Minimal free resolution of a finitely generated module over a regular local ring

TL;DR

This paper explores the numerical invariants of minimal free resolutions over regular local rings by comparing them with associated graded modules, extending known results from graded to local algebra settings.

Contribution

It introduces a method to study Betti numbers via filtrations and associated graded modules, extending rigidity results to local rings.

Findings

Provides upper bounds for Betti numbers of modules over regular local rings.

Identifies modules, including Koszul modules, that attain extremal Betti number values.

Extends rigidity results from graded algebras to local rings.

Abstract

Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{\mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting.