Noncommutative resolutions of AS-Gorenstein isolated singularites

Abstract

In this paper, we investigate noncommutative resolutions of (generalized) 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 is in fact a Morita-like theory at the quotient category level. Based on the equivalences, we propose a definition of noncommutative resolutions of generalized 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 generalized 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 dimension 2 and 3.