#### Vol. 15, No. 10, 2021

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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