Vol. 10, No. 3, 2016

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On twists of modules over noncommutative Iwasawa algebras

Somnath Jha, Tadashi Ochiai and Gergely Zábrádi

Vol. 10 (2016), No. 3, 685–694

It is well known that, for any finitely generated torsion module M over the Iwasawa algebra p[[Γ]], where Γ is isomorphic to 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 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 p[[Γ]] by certain torsion modules over p[[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 p-adic representations over certain p-adic Lie extensions.

Selmer group, noncommutative Iwasawa theory
Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 16S50
Received: 21 October 2015
Revised: 22 December 2015
Accepted: 1 February 2016
Published: 12 June 2016
Somnath Jha
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Kanpur 208016
Tadashi Ochiai
Department of Mathematics
Graduate School of Science
Osaka University
Machikaneyama 1-1
Osaka 5600043
Gergely Zábrádi
Department of Algebra and Number Theory
Mathematical Institute, Eötvös Loránd University
Bertalan Lajos utca 11
1111 Budapest