Non-Commutative Crepant Resolutions for Some Toric Singularities I

We give a criterion for the existence of noncommutative crepant resolutions (NCCRs) for certain toric singularities. In particular, we recover Broomhead’s result that a three-dimensional toric Gorenstein singularity has an NCCR. Our result also yields the existence of an NCCR for a four-dimensional toric Gorenstein singularity, which is known to have no toric NCCR.

Residues and duality for singularity categories of isolated Gorenstein singularities

We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen–Macaulay modules.