#### 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