Filter regular sequence under small perturbations

Abstract

We answer affirmatively a question of Srinivas–Trivedi (J Algebra 186(1):1–19, 1996): in a Noetherian local ring \((R,{{\,\mathrm{\mathfrak {m}}\,}})\), if \(f_1,\dots ,f_r\) is a filter-regular sequence and J is an ideal such that \((f_1, \ldots , f_r)+J\) is \({{\,\mathrm{\mathfrak {m}}\,}}\)-primary, then there exists \(N>0\) such that for any \(\varepsilon _1,\dots ,\varepsilon _r \in {{\,\mathrm{\mathfrak {m}}\,}}^N\), we have an equality of Hilbert functions: \(H(J, R/(f_1,\dots ,f_r))(n)=H(J, R/(f_1+\varepsilon _1,\dots , f_r+\varepsilon _r))(n)\) for all \(n\ge 0\). We also prove that the dimension of the non Cohen–Macaulay locus does not increase under small perturbations, generalizing another result of [20].