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Arithmetic in the Boij Söderberg Cone

Adam Boocher, Noah Huang, Harrison Wolf

TL;DR

This paper reframes two long-standing conjectures on Betti numbers of graded modules as arithmetic problems inside the $Boij-Soderberg$ cone, connecting commutative algebra with number theory. By exploiting pure resolutions and the $Herzog-Kühl$ equations, it identifies Diophantine obstructions and completely classifies them in codimension 3, illustrating how arithmetic constraints constrain Betti tables. It then establishes new cases in codimensions 5 and 6 for Gorenstein algebras, using a blend of theoretical reductions and computer-assisted algebra to rule out obstructions. Overall, the work demonstrates the efficacy of the Boij-Söderberg arithmetic framework for proving lower bounds on Betti numbers in higher codimensions and outlines a program with concrete computational and number-theoretic challenges for future work.

Abstract

We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic problems in the Boij-Söderberg cone. In this setting, potential counterexamples correspond to explicit Diophantine obstructions arising from the numerics of pure resolutions. Using number-theoretic methods, we completely classify these obstructions in the codimension three case revealing some delicate connections between Betti tables, commutative algebra and classical Diophantine equations. The new results in codimensions five and six concern Gorenstein algebras where a study of the variety determined by these Diophantine equations is sufficient to resolve the conjecture in this case.

Arithmetic in the Boij Söderberg Cone

TL;DR

This paper reframes two long-standing conjectures on Betti numbers of graded modules as arithmetic problems inside the cone, connecting commutative algebra with number theory. By exploiting pure resolutions and the equations, it identifies Diophantine obstructions and completely classifies them in codimension 3, illustrating how arithmetic constraints constrain Betti tables. It then establishes new cases in codimensions 5 and 6 for Gorenstein algebras, using a blend of theoretical reductions and computer-assisted algebra to rule out obstructions. Overall, the work demonstrates the efficacy of the Boij-Söderberg arithmetic framework for proving lower bounds on Betti numbers in higher codimensions and outlines a program with concrete computational and number-theoretic challenges for future work.

Abstract

We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic problems in the Boij-Söderberg cone. In this setting, potential counterexamples correspond to explicit Diophantine obstructions arising from the numerics of pure resolutions. Using number-theoretic methods, we completely classify these obstructions in the codimension three case revealing some delicate connections between Betti tables, commutative algebra and classical Diophantine equations. The new results in codimensions five and six concern Gorenstein algebras where a study of the variety determined by these Diophantine equations is sufficient to resolve the conjecture in this case.
Paper Structure (6 sections, 11 theorems, 78 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and History
  3. Codimension 3 Degree Sequences with $B_1$ or $B_2$ equal to $2$
  4. Extension to Higher Codimension
  5. Codimension 3 Degree Sequences with $\sum B_i < 12$
  6. Gorenstein Rings in Codimension 5 and 6

Key Result

Theorem 1

Suppose that $M = R/I$ is a Gorenstein ring of codimension $c$, not isomorphic to a complete intersection.

Figures (1)

  • Figure 1: The graph of $2(y-x)(1-x)=y$. Rational points, which are parametrized by lines with rational slopes through $(0,0)$, correspond to integer solutions to equation \ref{['eq:1']}.

Theorems & Definitions (27)

  • Theorem 1
  • Example 2
  • Theorem 3
  • Example 9
  • Lemma 10
  • proof
  • proof
  • Proposition 12
  • Theorem 13
  • proof
  • ...and 17 more