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Minimal free resolutions of numerical semigroup algebras via Apéry specialization

Benjamin Braun, Tara Gomes, Ezra Miller, Christopher O'Neill, Aleksandra Sobieska

TL;DR

The paper develops Apéry resolutions for numerical semigroup ideals by introducing the Apéry toric ideal $J_S$ in $R=\Bbbk[x_0,\dots,x_{m-1}]$ and proving a monomial-entry free resolution whose structure depends only on the multiplicity $m$. It shows the resolution is minimal precisely for MED semigroups, i.e., points in the interior of the Kunz cone $C_m$, and that within the relative interior of any fixed face of $C_m$ the same construction specializes uniformly to minimal resolutions. Furthermore, for arbitrary semigroups corresponding to points in a face, there exists a uniform method (a face-dependent change of basis) to obtain minimal free resolutions of $J_S$, with Betti numbers depending only on $m$ and the face. The work connects Apéry data, Kunz geometry, and toric ideals to produce explicit, geometry-aware resolutions and highlights avenues for constructive and topological refinement of Betti-number formulas.

Abstract

Numerical semigroups with multiplicity $m$ are parameterized by integer points in a polyhedral cone $C_m$, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative interior of a fixed face of $C_m$. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in $C_m$, and minimality of the resolution is achieved when the semigroup is maximal embedding dimension, which is the case parametrized by the interior of $C_m$ itself.

Minimal free resolutions of numerical semigroup algebras via Apéry specialization

TL;DR

The paper develops Apéry resolutions for numerical semigroup ideals by introducing the Apéry toric ideal in and proving a monomial-entry free resolution whose structure depends only on the multiplicity . It shows the resolution is minimal precisely for MED semigroups, i.e., points in the interior of the Kunz cone , and that within the relative interior of any fixed face of the same construction specializes uniformly to minimal resolutions. Furthermore, for arbitrary semigroups corresponding to points in a face, there exists a uniform method (a face-dependent change of basis) to obtain minimal free resolutions of , with Betti numbers depending only on and the face. The work connects Apéry data, Kunz geometry, and toric ideals to produce explicit, geometry-aware resolutions and highlights avenues for constructive and topological refinement of Betti-number formulas.

Abstract

Numerical semigroups with multiplicity are parameterized by integer points in a polyhedral cone , according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative interior of a fixed face of . The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in , and minimality of the resolution is achieved when the semigroup is maximal embedding dimension, which is the case parametrized by the interior of itself.
Paper Structure (10 sections, 11 theorems, 47 equations, 4 figures)

This paper contains 10 sections, 11 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kunz polyhedra and Apéry resolutions
  3. Semigroups and toric ideals
  4. Kunz cone
  5. Modules and maps for the Apéry resolution
  6. Modules
  7. Maps
  8. Maximal embedding dimension numerical semigroups
  9. Specialization for arbitrary numerical semigroups
  10. Open questions

Key Result

Lemma 2.4

If $S$ is a numerical semigroup of multiplicity $m$, then $b_{i,j} = 0$ if and only if $a_i+a_j = a_{i+j}$. Hence the Apéry coordinate vector of $S$ lies on the boundary of $C_m$ if and only if $b_{i,j} = 0$ for some $i,j$.

Figures (4)

  • Figure 1: The Apéry resolution for $m = 4$. The exponents $b_{i,j}$ and $c_{i,j}$ are constants depending on the particular numerical semigroup $S$.
  • Figure 2: The Apéry resolution for $m = 3$.
  • Figure 3: The Apéry resolution (above) and Eagon--Northcott resolution (below) for $I_S$ where $S = \langle4,9,10,11\rangle$
  • Figure 4: A specialization of the $m = 4$ Apéry resolution where $b_{11} = 0$, so $a_2 = 2a_1$. Note this forces $b_{13} = b_{23}$.

Theorems & Definitions (32)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Example 2.6
  • Theorem 2.7: kunz
  • Example 2.8
  • Example 2.9
  • ...and 22 more