Djament's Problem on Locally Noetherian Grothendieck Categories
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.
- Result
- Disproved(see note)
- Status
- Partial result
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Category theory
- Posed by
- Aurelien Djament
- Year posed
- —
- Years open
- —
- Solved
- 2026-07-26
- Model
- ChatGPT (GPT-5.5, GPT-5.6)
- Vendor
- OpenAI
- Collaborators
- Ryo Kanda
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3
What the AI did
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.
Verification
Single-author arXiv preprint; not yet peer-reviewed.
Source
arXiv:2607.23520 - A Grothendieck category with a noetherian generator and exact products