Computational Aspects of the Short Resolution
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arXiv
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| Format: | Preprint |
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2025
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| author | García-Marco, Ignacio Gimenez, Philippe González-Sánchez, Mario |
| author_facet | García-Marco, Ignacio Gimenez, Philippe González-Sánchez, Mario |
| contents | Let $R:= \Bbbk[x_1,\ldots,x_{n}]$ be a polynomial ring over a field $\Bbbk$, $I \subset R$ be a homogeneous ideal with respect to a weight vector $ω= (ω_1,\ldots,ω_n) \in (\mathbb{Z}^+)^n$, and denote by $d$ the Krull dimension of $R/I$. In this paper we study graded free resolutions of $R/I$ as $A$-module whenever $A :=\Bbbk[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. We exhibit a Schreyer-like method to compute a (non-necessarily minimal) graded free resolution of $R/I$ as $A$-module. When $R/I$ is a $3$-dimensional simplicial toric ring, we describe how to prune the previous resolution to obtain a minimal one. We finally provide an example of a $6$-dimensional simplicial toric ring whose Betti numbers, both as $R$-module and as $A$-module, depend on the characteristic of $\Bbbk$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computational Aspects of the Short Resolution García-Marco, Ignacio Gimenez, Philippe González-Sánchez, Mario Commutative Algebra Algebraic Geometry 14Q15, 13D02, 14Q10, 13F65, 20M50 Let $R:= \Bbbk[x_1,\ldots,x_{n}]$ be a polynomial ring over a field $\Bbbk$, $I \subset R$ be a homogeneous ideal with respect to a weight vector $ω= (ω_1,\ldots,ω_n) \in (\mathbb{Z}^+)^n$, and denote by $d$ the Krull dimension of $R/I$. In this paper we study graded free resolutions of $R/I$ as $A$-module whenever $A :=\Bbbk[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. We exhibit a Schreyer-like method to compute a (non-necessarily minimal) graded free resolution of $R/I$ as $A$-module. When $R/I$ is a $3$-dimensional simplicial toric ring, we describe how to prune the previous resolution to obtain a minimal one. We finally provide an example of a $6$-dimensional simplicial toric ring whose Betti numbers, both as $R$-module and as $A$-module, depend on the characteristic of $\Bbbk$. |
| title | Computational Aspects of the Short Resolution |
| topic | Commutative Algebra Algebraic Geometry 14Q15, 13D02, 14Q10, 13F65, 20M50 |
| url | https://arxiv.org/abs/2504.12019 |