On the Noncommutative Bondal-Orlov Conjecture

TL;DR

This paper establishes a criterion for derived equivalence of noncommutative crepant resolutions of certain Cohen-Macaulay rings, confirming the conjecture in dimensions up to three using cluster tilting theory.

Contribution

It provides a sufficient condition for derived equivalence of noncommutative crepant resolutions, extending the noncommutative Bondal-Orlov conjecture to rings of dimension up to three.

Findings

All noncommutative crepant resolutions of R are derived equivalent when dimension ≤ 3.

The criterion is based on cluster tilting theory for commutative algebras.

The method applies to normal, equi-codimensional Cohen-Macaulay rings with a canonical module.

Abstract

Let R be a normal, equi-codimensional Cohen-Macaulay ring of dimension $d\geq 2$ with a canonical module. We give a sufficient criterion that establishes a derived equivalence between the noncommutative crepant resolutions of R. When $d\leq 3$ this criterion is always satisfied and so all noncommutative crepant resolutions of R are derived equivalent. Our method is based on cluster tilting theory for commutative algebras, developed in [IW10].