arxiv: 2604.20177 · v2 · submitted 2026-04-22 · 🧮 math.RT

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Koszul Duality for Quadratic Monomial Algebras

M. Bouhada

Pith reviewed 2026-05-09 23:21 UTC · model grok-4.3

classification 🧮 math.RT

keywords quadratic monomial algebrasKoszul dualityleft coherenceperfect modulestails categorieslinearity defectBGG correspondence

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The pith

For finite-dimensional quadratic monomial algebras, their Koszul duals are left coherent and left co-coherent, with finitely presented modules coinciding with perfect modules.

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

The paper supplies a concrete class of examples for Koszul dualities by examining quadratic monomial algebras. It proves that when Λ is finite-dimensional and quadratic monomial, its Koszul dual Λ! is left coherent as well as left co-coherent. Finitely presented modules over Λ! are precisely the perfect modules, and finitely copresented modules are precisely the coperfect ones. These identifications imply that the associated tails and cotails categories are abelian and hereditary, and they admit explicit structural descriptions. The work further shows that quadratic monomial algebras are absolutely Koszul with global linearity defect at most one, so every finitely presented module possesses rational Poincaré and Hilbert series, and the graded derived, singular, and BGG dualities receive explicit refinements.

Core claim

For a finite-dimensional quadratic monomial algebra Λ, the Koszul dual Λ! is both left coherent and left co-coherent, and finitely presented modules coincide with perfect modules while finitely copresented modules coincide with coperfect modules. As a consequence the tails and cotails categories are abelian and hereditary. Quadratic monomial algebras are absolutely Koszul and have global linearity defect at most one, so every finitely presented module has rational Poincaré and Hilbert series. These facts yield explicit realizations of the graded derived and singular Koszul dualities as well as the graded BGG correspondence, inducing nonstandard t-structures whose hearts are described in the

What carries the argument

The Koszul dual algebra Λ! of a quadratic monomial algebra Λ, together with the identification of finitely presented modules with perfect modules over Λ!.

If this is right

The tails category is abelian and hereditary with an explicit description in terms of linear complexes.
Every finitely presented module over the Koszul dual has a rational Poincaré series and a rational Hilbert series.
The graded BGG correspondence receives an explicit triangulated equivalence realized by linear complexes.
Nonstandard t-structures appear on the derived category whose hearts consist of modules and linear complexes.
The singular Koszul duality is refined to an explicit equivalence in both graded and ungraded settings.

Where Pith is reading between the lines

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

These algebras supply concrete, computable instances in which the hearts of the nonstandard t-structures can be listed by hand.
The rationality of series for all finitely presented modules opens direct calculations of the Grothendieck group and Euler characteristics.
The hereditary property of the tails category implies that every short exact sequence of linear complexes splits in a controlled way.

Load-bearing premise

The framework of Koszul duality, coherence, and perfect modules from the cited prior work applies directly and without change to quadratic monomial algebras.

What would settle it

An explicit computation of the Koszul dual for a specific finite-dimensional quadratic monomial algebra such as k⟨x,y⟩/(xy) that checks whether the dual is left coherent and whether its finitely presented modules equal its perfect modules.

read the original abstract

This paper provides a new class of examples for the Koszul dualities established in~\cite{5}. We study quadratic monomial algebras from the perspective of Koszul duality, with particular emphasis on finitely presented and finitely copresented graded modules over the Koszul dual algebra. For a finite-dimensional quadratic monomial algebra \(\Lambda\), we prove that the Koszul dual \(\Lambda^{!}\) is both left coherent and left co-coherent, and that finitely presented (resp.\ finitely copresented) modules coincide with perfect (resp.\ coperfect) modules. As a consequence, the associated tails and cotails categories are abelian and hereditary, and admit explicit structural descriptions. We further show that quadratic monomial algebras are absolutely Koszul and have global linearity defect at most one. In particular, every finitely presented module has rational Poincar\'e and Hilbert series. Building on these results, we refine the graded derived and singular Koszul dualities, as well as the graded BGG correspondence, by giving explicit realizations of the associated triangulated equivalences. These equivalences induce nonstandard \(t\)-structures whose hearts admit concrete descriptions in terms of linear complexes and modules. We also obtain corresponding refinements in the ungraded setting.

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

This paper works out explicit Koszul dualities and t-structures for quadratic monomial algebras, showing their duals are coherent and fp modules match perfect ones.

read the letter

The main point is that finite-dimensional quadratic monomial algebras give a workable class where the Koszul dual is left coherent and left co-coherent. Finitely presented modules over the dual coincide with perfect modules, so the tails and cotails categories become abelian and hereditary. The paper also shows these algebras are absolutely Koszul with linearity defect at most one, which yields rational Poincaré and Hilbert series for every finitely presented module. It then refines the graded derived and singular Koszul dualities plus the BGG correspondence by giving explicit triangulated equivalences and nonstandard t-structures whose hearts are described in terms of linear complexes. The ungraded case gets similar but lighter treatment. What stands out is the concrete structural descriptions rather than just existence statements. The coherence claims and the fp-perfect equality organize the module categories in a usable way, and the low linearity defect is a clean consequence that feeds the rational series result. The work sits squarely on the general Koszul duality setup from the cited reference, so the new content is the verification and explicitness for this monomial class. That explicitness is useful and goes beyond a routine check. A soft spot is the inherited dependence on the prior framework; any gaps there would affect these results too, though the monomial restrictions seem chosen to make direct verification feasible. The ungraded refinements feel thinner and might not hold up as strongly on their own. This is for specialists already comfortable with Koszul duality, graded algebras, and t-structures on derived categories. Someone looking for concrete examples to compute with or to test general conjectures would get value from the heart descriptions and the coherence statements. It deserves a serious referee because the claims are specific, the class is natural, and the derivations can be checked directly.

Referee Report

0 major / 4 minor

Summary. The manuscript provides a new class of examples for the Koszul dualities established in the cited work [5] by studying quadratic monomial algebras. For a finite-dimensional quadratic monomial algebra Λ, it proves that the Koszul dual Λ! is both left coherent and left co-coherent, that finitely presented modules coincide with perfect modules (and dually for finitely copresented and coperfect modules), and that the associated tails and cotails categories are abelian and hereditary. It further shows that quadratic monomial algebras are absolutely Koszul with global linearity defect at most one (implying rational Poincaré and Hilbert series for finitely presented modules) and refines the graded derived and singular Koszul dualities as well as the BGG correspondence via explicit realizations of triangulated equivalences that induce nonstandard t-structures with hearts described in terms of linear complexes and modules. Corresponding refinements are obtained in the ungraded setting.

Significance. If the central claims hold, the paper supplies concrete, computable examples where the general Koszul duality framework applies, yielding explicit structural descriptions of module categories, abelian hereditary tails, and refined equivalences with concrete t-structures. The verification of absolute Koszulness, bounded linearity defect, and rationality of series for this class strengthens the theory by adding a broad family of algebras (including many finite-dimensional ones) where these advanced results are directly applicable, facilitating further explicit computations in graded representation theory.

minor comments (4)

[Abstract] The abstract and introduction state the main results clearly but would benefit from explicit cross-references to the theorem numbers (e.g., the coherence result and the absolute Koszulness statement) to help readers navigate the proofs.
[Introduction] While the reliance on definitions and results from [5] is appropriate, a short paragraph in the introduction or §2 recalling the precise statements of coherence, perfect modules, and the tails category from [5] would improve accessibility without re-deriving the general theory.
[Throughout] Notation for the Koszul dual (Λ! vs. Λ^!) and for tails/cotails categories should be checked for consistency across sections; minor inconsistencies in subscripts or superscripts appear in the abstract and later statements.
[Final section on refinements] The explicit structural descriptions of the hearts of the nonstandard t-structures are a strength; adding a small illustrative example (e.g., for a 2-generated quadratic monomial algebra) in the final section would make the refinements more concrete.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recognizing the value of quadratic monomial algebras as a concrete class of examples for the Koszul duality results in [5]. The recommendation of minor revision is noted, and we will incorporate any editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; explicit verifications add independent content

full rationale

The paper supplies a new class of examples and states explicit proofs that quadratic monomial algebras are absolutely Koszul with linearity defect at most 1, that their Koszul duals are left coherent and left co-coherent, and that finitely presented modules coincide with perfect modules (and dually). These claims rest on direct verifications for the new class rather than reducing by construction to the inputs or to a self-citation chain. The cited framework in [5] is external setup; the central results are presented as independent checks that do not rename or refit prior quantities. No load-bearing step collapses to a definition or fitted input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard framework of graded algebras, Koszul duality, and coherence as defined in the cited reference [5]; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond those already present in the literature on quadratic monomial algebras.

axioms (2)

domain assumption Koszul duality and associated notions of coherence, perfect modules, and derived equivalences hold as established in the cited work [5]
Invoked throughout the abstract when stating proofs for the quadratic monomial case and refinements of dualities.
standard math Quadratic monomial algebras are finite-dimensional graded algebras with monomial relations of degree two
Basic definition assumed when proving coherence and module coincidences.

pith-pipeline@v0.9.0 · 5509 in / 1566 out tokens · 54812 ms · 2026-05-09T23:21:37.855185+00:00 · methodology

discussion (0)

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