{"paper":{"title":"Generators versus projective generators in abelian categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Charles Paquette","submitted_at":"2017-10-19T16:43:39Z","abstract_excerpt":"Let $\\mathcal{A}$ be an essentially small abelian category. We prove that if $\\mathcal{A}$ admits a generator $M$ with ${\\rm End}_{\\mathcal{A}}(M)$ right artinian, then $\\mathcal{A}$ admits a projective generator. If $\\mathcal{A}$ is further assumed to be Grothendieck, then this implies that $\\mathcal{A}$ is equivalent to a module category. When $\\mathcal{A}$ is Hom-finite over a field $k$, the existence of a generator is the same as the existence of a projective generator, and in case there is such a generator, $\\mathcal{A}$ has to be equivalent to the category of finite dimensional right mod"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07239","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"}