On twists of modules over noncommutative Iwasawa algebras

Jha, Somnath; Ochiai, Tadashi; Zábrádi, Gergely

doi:10.2140/ant.2016.10.685

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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

DOI: 10.2140/ant.2016.10.685

Abstract

It is well known that, for any finitely generated torsion module $M$ over the Iwasawa algebra $ℤ_{p} [[Γ]]$ , where $Γ$ is isomorphic to $ℤ_{p}$ , there exists a continuous $p$ -adic character $ρ$ of $Γ$ such that, for every open subgroup $U$ of $Γ$ , the group of $U$ -coinvariants $M {(ρ)}_{U}$ is finite; here $M (ρ)$ denotes the twist of $M$ by $ρ$ . 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 $ℤ_{p} [[Γ]]$ by certain torsion modules over $ℤ_{p} [[G]]$ 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

Mathematical Subject Classification 2010

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

Vol. 10, No. 3, 2016