We study the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex. Some results that track the behavior of Gorenstein properties over local ring homomorphisms under composition and decomposition are given. As an application, we characterize a dualizing complex for $R$ in terms of the finiteness of the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex.

Let $\Gamma$ be a ring extension of $R$. We show the left $\Gamma$-module $U=\Gamma {\otimes }_{R}C$ with the endmorphism ring End${}_{\Gamma }U=\Delta$ is a generalized tilting module when ${}_{R}C$ is a generalized tilting module under some conditions.