Abstract: Let (L,A) be a fixed complete duality pair. Firstly, the author introduced the Gorenstein (L,A)-injective dimension of complexes, gave its  characterization, and proved that Gorenstein (L,A)-injective dimension of complexes was not larger than injective dimension. Secondly, the author also discussed relative cohomology and  Tate cohomology of complexes, and obtained the long exact sequence connecting absolute, relative and Tate cohomology.

Key words: duality pair, Gorenstein(L,A)-injective dimension, Tate cohomology, long exact sequence