Gorenstein projective and injective dimensions over Frobenius extensions

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, ${}_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module ${}_{R}M$ is Gorenstein projective (injective). (2) if $\mathrm{G}\text{-}{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{.}\mathrm{d}\mathrm{i}\mathrm{m}}_{A}M<\mathrm{\infty }$, then $$\mathrm{G}$$\text{-}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{.}\mathrm{d}\mathrm{i}\mathrm{m}${A}M = $\mathrm{G}$$\text{-}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{.}\mathrm{d}\mathrm{i}\mathrm{m}${R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\mathrm{G}\text{-}\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}(A)=\mathrm{G}\text{-}\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}(R)$.