On twists of modules over non-commutative Iwasawa algebras

Abstract

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