On the noncommutative Bondal–Orlov conjecture

Let R be a normal, equi-codimensional Cohen–Macaulay ring of dimension d ≥ 2 with a canonical module ωR. We give a sufficient criterion that establishes a derived equivalence between the noncommutative crepant resolutions of R. When d ≤ 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 by Iyama and Wemyss (2010)