MINIMAL FREE RESOLUTIONS OF NUMERICAL SEMIGROUP ALGEBRAS VIA APÉRY SPECIALIZATION

Benjamin Braun; Tara Gomes; Ezra Miller; Christopher O’Neill; Aleksandra Sobieska

doi:10.2140/pjm.2025.334.211

MINIMAL FREE RESOLUTIONS OF NUMERICAL SEMIGROUP ALGEBRAS VIA APÉRY SPECIALIZATION

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

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Numerical semigroups with multiplicity m are parametrized 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 of maximal embedding dimension, which is the case when it is parametrized by the interior of C_m itself.

Original language	English
Pages (from-to)	211-231
Number of pages	21
Journal	Pacific Journal of Mathematics
Volume	334
Issue number	2
DOIs	https://doi.org/10.2140/pjm.2025.334.211
State	Published - 2025

Bibliographical note

Publisher Copyright:
© 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

Keywords

free resolution
numerical semigroup
toric ideal

ASJC Scopus subject areas

General Mathematics

Access to Document

10.2140/pjm.2025.334.211

Cite this

@article{3b364d1ce4704d399f32565574188464,

title = "MINIMAL FREE RESOLUTIONS OF NUMERICAL SEMIGROUP ALGEBRAS VIA AP{\'E}RY SPECIALIZATION",

abstract = "Numerical semigroups with multiplicity m are parametrized by integer points in a polyhedral cone Cm, 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 Cm. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in Cm, and minimality of the resolution is achieved when the semigroup is of maximal embedding dimension, which is the case when it is parametrized by the interior of Cm itself.",

keywords = "free resolution, numerical semigroup, toric ideal",

author = "Benjamin Braun and Tara Gomes and Ezra Miller and Christopher O{\textquoteright}Neill and Aleksandra Sobieska",

note = "Publisher Copyright: {\textcopyright} 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).",

year = "2025",

doi = "10.2140/pjm.2025.334.211",

language = "English",

volume = "334",

pages = "211--231",

number = "2",

}

TY - JOUR

T1 - MINIMAL FREE RESOLUTIONS OF NUMERICAL SEMIGROUP ALGEBRAS VIA APÉRY SPECIALIZATION

AU - Braun, Benjamin

AU - Gomes, Tara

AU - Miller, Ezra

AU - O’Neill, Christopher

AU - Sobieska, Aleksandra

PY - 2025

Y1 - 2025

N2 - Numerical semigroups with multiplicity m are parametrized by integer points in a polyhedral cone Cm, 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 Cm. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in Cm, and minimality of the resolution is achieved when the semigroup is of maximal embedding dimension, which is the case when it is parametrized by the interior of Cm itself.

AB - Numerical semigroups with multiplicity m are parametrized by integer points in a polyhedral cone Cm, 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 Cm. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in Cm, and minimality of the resolution is achieved when the semigroup is of maximal embedding dimension, which is the case when it is parametrized by the interior of Cm itself.

KW - free resolution

KW - numerical semigroup

KW - toric ideal

UR - http://www.scopus.com/inward/record.url?scp=85219734055&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85219734055&partnerID=8YFLogxK

U2 - 10.2140/pjm.2025.334.211

DO - 10.2140/pjm.2025.334.211

M3 - Article

AN - SCOPUS:85219734055

SN - 0030-8730

VL - 334

SP - 211

EP - 231

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -

MINIMAL FREE RESOLUTIONS OF NUMERICAL SEMIGROUP ALGEBRAS VIA APÉRY SPECIALIZATION

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this