Frobenius splitting of valuation rings and F-singularities of centers

Datta, Rankeya

doi:10.2140/ant.2021.15.2485

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

Rankeya Datta

Vol. 15 (2021), No. 10, 2485–2512

DOI: 10.2140/ant.2021.15.2485

Abstract

Using a local monomialization result of Knaf and Kuhlmann, which was generalized by Cutkosky, we prove that the valuation ring of an Abhyankar valuation of a function field over an $F$ -finite 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 our work with Smith (Algebra Number Theory 10:5 (2016), 1057–1090 and its correction in 11:4 (2017), 1003–1007), this paper emphasizes the role of centers in controlling properties of valuation rings in prime characteristic.

Keywords

Frobenius splitting, Abhyankar valuations, valuation rings, $F$-singularities, $F$-finiteness

Mathematical Subject Classification 2010

Primary: 13F30

Secondary: 13A35, 14B05

Milestones

Received: 16 January 2020

Accepted: 7 March 2021

Published: 8 February 2022

Authors

Rankeya Datta
Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL United States

Vol. 15, No. 10, 2021