{"paper":{"title":"Viewing finite dimensional representations through infinite dimensional ones","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Birge Huisgen-Zimmermann, Dieter Happel","submitted_at":"2014-07-09T01:05:18Z","abstract_excerpt":"We develop criteria for deciding the contravariant finiteness status of a subcategory $A \\subseteq \\Lambda\\text{-mod}$, where $\\Lambda$ is a finite dimensional algebra. In particular, given a finite dimensional $\\Lambda$-module $X$, we introduce a certain class of modules -- we call them $A$-phantoms of $X$ -- which indicate whether or not $X$ has a right $A$-approximation: We prove that $X$ fails to have such an approximation if and only if $X$ has infinite-dimensional $A$-phantoms. Moreover, we demonstrate that large phantoms encode a great deal of additional information about $X$ and $A$ an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2327","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"}