{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:UJUU7JWF2J5CHJH2VXEU7OPJK7","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":"70b02fdc8b0b80b8df4f50c2ed947e6ab736ee80f0843e40e6a7de0a37a33050","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-10T05:46:07Z","title_canon_sha256":"2f1fd1058f63766ee72bb9b1608637230980de8be0172ad5b3ce3b215c5dba9c"},"schema_version":"1.0","source":{"id":"1008.1637","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1637","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1637v1","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1637","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"UJUU7JWF2J5C","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UJUU7JWF2J5CHJH2","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UJUU7JWF","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:3fb24564a044f2b32cb94b1bb010e3d0aba718fdf74c50a4efccc88614013efa","target":"graph","created_at":"2026-05-18T04:42:23Z","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":"We prove the depth formula, for homologically bounded complexes $X, Y$ provided that the complete intersection flat dimension of $X$ is finite and $\\sup(X\\utp_RY)<\\infty$. In particular, let $M$ and $N$ are two $R$-modules and the complete intersection flat dimension of $M$ is finite. Then $M$ and $N$ satisfies the depth formula, provided $\\Tor^R_i(M,N)=0$ for all $i\\ge 1$.","authors_text":"Parviz Sahandi, Siamak Yassemi, Tirdad Sharif","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-10T05:46:07Z","title":"Depth formula via complete intersection flat dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1637","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":"sha256:2e730e4b2a160fb1b103dc4f8b648af4488063585e9f5ba1d80fad30dff9b7ee","target":"record","created_at":"2026-05-18T04:42:23Z","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":"70b02fdc8b0b80b8df4f50c2ed947e6ab736ee80f0843e40e6a7de0a37a33050","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-10T05:46:07Z","title_canon_sha256":"2f1fd1058f63766ee72bb9b1608637230980de8be0172ad5b3ce3b215c5dba9c"},"schema_version":"1.0","source":{"id":"1008.1637","kind":"arxiv","version":1}},"canonical_sha256":"a2694fa6c5d27a23a4faadc94fb9e957fd49449fc25e9b8eb3701cd81e94964d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2694fa6c5d27a23a4faadc94fb9e957fd49449fc25e9b8eb3701cd81e94964d","first_computed_at":"2026-05-18T04:42:23.662918Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:23.662918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GR7JKz2XDZh4S8nLTvJuvpj7K3pmKykgeCjzhC37lO35SqOgO7iaOumPxxzn7YF7CJWHSc9mlaA7yeKTKF4+Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:23.663305Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.1637","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e730e4b2a160fb1b103dc4f8b648af4488063585e9f5ba1d80fad30dff9b7ee","sha256:3fb24564a044f2b32cb94b1bb010e3d0aba718fdf74c50a4efccc88614013efa"],"state_sha256":"224e09f6be0cb1aa13426e021c973dcaac72cd8e95c53c9e14df82dc190faded"}