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The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\\displaystyle H^j_I(M,N)=\\dlim\\Ext^j_R(M/I^nM,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal then $H^j_I(M,N)$ is $I$-cofinite for all $M, N$ and all $j$. Secondly, let $t$ be a non-negative integer such that $\\dim\\Supp(H^j_I(M,N))\\le 1 \\text{for all} j<t.$ Then $H^j_I(M,N)$ is $I$-cofinite for all $j<t$ and $\\Hom(R/I,H^t_I(M,N))$ is finitely g"},"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":"1207.0703","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-06-28T03:22:51Z","cross_cats_sorted":[],"title_canon_sha256":"e32faad9a6b6e08a6baa236d0dfd4897872576b7f7d56a7d111cb22a7a05dc31","abstract_canon_sha256":"359dcec9ca4942388c7a4347d5488f68e378bf693f714328633f19162c06c4bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:21.677269Z","signature_b64":"RzJUYdnVs7vgXxLTxvMwv2odcRKXrSHtIj+eKBqicUGAkx3zCcLTYtLsVlI6cJeMP2xgAF4aXdKuArBvqMQtDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9cbbe5d8f56110d244851647e66a5e129690990e454bfb487c85605c9859f79","last_reissued_at":"2026-05-18T01:28:21.676607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:21.676607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the cofiniteness of generalized local cohomology modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Nguyen Tu Cuong, Nguyen Van Hoang, Shiro Goto","submitted_at":"2012-06-28T03:22:51Z","abstract_excerpt":"Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\\displaystyle H^j_I(M,N)=\\dlim\\Ext^j_R(M/I^nM,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal then $H^j_I(M,N)$ is $I$-cofinite for all $M, N$ and all $j$. Secondly, let $t$ be a non-negative integer such that $\\dim\\Supp(H^j_I(M,N))\\le 1 \\text{for all} j<t.$ Then $H^j_I(M,N)$ is $I$-cofinite for all $j<t$ and $\\Hom(R/I,H^t_I(M,N))$ is finitely g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0703","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":"1207.0703","created_at":"2026-05-18T01:28:21.676729+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.0703v1","created_at":"2026-05-18T01:28:21.676729+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.0703","created_at":"2026-05-18T01:28:21.676729+00:00"},{"alias_kind":"pith_short_12","alias_value":"XHF34XMPKYIQ","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"XHF34XMPKYIQ2JCI","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"XHF34XMP","created_at":"2026-05-18T12:27:27.928770+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/XHF34XMPKYIQ2JCIKFSH4ZVF4E","json":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E.json","graph_json":"https://pith.science/api/pith-number/XHF34XMPKYIQ2JCIKFSH4ZVF4E/graph.json","events_json":"https://pith.science/api/pith-number/XHF34XMPKYIQ2JCIKFSH4ZVF4E/events.json","paper":"https://pith.science/paper/XHF34XMP"},"agent_actions":{"view_html":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E","download_json":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E.json","view_paper":"https://pith.science/paper/XHF34XMP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.0703&json=true","fetch_graph":"https://pith.science/api/pith-number/XHF34XMPKYIQ2JCIKFSH4ZVF4E/graph.json","fetch_events":"https://pith.science/api/pith-number/XHF34XMPKYIQ2JCIKFSH4ZVF4E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E/action/storage_attestation","attest_author":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E/action/author_attestation","sign_citation":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E/action/citation_signature","submit_replication":"https://pith.science/pith/XHF34XMPKYIQ2JCIKFSH4ZVF4E/action/replication_record"}},"created_at":"2026-05-18T01:28:21.676729+00:00","updated_at":"2026-05-18T01:28:21.676729+00:00"}