Frobenius splitting of valuation rings revisited

Abstract: Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $ν$ be a valuation of $K/k$. We show $ν$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.