Bondal–Orlov–Van den Bergh conjecture on derived equivalence of crepant resolutions and NCCRs

Let $R$ be a Gorenstein $coperatorname{\mathbb{C}}$-algebra. A crepant resolution of $R$ is a crepant resolution of the corresponding singularity, and an NCCR of $R$ is a non-commutative crepant resolution. Bondal–Orlov–Van den Bergh conjecture. All crepant resolutions of $R$ and all NCCRs of $R$ are derived equivalent. This conjecture proposes a common derived-categorical structure for commutative crepant resolutions and non-commutative crepant resolutions; its status is not determined by the supplied source context.