Frobenius splitting of valuation rings and $F$-singularities of centers

TL;DR

This paper proves that valuation rings of Abhyankar valuations over perfect fields are Frobenius split and explores how Frobenius splittings of centers influence those of valuation rings, highlighting the role of centers in Frobenius properties.

Contribution

It establishes Frobenius splitting of valuation rings using monomialization and shows how splittings of centers extend to valuation rings, emphasizing the importance of centers in Frobenius properties.

Findings

Valuation rings of Abhyankar valuations are Frobenius split.

Frobenius splitting of centers lifts to valuation rings.

Centers play a crucial role in controlling Frobenius properties.

Abstract

Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to [arXiv:1507.06009], this paper emphasizes the role of centers in controlling Frobenius properties of valuations rings in prime characteristic.