Frobenius split subvarieties pull back in almost all characteristics

Abstract

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