Noetherianity of polynomial rings up to group actions

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 $⩽$ on infinite $S$ such that $k[S]$ is Noetherian up to actions of $\mathrm{Aut}(S,⩽)$, 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-Römer, and a classification of highly homogenous groups by Cameron.

arXiv commentA major revision following reviewer's suggestions