Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the computation of Kähler differentials and characterizations of Galois extensions with independent defect

Steven Dale Cutkosky, Franz-Viktor Kuhlmann, Anna Rzepka

TL;DR

This work develops presentations and annihilator descriptions for the Kähler differentials $\Omega_{\mathcal{O}_L|\mathcal{O}_K}$ in algebraic extensions of valued fields, without rank restrictions on valuations, and connects these to ramification, differents, and defect phenomena. A central theme is the classification of Galois defect extensions of prime degree into independent vs dependent defect, with equivalent characterizations in terms of ramification ideals, annihilators of $\Omega$, and trace instructions. The authors provide a robust framework for generating immediate unibranched extensions, derive explicit presentations for $\Omega_{\mathcal{O}_L|\mathcal{O}_K}$ (including via the ideals $I_{\mathcal{E}}^{p-1}$), and specialize to Artin–Schreier and Kummer defect extensions to obtain concrete equalities such as $\Omega_{\mathcal{O}_L|\mathcal{O}_K} \cong I_{\mathcal{E}}/I_{\mathcal{E}}^p$ and $\mathcal{D}(\mathcal{O}_L|\mathcal{O}_K)$-relations. The results unify and extend ramification, trace, and differential viewpoints for defect phenomena, with consequences for deeply ramified and perfectoid fields where independent defect prevails. Theoretical tools include final segments, ramification jumps, Hasse–Schmidt derivatives, and detailed AS/Kummer analyses, all formulated for arbitrary valuations.

Abstract

For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler 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, Kähler 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.

On the computation of Kähler differentials and characterizations of Galois extensions with independent defect

TL;DR

This work develops presentations and annihilator descriptions for the Kähler differentials in algebraic extensions of valued fields, without rank restrictions on valuations, and connects these to ramification, differents, and defect phenomena. A central theme is the classification of Galois defect extensions of prime degree into independent vs dependent defect, with equivalent characterizations in terms of ramification ideals, annihilators of , and trace instructions. The authors provide a robust framework for generating immediate unibranched extensions, derive explicit presentations for (including via the ideals ), and specialize to Artin–Schreier and Kummer defect extensions to obtain concrete equalities such as and -relations. The results unify and extend ramification, trace, and differential viewpoints for defect phenomena, with consequences for deeply ramified and perfectoid fields where independent defect prevails. Theoretical tools include final segments, ramification jumps, Hasse–Schmidt derivatives, and detailed AS/Kummer analyses, all formulated for arbitrary valuations.

Abstract

For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler 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, Kähler 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.
Paper Structure (27 sections, 37 theorems, 142 equations)

This paper contains 27 sections, 37 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Defect
  4. Final segments
  5. Strongly convex subgroups
  6. Immediate and unibranched extensions
  7. Artin-Schreier and Kummer extensions
  8. Ideals and final segments
  9. Ramification jumps and ramification ideals
  10. Galois defect extensions of prime degree
  11. Generation of immediate unibranched extensions of valuation rings
  12. The general case
  13. The case of immediate Artin-Schreier and Kummer extensions
  14. Values of derivatives of minimal polynomials
  15. Artin-Schreier extensions
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $L|K$ be an algebraic field extension of degree $n$ and suppose that $A$ is a normal domain with quotient field $K$ and $B$ is a domain with quotient field $L$ such that $A\subset B$ is an integral extension. Suppose that there exist generators $b_{\alpha}\in B$ of $L|K$, which are indexed by a Further suppose that there exist $a_{\alpha},a_\beta\in A$ such that $a_{\beta}\mid a_{\alpha}$ if

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Example 1.10
  • ...and 51 more