On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras II
Ben Forrás
TL;DR
This work delivers a complete Wedderburn-type description of the total ring of quotients Q^F(G) for G = H ⋊ Γ with Γ ≅ Z_p, in terms of the Wedderburn components of F[H]. It introduces the key invariants v_{F,χ} and τ_F, and shows that each χ-component is a matrix algebra over a skew field D_{F,χ} which, in turn, is realized as the quotient of a skew power series ring by τ_F, without requiring total ramification. The authors unify ramified and unramified cases via unramified base change and produce an explicit decomposition D_{F,χ} ≅ Quot(O_{D_η}[[X; τ_F, τ_F−id]]) with a precise center description, enabling classification of maximal orders and applications to noncommutative Iwasawa theory. The paper also provides concrete unramified examples illustrating the general theory and extends the scope beyond earlier ramification-restricted results. Overall, the work gives a fully general, computable framework for the semisimple structure of completed Iwasawa algebras in this setting, with implications for arithmetic applications.
Abstract
Let $\mathcal G\simeq H\rtimesΓ$ be the semidirect product of a finite group $H$ and $Γ\simeq\mathbb Z_p$. 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.