On the computation of K\"ahler differentials and characterizations of Galois extensions with independent defect

TL;DR

This paper develops methods to compute K"ahler differentials in valued field extensions and applies these to classify Galois defect extensions of prime degree, especially focusing on independent defect in positive characteristic.

Contribution

It introduces new presentations and computations of K"ahler differentials for valued field extensions and characterizes independent defect using ramification ideals, traces, and differentials.

Findings

Characterizations of independent defect using K"ahler differentials

Extensions with independent defect only occur in perfectoid and deeply ramified fields

No restrictions on valuation rank or value groups in the results

Abstract

For important cases of algebraic extensions of valued fields, we develop presentations of the associated K\"ahler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated Dedekind differentials. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs.\ independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, K\"ahler differentials and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their rank or value groups.