Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs

TL;DR

This paper studies the class groups and conic divisorial ideals of edge rings of complete multipartite graphs, and constructs non-commutative crepant resolutions for Gorenstein cases, advancing understanding of their algebraic and geometric properties.

Contribution

It characterizes the class groups of these edge rings, describes conic divisorial ideals, and constructs NCCRs for Gorenstein instances, providing new insights into their structure.

Findings

Class group is isomorphic to Z^n for certain cases.

Explicit description of conic divisorial ideals for specific graphs.

Construction of non-commutative crepant resolutions for Gorenstein edge rings.

Abstract

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.