Filter regular sequence under small perturbations

TL;DR

This paper proves that filter-regular sequences in Noetherian local rings maintain their Hilbert functions under small perturbations, and that the non Cohen--Macaulay locus dimension remains stable under such perturbations.

Contribution

It establishes the stability of Hilbert functions and the non Cohen--Macaulay locus dimension under small perturbations of filter-regular sequences in local rings.

Findings

Hilbert functions are invariant under small perturbations of filter-regular sequences.

The dimension of the non Cohen--Macaulay locus does not increase with small perturbations.

Affirmative answer to Srinivas--Trivedi's question on stability of Hilbert functions.

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 $\varepsilon_1,\dots,\varepsilon_r \in \mathfrak{m}^N$, we have an equality of Hilbert functions: $H(J, R/(f_1,\dots,f_r))(n)=H(J, R/(f_1+\varepsilon_1,\dots, f_r+\varepsilon_r))(n)$ for all $n\geq 0$. We also prove that the dimension of the non Cohen--Macaulay locus does not increase under small perturbations.