{"paper":{"title":"Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.RT"],"primary_cat":"math.AC","authors_text":"Akihiro Higashitani, Koji Matsushita","submitted_at":"2020-11-16T04:19:41Z","abstract_excerpt":"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 fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.07714","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2011.07714/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}