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*.
Cite this article
Ryo Kanda, Non-Exactness of Direct Products of Quasi-Coherent Sheaves. Doc. Math. 24 (2019), pp. 2037–2056DOI 10.4171/DM/720