Simplicial Resolutions of Powers of Square-free Monomial Ideals
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| author | Cooper, Susan M. Khoury, Sabine El Faridi, Sara Mayes-Tang, Sarah Morey, Susan Sega, Liana M. Spiroff, Sandra |
| author_facet | Cooper, Susan M. Khoury, Sabine El Faridi, Sara Mayes-Tang, Sarah Morey, Susan Sega, Liana M. Spiroff, Sandra |
| contents | The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_03136 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Simplicial Resolutions of Powers of Square-free Monomial Ideals Cooper, Susan M. Khoury, Sabine El Faridi, Sara Mayes-Tang, Sarah Morey, Susan Sega, Liana M. Spiroff, Sandra Commutative Algebra Combinatorics 13D02, 13F55 The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution. |
| title | Simplicial Resolutions of Powers of Square-free Monomial Ideals |
| topic | Commutative Algebra Combinatorics 13D02, 13F55 |
| url | https://arxiv.org/abs/2204.03136 |