Betti numbers of powers of ideals

Abstract

Let A=K[x1, ..... ,xn] be a standard graded polynomial ring over a field K, let M = (x_1, .... , x_n) be the graded maximal ideal and I a graded ideal of A. For each i the Betti numbers b_i(I^k ) of I^k are polynomial functions for k>>0. We show that if I is M-primary, then these polynomial functions have the same degree for all i .