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

This paper describes an application of research that sits at the intersection of commutative algebra and combinatorics: Betti numbers of graphs. In particular, we describe a correspondence between simple undirected graphs and a class of ideals in a polynomial ring. We then briefly introduce some of the algebraic invariants that can be associated to the ideal and the relation of these invariants...

We prove an analogue of the Approximation Theorem of L-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.