{"paper":{"title":"Finiteness dimensions and cofiniteness of generalized local cohomology modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alireza Vahidi, Elahe Mahmoudi Renani, Moharram Aghapournahr","submitted_at":"2018-10-24T07:39:15Z","abstract_excerpt":"Let $R$ be a commutative Noetherian ring with non-zero identity, $\\mathfrak{a}$ and ideal of $R$, $M$ a finite $R$--module, and $n$ a non-negative integer. In this paper, for an arbitrary $R$--module $X$ which is not necessarily finite, we study the finiteness dimension $f_\\mathfrak{a}(M,X)$ and the $n$-th finiteness dimension $f^n_\\mathfrak{a}(M,X)$ of $M$ and $X$ with respect to $\\mathfrak{a}$. Assume that $\\operatorname{Ext}^{i}_{R}(R/\\mathfrak{a},X)$ is finite for all $i\\leq f^2_\\mathfrak{a}(M,X)$ (resp. $i< f^1_\\mathfrak{a}(M,X)$). We show that $\\operatorname{H}^{i}_{\\mathfrak{a}}(M,X)$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.10223","kind":"arxiv","version":2},"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"}