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We say a ring satisfies $(S_{\\ell}^j)$ when $\\depth R_P \\geq \\min \\{ \\ell , \\dim R_P -j \\}$ for all $P \\in \\Spec R$. We prove generalizations of results for rings satisfying Serre's condition."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1710.02631","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-10-07T03:59:54Z","cross_cats_sorted":[],"title_canon_sha256":"8a906203d0e48634cbb5b19c2ac1b4a0605f2b9de293eb7e1bb093a854039e51","abstract_canon_sha256":"320f17aa8a48270a29eac6a9334aa67b4bac25854d253857b79b531923974941"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:41.675563Z","signature_b64":"+8SFlZwWbuJh0lrtpP0YzJm1WwQJvlneqroqG1kJC4PAtHNLBi4rrOmb0mOz7A5PJIyrRVundKJkeYhBASuUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e58e7355a0e31e222d5369a87d6bad509a27164e93f77e09c335b52dc83862e","last_reissued_at":"2026-05-18T00:03:41.675146Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:41.675146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Generalized Serre's Condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Brent Holmes","submitted_at":"2017-10-07T03:59:54Z","abstract_excerpt":"Throughout, let $R$ be a commutative Noetherian ring. 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We prove generalizations of results for rings satisfying Serre's condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02631","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1710.02631","created_at":"2026-05-18T00:03:41.675212+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.02631v4","created_at":"2026-05-18T00:03:41.675212+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.02631","created_at":"2026-05-18T00:03:41.675212+00:00"},{"alias_kind":"pith_short_12","alias_value":"BZMOONK2BYY6","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BZMOONK2BYY6EIWV","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BZMOONK2","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U","json":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U.json","graph_json":"https://pith.science/api/pith-number/BZMOONK2BYY6EIWVG2NIPVV22U/graph.json","events_json":"https://pith.science/api/pith-number/BZMOONK2BYY6EIWVG2NIPVV22U/events.json","paper":"https://pith.science/paper/BZMOONK2"},"agent_actions":{"view_html":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U","download_json":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U.json","view_paper":"https://pith.science/paper/BZMOONK2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.02631&json=true","fetch_graph":"https://pith.science/api/pith-number/BZMOONK2BYY6EIWVG2NIPVV22U/graph.json","fetch_events":"https://pith.science/api/pith-number/BZMOONK2BYY6EIWVG2NIPVV22U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U/action/storage_attestation","attest_author":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U/action/author_attestation","sign_citation":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U/action/citation_signature","submit_replication":"https://pith.science/pith/BZMOONK2BYY6EIWVG2NIPVV22U/action/replication_record"}},"created_at":"2026-05-18T00:03:41.675212+00:00","updated_at":"2026-05-18T00:03:41.675212+00:00"}