Strongly Lech-independent ideals and Lech's conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866917948065579008 |
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| author | Meng, Cheng |
| author_facet | Meng, Cheng |
| contents | We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ is a flat local extension of local rings with $\dim R = \dim S$, the completion of $S$ is the completion of a standard graded ring over a field $k$ with respect to the homogeneous maximal ideal, and the completion of $\mathfrak{m}S$ is the completion of a homogeneous ideal, then $e(R) \leq e(S)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_09849 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Strongly Lech-independent ideals and Lech's conjecture Meng, Cheng Commutative Algebra 13H15 We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ is a flat local extension of local rings with $\dim R = \dim S$, the completion of $S$ is the completion of a standard graded ring over a field $k$ with respect to the homogeneous maximal ideal, and the completion of $\mathfrak{m}S$ is the completion of a homogeneous ideal, then $e(R) \leq e(S)$. |
| title | Strongly Lech-independent ideals and Lech's conjecture |
| topic | Commutative Algebra 13H15 |
| url | https://arxiv.org/abs/2112.09849 |