Non-exactness of direct products of quasi-coherent sheaves
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*.
- Publication:
-
arXiv e-prints
- 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)
- E-Print:
- 13 pages