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The paper claims a bounded Koszul duality for all infinite-dimensional Koszul algebras, yielding a BGG-type description of coherent sheaf derived categories on Koszul projective schemes.

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load-bearing objection Plausible program, but Theorem 3.2 relies on an unproved identification and Theorem 4.2 uses an unjustified equality; as written the proofs do not support the claims, though the gaps may be repairable. the 3 major comments →

arxiv 2607.14299 v2 pith:IYXH6OX6 submitted 2026-07-15 math.AG math.CTmath.RT

Koszul Duality for Coherent Sheaves

A. M. Bouhada This is my paper
classification math.AG math.CTmath.RT MSC 16E3018E3016G2016S3714F08
keywords Koszul dualitybounded derived categorycoherent sheavesBGG correspondencetails categoryalmost linear modulesalmost colinear modulessingularity category
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.

This paper tries to establish a bounded version of Koszul duality that works for infinite-dimensional Koszul algebras — graded algebras with especially regular homological behaviour — not just finite-dimensional ones. The central claim is that the usual Koszul duality functors restrict to mutually inverse equivalences between the bounded derived category of 'almost colinear' modules over the Koszul dual algebra and the bounded derived category of 'almost linear' modules over the original algebra. When the original algebra is a Koszul quotient of a polynomial ring, this yields a description of the bounded derived category of coherent sheaves on the associated projective scheme as a Verdier quotient, and hence a BGG-type correspondence for projective schemes beyond projective space. A sympathetic reader should care because it connects the geometry of coherent sheaves to purely algebraic categories over a usually noncommutative dual algebra, and it specializes to the classical BGG correspondence for projective space.

Core claim

The paper's central claim is Theorem 3.2: for every infinite-dimensional Koszul algebra Λ, the Koszul duality functors F and G give mutually inverse triangulated equivalences between the bounded derived category of almost colinear modules over the Koszul dual Λ! and the bounded derived category of almost linear modules over Λ. Theorem 4.2 then claims that when Λ is a Koszul quotient of a polynomial algebra and X = Proj(Λ), the bounded derived category of coherent sheaves is the quotient of D^b(Λ!-AclZ) by D^b(Λ!-CopZ) (a Verdier quotient), equivalently the stabilization S(Λ!-AclZ). Under this identification, coherent sheaves correspond to cohomological shifts of colinear Λ!-modules of infini

What carries the argument

The load-bearing machinery is the Koszul duality pair: the Koszul functor K sends a graded module over the Koszul dual Λ! to a linear complex of projectives over Λ, and the coKoszul functor K∨ goes the other way; their derived extensions F and G provide the equivalence. The boundedness is captured by the 'almost linear' and 'almost colinear' conditions: a module is almost linear (respectively almost colinear) if some degree truncation admits a linear projective (respectively colinear injective) resolution. These conditions mark exactly the modules whose Koszul complex has bounded cohomology, which is what lets the unbounded Koszul duality restrict to bounded derived categories.

Load-bearing premise

The proof of Theorem 3.2 relies on the unproved assertion that any module over the Koszul dual whose Koszul complex has bounded cohomology must be almost colinear (i.e., some truncation is colinear); if that fails, the bounded equivalence and the geometric description of coherent sheaves both collapse.

What would settle it

Exhibit a right-bounded graded module over the Koszul dual of an infinite-dimensional Koszul algebra (for instance, over the Koszul dual of the polynomial ring in two variables) whose minimal injective resolution is not colinear in any degree, yet whose Koszul complex has bounded cohomology. Alternatively, find an almost linear graded module over a Koszul quotient of a polynomial algebra that is not finitely generated, which would break the equality used in Theorem 4.2.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Projective schemes defined by Koszul quotients of polynomial rings get a fully algebraic, noncommutative model for their bounded derived category of coherent sheaves.
  • The classical BGG correspondence for projective space is recovered when Λ is a polynomial algebra and Λ! is the exterior algebra.
  • For quadratic monomial and certain absolutely Koszul algebras, the duality descends to singularity categories, giving D^b(cqgr(Λ!)) ≅ D_sg(Λ-FpZ).
  • For Iwanaga–Gorenstein quadratic monomial algebras, D^b(qgr(Λ!)) and D^b(cqgr(Λ!)) are both equivalent to the stable category of Gorenstein-projective modules.
  • The result ties commutative geometry to noncommutative projective geometry by realizing coherent sheaf categories as Verdier quotients of noncommutative derived categories.

Where Pith is reading between the lines

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

  • The identification Λ!-GMod^{-,b} = Λ!-AclZ is the hinge that turns unbounded Koszul duality into the bounded theorem; a natural test is whether a right-bounded non-almost-colinear module can have bounded Koszul cohomology, since that would force the theorem to hold for a strictly smaller category.
  • The geometric application relies on the standard theorem for commutative noetherian Koszul algebras that every finitely generated module is almost linear; the same reasoning would need a new input to generalize to noncommutative or non-noetherian Koszul algebras, where almost linear and finitely generated can diverge.
  • Because Theorem 4.2 expresses D^b(coh(X)) as a stabilization, it opens a route to computing morphism spaces in the derived category of coherent sheaves from the stable category of almost colinear modules over the Koszul dual.
  • The author notes the extension should be formulated in a more general framework without quiver presentations; checking whether the bounded duality survives outside that setting would determine how far the geometric description extends, for example to Grassmannians or Schubert varieties.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims a bounded derived Koszul duality for infinite-dimensional Koszul algebras: Theorem 3.2 asserts mutually inverse triangulated equivalences D^b(Λ!-AclZ) ↔ D^b(Λ-AlZ) for every infinite-dimensional Koszul algebra Λ. It then derives singular counterparts, specializes to quadratic monomial algebras (Theorem 3.6) and to absolutely Koszul algebras with an almost-linearity condition (Theorem 3.13), and applies the results to obtain a BGG-type description D^b(coh(X)) ≅ D^b(Λ!-AclZ)/D^b(Λ!-CopZ) ≅ S(Λ!-AclZ) for X=Proj(Λ) with Λ a Koszul quotient of a polynomial algebra (Theorem 4.2).

Significance. If the main equivalence were established, it would provide a uniform bounded Koszul duality beyond the finite-dimensional setting and a geometric BGG correspondence for projective schemes defined by Koszul quotients. The intended framework is natural and the paper is clearly organized. However, the central theorem is not actually proved: the key identification of modules whose Koszul complex has bounded cohomology with almost-colinear modules is asserted without proof, and Theorem 4.2 relies on an unjustified absolute co-Koszulity claim. The paper also depends heavily on the author's own unpublished preprints for several corollaries. These issues prevent the significance from being realized as stated.

major comments (3)
  1. [§3.1, Theorem 3.2] The proof reduces the theorem to the identification Λ!-GMod^{−,b} = Λ!-AclZ (and dually Λ-GMod^{+,b}=Λ-AlZ). The only cited support, Proposition 3.1, proves one direction: if M is coKoszul then K(M) is exact outside degree 0. It does not prove that H^r(K(M))=0 for r>N implies M_{\le N} is colinear, nor that bounded cohomology of K(M) forces some truncation to be colinear. This converse is exactly the bridge from the unbounded Koszul duality (Theorem 2.1) to the bounded statement. The sentence 'By Proposition 3.1, this is equivalent...' is therefore unsupported. Since Theorem 3.2 is the basis for Corollaries 3.3, 3.4, Theorem 3.13, and Theorem 4.2, this is a load-bearing gap.
  2. [§4.1, Theorem 4.2] The proof asserts that 'The absolute co-Koszulity of Λ! follows from Lemma 4.1 and Proposition 3.12.' But Proposition 3.12 requires Λ to satisfy both condition (i), absolute Koszulity, and condition (ii), almost-linearity of finitely presented modules. Lemma 4.1 supplies only condition (ii) (Avramov–Eisenbud). Absolute Koszulity of a Koszul quotient of a polynomial algebra is not established and is not a consequence of condition (ii). Thus the claim that Λ! is absolutely co-Koszul for every such Λ is unjustified; it is not known to hold in this generality and is not proved here. This also affects the later identification of objects in the Verdier quotient with weakly co-Koszul modules.
  3. [§3.1–3.2, Corollaries 3.4 and 3.9] Corollary 3.4 is dismissed with 'The proof is identical to the proof of [9, Theorem 3.16]', and Corollary 3.9 is stated to follow from [9, Theorem 3.17]. The present setting is infinite-dimensional Koszul algebras, whereas [9] treats the finite-dimensional case. No argument is given that the cited proofs carry over verbatim. Combined with the fact that [9]–[13] are the author's own preprints, the paper's main technical content is not self-contained. At minimum, the relevant arguments should be reproduced or the differences addressed.
minor comments (4)
  1. [§2.2] The notation 'coKoszul' and 'co-Koszul' is used inconsistently; please standardize.
  2. [Proposition 3.1] The first displayed isomorphism for H^n(K(M)) is stated as 'well known' without a reference; given its importance, a citation or derivation is needed.
  3. [Theorem 4.2] The phrase 'the category coh(X) identifies with the category consisting of cohomological shifts of colinear Λ!-modules' is imprecise: coh(X) is abelian, not a category of shifts in a triangulated category; clarify the intended embedding.
  4. [Introduction] The claim D^b(qgr(Λ)) ≃ D^b(coh(X)) by Serre's theorem requires the standing finiteness assumptions, which are not stated at that point in the introduction.

Circularity Check

4 steps flagged

Central bounded Koszul duality is assembled from an unproved identification asserted via Proposition 3.1; main specializations delegate proofs to same-author preprints.

specific steps
  1. self definitional [Section 3.1, proof of Theorem 3.2; identification of Λ!-GMod^{−,b} with Λ!-AclZ]
    "Let M∈Λ!-GMod^{−,b}. By definition, M is right bounded and the complex K(M) has bounded cohomology. Hence there exists an integer N such that H^r(K(M)) = 0 for all r > N. By Proposition 3.1, this is equivalent to saying that the truncation M^{≤N} is colinear."

    Proposition 3.1 proves only the forward direction that a coKoszul module has K(M) exact outside degree 0; it does not prove the bounded-converse criterion 'H^r(K(M))=0 for r>N implies M^{≤N} is colinear.' The proof labels this unproved equivalence as the bridge from the unbounded duality to the bounded one. The equality Λ!-GMod^{−,b}=Λ!-AclZ is therefore not derived; it is asserted, and the bounded equivalence D^b(Λ!-AclZ)↔D^b(Λ-AlZ) reduces to that asserted renaming.

  2. self citation load bearing [Section 3.1, Corollary 3.4 (singular Koszul duality), proof]
    "Proof.The proof is identical to the proof of [9, Theorem 3.16]. Indeed, under the graded derived Koszul duality, the bounded derived category of finite-dimensional graded Λ!-modules is sent to the full subcategory of perfect complexes in D^b(Λ-GMod^{+,b})."

    [9] is Bouhada's own preprint (arXiv:2604.16805) and the paper supplies no proof here; the singular Koszul duality of Corollary 3.4 is asserted to follow by copying that self-cited argument. This is a load-bearing step for the paper's claimed singular duality: without independent verification of [9, Theorem 3.16], the corollary is not derived in this paper.

  3. self citation load bearing [Section 3.2, Lemma 3.5, proof]
    "By [11, Lemme 3.2], the second syzygy of M decomposes as a finite direct sum Ω^2M ∼= ⊕_{j=1}^r Λα_j⟨i_j⟩ for suitable arrows α_j and integers i_j. Moreover, by [11, Proposition 3.2], each module Λα_j is linear."

    [11] is also by the same author and is cited for the two decisive facts (syzygy decomposition into line modules and linearity of those modules) that make every finitely presented module over a quadratic monomial algebra almost linear. This is the key input for Theorem 3.6, so the monomial specialization of the bounded duality is imported from the author's own preprint rather than proved here.

  4. self citation load bearing [Section 2.4, Theorem 2.1]
    "The following theorem is the fundamental result of Koszul duality. It was established in greater generality in [12, 13]; in our setting, it takes the following form. Theorem 2.1 ([12, Theorem 5.16], [13, Theorem 5.7])."

    Both [12] and [13] are preprints by the same author (with coauthors), and the proof of Theorem 3.2 starts with 'By Theorem 2.1'. The unbounded equivalence that serves as the base of the whole derivation chain is therefore accepted on the authority of same-author work; no independent proof or machine-check is offered in this paper for this load-bearing input.

full rationale

The paper is not empirically circular: there is no fitted parameter, and the geometric and monomial applications have independent mathematical content once the general bounded duality is granted. However, as written the central Theorem 3.2 is not derived. Its proof asserts the identification Λ!-GMod^{−,b}=Λ!-AclZ through an 'equivalent' that Proposition 3.1 does not establish, and the surrounding technical infrastructure is repeatedly cited to the same author's preprints ([9], [11], [12], [13]). Corollary 3.4's proof is explicitly 'identical to' [9], and Lemma 3.5 is delegated to [11]. Thus the bounded Koszul duality is, in effect, an assumed renaming plus an inheritance from a same-author citation chain, rather than a first-principles derivation. Score 6 reflects partial circularity: the central claim reduces to these inputs by construction, while the specialized applications still contain independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claims rest on the standard definition of Koszul duality, but the main load-bearing steps are imported from the author's own preprints [9, 11, 12, 13], which are not independently verified in this manuscript. A false equality (Λ-AlZ = Λ-FgZ) in the geometric application is used without proof. No new entities are introduced.

axioms (6)
  • domain assumption Λ is a Koszul algebra presented by a finite quiver with homogeneous ideal I (not necessarily admissible).
    Standing setup throughout; the Koszul dual Λ! is defined only for Koszul algebras.
  • ad hoc to paper The Koszul duality functors F, G of [12, Theorem 5.16] and [13, Theorem 5.7] are quasi-inverse equivalences D↓(Λ!-GMod−) ↔ D↑(Λ-GMod+).
    Relied on as a black box (Theorem 2.1); the cited references are by the same authors and are not reproduced or formally verified here.
  • ad hoc to paper The results of [9] (e.g., Theorem 2.18, Proposition 2.14, Theorem 3.8) are correct and apply in the present setting.
    Used in Section 3 (proofs of Theorem 3.2 and Corollaries 3.3, 3.4) as a black box; [9] is a preprint by the same author.
  • ad hoc to paper For quadratic monomial algebras, [11, Lemme 3.2 and Prop 3.2] hold: the second syzygy decomposes into shifts of linear modules Λα_j.
    Key input for Lemma 3.5; [11] is a preprint by the same author and its statements are not reproduced.
  • ad hoc to paper Lemma 3.10: the linear/colinear strands are described by applying the Koszul functor to cohomology; argued to extend [18] to the present setting.
    The paper says 'the same arguments extend' but gives no proof; this is an unproved assumption in the proof of Proposition 3.12.
  • standard math Lemma 4.1 (Avramov–Eisenbud): every finitely generated graded module over a commutative noetherian Koszul algebra is almost linear.
    Cited to [2]; standard published result used in Theorem 4.2.

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Cite this review

Pith. "Pith review of Koszul Duality for Coherent Sheaves." pith.science (2026). https://pith.science/paper/IYXH6OX6

@misc{pith2026260714299,
  author       = {Pith},
  title        = {Pith review of: Koszul Duality for Coherent Sheaves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYXH6OX6}},
  note         = {Machine review of arXiv:2607.14299}
}
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read the original abstract

We establish a bounded derived Koszul duality for infinite-dimensional Koszul algebras, and we obtain the corresponding singular Koszul duality. We then apply this framework to two classes of Koszul algebras, namely quadratic monomial algebras and absolutely Koszul algebras satisfying an additional homological condition. For these classes, the general duality specializes to particularly well-behaved forms. As an application to algebraic geometry, let \(\Lambda\) be a commutative noetherian Koszul algebra generated in degree \(1\), and let \(X=\operatorname{Proj}(\Lambda)\). We obtain a Koszul-dual description of the bounded derived category \(\mathsf{D}^{b}\!\bigl(\operatorname{coh}(X)\bigr)\). This gives a BGG-type correspondence for projective schemes defined by commutative noetherian Koszul algebras.

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