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arxiv: 2604.16805 · v1 · submitted 2026-04-18 · 🧮 math.RT · math.CT

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A Non-graded Koszul Duality and Its Applications

A. M. Bouhada
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Pith reviewed 2026-05-10 07:33 UTC · model grok-4.3

classification 🧮 math.RT math.CT
keywords Koszul dualitybounded derived categoriesKoszul algebrasorbit categoriescategory Operverse sheavesGorenstein-projective modules
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The pith

Finite-dimensional Koszul algebras admit derived Koszul dualities on bounded derived categories without finiteness conditions on the dual.

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

The paper proves a graded derived Koszul duality between the bounded derived categories of graded modules over a finite-dimensional Koszul algebra and over its Koszul dual. This equivalence requires no Noetherian or coherence assumptions on the dual. The ungraded bounded derived category is then recovered as the triangulated hull of a differential graded orbit category built from the graded theory. This yields a non-graded Koszul duality that applies directly to modules without grading. The results extend to singular and dg refinements and produce concrete descriptions for blocks of category O and certain perverse sheaves.

Core claim

Let Lambda be a finite-dimensional Koszul algebra with Koszul dual Lambda!. The bounded derived category D^b(Lambda-gmod) is equivalent to D^b(Lambda!-gmod) via a graded derived Koszul duality that holds without Noetherian or coherence assumptions on Lambda!. The ungraded bounded derived category D^b(Lambda-mod) arises as the triangulated hull of the differential graded orbit category associated to the graded duality. This construction yields a genuinely non-graded derived Koszul duality and extends to singular and dg versions, including a stable Koszul duality for Gorenstein-projective modules when Lambda is Iwanaga-Gorenstein.

What carries the argument

the triangulated hull of the differential graded orbit category of the graded derived category

Load-bearing premise

The construction of the ungraded duality via the triangulated hull of the dg orbit category of the graded theory holds for arbitrary finite-dimensional Koszul algebras with no additional Noetherian or coherence assumptions on the dual.

What would settle it

A concrete finite-dimensional Koszul algebra where the triangulated hull of its graded dg orbit category is not equivalent to its ungraded bounded derived category would falsify the reconstruction claim.

read the original abstract

Let \(\Lambda\) be a finite-dimensional Koszul algebra with Koszul dual \(\Lambda^!\). We establish derived Koszul dualities at the level of bounded derived categories, both in the graded setting \(\mathsf{D}^{b}(\Lambda\textup{-gmod})\) and in the ungraded setting \(\mathsf{D}^{b}(\Lambda\textup{-mod})\), without imposing finiteness conditions on \(\Lambda^!\). We first prove a graded derived Koszul duality for every finite-dimensional Koszul algebra, with no Noetherian or coherence assumptions on the Koszul dual. We then show that the bounded derived category \(\mathsf{D}^{b}(\Lambda\textup{-mod})\) can be reconstructed from the graded theory as the triangulated hull of a differential graded orbit category. This yields a genuinely non-graded derived Koszul duality. We further establish singular and dg refinements of these dualities. For Iwanaga--Gorenstein Koszul algebras, this gives a stable Koszul duality for graded Gorenstein-projective modules and their ungraded counterparts, providing a non-graded form of the Bernstein--Gel'fand--Gel'fand correspondence. As applications, we obtain new descriptions of the bounded derived categories \(\mathsf{D}^{b}(\mathcal{O}_{\lambda})\) for all integral blocks of category \(\mathcal{O}\), including singular blocks, as well as analogous dualities for certain categories of perverse sheaves arising in geometric representation theory. Finally, we formulate conjectural descriptions of bounded derived and singularity categories of finite-dimensional graded algebras in terms of dg orbit categories.

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 / 1 minor

Summary. The paper claims to establish derived Koszul dualities at the level of bounded derived categories for any finite-dimensional Koszul algebra Λ with Koszul dual Λ!, both in the graded setting D^b(Λ-gmod) and in the ungraded setting D^b(Λ-mod), without imposing finiteness conditions on Λ!. The ungraded duality is obtained by reconstructing D^b(Λ-mod) as the triangulated hull of a dg orbit category built from the graded duality. The work also provides singular and dg refinements (including a stable Koszul duality for Gorenstein-projective modules when Λ is Iwanaga-Gorenstein), applies the results to new descriptions of D^b(O_λ) for all integral blocks of category O (including singular blocks) and to certain categories of perverse sheaves, and formulates conjectures relating bounded derived and singularity categories of graded algebras to dg orbit categories.

Significance. If the central reconstruction of the ungraded derived category via the triangulated hull holds without additional coherence or Noetherian hypotheses, the result would meaningfully extend classical Koszul duality by removing finiteness restrictions on the dual and supplying a genuinely non-graded version. The applications to category O and geometric representation theory, together with the singular/dg refinements, would provide concrete new tools for describing derived categories in representation theory and algebraic geometry.

major comments (1)
  1. [paragraph after graded duality theorem] The paragraph following the graded duality theorem (and the subsequent development of the ungraded duality): the claim that D^b(Λ-mod) is recovered exactly as the triangulated hull of the dg orbit category constructed from D^b(Λ-gmod) ≃ D^b(Λ!-gmod) is asserted for arbitrary finite-dimensional Koszul Λ with no Noetherian or coherence assumptions on Λ!. When Λ! is infinite-dimensional and non-coherent, the dg Hom complexes in the orbit category can involve infinite projective resolutions or non-finitely-generated objects; it is not immediately clear from the argument that the triangulated hull nevertheless contains all objects of D^b(Λ-mod) and yields an equivalence. An explicit check or additional argument addressing this case is needed to support the load-bearing reconstruction step.
minor comments (1)
  1. Notation for the dg orbit category and its triangulated hull could be introduced with a short diagram or explicit universal property to improve readability for readers less familiar with dg enhancements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about the reconstruction of the ungraded derived category. We address the concern directly below and will revise the paper to include an explicit verification for the case of infinite-dimensional non-coherent Koszul duals.

read point-by-point responses

  1. Referee: The paragraph following the graded duality theorem (and the subsequent development of the ungraded duality): the claim that D^b(Λ-mod) is recovered exactly as the triangulated hull of the dg orbit category constructed from D^b(Λ-gmod) ≃ D^b(Λ!-gmod) is asserted for arbitrary finite-dimensional Koszul Λ with no Noetherian or coherence assumptions on Λ!. When Λ! is infinite-dimensional and non-coherent, the dg Hom complexes in the orbit category can involve infinite projective resolutions or non-finitely-generated objects; it is not immediately clear from the argument that the triangulated hull nevertheless contains all objects of D^b(Λ-mod) and yields an equivalence. An explicit check or additional argument addressing this case is needed to support the load-bearing reconstruction step.

    Authors: We appreciate the referee drawing attention to this subtlety in the ungraded reconstruction. The argument in the manuscript establishes the graded equivalence D^b(Λ-gmod) ≃ D^b(Λ!-gmod) using only finite-dimensionality of Λ together with the Koszul property (no coherence or Noetherian hypotheses on Λ! are used). The dg orbit category is obtained by imposing the grading shift as a dg automorphism, and the triangulated hull is identified with D^b(Λ-mod) via the universal property of hulls and the fact that every ungraded complex arises by forgetting the grading on a suitable graded complex whose homology is controlled by the Koszul duality. While the boundedness of the derived categories ensures that all objects remain within the hull even when dg Hom-spaces involve infinite resolutions, we agree that the non-coherent case was not isolated for separate verification. In the revised version we will insert a new lemma immediately after the main ungraded duality theorem. The lemma will explicitly construct the necessary cones and direct summands in the hull for an arbitrary object of D^b(Λ-mod) when Λ! is infinite-dimensional, confirming that no additional objects are generated and that the equivalence holds without coherence assumptions. A brief model example with an infinite-dimensional Koszul dual will also be included for illustration. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations rely on standard triangulated-category constructions and prior Koszul duality without reduction to inputs.

full rationale

The paper's central claims—graded derived Koszul duality for finite-dimensional Koszul algebras and reconstruction of D^b(Λ-mod) as the triangulated hull of the dg orbit category of the graded duality—are presented as new results built from standard operations on dg and triangulated categories. No quoted steps equate a prediction or theorem to its own inputs by definition, fitted parameters, or load-bearing self-citations. The abstract and described proofs invoke established Koszul duality theory as external support rather than circularly re-deriving it, and the ungraded reconstruction is explicitly constructed rather than presupposed. This is the typical self-contained case for a pure-mathematics extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard axioms of triangulated and derived categories together with the definition of Koszul algebras; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Bounded derived categories of modules over finite-dimensional algebras form triangulated categories with the usual shift and cone operations.
    Invoked when forming the triangulated hull of the dg orbit category.
  • domain assumption A finite-dimensional algebra is Koszul if its Ext algebra satisfies the standard quadratic duality relations.
    Central hypothesis on Λ throughout the dualities.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Koszul Duality for Quadratic Monomial Algebras

    math.RT 2026-04 unverdicted novelty 6.0

    Quadratic monomial algebras have Koszul duals that are coherent with finitely presented modules coinciding with perfect ones, enabling explicit triangulated equivalences and nonstandard t-structures in the graded and ...

Reference graph

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