Relative Cohomology And Tate Cohomology Of Complexes With Respect To Cotorsion Pairs

Let(x,y)be a complete and hereditary cotorsion pair in a bicomplete abelian category A.In this paper,we mainly study the relative and Tate cohomologies of complexes with respect to(x,y).Firstly,we consider the homological properties of Gorenstein complexes with respect to cotorsion pairs,and discuss the relationship between Gorenstein complexes and modules.Secondly,we introduce the definition of cohomology of Gorenstein complexes with respect to cotorsion pairs,and consider the balanceness of cohomology of Gorenstein complexes with respect to cotorsion pairs.Thirdly,we introduce the definition of Tate cohomology of complexes with respect to cotorsion pairs.In addition,the relationship between injective dimension and Tate cohomology of complexes is given.Finally,long exact sequences of Tate cohomology of complexes with respect to cotorsion pairs are established.