Flat covers in the category of quasi-coherent sheaves over the projective line

In this paper we prove the existence of a flat cover and a cotorsion :envelope for any quasi-coherent sheaf over the projective line P-1(R), where R is any commutative ring. We first prove a general result that guarantees the existence of F-covers and F-perpendicular to-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.