Twisting lemma for $Λ$-adic modules

A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[Γ]]$ with $Γ\cong \mathbb{Z}_p, \ \exists$ a continuous character $θ: Γ\rightarrow \mathbb{Z}_p^\times$ such that, the $Γ^{n}$-Euler characteristic of the twist $M(θ)$ 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 $Λ$-adic form over a $p$-adic Lie extension.

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