Covers and envelopes in Grothendieck categories: Flat covers of complexes with applications

In the general setting of Grothendieck categories with enough projectives, we prove theorems that make possible to restrict the study of the problem o f the existence of F-covers and envelopes to the study of some properties o f the class F. We then prove the existence of flat covers and cotorsion env elopes of complexes, giving some examples. This generalizes the earlier wor k (J. Algebra 201 (1998), 86-102) and finishes the problem of the existence of flat covers of complexes. (C) 2001 Academic Press.