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Simplicity of the automorphism group of fields with operators

Thomas Blossier, Zoé Chatzidakis, Charlotte Hardouin, Amador Martin-Pizarro

TL;DR

This work extends Lascar's simplicity phenomenon to uncountable models of stable theories, with a focus on fields equipped with operators. It introduces κ-tame models and a closure operator tied to a unique generic type $p_0$, and proves that every automorphism fixing $\\operatorname{cl}(Z)$ is a product of four conjugates of a maximally moving automorphism $\\tau$, yielding simplicity of $\\operatorname{Aut}(\\mathbb{U}/\\operatorname{cl}(Z))$ modulo bounded automorphisms. The authors develop a general back-and-forth framework, applicable to both saturated and κ-prime models, and verify the framework across a broad family of theories (ACF, DCF, ACFA, ACFP, SCF$_{p,e}$, SCF$_{p,\infty}$, etc.), including several with operators beyond pure algebraic fields. The results unify and extend classical normal-subgroup classifications for uncountable permutation and linear groups with model-theoretic methods, offering a blueprint for proving simplicity in new operator-field contexts and connecting to known Baer/Rosenberg-type outcomes. Practically, this yields simple automorphism groups for a wide range of uncountable models while clarifying when bounded automorphisms must vanish, highlighting the role of the closure operator and generic independence in determining group-theoretic structure.

Abstract

We adapt a proof of Lascar in order to show the simplicity of the group of automorphisms fixing pointwise all non-generic elements for a class of uncountable models of suitable theories, encompassing both strongly minimal theories as well as several theories of fields with operators.

Simplicity of the automorphism group of fields with operators

TL;DR

This work extends Lascar's simplicity phenomenon to uncountable models of stable theories, with a focus on fields equipped with operators. It introduces κ-tame models and a closure operator tied to a unique generic type , and proves that every automorphism fixing is a product of four conjugates of a maximally moving automorphism , yielding simplicity of modulo bounded automorphisms. The authors develop a general back-and-forth framework, applicable to both saturated and κ-prime models, and verify the framework across a broad family of theories (ACF, DCF, ACFA, ACFP, SCF, SCF, etc.), including several with operators beyond pure algebraic fields. The results unify and extend classical normal-subgroup classifications for uncountable permutation and linear groups with model-theoretic methods, offering a blueprint for proving simplicity in new operator-field contexts and connecting to known Baer/Rosenberg-type outcomes. Practically, this yields simple automorphism groups for a wide range of uncountable models while clarifying when bounded automorphisms must vanish, highlighting the role of the closure operator and generic independence in determining group-theoretic structure.

Abstract

We adapt a proof of Lascar in order to show the simplicity of the group of automorphisms fixing pointwise all non-generic elements for a class of uncountable models of suitable theories, encompassing both strongly minimal theories as well as several theories of fields with operators.
Paper Structure (6 sections, 24 theorems, 42 equations, 2 figures)

This paper contains 6 sections, 24 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tame models
  4. Automorphisms of $\kappa$-tame models
  5. General model-theoretic properties yield our properties
  6. Annex: The examples

Key Result

Lemma 3.4

Let $\mathbb U$ be a $\kappa$-tame model over $Z$ of the theory $T$ and consider three $\operatorname{cl}$-closed subsets $X$, $Y_1$ and $Y_2$ of $\mathbb U$ with $Z \subseteq X \subseteq Y_1 \cap Y_2$, each $\operatorname{cl}$-generated over $Z$ by subsets of size strictly less than $\kappa$. Give

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (74)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 64 more