  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,m) 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.
