Shapes of free resolutions over a local ring

TL;DR

This paper classifies the possible structures of minimal free resolutions over regular local rings and hypersurface rings, revealing complex behaviors and introducing asymptotic methods to analyze Betti sequences.

Contribution

It provides a comprehensive classification of free resolution shapes over regular and hypersurface rings, using novel asymptotic techniques for Betti sequence analysis.

Findings

Existence of free resolutions with pathological Betti number behaviors

Asymptotic characterization of resolution shapes over hypersurface rings

Introduction of asymptotic methods for studying formal Q-Betti sequences

Abstract

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences.