On twists of modules over noncommutative Iwasawa algebras

It is well known that, for any finitely generated torsion module M over the Iwasawa algebra ℤp[[Γ]], where Γ is isomorphic to ℤp, there exists a continuous pp-adic character p of Γ such that, for every open subgroup U of Γ, the group of U-coinvariants M(p)U is finite; here M(p) denotes the twist of M by pp. This twisting lemma was already used to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a noncommutative generalization of this twisting lemma, replacing torsion modules over ℤp[[Γ]] by certain torsion modules over ℤp[[G]] with more general p-adic Lie group G. In a forthcoming article, this noncommutative twisting lemma will be used to prove the functional equation of Selmer groups of general pp-adic representations over certain p-adic Lie extensions