A pr 2 00 9 Consecutive cancellations in Betti numbers of local rings ∗

Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulay’s Theorem, there exists a lexicographic ideal L = Lex(I) with the same Hilbert function as I. Peeva has proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R,n) by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaire’s construction, several applications are presented. This connection between the graded perspective and the local one is a new viewpoint and we hope it will be useful for studying the numerical invariants of classes of local rings.