Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

Abstract

2 Definitions, terminology and background 2.1 Algebras and modules Our coefficient ring R will always be a commutative pseudocompact ring. In later sections we will require further structure on R, the main coefficient rings of interest to us being complete discrete valuation rings.Let Λ be a pseudocompact R-algebra (we follow the treatments in [9] and [18]). Examples of particular interest are the completed group algebra R[[G]] of a profinite group G, or later the group algebra RG of a finite group G. We consider the following categories of modules for Λ:• Λ-Mod C : the category whose objects are pseudocompact left Λ-modules [9, §1]. Objects of this category are complete, Hausdorff topological Λ-modules U having a basis of open neighbourhoods of 0 consisting of submodules V for which U/V has finite length. In other words, the category of inverse limits of left Λ-modules of finite length over R.• Λ-Mod D : the category whose objects are those topological Λ-modules having the discrete topology. Such modules are precisely the direct limits of left Λ-modules of finite length over R. We will call such modules discrete (they are called "locally finite" in [18]).• Λ-Mod abs : the category of abstract left Λ-modules Proof. We mimic the proof of [38, Lem. 52.2]. Write X = I X i with each X i an RG-sublattice of X of R-rank 1. The sequence remains split over R and the X i are indecomposable R-modules, so by Proposition 3.5 applied to these R-modules there is a subset J of I such that X = s(X ′ ) ⊕ J X i , and this remains true over RG. Thus the RG-module homomorphism h ′ = h| J Xi is an isomorphism, where h is the map induced by h. By Nakayama's Lemma, h| J Xi = h ′ : J X i → X ′′ is an isomorphism. Now the composition of h ′−1 with the inclusion J X i → X is a splitting of h.

On the geometry of lattices and finiteness of Picard groups

“…Let M be an .OGb/ e -module representing an element of Pic.OGb/. By Weiss' criterion (see [21], and [14] for a version allowing O as a coefficient ring) our M has trivial source if and only if the lattice of P -fixed points taken on the left To finish, let us briefly mention the following nice consequence of Theorems A and B, even though it is implied by [20,Theorem (38.6)], a theorem due to Puig. To state it, we need the notion of a splendid Morita equivalence.…”

Infinitely generated virtually free pro‐p groups and p‐adic representations

“…Let be a finite ‐group and be a permutation pro‐ ‐module. (i)[, Corollary 2.3]: , where is some set of indices. (ii)[, Proposition 3.4]: Every direct summand of is a permutation module. Furthermore, there exists a subset for each such that complements in . (iii)[, Corollary 6.8]: If is also permutation ‐module, then an extension of by splits and so is a permutation ‐module. …”

“…Hence is an ‐decomposition and so by Proposition (ii) and are ‐permutation, since is. By [, Theorem 8.6] and are ‐monomial lattices, (that is, a direct product of lattices induced from a rank 1 lattices). Monomial ‐lattices are permutation ‐lattices for .…”

“…Monomial ‐lattices are permutation ‐lattices for . For we need to apply [, Lemma 8.4] to deduce that the ‐module decomposes as and hence by Proposition (ii) and are ‐permutation. Applying Theorem we deduce that is a ‐permutation module.…”

“…Namely, we prove that the abelianization is a permutation ‐module. Then we use the results of on infinitely generated ‐adic permutation modules and especially infinitely generated pro‐ version of the celebrated theorem of Weiss to prove the following theorem of independent interest. Theorem Let be a semidirect product of a finite ‐group and a free pro‐ group .…”