Homological dimensions of complexes of R-modules

Let R be an associative ring with identity, C(R) be the category of complexes of R-modules and Flat(C(R)) be the class of all flat complexes of Rmodules. We show that the flat cotorsion theory (Flat(C(R)), Flat(C(R))⊥) have enough injectives in C(R). As an application, we prove that for each flat complex F and each complex Y of R-modules, Exti (F, Y) = 0, whenever R is n-perfect and i > n.