Frobenius Split Subvarieties Pull Back In Almost All Characteristics

Let X and Y be schemes of finite type over Spec Z and let alpha : Y -> X be a finite map. We show the following holds for all sufficiently large primes p: If phi and psi are any splittings on X x Spec F-p and Y x Spec F-p, such that the restriction of alpha is compatible with phi and psi, and V is any compatibly split subvariety of (X x Spec F-p, phi), then the reduction alpha(-1)(V)(red) is a compatibly split subvariety of (Y x Spec F-p, psi). This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.