Simplicial Resolutions of the Quadratic Power of Monomial Ideals
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912378744995840 |
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| author | Cooper, Susan M. Faridi, Sara Mahmood, Hasan |
| author_facet | Cooper, Susan M. Faridi, Sara Mahmood, Hasan |
| contents | Given any monomial ideal $ I $ minimally generated by $ q $ monomials, we define a simplicial complex $\mathbb{M}_q^2$ that supports a resolution of $ I^2 $. We also define a subcomplex $\mathbb{M}^2(I)$, which depends on the monomial generators of $I$ and also supports the resolution of $ I^2 $. As a byproduct, we obtain bounds on the projective dimension of the second power of any monomial ideal. We also establish bounds on the Betti numbers of $ I^2 $, which are significantly tighter than those determined by the Taylor resolution of $ I^2 $. Moreover, we introduce the permutation ideal $\mathcal{T}_q$ which is generated by $q$ monomials. For any monomial ideal $I$ with $q$ generators, we establish that $β(I^2) \leq β({\mathcal{T}_q}^2)$. We show that the simplicial complex $\mathbb{M}_q^2$ supports the minimal resolution of ${\mathcal{T}_q}^2$. In fact, $\mathbb{M}_q^2$ is the Scarf complex of ${\mathcal{T}_q}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06751 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simplicial Resolutions of the Quadratic Power of Monomial Ideals Cooper, Susan M. Faridi, Sara Mahmood, Hasan Commutative Algebra Given any monomial ideal $ I $ minimally generated by $ q $ monomials, we define a simplicial complex $\mathbb{M}_q^2$ that supports a resolution of $ I^2 $. We also define a subcomplex $\mathbb{M}^2(I)$, which depends on the monomial generators of $I$ and also supports the resolution of $ I^2 $. As a byproduct, we obtain bounds on the projective dimension of the second power of any monomial ideal. We also establish bounds on the Betti numbers of $ I^2 $, which are significantly tighter than those determined by the Taylor resolution of $ I^2 $. Moreover, we introduce the permutation ideal $\mathcal{T}_q$ which is generated by $q$ monomials. For any monomial ideal $I$ with $q$ generators, we establish that $β(I^2) \leq β({\mathcal{T}_q}^2)$. We show that the simplicial complex $\mathbb{M}_q^2$ supports the minimal resolution of ${\mathcal{T}_q}^2$. In fact, $\mathbb{M}_q^2$ is the Scarf complex of ${\mathcal{T}_q}^2$. |
| title | Simplicial Resolutions of the Quadratic Power of Monomial Ideals |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2505.06751 |