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  "problem_id": "vibemathed:djament-problem-grothendieck-category",
  "title": "Djament's Problem on Locally Noetherian Grothendieck Categories",
  "canonical_statement": "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.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
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      "type": "disproved",
      "status": "partial",
      "occurred_at": "2026-07-26T00:00:00.000Z",
      "title": "arXiv:2607.23520 - A Grothendieck category with a noetherian generator and exact products",
      "summary": "the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3",
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      "attempted_at": "2026-07-26T00:00:00.000Z",
      "human_collaborators": [
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      "outcome": "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.",
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      "kind": "primary-mathematical-source",
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    {
      "label": "VibeMathed: Djament's Problem on Locally Noetherian Grothendieck Categories",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}