Twisting lemma for $\Lambda$-adic modules

Abstract

A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra ${\mathbb{Z}}_p[[\Gamma ]]$ with $\Gamma \cong {\mathbb{Z}}_p,\exists$ a continuous character $\theta :\Gamma \to {\mathbb{Z}}_p^{\times }$ such that, the ${\Gamma }^n$-Euler characteristic of the twist $M(\theta )$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$-adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $\mathcal{T}[[G]]$ modules, where $G$ is a compact $p$-adic Lie group and $\mathcal{T}$ is a finite extension of ${\mathbb{Z}}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic application by considering the twisted Euler Characteristic of the big Selmer (respectively fine Selmer) group of a $\Lambda$-adic form over a $p$-adic Lie extension.

14 pages