Noncommutative resolutions of noncommutative isolated singularities

Abstract

Noncommutative resolutions of AS-Gorenstein isolated singularities are investigated by Li--Shen--Wu. However, establishing their existence and constructing such resolutions are generally difficult, even when they exist. In this paper, we study conditions under which a commonly graded AS-regular algebra serves as a noncommutative resolution of an AS-Gorenstein isolated singularity. We investigate projective modules over a noetherian commonly graded AS-regular algebra whose endomorphism rings admit resolutions by the underlying regular algebra. This leads to a more general definition of noncommutative resolutions of balanced Cohen--Macaulay isolated singularities. We show that the existence of such resolutions is equivalent to the existence of cluster tilting modules over balanced CM isolated singularities. The corresponding noncommutative analogue of the Bondal-Orlov conjecture is established in dimensions $2$ and $3$. As an application, we study Hopf actions on commonly graded AS-Gorenstein algebras and investigate noncommutative resolutions of invariant rings. We present three examples of noncommutative resolutions, including one in which the noncommutative isolated singularity is not connected graded.