Azumaya skew group algebras and an application to quantum Kleinian singularities

TL;DR

This paper characterizes when skew group rings are Azumaya algebras, applies this to quantum Kleinian singularities, and proves these algebras are maximal orders, also providing a new proof of Auslander's Theorem.

Contribution

It provides necessary and sufficient conditions for skew group rings to be Azumaya and applies these to quantum Kleinian singularities, establishing their maximal order property.

Findings

Skew group rings associated with quantum Kleinian singularities are Azumaya after suitable localization.

These Azumaya skew group rings are maximal orders.

A new proof of Auslander's Theorem for quantum Kleinian singularities is presented.

Abstract

We provide easily-verified necessary and sufficient conditions for a skew group ring, or more generally, a crossed product ring, to be an Azumaya algebra. We use our results to show that (suitable localisations of) skew group rings associated to the quantum Kleinian singularities introduced by Chan, Kirkman, Walton, and Zhang in [CKWZ16a] are Azumaya, and use this to show that these algebras are maximal orders. We also give a new proof of Auslander's Theorem for quantum Kleinian singularities.