First Coefficient ideals and $R_1$ property of Rees algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909284397219840 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be an excellent normal local ring of dimension $d \geq 2$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal. Then the following assertions are equivalent:
(i) The extended Rees algebra $A[It, t^{-1}]$ is $R_1$.
(ii) The Rees algebra $A[It]$ is $R_1$.
(iii) $Proj(A[It])$ is $R_1$.
(iv) $(I^n)^* = (I^n)_1$ for all $n \geq 1$.
Here $(I^n)^*$ is the integral closure of $I^n$ and $(I^n)_1$ is the first coefficient ideal of $I^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First Coefficient ideals and $R_1$ property of Rees algebras Puthenpurakal, Tony J. Commutative Algebra Primary 13A30, 13B22, Secondary 13D40, 13H10 Let $(A,\mathfrak{m})$ be an excellent normal local ring of dimension $d \geq 2$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal. Then the following assertions are equivalent: (i) The extended Rees algebra $A[It, t^{-1}]$ is $R_1$. (ii) The Rees algebra $A[It]$ is $R_1$. (iii) $Proj(A[It])$ is $R_1$. (iv) $(I^n)^* = (I^n)_1$ for all $n \geq 1$. Here $(I^n)^*$ is the integral closure of $I^n$ and $(I^n)_1$ is the first coefficient ideal of $I^n$. |
| title | First Coefficient ideals and $R_1$ property of Rees algebras |
| topic | Commutative Algebra Primary 13A30, 13B22, Secondary 13D40, 13H10 |
| url | https://arxiv.org/abs/2408.05532 |