Filter regular sequence under small perturbations

Abstract

We answer affirmatively a question of Srinivas--Trivedi: in a Noetherian local ring $(R,\mathfrak{m})$, if $I=(f_1,\dots ,f_r)$ is an ideal generated by a filter-regular sequence and $J$ is an ideal such that $I+J$ is $\mathfrak{m}$-primary, then there exists $N>0$ such that for any ${\epsilon }_1,\dots ,{\epsilon }_r\in {\mathfrak{m}}^N$, we have an equality of Hilbert functions: $H(J,R/(f_1,\dots ,f_r))(n)=H(J,R/(f_1+{\epsilon }_1,\dots ,f_r+{\epsilon }_r))(n)$ for all $n\ge 0$. We also prove that the dimension of the non Cohen--Macaulay locus does not increase under small perturbations.

11 pages, final version