Correction to the article “Frobenius and
valuation rings”

Datta, Rankeya; Smith, Karen E.

doi:10.2140/ant.2017.11.1003

Cited by 6 publications

(6 citation statements)

References 1 publication

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“…maps c (p e−1 −1)p → cϕ(1) = 0, showing that Hom R (F e * R, R) is non-trivial. Obviously, (4) implies (5). We finish the proof by proving that (5) implies (2).…”

Section: Maps Inverse To Frobeniusmentioning

confidence: 73%

“…Then the p −e -linear map R) is non-trivial. Obviously, (4) implies (5). We finish the proof by proving that (5) implies (2).…”

mentioning

confidence: 73%

“…Remark 4.2. The paper [5], with corrections in [6], shows more generally that a (not necessarily Noetherian) valuation ring with F-finite function field will always be divisorial if it is F-finite; see [6, Thm 0.1]. Thus, in the class of valuation rings of an F-finite function field, F-finiteness implies Noetherian.…”

Section: Examples Of Non-excellent Ringsmentioning

confidence: 99%

“…In Section 4, we consider excellence in the setting of discrete valuation rings of a function field of characteristic p. Excellence in this case is equivalent to the DVR being divisorial, a topic explored in [5]. We show here how this makes it is easy to write down explicit examples of non-excellent discrete valuation rings.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, a simple countability argument shows that among domains whose fraction field is, say the function field of P 2 , non-excellent regular local rings of dimension one are far more abundant than the excellent ones. This paper is largely expository, drawing heavily on the work in [5] (and the corrections in [6]) where most of the harder proofs of the results discussed here appear. However, we are emphasizing a different aspect of the subject than in that paper, drawing conclusions not explicit there.…”

Section: Introductionmentioning

confidence: 99%

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Excellence in prime characteristic

Datta¹,

Smith²

2018

Local and Global Methods in Algebraic Geometry

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Fix any field K of characteristic p such that [K : K p ] is finite. We discuss excellence for domains whose fraction field is K, showing for example, that R is excellent if and only if the Frobenius map is finite on R. Furthermore, we show R is excellent if and only if it admits some nonzero p −e -linear map for R (in the language of [3]), or equivalently, that the Frobenius map R → F * R defines a solid R-algebra structure on F * R (in the language of [11]). In particular, this means that generically Ffinite, Frobenius split Noetherian domains are always excellent. We also show that non-excellent rings are abundant and easy to construct in prime characteristic, even within the world of regular local rings of dimension one inside function fields.

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“…maps c (p e−1 −1)p → cϕ(1) = 0, showing that Hom R (F e * R, R) is non-trivial. Obviously, (4) implies (5). We finish the proof by proving that (5) implies (2).…”

Section: Maps Inverse To Frobeniusmentioning

confidence: 73%

“…Then the p −e -linear map R) is non-trivial. Obviously, (4) implies (5). We finish the proof by proving that (5) implies (2).…”

mentioning

confidence: 73%

Section: Examples Of Non-excellent Ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Excellence in prime characteristic

Datta¹,

Smith²

2018

Local and Global Methods in Algebraic Geometry

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Tight closure and strongly F-regular rings

Hochster

2022

Res Math Sci

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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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Correction to the article “Frobenius and valuation rings”

Abstract: 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices.ANT peer review and production are managed by EditFLOW ® from MSP.

Cited by 6 publications

References 1 publication

Excellence in prime characteristic

Excellence in prime characteristic

Tight closure and strongly F-regular rings

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

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