Morse resolutions of monomial ideals and Betti splittings

TL;DR

This paper applies discrete Morse theory to monomial ideals to construct minimal free resolutions, unifying classical methods and introducing new reduction techniques for powers of ideals.

Contribution

It introduces a unified framework for minimal resolutions of monomial ideals using Morse theory and splitting, including new reduction methods for subideals and powers.

Findings

Established minimality of pruned resolutions for stable and linear quotient ideals

Unified classical resolutions within the Morse-based framework

Developed reduction techniques to simplify minimality analysis

Abstract

We use discrete Morse theory to study free resolutions of monomial ideals in combination with splitting techniques. We establish the minimality of such pruned resolutions for several classes of ideals, including stable and linear quotient ideals. In particular, we unify classical constructions such as the Eliahou-Kervaire and Herzog-Takayama resolutions within the pruned resolution framework. Additionally, we introduce methods to reduce the minimality study of a pruned resolution for an ideal to that of a smaller subideal and present a variant of our pruned resolution for powers of monomial ideals.