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Stillman's question for twisted commutative algebras

Karthik Ganapathy

Abstract

Let $\mathbf{A}_{n, m}$ be the polynomial ring $\text{Sym}(\mathbf{C}^n \otimes \mathbf{C}^m)$ with the natural action of $\mathbf{GL}_m(\mathbf{C})$. We construct a family of $\mathbf{GL}_m(\mathbf{C})$-stable ideals $J_{n, m}$ in $\mathbf{A}_{n, m}$, each equivariantly generated by one homogeneous polynomial of degree $2$. Using the Ananyan-Hochster principle, we show that the regularity of this family is unbounded. This negatively answers a question raised by Erman-Sam-Snowden on a generalization of Stillman's conjecture.

Stillman's question for twisted commutative algebras

Abstract

Let be the polynomial ring with the natural action of . We construct a family of -stable ideals in , each equivariantly generated by one homogeneous polynomial of degree . Using the Ananyan-Hochster principle, we show that the regularity of this family is unbounded. This negatively answers a question raised by Erman-Sam-Snowden on a generalization of Stillman's conjecture.

Paper Structure

This paper contains 2 sections, 2 theorems.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:mainresult']}

Key Result

Theorem 1.2

There exists a function $g\colon \mathbf{N} \rightarrow \mathbf{N}$ such that $\limsup_{n \rightarrow \infty} g(n) = \infty$ and $J_{n,g(n)}$ is generated (as a usual ideal of $\mathbf{A}_{n, g(n)}$) by $\frac{g(n)(g(n)+1)}{2}$ homogeneous polynomials of degree $2$ which form a regular sequence.

Theorems & Definitions (7)

  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:mainresult']}
  • Remark 2.3