{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:TEHJHOYTMIH4ALXM7SZGWL5EQT","short_pith_number":"pith:TEHJHOYT","canonical_record":{"source":{"id":"2102.02349","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2021-02-04T00:36:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3b16ed671d7942d2716b62560290378ce3da5379448b9a284e6a61464ed3fb0e","abstract_canon_sha256":"6cc5490cc11d849a36454ee8a56d904f353de166de245c668219bdbeb1e9e02f"},"schema_version":"1.0"},"canonical_sha256":"990e93bb13620fc02eecfcb26b2fa484d886a3cb37e992f60b399ce4efb624eb","source":{"kind":"arxiv","id":"2102.02349","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.02349","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"arxiv_version","alias_value":"2102.02349v1","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.02349","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"pith_short_12","alias_value":"TEHJHOYTMIH4","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"pith_short_16","alias_value":"TEHJHOYTMIH4ALXM","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"pith_short_8","alias_value":"TEHJHOYT","created_at":"2026-07-05T02:12:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:TEHJHOYTMIH4ALXM7SZGWL5EQT","target":"record","payload":{"canonical_record":{"source":{"id":"2102.02349","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2021-02-04T00:36:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3b16ed671d7942d2716b62560290378ce3da5379448b9a284e6a61464ed3fb0e","abstract_canon_sha256":"6cc5490cc11d849a36454ee8a56d904f353de166de245c668219bdbeb1e9e02f"},"schema_version":"1.0"},"canonical_sha256":"990e93bb13620fc02eecfcb26b2fa484d886a3cb37e992f60b399ce4efb624eb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:12:47.292066Z","signature_b64":"KomPX8YigHcnqf0xyTQxozWRudslxZwDtI8Pdch8ZaLGAn6wlY65mAVX/cZTBK6lZVJza7VXeAos7idtzwcaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"990e93bb13620fc02eecfcb26b2fa484d886a3cb37e992f60b399ce4efb624eb","last_reissued_at":"2026-07-05T02:12:47.291704Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:12:47.291704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.02349","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:12:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uAEp78sP2TlvNe0jUBOybVPqDZfvOM6o9qXzoBwEQ85M6S9c1KhSZ70tW7Nx5hz6IjeKBWaJ2kwcX7+BvlC/BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:04:51.148922Z"},"content_sha256":"c41274c255abedba3fbe0f64feaa2060dbd93964c5341f4e654a98cd04a54bc0","schema_version":"1.0","event_id":"sha256:c41274c255abedba3fbe0f64feaa2060dbd93964c5341f4e654a98cd04a54bc0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:TEHJHOYTMIH4ALXM7SZGWL5EQT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Akihiro Higashitani, Koji Matsushita","submitted_at":"2021-02-04T00:36:20Z","abstract_excerpt":"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 $r_1,\\ldots,r_n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.02349","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/2102.02349/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:12:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bXu9qeFRxOaVoP+qLwWNV/bkSGMXtgTt+b6jSyOaMLv/R8N5lN4MxY1+0+uqIfmvpkUF2HgCkY1yisgL7/rjBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:04:51.149429Z"},"content_sha256":"ce88e0e4c98bc8b0b03d8b722e158d7067be02c8f6f4389018b866688af89886","schema_version":"1.0","event_id":"sha256:ce88e0e4c98bc8b0b03d8b722e158d7067be02c8f6f4389018b866688af89886"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT/bundle.json","state_url":"https://pith.science/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T08:04:51Z","links":{"resolver":"https://pith.science/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT","bundle":"https://pith.science/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT/bundle.json","state":"https://pith.science/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TEHJHOYTMIH4ALXM7SZGWL5EQT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:TEHJHOYTMIH4ALXM7SZGWL5EQT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6cc5490cc11d849a36454ee8a56d904f353de166de245c668219bdbeb1e9e02f","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2021-02-04T00:36:20Z","title_canon_sha256":"3b16ed671d7942d2716b62560290378ce3da5379448b9a284e6a61464ed3fb0e"},"schema_version":"1.0","source":{"id":"2102.02349","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.02349","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"arxiv_version","alias_value":"2102.02349v1","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.02349","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"pith_short_12","alias_value":"TEHJHOYTMIH4","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"pith_short_16","alias_value":"TEHJHOYTMIH4ALXM","created_at":"2026-07-05T02:12:47Z"},{"alias_kind":"pith_short_8","alias_value":"TEHJHOYT","created_at":"2026-07-05T02:12:47Z"}],"graph_snapshots":[{"event_id":"sha256:ce88e0e4c98bc8b0b03d8b722e158d7067be02c8f6f4389018b866688af89886","target":"graph","created_at":"2026-07-05T02:12:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2102.02349/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"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 $r_1,\\ldots,r_n$.","authors_text":"Akihiro Higashitani, Koji Matsushita","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2021-02-04T00:36:20Z","title":"Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.02349","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c41274c255abedba3fbe0f64feaa2060dbd93964c5341f4e654a98cd04a54bc0","target":"record","created_at":"2026-07-05T02:12:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6cc5490cc11d849a36454ee8a56d904f353de166de245c668219bdbeb1e9e02f","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2021-02-04T00:36:20Z","title_canon_sha256":"3b16ed671d7942d2716b62560290378ce3da5379448b9a284e6a61464ed3fb0e"},"schema_version":"1.0","source":{"id":"2102.02349","kind":"arxiv","version":1}},"canonical_sha256":"990e93bb13620fc02eecfcb26b2fa484d886a3cb37e992f60b399ce4efb624eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"990e93bb13620fc02eecfcb26b2fa484d886a3cb37e992f60b399ce4efb624eb","first_computed_at":"2026-07-05T02:12:47.291704Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:12:47.291704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KomPX8YigHcnqf0xyTQxozWRudslxZwDtI8Pdch8ZaLGAn6wlY65mAVX/cZTBK6lZVJza7VXeAos7idtzwcaBg==","signature_status":"signed_v1","signed_at":"2026-07-05T02:12:47.292066Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.02349","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c41274c255abedba3fbe0f64feaa2060dbd93964c5341f4e654a98cd04a54bc0","sha256:ce88e0e4c98bc8b0b03d8b722e158d7067be02c8f6f4389018b866688af89886"],"state_sha256":"79755085e12c210ae5690c6232ecaef61287fd8db3aa08115da61f6e392095ec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ycj2BYxz1uOAITgVXJwYexoxZn00aJZ2c4GTR/j1Dq1I7/DMWJixSP1/07PCmAe4LfjpNfuNgaZVQx83uENZCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:04:51.153810Z","bundle_sha256":"9d17ac3e318d01ebf0fd11023c8c839e333c9d45b39c9dd767be46c26cd11e42"}}