Luís Duarte: Perturbations of ideals in local rings

Abstract:

Let \(I\) be an ideal of a Noetherian local ring \(R\). We study how properties of the ideal change under small perturbations, that is, when \(I\) is replaced by an ideal \(J\) which is the same as \(I\) modulo a large power of the maximal ideal. In particular, assuming that \(R/J\) has the same Hilbert function as \(R/I\), we show that the Betti numbers of \(R/J\) coincide with those of \(R/I\). We also compare the local cohomology modules of \(R/J\) with those of \(R/I\).