Frobenius split subvarieties pull back in almost all characteristics

TL;DR

The paper proves that Frobenius split subvarieties pull back under finite maps for large primes, aiding the classification of compatibly split subvarieties in algebraic geometry.

Contribution

It establishes that compatibly split subvarieties are preserved under pullback via finite maps in large characteristic, extending understanding of Frobenius splitting behavior.

Findings

Compatibility of Frobenius splittings is preserved under pullback for large primes.

Reductions of compatibly split subvarieties remain compatibly split after pullback.

Provides a tool for classifying compatibly split subvarieties in algebraic geometry.

Abstract

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