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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1710.07239","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-10-19T16:43:39Z","cross_cats_sorted":[],"title_canon_sha256":"9187fa9771b55779ebf500e90ac3362044d9a31b802f752921cfd997690d7361","abstract_canon_sha256":"cead63a1b8e86d3dd03ef62f38d5027276990fbdd41c49d0f6264a591becfcea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:26.464938Z","signature_b64":"o0iwjf3hc4ZcGEryWYiKOSu6ydWe4Mjyn00J7ntXQUsR6Pq+tfJHHizvOfDR4YGXM9CSftylGvFZJJhKBEabDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd3dc193254fa63f8bc50400892676c8b9c054537d76449a41b669a28f85080f","last_reissued_at":"2026-05-18T00:32:26.464394Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:26.464394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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