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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.19419 |
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| _version_ | 1866917359055273984 |
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| author | Mantero, Paolo Nguyen, Vinh |
| author_facet | Mantero, Paolo Nguyen, Vinh |
| contents | In this paper we prove that the Stanley--Reisner ideal or cover ideal $I$ of a matroid is minimally resolvable by iterated mapping cones. As a technical tool for this purpose, we introduce and study focal matroids, which are submatroids of a matroid $\mathcal{M}$ that are constructed relative to minimal $\ell$-covers of $\mathcal{M}$.
Our second main result is that the monomial support of the multigraded Betti numbers of $I$ corresponds precisely to the squarefree minimal generators of the symbolic powers of $I$. In fact, we prove that matroidal ideals are the only squarefree ideals with this property, thus obtaining a new homological characterization of matroidal ideals.
These techniques are foundational for a follow-up paper, where we will show that all symbolic power of $I$ are minimally resolvable by iterated mapping cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19419 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Focal matroids of covers and homological properties of matroids Mantero, Paolo Nguyen, Vinh Commutative Algebra 13F55 In this paper we prove that the Stanley--Reisner ideal or cover ideal $I$ of a matroid is minimally resolvable by iterated mapping cones. As a technical tool for this purpose, we introduce and study focal matroids, which are submatroids of a matroid $\mathcal{M}$ that are constructed relative to minimal $\ell$-covers of $\mathcal{M}$. Our second main result is that the monomial support of the multigraded Betti numbers of $I$ corresponds precisely to the squarefree minimal generators of the symbolic powers of $I$. In fact, we prove that matroidal ideals are the only squarefree ideals with this property, thus obtaining a new homological characterization of matroidal ideals. These techniques are foundational for a follow-up paper, where we will show that all symbolic power of $I$ are minimally resolvable by iterated mapping cones. |
| title | Focal matroids of covers and homological properties of matroids |
| topic | Commutative Algebra 13F55 |
| url | https://arxiv.org/abs/2603.19419 |