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I would like to know if there is a description (or at least some sufficient condition known) of a (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products.

I would also like to know some example when this fails, i.e. when product of epimorphisms of quasi-coherent sheaves is not an epimorphism.

I only stumbled upon the fact that direct products of quasi-coherent sheaves are not exact in general in the introduction to this paper by L. Positselski (first paragraph of the introduction, in fact). However, I cannot find any further information, so any reference would be greatly appreciated.

Thanks in advance for any help.

EDIT: Of course, I know that products are exact over affine schemes (i.e. products of modules over a ring are exact). I mention this to emphasize that this is not a sufficient condition I have in mind.

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  • $\begingroup$ I have crossposted this question at MO to improve chances of some response. $\endgroup$ – Pavel Čoupek Aug 25 '15 at 14:44

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