Ramification of local rings along valuations

In this paper we discuss stable forms of extensions of algebraic local rings along a valuation in all dimensions over a field k of characteristic zero, and generalize a formula of Ghezzi, Hà and Kashcheyeva describing the extension of associated graded rings along the valuation for stable extensions of regular algebraic local rings of dimension two to arbitrary ground fields k of characteristic zero. We discuss the failure of this result in positive characteristic.The main technique used in this paper is the algorithm for constructing generating sequences of Cutkosky and Vinh (2014) [6]. In Theorem 1.4, we show that the stable forms of extensions of regular algebraic local rings of dimension two over arbitrary ground fields of characteristic zero have a particularly simple relation between their generating sequences.