On certain definable coarsenings of valuation rings and their applications

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.