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We study Tate-Vogel and relative cohomologies of complexes by applying themodel structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) ofmodules. We show first that the class of complexes admitting a complete $\A$resolution is exactly the class of complexes with finite Gorenstein $\A$dimension. This lets us give general techniques for computing Tate-Vogelcohomoloies of complexes with finite Gorenstein $\A$ dimension. As aconsequence, relative cohomology groups for complexes with finite Gorenstein$\A$ dimension are investigated. Finally, the relationships between Gorenstein$\A$ dimensions and $\A$ dimensions for complexes are given.

Tate-Vogel and relative cohomologies of complexes with respect to cotorsion pairs