Some remarks on projective generators and injective cogenerators

In this paper, we give a relationship between projective generators (resp., injective cogenerators) in the category of R-modules and the counterparts in the category of complexes of R-modules. As a consequence, we get that complexes of W-Gorenstein modules are actually -Gorenstein complexes whenever W is a subcategory of R-modules satisfying W ⊥ W, where is the subcategory of exact complexes with all cycles in W. We also study when all cycles of a -Gorenstein complexes are W-Gorenstein modules.