Bounding the Projective Dimension of a Square-Free Monomial Ideal via Domination in Clutters

TL;DR

This paper introduces edgewise domination in clutters to bound the projective dimension of square-free monomial ideals, providing recursive formulas and characterizations for chordal clutters and related graph classes.

Contribution

It defines a new domination concept in clutters and applies it to derive bounds and formulas for the projective dimension of associated monomial ideals, including chordal cases.

Findings

Edgewise domination bounds projective dimension.

Recursive formula for chordal clutter ideals.

Chordality characterized via associated graphs.

Abstract

We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then use a simple recursion to recover a formula for the projective dimension of a monomial ideal associated to a chordal clutter, as defined by Woodroofe in \cite{russ}. We also study a family of clutters associated to graphs, and show that these clutters are chordal if and only if the associated graph is. Finally, we compute domination parameters for certain classes of these clutters.