Ramification of valuations and local rings

TL;DR

This paper explores the birational properties of ramification in excellent local rings, extending previous results to show strong local monomialization in certain cases and analyzing invariants of extensions in characteristic p.

Contribution

It extends earlier work on ramification and local monomialization, providing new results on the structure of extensions and invariants in two-dimensional excellent local rings.

Findings

Strong local monomialization holds along certain valuations.

Invariants alpha and beta are not eventually constant in characteristic p.

General results on the structure of extension of associated graded rings.

Abstract

In this paper we consider birational properties of ramification in excellent local rings. We extend earlier results of the author with Olivier Piltant to show that strong local monomialization is true along a valuation dominating a defectless extension of two dimensional excellent local rings. We also obtain general results on the structure of the extension of associated graded rings along a valuation, and show that the invariants alpha and beta of stable forms of two dimensional extensions in characteristic p of the author and Olivier Piltant are not eventually constant.