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

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.08752
 arXiv:
 arXiv:1810.08752
 Bibcode:
 2018arXiv181008752K
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory;
 14F05 (Primary);
 18E20;
 16D90;
 16W50;
 13C60 (Secondary)
 EPrint:
 13 pages