Flat Covers in the Category of Quasi-coherent Sheaves Over the Projective Line#

Abstract

In this paper we prove the existence of a flat cover and a cotorsion envelope for any quasi-coherent sheaf over the projective line , where R is any commutative ring. We first prove a general result that guarantees the existence of ℱ-covers and ℱ⊥-envelopes in the general setting of a Grothendieck category (not necessarily with enough projectives) provided that the class ℱ satisfies some “standard” conditions. This will generalize some results of the earlier work. [Aldrich, S. T., Enochs, E., García Rozas, J. R., Oyonarte, L. (2001). Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications. J. Algebra 243:615–630].