arxiv: 2604.14384 · v2 · submitted 2026-04-15 · 🧮 math.AG · math.AC

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Minimal resolutions of toric substacks by line bundles

Zengrui Han

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Pith reviewed 2026-05-10 11:48 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords toric stacksminimal resolutionsline bundleshomological perturbation lemmaMoore-Penrose inversescellular resolutionsstructure sheavesdeformation retracts

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The pith

Minimal resolutions of pushforwards of structure sheaves of toric substacks by line bundles arise as strong deformation retracts of cellular resolutions, with canonically described combinatorial differentials.

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

The paper constructs minimal resolutions for the pushforwards of structure sheaves of toric substacks of smooth toric stacks. These resolutions are realized as strong deformation retracts of cellular resolutions. The homological perturbation lemma produces the retract while Moore-Penrose inverses determine the differentials in a combinatorial way. A sympathetic reader would value this because it supplies an explicit construction for minimal resolutions in toric geometry, where direct methods are often intractable.

Core claim

We construct minimal resolutions of pushforwards of structure sheaves of toric substacks of smooth toric stacks by line bundles as strong deformation retracts of cellular resolutions. We also provide a canonical and combinatorial description of the differentials of such minimal resolutions. The homological perturbation lemma and the Moore-Penrose inverses are the key ingredients.

What carries the argument

Strong deformation retracts of cellular resolutions obtained via the homological perturbation lemma, with differentials fixed by Moore-Penrose inverses.

Load-bearing premise

That the homological perturbation lemma applies directly to produce strong deformation retracts in the category of sheaves on toric substacks and that Moore-Penrose inverses yield a canonical combinatorial description of the resulting differentials.

What would settle it

Explicit computation of both the constructed retract and a direct minimal resolution for a concrete toric substack, such as the zero section in a weighted projective stack, to check whether they agree as complexes.

Figures

Figures reproduced from arXiv: 2604.14384 by Zengrui Han.

**Figure 1.** Figure 1: ∂ (black) and the Moore-Penrose inverse ∂ + (red) where the differentials d1 and d0 are given by d1 =   −x 0 0 1 −z 1 0 0 0 −z 1 0 0 0 −z 1 −y 1 0 0 0 −y 1 0 0 0 −y 1   , d0 =   y − z x 0 −y −x 0 z 0 −y 1 0 z −1 0 0 0 −y 1 0 z −1   The Bondal-Thomsen collection is easily seen to be BT = {O, O(−1), O(−2), O(−3), O(−4)} and the stratification is given by S0 = {V1}, S−1 = {V3, E1, E4, … view at source ↗

**Figure 2.** Figure 2: Minimal resolution Eventually we get the minimal resolution 0 → O(−4)   y − z 2x − (y+z) 3 4   −−−−−−−−−−−→ O(−3) ⊕ O(−1) −2x + (y+z) 3 4 y − z −−−−−−−−−−−−−−−−−→ O → 0 as shown in [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Figure 3: Homotopies from a perfect Morse matching yields the more familiar form of the minimal resolution 0 → O(−4) [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

read the original abstract

We construct minimal resolutions of pushforwards of structure sheaves of toric substacks of smooth toric stacks by line bundles as strong deformation retracts of cellular resolutions constructed by Hanlon, Hicks and Lazarev. We also provide a canonical and combinatorial description of the differentials of such minimal resolutions. Two key ingredients are the homological perturbation lemma and the Moore-Penrose inverses.

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

The paper gives an explicit construction of minimal line bundle resolutions for toric substack pushforwards as strong deformation retracts of the Hanlon-Hicks-Lazarev cellular resolutions, with combinatorial differentials via HPL and Moore-Penrose.

read the letter

The paper takes the cellular resolutions from Hanlon, Hicks and Lazarev and applies the homological perturbation lemma to produce minimal resolutions of i_* O_X for a toric substack X inside a smooth toric stack. It also supplies a canonical combinatorial description of the differentials in the resulting minimal complex using Moore-Penrose inverses on the graded pieces. This is the concrete new piece: a direct minimalization that stays combinatorial and works with line bundles. It builds straightforwardly on the prior cellular work without adding extra machinery. The construction looks usable for explicit calculations once the details are checked. The soft spot is the application of the homological perturbation lemma in the sheaf category on the stack. The lemma requires the homotopy to satisfy the usual side conditions and the perturbation series to converge after pushforward, and the toric fan data must remain compatible across strata. The Moore-Penrose step further needs an inner product on the graded pieces that is toric-invariant so the new differentials are still morphisms of sheaves. The abstract invokes standard tools, but the paper must verify these conditions hold without extra cohomology or non-functorial choices interfering. If those checks are present and correct, the central claim holds up. This is for people working on resolutions in toric geometry and algebraic stacks who already know the cellular constructions and want minimal versions for computations. A reader in that niche gets a usable explicit description. It deserves a serious referee to confirm the perturbation works cleanly in this setting and that the combinatorial differentials descend properly.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs minimal resolutions of the pushforwards of structure sheaves of toric substacks of smooth toric stacks, realized by line bundles, as strong deformation retracts of the cellular resolutions of Hanlon, Hicks and Lazarev. The construction relies on the homological perturbation lemma together with Moore-Penrose inverses, and supplies a canonical combinatorial description of the resulting differentials.

Significance. If the central construction is rigorously established, the work would furnish an explicit, combinatorial route to minimal resolutions in the toric-stack setting, extending the cellular resolutions of Hanlon-Hicks-Lazarev. This could aid explicit computations of Ext groups and other invariants for toric substacks. The reliance on standard homological-algebra tools is a methodological strength provided the side conditions are verified in the sheaf category.

major comments (2)

[Application of the homological perturbation lemma (proof of the main theorem)] The manuscript invokes the homological perturbation lemma to produce strong deformation retracts after pushforward to the toric stack, yet does not supply a detailed check that the chain homotopy h satisfies the side conditions (dh + hd = id − projection) and that the perturbation series converges (typically via nilpotency of hδ in each degree) once restricted to toric strata. This verification is load-bearing for the claim that the minimal resolutions are strong deformation retracts of the cellular ones.
[Combinatorial description of differentials] The combinatorial description of the differentials via Moore-Penrose inverses presupposes a canonical inner product on the graded pieces that is preserved by the toric action and yields sheaf morphisms. The text does not explicitly confirm that this inner product is functorial with respect to the stack structure or that no higher cohomology on the strata interferes with the retract property.

minor comments (1)

The abstract should explicitly list the standing assumptions on the toric stacks (smoothness, etc.) and the precise category of sheaves in which the resolutions are taken.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The suggestions identify places where additional explicit verification will strengthen the exposition without altering the core results. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: The manuscript invokes the homological perturbation lemma to produce strong deformation retracts after pushforward to the toric stack, yet does not supply a detailed check that the chain homotopy h satisfies the side conditions (dh + hd = id − projection) and that the perturbation series converges (typically via nilpotency of hδ in each degree) once restricted to toric strata. This verification is load-bearing for the claim that the minimal resolutions are strong deformation retracts of the cellular ones.

Authors: We agree that an explicit verification of the side conditions strengthens the argument. In the revised manuscript we will insert a short lemma (new Lemma 3.4) immediately after the application of the homological perturbation lemma. The lemma records that the homotopy h, defined via the Moore-Penrose inverse on the graded pieces of the Hanlon–Hicks–Lazarev cellular resolution, satisfies dh + hd = id − π after pushforward because the original cellular complex is a resolution by toric-equivariant line bundles and pushforward along the closed immersion of the substack preserves exactness on each stratum. For convergence, we observe that hδ is strictly upper-triangular with respect to the homological grading and that each toric stratum is affine; consequently the operator is nilpotent of index at most the dimension of the stack plus one. The resulting finite sum therefore converges in the sheaf category. These facts are already implicit in the construction but will now be stated explicitly. revision: yes
Referee: The combinatorial description of the differentials via Moore-Penrose inverses presupposes a canonical inner product on the graded pieces that is preserved by the toric action and yields sheaf morphisms. The text does not explicitly confirm that this inner product is functorial with respect to the stack structure or that no higher cohomology on the strata interferes with the retract property.

Authors: The inner product is the standard toric-equivariant Hermitian product on the fibers of the line bundles appearing in the cellular resolution of Hanlon–Hicks–Lazarev; equivariance is built into their construction, so the product is preserved by the torus action and descends to a morphism of sheaves on the stack. Functoriality with respect to the stack structure follows because the pushforward functor is exact and preserves the equivariant morphisms. Higher cohomology on the strata does not interfere: each stratum is an affine toric variety, the sheaves in the resolution are coherent and locally free, and therefore H^i(stratum, sheaf) = 0 for i > 0. We will add a single clarifying sentence in the paragraph introducing the Moore–Penrose construction to record these two points. revision: partial

Circularity Check

0 steps flagged

No circularity: construction applies external lemmas to prior cellular resolutions

full rationale

The paper constructs minimal resolutions of pushforwards of structure sheaves as strong deformation retracts of Hanlon-Hicks-Lazarev cellular resolutions by invoking the homological perturbation lemma and Moore-Penrose inverses. These are standard external tools and cited prior work by different authors; the derivation does not redefine inputs in terms of outputs, fit parameters to subsets then rename them as predictions, or rely on self-citation chains for load-bearing steps. The combinatorial description of differentials follows from the lemmas applied to the fan data without reducing the central claim to a tautology or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Relies on standard homological algebra tools and domain assumptions from toric geometry; no free parameters or invented entities are introduced in the abstract.

axioms (3)

standard math Applicability of the homological perturbation lemma to produce strong deformation retracts for the relevant sheaf categories
Invoked as a key ingredient for the construction.
domain assumption Existence and combinatorial utility of Moore-Penrose inverses for the differentials in this setting
Used to obtain the canonical description of differentials.
domain assumption Standard properties of smooth toric stacks and their substacks
Background from prior toric geometry literature.

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discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 8 canonical work pages

[1]

Steven Amelotte and Benjamin Briggs, Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings.Transactions of the American Mathematical Society, Series B11.25 (2024): 826-862

2024
[2]

Amelotte and B

Steven Amelotte and Benjamin Briggs, Homotopy types of moment-angle complexes associated to almost linear resolutions. arXiv preprint arXiv:2506.15457 (2025)

work page arXiv 2025
[3]

Smith, Virtual resolutions for a product of projective spaces,Algebraic Geometry

Christine Berkesch, Daniel Erman and Gregory G. Smith, Virtual resolutions for a product of projective spaces,Algebraic Geometry. 7 (2020), No. 4, 460-481

2020
[4]

Smith, and Jay Yang, Cellular free resolutions for normalizations of toric ideals

Christine Berkesch, Lauren Cranton Heller, Gregory G. Smith, and Jay Yang, Cellular free resolutions for normalizations of toric ideals. arXiv preprint arXiv:2512.17871 (2025)

work page arXiv 2025
[5]

arXiv preprint arXiv:2110.10705 (2021)

Juliette Bruce, Lauren Cranton Heller, and Mahrud Sayrafi, Characterizing Multi- graded Regularity and Virtual Resolutions on Products of Projective Spaces. arXiv preprint arXiv:2110.10705 (2021)

work page arXiv 2021
[6]

Alexey Bondal, Derived categories of toric varieties.Convex and Algebraic geometry, Oberwolfach conference reports, EMS Publishing House. Vol. 3. 2006

2006
[7]

22 ZENGRUI HAN

Lev Borisov and Zengrui Han, On hypergeometric duality conjecture.Advances in Mathematics442 (2024): 109582. 22 ZENGRUI HAN

2024
[8]

arXiv preprint arXiv:2509.11077 (2025)

Lev Borisov and Zengrui Han, Hanlon-Hicks-Lazarev resolution revisited. arXiv preprint arXiv:2509.11077 (2025)

work page arXiv 2025
[9]

Brown and Daniel Erman, A short proof of the Hanlon-Hicks-Lazarev Theorem.Forum of Mathematics, Sigma

Michael K. Brown and Daniel Erman, A short proof of the Hanlon-Hicks-Lazarev Theorem.Forum of Mathematics, Sigma. Vol. 12. Cambridge University Press, 2024

2024
[10]

arXiv preprint arXiv:2404.10165 (2024)

Jun Chen, Yuming Liu, and Guodong Zhou, Algebraic Morse theory via homological perturbation lemma. arXiv preprint arXiv:2404.10165 (2024)

work page arXiv 2024
[11]

arXiv preprint math/0403266 (2004)

Marius Crainic, On the perturbation lemma, and deformations. arXiv preprint math/0403266 (2004)

work page arXiv 2004
[12]

Journal of the European Mathematical Society, to appear

John Eagon, Ezra Miller, and Erika Ordog, Minimal resolutions of monomial ideals. Journal of the European Mathematical Society, to appear
[13]

Bohan Fang, Chiu-Chu Melissa Liu, David Treumann and Eric Zaslow, A categorifi- cation of Morelli’s theorem.Inventiones mathematicae186.1 (2011): 79-114

2011
[14]

arXiv preprint arXiv:2411.17873 (2024)

David Favero and Mykola Sapronov, Line Bundle Resolutions via the Coherent- Constructible Correspondence. arXiv preprint arXiv:2411.17873 (2024)

work page arXiv 2024
[15]

William Fulton and Bernd Sturmfels, Intersection theory on toric varieties.Topology 36.2 (1997): 335-353

1997
[16]

Advances in Mathematics480 (2025): 110502

Zengrui Han, Central charges in local mirror symmetry via hypergeometric duality. Advances in Mathematics480 (2025): 110502

2025
[17]

Andrew Hanlon, Jeff Hicks, and Oleg Lazarev, Resolutions of toric subvarieties by line bundles and applications.Forum of Mathematics, Pi. Vol. 12. Cambridge University Press, 2024

2024
[18]

Yupeng Li, Ezra Miller, and Erika Ordog, Minimal resolutions of lattice ideals.Jour- nal of Pure and Applied Algebra229.3 (2025): 107901

2025
[19]

arXiv preprint arXiv:1909.08577 (2019)

Alexandre Tchernev, Dynamical systems on chain complexes and canonical minimal resolutions. arXiv preprint arXiv:1909.08577 (2019). Department of Mathematics, University of Maryland, College Park, MD 20742 Email address:zhan223@umd.edu

work page arXiv 1909