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arXiv · 1810.08752

Non-exactness of direct products of quasi-coherent sheaves

Ryo Kanda
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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*.

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BibTeXRIS

Ryo Kanda. 2018-10-20. Non-exactness of direct products of quasi-coherent sheaves. https://doi.org/10.25537/dm.2019v24.2037-2056

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