Bass and Betti numbers of A/In

Abstract

Let (A,m,k) be a Gorenstein local ring of dimension d≥1. Let I be an ideal of A with ht(I)≥d−1. We prove that the numerical functionn↦ℓ(ExtAi(k,A/In+1)) is given by a polynomial of degree d−1 in the case when i≥d+1 and curv(In)>1 for all n≥1. We prove a similar result for the numerical functionn↦ℓ(ToriA(k,A/In+1)) under the assumption that A is a Cohen-Macaulay local ring. We note that there are many examples of ideals satisfying the condition curv(In)>1, for all n≥1. We also consider more general functions n↦ℓ(ToriA(M,A/In) for a filtration {In} of ideals in A. We prove similar results in the case when M is a maximal Cohen-Macaulay A-module and {In=In‾} is the integral closure filtration, I an m-primary ideal in A.