Non-commutative crepant resolutions of Hibi rings with small class group

TL;DR

This paper investigates non-commutative crepant resolutions of Gorenstein Hibi rings with small class groups, demonstrating their existence and exploring module mutations and exchange graph connectivity.

Contribution

It establishes the existence of splitting NCCRs for Gorenstein Hibi rings with class group rac12;^2 and analyzes module mutations and exchange graph structure in three dimensions.

Findings

Gorenstein Hibi rings with class group rac12;^2 have splitting NCCRs.

Mutations of modules produce connected exchange graphs in three dimensions.

Existence of splitting NCCRs is confirmed for Gorenstein toric rings with class group rac12;.

Abstract

In this paper, we study splitting (or toric) non-commutative crepant resolutions (= NCCRs) of some toric rings. In particular, we consider Hibi rings, which are toric rings arising from partially ordered sets, and show that Gorenstein Hibi rings with class group $\mathbb{Z}^2$ have a splitting NCCR. In the appendix, we also discuss Gorenstein toric rings with class group $\mathbb{Z}$, in which case the existence of splitting NCCRs is already known. We especially observe the mutations of modules giving splitting NCCRs for the three dimensional case, and show the connectedness of the exchange graph.