Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910150465421312 |
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| author | Kling, Filip Jonsson Lundqvist, Samuel Mohammadi, Fatemeh Orth, Matthias Sáenz-de-Cabezón, Eduardo |
| author_facet | Kling, Filip Jonsson Lundqvist, Samuel Mohammadi, Fatemeh Orth, Matthias Sáenz-de-Cabezón, Eduardo |
| contents | For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gröbner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra Kling, Filip Jonsson Lundqvist, Samuel Mohammadi, Fatemeh Orth, Matthias Sáenz-de-Cabezón, Eduardo Commutative Algebra Combinatorics 13P10, 13E10, 05E05, 13D02 For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gröbner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach. |
| title | Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra |
| topic | Commutative Algebra Combinatorics 13P10, 13E10, 05E05, 13D02 |
| url | https://arxiv.org/abs/2411.10209 |