Vol. 10, No. 5, 2016

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Frobenius and valuation rings

Rankeya Datta and Karen E. Smith

Vol. 10 (2016), No. 5, 1057–1090
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
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