Properties of LCM Lattices of Monomial Ideals

TL;DR

This paper systematically studies properties of LCM lattices of monomial ideals, characterizing their lattice properties in relation to associated graphs and exploring conditions for Cohen-Macaulayness and projective dimension.

Contribution

It provides a comprehensive analysis of lattice properties of LCM lattices, linking them to graph characteristics and establishing new conditions for algebraic properties of monomial ideals.

Findings

Characterization of lattice properties for edge ideals in terms of graph features

Proof that minimal monomial ideals with modular lattices are Cohen-Macaulay

Necessary and sufficient conditions for projective dimension to match lattice height

Abstract

LCM lattices were introduced by Gasharov, Peeva, and Welker as a way to study minimal free resolutions of monomial ideals. All LCM lattices are atomic and all atomic lattices arise as the LCM lattice of some monomial ideal. We systematically study other lattice properties of LCM lattices. For lattices associated to the edge ideal of a graph, we completely characterize the many standard lattice properties in terms of the associated graphs: Boolean, modular, upper semimodular, lower semimodular, supersolvable, coatomic, and complemented; edge ideals with graded LCM lattices were previously characterized by Nevo and Peeva as those associated to gap-free graphs. For arbitrary monomial ideals, we prove the Cohen-Macaulayness of minimal monomial ideals associated to modular lattices. We also prove separate necessary and sufficient lattice conditions for when the projective dimension of a monomial ideal matches the height of its LCM lattice. Finally, we show that LCM lattices of Gorenstein edge ideals are coatomic and raise questions about the lattice properties of arbitrary Gorenstein monomial ideals.