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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2506.17591 |
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| _version_ | 1866908415753715712 |
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| author | Trung, Van Duc |
| author_facet | Trung, Van Duc |
| contents | Let $\mathbb{M} = \{ M_n \}$ be a good $\mathfrak{q}$-filtration of a finitely generated $R$-module $M$ of dimension $d$, where $(R,\mathfrak{m})$ is a local ring and $\mathfrak{q}$ is an $\mathfrak{m}$-primary ideal of $R$. In case $depth(M) \geq d-1$, we give an upper bound for the second Hilbert coefficient $e_2(\mathbb{M})$ generalizing results by Huckaba-Marley and Rossi-Valla proved assuming that $M$ is Cohen-Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module $gr_{\mathbb{M}}(M)$. A lower bound on $e_2(\mathbb{M})$ is proved generalizing a result by Rees and Narita. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The second Hilbert coefficient of modules with almost maximal depth Trung, Van Duc Commutative Algebra 13H10 F.2.2; I.2.7 Let $\mathbb{M} = \{ M_n \}$ be a good $\mathfrak{q}$-filtration of a finitely generated $R$-module $M$ of dimension $d$, where $(R,\mathfrak{m})$ is a local ring and $\mathfrak{q}$ is an $\mathfrak{m}$-primary ideal of $R$. In case $depth(M) \geq d-1$, we give an upper bound for the second Hilbert coefficient $e_2(\mathbb{M})$ generalizing results by Huckaba-Marley and Rossi-Valla proved assuming that $M$ is Cohen-Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module $gr_{\mathbb{M}}(M)$. A lower bound on $e_2(\mathbb{M})$ is proved generalizing a result by Rees and Narita. |
| title | The second Hilbert coefficient of modules with almost maximal depth |
| topic | Commutative Algebra 13H10 F.2.2; I.2.7 |
| url | https://arxiv.org/abs/2506.17591 |