Azumaya skew group algebras and an application to quantum Kleinian singularities

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.