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Of model completeness and algebraic groups

Daniel Max Hoffmann, Piotr Kowalski, Chieu-Minh Tran, Jinhe Ye

TL;DR

The paper addresses the problem of model completeness for nonabelian groups arising from algebraic groups by proving that for a model complete field $K$ and a semisimple split group $G$ over $K$, both $G(K)$ and its commutator group $G(K)'$ are model complete. It develops a framework that combines Borel–Tits type decompositions with model-theoretic interpretability to reduce homomorphisms between rational points to field homomorphisms and isogenies, complemented by a multi-field 1-elementarity analysis. The authors provide a Chevalley-case proof for $G(K)'$ and then extend to $G(K)$, using a finite central kernel and an automorphism-extension lemma to lift elementary maps through algebraic closures. These results yield new infinite noncommutative examples of model-complete groups and reinforce a deep link between algebraic group structure and first-order model theory, with implications for the definable topology and reconstruction of geometric content from group-theoretic data.

Abstract

We show that if G is a split semisimple algebraic group over a model complete field K, then the groups G(K) and G(K)' (the commutator group which is a ``Chevalley group'' as for example the group PSL_2(K)) are model complete as well.

Of model completeness and algebraic groups

TL;DR

The paper addresses the problem of model completeness for nonabelian groups arising from algebraic groups by proving that for a model complete field and a semisimple split group over , both and its commutator group are model complete. It develops a framework that combines Borel–Tits type decompositions with model-theoretic interpretability to reduce homomorphisms between rational points to field homomorphisms and isogenies, complemented by a multi-field 1-elementarity analysis. The authors provide a Chevalley-case proof for and then extend to , using a finite central kernel and an automorphism-extension lemma to lift elementary maps through algebraic closures. These results yield new infinite noncommutative examples of model-complete groups and reinforce a deep link between algebraic group structure and first-order model theory, with implications for the definable topology and reconstruction of geometric content from group-theoretic data.

Abstract

We show that if G is a split semisimple algebraic group over a model complete field K, then the groups G(K) and G(K)' (the commutator group which is a ``Chevalley group'' as for example the group PSL_2(K)) are model complete as well.
Paper Structure (8 sections, 21 theorems, 119 equations)

This paper contains 8 sections, 21 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Homomorphism between rational points of algebraic groups
  3. Classical results on algebraic groups
  4. Consequences of Borel-Tits Theorem
  5. Miscellaneous model theory
  6. Several fields
  7. Automorphisms of finite commutative groups
  8. The main result

Key Result

Theorem 2.2

If $G$ is a semisimple and simply connected algebraic group over a field $K$, then there is a decomposition where $G_1,\ldots,G_l$ are simple and simply connected algebraic groups over $K$.

Theorems & Definitions (60)

  • Remark 2.1
  • Theorem 2.2: Chapter 24a in milnealggps
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Theorem 2.6: ros57 and sga33
  • Theorem 2.7: Theorem 1.3 in Steinberg
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 50 more