Betti numbers under small perturbations

We study how Betti numbers of ideals in a local ring change under small perturbations. Given p is an element of N and given an ideal I of a Noetherian local ring (R,m), our main result states that there exists N > 0 such that if J is an ideal with I equivalent to J mod m(N) and with the same Hilbert function as I, then the Betti numbers beta(R)(i) (R/I) and beta(R)(i) (R/J) coincide for 0 <= i <= p. Moreover, we present several cases in which an ideal J such that I equivalent to J mod m(N) is forced to have the same Hilbert function as I, and therefore the same Betti numbers. (C) 2021 Elsevier Inc. All rights reserved.