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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.01700 |
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| _version_ | 1866911864533811200 |
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| author | Gomes, Tara O'Neill, Christopher Sobieska, Aleksandra Dávila, Eduardo Torres |
| author_facet | Gomes, Tara O'Neill, Christopher Sobieska, Aleksandra Dávila, Eduardo Torres |
| contents | Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m$, known as the Kunz cone. The faces of $C_m$ form a stratification of numerical semigroups that has been shown to respect a number of algebraic properties of $S$, including the combinatorial structure of the minimal free resolution of the defining toric ideal $I_S$. In this work, we prove that the structure of the infinite free resolution of the ground field $\Bbbk$ over the semigroup algebra $\Bbbk[S]$ also respects this stratification, yielding a new combinatorial approach to classifying homological properties like Golodness and rationality of the poincare series in this setting. Additionally, we give a complete classification of such resolutions in the special case $m = 4$, and demonstrate that the associated graded algebras do not generally respect the same stratification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite free resolutions over numerical semigroup algebras via specialization Gomes, Tara O'Neill, Christopher Sobieska, Aleksandra Dávila, Eduardo Torres Commutative Algebra Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m$, known as the Kunz cone. The faces of $C_m$ form a stratification of numerical semigroups that has been shown to respect a number of algebraic properties of $S$, including the combinatorial structure of the minimal free resolution of the defining toric ideal $I_S$. In this work, we prove that the structure of the infinite free resolution of the ground field $\Bbbk$ over the semigroup algebra $\Bbbk[S]$ also respects this stratification, yielding a new combinatorial approach to classifying homological properties like Golodness and rationality of the poincare series in this setting. Additionally, we give a complete classification of such resolutions in the special case $m = 4$, and demonstrate that the associated graded algebras do not generally respect the same stratification. |
| title | Infinite free resolutions over numerical semigroup algebras via specialization |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2405.01700 |