Polarizations of Artin monomial ideals
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910133380972544 |
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| author | Fløystad, Gunnar Gabrielsen, Ine Mafi, Amir |
| author_facet | Fløystad, Gunnar Gabrielsen, Ine Mafi, Amir |
| contents | We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author.
Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible.
Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal.
We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09528 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Polarizations of Artin monomial ideals Fløystad, Gunnar Gabrielsen, Ine Mafi, Amir Commutative Algebra Combinatorics Primary: 13F55, Secondary: 05E40, 05E45 We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals. |
| title | Polarizations of Artin monomial ideals |
| topic | Commutative Algebra Combinatorics Primary: 13F55, Secondary: 05E40, 05E45 |
| url | https://arxiv.org/abs/2212.09528 |