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

Abstract: We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order. By this classification, we show that for such toric singularities, all toric NCCRs are connected by iterated Iyama-Wemyss mutations.