Twisting lemma for λ-adic modules

A classical twisting lemma says that given a finitely generated torsion module M over the Iwasawa algebra Zₚ[[Γ ]] with Γ Zₚ, ∃ a continuous character θ: Γ → Zₚ^× such that, the Γⁿ-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 T[[G]] modules, where G is a compact p-adic Lie group and T is a finite extension of Zₚ[[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.