A solution to Itoh's conjecture for integral closure filtration
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909166705049600 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring of dimension $d \geq 3$ and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let $G_{\mathfrak{a}}(A)^* = \bigoplus_{n\geq 0 }(\mathfrak{a}^n)^*/(\mathfrak{a}^{n+1})^*$ be the associated graded ring of the integral closure filtration of $\mathfrak{a}$. Itoh conjectured in 1992 that if third Hilbert coefficient of $G_{\mathfrak{a}}(A)^*$ , i.e., $e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then
$G_{\mathfrak{a}}(A)^*$ is Cohen-Macaulay. In this paper we prove Itoh's conjecture (more generally for analytically unramified Cohen-Macaulay local rings). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A solution to Itoh's conjecture for integral closure filtration Puthenpurakal, Tony J. Commutative Algebra Primary 13A30, 13D45, Secondary 13H10, 13H15 Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring of dimension $d \geq 3$ and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let $G_{\mathfrak{a}}(A)^* = \bigoplus_{n\geq 0 }(\mathfrak{a}^n)^*/(\mathfrak{a}^{n+1})^*$ be the associated graded ring of the integral closure filtration of $\mathfrak{a}$. Itoh conjectured in 1992 that if third Hilbert coefficient of $G_{\mathfrak{a}}(A)^*$ , i.e., $e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A)^*$ is Cohen-Macaulay. In this paper we prove Itoh's conjecture (more generally for analytically unramified Cohen-Macaulay local rings). |
| title | A solution to Itoh's conjecture for integral closure filtration |
| topic | Commutative Algebra Primary 13A30, 13D45, Secondary 13H10, 13H15 |
| url | https://arxiv.org/abs/2404.07638 |