{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:Y3F44FDDK7YAMZTHXQVT2A5KL7","short_pith_number":"pith:Y3F44FDD","canonical_record":{"source":{"id":"1008.2573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-16T05:11:59Z","cross_cats_sorted":[],"title_canon_sha256":"0599df8f9b5351e38219eb6eee673d022d3fd231b600046ef627f525ac368871","abstract_canon_sha256":"f70b3e93f381c7cb217b56318a3bbe2ec5fe1d10e9dcb088cad1a168b2ef7575"},"schema_version":"1.0"},"canonical_sha256":"c6cbce146357f0066667bc2b3d03aa5fe30866bfc42e122d699f347d2b496edc","source":{"kind":"arxiv","id":"1008.2573","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.2573","created_at":"2026-05-18T04:23:34Z"},{"alias_kind":"arxiv_version","alias_value":"1008.2573v2","created_at":"2026-05-18T04:23:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.2573","created_at":"2026-05-18T04:23:34Z"},{"alias_kind":"pith_short_12","alias_value":"Y3F44FDDK7YA","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Y3F44FDDK7YAMZTH","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Y3F44FDD","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:Y3F44FDDK7YAMZTHXQVT2A5KL7","target":"record","payload":{"canonical_record":{"source":{"id":"1008.2573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-16T05:11:59Z","cross_cats_sorted":[],"title_canon_sha256":"0599df8f9b5351e38219eb6eee673d022d3fd231b600046ef627f525ac368871","abstract_canon_sha256":"f70b3e93f381c7cb217b56318a3bbe2ec5fe1d10e9dcb088cad1a168b2ef7575"},"schema_version":"1.0"},"canonical_sha256":"c6cbce146357f0066667bc2b3d03aa5fe30866bfc42e122d699f347d2b496edc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:34.283753Z","signature_b64":"/shyyN2OasZEJhCWBYefg8oRtFMg60LJYh/T6omth9Z1jNXfAm164ZsvAmbqJKOzEZ6wdFf0Z9PLUOd40l1aDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6cbce146357f0066667bc2b3d03aa5fe30866bfc42e122d699f347d2b496edc","last_reissued_at":"2026-05-18T04:23:34.283335Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:34.283335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.2573","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VAmhW6bVc75qrKfF/Md8L/KAzRXl9Oz26mysTefVh+agblFQMSGyjwDum6PUSJur0IoCciP1HlF0AUVSfMBOBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:27:32.433676Z"},"content_sha256":"983a3b552346f93ed5b0bc754225b14d14d79845193c922b613846c9254c7dad","schema_version":"1.0","event_id":"sha256:983a3b552346f93ed5b0bc754225b14d14d79845193c922b613846c9254c7dad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:Y3F44FDDK7YAMZTHXQVT2A5KL7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Necessary conditions for the depth formula over Cohen-Macaulay local rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hailong Dao, Olgur Celikbas","submitted_at":"2010-08-16T05:11:59Z","abstract_excerpt":"Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\\depth(M)+\\depth(N)=\\depth(R)+\\depth(M\\otimes_{R}N)$ to hold. We show that, under certain conditions, $M$ and $N$ satisfy the depth formula if and only if $\\Tor_{i}^{R}(M,N)$ vanishes for all $i\\geq 1$. We also examine the relationship between good depth of $M\\otimes_RN$ and the vanishing of $\\Ext$ modules, with various applications."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2573","kind":"arxiv","version":2},"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":"2026-05-18T04:23:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HmnNvqTQP3hujpqTnTJNATURWkWguWGI+PwXIdZ/Zt02j7KaJAOc80YxLJcD0ZZiuqOkPv6iWweIhPaHV/gVCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:27:32.434416Z"},"content_sha256":"aca67590beafb9fdd7ed1dec65bb9fbcfb3af4825918f6eefad078004daa3d85","schema_version":"1.0","event_id":"sha256:aca67590beafb9fdd7ed1dec65bb9fbcfb3af4825918f6eefad078004daa3d85"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7/bundle.json","state_url":"https://pith.science/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7/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":"2026-06-08T02:27:32Z","links":{"resolver":"https://pith.science/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7","bundle":"https://pith.science/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7/bundle.json","state":"https://pith.science/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y3F44FDDK7YAMZTHXQVT2A5KL7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:Y3F44FDDK7YAMZTHXQVT2A5KL7","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":"f70b3e93f381c7cb217b56318a3bbe2ec5fe1d10e9dcb088cad1a168b2ef7575","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-16T05:11:59Z","title_canon_sha256":"0599df8f9b5351e38219eb6eee673d022d3fd231b600046ef627f525ac368871"},"schema_version":"1.0","source":{"id":"1008.2573","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.2573","created_at":"2026-05-18T04:23:34Z"},{"alias_kind":"arxiv_version","alias_value":"1008.2573v2","created_at":"2026-05-18T04:23:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.2573","created_at":"2026-05-18T04:23:34Z"},{"alias_kind":"pith_short_12","alias_value":"Y3F44FDDK7YA","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"Y3F44FDDK7YAMZTH","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"Y3F44FDD","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:aca67590beafb9fdd7ed1dec65bb9fbcfb3af4825918f6eefad078004daa3d85","target":"graph","created_at":"2026-05-18T04:23:34Z","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 $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\\depth(M)+\\depth(N)=\\depth(R)+\\depth(M\\otimes_{R}N)$ to hold. We show that, under certain conditions, $M$ and $N$ satisfy the depth formula if and only if $\\Tor_{i}^{R}(M,N)$ vanishes for all $i\\geq 1$. We also examine the relationship between good depth of $M\\otimes_RN$ and the vanishing of $\\Ext$ modules, with various applications.","authors_text":"Hailong Dao, Olgur Celikbas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-16T05:11:59Z","title":"Necessary conditions for the depth formula over Cohen-Macaulay local rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2573","kind":"arxiv","version":2},"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":"sha256:983a3b552346f93ed5b0bc754225b14d14d79845193c922b613846c9254c7dad","target":"record","created_at":"2026-05-18T04:23:34Z","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":"f70b3e93f381c7cb217b56318a3bbe2ec5fe1d10e9dcb088cad1a168b2ef7575","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-08-16T05:11:59Z","title_canon_sha256":"0599df8f9b5351e38219eb6eee673d022d3fd231b600046ef627f525ac368871"},"schema_version":"1.0","source":{"id":"1008.2573","kind":"arxiv","version":2}},"canonical_sha256":"c6cbce146357f0066667bc2b3d03aa5fe30866bfc42e122d699f347d2b496edc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6cbce146357f0066667bc2b3d03aa5fe30866bfc42e122d699f347d2b496edc","first_computed_at":"2026-05-18T04:23:34.283335Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:34.283335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/shyyN2OasZEJhCWBYefg8oRtFMg60LJYh/T6omth9Z1jNXfAm164ZsvAmbqJKOzEZ6wdFf0Z9PLUOd40l1aDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:34.283753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.2573","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:983a3b552346f93ed5b0bc754225b14d14d79845193c922b613846c9254c7dad","sha256:aca67590beafb9fdd7ed1dec65bb9fbcfb3af4825918f6eefad078004daa3d85"],"state_sha256":"4cc1127be7fd1f6e14a428eb5320a99631c7d8bae6bad391cd653fed5a460ea8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bVYVi9Cu0YjJKrp6zc12l9mm2hqkegSUL5LZWrMFHnbam/Ho7Sfi76UWcbzHvTy8iGIJrxXJInKWO0cm7T7FAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T02:27:32.438174Z","bundle_sha256":"35ff696da808904892eada1646cc06e3e4a30e1a2eef35527c9ca9922c8a09db"}}