Big polynomial rings and Stillman's conjecture

TL;DR

This paper demonstrates that certain limits of polynomial rings are polynomial rings and uses this to provide two new proofs of Stillman's conjecture, advancing understanding in commutative algebra.

Contribution

It introduces novel methods to prove Stillman's conjecture, including a streamlined proof and a completely different approach leveraging recent noetherianity results.

Findings

Established limits of polynomial rings are polynomial rings.

Provided two new proofs of Stillman's conjecture.

Connected polynomial ring limits to algebraic properties in commutative algebra.

Abstract

The purpose of this paper is to prove that certain limits of polynomial rings are themselves polynomial rings, and show how this observation can be used to deduce some interesting results in commutative algebra. In particular, we give two new proofs of Stillman's conjecture. The first is similar to that of Ananyan-Hochster, though more streamlined; in particular, it establishes the existence of small subalgebras. The second proof is completely different, and relies on a recent noetherianity result of Draisma.