Gorenstein homological invariant properties under Frobenius extensions

We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that the extension algebra is CM(Cohen-Macaulay)-finite(resp.CM-free) if and only if so is the base algebra. Furthermore, we prove that the representation dimension of Artin algebras is invariant under separable Frobenius extensions.