Quasi-finite modules and asymptotic prime divisors

Let A be a Noetherian ring, J⊆A an ideal and C a finitely generated A-module. In this note we would like to prove the following statement. Let {In}n⩾0 be a collection of ideals satisfying: (i) In⊇Jn, for all n, (ii) Js⋅Is⊆Ir+s, for all r,s⩾0 and (iii) In⊆Im, whenever m⩽n. Then AssA(InC/JnC) is independent of n, for n sufficiently large. Note that the set of prime ideals ⋃n⩾1AssA(InC/JnC) is finite, so the issue at hand is the realization that the primes in AssA(InC/JnC) do not behave periodically, as one might have expected, say if ⨁n⩾0In were a Noetherian A-algebra generated in degrees greater than one. We also give a multigraded version of our results.