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Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\\dim M\\leq 1$, then the $R$-module $\\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all $i\\geq 0$, if and only if the $R$-module $\\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for $i=0, 1$. Let $t\\in\\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $\\Ext^i_R(R/I,M)$ is weakly Laskerian for all $i\\leq t+1$. We prove that if the $R$-module $\\lc^{i}_\\Phi(M)$ is ${\\rm FD_{\\leq 1}}$ for all $i<t$, then $\\lc^{i}_\\Phi(M)$ is $\\Phi$-weakly cofinite "},"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":"1707.06795","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AC","submitted_at":"2017-07-21T08:32:10Z","cross_cats_sorted":[],"title_canon_sha256":"8b7e9a0705cdc2883fdeaf3aa4071171e056120f9467459ef9390dc84ab34982","abstract_canon_sha256":"5202275c7cb30ad4f4e4bee17838420385b91c0135615ff43c66ee22126ef0cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:51.619868Z","signature_b64":"wdJuNFv+kupDe278PTKLBQ8UlaXiY5BLsS7SVuATR43fE7oWtNyidtBdndfrMqdPDMVs4G3HinyqOmrJlZQOAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59b8b8a4338de3c74944c985e86895bc392f5d7eb56a63b5db5512595eec581b","last_reissued_at":"2026-05-18T00:39:51.619379Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:51.619379Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weakly cofiniteness of local cohomology modules","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Moharram Aghapournahr","submitted_at":"2017-07-21T08:32:10Z","abstract_excerpt":"Let $R$ be a commutative Noetherian ring, $\\Phi$ a system of ideals of $R$ and $I\\in \\Phi$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\\dim M\\leq 1$, then the $R$-module $\\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all $i\\geq 0$, if and only if the $R$-module $\\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for $i=0, 1$. Let $t\\in\\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $\\Ext^i_R(R/I,M)$ is weakly Laskerian for all $i\\leq t+1$. We prove that if the $R$-module $\\lc^{i}_\\Phi(M)$ is ${\\rm FD_{\\leq 1}}$ for all $i<t$, then $\\lc^{i}_\\Phi(M)$ is $\\Phi$-weakly cofinite "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06795","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.06795","created_at":"2026-05-18T00:39:51.619455+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.06795v1","created_at":"2026-05-18T00:39:51.619455+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.06795","created_at":"2026-05-18T00:39:51.619455+00:00"},{"alias_kind":"pith_short_12","alias_value":"LG4LRJBTRXR4","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"LG4LRJBTRXR4OSKE","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"LG4LRJBT","created_at":"2026-05-18T12:31:28.150371+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/LG4LRJBTRXR4OSKEZGC6Q2EVXQ","json":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ.json","graph_json":"https://pith.science/api/pith-number/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/graph.json","events_json":"https://pith.science/api/pith-number/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/events.json","paper":"https://pith.science/paper/LG4LRJBT"},"agent_actions":{"view_html":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ","download_json":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ.json","view_paper":"https://pith.science/paper/LG4LRJBT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.06795&json=true","fetch_graph":"https://pith.science/api/pith-number/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/graph.json","fetch_events":"https://pith.science/api/pith-number/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/action/storage_attestation","attest_author":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/action/author_attestation","sign_citation":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/action/citation_signature","submit_replication":"https://pith.science/pith/LG4LRJBTRXR4OSKEZGC6Q2EVXQ/action/replication_record"}},"created_at":"2026-05-18T00:39:51.619455+00:00","updated_at":"2026-05-18T00:39:51.619455+00:00"}