Counterexamples to local monomialization in positive characteristic

TL;DR

This paper demonstrates that local monomialization, a property known to hold in characteristic zero, can fail in positive characteristic for algebraic local rings of dimension two or more, through a specific counterexample.

Contribution

The paper provides the first known counterexample showing failure of local monomialization in positive characteristic for higher-dimensional algebraic function fields.

Findings

Counterexample showing failure of local monomialization in positive characteristic

Local monomialization holds in characteristic zero but not necessarily in positive characteristic

Highlights limitations of monomialization techniques in positive characteristic settings

Abstract

In this paper we consider birational properties of ramification in excellent local rings. We give an example showing that local monomialization (and weak local monomialization) can fail for extensions of algebraic local rings in algebraic function fields of dimension greater than or equal to two along a valuation over a field of positive characteristic. It was earlier proven by the author that local monomialization holds within characteristic zero algebraic function fields.