Gorenstein and totally reflexive orders

In this paper we study orders over Cohen-Macaulay rings. We discuss the properties needed for these orders to give noncommutative crepant resolutions of the base rings; namely, we want algebraic analogs of birationality, nonsingularity, and crepancy. While some definitions have been made, we discuss an alternate definition and obstructions to the existence of such objects. We then give necessary and sufficient conditions for an order to have certain desirable homological properties. We examine examples of rings satisfying these properties to prove that certain endomorphism rings over abelian quotient singularities have infinite global dimension. (C) 2016 Elsevier Inc. All rights reserved.