NOETHERIAN SKEW POWER SERIES RINGS

Abstract

This dissertation is concerned with noncommutative analogues of formal power series rings in multiple variables. Our motivating examples arises from quantized coordinate rings; the completions of these quantized coordinate rings are iterated noetherian skew power series rings. Our first focus is on $q$-commutative power series rings, having the following form: $R = k_q[[x_1,\ldots,x_n]]$, where $q = (q_{ij})_{n\times n}$ with $q_{ii}=1$ and $q_{ij} = q^{-1}_{ji} \in \k^{\times}$ and where $x_jx_i = q_{ji} x_i x_j$. The corresponding skew Laurent series ring is $L=k_q[[x_1^{\pm 1},\ldots,x_n^{\pm 1}]]$. We first study the ideal structure of $L$. We prove that extension and contraction of ideals produces a bijection between the set of ideals of $L$ and the set of ideals of the center $Z$ of $L$. This bijection further produces a homeomorphism between $\spec L$ and $\spec Z$. Applying the analysis of $L$ to $R$, we prove that the prime spectrum $\spec R$ can be partitioned into finitely many strata each homeomorphic to the prime spectrum of a commutative noetherian ring. The rings $R$ and $L$ are completions, respectively, of the quantum coordinate ring of $n$-space and of the $n$-torus. Our second focus is on power series completions of iterated skew polynomial rings with nonzero derivations. Given an iterated skew polynomial ring $C[y_1;\t_1,\d_1]\ldots [y_n;\t_n,\d_n]$ over a complete local ring $C$ with maximal ideal $\m$, we prove, under suitable assumptions, that the completion at the ideal $\m + \left\langle y_1,y_2,\ldots,y_n\right\rangle$ is an iterated skew power series ring. Under further conditions, this completion is a local, noetherian, Auslander regular domain. Applicable examples include the following quantized coordinate rings: quantum matrices, quantum symplectic spaces, and quantum Euclidean spaces. Results in this dissertation are included in the following two preprints: 1. Prime ideals of $q$-commutative power series rings (joint with E. S. Letzter), submitted for publication. 2. Completions of quantum coordinate rings, to appear in Proceedings of the American Mathematical Society.