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

In this paper, we study the existence of ⊥ -envelopes, -envelopes, ⊥ -envelopes, -covers, and -covers where and denote the classes of modules of injective and projective dimension less than or equal to a natural number n, respectively. We prove that over any ring R, special ⊥ -preenvelopes and special -precovers always exist. If the ring is noetherian, the same holds for ⊥ -envelopes, and for ⊥ -envelopes and -covers when the ring is perfect. When inj.dim R ≤ n then -covers exist, and if R is such that a given class of homomorphisms is closed under well ordered direct limits then -envelopes exist.

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