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

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 $x_1,\dots ,x_r$ be a $J$-filter regular sequence on $M$. We provide an explicit number $N$ such that the Bass and Betti numbers of $M\mathrm{/}(x_1,\dots ,x_r)M$ are preserved when we perturb the sequence $x_1,\dots ,x_r$ by ${\epsilon }_1,\dots ,{\epsilon }_r\in {\mathfrak{m}}^N$.