Generalized tilting modules and Frobenius extensions

Let $A/S$ be a ring extension with $S$ commutative. We prove that $\omega {\otimes }_S{A}_A$ is a generalized tilting module if ${\omega }_S$ is a generalized tilting module. In this case, we obtain that ${}^{\mathrm{\perp }}\omega$-resol.dim ${}_S(M)$ and ${}^{\mathrm{\perp }}(\omega {\otimes }_{S}A)$-resol.dim ${}_A(M)$ are identical for any $A$-module $M$. As an application, we show that $S$ satisfies gorenstein symmetric Conjecture if and only if so does $A$. Furthermore, we introduce the concept of ${}^{\mathrm{\perp }}\omega$-Gorenstein projective modules, and we obtain that the relative Gorenstein projectivity is invariant under Frobenius extensions.