{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:GBSFW6M36VXGRXVFHXZL5U3OUH","short_pith_number":"pith:GBSFW6M3","schema_version":"1.0","canonical_sha256":"30645b799bf56e68dea53df2bed36ea1c71eaf2c903ec33eef6e3f8c448e8237","source":{"kind":"arxiv","id":"2011.07714","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2011.07714","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AC","submitted_at":"2020-11-16T04:19:41Z","cross_cats_sorted":["math.AG","math.CO","math.RT"],"title_canon_sha256":"726dfa126e8ee852fc040cb2b9137be2f8d7b1be57f141d6477b2c0d8aa134bf","abstract_canon_sha256":"8bad3ebe87b50ed9b99395a70a07f9a99dee5da928ea13b513b72b5fc59bbdd0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:51:53.272823Z","signature_b64":"HRbeIRp3E0tOdcnfZMkIhXRLsVrx6W8BJkF3nxpjj2oP7Tu52lu1xQfTIVv6J+WpWjCUHcjd49ENXnGAkjDtDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30645b799bf56e68dea53df2bed36ea1c71eaf2c903ec33eef6e3f8c448e8237","last_reissued_at":"2026-07-05T01:51:53.272423Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:51:53.272423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2011.07714","created_at":"2026-07-05T01:51:53.272478+00:00"},{"alias_kind":"arxiv_version","alias_value":"2011.07714v1","created_at":"2026-07-05T01:51:53.272478+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.07714","created_at":"2026-07-05T01:51:53.272478+00:00"},{"alias_kind":"pith_short_12","alias_value":"GBSFW6M36VXG","created_at":"2026-07-05T01:51:53.272478+00:00"},{"alias_kind":"pith_short_16","alias_value":"GBSFW6M36VXGRXVF","created_at":"2026-07-05T01:51:53.272478+00:00"},{"alias_kind":"pith_short_8","alias_value":"GBSFW6M3","created_at":"2026-07-05T01:51:53.272478+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH","json":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH.json","graph_json":"https://pith.science/api/pith-number/GBSFW6M36VXGRXVFHXZL5U3OUH/graph.json","events_json":"https://pith.science/api/pith-number/GBSFW6M36VXGRXVFHXZL5U3OUH/events.json","paper":"https://pith.science/paper/GBSFW6M3"},"agent_actions":{"view_html":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH","download_json":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH.json","view_paper":"https://pith.science/paper/GBSFW6M3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2011.07714&json=true","fetch_graph":"https://pith.science/api/pith-number/GBSFW6M36VXGRXVFHXZL5U3OUH/graph.json","fetch_events":"https://pith.science/api/pith-number/GBSFW6M36VXGRXVFHXZL5U3OUH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH/action/storage_attestation","attest_author":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH/action/author_attestation","sign_citation":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH/action/citation_signature","submit_replication":"https://pith.science/pith/GBSFW6M36VXGRXVFHXZL5U3OUH/action/replication_record"}},"created_at":"2026-07-05T01:51:53.272478+00:00","updated_at":"2026-07-05T01:51:53.272478+00:00"}