{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BUSSBWIBLXSRMFOEQTGF65C56G","short_pith_number":"pith:BUSSBWIB","canonical_record":{"source":{"id":"1107.3102","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-07-15T16:28:34Z","cross_cats_sorted":[],"title_canon_sha256":"2345f5997b01f3a60eb90f5418ae2f29ea6d3fe53e748ee091829a3d8b5ec0e9","abstract_canon_sha256":"bcf336ffd2e28fb9b687a4f202d8f47bdb2223bf73475c4285f26de9edf64e90"},"schema_version":"1.0"},"canonical_sha256":"0d2520d9015de51615c484cc5f745df1afb1c89776da53f5356be9e3cf2e452e","source":{"kind":"arxiv","id":"1107.3102","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3102","created_at":"2026-05-18T03:04:38Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3102v2","created_at":"2026-05-18T03:04:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3102","created_at":"2026-05-18T03:04:38Z"},{"alias_kind":"pith_short_12","alias_value":"BUSSBWIBLXSR","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BUSSBWIBLXSRMFOE","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BUSSBWIB","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BUSSBWIBLXSRMFOEQTGF65C56G","target":"record","payload":{"canonical_record":{"source":{"id":"1107.3102","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-07-15T16:28:34Z","cross_cats_sorted":[],"title_canon_sha256":"2345f5997b01f3a60eb90f5418ae2f29ea6d3fe53e748ee091829a3d8b5ec0e9","abstract_canon_sha256":"bcf336ffd2e28fb9b687a4f202d8f47bdb2223bf73475c4285f26de9edf64e90"},"schema_version":"1.0"},"canonical_sha256":"0d2520d9015de51615c484cc5f745df1afb1c89776da53f5356be9e3cf2e452e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:38.830836Z","signature_b64":"KCu1GP7plENoI8I4wjtyEHlK2amlTOEkizgYWORzJepypPEyEyjRRcwXGI+u19Fvi7/NhHpSPeX55aHjUJs1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d2520d9015de51615c484cc5f745df1afb1c89776da53f5356be9e3cf2e452e","last_reissued_at":"2026-05-18T03:04:38.830031Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:38.830031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.3102","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":"2026-05-18T03:04:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eYWDzPVVYP1zUsl3aLsDoHPI28W1iZRDzjILvKWYmJQyk90vtX7k9ULkxIqtEaeKFvqv6CsFOumIALDZzvWnBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T19:56:42.151046Z"},"content_sha256":"1dc052a0c0738b15071c562fd25f629326178ed3a3c9d6be2fb504222488f9f9","schema_version":"1.0","event_id":"sha256:1dc052a0c0738b15071c562fd25f629326178ed3a3c9d6be2fb504222488f9f9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BUSSBWIBLXSRMFOEQTGF65C56G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Vanishing of Tate homology and depth formulas over local rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"David A. Jorgensen, Lars Winther Christensen","submitted_at":"2011-07-15T16:28:34Z","abstract_excerpt":"Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M \\otimes N) = depth(M) + depth(N) - depth(R), has been generalized in several directions over a span of four decades. In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes previous eneralizations of Auslander's formula and yields exact bounds for vanishing of cohomology over certain Gorenstein rings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3102","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":"2026-05-18T03:04:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sEUpqLadUHxVh690ABmKc6qVRFMT4OQcfGeDmJmZ5V/ndkVPjcZtR4nLfnRiRwrCPWQ17xwoIcuuzU+WkozABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T19:56:42.151415Z"},"content_sha256":"ba9255ddf752d5e8cb78412f17c51d021010ff965e04c7b24cc6a87eba57ffbc","schema_version":"1.0","event_id":"sha256:ba9255ddf752d5e8cb78412f17c51d021010ff965e04c7b24cc6a87eba57ffbc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BUSSBWIBLXSRMFOEQTGF65C56G/bundle.json","state_url":"https://pith.science/pith/BUSSBWIBLXSRMFOEQTGF65C56G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BUSSBWIBLXSRMFOEQTGF65C56G/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":"2026-06-23T19:56:42Z","links":{"resolver":"https://pith.science/pith/BUSSBWIBLXSRMFOEQTGF65C56G","bundle":"https://pith.science/pith/BUSSBWIBLXSRMFOEQTGF65C56G/bundle.json","state":"https://pith.science/pith/BUSSBWIBLXSRMFOEQTGF65C56G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BUSSBWIBLXSRMFOEQTGF65C56G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BUSSBWIBLXSRMFOEQTGF65C56G","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":"bcf336ffd2e28fb9b687a4f202d8f47bdb2223bf73475c4285f26de9edf64e90","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-07-15T16:28:34Z","title_canon_sha256":"2345f5997b01f3a60eb90f5418ae2f29ea6d3fe53e748ee091829a3d8b5ec0e9"},"schema_version":"1.0","source":{"id":"1107.3102","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3102","created_at":"2026-05-18T03:04:38Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3102v2","created_at":"2026-05-18T03:04:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3102","created_at":"2026-05-18T03:04:38Z"},{"alias_kind":"pith_short_12","alias_value":"BUSSBWIBLXSR","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BUSSBWIBLXSRMFOE","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BUSSBWIB","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:ba9255ddf752d5e8cb78412f17c51d021010ff965e04c7b24cc6a87eba57ffbc","target":"graph","created_at":"2026-05-18T03:04:38Z","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":"Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M \\otimes N) = depth(M) + depth(N) - depth(R), has been generalized in several directions over a span of four decades. In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes previous eneralizations of Auslander's formula and yields exact bounds for vanishing of cohomology over certain Gorenstein rings.","authors_text":"David A. Jorgensen, Lars Winther Christensen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-07-15T16:28:34Z","title":"Vanishing of Tate homology and depth formulas over local rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3102","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":"sha256:1dc052a0c0738b15071c562fd25f629326178ed3a3c9d6be2fb504222488f9f9","target":"record","created_at":"2026-05-18T03:04:38Z","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":"bcf336ffd2e28fb9b687a4f202d8f47bdb2223bf73475c4285f26de9edf64e90","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-07-15T16:28:34Z","title_canon_sha256":"2345f5997b01f3a60eb90f5418ae2f29ea6d3fe53e748ee091829a3d8b5ec0e9"},"schema_version":"1.0","source":{"id":"1107.3102","kind":"arxiv","version":2}},"canonical_sha256":"0d2520d9015de51615c484cc5f745df1afb1c89776da53f5356be9e3cf2e452e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d2520d9015de51615c484cc5f745df1afb1c89776da53f5356be9e3cf2e452e","first_computed_at":"2026-05-18T03:04:38.830031Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:38.830031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KCu1GP7plENoI8I4wjtyEHlK2amlTOEkizgYWORzJepypPEyEyjRRcwXGI+u19Fvi7/NhHpSPeX55aHjUJs1Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:38.830836Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.3102","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1dc052a0c0738b15071c562fd25f629326178ed3a3c9d6be2fb504222488f9f9","sha256:ba9255ddf752d5e8cb78412f17c51d021010ff965e04c7b24cc6a87eba57ffbc"],"state_sha256":"23d55b16b18011a3c7a4148fb79b1cbd8045eae8c40e74a9d95eddc14153fa0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QdXCogAwvt29LfIJGGm9EAA+sGS3MQywDO4m89y6cX6O4XmmYKX7XB38QbGtlnNr+m2eRTqtH6hf9ji/dP2mCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T19:56:42.153338Z","bundle_sha256":"c1706d87517aabdde0aecb25148c9ead1a4bbe20f4f484570c88e6955fe9e1b4"}}