Prime ideals of Q-commutative power series rings

Abstract

We study the "q-commutative" power series ring R:= k q[[x 1,···,x n]], defined by the relations x ixj = q ijx jx i, for mulitiplicatively 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). © Springer Science+Business Media B.V. 2010.