Free resolutions and marked families
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915586896822272 |
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| author | Bertone, Cristina Cioffi, Francesca Lella, Paolo |
| author_facet | Bertone, Cristina Cioffi, Francesca Lella, Paolo |
| contents | Let $\mathbb{K}$ be a field and $A$ a Noetherian $\mathbb{K}$-algebra. In a paper of 2020, M. Albert, C. Bertone, M. Roggero and W. M. Seiler proved that, given a quasi-stable module $U \subset R^m$ with $R=\mathbb{K}[x_0,\dots,x_n]$, any submodule $M\subseteq (R\otimes A)^m$ generated by a marked basis over $U$ admits a special free resolution described in terms of marked bases as well, called the {\em $U$-resolution of $M$}. In this paper, we first investigate the minimality of the $U$-resolution and its structure. When $M$ is an ideal and $A=\mathbb{K}$, we show that $M$ is componentwise linear if and only if its $U$-resolution is minimal, up to a linear change of variables. Then, adopting a functorial approach to the construction of the $U$-resolution, we prove that certain functors naturally associated with the resolution are isomorphic. These isomorphisms arise from the fact that the marked basis of the $i$-th syzygy module in the $U$-resolution can be expressed in terms of the coefficients of the marked basis of $M$. Moreover, when $M$ is an ideal of depth at least 2, this correspondence can be reversed: in this case, the marked basis of $M$ itself can be written in terms of the coefficients of the marked basis of its first syzygy module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free resolutions and marked families Bertone, Cristina Cioffi, Francesca Lella, Paolo Commutative Algebra Algebraic Geometry 13P10 (Primary) 13D02, 14C05, 14J10 (Secondary) Let $\mathbb{K}$ be a field and $A$ a Noetherian $\mathbb{K}$-algebra. In a paper of 2020, M. Albert, C. Bertone, M. Roggero and W. M. Seiler proved that, given a quasi-stable module $U \subset R^m$ with $R=\mathbb{K}[x_0,\dots,x_n]$, any submodule $M\subseteq (R\otimes A)^m$ generated by a marked basis over $U$ admits a special free resolution described in terms of marked bases as well, called the {\em $U$-resolution of $M$}. In this paper, we first investigate the minimality of the $U$-resolution and its structure. When $M$ is an ideal and $A=\mathbb{K}$, we show that $M$ is componentwise linear if and only if its $U$-resolution is minimal, up to a linear change of variables. Then, adopting a functorial approach to the construction of the $U$-resolution, we prove that certain functors naturally associated with the resolution are isomorphic. These isomorphisms arise from the fact that the marked basis of the $i$-th syzygy module in the $U$-resolution can be expressed in terms of the coefficients of the marked basis of $M$. Moreover, when $M$ is an ideal of depth at least 2, this correspondence can be reversed: in this case, the marked basis of $M$ itself can be written in terms of the coefficients of the marked basis of its first syzygy module. |
| title | Free resolutions and marked families |
| topic | Commutative Algebra Algebraic Geometry 13P10 (Primary) 13D02, 14C05, 14J10 (Secondary) |
| url | https://arxiv.org/abs/2510.26409 |