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Suppose that R\rightarrow S is an extension of local domains and \nu^* is avaluation dominating S. We consider the natural extension of associated gradedrings along the valuation gr_{\nu^*}(R)\rightarrow gr_{\nu^*}(S). We giveexamples showing that in general, this extension does not share good propertiesof the extension $\rightarrow S, but after enough blow ups above thevaluations, good properties of the extension R\rightarrow S are reflected inthe extension of associated graded rings. Stable properties of this extension(after blowing up) are much better in characteristic zero than in positivecharacteristic. Our main result is a generalization of the Abhyankar-Jungtheorem which holds for extensions of associated graded rings along thevaluation, after enough blowing up.

A generalization of the Abhyankar Jung theorem to associated graded rings of valuations