The Centre of Completed Group Algebras

of Pro-p Groups

Konstantin Ardakov

Received: December 13, 2004

Communicated by Peter Schneider

Abstract. We compute the centre of the completed group algebra of an arbitrary countably based pro-p group with coefficients in 𝔽_{p} or ℤ_{p}. Some other results are obtained.

2000 Mathematics Subject Classification: 16S34, 20E18

1 Introduction

Let G be a pro-p group. In this paper we investigate some rings related to the completed group algebra of G over 𝔽_{p}, which we denote by Ω_{G}:

Ω_{G} = 𝔽_{p}[[G]] := lim_{⟵}_{N◁_{o}G}𝔽_{p}[G/N].

When G is analytic in the sense of [3], Ω_{G} and its p-adic analogue Λ_{G} defined by

Λ_{G} = ℤ_{p}[[G]] := lim_{⟵}_{N◁_{o}G}ℤ_{p}[G/N]

are right and left Noetherian rings, which are in general noncommutative. If in addition G is torsion free, the results of Brumer, Neumann and others show that Ω_{G} and Λ_{G} have finite global dimension and have no zero-divisors; for an overview, see [1]. Moreover, under the name of Iwasawa algebras, these rings are frequently of interest to number theorists (see [2] for more details). Our main result is Theorem A. Let G be a countably based pro-p group. Then the centre of Ω_{G} is equal to the closure of the centre of 𝔽_{p}[G]:

Z(Ω_{G}) = \overline{Z(𝔽_{p}[G])}.

Similarly, the centre of Λ_{G} is equal to the closure of the centre of ℤ_{p}[G]:

Z(Λ_{G}) = \overline{Z(ℤ_{p}[G])}.

When G is p-valued in the sense of Lazard [5, III.2.1.2], we obtain a cleaner result.

Corollary A. Let G be a countably based p-valued pro-p group with centre Z. Then

Z(Ω_{G}) = Ω_{Z} and Z(Λ_{G}) = Λ_{Z}.

The class of p-valued pro-p groups is rather large; for example, every closed subgroup of a uniform pro-p group [3, 4,1] is p-valued. Also, any pro-p subgroup of GL_{n}(ℤ_{p}) is p-valued when p > n + 1 [5, p. 101]. We remark that when G is an open pro-p subgroup of GL_{2}(ℤ_{p}), a version of the above result was proved by Howson [4, 4.2] using similar techniques. We also use the method used in the proof of Theorem A to compute endomorphism rings of certain induced modules for Ω_{G}, when G is an analytic pro-p group.

Theorem B. Let H be a closed subgroup of an analytic pro-p group G. Let M = 𝔽_{p} ⊗_{Ω_{H}} Ω_{G} and write R = End_{Ω_{G}}(M). Then R is finite-dimensional over 𝔽_{p} if and only if N_{𝔤}(𝔥) = 𝔥, where 𝔥 and 𝔤 denote the ℚ_{p}-Lie algebras of H and G, respectively.

The author would like to thank Chris Brookes and Simon Wadsley for many valuable discussions. This research was financially supported by EPSRC grant number 00802002.

2 Fixed Points

Let X be a group. For any (right) X-space S, let S^{X} = {s ∈ S : s.X = s} denote the set of fixed points of X in S. Also, let O(S) denote the collection of all finite X-orbits on S, and for any orbit C ∈ O(S), let Ĉ denote the orbit sum

Ĉ = ∑_{s∈C} s,

viewed as an element of the permutation module 𝔽_{p}[S]. Thus, 𝔽_{p}[S]^{X} is spanned by all the Ĉ as C ranges over O(S): 𝔽_{p}[S]^{X} = 𝔽_{p}[Ô(S)]. Now let X be a pro-p group. Assume we are given an inverse system

… ↠^{π_{n+1}} A_{n} ↠^{π_{n}} A_{n−1} ↠^{π_{n−1}} … ↠^{π_{2}} A_{1}

of finite X-spaces. We can consider the natural inverse system of permutation modules associated with the A_{i}:

… ↠^{π_{n+1}} 𝔽_{p}[A_{n}] ↠^{π_{n}} 𝔽_{p}[A_{n−1}] ↠^{π_{n−1}} … ↠^{π_{2}} 𝔽_{p}[A_{1}]

where we keep the same notation for the connecting maps π_{n}. Now, form the inverse limit

Y = lim_{⟵} A_{n},

this is clearly an X-space. We can also form the inverse limit

Ω_{Y} = lim_{⟵} 𝔽_{p}[A_{n}],

which is easily seen to be an Ω_{X}-module. Note that Ω_{Y} is a compact metric space, with metric given by d(α, β) = ∥α − β∥, where ∥.∥ is a norm on Ω_{Y} given by

∥(α_{n})_{n}∥ = sup{p^{−n} : α_{n} ≠ 0} and ∥0∥ = 0.

We are interested in the fixed points of Ω_{Y}, viewed as an X-space. It is straightforward to see that there is a natural embedding of 𝔽_{p}[Y] into Ω_{Y} and that 𝔽_{p}[Y]^{X} ⊆ Ω_{Y}^{X}.

Proposition 2.1. With the notations above, Ω_{Y}^{X} = \overline{𝔽_{p}[Y]}^{X} = \overline{𝔽_{p}[Y]^{X}}.

Proof. Because the action of X on Ω_{Y} is continuous, it is clear that Ω_{Y}^{X} is a closed subset of Ω_{Y}, so by the above remarks \overline{𝔽_{p}[Y]^{X}} ⊆ Ω_{Y}^{X}. Let α = (α_{n})_{n} ∈ Ω_{Y}^{X}. Since the natural maps Ω_{Y} ↠ 𝔽_{p}[A_{n}] are maps of X-spaces, we see that each α_{n} lies in 𝔽_{p}[A_{n}]^{X}. Let the integer r be least such that α_{r} ≠ 0. Consider α_{r} ∈ 𝔽_{p}[A_{r}]^{X}; thus α_{r} = ∑_{C∈O(A_{r})} λ_{C}Ĉ and not all the λ_{C} are zero. Pick a C ∈ O(A_{r}) with λ_{C} ≠ 0. Since π_{r+1} is a map of X-spaces, π_{r+1}^{−1}(C) = D_{1}∪D_{2}∪…∪D_{k} is a union of X-orbits, with π_{r+1}(D_{j}) = C for j = 1, …, k and π_{r+1}(D_{j}) ∩ C = ∅ for j > k, if we let D_{k+1}, …, D_{m} denote the remaining elements of O(A_{r+1}). We claim we can find a D_{j} with 1 ≤ j ≤ k such that |D_{j}| = |C|. For, suppose not. Then |D_{j}| > |C| for each j = 1, …, k. As π_{r+1} : D_{j} ↠ C is a surjective map of finite transitive X-spaces, and because X is a pro-p group, we deduce that each fibre (π_{r+1}|D_{j})^{−1}(s) for s ∈ C has size a power of p greater than 1. But then, because we are working over 𝔽_{p}, we must have π_{r+1}(D̂_{j}) = 0, for each 1 ≤ j ≤ k. Now, since α_{r+1} ∈ 𝔽_{p}[A_{r+1}]^{X}, we can write α_{r+1} = ∑_{j=1}^{m} μ_{j}D̂_{j} for some μ_{j} ∈ 𝔽_{p}. So, α_{r} = π_{r+1}(α_{r+1}) = ∑_{j=k+1}^{m} μ_{j}π_{r+1}(D̂_{j}). But π_{r+1}(D_{j}) ∩ C = ∅ for all j > k, contradicting the fact that C ⊆ supp(α_{r}). Hence, we can find C_{r+1} ∈ O(A_{r+1}) with |C_{r+1}| = |C_{r}| and π_{r+1}(C_{r+1}) = C_{r}, where we set C_{r} to be C. It is clear that we can continue this process of “lifting” the X-orbits, without ever increasing the sizes. Thus, we get a sequence

⋯ ↠^{π_{n+2}} C_{n+1} ↠^{π_{n+1}} C_{n} ↠^{π_{n}} … ↠^{π_{r+1}} C_{r}

of X-orbits, each having the same size as C_{r}. Now, pick some s_{r} ∈ C_{r} and inductively choose lifts s_{n} ∈ C_{n} for each n ≥ r. Let s be the element of Y determined by these lifts. It is then straightforward to see that the X-orbit of s in Y is finite and that the image of this orbit in A_{r} equals C. Let F_{C} denote this element of O(Y). Finally, we can consider the element β = ∑_{C∈O(A_{r})} λ_{C}F̂_{C}. Obviously β lies in 𝔽_{p}[Y]^{X}, and the image of β in 𝔽_{p}[A_{r}] coincides with α_{r}. Hence, α − β has norm strictly smaller than that of α and also lies in Ω_{Y}^{X}. Applying the argument above to α − β instead of α and iterating, we see that α can be approximated arbitrarily closely by elements of 𝔽_{p}[Y]^{X}.

Next we turn to the analogous proposition over the p-adics. Let

Λ_{Y} = lim_{⟵} ℤ_{p}[A_{n}].

This is naturally a Λ_{X}-module and there is a natural isomorphism Λ_{Y} /pΛ_{Y} ≅ Ω_{Y} of Λ_{X}-modules. Moreover, since Λ_{Y} ≅ lim_{⟵}(ℤ/p^{n}ℤ)[A_{n}] is a countably based pro-p group, it is a compact metric space.

Proposition 2.2. With the notations above, Λ_{Y}^{X} = \overline{ℤ_{p}[Y]^{X}}.

Proof. As in the proof of Proposition 2.1, the inclusion \overline{ℤ_{p}[Y]^{X}} ⊆ Λ_{Y}^{X} is clear. Let¯ : Λ_{Y} → Ω_{Y} denote reduction mod p. Let α ∈ Λ_{Y}^{X} so that \overline{α} ∈ Ω_{Y}^{X}. By Proposition 2.1, \overline{α} = lim_{n→∞} u_{n} for some u_{n} ∈ 𝔽_{p}[Y]^{X}. Since 𝔽_{p}[Y]^{X} = 𝔽_{p}[Ô(Y)] and ℤ_{p}[Y]^{X} = ℤ_{p}[Ô(Y)], we can choose v_{n} ∈ ℤ_{p}[Y]^{X} such that \overline{v_{n}} = u_{n} for all n. Since Λ_{Y} is compact, by passing to a convergent subsequence we may assume that v_{n} converges to β_{0} ∈ \overline{ℤ_{p}[Y]^{X}}. Now \overline{α} = lim_{n→∞} \overline{v_{n}} = \overline{β_{0}}, so α − β_{0} ∈ pΛ_{Y} ∩ Λ_{Y}^{X} = pΛ_{Y}^{X}, since Λ_{Y} is ptorsion free. Hence we can write α = β_{0} + pα_{1} where α_{1} ∈ Λ_{Y}^{X}. Iterating the above argument, we obtain elements β_{1}, β_{2}, … ∈ \overline{ℤ_{p}[Y]^{X}} and α_{1}, α_{2}, … ∈ Λ_{Y}^{X} such that α_{n} = β_{n} + pα_{n+1} for all n ≥ 1. So α = ∑_{n=0}^{∞} β_{n}p^{n} ∈ \overline{ℤ_{p}[Y]^{X}}.

3 Main Results

We immediately make use of the above Propositions.

Proof of Theorem A. Since G is countably based, we can write G as an inverse limit of the countable system A_{n} = G/G_{n}, for some suitable open normal subgroups G_{1} ⊃ G_{2} ⊃ … ⊃ G_{n} ⊃ … of G. Each A_{n} is a finite G-space, where G acts by conjugation. Now apply Propositions 2.1 and 2.2.

Proof of Corollary A. Z(𝔽_{p}[G]) is spanned over 𝔽_{p} by all conjugacy class sums Ĉ, where C is a finite conjugacy class of G. Let C be such a conjugacy class and let x ∈ C. Then the centralizer C_{G}(x) of x in G is a closed subgroup of G of finite index. Let y ∈ G; then y^{p^{n}} ∈ C_{G}(x) for some n, so (x^{−1}yx)^{p^{n}} = y^{p^{n}}. Since G is p-valued, the map g ↦ g^{p} is injective on G by [5, Chapter III, Proposition 2.1.4], so x^{−1}yx = y and C ⊆ Z. Hence Z(𝔽_{p}[G]) = 𝔽_{p}[Z], and similarly Z(ℤ_{p}[G]) = ℤ_{p}[Z]. The result follows from Theorem A.

We remark that Corollary A does not extend to arbitrary torsion free analytic pro-p groups. This can be easily checked for the group given in [5, Chapter III, Example 3.2.5]. We now turn to the proof of Theorem B. Let G = lim_{⟵} G/G_{n} be a pro-p group, H a closed subgroup. Let M = 𝔽_{p} ⊗_{Ω_{H}} Ω_{G} be the induced module from the trivial module for Ω_{H}. G acts on the coset space Y = H\G by right translation and we can write Y = lim_{⟵} HG_{n}\G as an inverse limit of finite G-spaces. It is easy to see that Ω_{Y} is then naturally isomorphic to M. Let R denote the endomorphism ring End_{Ω_{G}} M of M. Each element f ∈ R gives rise to a trivial Ω_{H}−submodule of M generated by f(1 ⊗ 1), when we view M as an Ω_{H}−module by restriction. This gives an isomorphism of 𝔽_{p}−vector spaces

R = Hom_{Ω_{G}}(𝔽_{p} ⊗_{Ω_{H}} Ω_{G}, M) ≅ Hom_{Ω_{H}}(𝔽_{p}, M),

expressing the fact “induction is left adjoint to restriction”. Now Hom_{Ω_{H}}(𝔽_{p}, M) can be thought of as the sum of all trivial Ω_{H}−submodules of M, which is precisely the set M^{H} = Ω_{Y}^{H}, where H acts on Y by right translation. In view of Proposition 2.1, we are interested in the finite H−orbits on Y; these are given by those double cosets of H in G which are finite unions of left cosets of H. Suppose HxH is such a double coset; then Stab_{H}(Hx) = {h ∈ H : Hxh = Hx} = H ∩ H^{x} has finite index in H, so the set N_{G}(H) = {x ∈ G : H ∩ H^{x} ≤_{o} H} is of interest; we observe that it contains the usual normalizer N_{G}(H) of H in G. This set is sometimes called the commensurator of H in G.

Proof of Theorem B. Recall [3, 9.5] that G contains an open normal uniform subgroup K and that the ℚ_{p}-Lie algebra of G can be defined by

L(G) = K ⊗_{ℤ_{p}} ℚ_{p}

where K is viewed as a ℤ_{p}-module of finite rank [3, 4.17]. The conjugation action of G on K is ℤ_{p}-linear and therefore extends to an action of G on L(G), which is easily checked to be independent of the choice of K. This is just the adjoint action of G on L(G). Next, we observe that when x ∈ G,

\begin{aligned}H ∩ H^{x} ≤_{o} H & ⇔ & L(H ∩ H^{x}) = L(H) ∩ L(H)^{x} = L(H) \\ & ⇔ & L(H)^{x} = L(H), \end{aligned}

so N := N_{G}(H) = Stab_{G}𝔥 is a (closed) subgroup of G. By [3, Exercise 9.10], we see that the Lie algebra of N is equal to the normalizer N_{𝔤}(𝔥) of 𝔥 in 𝔤. We remark in passing that this implies that N_{G}(H) has finite index in N when dealing with analytic pro-p groups; this is not true in general. Now, by Proposition 2.1 and the above remarks, R is finite dimensional over 𝔽_{p} if and only if the number of finite H-orbits on Y = H\G is finite. Clearly {Hx : HxH is a finite H-orbit} = H\N, so the number of finite H-orbits on Y is finite if and only if H has finite index in N. This happens if and only if 𝔥 = L(H) = L(N) = N_{𝔤}(𝔥), as required.

References

[1] K. Ardakov, K.A. Brown, Primeness, semiprimeness and localisation in Iwasawa algebras, preprint.

[2] J. Coates, P. Schneider, R. Sujatha, Modules over Iwasawa algebras, Journal of the Inst. of Math. Jussieu (2003), 2(1), 73-108.

[3] J. D.Dixon, M.P.F. Du Sautoy, A.Mann, D.Segal, Analytic prop groups, 2nd edition, CUP (1999).

[4] S. Howson, Structure of central torsion Iwasawa modules, Bull. Soc. Math. France 130 (2002), no. 4, 507-535.

[5] M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965), 389-603.

Konstantin Ardakov Christ’s College Cambridge CB2 3BU k.ardakov@dpmms.cam.ac.uk