The centre of completed group algebras of pro-p groups

. 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.


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 : When G is analytic in the sense of [3], Ω G and its p-adic analogue Λ G defined by 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]: Similarly, the centre of Λ G is equal to the closure of the centre of 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 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
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.
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): ]. Now let X be a pro-p group.Assume we are given an inverse system . . .
of finite X-spaces.We can consider the natural inverse system of permutation modules associated with the A i : . . .
where we keep the same notation for the connecting maps π n .Now, form the inverse limit this is clearly an X-space.We can also form the inverse limit 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 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 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 ( Dj ) = 0, for each 1 ≤ j ≤ k.Now, since α r+1 ∈ F p [A r+1 ] X , we can write α r+1 = m j=1 µ j Dj for some µ j ∈ F p .So, α r = π r+1 (α r+1 ) = m j=k+1 µ j π r+1 ( Dj ).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 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(Ar) λ C FC .Obviously β lies in F p [Y ] X , and the image of

coincides with
The centre of completed group algebras of pro-p groups 5 α r .Hence, α − β has norm strictly smaller than that of α and also lies in Ω X Y .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.
This is naturally a Λ X -module and there is a natural isomor- Since Λ Y is compact, by passing to a convergent subsequence we may assume that v n converges to Hence we can write α = β 0 + pα 1 where α 1 ∈ Λ X Y .Iterating the above argument, we obtain elements

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 . 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. 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 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 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 ←− 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 [3, , 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. centre of completed group algebras of pro-p groups 7 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 byL(G) = K ⊗ Zp Q pwhere 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,H ∩ H x ≤ o H ⇔ L(H ∩ H x ) = L(H) ∩ L(H) x = L(H) ⇔ L(H) x = L(H), 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.