Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

TL;DR

This paper investigates the algebraic properties of edge ideals derived from comparability graphs and clique clutters, establishing their normality and torsion-free conditions through combinatorial and polyhedral methods.

Contribution

It proves that edge ideals of clique clutters of comparability graphs are normally torsion free and links normality to polyhedral properties, extending understanding of monomial ideal normality.

Findings

Edge ideals of clique clutters satisfy the max-flow min-cut property.

Edge ideals of complete admissible uniform clutters are normally torsion free.

Normality is characterized via blocking polyhedra and the integer decomposition property.

Abstract

Let (P,<) be a finite poset and let G be its comparability graph. If cl(G) is the clutter of maximal cliques of G, we prove that cl(G) satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. We prove that edge ideals of complete admissible uniform clutters are normally torsion free. The normality of a monomial ideal is expressed in terms of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property completely determine their normality