Commutative Algebra Seminar

Title

Valuations and Frobenius

Abstract

We examine some usual notions of singularities in characteristic p > 0 such as F-finiteness, F-splitting, F-purity etc., for valuation rings. The novelty of this approach stems from the fact that most of the characteristic p notions of singularities have been extensively studied in the Noetherian setting, whereas valuation rings are usually highly non-Noetherian.

There is a surprising connection between the well-studied class of Abhyankar valuations (these valuations may be viewed as higher dimensional analogues of valuations associated to prime divisors on a variety) and the class of F-finite valuations. The talk is based on joint work with Karen Smith, and our work attempts to answer some questions asked by Karl Schwede and Zsolt Patakfalvi.