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Register a new userConic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions
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In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
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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.
We introduce special classes of non-commutative crepant resolutions (= NCCR) which we call steady and splitting. We show that a singularity has a steady splitting NCCR if and only if it is a quotient singularity by a finite abelian group. We apply our results to toric singularities and dimer models.
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $Bbbk[K_{r_1,ldots,r_n}]$, where $1 leq r_1 leq cdots leq r_n$. More concretely, we prove that the class group of $Bbbk[K_{r_1,ldots,r_n}]$ is isomorphic to $mathbb{Z}^n$ if $n =3$ with $r_1 geq 2$ or $n geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $Bbbk[K_{r_1,ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals for certain $K_{r_1,ldots,r_n}$ including all cases where $Bbbk[K_{r_1,ldots,r_n}]$ is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of $Bbbk[K_{r_1,ldots,r_n}]$ in the case where it is Gorenstein.
A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a $3$-dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a nice class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we introduce the notion of semi-steady NCCRs, and show a consistent dimer model gives a semi-steady NCCR if and only if it is homotopy equivalent to a regular dimer model.
Recently, Segal constructed a derived equivalence for an interesting 5-fold flop that was provided by Abuaf. The aim of this article is to add some results for the derived equivalence for Abuafs flop. Concretely, we study the equivalence for Abuafs flop by using Toda-Ueharas tilting bundles and Iyama-Wemysss mutation functors. In addition, we observe a flop-flop=twist result and a multi-mutation=twist result for Abuafs flop.
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