JournalsdmVol. 24pp. 2037–2056

Non-Exactness of Direct Products of Quasi-Coherent Sheaves

  • Ryo Kanda

    Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
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Abstract

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. 2037–2056

DOI 10.4171/DM/720