#### Vol. 10, No. 3, 2016

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On twists of modules over noncommutative Iwasawa algebras

### Somnath Jha, Tadashi Ochiai and Gergely Zábrádi

Vol. 10 (2016), No. 3, 685–694
##### Abstract

It is well known that, for any finitely generated torsion module $M$ over the Iwasawa algebra ${ℤ}_{p}\left[\left[\Gamma \right]\right]$, where $\Gamma$ is isomorphic to ${ℤ}_{p}$, there exists a continuous $p$-adic character $\rho$ of $\Gamma$ such that, for every open subgroup $U$ of $\Gamma$, the group of $U$-coinvariants $M{\left(\rho \right)}_{U}$ is finite; here $M\left(\rho \right)$ denotes the twist of $M$ by $\rho$. This twisting lemma was already used to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a noncommutative generalization of this twisting lemma, replacing torsion modules over ${ℤ}_{p}\left[\left[\Gamma \right]\right]$ by certain torsion modules over ${ℤ}_{p}\left[\left[G\right]\right]$ with more general $p$-adic Lie group $G$. In a forthcoming article, this noncommutative twisting lemma will be used to prove the functional equation of Selmer groups of general $p$-adic representations over certain $p$-adic Lie extensions.

##### Keywords
Selmer group, noncommutative Iwasawa theory
Primary: 11R23
Secondary: 16S50
##### Milestones
Received: 21 October 2015
Revised: 22 December 2015
Accepted: 1 February 2016
Published: 12 June 2016
##### Authors
 Somnath Jha Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208016 India Tadashi Ochiai Department of Mathematics Graduate School of Science Osaka University Machikaneyama 1-1 Toyonaka Osaka 5600043 Japan Gergely Zábrádi Department of Algebra and Number Theory Mathematical Institute, Eötvös Loránd University Bertalan Lajos utca 11 1111 Budapest Hungary