Pith. sign in

REVIEW 3 major objections 5 minor 21 references

When are Morse resolutions polyhedral?

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper exhibits a six-generator monomial ideal whose minimal free resolution is supported on a Morse complex, yet no Morse matching yields a polyhedral complex, and the ideal has no polyhedral minimal resolution at all.

desk verdict A credible and interesting result with a real gap: the classification of maximal acyclic matchings in Theorem 5.4 is asserted, not proven. read the letter →

arxiv 2505.08580 v1 pith:WZTE23VG submitted 2025-05-13 math.AC

Louis Bu , Sara Faridi , Iresha Madduwe Hewalage , Thiago Holleben , Hasan Mahmood , Dharm Veer , Kyle Wang , Scott Wesley This is my paper · ORCID
classification math.AC MSC 13D02
keywords monomialidealfreeresolutionMorsematchingcomplexpolyhedralcellcellularTaylorScarf
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Every monomial ideal with $r$ generators has a free resolution built from the simplex on $r$ vertices, but that Taylor resolution is usually far from minimal. Discrete Morse theory shrinks the simplex by deleting pairs of faces whose least-common-multiple labels agree, producing a Morse complex that still resolves the ideal; the caveat is that the surviving complex need not be built from convex polytopes. This paper proves a boundary result: for ideals minimally generated by up to four monomials, some maximal acyclic deletion pattern yields a polyhedral Morse complex, whereas the six-generator ideal in (5.1) admits no maximal acyclic deletion pattern whose Morse complex is polyhedral. The paper further shows that this ideal has no minimal polyhedral free resolution at all, so the obstruction lies in the ideal itself, not in the choice of deletion pattern.

What carries the argument

The central object is the Taylor complex: the simplex whose faces are labeled by the least common multiples of subsets of the minimal generators, whose homogenized chain complex is the Taylor resolution. A homogeneous acyclic matching pairs faces of the simplex that carry the same least-common-multiple label, with no directed cycle in the resulting gradient graph, and the Morse complex $X_M$ is the cell complex whose cells are the unmatched faces, with face inclusions read off from gradient paths. The load-bearing step in Theorem 5.4 is a classification of all maximal homogeneous acyclic matchings of the six-vertex Taylor complex into four matchings $M_1,\dots,M_4$, together with a check that in each case the single extra 3-cell intersects the Scarf complex in a way that violates the defining property of a polyhedral complex: the intersection of two 3-cells is not a face of both.

What would settle it

An exhaustive computer enumeration of all maximal homogeneous acyclic matchings of the Taylor complex of the ideal in (5.1) would settle the main claim: a matching outside the four listed ones whose Morse complex is polyhedral would refute Theorem 5.4, and a polyhedral complex with the required f-vector would refute Remark 5.10.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5.4: for the squarefree monomial ideal $I=(m_1,\dots,m_6)$ defined in (5.1), every maximal homogeneous acyclic matching $M$ of the Taylor complex $X$ gives a Morse complex $X_M$ that supports the minimal free resolution of $I$, and $X_M$ is not a polyhedral cell complex. Remark 5.10 strengthens this: the ideal has no minimal polyhedral free resolution whatsoever. The reason is that any such resolution would have to be the Scarf complex of $I$ together with exactly one additional 3-cell carrying the multidegree $x_1x_2\cdots x_{11}$, and the two candidate cells $\{1,2,3,5\}$ and $\{2,3,4,5\}$ both force the attachment of a simplicial 3-polytope that cannot exist, since Steinitz's face-count relations are violated. The positive companion is Theorem 4.1: a monomial ideal minimally generated by at most four monomials always has a maximal homogeneous acyclic matching whose Morse complex is polyhedral.

Load-bearing premise

The load-bearing premise is that the four listed patterns are the only maximal ways to delete paired faces from the six-vertex simplex while keeping equal least-common-multiple labels; if a fifth pattern exists, the conclusion that every resulting complex is non-polyhedral does not follow.

Editorial extensions

If this is right

  • For monomial ideals with at most four generators, there is always a maximal Morse matching whose Morse complex is a polyhedral cell complex, so the Taylor simplex can be shrunk without losing convex geometry.
  • For the six-generator ideal in (5.1), non-polyhedrality is forced by the ideal itself: every maximal homogeneous acyclic matching produces a non-polyhedral Morse complex supporting the minimal free resolution.
  • The example has no minimal polyhedral resolution, so the number of generators alone does not guarantee polyhedral minimal resolutions; four generators always suffice, and six already fail.
  • Only one multidegree outside the Scarf complex is needed to destroy polyhedrality: the extra 3-cell in multidegree $x_1x_2\cdots x_{11}$ attaches to the Scarf complex along non-faces, which is why the resulting complex cannot be a polyhedral cell complex.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the non-polyhedrality argument depends only on the least-common-multiple labels and the face lattice of the Scarf complex, the same conclusion should hold over any field; the obstruction is not a characteristic-dependent phenomenon.
  • The classification of matchings in (5.5)–(5.9) suggests a general computational test: enumerate the minimal homogeneous pairs of a monomial ideal and inspect their overlap graph, since an overlap pattern like the one here may predict when every maximal matching yields a non-polyhedral Morse complex.
  • Perturbing the exponents of one generator in (5.1) should break some of the label equalities that force the four matchings, producing a family of ideals in which the non-polyhedral obstruction can be switched on or off; such a family would test how sharp the six-generator boundary is.
  • The Scarf-complex-plus-one-cell obstruction recasts part of the search for polyhedral minimal resolutions as a polytope-realizability problem: once the missing cell's boundary is forced to be simplicial, Steinitz's f-vector relations become a necessary condition that can rule out an attachment before any geometry is drawn.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies Morse matchings on the Taylor complex of a monomial ideal and asks when the associated Morse complex is a polyhedral cell complex. It proves that for ideals minimally generated by at most four monomials, there exists a maximal homogeneous acyclic matching whose Morse complex supports a polyhedral resolution (Theorem 4.1). Its main negative result, Theorem 5.4, is an explicit six-generator square-free monomial ideal (5.1) for which every maximal homogeneous acyclic matching is claimed to yield a Morse complex supporting the minimal free resolution, yet none of these Morse complexes is polyhedral; Remark 5.10 further claims the ideal has no minimal polyhedral resolution at all, using Steinitz's f-vector relations. The argument combines discrete Morse theory, the Scarf complex, and Macaulay2/SageMath computations for Betti numbers and homology.

Significance. If the main claim stands, the six-generator example is a significant and nearly optimal counterexample: the Scarf complex differs from the Morse complex by a single 3-cell, so the failure of polyhedrality is reduced to a concrete gluing obstruction rather than to counting Betti numbers. The positive result for up to four generators and the Steinitz-based argument in Remark 5.10 are also valuable. The paper is explicit about its computational inputs and uses established, credited tools. However, the universal statement in Theorem 5.4 depends on an exhaustive classification of maximal homogeneous acyclic matchings that is currently asserted rather than proved, so the significance is conditional on closing that gap.

major comments (3)
  1. [§5, proof of Theorem 5.4, after (5.9)] The assertion that every maximal homogeneous acyclic matching M is one of M1–M4 is not derived. The text states 'Summarizing the discussion above, we have M could be one of the following sets' and justifies the list by 'using (5.6) and (5.5) we see,' but it never proves that the only faces appearing in more than one homogeneous pair are γ\{1}, γ\{4}, γ\{6}, and γ\{1,4}, nor that M must contain the forced set M0. The claim that M contains M0 is argued from the fact that the endpoints of each M0 edge appear in no other pair in A; this is a matching-theoretic maximality argument that does not address acyclicity. A maximal acyclic matching can omit such an edge if adding it would create a directed cycle. The proof must either prove that every maximal homogeneous acyclic matching is a maximal matching of the graph of homogeneous pairs, or supply a separate acyclicity-preserving argument for each forced edge. Since Theorem 5.4 is a universal statement, an unclassified maximal homogeneous acyclic matching outside M1–M4 would invalidate it.
  2. [§5, acyclicity of M1–M4] The sentence 'Each of these matchings is, moreover, acyclic, because by [FFDGGYP24, Lemma 3.3], a directed cycle will have to go between at least 6 vertices all sharing the same monomial label' does not verify the hypotheses of that lemma for the specific matchings M1–M4. The authors do not identify the monomial label classes of size at least six in this example, nor do they show that no such cycle occurs in each of the four cases. Acyclicity is part of the definition of a Morse matching and therefore load-bearing; a short direct check or a verified application of the lemma is needed.
  3. [§4, proof of Theorem 4.1] The proof of the four-generator existence result also contains an asserted enumeration: after constructing the matching M, it says 'Let M1 be a homogeneous acyclic matching of the simplicial complex on the right. Then M1 is one of the following sets.' No case analysis or argument is given that rules out other homogeneous acyclic matchings. For the existence claim it would suffice to exhibit one maximal homogeneous acyclic extension and verify that its Morse complex is polyhedral, so this gap is less damaging than the one in Theorem 5.4, but it should still be repaired.
minor comments (5)
  1. [§2, Definition 2.3] The definition of a cell complex by attaching k-simplices assumes that the attaching map sends each face of Δk homeomorphically onto a cell; this is the definition of a regular CW complex, not a general cell complex. Since the paper later relies on non-polyhedral and possibly non-regular Morse complexes, the definition should be adjusted or the restriction stated.
  2. [§5, after (5.9)] The phrase 'M could be one of the following sets' should read 'M must be one of the following sets' if the intended claim is exhaustive classification; otherwise the universal conclusion in Theorem 5.4 does not follow from the displayed list.
  3. [§5, proof of Theorem 5.4] The incidence computation for the non-polyhedral gluing is summarized as 'following the gradient paths in Figure 8'; the text-only version of the figure does not make the verification transparent. Please list the relevant gradient paths explicitly or provide the computational data used for the SageMath/Macaulay2 checks.
  4. [Keywords] The keyword 'disrete Morse theory' contains a typo and should be 'discrete Morse theory.'
  5. [§5, after Theorem 5.4] The sentence 'The two Morse complex XM1 and XM3 contain a single non-Scarf 3-cell' has a subject-verb agreement error; it should be 'The two Morse complexes XM1 and XM3 contain...'

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; the asserted matching classification is a proof gap, not a circular reduction.

full rationale

The paper's central claims are (1) a positive theorem that ideals with at most four generators admit a maximal homogeneous acyclic matching whose Morse complex is polyhedral, and (2) a counterexample with six generators where every maximal homogeneous acyclic matching yields a non-polyhedral Morse complex and no polyhedral minimal resolution exists. Neither claim reduces to its own inputs by construction. The positive theorem is proved by a direct case analysis of minimal homogeneous pairs and the resulting simplicial/polyhedral complexes. The negative theorem rests on identifying the possible maximal homogeneous acyclic matchings of the Taylor complex as M1 through M4, then checking non-polyhedrality geometrically: the intersection of the unique non-Scarf 3-cell with a Scarf facet is not a face of the Morse complex, and Remark 5.10 excludes polyhedral resolutions independently via Steinitz's lemma and Stanley's f-vector theorem. The identification of M1 through M4 is asserted rather than fully derived, and the acyclicity of those matchings is justified by a cited lemma; these are potential correctness or completeness concerns, not circularity. In particular, the citation to [FFDGGYP24, Lemma 3.3] is a self-citation because Faridi is an author of both works, but it is a published lemma with an independent proof, it is used only to verify acyclicity of specific matchings, and its statement does not include the paper's target conclusion of non-polyhedrality. There is no fitted parameter later relabeled as a prediction, no quantity defined in terms of the desired conclusion, and no uniqueness theorem imported from the authors' prior work to force the choice of matching. The non-polyhedrality checks are explicit geometric verifications using the paper's own definitions of polyhedral cell complexes and the gradient-path criterion from Batzies--Welker. Accordingly, the paper is not circular; the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities appear. The central claim rests on standard theorems, on a cited lemma from an earlier paper by one of the authors, and on computational outputs that are not fully documented in the preprint.

assumptions (5)
  • domain assumption The Macaulay2-computed Betti table of S/I is (1,6,15,17,7) and SageMath gives H~_2(Scarf(I),K)=K.
    These computations are reported without code or transcripts and justify the statement that exactly one 3-cell must be attached to the Scarf complex.
  • domain assumption [FFDGGYP24, Lemma 3.3]: a directed cycle in a homogeneous matching requires at least six vertices sharing the same monomial label.
    Used to assert that the four matchings M1 through M4 are acyclic; the hypothesis is not checked explicitly for the ideal in (5.1).
  • standard math Every cell complex supporting a free resolution of I contains Scarf(I) as a subcomplex.
    Cited from Peeva [Pee11, Proposition 59.4]; it restricts possible minimal polyhedral resolutions to attachments of one cell to the Scarf complex.
  • standard math Steinitz's lemma for simplicial 3-polytopes (f1=3f0-6, f2=2f0-4) and Stanley's h-vector nonnegativity criterion.
    Used in Remark 5.10 to rule out the existence of a simplicial 3-polytope boundary with the required f-vector.
  • standard math The Morse complex theorem of Batzies and Welker: a homogeneous acyclic matching produces a cell complex supporting a multigraded free resolution.
    This is the foundational tool that defines the Morse complex X_M and guarantees it resolves the ideal.

how reviews work

0 comments
Cite this review

Pith. "Pith review of When are Morse resolutions polyhedral?." pith.science (2026). https://pith.science/paper/WZTE23VG

@misc{pith2026250508580,
  author       = {Pith},
  title        = {Pith review of: When are Morse resolutions polyhedral?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZTE23VG}},
  note         = {Machine review of arXiv:2505.08580}
}
abstract

It is known that the chain complex of a simplex on $q$ vertices can be used to construct a free resolution of any ideal generated by $q$ monomials, and as a direct result, the Betti numbers always have binomial upper bounds, given by the number of faces of a simplex in each dimension. It is also known that for most monomials the resolution provided by the simplex is far from minimal. Discrete Morse theory provides an algorithm called \say{Morse matchings} by which faces of the simplex can be removed so that the chain complex on the remaining faces is still a free resolution of the same ideal. An immediate positive effect is an often considerable improvement on the bounds on Betti numbers. A caveat is the loss of the combinatorial structure of the simplex we started with: the output of the Morse matching process is a cell complex with no obvious structure besides an \say{address} for each cell. The main question in this paper is: which Morse matchings lead to Morse complexes that are polyhedral cell complexes? We prove that if a monomial ideal is minimally generated by up to four generators, then there is a maximal Morse matching of the simplex such that the resulting cell complex is a polyhedral cell complex. We then give an example of a monomial ideal minimally generated by six generators whose minimal free resolution is supported on a Morse complex and the Morse complex cannot be polyhedral no matter what Morse matching is chosen, and we go further to show that this ideal cannot have any polyhedral minimal free resolution.

Figures

Figures reproduced from arXiv: 2505.08580 by the authors.

Figure 1
Figure 1. Figure for Example 2.5 If we remove the central triangle and one of the edges, we will have the middle picture in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Homogeneous acyclic matching M. The vertices corresponding to the faces of Scarf(I) are filled with black. 2 6 4 1 3 5 2 6 4 1 3 5 4. When are Morse complexes polyhedral? Given the algorithmic nature of the construction of the Morse complex, which is done via pairwise elimination of face and facets from the Taylor simplex, a natural question is: is there always a way to do the matchings so that the Morse complex is … view at source ↗
Figure 4
Figure 4. Illustrative sub-complexes of Scarf(I). Computations conducted using SageMath [The20] show that He2(Scarf(I), K) ∼= K; thus, Scarf(I) is not acyclic and therefore cannot support a free resolution of I. In terms of its 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: A visualization of Scarf(I) in R 3 . Note that three of the tetrahedra in this diagram appear twice, to avoid self-inversion. This should be thought of as a quotient relation on the space. For ease of viewing, the three pairs of 3-cells have been highlighted in red, gr…
Figure 6
Figure 6. Figure 6: The matching M0, Scarf faces are highlighted with black filled dots {123456} {12345} {12346} {12356} {12456} {13456} {23456} {1234}{1235}{1236}{1245}{1356}{2345}{2356}{2456}{3456} {123} {245} {356} {123456} {12345} {12346} {12356} {12456} {13456} {23456} {1234}{1235}{1…
Figure 7
Figure 7. Figure 7: The matchings M1, . . . , M4, excluding Scarf faces, edges in M0 are blue The homogeneous matchings M1, . . . , M4 are shown in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Gradient paths By Proposition 3.3 and following the gradient paths in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 14 canonical work pages

  1. [1]

    Computational synthetic geometry , volume 1355 of Lecture Notes in Mathematics

    J\"urgen Bokowski and Bernd Sturmfels. Computational synthetic geometry , volume 1355 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1989

    work page 1989

  2. [2]

    Cellular resolutions of monomial modules

    Dave Bayer and Bernd Sturmfels. Cellular resolutions of monomial modules. J. Reine Angew. Math. , 502:123--140, 1998

    1998

  3. [3]

    Batzies and V

    E. Batzies and V. Welker. Discrete M orse theory for cellular resolutions. J. Reine Angew. Math. , 543:147--168, 2002

    work page 2002

  4. [4]

    Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Susan Morey, Liana M

    Susan M. Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Susan Morey, Liana M. Sega, and Sandra Spiroff. Simplicial resolutions for the second power of square-free monomial ideals. In Women in commutative algebra , volume 29 of Assoc. Women Math. Ser. , pages 193--205. Springer, Cham, [2021] 2021

    work page 2021

  5. [5]

    Minimal cellular resolutions of monomial ideals with five generators and their Artinian reductions

    Trung Chau. Minimal cellular resolutions of monomial ideals with five generators and their artinian reductions. arXiv preprint arXiv:2502.06182 , 2025

    work page Pith review arXiv 2025

  6. [6]

    Barile- M acchia resolutions

    Trung Chau and Selvi Kara. Barile- M acchia resolutions. J. Algebraic Combin. , 59(2):413--472, 2024

    2024

  7. [7]

    Faridi, M

    S. Faridi, M. Farrokhi D. G., R. Ghorbani, and A. Yazdan Pour. Cellular resolutions of monomial ideals and their A rtinian reductions. J. Pure Appl. Algebra , 228(6):Paper No. 107608, 28, 2024

    work page 2024

  8. [8]

    Grayson and Michael E

    Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/

Show all 21 references
  1. [9]

    Gr\"unbaum and G

    B. Gr\"unbaum and G. C. Shephard. Convex polytopes. Bull. London Math. Soc. , 1:257--300, 1969

    1969

  2. [10]

    Combinatorial algebraic topology , volume 21 of Algorithms and Computation in Mathematics

    Dmitry Kozlov. Combinatorial algebraic topology , volume 21 of Algorithms and Computation in Mathematics . Springer, Berlin, 2008

    2008

  3. [11]

    Morse resolutions of monomial ideals and betti splittings

    Josep \`A lvarez Montaner, Mar \' a Luc \' a Aparicio Garc \' a, and Amir Mafi. Morse resolutions of monomial ideals and betti splittings. arXiv preprint arXiv:2502.02122 , 2025

    2025 arXiv

  4. [12]

    Combinatorial commutative algebra , volume 227 of Graduate Texts in Mathematics

    Ezra Miller and Bernd Sturmfels. Combinatorial commutative algebra , volume 227 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2005

    2005

  5. [13]

    James R. Munkres. Elements of algebraic topology . Addison-Wesley Publishing Company, Menlo Park, CA, 1984

    1984

  6. [14]

    Graded syzygies , volume 14 of Algebra and Applications

    Irena Peeva. Graded syzygies , volume 14 of Algebra and Applications . Springer-Verlag London, Ltd., London, 2011

    2011

  7. [15]

    Linear syzygies of stanley-reisner ideals

    V Reiner and V Welker. Linear syzygies of stanley-reisner ideals. Mathematica Scandinavica , 89(1):117–132, Sep. 2001

    2001

  8. [16]

    Richard P. Stanley. The number of faces of a simplicial convex polytope. Adv. in Math. , 35(3):236--238, 1980

    1980

  9. [17]

    Ideals generated by monomials in an R-sequence

    Diana Kahn Taylor. Ideals generated by monomials in an R-sequence . ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)--The University of Chicago

    1966

  10. [18]

    S ageMath, the S age M athematics S oftware S ystem ( V ersion 9.2) , 2020

    The Sage Developers . S ageMath, the S age M athematics S oftware S ystem ( V ersion 9.2) , 2020. https://www.sagemath.org

    2020

  11. [19]

    Minimal free resolutions that are not supported by a CW -complex

    Mauricio Velasco. Minimal free resolutions that are not supported by a CW -complex. J. Algebra , 319(1):102--114, 2008

    2008

  12. [20]

    G\"unter M. Ziegler. Lectures on polytopes , volume 152 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995

    1995

  13. [21]

    G\"unter M. Ziegler. Convex polytopes: extremal constructions and f -vector shapes. In Geometric combinatorics , volume 13 of IAS/Park City Math. Ser. , pages 617--691. Amer. Math. Soc., Providence, RI, 2007

    2007

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.