On unmixed and equi-dimensional associated graded rings
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| Format: | Preprint |
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2024
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| _version_ | 1866913005878378496 |
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| author | Puthenpurakal, Tony J. Sahoo, Samarendra |
| author_facet | Puthenpurakal, Tony J. Sahoo, Samarendra |
| contents | Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p > 0$ then let $I^*$ denote the tight closure of $I$. Let $G_I(A)=\bigoplus_{n\geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$. Assume $G_I(A)$ is unmixed and equi-dimensional. We show that either the function $P_{\overline{I}} :\,n\mapsto λ(\overline{I^n}/I^n)$ is a polynomial type of degree $d-1$ or $\overline{I^n}=I^n$ for all $n\geq 1.$ We prove an analogus result for the tight closure filtration if $A$ is of characteristic $p > 0$. When $A$ is generalized Cohen-Macaulay and $I$ is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of $I$ and the tight closure filtration of $I$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_20647 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On unmixed and equi-dimensional associated graded rings Puthenpurakal, Tony J. Sahoo, Samarendra Commutative Algebra Primary 13A30, 13D40 Secondary 13B22 Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p > 0$ then let $I^*$ denote the tight closure of $I$. Let $G_I(A)=\bigoplus_{n\geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$. Assume $G_I(A)$ is unmixed and equi-dimensional. We show that either the function $P_{\overline{I}} :\,n\mapsto λ(\overline{I^n}/I^n)$ is a polynomial type of degree $d-1$ or $\overline{I^n}=I^n$ for all $n\geq 1.$ We prove an analogus result for the tight closure filtration if $A$ is of characteristic $p > 0$. When $A$ is generalized Cohen-Macaulay and $I$ is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of $I$ and the tight closure filtration of $I$. |
| title | On unmixed and equi-dimensional associated graded rings |
| topic | Commutative Algebra Primary 13A30, 13D40 Secondary 13B22 |
| url | https://arxiv.org/abs/2405.20647 |