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On residual domination and types orthogonal to the value group

Pablo Cubides Kovacsics, Silvain Rideau-Kikuchi, Mariana Vicaría

TL;DR

The paper extends Hrushovski–Loeser style characterizations from ACVF to equicharacteristic zero henselian fields by introducing residual domination and proving its equivalence with orthogonality to the value group and with ACVF-reducts that are generically stable. It develops a robust framework linking stable and simple residue-field behavior to tameness of type spaces via the Lin sorts, and shows these results persist in valued fields with operators such as diffRV and derivations. The main theorem establishes that residually dominated types coincide with those orthogonal to the value group and with reducts that are generically stable (or generically simple under simple residue fields), enabling base-change results and lifting tameness to broader settings. The framework applies to ultraproducts of $p$-adics and the limit theory $VFA_0$, and provides a path toward analyzing homotopy-type questions for type spaces in real-closed valued fields. Overall, the work unifies domination phenomena across expansions and operators, tying value-group orthogonality to residue-field tameness in a broad equicharacteristic zero landscape.

Abstract

We present a unifying framework of residual domination for (expansions of) henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to ACVF (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. When the residue field is stable (resp. simple) we relate these equivalent notions to generic stability (resp. simplicity). Those results apply in particular to ultraproducts of $p$-adic fields and to the limit theory VFA$_{0}$ of algebraically closed valued fields of characteristic $p$ with the Frobenius automorphism (as $p$ tends to infinity).

On residual domination and types orthogonal to the value group

TL;DR

The paper extends Hrushovski–Loeser style characterizations from ACVF to equicharacteristic zero henselian fields by introducing residual domination and proving its equivalence with orthogonality to the value group and with ACVF-reducts that are generically stable. It develops a robust framework linking stable and simple residue-field behavior to tameness of type spaces via the Lin sorts, and shows these results persist in valued fields with operators such as diffRV and derivations. The main theorem establishes that residually dominated types coincide with those orthogonal to the value group and with reducts that are generically stable (or generically simple under simple residue fields), enabling base-change results and lifting tameness to broader settings. The framework applies to ultraproducts of -adics and the limit theory , and provides a path toward analyzing homotopy-type questions for type spaces in real-closed valued fields. Overall, the work unifies domination phenomena across expansions and operators, tying value-group orthogonality to residue-field tameness in a broad equicharacteristic zero landscape.

Abstract

We present a unifying framework of residual domination for (expansions of) henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to ACVF (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. When the residue field is stable (resp. simple) we relate these equivalent notions to generic stability (resp. simplicity). Those results apply in particular to ultraproducts of -adic fields and to the limit theory VFA of algebraically closed valued fields of characteristic with the Frobenius automorphism (as tends to infinity).

Paper Structure

This paper contains 16 sections, 27 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stably embedded sets
  4. Domination
  5. Equicharacteristic zero henselian fields
  6. The structure LinA
  7. Setup
  8. Residual domination
  9. Algebraically closed residue fields and dense value groups
  10. Between M and Mur
  11. The main equivalence
  12. Valued fields with operators
  13. Lifting tameness
  14. Generic stability
  15. Generic simplicity
  16. ...and 1 more sections

Key Result

Theorem 1

Let $M$ be an equicharacteristic zero henselian valued field such that the value group $\Gamma(M)$ is either: Then, the following are equivalent for a type $p$ of field points over an algebraically closed set of imaginary elements:

Theorems & Definitions (82)

  • Theorem
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7: 3dividing
  • ...and 72 more