Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

AKE principles for deeply ramified fields

Franziska Jahnke, Jonas van der Schaaf

Abstract

We study the model theory of deeply ramified fields of positive characteristic. Generalizing the perfect case treated in work by Jahnke and Kartas on the model theory of perfectoid fields, we obtain Ax-Kochen/Ershov principles for certain deeply ramified fields of positive characteristic and fixed degree of imperfection. Our results apply in particular to all deeply ramified henselian valued fields of rank 1.

AKE principles for deeply ramified fields

Abstract

We study the model theory of deeply ramified fields of positive characteristic. Generalizing the perfect case treated in work by Jahnke and Kartas on the model theory of perfectoid fields, we obtain Ax-Kochen/Ershov principles for certain deeply ramified fields of positive characteristic and fixed degree of imperfection. Our results apply in particular to all deeply ramified henselian valued fields of rank 1.

Paper Structure

This paper contains 7 sections, 18 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Deeply ramified and almost separably tame fields
  3. Separably tame valued fields
  4. Deeply ramified valued fields
  5. Unramified extensions
  6. Almost separably tame valued fields
  7. AKE Principles for deeply ramified fields and applications

Key Result

Theorem 1

Fix a prime $p$ and $e\in\mathbb{N} \cup\{\infty\}$. The class $\mathcal{C}_{p,e}$ of pointed valued fields $(K,v,t)$ satisfying the following properties is elementary:

Theorems & Definitions (42)

  • Theorem : see \ref{['cor:elem']}
  • Theorem : see \ref{['thm:ake']}
  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2: Kuhlmann for $e=0$, Kuhlmann--Pal for $e<\infty$, Anscombe for $e=\infty$
  • Definition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 32 more