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

Rankeya Datta

doi:10.2140/ant.2021.15.2485

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arXiv · 1707.01649

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

Rankeya Datta

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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.

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Rankeya Datta. 2020-03-02. Frobenius splitting of valuation rings and $F$-singularities of centers. https://doi.org/10.2140/ant.2021.15.2485

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