Bass and Betti Numbers of $A/I^n.$

Abstract

Let $(A, \m, k)$ be a Gorenstein local ring of dimension $d\ge 1.$ 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 $i\ge d+1$ and $\curv(I^n) > 1$ for all $n\ge 1.$ 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. \noindent We note that there are many examples of ideals satisfying the condition $\curv(I^n) > 1,$ for all $n\ge 1.$ 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 $A.$