Elimination of Ramification I: The Generalized Stability Theorem

TL;DR

This paper proves a generalized stability theorem for valued fields, showing that trivial ramification defect in a base field extends to certain finitely generated extensions, with implications for ramification elimination and model theory.

Contribution

It introduces a broad version of the Stability Theorem linking ramification defect triviality from base fields to their finitely generated extensions under specific conditions.

Findings

Proves the generalized Stability Theorem for valued fields.

Demonstrates elimination of ramification in valued function fields.

Provides applications to local uniformization and model theory in positive characteristic.

Abstract

We prove a general version of the "Stability Theorem": if $K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $F|K$ is a finitely generated (transcendental) extension of valued fields for which equality holds in the Abhyankar inequality, then the defect is also trivial for all finite extensions of $F$. This theorem is applied to eliminate ramification in such valued function fields. It has applications to local uniformization and to the model theory of valued fields in positive characteristic.