Asymptotic prime divisors and Vasconcelos invariant

TL;DR

This paper studies the asymptotic behavior of prime divisors and Vasconcelos invariants of modules over graded rings, revealing a dichotomy in their growth patterns for large powers of ideals.

Contribution

It establishes the asymptotic structure of associated primes and Vasconcelos invariants for modules over graded rings, extending previous results to more general cases.

Findings

Associated primes stabilize for large n.

Vasconcelos invariant either stabilizes or grows linearly with degree one.

Results strengthen previous linearity findings under broader conditions.

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