Gorenstein injective complexes of modules over Noetherian rings

A complex C is called Gorenstein injective if there exists an exact sequence of complexes $?\to I_{\mathrm{?1}}\to I_0\to I_1\to ?$ such that each $I_i$ is injective, $C=\mathrm{Ker}(I_0\to I_1)$ and the sequence remains exact when $\mathrm{Hom}(E,?)$ 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 $C^m$ is Gorenstein injective in R-Mod for all $m\in \mathbb{Z}$. Also Gorenstein injective dimensions of complexes are considered.