Canonical singularities of orders over surfaces

Abstract

We define and study canonical singularities of orders over surfaces. These are non-commutative analogues of Kleinian singularities that arise naturally in the minimal model program for orders over surfaces D. Chan and C. Ingalls, Invent. Math. 161 (2005) 427–452. We classify canonical singularities of orders using their minimal resolutions (which we define). We describe them explicitly as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring. We also prove that they are Gorenstein, describe their Auslander–Reiten quivers, and note a simple version of the McKay correspondence.