Abstract

Abstract. Let R be a Noetherian ring, I1,...,Ir be ideals of R, and N ⊆ M be finitely generated R-modules. Let S = n∈Nr Sn be a Noetherian standard Nr-graded ring with S0 = R, and M be a f initely generated Zr-graded S-module. For n = (n1,...,nr) ∈ Nr, set Gn := Mn or Gn := M/InN, where In = In1 1 ···Inr r . Suppose F is a coherent functor on the category of finitely generated R-modules. We prove that the set AssR F(Gn) of associate primes and grade J,F(Gn) stabilize for all n ≫ 0, where J is a non-zero ideal of R. Furthermore, if the length λR(F(Gn)) is finite for all n ≫ 0, then there exists a polynomial P in r variables over Q such that λR(F(Gn)) = P(n) for all n ≫ 0. When R is a local ring, and Gn = M/InN, 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 βR i (F(Gn)) and Bass number µi R(F(Gn)) are given by polynomials in n for all n ≫ 0. Thus, in particular, the projective dimension pdR(F(Gn)) (resp., injective dimension idR(F(Gn))) is constant for all n ≫ 0