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 $L=\operatorname{Lex}(I)$ 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 $(R,\mathfrak{n})$ 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, we present several applications. 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.