{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CJZTURMCYISYJ2GEIWSFC2XT6W","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":"2dde54a37bb6139d2babf4d5caf89da7c864e6b672f48531af2015b043854b1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-15T12:52:02Z","title_canon_sha256":"f7da406dbcce8bd336ebab49c3a86b1e938c4510e59c4eab64b81803e42890d3"},"schema_version":"1.0","source":{"id":"1407.3967","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.3967","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"arxiv_version","alias_value":"1407.3967v4","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.3967","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"pith_short_12","alias_value":"CJZTURMCYISY","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CJZTURMCYISYJ2GE","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CJZTURMC","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:fe7f12b7e9eac7919366d7895ae25e36ce81a9ed70c3a0c088f4fca40719ffef","target":"graph","created_at":"2026-05-18T01:33:57Z","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":"In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger than the projective dimension of S/I and (iii) the K-algebra generated by some generators of I is a direct summand of S, then f(k) is constant. When I is a square-free monomial ideal, the above criterion includes as special cases all the results of a recent paper by Herzog and Vladoiu. In this combinatorial setting there is a chance that the converse of the ","authors_text":"Le Dinh Nam, Matteo Varbaro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-15T12:52:02Z","title":"When does depth stabilize early on?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3967","kind":"arxiv","version":4},"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":"sha256:15e716a5c10e929aafb5439ccbdaf65bcef6bb8b837db1cb62a35109560a7163","target":"record","created_at":"2026-05-18T01:33:57Z","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":"2dde54a37bb6139d2babf4d5caf89da7c864e6b672f48531af2015b043854b1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-15T12:52:02Z","title_canon_sha256":"f7da406dbcce8bd336ebab49c3a86b1e938c4510e59c4eab64b81803e42890d3"},"schema_version":"1.0","source":{"id":"1407.3967","kind":"arxiv","version":4}},"canonical_sha256":"12733a4582c22584e8c445a4516af3f59127f78284af0f548ae8c0f425aedc93","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"12733a4582c22584e8c445a4516af3f59127f78284af0f548ae8c0f425aedc93","first_computed_at":"2026-05-18T01:33:57.023637Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:57.023637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q4PdiuySmHoEOhFV81a+gNiMrxzZrg1c/BdpyvwMNlrGZqxgnug0GnZMJXARKVqaNoAw/RZMGV4dHgkgq8c7CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:57.023987Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.3967","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15e716a5c10e929aafb5439ccbdaf65bcef6bb8b837db1cb62a35109560a7163","sha256:fe7f12b7e9eac7919366d7895ae25e36ce81a9ed70c3a0c088f4fca40719ffef"],"state_sha256":"c455881adb869e346363f295de5db914e550615f00c2759c378f79c6fd1a6626"}