  For a noetherian scheme that has an ample family of invertible sheaves, we
prove that direct products in the category of quasi-coherent sheaves are not
exact unless the scheme is affine. This result can especially be applied to all
quasi-projective schemes over commutative noetherian rings. The main tools of
the proof are the Gabriel-Popescu embedding and Roos' characterization of
Grothendieck categories satisfying Ab6 and Ab4*.
