Noncommutative (crepant) desingularizations and the global spectrum of commutative rings

TL;DR

This paper explores noncommutative resolutions of singularities over commutative rings, introduces a new notion of NCCR, and studies the global spectrum of rings to understand their endomorphism rings' global dimensions.

Contribution

It proposes a weaker, more inclusive definition of NCCR, provides conditions for their existence, and introduces the concept of the global spectrum of a ring.

Findings

New examples of NCRs for various singularities

Necessary and sufficient conditions for NCCR existence

Computed global dimensions of endomorphism rings

Abstract

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but subsumes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring $R$, that is, the set of all possible finite global dimensions of endomorphism rings of MCM $R$-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.