Let R be a Noetherian ring, I₁, , Iᵣ be ideals of R, and N M be finitely generated R-modules. Let S = ₍ ₍⋒ S₍ be a Noetherian standard Nʳ-graded ring with S₀ = R, and M be a finitely generated Zʳ-graded S-module. For n = (n₁, , nᵣ) Nʳ, set G₍: = M₍ or G₍: = M/ I^n N, where I^n = I₁^n₁ Iᵣ^nᵣ. Suppose F is a coherent functor on the category of finitely generated R-modules. We prove that the set AssR (F (G₍) ) of associate primes and grade (J, F (G₍) ) stabilize for all n 0, where J is a non-zero ideal of R. Furthermore, if the length R (F (G₍) ) is finite for all n 0, then there exists a polynomial P in r variables over Q such that R (F (G₍) ) = P (n) for all n 0. When R is a local ring, and G₍ = M/ I^n N, we give a sharp upper bound of the total degree of P. As applications, when R is a local ring, we show that for each fixed i 0, the ith Betti number ᵢR (F (G₍) ) and Bass number ⁱR (F (G₍) ) are given by polynomials in n for all n 0. Thus, in particular, the projective dimension pdR (F (G₍) ) (resp. , injective dimension idR (F (G₍) ) ) is constant for all n 0.
Dey et al. (Sat,) studied this question.