Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra
Abstract
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.
Fields of Study & Subjects
Advanced Topics in Algebra
Algebraic structures and combinatorial models
Rings
Modules
and Algebras
Publication details
- Journal / source
- Journal of Algebra
- Volume
- 700
- Pages
- 146–184
- Publication date
- 15 April 2026
- Language
- EN