Twisting lemma for $Λ$-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 $\theta: \Gamma \rightarrow \mathbb{Z}_p^\times$ 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.