Barile-Macchia 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.
Keywords
Cite
@article{arxiv.2211.04640,
title = {Barile-Macchia resolutions},
author = {Trung Chau and Selvi Kara},
journal= {arXiv preprint arXiv:2211.04640},
year = {2024}
}
Comments
49 pages. Updated version: Adding a result saying that being bridge-friendly implies having a minimal Barile-Macchia resolution and adding the notion of Batzies-Welker matchings. Comments are welcome!!!