A note on the $v$-invariant
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911744098566144 |
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| author | Conca, Aldo |
| author_facet | Conca, Aldo |
| contents | Let $R$ be a finitely generated $\mathbb N$-graded algebra domain over a Noetherian ring and let $I$ be a homogeneous ideal of $R$. Given $P\in Ass(R/I)$ one defines the $v$-invariant $v_P(I)$ of $I$ at $P$ as the least $c\in \mathbb N$ such that $P=I:f$ for some $f\in R_c$. A classical result of Brodmann asserts that $Ass(R/I^n)$ is constant for large $n$. So it makes sense to consider a prime ideal $P\in Ass(R/I^n)$ for all the large $n$ and investigate how $v_P(I^n)$ depends on $n$. We prove that $v_P(I^n)$ is eventually a linear function of $n$. When $R$ is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in a recent preprint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00022 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the $v$-invariant Conca, Aldo Commutative Algebra 13A30 Let $R$ be a finitely generated $\mathbb N$-graded algebra domain over a Noetherian ring and let $I$ be a homogeneous ideal of $R$. Given $P\in Ass(R/I)$ one defines the $v$-invariant $v_P(I)$ of $I$ at $P$ as the least $c\in \mathbb N$ such that $P=I:f$ for some $f\in R_c$. A classical result of Brodmann asserts that $Ass(R/I^n)$ is constant for large $n$. So it makes sense to consider a prime ideal $P\in Ass(R/I^n)$ for all the large $n$ and investigate how $v_P(I^n)$ depends on $n$. We prove that $v_P(I^n)$ is eventually a linear function of $n$. When $R$ is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in a recent preprint. |
| title | A note on the $v$-invariant |
| topic | Commutative Algebra 13A30 |
| url | https://arxiv.org/abs/2401.00022 |