On the Noncommutative Bondal-Orlov Conjecture

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].