Gorenstein injective complexes of modules over Noetherian rings

A complex C is called Gorenstein injective if there exists an exact sequence of complexes ⋯→I−1→I0→I1→⋯ such that each Ii is injective, C=Ker(I0→I1) and the sequence remains exact when Hom(E,−) is applied to it for any injective complex E. We show that over a left Noetherian ring R, a complex C of left R-modules is Gorenstein injective if and only if Cm is Gorenstein injective in R-Mod for all m∈Z. Also Gorenstein injective dimensions of complexes are considered.