Integral closure and local cohomology
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910399447695360 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{F} = \{ J_n \}_{n \geq 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F}) = \bigoplus_{n \geq 0} J_n$ is a finitely generated $A$-algebra. Let $\mathcal{R} = A[It]$ and assume $I^n \subseteq J_n$ for all $n \geq 1$. We show the following two assertions are equivalent:
(1) For all $i \geq 0$ we have $H^i_{\mathcal{R}_+}(\mathcal{R}(\mathcal{F}))_n = 0$ for all $n \gg 0$.
(2) $J_n \subseteq \overline{I^n}$ for all $n \geq 1$.
Here $\overline{I^n}$ is the integral closure of $I^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03841 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral closure and local cohomology Puthenpurakal, Tony J. Commutative Algebra Primary 13A30, 13D45, Secondary 13D40 Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{F} = \{ J_n \}_{n \geq 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F}) = \bigoplus_{n \geq 0} J_n$ is a finitely generated $A$-algebra. Let $\mathcal{R} = A[It]$ and assume $I^n \subseteq J_n$ for all $n \geq 1$. We show the following two assertions are equivalent: (1) For all $i \geq 0$ we have $H^i_{\mathcal{R}_+}(\mathcal{R}(\mathcal{F}))_n = 0$ for all $n \gg 0$. (2) $J_n \subseteq \overline{I^n}$ for all $n \geq 1$. Here $\overline{I^n}$ is the integral closure of $I^n$. |
| title | Integral closure and local cohomology |
| topic | Commutative Algebra Primary 13A30, 13D45, Secondary 13D40 |
| url | https://arxiv.org/abs/2404.03841 |