Properties of LCM Lattices of Monomial Ideals

TL;DR

The paper develops a comprehensive lattice-theoretic framework for the LCM lattices of monomial ideals, linking algebraic invariants to combinatorial graph structures via edge ideals. It establishes that every LCM lattice is atomic and that modular lattices yield Cohen–Macaulay minimal monomial ideals, while providing sharp criteria for when projective dimension equals the lattice height. For edge ideals, it provides complete classifications of graded, modular, Boolean, supersolvable, lower semimodular, coatomic, and complemented lattices in terms of graph properties, and analyzes LCM lattices for standard graph families to illustrate these correspondences. The work thereby maps precise graph-theoretic conditions to homological and lattice-theoretic properties, enabling targeted analysis of monomial resolutions and their combinatorial underpinnings.

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. Such a characterization for arbitrary monomial ideals appears to be out of reach. However, 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.

Figures (9)

• Figure 2.1: Diagram of Implications for Finite Atomic Lattices
• Figure 4.1: Fano Plane
• Figure 4.2: Lattice of Flats for the Fano Plane
• Figure 4.3: Lattice of flats of graphic matroid associated to $G$ (labeled by edges of $G$)
• Figure 5.1: A graph $G$ whose LCM lattice is not geometric but for with $\mathop{\mathrm{pd}}\nolimits_S(S/I(G)) = \mathop{\mathrm{ht}}\nolimits(\mathcal{L}_{I(G)})$.

Theorems & Definitions (52)

• Theorem 2.1: Phan Phan06, see also Peeva11
• Theorem 2.2: Gasharov-Peeva-Welker GPW99, see Peeva11
• Definition 3.1
• Theorem 3.2: Peeva11
• Theorem 3.3
• proof
• Theorem 4.1
• Theorem 4.2: Birkhoff48
• Definition 4.3
• Lemma 4.4