Noetherianity of polynomial rings up to group actions

TL;DR

This paper investigates conditions under which polynomial rings parameterized by infinite sets are Noetherian when considering highly homogeneous group actions, linking algebraic properties to set-theoretic axioms.

Contribution

It establishes that polynomial rings are Noetherian up to certain group actions and connects the existence of a linear order on infinite sets to the axiom of choice.

Findings

Polynomial rings are Noetherian up to group actions.

Existence of a linear order on infinite sets is equivalent to the axiom of choice.

A sheaf theoretic approach is used to prove Noetherian properties.

Abstract

Let $k$ be a commutative Noetherian ring, and $k[S]$ the polynomial ring whose indeterminates are parameterized by elements in a set $S$. We show that $k[S]$ is Noetherian up to highly homogenous actions of groups. In particular, there is a special linear order $\leqslant$ on infinite $S$ such that $k[S]$ is Noetherian up to actions of $\mathrm{Aut}(S, \leqslant)$, and the existence of such a linear order for every infinite set is equivalent to the axiom of choice. These Noetherian results are proved via a sheaf theoretic approach based on Artin's theorem, the work of Nagel-R\"{o}mer, and a classification of highly homogenous groups by Cameron.