary abelian of rank 12. Hence H_{1}F_{i}S_{n} and H^{1}F_{i}S_{n} are also of dimension 12 only.

\underline{3.2} The algebras H^{∗}(O_{n}^{×}) and H^{∗}S_{n} are known to be finitely generated 𝔽_{p} - algebras (e.g because they have a finite index normal subgroup, say F_{1}S_{n}, whose cohomology algebra is even finite), hence Theorem 1.4 and its consequences can be applied to both groups. In order to do this we need to determine the categories A_{∗}(G), G = O_{n}^{×} or G = S_{n}. The first step to determine these categories is given by the following well known theorem. For the convenience of the reader we repeat its short proof.

THEOREM 3.2.1. The groups O_{n}^{×} resp. S_{n} have elements of order p iff n ≡ 0 mod(p − 1) in which case both groups have p - rank 1.

\underline{Proof:} If A is any finite abelian subgroup of 𝔻_{n}^{×} then A generates a commutative subfield K of 𝔻_{n} and A is a finite subgroup of its roots of unity. However, for any commutative field the roots of unity form a cyclic subgroup and hence the p - rank is at most 1. Furthermore, the p - rank of the units 𝔻_{n}^{×} is 1 if and only if 𝔻_{n} contains the cyclotomic field ℚ_{p}(ζ_{p}) of degree p − 1 over ℚ_{p} which happens if and only if n is a multiple of p − 1. Finally any element of finite order in 𝔻_{n}^{×} must have valuation 0, i.e. is contained in O_{n}^{×}; furthermore, if the order of the element is a power of p it must be in the p - Sylow subgroup S_{n}.

\underline{Remark:} Lemma 3.1.4.b) implies directly that an element of order p can exist in S_{n} only if \frac{1}{p−1} is of the form \frac{k}{n}, i.e. if n = k(p − 1). In fact, in this case any non-trivial element x of order p is necessarily contained in F_{\frac{1}{p−1}} S_{n} and x = 1 + aS^{k} satisfies ā + ā^{1+p^{k}+...+p^{(p−1)k}} = 0

with ā ≠ 0. One can show that for any ā ≠ 0 satisfying ā + ā^{1+p^{k}+...+p^{(p−1)k}} = 0 one can find a ∈ O_{n} such that 1 + aS^{k} is of order p.

The following more precise result gives the complete description of the categories A_{∗}(G) and of the centralizers of each object.

THEOREM 3.2.2. Let n = k(p − 1).

a) Any two subgroups of O_{n}^{×} which are isomorphic to ℤ/p are conjugate in O_{n}^{×} and each abstract automorphism of such a subgroup E is induced by conjugation by an element in O_{n}^{×}, i.e. Aut_{A_{∗}(O_{n}^{×})}(E) ≅ Aut(E) ≅ ℤ/(p − 1). Furthermore the centralizer C_{O_{n}^{×}}(E) is given as the group of units in the maximal order in the central division algebra over ℚ_{p}(ζ_{p}) of dimension k^{2} and invariant \frac{1}{k}.

b) There are exactly \frac{p^{n}−1}{(p−1)(p^{k}−1)} conjugacy classes of subgroups isomorphic to ℤ/p in S_{n} and for each such subgroup E the group Aut_{A_{∗}(S_{n})}(E) is trivial. Furthermore the centralizer C_{S_{n}}(E) is normal in C_{O_{n}^{×}}(E) with cyclic quotient of order p^{k} − 1.

\underline{Proof:} a) First note that the group of field automorphisms of ℚ_{p}(ζ_{p}) maps via restriction isomorphically to the group of all abstract automorphisms of the multiplicative subgroup of order p generated by ζ_{p}. Furthermore, by the Skolem - Noether Theorem any two embeddings φ, φ′ of ℚ_{p}(ζ_{p}) in 𝔻_{n} are conjugate, i.e. there exists u ∈ 𝔻_{n}^{×} such that φ(x) = uφ′(x)u^{−1} for each x ∈ ℚ_{p}(ζ_{p}). These two facts imply immediately that the assertion on conjugacy classes and automorphisms in part a) hold if O_{n}^{×} is replaced by 𝔻_{n}^{×}. To show them for O_{n}^{×} it suffices therefore to show that in the Skolem - Noether Theorem one can take u of valuation 0. To see this it is enough to note that the valuation on 𝔻_{n} and its restriction to the centralizer C_{𝔻_{n}}(φ(ℚ_{p}(ζ_{p}))) takes the same values. Now C_{𝔻_{n}}(φ(ℚ_{p}(ζ_{p}))) is again a division algebra which is a central division algebra over ℚ_{p}(ζ_{p}) of dimension k^{2} and of invariant \frac{1}{k} (s. [Ha, 20.2.16, 23.1.4]). Then the required property of the value groups follows easily from Theorem 14.3 in [Rn]. If E ≅ ℤ/p ⊂ O_{n}^{×} then E generates a subfield which we can identify with ℚ_{p}(ζ_{p}). Furthermore C_{O_{n}^{×}}(E) is just the intersection O_{n}^{×} ∩ C_{𝔻_{n}}(E) ≅ O_{n}^{×} ∩ C_{𝔻_{n}}(ℚ_{p}(ζ_{p})) which is precisely the group of units in the maximal order of C_{𝔻_{n}}(ℚ_{p}(ζ_{p})). Thus the proof of a) is complete.

b) First we observe that for any E ≅ ℤ/p in S_{n} the group Aut_{A_{∗}(S_{n})}(E) is isomorphic to N_{S_{n}}(E)/C_{S_{n}}(E), so it is a subquotient of a profinite p - group and hence a p - group. However, the abstract isomorphism group is of order p − 1, i.e. of order prime to p, so we see that Aut_{A_{∗}(S_{n})}(E) is trivial.

The remaining parts of b) are now deduced from a). We know from (a) that 𝔽_{q}^{×} acts transitively on the set of S_{n} - conjugacy classes of subgroups E ≅ ℤ/p. So in order to determine the number of S_{n} - conjugacy classes it suffices to show that the order of the isotropy group of this action is (p − 1)(p^{k} − 1). Now the isotropy subgroup of E is equal to the image of the normalizer N_{O_{n}^{×}}(E) under the quotient map O_{n}^{×} ⟶ 𝔽_{q}^{×}. The image of the centralizer is of order p^{k} − 1 because C_{𝔻_{n}}(E) is of dimension k^{2} over ℚ_{p}(ζ_{p}) and hence the residue field of the maximal order of C_{𝔻_{n}}(E) has order p^{k}. Again by a) the quotient N_{O_{n}^{×}}(E)/C_{O_{n}^{×}}(E) has order p − 1. We claim that the image of the C_{O_{n}^{×}}(E) in 𝔽_{q}^{×} continues to have index p − 1 in the image of N_{O_{n}^{×}}(E) and therefore the isotropy group has order as claimed. Assume the image of C_{O_{n}^{×}}(E) in 𝔽_{q}^{×} had index less than p−1 in the image of N_{O_{n}^{×}}(E). By a) again there would be a nontrivial automorphism α of E induced by conjugation by an element y ∈ N_{O_{n}^{×}}(E), and y = zx with z ∈ C_{O_{n}^{×}}(E) and x ∈ S_{n}. However, then we would even have x ∈ S_{n} ∩ N_{O_{n}^{×}}(E) = N_{S_{n}}(E) = C_{S_{n}}(E), hence y ∈ C_{O_{n}^{×}}(E) and this is in contradiction to the non-triviality of α. Finally, C_{S_{n}}(E) is normal in C_{O_{n}^{×}}(E) with cyclic quotient of order p^{k−1} because C_{S_{n}}(E) is the kernel of the surjective map from the units in C_{O_{n}}(E) to the units in the residue field of C_{O_{n}}(E) which is of order p^{k}.

\underline{Remark:} We remark that in case p = 2 Theorem 3.2.2 becomes trivial because there is a unique central element of order 2, namely the element −1 ∈ O_{n}. However, in this case Proposition 2.10 implies that H^{∗}S_{n} is finitely generated and free over a polynomial subalgebra on one generator. If p is odd, the number of conjugacy classes grows quickly with p and n, e.g. if p = 3 and n = 2 we get 2 classes, and if p = 5 and n = 4 we get 39 classes. We will see in the remark after the proof of 4.3 below that for p > 3 and n = p − 1 the map ρ of 2.10 is not mono in dimension 2 and hence H^{∗}S_{n} cannot be free over a polynomial subalgebra on one generator as it was claimed in [Ra 2, 6.2.10b)].

\underline{3.3.} We will now consider our main theorem for S_{n} in case n = p − 1 and we will assume that p is odd, the case p = 2 being trivial. Let E be a cyclic subgroup of order p in S_{n}. By 3.2.2 we know that C_{O_{n}^{×}}(E) is given as the group of units in the maximal order of ℚ_{p}(ζ_{p}), which is equal to ℤ_{p}[ζ_{p}]; in particular C_{O_{n}^{×}}(E) is abelian. In fact, the units ℤ_{p}[ζ_{p}]^{×} are well known to be (non-canonically) isomorphic to ℤ/p × (ℤ_{p})^{n} × ℤ/n and hence 3.2.2 yields C_{S_{n}}(E) ≅ ℤ/p × (ℤ_{p})^{n} (see also the remark after 3.4 below). So the cohomology of C_{S_{n}}(E) is isomorphic to H^{∗}(ℤ/p) ⊗ H^{∗}(ℤ_{p})^{⊗n} and can be written as 𝔽_{p}[y] ⊗ E(x) ⊗ E(a_{1}, ⋯, a_{n}) where y has degree 2 and x and the elements a_{j} have degree 1. The following result (Theorem 1.9 of the introduction) is now an immediate consequence of Corollary 1.7 and of Theorem 3.3.2.

THEOREM 3.3. Let p be an odd prime and n = p − 1. Then S_{n} has \frac{p^{n}−1}{(p−1)^{2}} conjugacy classes of subgroups which are isomorphic to ℤ/p and whose centralizers are all isomorphic to ℤ/p×(ℤ_{p})^{n}. Choose representatives E_{i}, i = 1, …, \frac{p^{n}−1}{(p−1)^{2}} from each conjugacy class. Then the map

ρ : H^{∗}S_{n} ⟶ ∏_{i} H^{∗}C_{S_{n}}(E_{i}) ≅ ∏_{i} 𝔽_{p}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i,1}, ⋯, a_{i,n})

induced by the restriction maps has finite kernel and cokernel.

\underline{3.4.} Now assume again that n = k(p − 1) with k arbitrary. The conjugation action of O_{n}^{×} on S_{n} induces an action of (𝔽_{q})^{×} on H^{∗}S_{n} which is important in applications in homotopy theory. The group (𝔽_{q})^{×} acts also on ∏_{(E)} H^{∗}C_{S_{n}}(E) in such a way that the map ρ is (𝔽_{q})^{×} - linear. This action can be described as follows: Let E_{1} be a fixed cyclic subgroup of S_{n} of order p. The action of N_{O_{n}^{×}}(E_{1}) on H^{∗}C_{S_{n}}(E_{1}) induces an action of the image of N_{O_{n}^{×}}(E_{1}) in (𝔽_{q})^{×}. We denote this image by N̄. The product ∏_{(E)} H^{∗}C_{S_{n}}(E) can be identified with the representation of (𝔽_{q})^{×} which is induced from that of N̄ on H^{∗}C_{S_{n}}(E_{1}). With this 𝔽_{q}^{×} - action on its target the map ρ is linear. The following result explicitly describes the action of N̄ on C_{S_{n}}(E_{1}) in case n = p − 1, and hence gives an explicit description of the action of 𝔽_{q}^{×} on ∏_{(E)} H^{∗}C_{S_{n}}(E) in this case. Note that in case n = (p − 1) we have isomorphisms N̄ ≅ ℤ/n^{2} and C_{S_{n}}(E_{1}) ≅ ℤ/p × (ℤ_{p})^{n}.

PROPOSITION 3.4. Let n = p − 1. The action of N̄ ≅ ℤ/n^{2} on C_{S_{n}}(E_{1}) factors through an action of the quotient ℤ/n. As a ℤ/n - module C_{S_{n}}(E_{1}) ≅ ℤ/p × (ℤ_{p})^{n} splits as the direct sum of the module ℤ/p (with the natural action of ℤ/n ≅ Aut(Z/p)) and the n different one-dimensional representation of ℤ/n over the ring ℤ_{p}.

\underline{Proof:} The first statement follows because the image C̄ of C_{O_{n}^{×}}(E_{1}) (which is isomorphic to ℤ/n) acts clearly trivially and hence the action factors through N̄/C̄ ≅ ℤ/n. This action agrees by the Skolem - Noether Theorem with the action of the Galois group of the cyclotomic extension which is well understood in number theory (see [W. p.301]).

\underline{Remark:} By Proposition 3.4 it is clear that for a suitable choice of an isomorphism τ : ℤ/n ⟶ 𝔽_{p}^{×} and of elements y, x and a_{j} the action of ℤ/n on H^{∗}(ℤ/p × (ℤ_{p})^{n}) ≅ 𝔽_{p}[y] ⊗ E(x) ⊗ E(a_{1}, ⋯, a_{n}) is described by the formula given in Theorem 1.10 of the introduction. In the next section we will have to be even more specific with the choice of these generators so we take the time now to explain this. The valuation on 𝔻_{n} restricts to one on C_{𝔻_{n}}(E) and as in 3.1.2 we get a filtration on the group C_{S_{n}}(E). If we identify C_{𝔻_{n}}(E) with ℚ_{p}(ζ_{p}) then the maximal ideal in the maximal order of C_{𝔻_{n}}(E) is generated by the element ζ_{p} − 1 of valuation \frac{1}{p−1}. Using the p - th power map on the associated graded of this filtration one sees that a minimal set of topological generators for C_{S_{n}}(E) is given by the element ζ_{p} of order p and any choice of elements η_{j}, j = 2, …, n + 1 with the property that η_{j} ≡ 1 + (ζ_{p} − 1)^{j} mod(ζ_{p} − 1)^{j+1}. Furthermore the filtration is ℤ/n - invariant and because n is prime to p the elements η_{j} can be chosen to generate the different one dimensional representations of ℤ/n over ℤ_{p}. With such a choice mod p reduction gives a set of generators \overline{ζ}_{p}, \overline{η}_{2}, …, \overline{η}_{n+1} of H_{1}(ℤ_{p}[ζ_{p}]^{×}). If we take for x, a_{2}, …, a_{n}, a_{n+1} =: a_{1} the dual basis in H^{1} and for y the Bockstein of x then the formula given in 1.10 holds.

4. The case p = 3 and n = 2

\underline{4.1.} In this section we will consider the case p = 3 and n = 2 in fair detail. This is the first non-trivial case where our main theorem can be applied to get information on H^{∗}S_{n}. By Theorem 3.3. we find two conjugacy classes of ℤ/3’s in S_{2} whose centralizers are isomorphic to ℤ/3 × (ℤ_{3})^{2}. We will compute H^{∗}S_{2}, in particular we will show that the map ρ of 2.10 is a monomorphism and we will describe H^{∗}S_{2} as a subalgebra of ∏_{i=1}^{2} 𝔽_{3}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i,1}, a_{i,2}). First we recall a product decomposition of the group S_{n}. The algebra O_{n} has ℤ_{p} as its center, hence ℤ_{p}^{×} ≅ ℤ/(p−1)×ℤ_{p} is central in O_{n}^{×}, i.e. ℤ_{p} is central in S_{n}. Furthermore the reduced norm which is a homomorphism from 𝔻_{n}^{×} to ℚ_{p}^{×} induces a homomorphism from S_{n} back to ℤ_{p} which is left inverse to the inclusion of the central ℤ_{p} as long as n ≢ 0 mod p. In other words the group S_{n} splits as a product ℤ_{p} × S_{n}^{1} of the central ℤ_{p} with the kernel of the reduced norm. Following Ravenel [Ra 2] we will call this kernel S_{n}^{1}. Similarly the centralizers C_{S_{n}}(E), E ≅ ℤ/p, split as C_{S_{n}}(E) ≅ C_{S_{n}^{1}}(E) × ℤ_{p}, and C_{S_{n}^{1}}(E) ≅ ℤ/3 × ℤ_{3} if n = p − 1 = 2. The action of the group ℤ/2 of Proposition 3.4 respects this splitting. In fact, ℤ/2 acts trivially on the central ℤ_{3} and by −1 on C_{S_{n}^{1}}(E). We need to specify the elements y_{i}, x_{i}, a_{i,1} and a_{i,2}. For this we pick a representative E_{1} of one of the two conjugacy classes and choose elements y_{1}, x_{1}, a_{1,1}, a_{1,2} of H^{∗}C_{S_{2}}(E_{1}) as in the remark after 3.4. If ω generates 𝔽_{9}^{×} then ω^{2} generates the group N̄ of 3.4 and acts on H^{∗}C_{S_{2}}(E_{1}) by ω^{2}(y_{1}) = −y_{1}, ω^{2}(x_{1}) = −x_{1}, ω^{2}(a_{1,1}) = −a_{1,1} and ω^{2}(a_{1,2}) = a_{1,2}. Furthermore, ρ will be linear with respect to the action of 𝔽_{9}^{×} if we choose the classes y_{2}, x_{2}, a_{2,1} and a_{2,2} such that ω(y_{1}) = y_{2}, ω(x_{1}) = x_{2}, ω(a_{1,1}) = a_{2,1} and ω(a_{1,2}) = a_{2,2}. In H^{∗}C_{S_{2}^{1}}(E_{i}) the class a_{i,2} is missing but otherwise the same formula holds. In the discussion below we will change notation and write a_{i} instead of a_{i,1} and a_{i}′ instead of a_{i,2}. With these preparations we can finally formulate the main result of this section.

THEOREM 4.2. a) Let p = 3 and n = 2. Then the map ρ of 2.10 is a monomorphism and identifies H^{∗}S_{2} with the subalgebra of ∏_{i=1}^{2} 𝔽_{3}[y_{i}]⊗E(x_{i})⊗E(a_{i}, a_{i}′) generated by the classes x_{1}, x_{2}, y_{1}, y_{2}, a_{1}′ + a_{2}′, x_{1}a_{1} − x_{2}a_{2}, y_{1}a_{1} and y_{2}a_{2}.

b) In particular H^{∗}S_{2} is a finitely generated free module over 𝔽_{3}[y_{1} + y_{2}] ⊗ E(a_{1}′ + a_{2}′) with 8 generators which we can choose as follows: 1, x_{1}, x_{2}, y_{1}, x_{1}a_{1} − x_{2}a_{2}, y_{1}a_{1}, y_{2}a_{2} and y_{1}x_{1}a_{1}.

Before we begin with the proof we compare this calculation with the one of Ravenel in [Ra 1, Theorem 3.3]. After extension of scalars to 𝔽_{9} the two results have to agree. However, they do not. For example, according to Ravenel H^{∗}S_{2} would be multiplicatively generated by classes in degree 1 and 2 while in our computation the classes y_{1}a_{1} and y_{2}a_{2} are indecomposable classes of degree 3. Furthermore, it is easy to see, that if one extends scalars to 𝔽_{9} in Ravenel’s calculation, then the resulting algebra cannot be embedded into ∏_{i=1}^{2} 𝔽_{9}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i}, a_{i}′). The two computations both give the same Poincaré series, however. Furthermore both computations give free modules over a polynomial generator of degree 2 which means that Ravenel’s computation is not compatible with 2.10.

Because of the decomposition S_{n} ≅ ℤ_{p} × S_{n}^{1} it suffices to prove the following analogous result for the group S_{n}^{1}.

PROPOSITION 4.3. Assume p = 3 and n = 2.

a) The map ρ of 2.10 is a monomorphism and identifies H^{∗}S_{2}^{1} with the subalgebra of ∏_{i=1}^{2} 𝔽_{3}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i}) generated by the classes x_{1}, x_{2}, y_{1}, y_{2}, x_{1}a_{1} − x_{2}a_{2}, y_{1}a_{1} and y_{2}a_{2}.

b) In particular H^{∗}S_{2}^{1} is a finitely generated free module over 𝔽_{3}[y_{1} + y_{2}] with 8 generators which we can choose as follows: 1, x_{1}, x_{2}, y_{1}, x_{1}a_{1} − x_{2}a_{2}, y_{1}a_{1}, y_{2}a_{2} and y_{1}x_{1}a_{1}.

The crucial step in the proof of 4.3 is given by the following proposition.

PROPOSITION 4.4. Assume p = 3 and n = 2. There is a homomorphism S_{2}^{1} ⟶ ℤ/3 whose kernel K is torsion free and such that S_{2}^{1} is isomorphic to the semidirect product K × ℤ/3. Furthermore H^{∗}K is a Poincaré duality algebra of dimension 3 and as a ℤ/3 module H^{1}K ≅ (𝔽_{3})^{2} is isomorphic to the augmentation ideal I(ℤ/3) in the group algebra 𝔽_{3}[ℤ/3].

\underline{Proof of 4.4:} We will make use of the filtration on S_{n}^{1} which is induced from the filtration on S_{n} that we discussed in 3.1. The central ℤ_{p} in S_{n} is topologically generated by 1 + p ∈ F_{1}S_{n}. Furthermore an inspection of the formula for the reduced norm (cf. [M]) shows that it sends F_{i}S_{n} onto p^{[i]}ℤ_{p} where [i] denotes the smallest integer which is bigger or equal to i, and that it induces the trace map T r : gr_{i}S_{n} ≅ 𝔽_{q} ⟶ 𝔽_{p} ≅ p^{[i]}ℤ_{p}/p^{[i]+1}ℤ_{p} if i is an integer. In particular we obtain

gr_{i}S_{n}^{1} ≅ \{ \begin{matrix}𝔽_{q} & if i ∉ ℕ \\ Ker T r : 𝔽_{q} ⟶ 𝔽_{p} & if i ∈ ℕ . \end{matrix}

Furthermore the Lie bracket as well as the map P are given on grS_{n}^{1} by the formula of Lemma 3.1.4. So far p and n were general. Now assume p = 3 and n = 2. As we have noted in the remark after the proof of 3.2.1, all non-trivial elements of order 3 in S_{2} have the form 1 + aS with ā ≠ 0 and ā + ā^{1+3+9} = 0, i.e ā^{4} = −1. (Recall that ā ∈ 𝔽_{9} denotes the residue class of a ∈ O_{2}.) In particular there is no ā ∈ 𝔽_{3} with this property. Therefore, if we identify 𝔽_{9} with (ℤ/3)^{2}, and if we divide out by 𝔽_{3}, we get a homomorphism S_{2}^{1} ⟶ gr_{\frac{1}{2}} S_{2}^{1} ⟶ ℤ/3 whose kernel K is torsionfree. Furthermore, as S_{2}^{1} contains elements of order 3 the group S_{2}^{1} is isomorphic to the semidirect product K × ℤ/3. The group S_{2} is an analytic pro 3 - group of dimension 4 and hence S_{2}^{1} and K are analytic pro 3 - groups of dimension 3, so by [La, V.2.5.8] H^{∗}K is a Poincaré duality algebra of dimension 3. To finish it suffices to show that H_{1}K ≅ (ℤ/3)^{2} and that there are elements z_{1} and z_{2} in K which project to a basis z̄_{1} and z̄_{2} of H_{1}K and such that, if x ∈ S_{2}^{1} is a suitable non-trivial element of order 3, then xz_{1}x^{−1} = z_{1}z_{2} mod Φ(K) and xz_{2}x^{−1} = z_{2} mod Φ(K) where Φ denotes the Frattini subgroup. Now it follows from Lemma 3.1.4 (by using [La, III.2.1.8]) that all elements in F_{2}S_{2} ∩ K are third powers and hence H_{1}K ≅ H_{1}K̃ where K̃ denotes the group K/F_{2}S_{2} ∩K. The filtration of S_{2} induces one on K̃ and we obtain

gr_{i}K̃ ≅ \{ \begin{matrix}𝔽_{3} & if i = \frac{1}{2} \\ Ker T r : 𝔽_{9} ⟶ 𝔽_{3} & if i = 1 \\ 𝔽_{9} & if i = \frac{3}{2} \\ 0 & otherwise . \end{matrix}

Furthermore the Lie bracket gr_{\frac{1}{2}} K̃ × gr_{1}K̃ ⟶ gr_{\frac{3}{2}} K̃ and the map P : gr_{\frac{1}{2}} K̃ ⟶ gr_{\frac{3}{2}} K̃ are given by [ā, b̄] = āb̄^{3} − b̄ā and P ā = ā + ā^{1+3+9}. With this it is easy to check that gr_{\frac{3}{2}} K̃ is generated by commutators and third powers and H_{1}K ≅ K̃/gr_{\frac{3}{2}} K̃ ≅ (ℤ/3)^{2}. The action of an element x ∈ ℤ/3 on H_{1}K can now be read off from the commutator formula in 3.1.4. Let z_{1} ∈ K and z_{2} ∈ K be elements in the appropriate filtration which project nontrivially to gr_{\frac{1}{2}} K resp. gr_{1}K. The element x is represented by an element x̄ ∈ gr_{\frac{1}{2}} S_{2} with x̄^{4} = −1. Then xz_{2}x^{−1}z_{2}^{−1} ∈ K ∩ F_{\frac{3}{2}} S_{2} and hence gives zero in H_{1}K. Furthermore [x̄, z̄_{1}] = x̄z̄_{1}^{3} − z̄_{1}x̄^{3}. This is non-trivial and can be made equal to z̄_{2} if x is chosen appropriately. In other words xz_{1}x^{−1} = z_{1}z_{2} mod Φ(K) and we are done.

\underline{Proof of 4.3:} We consider the spectral sequence of the group extension 1 → K → S_{2}^{1} → ℤ/3 → 1 with E_{2} - term E_{2}^{∗,q} ≅ H^{∗}(ℤ/3; H^{q}K). This is a spectral sequence of modules over H^{∗}ℤ/3 and the lines q = 0 and q = 3 are free H^{∗}ℤ/3 - modules on one generator on the vertical edge. Furthermore H^{2}K ≅ H^{1}K as ℤ/3 - module by Poincaré duality. The exact sequence 0 → I(ℤ/3) → 𝔽_{3}[ℤ/3] → 𝔽_{3} → 0 shows that for 0 ≤ q ≤ 3 the graded vector space H^{∗}(ℤ/3; H^{q}K) is additively independent of q, and in fact this is true even as modules over the polynomial subalgebra of H^{∗}ℤ/3 generated by the periodicity generator in degree 2. Therefore the spectral sequence collapses because by 1.9 we have H^{∗}S_{2}^{1} ≅ ∏_{i=1}^{2} 𝔽_{3}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i}) in large degrees, and a non-trivial differential would give too small a result. In particular we see that H^{∗}S_{2}^{1} is a free module of rank 8 over the polynomial subalgebra of H^{∗}ℤ/3 generated in degree 2. The module generators have degree 0, 1, 1, 2, 2, 3, 3, 4. By 2.10 we conclude that ρ is a monomorphism, and we compute the Poincaré series of the cokernel of ρ to be 1 + 2t + t^{2}. It remains to identify the image of ρ.

Let us first consider H^{1}; it is two dimensional and can be identified with the dual of gr_{\frac{1}{2}} S_{2}^{1}. For each ℤ/3 ⊂ S_{2} the basis element \overline{η}_{3} in H_{1}C_{S_{2}}(ℤ/3) (which was defined in the remark after 3.4) maps trivially to gr_{\frac{1}{2}} S_{2}. Therefore the image of ρ in dimension 1 is contained in the linear span of the elements x_{i} and by a dimension argument is equal to this span. The Bocksteins of x_{1} and x_{2} give the elements y_{1} and y_{2}. However, H^{2}S_{2}^{1} has dimension 3, so we need one more element. To identify it we consider the action of 𝔽_{9}^{×} on H^{∗}S_{2} and ∏_{i=1}^{2} 𝔽_{3}[y_{i}] ⊗ E(x_{i}) ⊗ E(a_{i}) (as described in 4.1) and use the linearity of ρ. The subspace generated by y_{1} and y_{2} is invariant under this action and because the order of 𝔽_{9}^{×} is prime to the characteristic we can assume that this last element is an eigenvector for the action of the generator ω ∈ 𝔽_{9}^{×}. Now there are only two eigenspaces of ω, with eigenvalue 1 resp. −1 and eigenvectors x_{1}a_{1} + x_{2}a_{2} resp. x_{1}a_{1} − x_{2}a_{2}. By Lemma 4.5 below it is an eigenvector with eigenvalue −1. Finally ρ is onto in degrees 3, so we have determined the image of ρ and the missing parts of the proposition follow easily.

\underline{Remark:} We have remarked after 3.2.2 that the map ρ is not a monomorphism if n = p − 1 and p > 3. In fact, in this case we have again a decomposition S_{n} ≅ ℤ_{p} × S_{n}^{1} and H^{1}S_{n}^{1} ≅ gr_{\frac{1}{n}} S_{n}^{1} ≅ (ℤ/p)^{n}. As above one sees that for any E ≅ ℤ/p ⊂ S_{n}^{1} the image of the restriction map H^{1}S_{n}^{1} ⟶ H^{1}C_{S_{n}^{1}}(E) ≅ 𝔽_{p}[y] ⊗ E(x) ⊗ E(a_{1}, … a_{p−1}) is spanned by the class x. In particular the product of any two one dimensional classes restricts trivially to all centralizers. According to [Ra 2, Theorem 6.3.14] there are non-trivial products of one dimensional classes as soon as p > 3 and hence ρ is not injective in degree 2.

LEMMA 4.5. Let p = 3 and n = 2. The action of 𝔽_{9}^{×} on H^{2}S_{2}^{1} ≅ (ℤ/3)^{3} decomposes into a direct sum of the subspace generated by y_{1} and y_{2} with ωy_{1} = y_{2}, ωy_{2} = −y_{1} and a one dimensional subspace on which ω acts by multiplication with −1.

\underline{Proof:} The element ω ∈ 𝔽_{9}^{×} can be lifted to a primitive 8-th root of unity in 𝕎_{2} ⊂ O_{2} which we will still call ω. The group extension that we used in 4.4 to investigate H^{∗}S_{2}^{1} is not invariant under the conjugation action x ↦ ωxω^{−1} and hence not suited for the problem that we are considering here. Therefore we consider the subgroup F_{1}S_{2}^{1} := S_{2}^{1} ∩ F_{1}S_{2} which is invariant under the action of ω and normal in S_{2}^{1} with quotient gr_{\frac{1}{2}} S_{2}^{1} ≅ 𝔽_{9} ≅ (ℤ/3)^{2}. It follows easily from [Lazard, V.2.2.7] that H^{∗}F_{1}S_{2}^{1} is an exterior algebra on 3 classes in degree 1. We will prove the lemma by inspecting the spectral sequence

E_{2}^{p,q} ≅ H^{p}((ℤ/3)^{2}, H^{q}F_{1}S_{2}^{1}) ⟹ H^{p+q}S_{2}^{1}

and for this we need to understand the action of (ℤ/3)^{2} on H^{∗}F_{1}S_{2}^{1} and the action of 𝔽_{9}^{×} on (ℤ/3)^{2} and on H^{∗}F_{1}S_{2}^{1} The element ω^{2} is a primitive fourth root of unity for which we will write i. Together with the unit element it forms a basis of 𝔽_{9} as an 𝔽_{3} vector space. Therefore the elements a = 1 + S and b = 1 + iS project to a basis \{ā, b̄\} of gr_{\frac{1}{2}} S_{n}^{1}, while the elements c = 1 + iS^{2} = 1 + 3i, d = 1 + S^{3} and e = 1 + iS^{3} project to a basis {c̄, d̄, ē} in H_{1}F_{1}S_{2}^{1}. Furthermore the relation Sω = ω^{3}S in O_{2} implies that the conjugation action of ω on gr_{\frac{1}{2}} S_{2}^{1} resp. on H_{1}F_{1}S_{2}^{1} is given by the following formula:

ω_{∗}ā = −b̄ , ω_{∗}b̄ = ā
ω_{∗}c̄ = c̄ , ω_{∗}d̄= −ē , ω_{∗}ē = d̄ .

The action of ā and b̄ on H_{1}F_{1}S_{2}^{1} can be read off from the commutator formula of 3.1.4 and we obtain:

ā_{∗}(d̄) = d̄, ā_{∗}(ē) = ē , ā_{∗}(c̄) = c̄ + ē
b̄_{∗}(d̄) = d̄, b̄_{∗}(ē) = ē , b̄_{∗}(c̄) = c̄ − d̄ .

With this information at hand we can look at the spectral sequence. The important groups for us are E_{2}^{1,1} and E_{2}^{3,0} as modules over 𝔽_{9}^{×}. A straightforward computation gives E_{2}^{1,1} ≅ (ℤ/3)^{3} and E_{2}^{3,0} ≅ (ℤ/3)^{4}. Furthermore with respect to the ω-action E_{2}^{1,1} decomposes as a sum of a two-dimensional eigenspace with eigenvalue −1 and a one-dimensional eigenspace with eigenvalue 1 while E_{2}^{3,0} decomposes into a direct sum of two two-dimensional eigenspaces with respective eigenvalues 1 and −1. From the proof of 4.3 we know already that the classes x_{1} and x_{2} are represented on E_{2}^{1,0} and consequently the classes y_{1} and y_{2} are represented on E_{2}^{2,0}. Because we know already that H^{2}S_{n}^{1} is of dimension 3 it follows that the kernel of the differential d_{2} : E_{2}^{1,1} ⟶ E_{2}^{3,0} is at most one-dimensional, and because we also know that the classes x_{1}y_{1} and x_{2}y_{2} in E_{2}^{3,0} survive to E_{∞} the kernel is precisely one dimensional and gives the missing class in H^{2}S_{2}^{1}. We have to show that this kernel is contained in the −1 eigenspace of ω. In fact, E_{3}^{3,0}, the quotient of E_{2}^{3,0} by the image of d_{2} is generated by x_{1}y_{1} and x_{2}y_{2} and is a direct sum of two one-dimensional eigenspaces with eigenvalues 1 resp. −1. The decomposition of E_{2}^{3,0} implies that the image of d_{2} is also a direct sum of two one-dimensional eigenspaces with eigenvalues 1 resp. −1, and hence the decomposition of E_{2}^{1,1} gives that ω acts by multiplication by −1 on the kernel of d_{2}.

5. The case GL(n, ℤ_{p})

\underline{5.1.} We start with a few general remarks on the continuous cohomology of the groups GL(n, ℤ_{p}). Mod p^{r} - reduction defines maps from GL(n, ℤ_{p}) to GL(n, ℤ/p^{r}) with kernel Γ(p^{r}). The groups Γ(p^{r}) form a decreasing sequence of closed subgroups with GL(n, ℤ_{p}) ≅ lim GL(n, ℤ_{p})/Γ(p^{r}). Furthermore, the quotients Γ(p^{r})/Γ(p^{r+1}) are elementary abelian p - groups (of rank n^{2}) and hence Γ(p) ≅ lim Γ(p)/Γ(p^{r}) is a profinite p - group. In particular, for a prime l ≠ p, mod p - reduction induces an isomorphism in continuous cohomology H^{∗}(GL(n, ℤ/p); 𝔽_{l}) ⟶ H^{∗}(GL(n, ℤ_{p}); 𝔽_{l}) and hence H^{∗}(GL(n, ℤ_{p}); 𝔽_{l}) is known by the work of Quillen [Q2]. However, if p = l, then very little seems to be known about H^{∗}(GL(n, ℤ_{p}); 𝔽_{p}) = H^{∗}GL(n, ℤ_{p}) (from now on we will omit the coefficients again in our notation). For example, studying mod p - reduction does not lead very far, because the mod - p - cohomology of the quotient GL(n, 𝔽_{p}) is not known unless n is very small. We will use our centralizer approach to compute H^{∗}GL(n, ℤ_{p}) in large dimensions in case n = p − 1, p odd. The following result gives the necessary group theoretic information to apply Theorem 1.4 resp. Corollary 1.7.

THEOREM 5.2. Let p be odd and n = p − 1.

a) The p - rank of GL(n, ℤ_{p}) is equal to one and, up to conjugacy, there is a unique subgroup E of GL(n, ℤ_{p}) which is isomorphic to ℤ/p.

b) The centralizer C_{GL(n,ℤ_{p})}(E) is isomorphic to ℤ_{p}[ζ_{p}]^{×} ≅ ℤ/p × (ℤ_{p})^{n} × ℤ/(p − 1).

c) Aut_{A_{∗}(GL(n,ℤ_{p}))}(E) ≅ ℤ/n and the action of ℤ/n on C_{GL(n,ℤ_{p})}(E) corresponds via the isomorphism of (b) to the Galois action on ℤ_{p}[ζ_{p}]^{×} which was explicitly described in 3.4.

The crucial input for 5.2 is the following p-adic version of the theorem of Diederichsen and Reiner [CR, Theorem (74.3)]. It can be proved in the same way as the integral version except that some of the details simplify because class group phenomena dissappear in the p - adic version.

THEOREM 5.3. Let G = ℤ/p and M a ℤ_{p}[G] - module which is finitely generated and free as ℤ_{p} - module. Let F = ℤ_{p}[G] be the free ℤ_{p}[G]- module on one generator, T ≅ ℤ_{p} the trivial one dimensional module, and R = ℤ_{p}[ζ_{p}] the ring of integers in the cyclotomic extension ℚ_{p}(ζ_{p}) with action of a generator g ∈ G given by gr = ζ_{p}r for r ∈ R. Then

M ≅ F^{k} ⊕ T^{l} ⊕ R^{m}

for a unique triple (k, l, m) of non-negative numbers.

\underline{Proof of 5.2:} a) The ℤ_{p}[G] - module R is isomorphic to (ℤ_{p})^{n} as ℤ_{p} - module which shows that there is an embedding of G into GL(n, ℤ_{p}). By Theorem 5.2 there is a unique ℤ_{p}[G] - module structure on (ℤ_{p})^{n} for which the action is faithful which means that all subgroups E of order p in GL(n, ℤ_{p}) are conjugate. That the p - rank is not bigger than 1 will follow from part b) because the p - rank of the centralizer of E is only 1.

b) We have isomorphisms C_{GL(n,ℤ_{p})}(E) ≅ Aut_{ℤ_{p}[E]}(R) and because the ℤ_{p}[E] module structure on the ring R is pulled back from the R - module structure we also get Aut_{ℤ_{p}[E]}(R) ≅ Aut_{R}(R) ≅ R^{×}.

c) The group Aut_{A_{∗}(GL(n,ℤ_{p}))}(E) is a subgroup of the group of all abstract automorphisms of E and the Galois group of the cyclotomic extension realizes all of them through conjugations in GL(n, ℤ_{p}).

Using the notation of 3.3 and 3.4 we write H^{∗}C_{GL(n,ℤ_{p})}(E) ≅ 𝔽_{p}[y] ⊗ E(x) ⊗ E(a_{1}, ⋯, a_{n}). The action of the Galois group is determined by 3.4. Combining Theorem 5.2 with Corollary 1.7 leads to the following result (Theorem 1.10 of the introduction).

THEOREM 5.4. Let p be an odd prime, n = p − 1 and E ≅ ℤ/p ⊂ GL(n, ℤ_{p}). Then the restriction map

ρ : H^{∗}GL(n, ℤ_{p}) ⟶ (H^{∗}C_{GL(n,ℤ_{p})}(E))^{ℤ/n} ≅ (𝔽_{p}[y] ⊗ E(x) ⊗ E(a_{1}, ⋯, a_{n}))^{ℤ/n}

has finite kernel and cokernel.

As in [A] one can analyze all p - rank one cases p − 1 ≤ n ≤ 2p − 3 further and reduce the computation of H^{∗}GL(n, ℤ_{p}) in large dimensions to the computation of the cohomology of GL(n − p + 1, ℤ_{p}) and of appropriate congruence subgroups thereof. We leave the details to the interested reader.

\underline{5.5.} We finish with a brief discussion of the case p = 3 and n = 2. This case is simple enough that one could do it directly with standard methods. However, we include it here as another example illustrating our theory and how the map ρ of 1.7 may fail to be an isomorphism in small dimensions. The situation is very similar to that of the group S_{2} for the prime 3 that we discussed in section 4. Using the same notation as in 4.1 we write H^{∗}C_{GL(2,ℤ_{3})}(ℤ/3) ≅ 𝔽_{p}[y] ⊗ E(x) ⊗ E(a, a′) where y is of degree 2 and all the other classes are of degree 1 and the action of the non-trivial element g in the Galois group is trivial on a′ and multiplies all other generators by −1. In particular, the ring of invariants (𝔽_{3}[y]⊗E(x)⊗E(a, a′))^{ℤ/2} is equal to the subring generated by the elements y^{2}, yx, ya, xa and a′. This is a free module of rank 4 over 𝔽_{3}[y^{2}] ⊗ E(a′) on generators 1, yx, ya and xa.

PROPOSITION 5.5. The restriction map ρ : H^{∗}GL(2, ℤ_{3}) ⟶ (H^{∗}C_{GL(2,ℤ_{3})}(ℤ/3))^{ℤ/2} ≅ (𝔽_{3}[y]⊗E(x)⊗E(a, a′))^{ℤ/2} is a monomorphism and identifies H^{∗}GL(2, ℤ_{3}) with the subalgebra 𝔽_{3}[y^{2}] ⊗ E(yx, ya) ⊗ E(a′) of the invariants.

\underline{Proof:} First we note that GL(n, ℤ_{p}) is isomorphic to ℤ_{p} × GL^{1}(n, ℤ_{p}) where GL^{1}(n, ℤ_{p}) denotes the subgroup of GL(n, ℤ_{p}) which is the preimage of ℤ/p−1 ⊂ ℤ_{p}^{×} under the determinant map. In fact, ℤ_{p}^{×} ≅ ℤ_{p} × ℤ/p − 1 identifies with the center of GL(n, ℤ_{p}) and the composition with the determinant is multiplication by n, hence an isomorphism on the ℤ_{p} summand of ℤ_{p}^{×} if n ≢ 0 mod p. Furthermore, H^{∗}GL^{1}(n, ℤ_{p}) ≅ (H^{∗}SL(n, ℤ_{p}))^{ℤ/p−1}, and hence 5.5 will follow from the following result for the special linear group.

First note that if one works with SL(2, ℤ_{3}) instead of GL(2, ℤ_{3}) then one still has a unique subgroup ℤ/3 up to conjugacy with centralizer C_{SL(2,ℤ_{3})} ≅ ℤ/2×ℤ/3×ℤ_{3} and H^{∗}C_{SL(2,ℤ_{3})}(E) ≅ 𝔽_{3}[y] ⊗ E(x) ⊗ E(a). Furthermore, in this case Aut_{A_{∗}(SL(2,ℤ_{3}))}(ℤ/3) is the trivial group so that in large degrees H^{∗}SL(2, ℤ_{3}) ≅ 𝔽_{3}[y] ⊗ E(x) ⊗ E(a) by 1.7.

PROPOSITION 5.6. The restriction map ρ : H^{∗}SL(2, ℤ_{3}) ⟶ H^{∗}C_{SL(2,ℤ_{3})}(ℤ/3)) ≅ 𝔽_{3}[y] ⊗ E(x, a) is a monomorphism and identifies H^{∗}SL(2, ℤ_{3}) with 𝔽_{3}[y] ⊗ E(x, ya).

\underline{Proof:} We consider the mod 3 reduction map SL(2, ℤ_{3}) ⟶ SL(2, 𝔽_{3}). The kernel K is a torsion free 3 - dimensional analytic pro 3 - group and hence H^{∗}K is a 3 - dimensional Poincaré duality algebra by [La,V.2.5.8]. Now consider the graded Lie algebra associated to the decreasing filtration of SL(2, ℤ_{3}) by the kernels of mod 3^{k} reduction, k = 1, 2, …. It is easy to see from this Lie algebra, say as in the proof of 4.4, that H_{1}K ≅ (𝔽_{3})^{3}. Furthermore, using [La,V.2.2.7] we see that H^{∗}K is exterior on the 3 generators in degree 1. Now we consider the spectral sequence of the extension

1 ⟶ K ⟶ SL(2, ℤ_{3}) ⟶ SL(2, 𝔽_{3}) ⟶ 1

with E_{2}^{∗,q} ≅ H^{∗}(SL(2, 𝔽_{3}); H^{q}K). The group SL(2, 𝔽_{3}) acts necessarily trivial on H^{3}K because there are no non-trivial homomomorphisms from SL(2, 𝔽_{3}) to GL(1, 𝔽_{3}). Next one can check that the invariants of H^{1}K with respect to the action of the 2 - Sylow subgroup Q_{8} of SL_{2}(𝔽_{3}) are trivial. Because Q_{8} is normal in SL(2, 𝔽_{3}) this implies H^{∗}(SL(2, 𝔽_{3}); H^{1}K) = 0 and by Poincaré duality we also have H^{∗}(SL(2, 𝔽_{3}); H^{2}K) = 0. Finally the restriction map to the 3 - Sylow subgroup is well known to induce an isomorphism H^{∗}SL(2, 𝔽_{3}) ≅ H^{∗}ℤ/3 and so our spectral sequencs has just 2 non-trivial rows at E_{2}, which are both isomorphic to H^{∗}ℤ/3. As in the proof of 4.3 we see now that the spectral sequence has to collapse; a non-trivial differential would lead to a result which is too small to be compatible with 1.7. Then the spectral sequence shows that H^{∗}SL(2, ℤ_{3}) is free over 𝔽_{3}[y] and hence ρ is injective. It is clear that the elements x and y are in the image of ρ and by a counting dimensions one sees that ρ is an isomorphism in degrees 3 and bigger, in particular the image of ρ is 𝔽_{3}[y] ⊗ E(x, ya).

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Hans–Werner Henn Mathematisches Institut der Universität Im Neuenheimer Feld 288 D–69120 Heidelberg Fed. Rep. of Germany

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