A Tale of Valuation Rings in Prime Characteristic

Abstract

We examine valuation rings in prime characteristic from the lens of singularity theory defined using the Frobenius map. We show that valuation rings are always F-pure, while the question of Frobenius splitting is more mysterious. Using a characteristic-independent local monomialization result of Knaf and Kuhlmann [KK05], we are able to prove that Abhyankar valuations of functions fields over perfect ground fields are always Frobenius split. At the same time, we construct discrete valuation rings of function fields that do not admit any Frobenius splittings. Connections between F-singularities of valuation rings and the notion of defect of an extension of valuations are established. Our examination reveals that there is an intimate relationship between defect and Abhyankar valuations. We study tight closure of ideals of valuation rings, establishing a link between tight closure and Huber's notion of f-adic valued fields. Tight closure turns out to be an interesting closure operation only for those valued fields that are f-adic in the valuation topology. We also introduce a variant of Hochster and Huneke's notion of strong F-regularity [HH89], calling it F-pure regularity. F-pure regularity is a better notion of singularity in the absence of finiteness hypotheses, and we use it to recover an analogue of Aberbach and Enescu's splitting prime [AE05] in the valuative setting. We show that weak F-regularity and F-pure regularity coincide for a valuation ring, and both notions are equivalent to the ring being Noetherian. Thus, the various variants of F-regularity are perhaps reasonable notions of singularity only in the world of Noetherian rings. In the final chapter, we prove a prime characteristic analogue of a result of Ein, Lazarsfeld and Smith [ELS03] on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations. As a consequence, we deduce a prime characteristic Izumi theorem for real-valued Abhyankar valuations that admit a common smooth center.