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arXiv · 2310.03612

Minimal free resolutions of numerical semigroup algebras via Ap\'ery specialization

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

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Benjamin Braun, Tara Gomes, Ezra Miller, Christopher O'Neill, Aleksandra Sobieska. 2023-10-05. Minimal free resolutions of numerical semigroup algebras via Ap\'ery specialization. https://doi.org/10.2140/pjm.2025.334.211

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