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Pseudo-Frobenius numbers and defining ideals in stretched numerical semigroup rings

Do Van Kien, Naoyuki Matsuoka, Taiga Ozaki

TL;DR

The paper addresses how pseudo-Frobenius numbers $ ext{PF}(H)$ control the defining ideal $I_H$ of the numerical semigroup ring $k[H]$, proving a conjecture under the stretchedness condition that $k[H]/(t^{a_1})$ is stretched. The authors show that if $ ext{PF}(H)$ forms an arithmetic sequence of length $n-1$, then, after a suitable permutation of generators, $I_H$ can be realized as a determinantal ideal $ ext{I}_2(M)$ of a monomial matrix $M$, with the graded minimal free resolution given by the Eagon–Northcott complex. They further derive precise Cohen–Macaulay criteria for the tangent cone $ ext{gr}(k[H])$ in this stretched setting, expressed as explicit inequalities involving parameters $h_1$, $ ext{ell}$, and $eta$, and discuss special cases such as $ ext{ell}=2$ where CM holds automatically. The results illuminate a deep link between the combinatorial invariant $ ext{PF}(H)$ and the algebraic structure of $k[H]$, offering concrete tools for determining the defining equations and tangent-cone properties, as demonstrated by detailed examples.

Abstract

The pseudo-Frobenius numbers of a numerical semigroup $H$ are deeply connected to the structure of the defining ideal of its semigroup ring $k[H]$. In this paper, we resolve a certain conjecture related to this connection under the assumption that $k[H]/(t^a)$ is stretched, where $a$ is the multiplicity of $H$. Furthermore, we provide numerical conditions for the tangent cone of $k[H]$ to be Cohen-Macaulay.

Pseudo-Frobenius numbers and defining ideals in stretched numerical semigroup rings

TL;DR

The paper addresses how pseudo-Frobenius numbers control the defining ideal of the numerical semigroup ring , proving a conjecture under the stretchedness condition that is stretched. The authors show that if forms an arithmetic sequence of length , then, after a suitable permutation of generators, can be realized as a determinantal ideal of a monomial matrix , with the graded minimal free resolution given by the Eagon–Northcott complex. They further derive precise Cohen–Macaulay criteria for the tangent cone in this stretched setting, expressed as explicit inequalities involving parameters , , and , and discuss special cases such as where CM holds automatically. The results illuminate a deep link between the combinatorial invariant and the algebraic structure of , offering concrete tools for determining the defining equations and tangent-cone properties, as demonstrated by detailed examples.

Abstract

The pseudo-Frobenius numbers of a numerical semigroup are deeply connected to the structure of the defining ideal of its semigroup ring . In this paper, we resolve a certain conjecture related to this connection under the assumption that is stretched, where is the multiplicity of . Furthermore, we provide numerical conditions for the tangent cone of to be Cohen-Macaulay.
Paper Structure (8 sections, 19 theorems, 28 equations)

This paper contains 8 sections, 19 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental notation in numerical semigroups
  4. Stretched local rings and stretched numerical semigroup rings
  5. Defining ideals of numerical semigroup rings
  6. Proof of Theorem \ref{['main theorem']}
  7. Cohen-Macaulayness of $\hbox{\rm gr}(k[H])$
  8. Examples

Key Result

Theorem 1.2

Let $H = \left<a_1, a_2,\ldots , a_n\right>$ be an $n(\ge 3)$-generated numerical semigroup with multiplicity $a_1$, and suppose that $k[H]/(t^{a_1})$ is stretched. Then the following are equivalent. When this is the case, the graded minimal $S$-free resolution of $k[H]$ is given by the Eagon-Northcott complex eagon of the matrix appearing in the conditions (1) and (2).

Theorems & Definitions (42)

  • Conjecture 1.1: by discussion between the the first author, the second author, D. T. Cuong, and H. L. Truong
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6: sally1
  • Theorem 2.7: sally1 for Gorenstein case, elias for general case
  • ...and 32 more