On twists of modules over noncommutative Iwasawa algebras

Abstract

It is well known that, for any finitely generated torsion module $M$ over the Iwasawa algebra ${\mathbb{Z}}_p\left[\left[\Gamma \right]\right]$, where $\Gamma$ is isomorphic to ${\mathbb{Z}}_p$, 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 ${\mathbb{Z}}_p\left[\left[\Gamma \right]\right]$ by certain torsion modules over ${\mathbb{Z}}_p\left[\left[G\right]\right]$ 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.