  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, H\`a 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.
