A Grothendieck category with a noetherian generator and exact products that is not a module category

Ryo Kanda

arXiv:2607.23520v1 Primary: math.CT Secondary: math.RA, math.RT

Abstract

We construct a Grothendieck category that has a noetherian generator, satisfies AB4*, and is not equivalent to a module category. This gives a negative answer to Djament's problem. The category is obtained as a Gabriel quotient of a module category over the endomorphism ring appearing in the work of Herbera, Příhoda, and Wiegand. The proof relies on the trace criterion established by Martini, Parra, Saorín, and Virili.

Disclosure

“(see Remark 3.1), the following problem remains open: Problem 1.3. Is there a locally noetherian Grothendieck category satisfying AB4∗ that does not admit a projective generator? Tool and Computational Resource Disclosure. The author used OpenAI’s ChatGPT, with models from the GPT-5.5 and GPT-5.6 families, as a general-purpose tool for research assistance and manuscript preparation, including exploratory assistance and suggestions concerning exposi- tion and language revision. In particular”
PDF page 2

Classification: Drafting limited passages
Multiplier: 5
Verified: 2026-08-23T19:08:19.297393Z

Structural counts

Pages 4 pdf

Theorems 2 source

Lemmas 0 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 10 source

Bibliography entries 6 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file GrothCatExample.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.