Linear syzygy graph and linear resolution

TL;DR

This paper explores the relationship between the structure of certain graphs derived from monomial ideals and the properties of their linear resolutions, providing characterizations for specific classes of ideals and complexes.

Contribution

It introduces a graph-based approach to characterize when monomial ideals have linear resolutions, especially for cycle or tree graphs, and applies these results to Cohen-Macaulay complexes.

Findings

Equivalence of linear resolution, linear quotients, and variable-decomposability for certain ideals.

Complete characterization of monomial ideals with linear resolution when associated graph is a cycle or a tree.

Characterization of Cohen-Macaulay simplicial complexes with specific associated graphs.

Abstract

For each squarefree monomial ideal $I\subset S = k[x_{1},\ldots, x_{n}] $, we associate a simple graph $G_I$ by using the first linear syzygies of $I$. In cases, where $G_I$ is a cycle or a tree, we show the following are equivalent: (a) $ I $ has a linear resolution (b) $ I $ has linear quotients (c) $ I $ is a variable-decomposable ideal In addition, with the same assumption on $G_I$, we characterize all monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension $2$ monomial ideals with a linear resolution. As an other application of our results, we also characterize all Cohen-Macaulay simplicail complexes in cases that $G_{\Delta}\cong G_{I_{\Delta^{\vee}}}$ is a cycle or a tree.