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Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs

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 Added by Akihiro Higashitani
 Publication date 2021
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and research's language is English




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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 $Bbbk[K_{r_1,ldots,r_n}]$ with $1 leq r_1 leq cdots leq r_n$. We give the complete characterization of which $Bbbk[K_{r_1,ldots,r_n}]$ is level in terms of $n$ and $r_1,ldots,r_n$. Similarly, we also give the complete characterization of which $Bbbk[K_{r_1,ldots,r_n}]$ is almost Gorenstein in terms of $n$ and $r_1,ldots,r_n$.



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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.
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