On graded Gorenstein injective dimension

There are nice relations between graded homological dimensions and ordinary homological dimensions. We study the Gorenstein injective dimension of a complex of graded modules denoted by ${}^{\ast }\text{Gid}$, and derive its properties. In particular we prove the Chouinard’s like formula for ${}^{\ast }\text{Gid}$, and compare it with the usual Gorenstein injective dimension.