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Elements of Cohomology for model-theorically finite-dimensional groups and Lie algebras

Samuel Zamour

TL;DR

This paper develops a model-theoretic version of group and Lie algebra cohomology in finite-dimensional theories, focusing on the first cohomology group and its structural implications. It uses inflation-restriction and exact sequences (avoiding spectral sequences) to obtain vanishing results, starting with $H^1(G,A)=0$ for definable connected nilpotent $G$ with $A^G=0$, and extends to decompositions and Frattini-type arguments for Cartan subgroups. It provides a definable Maschke theorem for abelian groups without $p$-torsion and shows that certain cohomological vanishing statements yield strong decomposition and normalizer results. An appendix translates the whole framework to Lie algebras, showing the parallel validity of the results in that setting and establishing the broader utility of definable cohomology in model theory.

Abstract

We use the cohomology theory for groups invented by Hochshild and Serre to compute the first cohomology group for nilpotent groups that are definable in a finite-dimensional theory. Based on the established cohomological results, we derive some structural results : we prove a weak form of the Frattini argument for definable connected Cartan subgroups and we give a definable version of Maschke's theorem in the case of a definable abelian group without p-torsion. The same results hold when one works with Lie algebras instead of groups.

Elements of Cohomology for model-theorically finite-dimensional groups and Lie algebras

TL;DR

This paper develops a model-theoretic version of group and Lie algebra cohomology in finite-dimensional theories, focusing on the first cohomology group and its structural implications. It uses inflation-restriction and exact sequences (avoiding spectral sequences) to obtain vanishing results, starting with for definable connected nilpotent with , and extends to decompositions and Frattini-type arguments for Cartan subgroups. It provides a definable Maschke theorem for abelian groups without -torsion and shows that certain cohomological vanishing statements yield strong decomposition and normalizer results. An appendix translates the whole framework to Lie algebras, showing the parallel validity of the results in that setting and establishing the broader utility of definable cohomology in model theory.

Abstract

We use the cohomology theory for groups invented by Hochshild and Serre to compute the first cohomology group for nilpotent groups that are definable in a finite-dimensional theory. Based on the established cohomological results, we derive some structural results : we prove a weak form of the Frattini argument for definable connected Cartan subgroups and we give a definable version of Maschke's theorem in the case of a definable abelian group without p-torsion. The same results hold when one works with Lie algebras instead of groups.

Paper Structure

This paper contains 7 sections, 15 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the cohomology of groups
  3. The case of finite groups
  4. Finite-dimensional nilpotent groups
  5. A weak Frattini argument for definable connected Cartan subgroups
  6. Definable Maschke's theorem
  7. Appendix : Lie algebras

Key Result

Lemma 2.11

Let $(G,A)$ be a $G$-module and let $H$ be a central subgroup of $G$, such that $H\leq C_{G}(A)$. Suppose that $A^{G}=0$. Then $H^1(H,A)^{G/H}=0$ and $H^1(G/H, A)\simeq H^1(G, A)$.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.5
  • proof
  • Definition 2.8
  • proof
  • proof
  • Lemma 2.11
  • proof
  • ...and 36 more