Noncommutative resolutions of AS–Gorenstein isolated singularities

Abstract

In this paper, we investigate noncommutative resolutions of AS–Gorenstein isolated singularities. Noncommutative resolutions in graded case are achieved as the graded endomorphism rings of some finitely generated graded modules, which are seldom $\mathbb{N}$-graded algebras but bounded-below $\mathbb{Z}$-graded algebras. So the paper works on locally finite bounded-below $\mathbb{Z}$-graded algebras. We first define and study noncommutative projective schemes after Artin–Zhang and define noncommutative quasi-projective spaces as the base spaces of noncommutative projective schemes. The equivalences between noncommutative quasi-projective spaces are proved to be induced by so-called modulo-torsion-invertible bimodules, which are in fact a Morita-like theory at the quotient category level. Based on the equivalences, we propose a definition of noncommutative resolutions of AS–Gorenstein isolated singularities and prove that such noncommutative resolutions are generalized AS-regular algebras. The center of any noncommutative resolution is isomorphic to the center of the original AS–Gorenstein isolated singularity. In the final part, we prove that a noncommutative resolution of an AS–Gorenstein isolated singularity of dimension $d$ is given by an MCM generator $M$ if and only if $M$ is a $(d-1)$-cluster tilting module. A noncommutative version of the Bondal–Orlov conjecture is also proved to be true in dimensions $2$ and $3$.