Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs

Abstract

Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by $\mathbb{k}[K_{r_1,\dots ,r_n}]$ with $1\le r_1\le \cdots \le r_n.$ We give the complete characterization of which $\mathbb{k}[K_{r_1,\dots ,r_n}]$ is level in terms of n and $r_1,\dots ,r_n.$ Similarly, we also give the complete characterization of which $\mathbb{k}[K_{r_1,\dots ,r_n}]$ is almost Gorenstein in terms of n and $r_1,\dots ,r_n.$

Keywords:

• Level
• almost Gorenstein
• edge rings
• complete multipartite graphs

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13H10
• Secondary: 52B20
• 13F65
• 13A02
• 05C25

Acknowledgement

The first named author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03513.