Non-commutative crepant resolutions of toric singularities with divisor class group of rank one

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities whose divisor class group has rank one. More precisely, such toric NCCRs are in bijection with non-trivial upper sets in a certain quotient of the divisor class group equipped with a natural partial order. This classification allows us to prove that all toric NCCRs of such toric singularities are connected by iterated Iyama–Wemyss mutations, and hence are derived equivalent to one another. We further give a dimer-model realization of this classification in the non-pyramidal case. More precisely, we construct periodic quivers with cuts on a $d$-dimensional torus, establish a cut-upper set correspondence, and prove that the resulting cut quiver with relations presents the corresponding toric NCCR. For $d=2$, this recovers the quiver-theoretic part of the usual dimer-model construction. In the appendix, we give an explicit formula for the volume of $d$-dimensional lattice polytopes with $d+2$ vertices. As an application, we verify Van den Bergh’s conjectural equality, for Gorenstein toric singularities with divisor class group of rank one, between the number of indecomposable direct summands of a toric NCCR and the normalized volume of the corresponding lattice polytope.