Vanishing and non-negativity of the first normal Hilbert coefficient
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914921029042176 |
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| author | Ma, Linquan Quy, Pham Hung |
| author_facet | Ma, Linquan Quy, Pham Hung |
| contents | Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and $ν(\mathfrak{m}/Q)\leq 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if $\widehat{R}$ is equidimensional, then $\bar{e}_1(Q)\geq 0$ for all parameter ideals $Q\subseteq R$, and in characteristic $p>0$, we actually have $e_1^*(Q)\geq 0$. Our proofs rely on the existence of big Cohen-Macaulay algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13084 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Vanishing and non-negativity of the first normal Hilbert coefficient Ma, Linquan Quy, Pham Hung Commutative Algebra Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and $ν(\mathfrak{m}/Q)\leq 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if $\widehat{R}$ is equidimensional, then $\bar{e}_1(Q)\geq 0$ for all parameter ideals $Q\subseteq R$, and in characteristic $p>0$, we actually have $e_1^*(Q)\geq 0$. Our proofs rely on the existence of big Cohen-Macaulay algebras. |
| title | Vanishing and non-negativity of the first normal Hilbert coefficient |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2301.13084 |