On certain definable coarsenings of valuation rings and their applications

TL;DR

This paper explores how certain field extensions induce definable coarsenings of valuation rings, with applications to classifying defects, understanding ramification, and constructing specific convex subgroups in valued field extensions.

Contribution

It introduces methods to define coarsenings of valuation rings from prime degree extensions, including wild ramification cases, and applies these to classify defects and construct convex subgroups.

Findings

Definable coarsenings correspond to prime degree extensions.

Ramification ideals can be defined in wild ramification cases.

Constructed convex subgroups relate to Galois extensions with independent defect.

Abstract

We show how suitable extensions $(L|K,v)$ of prime degree of valued fields give rise to definable coarsenings of the valuation rings of $L$ and $K$. In the case of Artin-Schreier and Kummer extensions with wild ramification, we can also define the ramification ideal. We demonstrate the use of the coarsenings on $L$, their maximal ideals, and the ramification ideals for the classification of defects and for the presentation of the K\"ahler differentials of the extension of the valuation rings of $(L|K,v)$, and their annihilators. Finally, we give a construction that realizes predescribed convex subgroups of suitable value groups as those that are associated with Galois extensions of degree $p$ with independent defect, which in turn give rise to definable coarsenings.