Barile-Macchia resolutions

TL;DR

This paper introduces Barile-Macchia resolutions, a new method for constructing cellular resolutions of monomial ideals using discrete Morse theory, providing minimal resolutions for certain classes of ideals and recursive formulas for algebraic invariants.

Contribution

The paper develops an algorithm for creating homogeneous acyclic matchings leading to Barile-Macchia resolutions, and compares these to existing resolutions, establishing minimality under specific conditions.

Findings

Minimal resolutions for edge ideals of weighted oriented forests and most cycles.

Recursive formulas for graded Betti numbers and projective dimension.

Comparison showing when Barile-Macchia resolutions are minimal, matching existing resolutions.

Abstract

We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile-Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile-Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile-Macchia resolutions.