Abstract
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by k[Kr1,…,rn], where 1≤r1≤⋯≤rn. More concretely, we prove that the class group of k[Kr1,…,rn] is isomorphic to Zn if n=3 with r1≥2 or n≥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 k[Kr1,…,rn], called conic divisorial ideals. We describe conic divisorial ideals for certain Kr1,…,rn including all cases where k[Kr1,…,rn] is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of k[Kr1,…,rn] in the case where it is Gorenstein.
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