On Prime Ideals of Noetherian Skew Power Series Rings

TL;DR

This paper investigates the structure of prime ideals in skew power series rings over certain noetherian rings, providing conditions for prime ideal contraction and extension, especially under specific derivation assumptions relevant to noncommutative Iwasawa theory.

Contribution

It characterizes prime ideals in skew power series rings with a focus on contraction and extension properties under $ au$-stability and derivation conditions, extending previous results to a noncommutative setting.

Findings

Prime ideals correspond to $ au$-prime ideals stable under $ au$ and $ au - id$.

Existence of prime ideals contracting to a given ideal depends on $ au$-stability.

Use of topological methods and zariskian properties to analyze prime ideals.

Abstract

We study prime ideals in skew power series rings $T:=R[[y;\tau,\delta]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $\tau$ of $R$, and $\tau$-derivations $\delta$ of $R$. These rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, under the additional assumption that $\delta = \tau - id$ (a basic feature of the Iwasawa-theoretic context), we prove: If $I$ is an ideal of $R$, then there exists a prime ideal $P$ of $S$ contracting to $I$ if and only if $I$ is a $\delta$-stable $\tau$-prime ideal of $R$. Our approach essentially depends on two key ingredients: First, the algebras considered are zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings.