Properties of LCM Lattices of Monomial Ideals

Jason McCullough; Matthew Dorang

arxiv: 2505.08722 · v2 · submitted 2025-05-13 · 🧮 math.AC · math.CO

Properties of LCM Lattices of Monomial Ideals

Matthew Dorang , Jason McCullough This is my paper

Pith reviewed 2026-05-22 15:27 UTC · model grok-4.3

classification 🧮 math.AC math.CO

keywords LCM latticemonomial idealedge idealmodular latticeCohen-Macaulayprojective dimensionGorenstein idealatomic lattice

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The pith

LCM lattices of edge ideals have Boolean, modular, semimodular, supersolvable, coatomic and complemented properties exactly when the graphs satisfy the matching conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines lattice properties of LCM lattices of monomial ideals beyond the known fact that they are precisely the atomic lattices. For those coming from edge ideals of graphs it gives complete characterizations of when the lattice is Boolean, modular, upper semimodular, lower semimodular, supersolvable, coatomic or complemented, all expressed as conditions on the graph. These characterizations matter because the LCM lattice determines the minimal free resolution of the ideal, so the results supply combinatorial tests for algebraic features such as Cohen-Macaulayness and the value of projective dimension. The work also proves that minimal monomial ideals attached to modular lattices are Cohen-Macaulay and supplies necessary and sufficient lattice conditions under which projective dimension equals the height of the LCM lattice.

Core claim

LCM lattices are always atomic and every atomic lattice arises as the LCM lattice of some monomial ideal. For the LCM lattice of an edge ideal of a graph the properties Boolean, modular, upper semimodular, lower semimodular, supersolvable, coatomic and complemented are completely characterized in terms of the graph. Minimal monomial ideals associated to modular lattices are Cohen-Macaulay. Necessary and sufficient conditions on the lattice determine when the projective dimension of the monomial ideal equals the height of its LCM lattice. The LCM lattice of any Gorenstein edge ideal is coatomic.

What carries the argument

The LCM lattice of a monomial ideal, formed by the least common multiples of subsets of its minimal generators and used to build the minimal free resolution.

If this is right

Minimal monomial ideals whose LCM lattices are modular are Cohen-Macaulay.
Projective dimension equals the height of the LCM lattice precisely when the lattice satisfies the identified necessary and sufficient conditions.
Gorenstein edge ideals always produce coatomic LCM lattices.
Graded LCM lattices of edge ideals arise exactly from gap-free graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The graph-to-lattice dictionary may let combinatorial algorithms compute minimal resolutions for edge ideals without building the full resolution.
The open questions about lattice properties of arbitrary Gorenstein monomial ideals point toward using coatomicity as a test case for broader Gorenstein classifications.
Similar characterizations might exist for other families of monomial ideals whose generators have clear combinatorial meaning.

Load-bearing premise

The characterizations assume that the monomial ideal is generated exactly by the monomials corresponding to the edges of the graph and that the lattice is built from the standard LCM construction without extra generators or relations that would change its structure.

What would settle it

A concrete graph whose edge ideal yields an LCM lattice that is modular yet the graph fails the combinatorial condition given for modularity would show the claimed characterization is incomplete.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit graph conditions for when the LCM lattice of an edge ideal is Boolean, modular, semimodular, supersolvable, coatomic, or complemented, plus Cohen-Macaulay and projective-dimension results for modular and general cases.

read the letter

The main point is that for edge ideals the lattice properties translate directly into graph properties. Boolean lattices come from disjoint unions of edges and isolates, modular ones from certain complete multipartite graphs, and the other properties follow similar forbidden-configuration rules. This extends the graded case from Nevo-Peeva with full lists rather than just one property at a time. The approach sticks to the standard Gasharov-Peeva-Welker LCM construction, where joins are LCMs and meets are GCDs, then converts each lattice axiom into edge-set conditions. That dictionary looks clean and avoids circularity. The Cohen-Macaulay theorem for minimal monomial ideals over modular lattices and the separate necessary and sufficient conditions for projective dimension equaling height are practical additions. The observation that Gorenstein edge ideals have coatomic LCM lattices is a quick consequence of the earlier work. One minor soft spot is that the Gorenstein section mostly raises questions instead of settling them, but that does not affect the main characterizations. The paper stays within its subfield and does not overclaim. It is aimed at people working on resolutions of monomial ideals or combinatorial commutative algebra. Anyone who already uses LCM lattices for homological invariants will find concrete checks they can apply to graphs. The results rest on reproducible lattice-to-graph translations and build on cited prior results without fitting or self-reference issues. I would send this to peer review.

Referee Report

0 major / 2 minor

Summary. The paper studies lattice properties of LCM lattices of monomial ideals, building on the Gasharov-Peeva-Welker construction. For edge ideals I(G) of a graph G, it gives complete characterizations of when the associated LCM lattice is Boolean, modular, upper semimodular, lower semimodular, supersolvable, coatomic, or complemented, expressed directly in terms of forbidden configurations or structural properties of G. It proves that minimal monomial ideals whose LCM lattice is modular are Cohen-Macaulay, establishes separate necessary and sufficient lattice conditions for the projective dimension to equal the height of the LCM lattice, shows that LCM lattices of Gorenstein edge ideals are coatomic, and poses questions about lattice properties of arbitrary Gorenstein monomial ideals.

Significance. The explicit graph-theoretic characterizations for the listed lattice properties constitute a clear advance in linking combinatorial commutative algebra with lattice theory. The Cohen-Macaulayness result for modular LCM lattices and the projective-dimension criteria are load-bearing contributions that follow from the same dictionary of translations. The work is strengthened by its reliance on the standard LCM construction without additional parameters and by its precise open questions on Gorenstein monomial ideals.

minor comments (2)

The introduction would benefit from a brief explicit list of the lattice properties characterized for edge ideals, rather than referring to 'many standard lattice properties'.
In the section on projective dimension, the statement of the necessary and sufficient lattice conditions could include a short reminder of the definition of height of the LCM lattice for readers less familiar with the Gasharov-Peeva-Welker setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately reflects our main results on the lattice properties of LCM lattices for monomial ideals, including the complete characterizations for edge ideals in terms of graph structure and the Cohen-Macaulayness theorem for minimal monomial ideals with modular LCM lattices. We appreciate the recognition of the advance in connecting combinatorial commutative algebra with lattice theory and the note on open questions for Gorenstein monomial ideals.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivations rest on the external Gasharov-Peeva-Welker definition of the LCM lattice (elements are LCMs of subsets of minimal generators, ordered by divisibility) and translate standard lattice axioms (Boolean, modular, semimodular, etc.) into explicit graph-theoretic forbidden configurations for edge ideals I(G). These translations are direct and do not reduce any claimed equivalence to a fitted parameter or self-referential definition. Cited results (Nevo-Peeva on graded LCM lattices; prior work on Gorenstein edge ideals) are independent external references, not self-citations by the present authors. Cohen-Macaulayness and height-vs-projective-dimension criteria follow from the same non-circular dictionary without invoking uniqueness theorems or ansatzes from the authors' own prior work. The derivation chain is therefore self-contained against external lattice theory and graph combinatorics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established definition of LCM lattices and standard facts from lattice theory and commutative algebra; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math LCM lattice of a monomial ideal is defined via the poset of least common multiples of subsets of generators (Gasharov-Peeva-Welker).
Invoked throughout as the central object of study.
standard math Standard definitions and equivalences for Boolean, modular, semimodular, supersolvable, coatomic, and complemented lattices.
Used to state the characterizations.

pith-pipeline@v0.9.0 · 5714 in / 1463 out tokens · 44505 ms · 2026-05-22T15:27:51.290852+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On Some Properties of LCM-Lattices of Edge Ideals of k-Uniform Hypergraphs
math.AC 2026-05 unverdicted novelty 5.0

Conditions are established for lcm-lattices of edge ideals of k-uniform hypergraphs to be Boolean, modular, or complemented, with extensions to products and polarization effects.