  Let $(R,\frak m, k)$ be a noetherian local ring. It is well-known that $R$ is
regular if and only if the injective dimension of $k$ is finite. In this paper
it is shown that $R$ is Gorenstein if and only if the Gorenstein injective
dimension of $k$ is finite. On the other hand a generalized version of the
so-called Bass formula is proved for finitely generated modules of finite
Gorenstein injective dimension. It also improves Christensen's generalized Bass
formula (cf. "Gorenstein dimensions", volume 1747 of Lecture Notes in
Mathematics, Springer-Verlag, Berlin, 2000).
