# Resolutions of Monomial Ideals of Projective Dimension 1

## TL;DR

This paper characterizes monomial ideals with projective dimension at most 1 by linking their minimal free resolutions to tree-supported graphs and provides a new proof for a known characterization involving quasi-trees.

## Contribution

It establishes a new graph-theoretic characterization of monomial ideals with projective dimension 1 and offers a novel proof of existing results using this framework.

## Key findings

- Minimal free resolution supported on a tree graph for projective dimension ≤ 1
- New characterization of quasi-trees in the context of monomial ideals
- Alternative proof of Herzog, Hibi, and Zheng's characterization

## Abstract

We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the $S/I$. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.