Consecutive cancellations in Betti numbers of local rings

TL;DR

This paper extends Peeva's result on Betti number cancellations from polynomial rings to regular local rings using associated graded rings, providing new insights into local ring invariants.

Contribution

It generalizes the concept of consecutive Betti number cancellations from polynomial to local rings via associated graded rings, broadening the theoretical framework.

Findings

Betti numbers of local rings can be derived through cancellations from graded Betti numbers.

The approach uses the associated graded ring to extend polynomial ring results.

Applications demonstrate the utility of the new viewpoint in studying local ring invariants.

Abstract

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.