Asymptotic prime divisors and Vasconcelos invariant
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arXiv
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2026
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| _version_ | 1866915857472421888 |
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| author | Ghosh, Dipankar Nanduri, Ramakrishna Pramanik, Siddhartha |
| author_facet | Ghosh, Dipankar Nanduri, Ramakrishna Pramanik, Siddhartha |
| contents | Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. In this article, we prove that $$\mathrm{Ass}_R(M/I^{n} M) = \mathrm{Ass}_R(0:_{M} I) \cup \mathrm{Ass}_R(I^{n-1} M/I^{n} M) \text{ for all } n \gg 0.$$ We then investigate the asymptotic behaviour of the (local) Vasconcelos invariant of $M/I^{n} M$ as a function of $n$, when $R$ is $\mathbb{N}$-graded, $I$ is homogeneous, and $M$ is $\mathbb{Z}$-graded. When $I$ is generated by elements of positive degree, we show that, for sufficiently large n, the (local) Vasconcelos invariant of $M/I^{n} M$ either coincides with that of the colon submodule $(0 :_{M} I)$, or is a polynomial in $n$ of degree one whose leading coefficient is one of the degrees of the generators of $I$. This dichotomy depends exclusively on two cases determined by $(0:_{M} I)$. Thus, we recover and considerably strengthen the main results of Fiorindo-Ghosh [Nagoya Math. J. 258 (2025), 296-310.], where asymptotic linearity was shown under the additional assumption that $(0:_{M} I)=0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11961 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic prime divisors and Vasconcelos invariant Ghosh, Dipankar Nanduri, Ramakrishna Pramanik, Siddhartha Commutative Algebra Primary: 13A15, Secondary: 13E05, 13A05, 13A02 Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. In this article, we prove that $$\mathrm{Ass}_R(M/I^{n} M) = \mathrm{Ass}_R(0:_{M} I) \cup \mathrm{Ass}_R(I^{n-1} M/I^{n} M) \text{ for all } n \gg 0.$$ We then investigate the asymptotic behaviour of the (local) Vasconcelos invariant of $M/I^{n} M$ as a function of $n$, when $R$ is $\mathbb{N}$-graded, $I$ is homogeneous, and $M$ is $\mathbb{Z}$-graded. When $I$ is generated by elements of positive degree, we show that, for sufficiently large n, the (local) Vasconcelos invariant of $M/I^{n} M$ either coincides with that of the colon submodule $(0 :_{M} I)$, or is a polynomial in $n$ of degree one whose leading coefficient is one of the degrees of the generators of $I$. This dichotomy depends exclusively on two cases determined by $(0:_{M} I)$. Thus, we recover and considerably strengthen the main results of Fiorindo-Ghosh [Nagoya Math. J. 258 (2025), 296-310.], where asymptotic linearity was shown under the additional assumption that $(0:_{M} I)=0$. |
| title | Asymptotic prime divisors and Vasconcelos invariant |
| topic | Commutative Algebra Primary: 13A15, Secondary: 13E05, 13A05, 13A02 |
| url | https://arxiv.org/abs/2603.11961 |