The Generators of a Colon Ideal with an Application to the Weak Lefschetz Property for Monomial Almost Complete Intersections in Three Variables
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908881601429504 |
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| author | Booth, Matthew Davidson Vraciu, Adela |
| author_facet | Booth, Matthew Davidson Vraciu, Adela |
| contents | Much progress has been made in classifying when the weak Lefschetz property holds for $A=\mathbb{F}[x,y,z]/I$ where $\text{char}(\mathbb{F})=0$ and $I=(x_{1}^{d_{1}},y^{d_{2}},z^{d_{3}},x^{a_{1}}y^{a_{2}}z^{a_{3}})$ is a monomial almost complete intersection. We connect this problem to the setting of two variables through a certain relation. In so doing, we are led to determine explicit formulas for the generators of the colon ideal $(x^{d_{1}},y^{d_{2}}):(x+y)^{a_{3}}$. With these generators in hand, we construct a matrix and show that failure of WLP for $A$ is dictated by the vanishing of a certain polynomial (namely the determinant of our matrix) when $A$ is level. We further show in the level case that a conjecture first posed by Migliore, Miró-Roig, and Nagel is true in a few new cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11491 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Generators of a Colon Ideal with an Application to the Weak Lefschetz Property for Monomial Almost Complete Intersections in Three Variables Booth, Matthew Davidson Vraciu, Adela Commutative Algebra 13C05, 13E10, 05E40, 13D40 Much progress has been made in classifying when the weak Lefschetz property holds for $A=\mathbb{F}[x,y,z]/I$ where $\text{char}(\mathbb{F})=0$ and $I=(x_{1}^{d_{1}},y^{d_{2}},z^{d_{3}},x^{a_{1}}y^{a_{2}}z^{a_{3}})$ is a monomial almost complete intersection. We connect this problem to the setting of two variables through a certain relation. In so doing, we are led to determine explicit formulas for the generators of the colon ideal $(x^{d_{1}},y^{d_{2}}):(x+y)^{a_{3}}$. With these generators in hand, we construct a matrix and show that failure of WLP for $A$ is dictated by the vanishing of a certain polynomial (namely the determinant of our matrix) when $A$ is level. We further show in the level case that a conjecture first posed by Migliore, Miró-Roig, and Nagel is true in a few new cases. |
| title | The Generators of a Colon Ideal with an Application to the Weak Lefschetz Property for Monomial Almost Complete Intersections in Three Variables |
| topic | Commutative Algebra 13C05, 13E10, 05E40, 13D40 |
| url | https://arxiv.org/abs/2603.11491 |