{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7NJVUT4TAPPCLRTHQWP2G772AO","short_pith_number":"pith:7NJVUT4T","canonical_record":{"source":{"id":"1701.06475","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-01-23T16:12:55Z","cross_cats_sorted":[],"title_canon_sha256":"f84e371468eca78fe9cd0000e2a89cd79fed8b09706f7ca6622d9772d7e61669","abstract_canon_sha256":"afe0081ef6d2d3b0ab306a4c2643295d3e0e360709a1096f56987671d428caa6"},"schema_version":"1.0"},"canonical_sha256":"fb535a4f9303de25c667859fa37ffa0383a87d748eb57d306225cf05af15b599","source":{"kind":"arxiv","id":"1701.06475","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.06475","created_at":"2026-05-18T00:52:18Z"},{"alias_kind":"arxiv_version","alias_value":"1701.06475v1","created_at":"2026-05-18T00:52:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.06475","created_at":"2026-05-18T00:52:18Z"},{"alias_kind":"pith_short_12","alias_value":"7NJVUT4TAPPC","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7NJVUT4TAPPCLRTH","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7NJVUT4T","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7NJVUT4TAPPCLRTHQWP2G772AO","target":"record","payload":{"canonical_record":{"source":{"id":"1701.06475","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-01-23T16:12:55Z","cross_cats_sorted":[],"title_canon_sha256":"f84e371468eca78fe9cd0000e2a89cd79fed8b09706f7ca6622d9772d7e61669","abstract_canon_sha256":"afe0081ef6d2d3b0ab306a4c2643295d3e0e360709a1096f56987671d428caa6"},"schema_version":"1.0"},"canonical_sha256":"fb535a4f9303de25c667859fa37ffa0383a87d748eb57d306225cf05af15b599","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:18.255547Z","signature_b64":"Tr2mUWzfgepjbJKiniwkXoyhgxffvR7OKKuoSS2IXCsrgR3qTcEfzS3p1OPxc/FLosQgQIZISna1Jk9SYq84Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb535a4f9303de25c667859fa37ffa0383a87d748eb57d306225cf05af15b599","last_reissued_at":"2026-05-18T00:52:18.254995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:18.254995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.06475","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":"2026-05-18T00:52:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VC37VoC8dKnpn8gnZbV9LwFL4Bj5xPXypp0dU97vMH995EWvtRpkmjbOgpDz6Rz1UP+Eij+rJ9oIly3NGmdjDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:58:24.301421Z"},"content_sha256":"781304abcd993fc6230d24802d51e9de74ba8256f9de2634763161e5be801b32","schema_version":"1.0","event_id":"sha256:781304abcd993fc6230d24802d51e9de74ba8256f9de2634763161e5be801b32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7NJVUT4TAPPCLRTHQWP2G772AO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modules with Pure Resolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"H. Ananthnarayan, Rajiv Kumar","submitted_at":"2017-01-23T16:12:55Z","abstract_excerpt":"We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06475","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":"2026-05-18T00:52:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XcmnHHVkNu+THgBAcMbII860Sbd4jj7L8VhmF28NoMmr8tMmu3fnMAxSbC2hiTEz5iIhalV3hM0UR7aen5+DAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:58:24.301778Z"},"content_sha256":"1c7bcfc32278d3009869e87dbd5ff5068f44d9d9f268705a632bfec05b747a88","schema_version":"1.0","event_id":"sha256:1c7bcfc32278d3009869e87dbd5ff5068f44d9d9f268705a632bfec05b747a88"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7NJVUT4TAPPCLRTHQWP2G772AO/bundle.json","state_url":"https://pith.science/pith/7NJVUT4TAPPCLRTHQWP2G772AO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7NJVUT4TAPPCLRTHQWP2G772AO/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":"2026-06-03T18:58:24Z","links":{"resolver":"https://pith.science/pith/7NJVUT4TAPPCLRTHQWP2G772AO","bundle":"https://pith.science/pith/7NJVUT4TAPPCLRTHQWP2G772AO/bundle.json","state":"https://pith.science/pith/7NJVUT4TAPPCLRTHQWP2G772AO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7NJVUT4TAPPCLRTHQWP2G772AO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7NJVUT4TAPPCLRTHQWP2G772AO","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":"afe0081ef6d2d3b0ab306a4c2643295d3e0e360709a1096f56987671d428caa6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-01-23T16:12:55Z","title_canon_sha256":"f84e371468eca78fe9cd0000e2a89cd79fed8b09706f7ca6622d9772d7e61669"},"schema_version":"1.0","source":{"id":"1701.06475","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.06475","created_at":"2026-05-18T00:52:18Z"},{"alias_kind":"arxiv_version","alias_value":"1701.06475v1","created_at":"2026-05-18T00:52:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.06475","created_at":"2026-05-18T00:52:18Z"},{"alias_kind":"pith_short_12","alias_value":"7NJVUT4TAPPC","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7NJVUT4TAPPCLRTH","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7NJVUT4T","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:1c7bcfc32278d3009869e87dbd5ff5068f44d9d9f268705a632bfec05b747a88","target":"graph","created_at":"2026-05-18T00:52:18Z","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 show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.","authors_text":"H. Ananthnarayan, Rajiv Kumar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-01-23T16:12:55Z","title":"Modules with Pure Resolutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06475","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":"sha256:781304abcd993fc6230d24802d51e9de74ba8256f9de2634763161e5be801b32","target":"record","created_at":"2026-05-18T00:52:18Z","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":"afe0081ef6d2d3b0ab306a4c2643295d3e0e360709a1096f56987671d428caa6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-01-23T16:12:55Z","title_canon_sha256":"f84e371468eca78fe9cd0000e2a89cd79fed8b09706f7ca6622d9772d7e61669"},"schema_version":"1.0","source":{"id":"1701.06475","kind":"arxiv","version":1}},"canonical_sha256":"fb535a4f9303de25c667859fa37ffa0383a87d748eb57d306225cf05af15b599","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb535a4f9303de25c667859fa37ffa0383a87d748eb57d306225cf05af15b599","first_computed_at":"2026-05-18T00:52:18.254995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:18.254995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Tr2mUWzfgepjbJKiniwkXoyhgxffvR7OKKuoSS2IXCsrgR3qTcEfzS3p1OPxc/FLosQgQIZISna1Jk9SYq84Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:18.255547Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.06475","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:781304abcd993fc6230d24802d51e9de74ba8256f9de2634763161e5be801b32","sha256:1c7bcfc32278d3009869e87dbd5ff5068f44d9d9f268705a632bfec05b747a88"],"state_sha256":"4931a0a35c77832bfbb7d0c935e80f4e8267b60323b1d58c78bce011cfbb8692"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GsmurzCICRuSBc2w2kG5ED97yjBWJx0xqGxGS94PC7rQrqcso/xQk0qN6FAgfKs94MDM6MWRziF3JwX12awRCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T18:58:24.303937Z","bundle_sha256":"c1b2519ea7a85d0c327fcc60500819de750c89be677bec23ab3ab65a7da8ab8f"}}