Problem detail · source-aware
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Djament asked whether a Grothendieck category satisfying suitable finiteness and exactness conditions must be equivalent to a module category. In the locally noetherian case the answer is no: there is a Grothendieck category with a noetherian generator satisfying AB4* that is not equivalent to a module category, built as a Gabriel quotient of a module category over an endomorphism ring of Herbera, Prihoda and Wiegand.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The disclosure attributes two specific things. The tool suggested examining the example in Section 8 of Herbera-Prihoda-Wiegand as a possible source of a negative answer, which is the example the paper is built on, and it suggested a proof strategy. The author independently checked that strategy and supplied the proof.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Single-author arXiv preprint; not yet peer-reviewed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.