Coherent functors, powers of ideals, and asymptotic stability

Let R be a Noetherian ring, I₁,…,Iᵣ be ideals of R, and N⊆ M be finitely generated R-modules. Let S = _n ∈ Nʳ S_n be a Noetherian standard Nʳ-graded ring with S_0 = R, and M be a finitely generated Zʳ-graded S-module. For n = (n₁,,nᵣ) ∈ Nʳ, set G_n := M_n or G_n := M/ I^n N, where I^n = I₁n₁ ⋯ Iᵣnᵣ. Suppose F is a coherent functor on the category of finitely generated R-modules. We prove that the set AssR (F(G_n) ) of associate primes and grade(J, F(G_n)) stabilize for all n 0, where J is a non-zero ideal of R. Furthermore, if the length λR(F(G_n)) is finite for all n 0, then there exists a polynomial P in r variables over Q such that λR(F(G_n)) = P(n) for all n 0. When R is a local ring, and G_n = 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 βᵢR(F(G_n)) and Bass number μⁱR(F(G_n)) are given by polynomials in n for all n 0. Thus, in particular, the projective dimension pdR(F(G_n)) (resp., injective dimension idR(F(G_n))) is constant for all n 0.