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arxiv: 2309.17416 · v2 · submitted 2023-09-29 · 🧮 math.AC · math.AG· math.CO

A Uniform Identification of Stable Sheaf Cohomology

Luca Fiorindo , Ethan Reed , Shahriyar Roshan Zamir , Hongmiao Yu This is my paper

Pith reviewed 2026-05-24 06:35 UTC · model grok-4.3

classification 🧮 math.AC math.AGmath.CO
keywords stable sheaf cohomologyarithmetic complexesSchur functorscotangent sheafprojective spaceinteger-valued polynomialsisomorphismflag varieties
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The pith

Generalizing arithmetic complexes over integer-valued polynomials proves a conjectured isomorphism that unifies stable sheaf cohomology identifications for hook and two-column Schur functors.

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

The paper generalizes certain arithmetic complexes from earlier work to the ring of integer-valued polynomials. It establishes an isomorphism between these complexes that matches a prior conjecture. This unification makes the identification of stable sheaf cohomology for hook and two-column partition Schur functors on the cotangent sheaf of projective space consistent across the complexes. The same results hold when projective space is defined over the integers. A reader would care because the isomorphism provides a single framework for these cohomology calculations instead of separate case-by-case arguments.

Core claim

Defined over the ring of integer valued polynomials, we prove an isomorphism of these complexes as conjectured by Gao, Raicu, and VandeBogert. In particular, this shows that a previously made identification between the stable sheaf cohomology of hook and two column partition Schur functors applied to the cotangent sheaf of projective space can be made to be uniform with respect to these complexes. These results are extended to the projective space defined over the integers.

What carries the argument

The arithmetic complexes generalized over the ring of integer-valued polynomials, whose proved isomorphism unifies the stable sheaf cohomology identifications for the relevant Schur functors.

If this is right

  • The stable sheaf cohomology identification for hook and two-column Schur functors becomes uniform rather than case-specific.
  • The isomorphism and resulting identifications extend directly to projective space defined over the integers.
  • Computations of stable sheaf cohomology on flag varieties can now rely on a single pair of isomorphic complexes instead of separate constructions.

Where Pith is reading between the lines

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

  • The same generalization technique might produce uniform identifications for Schur functors corresponding to other partitions.
  • Similar arithmetic complexes could be defined over other polynomial rings to handle cohomology on additional varieties.
  • The integer-valued polynomial setting may reveal integrality properties of the cohomology groups that were not visible before.

Load-bearing premise

The arithmetic complexes from prior work can be extended over the ring of integer-valued polynomials so that the conjectured isomorphism between them holds.

What would settle it

An explicit computation showing that the two generalized complexes over the ring of integer-valued polynomials are not isomorphic in some degree would falsify the central claim.

Figures

Figures reproduced from arXiv: 2309.17416 by Ethan Reed, Hongmiao Yu, Luca Fiorindo, Shahriyar Roshan Zamir.

Figure 1
Figure 1. Figure 1: Undirected Weighted Path Graph with Vertex Weights w where wi ’s denote weights on the vertices. Given d, t P N such that 1 ď t ď d ` 1 for any weak composition of d ´ t ` 1, that is non-negative integers pλt, . . . , λ1q such that řt i“1 λi “ d ´ t ` 1, there is a unique decomposition of the above path graph into t disjoint intervals Iλi of length λi . If λi “ 0 then Iλi consists of a single vertex. The w… view at source ↗
Figure 2
Figure 2. Figure 2: A decomposition for d “ 6, t “ 4, and pλ4, λ3, λ2, λ1q “ p0, 2, 0, 1q. Definition 2.1. Define the arithmetic complex, C‚pwq “ C‚pw0, . . . , wdq, as fol￾lows: for each k P Z, Ckpwq :“ à řt i“1 λi“k R ¨ fpλt,...,λ1q – R ‘p d kq , where t “ d´k`1 ě 1, and fpλt,...,λ1q corresponds to a basis element of Ckpwq. Note the basis elements of Ckpwq are in bijective correspondence with decompositions of the path grap… view at source ↗
read the original abstract

This paper considers generalizations of certain arithmetic complexes appearing in the work of Raicu and VandeBogert in connection with the study of stable sheaf cohomology on flag varieties. Defined over the ring of integer valued polynomials, we prove an isomorphism of these complexes as conjectured by Gao, Raicu, and VandeBogert. In particular, this shows that a previously made identification between the stable sheaf cohomology of hook and two column partition Schur functors applied to the cotangent sheaf of projective space can be made to be uniform with respect to these complexes. These results are extended to the projective space defined over the integers.

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

1 major / 0 minor

Summary. The paper generalizes arithmetic complexes appearing in Raicu-VandeBogert to the ring of integer-valued polynomials, proves the isomorphism of these complexes conjectured by Gao-Raicu-VandeBogert, and thereby obtains a uniform identification of the stable sheaf cohomology of hook and two-column partition Schur functors on the cotangent sheaf of projective space. The results are extended to projective space over the integers.

Significance. If the claimed isomorphism holds, the work supplies a uniform, base-ring-independent identification that removes case distinctions between previously treated situations and extends the setting to Z. This strengthens the link between stable cohomology computations and arithmetic complexes, and the integer-valued polynomial generalization may enable new integral results.

major comments (1)
  1. [Abstract] The abstract asserts that an isomorphism is proved, yet supplies no derivation steps, lemmas, or verification details. Without the explicit construction of the generalized complexes or the chain maps realizing the isomorphism, it is impossible to check whether the argument is free of post-hoc choices or hidden base-ring assumptions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts that an isomorphism is proved, yet supplies no derivation steps, lemmas, or verification details. Without the explicit construction of the generalized complexes or the chain maps realizing the isomorphism, it is impossible to check whether the argument is free of post-hoc choices or hidden base-ring assumptions.

    Authors: Abstracts are concise summaries by design and do not contain full derivations. The manuscript supplies the explicit construction of the generalized arithmetic complexes over the ring of integer-valued polynomials in Section 2, together with the chain maps and the proof of the conjectured isomorphism in Section 3 (Theorem 3.1). These constructions are formulated to be independent of the base ring from the outset, extending the earlier work of Raicu-VandeBogert without introducing post-hoc choices or hidden assumptions on the base. revision: no

Circularity Check

0 steps flagged

No circularity: proof of external conjecture

full rationale

The paper's central claim is a proof of an isomorphism of arithmetic complexes conjectured by Gao, Raicu, and VandeBogert (distinct from the current authors). It generalizes prior complexes over the ring of integer-valued polynomials and extends results to projective space over Z. No load-bearing steps reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains; the derivation is presented as an independent proof against an external conjecture. This matches the default expectation of a self-contained mathematical result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the result rests on the definitions of the arithmetic complexes and the ring of integer-valued polynomials from prior work.

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Reference graph

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