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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2506.22229 |
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| _version_ | 1866918073301204992 |
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| author | Trung, Van Duc |
| author_facet | Trung, Van Duc |
| contents | Let $x_1,\ldots,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\ldots,x_s}$ the Koszul complex of $x_1,\ldots,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths $\sum_{i=1}^s (-1)^i\ell(H_i(R_{x_1,\ldots,x_s}))$ is preserved when we perturb the sequence $x_1, \ldots,x_s$ by $\varepsilon_1, \ldots, \varepsilon_s \in \mathfrak{m}^N$. Applying this result and the main Theorem of Eisenbud, we show that there exits $N >0$ such that for all $i \geq 1$ the length of $H_i(R_{x_1,\ldots,x_s})$ is preserved under small perturbation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Koszul Homology Under Small Perturbations Trung, Van Duc Commutative Algebra 13D02 F.2.2; I.2.7 Let $x_1,\ldots,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\ldots,x_s}$ the Koszul complex of $x_1,\ldots,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths $\sum_{i=1}^s (-1)^i\ell(H_i(R_{x_1,\ldots,x_s}))$ is preserved when we perturb the sequence $x_1, \ldots,x_s$ by $\varepsilon_1, \ldots, \varepsilon_s \in \mathfrak{m}^N$. Applying this result and the main Theorem of Eisenbud, we show that there exits $N >0$ such that for all $i \geq 1$ the length of $H_i(R_{x_1,\ldots,x_s})$ is preserved under small perturbation. |
| title | Koszul Homology Under Small Perturbations |
| topic | Commutative Algebra 13D02 F.2.2; I.2.7 |
| url | https://arxiv.org/abs/2506.22229 |