TITLE: Arbitrary products of quasi-coherent sheaves?
QUESTION [11 upvotes]: I have a short question:
Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to the product as $\mathcal{O}_{X}$-modules, but I haven't been able to find a counterexample.
Any help is appreciated!

REPLY [3 votes]: This was already discussed here:
https://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set
https://mathoverflow.net/questions/40587/quasi-coherent-envelope-of-a-module
It should be remarked that infinite products in categories of quasi-coherent sheaves are not well-behaved at all. For example, they are not compatible with restriction maps of open sets. While it is true that this category is presentable, I don't think that this will be really useful for schemes which are not concentrated.