On the Stability of Bass and Betti Numbers under Ideal Perturbations in a Local Ring

TL;DR

This paper establishes conditions under which the Bass and Betti numbers of a module over a local ring remain stable when a regular sequence is perturbed within a certain power of the maximal ideal.

Contribution

It provides an explicit bound on perturbations of a regular sequence that guarantees the invariance of Bass and Betti numbers in a local ring setting.

Findings

Bass and Betti numbers are stable under specific ideal perturbations.

An explicit number N bounds the perturbations preserving these invariants.

The results apply to modules over Noetherian local rings with filter regular sequences.

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $J$ be an arbitrary ideal of $R$. Suppose $M$ is a finitely generated $R$-module. Let $x_1,\ldots,x_r$ be a $J$-filter regular sequence on $M$. We provide an explicit number $N$ such that the Bass and Betti numbers of $M/(x_1, \ldots, x_r)M$ are preserved when we perturb the sequence $x_1, \ldots,x_r$ by $\varepsilon_1, \ldots, \varepsilon_r \in \mathfrak{m}^N$.