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Semigroup Graded Stillman's Conjecture

John Cobb, Nathaniel Gallup, John Spoerl

TL;DR

The paper extends Stillman's conjecture to families of Gamma-graded polynomial rings, showing that the existence of a uniform bound for projective dimension hinges on the combinatorial property of the grading monoid Lambda having bounded factorization. The main result proves an iff condition: bounded factorization of Lambda guarantees a Stillman bound across all connected Lambda-graded rings, while lacking it yields explicit counterexamples with unbounded pdim; the proof leverages regrading to Z_+ when possible and builds explicit counterexamples otherwise. It then discusses corollaries and applications, including sharper bounds through finer gradings, and situates the work in toric geometry via effective cones and in the broader context of Stillman uniformity. The work broadens the program to multigraded and nonstandard gradings, identifies precise structural criteria for uniformity, and highlights interactions with toric geometry and graded invariants.

Abstract

We resolve Stillman's conjecture for families of polynomial rings that are graded by any semigroup under mild conditions. Conversely, we show that these conditions are necessary for the existence of a Stillman bound. This has applications even for the well-known standard graded case.

Semigroup Graded Stillman's Conjecture

TL;DR

The paper extends Stillman's conjecture to families of Gamma-graded polynomial rings, showing that the existence of a uniform bound for projective dimension hinges on the combinatorial property of the grading monoid Lambda having bounded factorization. The main result proves an iff condition: bounded factorization of Lambda guarantees a Stillman bound across all connected Lambda-graded rings, while lacking it yields explicit counterexamples with unbounded pdim; the proof leverages regrading to Z_+ when possible and builds explicit counterexamples otherwise. It then discusses corollaries and applications, including sharper bounds through finer gradings, and situates the work in toric geometry via effective cones and in the broader context of Stillman uniformity. The work broadens the program to multigraded and nonstandard gradings, identifies precise structural criteria for uniformity, and highlights interactions with toric geometry and graded invariants.

Abstract

We resolve Stillman's conjecture for families of polynomial rings that are graded by any semigroup under mild conditions. Conversely, we show that these conditions are necessary for the existence of a Stillman bound. This has applications even for the well-known standard graded case.
Paper Structure (4 sections, 4 theorems, 5 equations)

This paper contains 4 sections, 4 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. $\Gamma$-graded Commutative Algebra
  3. Main Results and Corollaries
  4. Applications and Examples

Key Result

Theorem 1.2

For any degree sequence $\bm{d}$ from $\Lambda$, there is a number $N(\Lambda, \bm{d})$ depending only on $\Lambda$ and $\bm{d}$ bounding the projective dimension of any ideal with degree sequence bounded by $\bm{d}$ in any connected $\Gamma$-graded polynomial ring with support contained in $\Lambda

Theorems & Definitions (15)

  • Example 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Example 2.2: Well-founded but no bounded factorization
  • Lemma 3.1
  • proof
  • Example 3.2
  • proof : Proof of Theorem \ref{['thm:polynomial_stillman']}
  • Corollary 3.3
  • ...and 5 more