{"paper":{"title":"A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"B. Huisgen-Zimmermann, S. O. Smal\\o","submitted_at":"2014-07-09T00:53:12Z","abstract_excerpt":"Given a finite dimensional algebra $\\Lambda$, we show that a frequently satisfied finiteness condition for the category ${\\cal P}^{\\infty}(\\Lambda\\rm{-mod})$ of all finitely generated (left) $\\Lambda$-modules of finite projective dimension, namely contravariant finiteness of ${\\cal P}^{\\infty}(\\Lambda\\rm{-mod})$ in $\\Lambda\\rm{-mod}$, forces arbitrary modules of finite projective dimension to be direct limits of objects in ${\\cal P}^{\\infty}(\\Lambda\\rm{-mod})$. Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimension conje"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2326","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"}