On Finite and Nonfinite Generation of Associated Graded Rings of Abhyankar Valuations

Abstract

We consider the condition that the associated graded ring of a local ring along an Abhyankar valuation is finitely generated. We characterize when this holds for regular local rings of domension two, and give some examples of maximal rank Abhyankar valuations for which the associated graded ring is not finitely generated on two dimensional normal local rings and regular local rings of dimension larger than two. We characterize two dimensional normal local rings for which all divisorial valuations which dominate the ring have a finitely generated associated graded ring.