Betti numbers under small perturbations

TL;DR

This paper investigates how Betti numbers of ideals in a local ring are affected by small perturbations, establishing conditions under which Betti numbers remain unchanged when ideals are close in the m-adic topology.

Contribution

It proves that Betti numbers are stable under small perturbations of ideals with the same Hilbert function in a Noetherian local ring.

Findings

Betti numbers remain invariant under certain small ideal perturbations.

Existence of an N such that perturbations within m^N preserve Betti numbers.

Identification of cases where perturbed ideals share the same Hilbert function.

Abstract

We study how Betti numbers of ideals in a local ring change under small perturbations. Given $p\in\mathbb N$ and given an ideal $I$ of a Noetherian local ring $(R,\mathfrak m)$, our main result states that there exists $N>0$ such that if $J$ is an ideal with $I\equiv J\bmod \mathfrak m^N$ and with the same Hilbert function as $I$, then the Betti numbers $\beta_i^R(R/I)$ and $\beta_i^R(R/J)$ coincide for $0\le i\le p$. Moreover, we present several cases in which an ideal $J$ such that $I \equiv J \bmod \mathfrak m^N$ is forced to have the same Hilbert function as $I$, and therefore the same Betti numbers.