Extensions of almost faithful prime ideals in virtually nilpotent mod-$p$ Iwasawa algebras

TL;DR

This paper investigates the extension of prime ideals in completed group rings of certain $p$-adic analytic groups, showing that almost faithful $G$-stable primes in a subgroup extend to the larger group ring.

Contribution

It proves that almost faithful $G$-stable prime ideals in $kH$ extend to prime ideals in $kG$ for a class of nilpotent-by-finite $p$-adic groups.

Findings

Almost faithful $G$-stable prime ideals extend to $kG$.

Extension results hold for groups with specific nilpotent-by-finite structures.

Provides new insights into the structure of prime ideals in $p$-adic group rings.

Abstract

Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.