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Let A be a Grothendieck category satisfying (AB4*) (that is, with exact products).

Assuming that, moreover, A is locally noetherian, must this category have enough small projective objects (so, to be equivalent to a category of additive functors from a small preadditive category to abelian groups)?

If it is, may the "locally noetherian" assumption be relaxed into "locally finitely presented", or even "locally finitely generated" of (AB6)?

Thanks!

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    $\begingroup$ In this paper: academia.edu/31035256/… one finds the Theorem that, for a locally finite Grothendieck category, (AB4*) is equivalent to having enough projectives. It is a first step... $\endgroup$
    – Aurélien Djament
    Commented Jun 3, 2019 at 8:59
  • $\begingroup$ If you look at the edit history to this answer you'll see I spent some time trying to prove this a few years ago, without success. $\endgroup$
    – Tim Campion
    Commented Jun 3, 2019 at 17:34
  • $\begingroup$ Thank you Tim, I was not aware of your post. So, the question seems really hard. $\endgroup$
    – Aurélien Djament
    Commented Jun 4, 2019 at 6:50
  • $\begingroup$ You're probably already aware, but for context I find this paper of Roos very relevant. In particular, there is Cor 1.4 which says that for a Grothendieck category, $AB4^\ast$ is equivalent to the existence of projective effacements, and the results of Section 4, where Roos gives examples of nonzero Grothendieck categories satisfying $AB4^\ast$ and $AB6$ which have no nonzero projectives. Thus certainly some condition is needed. $\endgroup$
    – Tim Campion
    Commented Jun 4, 2019 at 20:14
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    $\begingroup$ Indeed Roos's paper contains a lot of very interesting results and couter-examples. But I guess that all the counter-examples are far from being locally noetherian... It answers my last question (with only (AB6)) anyway. Even with "locally finitely generated", it is not so clear for me what happens. $\endgroup$
    – Aurélien Djament
    Commented Jun 5, 2019 at 7:07

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