On the Lefschetz Property for quotients by monomial ideals containing squares of variables
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| Format: | Preprint |
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2021
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| _version_ | 1866913258900815872 |
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| author | Dao, Hailong Nair, Ritika |
| author_facet | Dao, Hailong Nair, Ritika |
| contents | Let $Δ$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(Δ) = \Bbbk[x_1, \ldots, x_n]/ \langle x_1^2, \ldots, x_n^2, I_Δ \rangle$, where $\Bbbk$ is a field of characteristic $0$ and $I_Δ$ is the Stanley-Reisner ideal associated to $Δ$. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of $A(Δ)$ in terms of the simplicial complex $Δ$. We are able to completely analyze when WLP holds in degree $1$, complementing work by Migliore, Nagel and Schenck in [MNS2020]. We give a complete characterization of all $2$-dimensional pseudomanifolds $Δ$ such that $A(Δ)$ satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_09434 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Lefschetz Property for quotients by monomial ideals containing squares of variables Dao, Hailong Nair, Ritika Commutative Algebra Combinatorics 05C25, 05E40, 13E10, 13F55, 13H10 Let $Δ$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(Δ) = \Bbbk[x_1, \ldots, x_n]/ \langle x_1^2, \ldots, x_n^2, I_Δ \rangle$, where $\Bbbk$ is a field of characteristic $0$ and $I_Δ$ is the Stanley-Reisner ideal associated to $Δ$. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of $A(Δ)$ in terms of the simplicial complex $Δ$. We are able to completely analyze when WLP holds in degree $1$, complementing work by Migliore, Nagel and Schenck in [MNS2020]. We give a complete characterization of all $2$-dimensional pseudomanifolds $Δ$ such that $A(Δ)$ satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization. |
| title | On the Lefschetz Property for quotients by monomial ideals containing squares of variables |
| topic | Commutative Algebra Combinatorics 05C25, 05E40, 13E10, 13F55, 13H10 |
| url | https://arxiv.org/abs/2112.09434 |