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2016 Frobenius and valuation rings
Rankeya Datta, Karen Smith
Algebra Number Theory 10(5): 1057-1090 (2016). DOI: 10.2140/ant.2016.10.1057

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 Abhyankar; 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.

Citation

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Rankeya Datta. Karen Smith. "Frobenius and valuation rings." Algebra Number Theory 10 (5) 1057 - 1090, 2016. https://doi.org/10.2140/ant.2016.10.1057

Information

Received: 30 July 2015; Accepted: 18 January 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1348.13007
MathSciNet: MR3531362
Digital Object Identifier: 10.2140/ant.2016.10.1057

Subjects:
Primary: 13A35
Secondary: 13F30, 14B05

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.10 • No. 5 • 2016
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