Cohen-Macaulay Monomial Ideals of Codimension 2

TL;DR

This paper provides a structural characterization of Cohen-Macaulay monomial ideals of codimension 2, including their relation matrices, especially focusing on those with linear resolutions and their connection to chordal graphs.

Contribution

It introduces a comprehensive structure theorem for these ideals and links their relation matrices to spanning trees of specific chordal graphs, advancing understanding of their algebraic and combinatorial properties.

Findings

Characterization of relation matrices for Cohen-Macaulay monomial ideals of codimension 2

Identification of relation matrices with spanning trees of chordal graphs

Description of conditions for ideals with linear resolutions

Abstract

We give a structure theorem for Cohen Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common.