Covariant functors and asymptotic stability

Let R be a commutative Noetherian ring, I,J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then the sets AssRF(M/InM), AssRF(In−1M/InM) and the values depthJF(M/InM), depthJF(In−1M/InM) become independent of n for large n. Next, we consider several examples in which F is a rather familiar functor, but is not coherent or not even finitely generated in general. In these cases, the sets AssRF(M/InM) still become independent of n for large n. We then show one negative result where F is not finitely generated. Finally, we give a positive result where F belongs to a special class of functors which are not finitely generated in general, an example of which is the zeroth local cohomology functor.