Frobenius and valuation rings

Datta, Rankeya; Smith, Karen

doi:10.2140/ant.2016.10.1057

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Frobenius and valuation rings

Rankeya Datta and Karen E. Smith

Vol. 10 (2016), No. 5, 1057–1090

DOI: 10.2140/ant.2016.10.1057

Correction: 10.2140/ant.2017.11.1003

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.

Keywords

valuation rings, Abhyankar valuations, characteristic $p$ commutative algebra, F-pure, F-regular, Frobenius split

Mathematical Subject Classification 2010

Primary: 13A35

Secondary: 13F30, 14B05

Milestones

Received: 30 July 2015

Accepted: 18 January 2016

Published: 28 July 2016

Correction: 18 June 2017

Authors

Rankeya Datta
Mathematics Department University of Michigan 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States

Karen E. Smith
Department of Mathematics University of Michigan 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States

Vol. 10, No. 5, 2016