Complete cotorsion pairs in the category of complexes

In this paper, we study completeness of cotorsion pairs in the category of complexes of R-modules. Let (A, B) be a cotorsion pair in R-Mod. It is shown that the cotorsion pair (widetilde{A}, dgwidetilde{B}) and (overline{A}, overline{A}^{perp}) are complete if A is closed under pure submodules and cokernels of pure monomorphisms, where in Gillespie s definitions widetilde{A} is the class of exact complexes with cycles in A and dgwidetilde{B} is the class of complexes X with components in B such that the complex Hom(A, X) is exact for every complex A in widetilde{A}; and overline{A} is the class of all complexes with components in A. Furthermore, they are perfect. As an application, we get that every complex over a right coherent ring has a Gorenstein flat cover, which generalizes the well-known results on the existence of Gorenstein flat covers.