TL;DR
This paper investigates orders over Cohen-Macaulay rings, exploring their properties related to NCCRs, and provides conditions for desirable homological features, including examples with infinite global dimension.
Contribution
It introduces an alternative definition for orders representing NCCRs and establishes necessary and sufficient conditions for their homological properties.
Findings
Certain endomorphism rings over abelian quotient singularities have infinite global dimension.
Some definitions of orders for NCCRs are shown to be non-existent or incomplete.
Conditions for homological properties of orders are characterized.
Abstract
In this paper we study orders over Cohen-Macaulay rings. We discuss desirable properties for these orders if they are to represent NCCRs of the base rings. While some definitions have been made, we discuss an alternate definition and the non-existence of examples. 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 endormorphism rings over abelian quotient singularities have infinite global dimension.
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