Quasi-finite modules and asymptotic prime divisors

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 $\{I_n{\}}_{n\ge 0}$ 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 $r,s\ge 0$ and (iii) $I_n\subseteq I_m$, whenever $m\le n$. Then ${\text{Ass}}_A(I_{n}C\mathrm{/}{J}^{n}C)$ is independent of $n$, for $n$ sufficiently large. Note that the set of prime ideals ${\cup }_{n\ge 1}{\text{Ass}}_A(I_{n}C\mathrm{/}{J}^{n}C)$ is finite, so the issue at hand is the realization that the primes in ${\text{Ass}}_A(I_{n}C\mathrm{/}{J}^{n}C)$ $\mathit{\text{do}}\mathit{\text{not}}$ behave periodically, as one might have expected, say if ${⨁}_{n\ge 0}{I}_n$ were a Noetherian $A$-algebra generated in degrees greater than one. We also give a multigraded version of our results.