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arxiv: 2605.19973 · v1 · pith:SMLLVOQTnew · submitted 2026-05-19 · 🧮 math.AC

Golod ideals in combinatorial commutative algebra

Benjamin Briggs , Trung Chau , Alessandro De Stefani This is my paper

Pith reviewed 2026-05-20 03:34 UTC · model grok-4.3

classification 🧮 math.AC
keywords Golod idealsdeterminantal idealsbinomial edge idealspermanental idealscover idealsKoszul cyclesMassey productsstrongly Golod ideals
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The pith

Determinantal ideals are Golod if and only if they have a linear resolution.

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

This paper investigates the Golod property for standard graded algebras arising from combinatorial ideals. It proves that determinantal ideals, binomial edge ideals, and permanental ideals satisfy the Golod condition exactly when they possess linear resolutions. It supplies a characterization for cover ideals that turns on multidegrees of Koszul cycles and the vanishing of Massey products. It further shows that squarefree strongly Golod ideals, and more generally lcm-strongly Golod ideals, are Golod rather than only weakly Golod. These equivalences and characterizations clarify when the homological algebra of the quotient ring simplifies in concrete combinatorial families.

Core claim

Determinantal ideals, binomial edge ideals, and permanental ideals are Golod if and only if they have a linear resolution. Cover ideals define Golod rings under a characterization that uses multidegrees of Koszul cycles and Massey products. Squarefree strongly Golod ideals and lcm-strongly Golod ideals are Golod.

What carries the argument

The equivalence between the Golod property and the existence of a linear resolution, together with multidegree conditions on Koszul cycles that control Massey products for cover ideals and the strongly Golod condition for squarefree ideals.

Load-bearing premise

The algebras under study are standard graded over a field, which makes linear resolutions and multidegrees of Koszul cycles well-defined.

What would settle it

A determinantal ideal possessing a linear resolution whose quotient ring fails to be Golod would refute the claimed equivalence.

read the original abstract

In this article we study the Golod property of standard graded algebras. We show that determinantal ideals, binomial edge ideals, and permanental ideals are Golod if and only if they have a linear resolution. Next, we give a characterization of when cover ideals define Golod rings, exploiting some considerations on multidegrees of Koszul cycles and Massey products. Finally, we show that squarefree strongly Golod ideals (and, more generally, lcm-strongly Golod ideals) are Golod, and not just weakly Golod.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies the Golod property of standard graded algebras over a field. It proves that determinantal ideals, binomial edge ideals, and permanental ideals are Golod if and only if they have a linear resolution. It gives a characterization of cover ideals that define Golod rings using multidegrees of Koszul cycles and Massey products. It also shows that squarefree strongly Golod ideals (and more generally lcm-strongly Golod ideals) are Golod.

Significance. If the central claims hold, the results provide concrete equivalences and characterizations connecting the Golod property to linear resolutions and explicit homological data for well-studied combinatorial ideal classes. This strengthens the toolkit for detecting Golod rings via Koszul homology and Massey products, with potential applications to determinantal and edge ideals.

major comments (2)
  1. [§3] §3: The converse (Golod implies linear resolution) for determinantal, binomial edge, and permanental ideals is load-bearing and rests on showing that any non-linear syzygy produces a non-vanishing Massey product. This step uses the explicit form of the minimal free resolutions (e.g., Eagon-Northcott type); if higher-degree generators outside the combinatorial description exist, the implication fails. An explicit verification or counterexample check for at least one family is needed.
  2. [§4] §4: The characterization of Golod rings for cover ideals via multidegrees of Koszul cycles assumes all relevant Massey products are detected by the stated multidegree conditions. This is central to the if-and-only-if claim; a concrete computation exhibiting a Massey product missed by the multidegree criterion would undermine the result.
minor comments (2)
  1. [§2] The notation for multidegrees and the precise definition of lcm-strongly Golod could be illustrated with a small explicit example early in the paper for clarity.
  2. [References] A few citations to foundational works on Massey products in the Koszul complex appear to be missing from the references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The suggestions highlight important aspects of the proofs that we will clarify in the revision. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3: The converse (Golod implies linear resolution) for determinantal, binomial edge, and permanental ideals is load-bearing and rests on showing that any non-linear syzygy produces a non-vanishing Massey product. This step uses the explicit form of the minimal free resolutions (e.g., Eagon-Northcott type); if higher-degree generators outside the combinatorial description exist, the implication fails. An explicit verification or counterexample check for at least one family is needed.

    Authors: We agree that the converse relies on the completeness of the known minimal free resolutions for these classes. For determinantal ideals the Eagon-Northcott complex is minimal and exhaustive; any syzygy outside the expected degrees would violate the known Betti numbers. For binomial edge ideals the resolution is given by the Herzog-Huneke-Srinivasan complex, which likewise accounts for all generators. To make this explicit we will add, in the revised §3, a short verification for the binomial edge ideal of a path graph, confirming that a non-linear syzygy in degree 2 produces a non-vanishing Massey product of the expected multidegree. This concrete check supports the general argument without altering the statement. revision: yes

  2. Referee: [§4] §4: The characterization of Golod rings for cover ideals via multidegrees of Koszul cycles assumes all relevant Massey products are detected by the stated multidegree conditions. This is central to the if-and-only-if claim; a concrete computation exhibiting a Massey product missed by the multidegree criterion would undermine the result.

    Authors: The multidegree criterion in §4 arises because, for square-free monomial ideals, every Massey product is represented by a Koszul cycle whose multidegree is fixed by the lcm of the generators involved; any product outside these degrees is necessarily zero by the grading. We therefore believe the stated conditions capture all non-vanishing products. Nevertheless, to address the concern we will include in the revision a small explicit computation for the cover ideal of a triangle, exhibiting both a vanishing and a non-vanishing Massey product and verifying that the multidegree test correctly predicts the Golod property in each case. revision: partial

Circularity Check

0 steps flagged

No circularity: equivalences and characterizations derived from standard Koszul homology and external classical resolutions.

full rationale

The paper proves that determinantal, binomial edge, and permanental ideals are Golod precisely when they admit linear resolutions by invoking the general fact that linear resolutions imply vanishing of higher Koszul homology (hence Golod) together with the explicit form of minimal free resolutions for these classes, which are classical external results such as the Eagon-Northcott complex. The cover-ideal characterization via multidegrees of Koszul cycles and Massey products is obtained by direct computation of the relevant products in the Koszul complex. The statement that squarefree strongly Golod ideals are Golod follows from a direct verification that the defining condition forces the Massey products to vanish. None of these steps reduces by definition or by self-citation to the paper's own inputs; all rely on independently established homological machinery. No fitted parameters, self-definitional loops, or load-bearing self-citations appear.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Results rest on standard definitions of Golod rings, linear resolutions, Koszul homology, and Massey products in graded commutative algebra; no free parameters or invented entities introduced.

axioms (1)
  • domain assumption Algebras under consideration are standard graded over a field
    Abstract opens with study of Golod property of standard graded algebras; this underpins all resolution and multidegree statements.

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