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

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.

The latest version shows that all spherically complete non-Archimedean fields of prime characteristic have F-split valuation rings; bibliography has been updated; other minor changes; comments welcome

References in corpus (2)

• Existence of valuations with smallest normalized volume
• Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic

Cited by in corpus (1)

• Tate algebras and Frobenius non-splitting of excellent regular rings