{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:4YO6FQAW52TGN7YQMXRTFDXF5T","short_pith_number":"pith:4YO6FQAW","canonical_record":{"source":{"id":"1711.06931","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-11-18T22:14:42Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"61b2c69decd6ba0adec6676880ebad9f330e4946c1f7cf81728163b314f9d378","abstract_canon_sha256":"4ad8110adc1f9aef849452a4e2b1057f72f78c4b80b35a86893f040138f4fb98"},"schema_version":"1.0"},"canonical_sha256":"e61de2c016eea666ff1065e3328ee5ecf0c654108bb9c73bff0a1f76f80ff81a","source":{"kind":"arxiv","id":"1711.06931","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.06931","created_at":"2026-05-18T00:30:15Z"},{"alias_kind":"arxiv_version","alias_value":"1711.06931v1","created_at":"2026-05-18T00:30:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.06931","created_at":"2026-05-18T00:30:15Z"},{"alias_kind":"pith_short_12","alias_value":"4YO6FQAW52TG","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4YO6FQAW52TGN7YQ","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4YO6FQAW","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:4YO6FQAW52TGN7YQMXRTFDXF5T","target":"record","payload":{"canonical_record":{"source":{"id":"1711.06931","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-11-18T22:14:42Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"61b2c69decd6ba0adec6676880ebad9f330e4946c1f7cf81728163b314f9d378","abstract_canon_sha256":"4ad8110adc1f9aef849452a4e2b1057f72f78c4b80b35a86893f040138f4fb98"},"schema_version":"1.0"},"canonical_sha256":"e61de2c016eea666ff1065e3328ee5ecf0c654108bb9c73bff0a1f76f80ff81a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:15.146062Z","signature_b64":"kc4S6CQ2xECRAwS3erRPt5zpFYY+XPFygEtnBcq6mVO8OaGFXAdrgS1jf7h+X4feet9WO9EfynIBfqwIE6iHAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e61de2c016eea666ff1065e3328ee5ecf0c654108bb9c73bff0a1f76f80ff81a","last_reissued_at":"2026-05-18T00:30:15.145338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:15.145338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.06931","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":"2026-05-18T00:30:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"flL5vdkg7++F/jLJKwFox4bBtdmYQM+Nb24cIopOH37CsVnCyOz8BbHf9Z/FdbfbX5ff3FETIgADUrgX+PylDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:37:59.747980Z"},"content_sha256":"71061c1b291f1bc7b1fc49d62ae03ad7cec8b350a0f66e9767230b408ec5982b","schema_version":"1.0","event_id":"sha256:71061c1b291f1bc7b1fc49d62ae03ad7cec8b350a0f66e9767230b408ec5982b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:4YO6FQAW52TGN7YQMXRTFDXF5T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Koszul duality between Betti and Cohomology numbers in Calabi-Yau case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Alexander Pavlov","submitted_at":"2017-11-18T22:14:42Z","abstract_excerpt":"Let $X$ be a smooth projective Calabi-Yau variety and $L$ a Koszul line bundle on $X$. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring $A$ of $X$ there are formulas similar to the formulas for cohomology number. This similarity is realized via the box-product resolution of the diagonal $\\Delta_X \\subset X \\times X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06931","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":""},"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":"2026-05-18T00:30:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GAZoYs88xFbEPgwk/uXBsfeIQM/CIw+47jbxbAm1rjdcIx2fJMv0/5Hs1W6jcnM7Fz1mLPfSsdaSaKaoOrjsAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:37:59.748334Z"},"content_sha256":"b18e390c82d90ec6139edc34aa48c402e901533d93b5c4f022cd2ef76ca8dc3e","schema_version":"1.0","event_id":"sha256:b18e390c82d90ec6139edc34aa48c402e901533d93b5c4f022cd2ef76ca8dc3e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4YO6FQAW52TGN7YQMXRTFDXF5T/bundle.json","state_url":"https://pith.science/pith/4YO6FQAW52TGN7YQMXRTFDXF5T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4YO6FQAW52TGN7YQMXRTFDXF5T/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":"2026-05-28T12:37:59Z","links":{"resolver":"https://pith.science/pith/4YO6FQAW52TGN7YQMXRTFDXF5T","bundle":"https://pith.science/pith/4YO6FQAW52TGN7YQMXRTFDXF5T/bundle.json","state":"https://pith.science/pith/4YO6FQAW52TGN7YQMXRTFDXF5T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4YO6FQAW52TGN7YQMXRTFDXF5T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4YO6FQAW52TGN7YQMXRTFDXF5T","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":"4ad8110adc1f9aef849452a4e2b1057f72f78c4b80b35a86893f040138f4fb98","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-11-18T22:14:42Z","title_canon_sha256":"61b2c69decd6ba0adec6676880ebad9f330e4946c1f7cf81728163b314f9d378"},"schema_version":"1.0","source":{"id":"1711.06931","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.06931","created_at":"2026-05-18T00:30:15Z"},{"alias_kind":"arxiv_version","alias_value":"1711.06931v1","created_at":"2026-05-18T00:30:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.06931","created_at":"2026-05-18T00:30:15Z"},{"alias_kind":"pith_short_12","alias_value":"4YO6FQAW52TG","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4YO6FQAW52TGN7YQ","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4YO6FQAW","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:b18e390c82d90ec6139edc34aa48c402e901533d93b5c4f022cd2ef76ca8dc3e","target":"graph","created_at":"2026-05-18T00:30:15Z","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"},"paper":{"abstract_excerpt":"Let $X$ be a smooth projective Calabi-Yau variety and $L$ a Koszul line bundle on $X$. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring $A$ of $X$ there are formulas similar to the formulas for cohomology number. This similarity is realized via the box-product resolution of the diagonal $\\Delta_X \\subset X \\times X$.","authors_text":"Alexander Pavlov","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-11-18T22:14:42Z","title":"Koszul duality between Betti and Cohomology numbers in Calabi-Yau case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06931","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":"sha256:71061c1b291f1bc7b1fc49d62ae03ad7cec8b350a0f66e9767230b408ec5982b","target":"record","created_at":"2026-05-18T00:30:15Z","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":"4ad8110adc1f9aef849452a4e2b1057f72f78c4b80b35a86893f040138f4fb98","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-11-18T22:14:42Z","title_canon_sha256":"61b2c69decd6ba0adec6676880ebad9f330e4946c1f7cf81728163b314f9d378"},"schema_version":"1.0","source":{"id":"1711.06931","kind":"arxiv","version":1}},"canonical_sha256":"e61de2c016eea666ff1065e3328ee5ecf0c654108bb9c73bff0a1f76f80ff81a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e61de2c016eea666ff1065e3328ee5ecf0c654108bb9c73bff0a1f76f80ff81a","first_computed_at":"2026-05-18T00:30:15.145338Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:30:15.145338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kc4S6CQ2xECRAwS3erRPt5zpFYY+XPFygEtnBcq6mVO8OaGFXAdrgS1jf7h+X4feet9WO9EfynIBfqwIE6iHAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:30:15.146062Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.06931","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71061c1b291f1bc7b1fc49d62ae03ad7cec8b350a0f66e9767230b408ec5982b","sha256:b18e390c82d90ec6139edc34aa48c402e901533d93b5c4f022cd2ef76ca8dc3e"],"state_sha256":"366269dcb34c0e487e25bf79c22975e2fdd18db640e67a1df3d854bbb0987d2a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C5MX8nLWe+Txi7AywQQ2QaO0EEK0YFl+zSZDGjK+Y+a394TEpcT5d1FFEY/+s3WIZDAjgvLqyMH+8uKx6vc6Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T12:37:59.750279Z","bundle_sha256":"7e81c7007cdd719cf68ebae13d9f6b3bc671c0ba55ce2c7a3167e22702b0f6cf"}}