{"paper":{"title":"Local homology, finiteness of Tor modules and cofiniteness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hossein Faridian, Kamran Divaani-Aazar, Massoud Tousi","submitted_at":"2017-01-26T14:43:20Z","abstract_excerpt":"Let $\\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\\V(\\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\\fa}_i(M)\\neq 0$. We show that $M$ is $\\fa$-cofinite if and only if the $R$-module $\\Tor^R_i(R/\\fa,M)$ is finitely generated for every $0\\leq i\\leq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $\\mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07721","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"}