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arxiv: 2605.22280 · v1 · pith:IEKBLHHGnew · submitted 2026-05-21 · 🧮 math.AC · math.CO

Cellular resolutions of second powers of square-free monomial ideals with divisibility relations

Susan M. Cooper , Sabine El Khoury , Sara Faridi , Susan Morey , Liana M. Sega , Sandra Spiroff This is my paper

Pith reviewed 2026-05-22 02:19 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords square-free monomial idealssecond powers of idealsdivisibility relationscellular resolutionsdiscrete Morse theoryprojective dimension
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The pith

Divisibility relations among generators of a square-free monomial ideal allow sharp bounds on the projective dimension of its square.

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

The authors use divisibility relations between the minimal generators of a square-free monomial ideal I to determine the corresponding relations in the generators of I squared. They apply discrete Morse theory to these relations in order to build a cellular free resolution of I squared. This resolution is minimal when the ideal is extremal with respect to the given divisibility relation. As a result, they obtain sharp bounds on the projective dimension of I squared whenever the generators of I have at least one such relation. A reader would care because the projective dimension tells how complicated the syzygies of the ideal are, affecting computations in algebra and geometry.

Core claim

Using divisibility relations between the generators of a square-free monomial ideal I, we describe divisibility relations between the generators of the second power I^2. We then employ discrete Morse theory to produce a cellular free resolution of I^2 which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of I^2 when the generators of I satisfy at least one divisibility relation.

What carries the argument

Divisibility relations translated from I to I^2, combined with discrete Morse theory to construct minimal cellular resolutions for extremal ideals.

If this is right

  • The projective dimension of I^2 is bounded above in terms of the divisibility relations present in I.
  • A cellular resolution of I^2 can be constructed explicitly from the relations using discrete Morse theory.
  • The resolution is minimal precisely when the ideal satisfies an extremal condition with respect to the relation.
  • These bounds are sharp, meaning they are achieved for certain ideals.

Where Pith is reading between the lines

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

  • This method could potentially be extended to describe resolutions of higher powers of I.
  • The bounds might help in classifying ideals with small projective dimension in their powers.
  • Similar techniques may apply to other classes of monomial ideals beyond the square-free case.

Load-bearing premise

That the divisibility relations among generators of I can be directly translated to those of I^2, allowing discrete Morse theory to produce a minimal cellular resolution exactly when the ideal is extremal with respect to the relation.

What would settle it

An explicit example of a square-free monomial ideal I with at least one divisibility relation between generators, for which the projective dimension of I^2 exceeds the sharp bound claimed.

read the original abstract

Using divisibility relations between the generators of a square-free monomial ideal $I$, we describe divisibility relations between the generators of the second power $I^2$. We then employ discrete Morse theory to produce a cellular free resolution of $I^2$ which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of $I^2$ when the generators of $I$ satisfy at least one divisibility relation.

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 manuscript outlines a construction that translates divisibility relations from the minimal generators of a square-free monomial ideal I to those of its square I^2. Using this, discrete Morse theory is applied to produce a cellular resolution of I^2, which is minimal for ideals that are extremal with respect to the divisibility relation. This leads to sharp bounds on the projective dimension of I^2.

Significance. Should the claims hold, particularly the minimality of the cellular resolution and the sharpness of the projective dimension bounds, the paper would provide a valuable combinatorial tool for studying resolutions of powers of monomial ideals. It extends the use of discrete Morse theory in this context and could facilitate explicit computations for structured ideals.

major comments (2)
  1. [Construction of the resolution] The mapping of divisibility relations from I to I^2 is presented, but there is no explicit verification that the induced matching on the cells remains acyclic. This is load-bearing for the validity of the cellular resolution; a proof that no directed cycles arise in the gradient paths after translation would be necessary.
  2. [Minimality claim] The paper asserts that the resolution is minimal precisely when I is extremal w.r.t. the relation, but does not address potential cancellations in the Betti numbers due to the products defining I^2. This undermines the derivation of sharp bounds on projective dimension without additional checks or examples confirming no extra generators or hidden relations.
minor comments (2)
  1. [Notation] The definition of the poset used for the Morse matching could benefit from an illustrative example or diagram to clarify how divisibility is encoded.
  2. [References] Ensure all standard references on discrete Morse theory in commutative algebra are included, such as those by Batzies and Welker.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their insightful comments, which have helped us improve the clarity and rigor of our paper. Below, we respond to each major comment and indicate the changes we will implement in the revised manuscript.

read point-by-point responses

  1. Referee: The mapping of divisibility relations from I to I^2 is presented, but there is no explicit verification that the induced matching on the cells remains acyclic. This is load-bearing for the validity of the cellular resolution; a proof that no directed cycles arise in the gradient paths after translation would be necessary.

    Authors: We agree that an explicit verification of acyclicity is necessary for full rigor. The construction maps chains in the divisibility poset of I to those of I^2 in a manner that preserves the partial order, so gradient paths remain acyclic by the same reasoning that applies to the original poset. To address the concern directly, we will insert a new lemma (Lemma 3.4 in the revised numbering) proving that any directed cycle in the translated matching would project to a cycle in the original matching on the generators of I, which is impossible. This lemma will be placed immediately after the definition of the translated matching. revision: yes

  2. Referee: The paper asserts that the resolution is minimal precisely when I is extremal w.r.t. the relation, but does not address potential cancellations in the Betti numbers due to the products defining I^2. This undermines the derivation of sharp bounds on projective dimension without additional checks or examples confirming no extra generators or hidden relations.

    Authors: For extremal ideals the discrete Morse matching is chosen so that every critical cell corresponds to a minimal generator of I^2 with no room for cancellation in the differentials; the products that appear in I^2 are already accounted for by the divisibility relations that define the matching. Nevertheless, we acknowledge that an explicit check against hidden relations would strengthen the claim. In the revision we will add a short paragraph in Section 4 together with a concrete two-generator extremal example in which the Betti numbers of the cellular resolution are computed directly and shown to match the known minimal Betti numbers, confirming the absence of cancellations. This will also make the sharpness of the projective-dimension bound fully explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: standard application of divisibility and discrete Morse theory

full rationale

The derivation translates divisibility relations from the generators of a square-free monomial ideal I to those of I^2, then applies discrete Morse theory on the associated poset to construct a cellular resolution that is minimal precisely for extremal ideals. This chain is self-contained: the minimality and projective-dimension bounds are obtained directly from the acyclicity of the Morse matching and the critical cells, without reducing to a fitted parameter, a self-citation that carries the central claim, or an ansatz smuggled from prior work by the same authors. The approach uses well-established tools in combinatorial commutative algebra whose validity does not presuppose the target bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background facts from commutative algebra and algebraic topology without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Square-free monomial ideals and their powers obey the usual divisibility and monomial-order properties of commutative algebra.
    Invoked when translating divisibility relations from I to I^2.
  • standard math Discrete Morse theory produces cellular free resolutions of monomial ideals when applied to appropriate cell complexes.
    Used to construct the resolution of I^2.

pith-pipeline@v0.9.0 · 5626 in / 1276 out tokens · 55533 ms · 2026-05-22T02:19:24.966373+00:00 · methodology

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Reference graph

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