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Reading the basics of $\mathcal{O}_{X}$ modules, where $\left(X,\mathcal{O}_{X}\right)$ is a fixed ringed space, i understood that arbitrary direct products of quasi-coherent $\mathcal{O}_{X}$ modules are quasi-coherent $\mathcal{O}_{X}$ modules. I also know by a result of Gabber that $\mathsf{Qcoh}\left(X\right)$ is a Grothendieck category for an arbitrary scheme $X$ and consequently by (iV) theorem 48 of section 2.3 it is a complete category. But, i was wondering: is the product of quasi-coherent sheaves coincide with the product in the category of $\mathcal{O}_{X}$ modules? Or there is a counter-example that arbitrary products of quasi-coherent sheaves are not quasi-coherent?

T. Wildwolf
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    I don't know why this question is closed. The OP asks for a counter-example, while the linked question contains no concrete examples. Still, see Lem. 2.1.6 in https://webusers.imj-prg.fr/~haohao.liu/OFM.pdf, which says every integral scheme other than the spectrum of fields gives a counter-exmaple. – Doug Jul 09 '23 at 08:10

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