On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras II

TL;DR

This paper provides a comprehensive description of the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras, extending previous results to all components without restrictions on ramification.

Contribution

It generalizes the Wedderburn decomposition of the total ring of quotients for completed group rings of semidirect products, removing previous ramification restrictions.

Findings

Complete Wedderburn decomposition in terms of $F[H]$

Extension of previous partial results to all simple components

Semisimplicity of the total ring of quotients

Abstract

Let $\mathcal G\simeq H\rtimes\Gamma$ be the semidirect product of a finite group $H$ and $\Gamma\simeq\mathbb Z_p$. Let $ F/\mathbb Q_p$ be a finite extension with ring of integers $\mathcal O_F$. Then the total ring of quotients $\mathcal Q^F(\mathcal G)$ of the completed group ring $\mathcal O_F[[\mathcal G]]$ is semisimple artinian. We determine its Wedderburn decomposition in full generality in terms of the Wedderburn decomposition of the group ring $ F[H]$. Such a description was previously available only for those simple components for which a certain associated field extension is totally ramified.