Big polynomial rings and Stillman's conjecture

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.

21 pages. v4: removes all hypotheses that we are working over an infinite ground field. v5: Fixes a gap in the proof of Lemma 5.11

References in corpus (2)

• Big polynomial rings with imperfect coefficient fields
• Generalizations of Stillman's conjecture via twisted commutative algebras

Cited by in corpus (16)

• The geometry of polynomial representations
• Big polynomial rings with imperfect coefficient fields
• An equivariant Hochster's formula for ${\mathfrak{S}}_n$-invariant monomial ideals
• Universality of high-strength tensors
• Generalizations of Stillman's conjecture via twisted commutative algebras
• The set of forms with bounded strength is not closed
• Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum
• Strength conditions, small subalgebras, and Stillman bounds in degree $\le 4$
• Structures in representation stability
• Small projective spaces and Stillman uniformity for sheaves
• GL-algebras in positive characteristic II: the polynomial ring
• Explicit Stillman bounds for all degrees
• Free limits of free algebras
• Semigroup Graded Stillman's Conjecture
• Stillman's question for twisted commutative algebras
• Equivariant algebraic and semi-algebraic geometry of infinite affine space