On twists of modules over non-commutative Iwasawa algebras

TL;DR

This paper generalizes a known twisting lemma for modules over cyclotomic Iwasawa algebras to the non-commutative setting, enabling broader applications in arithmetic geometry.

Contribution

It extends the twisting lemma from commutative to non-commutative Iwasawa algebras, broadening its applicability to modules over p-adic Lie groups.

Findings

Established a non-commutative twisting lemma for torsion modules over Z_p[[G]]

Demonstrated the lemma's potential for studying arithmetic properties in non-commutative Iwasawa theory

Provided foundational tools for future research in non-commutative arithmetic geometry

Abstract

It is well known that, for any finitely generated torsion module M over the Iwasawa algebra Z_p [[{\Gamma} ]], where {\Gamma} is isomorphic to Z_p, there exists a continuous p-adic character {\rho} of {\Gamma} such that, for every open subgroup U of {\Gamma}, the group of U-coinvariants M({\rho})_U is finite; here M( {\rho}) denotes the twist of M by {\rho}. 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 [[ {\Gamma} ]] by certain torsion modules over Z_p [[G]] with more general p-adic Lie group G.