Gorenstein projective and injective dimensions over Frobenius extensions

Abstract: 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 $\mathrm{G}\text{-}\mathrm{proj.dim}{A}M<\infty$, then $\mathrm{G}\text{-}\mathrm{proj.dim}{A}M = \mathrm{G}\text{-}\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\mathrm{G}\text{-}\mathrm{gldim}(A)= \mathrm{G}\text{-}\mathrm{gldim}(R)$.