Noncommutative resolutions of AS-Gorenstein isolated singularites

TL;DR

This paper explores noncommutative resolutions of AS-Gorenstein isolated singularities, defining noncommutative projective schemes and proving their properties, including a noncommutative Bondal-Orlov conjecture in low dimensions.

Contribution

It introduces a new framework for noncommutative resolutions using noncommutative projective schemes and establishes their properties and equivalences, extending classical singularity theory.

Findings

Noncommutative resolutions are generalized AS regular algebras.

Centers of resolutions are isomorphic to the original singularity centers.

Noncommutative Bondal-Orlov conjecture holds in dimensions 2 and 3.

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.