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

TL;DR

This paper extends the Abhyankar-Jung theorem to associated graded rings of valuations, showing that after sufficient blow-ups, the extension properties are well-behaved, especially in characteristic zero.

Contribution

It generalizes the Abhyankar-Jung theorem to associated graded rings of valuations, demonstrating stable properties after blow-ups, with a focus on characteristic zero.

Findings

Extensions of associated graded rings do not always share good properties.

Blowing up above valuations improves extension properties.

The main theorem generalizes Abhyankar-Jung for these extensions after enough blow-ups.

Abstract

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