Invariants of finite groups acting on (free) skew fields

TL;DR

This paper studies the structure of invariant subfields under finite group actions on free skew fields, proving finite generation and solving the free Noether problem in certain cases, while providing counterexamples to related conjectures.

Contribution

It proves finite generation of invariants in skew fields and solves the free Noether problem for linear actions, also providing counterexamples to conjectures on invariants and centralizers.

Findings

$M^G$ is finitely generated for finite groups $G$

The free Noether problem is positively solved for linear actions when char$(k)$ does not divide $|G|$

Counterexamples show $M^{Z_2}$ need not be a free skew field and refute Cohn's conjecture

Abstract

Let $M$ be a finitely generated skew field over a ground field $k$, and let $G$ be a finite group of $k$-linear automorphisms of $M$. This paper investigates finite generation of the skew subfield $M^G$ of $G$-invariants in $M$, and relations between the generators. The first main result shows that $M^G$ is finitely generated. Stronger conclusions hold when $M$ is a free skew field, i.e., the universal skew field of fractions of a free algebra. The second main result is the solution of the free Noether problem for non-modular linear group actions: if $G$ acts linearly on the free skew field $M$ on $m$ generators and the characteristic of $k$ does not divide $|G|$, then $M^G$ is the free skew field on $|G|(m-1)+1$ generators. In contrast, a nonlinear action of $Z_2$ on the free skew field $M$ on two generators is presented such that $M^{Z_2}$ is not a free skew field, resolving the free L\"uroth problem. This action also exposes a non-scalar element of $M$ whose centralizer is not a rational field, refuting a conjecture of P. M. Cohn from 1978.