Maximal prime homomorphic images of mod-$p$ Iwasawa algebras

TL;DR

This paper characterizes the structure of maximal prime homomorphic images of mod-$p$ Iwasawa algebras for compact $p$-adic analytic groups, revealing their isomorphism to matrix rings over twisted group rings with applications to ideal correspondences.

Contribution

It provides a detailed description of the structure of prime homomorphic images of mod-$p$ Iwasawa algebras, including explicit isomorphisms and applications to ideal theory.

Findings

Isomorphism between $kG/P$ and matrix rings over twisted group rings

Identification of subquotients with no finite normal subgroups

Preservation of properties like almost-faithfulness and control of ideals

Abstract

Let $k$ be a finite field of characteristic $p$, and $G$ a compact $p$-adic analytic group. Write $kG$ for the completed group ring of $G$ over $k$. In this paper, we describe the structure of the ring $kG/P$, where $P$ is a minimal prime ideal of $kG$. We give an isomorphism between $kG/P$ and a matrix ring with coefficients in the ring $(k'G')_\alpha$, where $k'/k$ is a finite field extension, $G'$ is a large subquotient of $G$ with no finite normal subgroups, and $(-)_\alpha$ is a "twisting" operation that preserves several desirable properties of the ring structure. We demonstrate an application of this isomorphism by setting up correspondences between certain ideals and subrings of $kG$ and those of $(k'G')_\alpha$, and showing that these correspondences often preserve some useful properties, such as almost-faithfulness of an ideal, or control of an ideal by a closed normal subgroup.