Quasi-finite modules and asymptotic prime divisors

TL;DR

This paper proves that the associated primes of certain modules stabilize for large n under specific ideal conditions, revealing non-periodic behavior and providing a multigraded extension.

Contribution

It establishes the asymptotic stability of associated primes for modules defined by a family of ideals satisfying particular properties, including a multigraded generalization.

Findings

Associated primes stabilize for large n.

The set of associated primes is finite.

Primes do not behave periodically as previously expected.

Abstract

Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i) $I_n\supseteq J^n$, for all $n$, (ii) $J^s\cdot I_s \subseteq I_{r+s}$, for all $r,s\geq 0$ and (iii) $I_n\subseteq I_m$, whenever $m\leq n$. Then $\Ass_A(I_nC/J^nC)$ is independent of $n$, for $n$ sufficiently large. Note that the set of prime ideals $\cup_{n\geq 1} \Ass_A(I_nC/J^nC)$ is finite, so the issue at hand is the realization that the primes in $\Ass_A(I_nC/J^nC)$ \textit{do not} behave periodically, as one might have expected, say if $\bigoplus _{n\geq 0}I_n$ were a Noetherian $A$-algebra generated in degrees greater than one. We also give a multigraded version of our results.