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Sheaf homology of Orlik–Solomon algebras on a geometric lattice concentrates in top degree and splits as a convolution of local OS groups with filter-complement poset homology.

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load-bearing objection Clean, correctly derived computation of OS-sheaf homology with a usable convolution decomposition and Specht formulas for uniforms.

arxiv 2607.08494 v1 pith:UPSQYTBW submitted 2026-07-09 math.CO

Orlik--Solomon sheaf homology of geometric lattices

Ye Liu This is my paper
classification math.CO MSC 55N3505B35
keywords geometric latticeOrlik–Solomon algebrasheaf homologygeometric semilatticeNBC basesuniform matroidsymmetric group representations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper attaches to every finite geometric lattice a sheaf whose stalk at a flat is the Orlik–Solomon algebra of the principal ideal below that flat. It then computes the homology of the lattice (minus its bottom element) with coefficients in each graded piece of this sheaf. The main theorem states that the homology vanishes except in the single top degree, and that the top group decomposes as a direct sum, over all flats of a fixed rank, of the corresponding local Orlik–Solomon group tensored with the top reduced homology of the order complex of the complementary filter. Both factors are free abelian, with ranks given by Möbius numbers of the lattice. Explicit bases are constructed from no-broken-circuit sets and from Ziegler’s β-NBC bases. When the lattice is the flat lattice of a uniform matroid, the resulting homology groups become concrete representations of the symmetric group, expressed via induction, Littlewood–Richardson coefficients, and Kronecker products of hook Specht modules. The calculation therefore supplies both a new structural description of Orlik–Solomon data and a source of interesting symmetric-group representations.

Core claim

For a geometric lattice L of rank ℓ and p > 0, the sheaf homology H_i(P; OS^p) of P = L \ {0̂} vanishes for all i ≠ ℓ−1, while the top group is free abelian and isomorphic to the direct sum over z of rank p of OS^p(L ≤ z) tensor the reduced top homology of the order complex of the principal-filter complement Q_z. The ranks of the two factors are |μ(0̂,z)| and the absolute value of the corresponding Möbius sum over the filter, respectively.

What carries the argument

The Orlik–Solomon sheaf OS^•, whose stalk at a flat x is the Orlik–Solomon algebra of the principal ideal L ≤ x and whose restriction maps are the natural surjections that kill generators outside that ideal; the Brieskorn decomposition of its graded pieces then reduces the sheaf homology to ordinary poset homology of the complementary geometric semilattices.

Load-bearing premise

The argument relies on every geometric semilattice obtained by deleting a principal filter being shellable, so that its order complex is a wedge of spheres of a single dimension; without that shellability the freeness and concentration claims would not follow from the paper’s reasoning.

What would settle it

Compute the sheaf chain complex T_*(P; OS^p) for a concrete geometric lattice of small rank (for example the Boolean lattice of rank 3 or the lattice of U_{3,5}) and check whether its homology is zero outside degree ℓ−1 and whether the free rank of the top group equals the Möbius formula of Theorem 3.1.

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Editorial analysis

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Referee Report

0 major / 4 minor

Summary. The paper defines the Orlik–Solomon sheaf OS• on a finite geometric lattice L of rank ℓ by assigning to each flat x the OS algebra of the principal ideal L≤x, with structure maps the natural projections that kill atom generators not below the lower flat. For P = L \ {0̂} and p > 0 it proves that the sheaf homology H_i(P; OS^p) vanishes except in degree i = ℓ-1, where it decomposes as ⊕_{z∈L_p} OS^p(L≤z) ⊗ ḔH_{ℓ-2}(ΔQ_z; ℤ) with Q_z = {x ∈ P | z ≰ x}. Both factors are free abelian of ranks given by Möbius numbers, and an explicit basis is constructed from NBC sets and Ziegler’s β-NBC cycles. An Euler-characteristic formula and representation-theoretic specializations for Boolean lattices, rank-2 lattices, and uniform matroids U_{r,n} are derived as corollaries.

Significance. The result cleanly extends the Everitt–Turner program of sheaf homology on geometric lattices from the natural sheaf (and its exterior powers) to the local Orlik–Solomon algebras that encode the cohomology of arrangement complements. The convolution decomposition of Theorem 3.1 is new and gives a uniform source of free abelian groups and of interesting S_n-representations (hooks, Littlewood–Richardson coefficients, Kronecker squares of hooks) for uniform matroids. The argument is self-contained once classical shellability of geometric semilattices is granted, and the basis construction via β-NBC sets makes the homology groups combinatorially explicit. These features make the paper a solid contribution to the combinatorial topology of matroids and arrangements.

minor comments (4)
  1. In the proof of Lemma 3.3 the identification T_*(P; E_z) ≅ C_*(ΔP, ΔQ_z; ℤ) is clear for the chain groups, but a one-sentence verification that the sheaf differential on the top vertex matches the relative boundary when the new top leaves the filter would make the argument fully self-contained.
  2. Section 3.3 invokes Ziegler’s EL-shelling and β-NBC bases without restating the precise edge-labeling; a short parenthetical reminder of the atom order (γ_z first) would help readers who do not have [Zie92] open.
  3. In Example 4.3 the short exact sequence of S_p imes S_q-modules is asserted to split; while the subsequent multiplicity formula is unaffected, a brief remark that the sequence splits because both ends are free (or by complete reducibility over ℂ) would remove any residual doubt.
  4. Typographical: the abstract and introduction use both “Orlik–Solomon” and “Orlik--Solomon”; standardize the dash. Also, “Poincaré” appears with an accent in some places and without in others.

Circularity Check

0 steps flagged

No circularity: the main isomorphism is derived from the OS-sheaf definition, Brieskorn decomposition, relative chains, and classical shellability/Möbius facts.

full rationale

Theorem 3.1 is obtained by writing the degree-p OS sheaf as a direct sum of constant modules B_z tensored with indicator sheaves E_z (via the Brieskorn decomposition of Proposition 2.11(3)), identifying the resulting chain complexes with relative simplicial chains of (ΔP, ΔQ_z) (Lemma 3.3), proving purity of the geometric semilattice I_z by an elementary maximality argument (Lemma 3.4), and invoking the classical CL-shellability of geometric semilattices (Wachs–Walker) together with the Möbius formula for the reduced Euler characteristic (Lemma 3.5). None of these steps defines a quantity in terms of the target homology, fits a parameter to data, or relies on a uniqueness theorem or ansatz from the present author. The background citations (Everitt–Turner for sheaf homology, Orlik–Solomon/Orlik–Terao for OS algebras, Wachs–Walker and Ziegler for shellability and β-NBC bases) are independent external results; the representation-theoretic examples in Section 4 are merely specializations of the same isomorphism. The derivation is therefore self-contained against its stated inputs and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 1 invented entities

The paper is pure combinatorial algebra. It introduces one new mathematical object (the Orlik–Solomon sheaf) and otherwise relies entirely on standard definitions and published theorems about geometric lattices, OS algebras, poset sheaf homology, and shellability. No free parameters or physical postulates appear.

axioms (6)
  • standard math A geometric lattice is a finite ranked lattice that is atomic and semimodular; equivalently the flat lattice of a simple matroid.
    Used throughout as the ambient category; Definition 2.2 and Example 2.3 cite Oxley/Stanley.
  • standard math The Orlik–Solomon algebra of a geometric lattice is free on the NBC basis and its Hilbert series equals the Poincaré polynomial of the lattice.
    Proposition 2.11, cited from Orlik–Terao; supplies freeness and rank of the local OS factors B_z.
  • standard math Sheaf homology of a poset equals the homology of the chain complex T_*(P;F) of chains with coefficients in the sheaf.
    Proposition 2.8, cited from Gabriel–Zisman / Everitt–Turner; the computational engine of the whole paper.
  • standard math Every geometric semilattice is CL-shellable (hence its order complex is a wedge of spheres of top dimension).
    Invoked after Lemma 3.4 via Wachs–Walker Corollary 7.3 and Ziegler’s EL-shelling; needed for freeness and single-degree concentration of ẽH_*(ΔQ_z).
  • standard math For a geometric lattice L and flat x, the complement L\L≥x is a geometric semilattice.
    Proposition 2.7, cited from Wachs–Walker; identifies I_z with a geometric semilattice so shellability applies.
  • standard math Philip Hall’s theorem: μ(0̂,1̂) equals the reduced Euler characteristic of the order complex of the open interval.
    Theorem 2.1; used in Lemma 3.5 to convert the Möbius sum into the top Betti number b_z.
invented entities (1)
  • Orlik–Solomon sheaf OS^• on a geometric lattice independent evidence
    purpose: Supplies the coefficient system whose sheaf homology is the object of the main theorem; local value at x is OS(L≤x) with projection structure maps.
    Definition 2.12 is the paper’s central new construction. It is a purely mathematical object with no external physical prediction, but it is independently studyable once defined.

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read the original abstract

We associate the Orlik--Solomon sheaf with a finite geometric lattice and compute its sheaf homology. We show that this homology concentrates in top degree, admitting a convolution-type decomposition into a principal ideal OS piece tensoring with a principal filter complement poset homology. Applications to uniform matroids provide interesting representations of symmetric groups.

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