Prime Ideals of q-Commutative Power Series Rings

TL;DR

This paper investigates the prime ideal structure of q-commutative power series rings, revealing their stratification, catenarity, and conditions under which they are UFDs, thus advancing understanding of their noncommutative algebraic properties.

Contribution

It provides a detailed analysis of prime ideals in q-commutative power series rings, including stratification, normal separation, catenarity, and conditions for being a UFD, extending prior work on noncommutative domains.

Findings

Prime spectrum is normally separated and finitely stratified.

R is catenary under certain conditions.

For generic q_{ij}, R has finitely many primes and is a UFD.

Abstract

We study the "q-commutative" power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters).