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Beyond the Fontaine-Wintenberger theorem

Franziska Jahnke, Konstantinos Kartas

TL;DR

The work develops a model-theoretic framework for perfectoid fields and their tilts, constructing an elementary extension and a henselian defectless valuation with divisible value group and a residue field linked to the tilt. A central Ax–Kochen–Ershov principle is established for deeply ramified fields, enabling a detailed embedding of the tilt into the residue field and revealing new phenomena in positive characteristic. The authors prove a non-standard version of almost purity that, via ultraproducts and taming of valuations, yields elementary equivalences and transfer results between $K$ and $K^{\flat}$, as well as corollaries about Galois groups and elementary substructures. Tilting thus acts as a bridge for transferring first-order information, with the perfect hull of $\mathbb{F}_p(t)^h$ identified as an elementary substructure of the perfect hull of $\mathbb{F}_p((t))$ in a refined setting. The framework unifies almost mathematics with standard valuation theory, offering new tools for understanding the Fontaine–Wintenberger correspondence in a model-theoretic light and enabling applications to decidability and elementary equivalence across bases.

Abstract

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of $\mathbb{F}_p(t)^h$ is an elementary substructure of the perfect hull of $\mathbb{F}_p(\!(t)\!)$.

Beyond the Fontaine-Wintenberger theorem

TL;DR

The work develops a model-theoretic framework for perfectoid fields and their tilts, constructing an elementary extension and a henselian defectless valuation with divisible value group and a residue field linked to the tilt. A central Ax–Kochen–Ershov principle is established for deeply ramified fields, enabling a detailed embedding of the tilt into the residue field and revealing new phenomena in positive characteristic. The authors prove a non-standard version of almost purity that, via ultraproducts and taming of valuations, yields elementary equivalences and transfer results between and , as well as corollaries about Galois groups and elementary substructures. Tilting thus acts as a bridge for transferring first-order information, with the perfect hull of identified as an elementary substructure of the perfect hull of in a refined setting. The framework unifies almost mathematics with standard valuation theory, offering new tools for understanding the Fontaine–Wintenberger correspondence in a model-theoretic light and enabling applications to decidability and elementary equivalence across bases.

Abstract

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of is an elementary substructure of the perfect hull of .
Paper Structure (57 sections, 48 theorems, 130 equations)

This paper contains 57 sections, 48 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. The method of letting $p\to \infty$
  3. The method of letting $e\to \infty$
  4. Motivating question
  5. Main theorem
  6. Part I
  7. Positive characteristic
  8. Characteristic zero
  9. Almost mathematics
  10. Part II: Embedding the tilt in the residue field
  11. Other applications
  12. Preliminaries
  13. Ultraproducts
  14. Ł oś' Theorem
  15. Saturation
  16. ...and 42 more sections

Key Result

Theorem 1

Let $(K,v)$ be a perfectoid field and $\varpi\in \mathfrak{m}\backslash \{0\}$. Let $U$ be a non-principal ultrafilter on $\mathbb{N}$ and $(K_U,v_U)$ be the corresponding ultrapower. Let $w$ be the coarsest coarsening of $v_U$ such that $w\varpi>0$. Then: Moreover, the isomorphism $G_{K_U}\cong G_{k_w}$ restricts to $G_K\cong G_{K^{\flat}}$.

Theorems & Definitions (141)

  • Theorem : see Theorem \ref{['modeltheoreticFont']}
  • Theorem : Almost purity
  • Theorem 1.7.3
  • Corollary 1.7.5
  • Corollary 1.7.6
  • Corollary 1.7.7
  • Definition 2.1.4
  • Remark 2.1.6
  • Example 2.1.7
  • Remark 2.1.10
  • ...and 131 more