Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

TL;DR

This paper extends the topological Noetherianity of polynomial functors to rings with Noetherian spectrum, enabling characteristic-independent proofs of algebraic conjectures and developing the theory of polynomial laws.

Contribution

It proves topological Noetherianity for polynomial functors over rings with Noetherian spectrum, generalizing previous results and advancing the theory of polynomial laws.

Findings

Topological Noetherianity holds for polynomial functors over rings with Noetherian spectrum.

Associated topological spaces of finitely generated modules are Noetherian under these conditions.

Application to characteristic-independent proofs of algebraic conjectures like Stillman's conjecture.

Abstract

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when $\operatorname{Spec}(R)$ is; this is the degree-zero case of our result on polynomial functors.