Frobenius and valuation rings

Datta, Rankeya; Smith, Karen E.

doi:10.2140/ant.2016.10.1057

Cited by 15 publications

(12 citation statements)

References 47 publications

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“…On the contrary, [DS17, Theorem 0.1] shows that finiteness of Frobenius for valuation rings of function fields (a function field is a finitely generated extension of a base field) is equivalent to the associated valuation being divisorial, and so non F-finite Abhyankar valuations are abundant. The current paper rectifies the error in [DS16].…”

Section: Introductionmentioning

confidence: 77%

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

Datta

2017

Preprint

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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 [DS16, DS17], this paper emphasizes the role of centers in controlling Frobenius properties of valuation rings in prime characteristic.

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Section: Introductionmentioning

confidence: 77%

“…then ν is centered on an excellent, local domain; the latter assertion is false even when K is not perfect. Indeed, using the theory of F -singularities of valuations developed in [DS16,DS17], one can show that if the valuation ring of ν is F -finite, then the identity…”

Section: Introductionmentioning

confidence: 99%

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

Datta

2017

Preprint

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“…For the rest of the proof, our goal is to show that (11) h 1 ( X, ν * L − δν * A) = 0 for 0 < δ 1, contradicting (10).…”

Section: Proof Of Theorem Bmentioning

confidence: 98%

The Gamma Construction and Asymptotic Invariants of Line Bundles Over Arbitrary fields

Murayama

2019

Nagoya Math. J.

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We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is F -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama's description of the restricted base locus to klt or strongly F -regular varieties over arbitrary fields. X ; see [PST17]. For imperfect fields, however, this approach runs into another problem:(III) Most applications of Frobenius techniques require k to be F -finite, i.e., satisfy [k : k p ] < ∞.This last issue arises since Grothendieck duality cannot be applied to the Frobenius if it is not finite. Recent advances in the minimal model program over imperfect fields due to Tanaka [Tan18; Tan] suggest that it would be worthwhile to develop a systematic way to deal with (III).Our first goal is to provide such a systematic way to reduce to the case when k is F -finite. While passing to a perfect closure of k fixes the F -finiteness issue, this operation can change the singularities of X drastically. To preserve singularities, we prove the following scheme-theoretic version of the gamma construction of Hochster and Huneke [HH94].Theorem A. Let X be a scheme essentially of finite type over a field k of characteristic p > 0, and let Q be a set of properties in the following list: local complete intersection, Gorenstein, Cohen-Macaulay, (S n ), regular, (R n ), normal, weakly normal, reduced, strongly F -regular, F -pure, Frational, F -injective. Then, there exists a purely inseparable field extension k ⊆ k Γ such that k Γ is

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“…One of the main results of the paper [4] states that if V is a valuation domain such that F p ⊂ V , then the Frobenius endomorphism on V is faithfully flat. We remark that any perfect F p -algebra has this property.…”

Section: Frobenius Map On Valuation Ringsmentioning

confidence: 99%