{"paper":{"title":"Homological Dimensions in Cotorsion Pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Lidia Angeleri Hugel, Octavio Mendoza Hernandez","submitted_at":"2008-08-11T21:16:56Z","abstract_excerpt":"Two classes $\\mathcal A$ and $\\mathcal B$ of modules over a ring $R$ are said to form a cotorsion pair $(\\mathcal A, \\mathcal B)$ if $\\mathcal A={\\rm Ker Ext}^1_R(-,\\mathcal B)$ and $\\mathcal B={\\rm Ker Ext}^1_R(\\mathcal A,-)$. We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of $R$. We show that $\\Findim R<\\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\\mathcal A, \\mathcal B)$ in $\\mathrm{Mod} R$: the relative projective dimension of $\\A$ with respect to itself, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1585","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0808.1585/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}