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Asymptotic prime divisors and Vasconcelos invariant

Dipankar Ghosh, Ramakrishna Nanduri, Siddhartha Pramanik

Abstract

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 $(0:_{M} I)=0$.

Asymptotic prime divisors and Vasconcelos invariant

Abstract

Let be a Noetherian ring, an ideal of , and a finitely generated -module. In this article, we prove that We then investigate the asymptotic behaviour of the (local) Vasconcelos invariant of as a function of , when is -graded, is homogeneous, and is -graded. When is generated by elements of positive degree, we show that, for sufficiently large n, the (local) Vasconcelos invariant of either coincides with that of the colon submodule , or is a polynomial in of degree one whose leading coefficient is one of the degrees of the generators of . This dichotomy depends exclusively on two cases determined by . 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 .
Paper Structure (4 sections, 18 theorems, 65 equations)

This paper contains 4 sections, 18 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Asymptotic prime divisors
  3. Asymptotic Vasconcelos invariant
  4. Examples

Key Result

Theorem 1.2

Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. Then, In particular, $\mathcal{A}_M(I) = \mathop{\mathrm{Ass}}\nolimits_R(0:_M I) \cup \mathcal{B}_M(I)$, cf. Notation notation.

Theorems & Definitions (49)

  • Theorem 1.2: See Theorem \ref{['th:ass']} for stronger results
  • Definition 1.3
  • Theorem 1.5: See Theorem \ref{['th:main']} for stronger results
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 39 more