Betti numbers under small perturbations

Abstract

We study how Betti numbers of ideals in a local ring change under small perturbations. Given p∈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≡JmodmN and with the same Hilbert function as I, then the Betti numbers βiR(R/I) and βiR(R/J) coincide for 0≤i≤p. Moreover, we present several cases in which an ideal J such that I≡JmodmN is forced to have the same Hilbert function as I, and therefore the same Betti numbers.