Multigraded strong Lefschetz property for balanced simplicial complexes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916527578546176 |
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| author | Oba, Ryoshun |
| author_facet | Oba, Ryoshun |
| contents | Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. We show that, for $\mathbf{a} \in \mathbb{N}^m_+$, the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner ring of an $\mathbf{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property. A corollary is the inequality $h_{\mathbf{b}} \leq h_{\mathbf{c}}$ for $\mathbf{b} \leq \mathbf{c} \leq \mathbf{a}-\mathbf{b}$ among the flag $h$-numbers of an $\mathbf{a}$-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to $\mathbf{a}$-balanced homology manifolds and $\mathbf{a}$-balanced simplicial cycles over a field of characteristic $2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_17110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multigraded strong Lefschetz property for balanced simplicial complexes Oba, Ryoshun Combinatorics Commutative Algebra Algebraic Geometry Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. We show that, for $\mathbf{a} \in \mathbb{N}^m_+$, the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner ring of an $\mathbf{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property. A corollary is the inequality $h_{\mathbf{b}} \leq h_{\mathbf{c}}$ for $\mathbf{b} \leq \mathbf{c} \leq \mathbf{a}-\mathbf{b}$ among the flag $h$-numbers of an $\mathbf{a}$-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to $\mathbf{a}$-balanced homology manifolds and $\mathbf{a}$-balanced simplicial cycles over a field of characteristic $2$. |
| title | Multigraded strong Lefschetz property for balanced simplicial complexes |
| topic | Combinatorics Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2408.17110 |