Completions of quantum coordinate rings

TL;DR

This paper proves that certain quantum coordinate rings, when completed at specific ideals, form iterated skew power series rings and can be noetherian, regular domains, with applications to quantum matrices and spaces.

Contribution

It establishes conditions under which quantum coordinate rings' completions are iterated skew power series rings and noetherian regular domains.

Findings

Completion at specific ideals yields iterated skew power series rings.

Under additional conditions, these completions are noetherian, Auslander regular domains.

Applicable to quantum matrices, symplectic spaces, and Euclidean space.

Abstract

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 + < y_1,y_2,ldots,y_n> is an iterated skew power series ring. Under further conditions, this completion is a local, noetherian, Auslander regular domain. Applicable examples include quantum matrices, quantum symplectic spaces, and quantum Euclidean space.