For a Gorenstein normal singularity R, a non-commutative crepant resolution (NCCR), introduced by Van den Bergh, is a non-commutative analogue of a crepant resolution and provides a framework for generalizing the derived McKay correspondence. For Gorenstein toric singularities, it is natural to focus on toric NCCRs, namely NCCRs arising from direct sums of divisorial modules. The existence of toric NCCRs has been established in several cases, including when dimR≤3, when Cl(R) is torsion, when Cl(R)≅Z, and in some other cases.

In this talk, we prove the existence of toric NCCRs for Gorenstein toric singularities R whose divisor class group Cl(R) has rank one. Moreover, we classify all toric NCCRs: we show that they are in bijection with the non-trivial upper sets of a certain poset. This classification is new even when Cl(R)≅Z. Using this classification, we prove that all toric NCCRs of such toric singularities are connected by iterated Iyama--Wemyss mutations, and hence are derived equivalent to one another.

If time permits, we will also describe explicitly the quivers with relations of our toric NCCRs from the viewpoint of higher-dimensional analogues of dimer models. More precisely, although we do not propose a general definition of higher-dimensional dimer models, we describe, in some special cases corresponding to our toric singularities, the quivers that would be expected to arise as dual quivers of such objects, should they exist.

This talk will take place in hybrid mode at the Institut Henri Poincaré.