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A classical twisting lemma says that given a finitely generated torsionmodule $M$ 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}$-Eulercharacteristic of the twist $M(\theta)$ is finite for every $n$. This twistinglemma has been generalized for the Iwasawa algebra of a general compact$p$-adic Lie group $G$. In this article, we consider a further generalizationof 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 alsoindicate arithmetic application by considering the twisted Euler Characteristicof the big Selmer (respectively fine Selmer) group of a $\Lambda$-adic formover a $p$-adic Lie extension.

Twisting lemma for $\Lambda$-adic modules