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Then $\\cograde_{R_{\\p}}\\Hom_{R}(R_{\\p},M)=\\inf\\{i | \\pi_{i}(\\p,M)>0\\}$ and $$\\pi_{i}(\\p,M)>0\\Rightarrow\\cograde_{R_{\\p}}\\Hom_{R}(R_{\\p},M)\\leq i\\leq\\fd_{R_{\\p}}\\Hom_{R}(R_{\\p},M),$$ where $\\pi_{i}(\\p,M)$ is the $i$-th dual Bass number of $M$ with respect to $\\p$, the integer $\\cograde_{R_{\\p}}\\Hom_{R}(R_{\\p},M)$ is the common length of any maximal $\\Hom_{R}(R_{\\p},M)$-quasi co-regular sequence contained in $\\p R_{\\p}$, and $\\fd_{R_{\\p}}\\Hom_{R}(R_{\\p},M)$ is the flat dimension of $R_{\\p}$-module $\\Hom_{R}(R_{\\p},M)$ (The"},"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":"1005.1754","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-05-11T08:37:17Z","cross_cats_sorted":[],"title_canon_sha256":"496d7b83c44c83543d78e000c71360ca051043db2d497d4290a5291e59d6d636","abstract_canon_sha256":"10e88b0aa171b0231c6c0f12a7dd7e1f93603881f1554e9f0a4f1b0306a1e84c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:32.237472Z","signature_b64":"EKKSJSELqM11ZwYojD0QcA5tNTUvf+cGLntGJU+J+4iWfQQQeWd2dcH0LU2J9GXdugvGDe5Oh9Mufg0Si68eDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"619455cfc87145e9a9b590739eb618fb4bedafb26cd09d8bdce3f1f049d313ff","last_reissued_at":"2026-05-18T03:18:32.236606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:32.236606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vanishing Properties of Dual Bass numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Lingguang Li","submitted_at":"2010-05-11T08:37:17Z","abstract_excerpt":"Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module, $\\p\\in\\Cos_RM$. Then $\\cograde_{R_{\\p}}\\Hom_{R}(R_{\\p},M)=\\inf\\{i | \\pi_{i}(\\p,M)>0\\}$ and $$\\pi_{i}(\\p,M)>0\\Rightarrow\\cograde_{R_{\\p}}\\Hom_{R}(R_{\\p},M)\\leq i\\leq\\fd_{R_{\\p}}\\Hom_{R}(R_{\\p},M),$$ where $\\pi_{i}(\\p,M)$ is the $i$-th dual Bass number of $M$ with respect to $\\p$, the integer $\\cograde_{R_{\\p}}\\Hom_{R}(R_{\\p},M)$ is the common length of any maximal $\\Hom_{R}(R_{\\p},M)$-quasi co-regular sequence contained in $\\p R_{\\p}$, and $\\fd_{R_{\\p}}\\Hom_{R}(R_{\\p},M)$ is the flat dimension of $R_{\\p}$-module $\\Hom_{R}(R_{\\p},M)$ (The"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1754","kind":"arxiv","version":5},"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":"1005.1754","created_at":"2026-05-18T03:18:32.236755+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.1754v5","created_at":"2026-05-18T03:18:32.236755+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1754","created_at":"2026-05-18T03:18:32.236755+00:00"},{"alias_kind":"pith_short_12","alias_value":"MGKFLT6IOFC6","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"MGKFLT6IOFC6TKNV","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"MGKFLT6I","created_at":"2026-05-18T12:26:10.704358+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/MGKFLT6IOFC6TKNVSBZZ5NQY7N","json":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N.json","graph_json":"https://pith.science/api/pith-number/MGKFLT6IOFC6TKNVSBZZ5NQY7N/graph.json","events_json":"https://pith.science/api/pith-number/MGKFLT6IOFC6TKNVSBZZ5NQY7N/events.json","paper":"https://pith.science/paper/MGKFLT6I"},"agent_actions":{"view_html":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N","download_json":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N.json","view_paper":"https://pith.science/paper/MGKFLT6I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.1754&json=true","fetch_graph":"https://pith.science/api/pith-number/MGKFLT6IOFC6TKNVSBZZ5NQY7N/graph.json","fetch_events":"https://pith.science/api/pith-number/MGKFLT6IOFC6TKNVSBZZ5NQY7N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N/action/storage_attestation","attest_author":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N/action/author_attestation","sign_citation":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N/action/citation_signature","submit_replication":"https://pith.science/pith/MGKFLT6IOFC6TKNVSBZZ5NQY7N/action/replication_record"}},"created_at":"2026-05-18T03:18:32.236755+00:00","updated_at":"2026-05-18T03:18:32.236755+00:00"}