Let $(A, m, k)$ be a Gorenstein local ring of dimension $ dgeq 1.$ Let $I$ be an ideal of $A$ with $htt(I) geq d-1.$ We prove that the numerical function [ n mapsto ell(ext_A^i(k, A/I^{n+1}))] is given by a polynomial of degree $d-1 $ in the case when $ i geq d+1 $ and $curv(I^n) > 1$ for all $n geq 1.$ We prove a similar result for the numerical function [ n mapsto ell(Tor_i^A(k, A/I^{n+1}))] under the assumption that $A$ is a CM ~ local ring. oindent We note that there are many examples of ideals satisfying the condition $curv(I^n) > 1,$ for all $ n geq 1.$ We also consider more general functions $n mapsto ell(Tor_i^A(M, A/I_n)$ for a filtration ${I_n }$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal CM ~ $A$-module and ${I_n=overline{I^n} }$ is the integral closure filtration, $I$ an $m$-primary ideal in $A.$
Download