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

TL;DR

This paper investigates the polynomial behavior of certain numerical functions related to Ext and Tor modules over Gorenstein and Cohen-Macaulay local rings, focusing on powers of ideals with specific curvature conditions.

Contribution

It establishes that these numerical functions are given by polynomials of degree d-1 under specific conditions on the ideal and ring, extending understanding of asymptotic invariants.

Findings

The functions n o ext{length}( ext{Ext}_A^i(k, A/I^{n+1})) are polynomial of degree d-1 for i ≥ d+1.

The functions n o ext{length}( ext{Tor}_i^A(k, A/I^{n+1})) are polynomial under Cohen-Macaulay assumptions.

Many ideals satisfy the curvature condition ext{curv}(I^n) > 1 for all n}.

Abstract

Let $(A, \m, k)$ be a Gorenstein local ring of dimension $ d\geq 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. \noindent 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 $A.$