When are direct products exact in the category of quasi-coherent sheaves?

Question

(This question is crossposted from MSE, since there the question did not recieve any attention whatsoever.)

I would like to know if there is a description (or at least some sufficient condition known) of (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products. Of course, I do not mean affine schemes since for affine schemes, direct products are exact (kind of) trivially.

Seems to me that viable candidates would be projective schemes, since direct products in the category of graded modules are exact, and the well-known functor $M \mapsto \tilde{M}$ is not far from being an equivalence (at least in my mind) of the category of graded modules over a graded ring and the category of quasi-coherent sheaves over the projective scheme. However, I was told this is not true, i.e. that the direct products are not exact even on simple projective schemes.

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 (especially if the scheme is projective).

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.

A reference to some text treating these question I would find most helpful.

Thanks in advance for any help.

Answer 1

A counterexample showing that direct products in the category of quasi-coherent sheaves over the projective line $\mathbb P_k^1$ over a field $k$ are not exact functors can be found in the paper "The stable derived category of a Noetherian scheme", by H. Krause, http://arxiv.org/abs/math.AG/0403526 , Example 4.9 (attributed to B. Keller).

The point is that the functor $M\longmapsto\widetilde M$ from the category of graded modules over the projective coordinate ring $R$ to the category of quasi-coherent sheaves on the projective spectrum $X$ establishes an equivalence between the category of quasi-coherent sheaves on $X$ and the quotient category of the category of graded $R$-modules by the Serre/localizing subcategory of torsion modules (i.e., modules in which every element is annihilated by all the elements of high enough degree in $R$).

It is important that the full subcategory of torsion modules is not preserved by direct products in the category of graded $R$-modules (so the functor $M\longmapsto\widetilde M$ cannot possibly preserve direct products). One proceeds from this observation in order to construct the counterexample in the paper under the link, which likely can be generalized to show that direct products in the category of quasi-coherent sheaves on a projective scheme $X$ over the spectrum of a Noetherian ring $A$ are not exact unless $X$ is finite over $\operatorname{Spec}A$ (and consequently affine), or something close to that.

It would be interesting to know whether direct products in the category of quasi-coherent sheaves over the affine plane without point $X=\mathbb A_k^2\setminus(0,0)$ are exact. My guess would be that they are not.