Valuations and Frobenius

Abstract

The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is divisorial; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split if and only if Frobenius is finite, or equivalently, if and only if the valuation ring is excellent.

arXiv commentAn Erratum has been added correcting Theorem 5.1 on the equivalence of F-finite and Abhyankar valuations. Comments are welcome

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Cited by in corpus (9)

• Valuations and Frobenius
• Mayer-Vietoris property for relative symplectic cohomology
• The gamma construction and asymptotic invariants of line bundles over arbitrary fields
• Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
• Frobenius splitting of valuation rings and $F$-singularities of centers
• Excellence, F-singularities, and solidity
• Another proof of the almost purity theorem for perfectoid valuation rings
• Essential finite generation of extensions of valuation rings
• Uniform approximation of Abhyankar valuation ideals in function field of prime characteristic