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

Abstract

Let $\mathcal{G}\simeq H⋊\Gamma$ 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.

12 pages. Rewritten version of §7 of arXiv:2403.04663v3 reorganised into a separate manuscript, v2: added §6 discussing the 2-adic case