Ramification of local rings along valuations

TL;DR

This paper explores how algebraic local rings extend along valuations in all dimensions over characteristic zero fields, generalizing known formulas and examining failures in positive characteristic.

Contribution

It generalizes a formula for the extension of associated graded rings along valuations from dimension two to arbitrary dimensions over characteristic zero fields.

Findings

Generalization of the extension formula to all dimensions in characteristic zero

Identification of failures of the formula in positive characteristic

Analysis of stable forms of local ring extensions along valuations

Abstract

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.