On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras
Ben Forrás
TL;DR
The paper determines the Wedderburn decomposition of the total ring of quotients of the completed Iwasawa algebra for G = H ⋊ Γ with H finite and Γ ≅ Z_p, under a ramification hypothesis. It extends Galois actions to skew fields, describes the center and Schur indices via skew power series rings, and expresses each Wedderburn component D_χ as the quotient of a skew power series ring when the extensions F(η)/F_χ are totally ramified. The key technical advancement is showing D_χ ≅ Quot(O_{D_η}[[X;τ,τ-id]]) and, in the totally ramified case, providing explicit dimension and conjugacy results that yield a complete description of the χ-part of the decomposition, including the construction of the A_χ cross-product algebra. The results generalize prior work for pro-p G and connect Nickel’s divisibility relations with an explicit skew-power-series realization, offering a precise, ramification-sensitive description of the semisimple components and their maximal orders.
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 a semisimple ring. We determine its Wedderburn decomposition under a ramification hypothesis by relating it to the Wedderburn decomposition of the group ring $F[H]$.