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

Zalesskiĭ, Pavel

doi:10.1112/topo.12086

Cited by 2 publications

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“…This theorem plays a central role in the proof of the pro-p version [39] of the theorem of Karras, Pietrowski, Solitar, Cohen and Scott [19,12,31], which states that a virtually free group acts on a tree with finite vertex stabilizers. Indeed, in the pro-p case Theorem 1.2 replaces Stallings' theory of ends, crucial in the proof of the original result.…”

Section: Introductionmentioning

confidence: 99%

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

MacQuarrie

Symonds

Zalesskiĭ

2020

Advances in Mathematics

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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.

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