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arXiv · 2411.10209

Gr\"obner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra

Filip Jonsson Kling·Samuel Lundqvist·Fatemeh Mohammadi·Matthias Orth·Eduardo Sáenz-de-Cabezón
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Abstract

For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gr\"obner 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.

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Filip Jonsson Kling, Samuel Lundqvist, Fatemeh Mohammadi, Matthias Orth, Eduardo Sáenz-de-Cabezón. 2024-11-15. Gr\"obner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra. https://arxiv.org/abs/2411.10209

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