Kanda, Ryo

Non-Exactness of Direct Products of Quasi-Coherent Sheaves

Doc. Math. 24, 2037-2056 (2019)
DOI: 10.25537/dm.2019v24.2037-2056
Communicated by Henning Krause


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*.

Mathematics Subject Classification

14F05, 18E20, 16D90, 16W50, 13C60


quasi-coherent sheaf, divisorial scheme, invertible sheaf, direct product, Gabriel-Popescu embedding, Grothendieck category


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