The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures

Fernando Muro; Gustavo Jasso

REVIEW 3 major objections 3 minor 34 references

A single cohomological vanishing condition decides when a cluster-tilting dg algebra is bimodule Calabi–Yau.

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load-bearing objection A genuinely new Calabi–Yau refinement of the Auslander–Iyama correspondence, but the paper is not self-contained: the key lifting theorem is quoted from a companion paper and the abstract's advertised first non-liftable example is never constructed in the body. the 3 major comments →

arxiv 2509.22625 v4 pith:VGVUS35O submitted 2025-09-26 math.RT

The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures

Gustavo Jasso , Fernando Muro This is my paper

classification math.RT MSC 18G8018N40

keywords triangulated categoriesdifferential graded algebrasHochschild cohomologyA∞-algebrasA∞-bimodulesMassey productsCalabi–Yau algebrascluster tilting objects

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 aims to detect the bimodule right Calabi–Yau (CY) property of a dg algebra purely from its cohomology, in the setting where its perfect derived category has a basic dZ-cluster tilting object. It proves that, apart from a graded bimodule isomorphism A(n)≅DA, the only obstruction is the vanishing of a Batalin–Vilkovisky operator applied to the universal Massey product of length d+2. This yields a bijective correspondence, Theorem D, between such CY dg algebras and pairs consisting of a basic Frobenius algebra that is twisted (d+2)-periodic with an invertible bimodule, satisfying the same BV-obstruction vanishing. The result matters because it reduces a higher-homotopy property to a checkable cohomological condition, and because it produces the first algebraic triangulated category with a CY structure that cannot be lifted to any dg enhancement.

Core claim

On the paper's own terms: Theorem D establishes a bijection between (1) quasi-isomorphism classes of dg algebras A whose H^0(A) is basic finite-dimensional, whose free module A is a dZ-cluster tilting object in the perfect derived category, and which are bimodule right n-CY for n=md, and (2) equivalence classes of pairs (Λ,I) where Λ is a basic Frobenius algebra, twisted (d+2)-periodic via an automorphism σ, I is an invertible Λ-bimodule with I≅Λ_σ, and the pair satisfies: a graded Λ-bimodule isomorphism Λ(n)≅DΛ and the vanishing of the BV-operator obstruction Δ({{m^{d+2}_η}}) in HH^{d+1,-d}(Λ). The correspondence sends A to (H^0(A), H^{-d}(A)). The engine behind the theorem is Theorem E, a

What carries the argument

The novel object is Massey bimodule cohomology EM, defined from the bimodule Hochschild cochain complex: it measures obstructions to existence and uniqueness of minimal A∞-bimodule structures. The bimodule universal Massey product of length d+2, {{m^{A⋉M}_{d+2}}}, is the first higher operation in a d-sparse minimal model and lives in bimodule Hochschild cohomology HH^{d+2,-d}(A|M). When M=A is the diagonal, HH(A|A) is isomorphic to HH(A)[ε]/(ε²), and under a graded isomorphism φ:A(n)≅DA this isomorphism transports the obstruction to the BV operator Δ_φ on ordinary Hochschild cohomology. Vanishing of Δ_φ({{m^A_{d+2}}}) is the single non-formality condition that, together with vanishing of EM

Load-bearing premise

The whole argument hinges on a theorem imported from a companion article: that vanishing of Massey bimodule cohomology EM^{p+1,-p} for p>d plus equality of the bimodule universal Massey product forces a gauge A∞-isomorphism of minimal A∞-bimodules.

What would settle it

Exhibit a dg algebra A with d-sparse cohomology, H^0(A) basic, A a dZ-cluster tilting object, and a graded bimodule isomorphism H^•(A)(n)≅DH^•(A), but with Δ({{m^A_{d+2}}})≠0, that nevertheless admits a quasi-isomorphism A[n]≃DA; Theorem C says no such example exists. Alternatively, produce a pair (Λ,I) satisfying all conditions of Theorem D(2) except the BV-vanishing whose associated dg algebra is still bimodule right CY.

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

If this is right

For d=1 the correspondence restricts to the previously studied additively-finite Calabi–Yau triangulated categories, recovering classifications in the graded setting.
Theorem C gives a practical criterion: a dg algebra with d-sparse cohomology is bimodule right n-CY iff the obvious graded isomorphism exists and the BV-operator class vanishes.
The BV-vanishing condition is necessary, not just sufficient: every bimodule right CY dg algebra with d-sparse cohomology satisfies it (Proposition 6.27).
As an application, there exists an algebraic triangulated category carrying a triangulated Calabi–Yau structure that cannot be lifted to a bimodule right Calabi–Yau structure on any dg enhancement—the first such example.
The correspondence is bijective on equivalence classes, so the abstract classification of such dg algebras is reduced to the representation theory of twisted periodic Frobenius algebras.

Where Pith is reading between the lines

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

If the BV-vanishing condition is genuinely independent of the graded isomorphism, then varying φ might give different CY structures on the same dg algebra; the paper does not address this, but the bijection is on isomorphism classes of a single structure.
The d=2, m=1 case is directly relevant to Hua–Keller's conjecture on contractible curves in Calabi–Yau threefolds: the paper provides a sufficient criterion for k[u,u^{-1}]-enhancement once a further 2-periodicity condition holds, so a natural test is to run the criterion on the deformation algebra of a non-contractible rigid curve.
A testable extension: compute EM for a family of d=2 cluster-tilted algebras whose UMP BV-class is known, and compare the vanishing with the existence of dg enhancements of the module category; this would probe whether the EM vanishing is also necessary.
The first non-liftable example suggests that triangulated CY structures are strictly more flexible than bimodule right CY structures; one might expect similar non-liftability in higher dimensions by taking products of the example with other triangulated categories.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A genuinely new Calabi–Yau refinement of the Auslander–Iyama correspondence, but the paper is not self-contained: the key lifting theorem is quoted from a companion paper and the abstract's advertised first non-liftable example is never constructed in the body.

read the letter

The genuinely new content here is the Calabi–Yau variant of the Auslander–Iyama correspondence, with the bimodule universal Massey product and the BV-operator obstruction as new tools. The correspondence is a natural but non-trivial extension of JKM22, and the proof is detailed with careful attention to signs. The Massey bimodule cohomology and its use as an obstruction theory are a real contribution, and the reduction of Theorem D to Theorem E plus the earlier correspondence is plausible and well explained.

The soft spots are the ones you'd expect. First, the load-bearing Theorem 6.16 is quoted verbatim from the companion [JM25] without proof. That is acceptable in a sequel, but it means the central lifting step is conditional on an external paper. Second, the abstract promises the first example of a triangulated CY category not liftable to a bimodule right CY structure, but the body never constructs it. That mismatch needs fixing before publication. Third, there is a sign typo in the null-homotopy display in §7.2: the first equation has a minus sign where the following 'rewrite' has a plus sign. I checked the Gerstenhaber relation; the plus version is correct, and the proof goes through with the rewritten equation. So it is a presentation slip, not a mathematical gap. There are also a handful of typos ('a easier', 'condtions') that a referee should ask to be cleaned up.

The stress-test note's worry about the sign is real but does not land as a fatal flaw, because the paper immediately states the correct equation. The bigger question is whether Theorem 6.16 holds in the required generality, especially in characteristic 2 at the delicate bidegree; that has to be verified in the companion paper, and a referee should have access to it.

This paper is for specialists in representation theory, noncommutative geometry, and dg category theory. It deserves a serious referee because the potential significance is high and the core argument is coherent. I would send it to review, with the instruction that the referee must check the companion paper and that the authors must either supply the advertised example or qualify the abstract.

Referee Report

3 major / 3 minor

Summary. The paper proves a Calabi–Yau refinement of the derived Auslander–Iyama correspondence established in [JKM22]. The main result, Theorem D, asserts a bijection between quasi-isomorphism classes of dg algebras A with basic finite-dimensional H^0(A), with A a dZ-cluster tilting object in D_c(A), and which are bimodule right n-CY for n=md, and equivalence classes of pairs (Λ,I) consisting of a basic Frobenius algebra Λ twisted (d+2)-periodic with invertible bimodule I, satisfying a graded bimodule isomorphism Λ(n) ≅ DΛ and a BV-operator vanishing condition for the associated universal Massey product. Theorem C gives an equivalent cohomological criterion for a single such dg algebra to be bimodule right n-CY, and Theorem E is a Kadeishvili-type lifting theorem from cohomological data to a quasi-isomorphism A[n] ≃ DA. The paper also develops a substantial amount of foundational material on bimodule Hochschild cohomology and A∞-bimodules, with detailed sign conventions, and the abstract announces an application to produce an algebraic triangulated category with a Calabi–Yau structure not liftable to any dg enhancement.

Significance. If the main results are correct, they provide a complete cohomological characterisation of bimodule right Calabi–Yau structures in the cluster-tilting setting, extending the earlier correspondence of [JKM22] and giving a precise obstruction-theoretic meaning to the BV-operator condition. The paper is careful and systematic about signs and about the structure of bimodule Hochschild cohomology, and the overall strategy is plausible. The main caveat is that the decisive lifting step is imported as Theorem 6.16 from the companion paper [JM25] without proof, and the abstract's advertised first example is not constructed in the body. The significance is therefore conditional on the companion theorem and on the completeness of the presented arguments.

major comments (3)

[§6.4, Theorem 6.16; §6.7, proof of Theorem E] The central lifting step is quoted verbatim from [JM25, Theorem 6.2.7] and is not proved here. Theorem E and hence Theorem D depend on this theorem to pass from cohomological data, including the BV-operator vanishing, to an A∞-isomorphism of minimal A∞-bimodules. The present manuscript does not verify the hypotheses of Theorem 6.16 in the relevant d-sparse diagonal bimodule setting, especially in characteristic 2 at the bidegree (d+1,-d), where Remark 6.22 notes a discrepancy between the algebra and bimodule Massey differentials. Please either include a proof or a precise verification of the hypotheses, or state explicitly and prominently that the result is assumed from the companion article and ensure that companion is accessible.
[Abstract; §1.3–§1.4] The abstract claims: 'we obtain, to our knowledge, the first example of an algebraic triangulated category with a triangulated Calabi–Yau structure that cannot be lifted to a bimodule right Calabi–Yau structure on any of its dg enhancements.' I could not locate this example in the body. Section 1.3 gives standard motivating examples (cluster categories, AGK categories) and Section 1.4 describes potential applications, but no explicit construction or proof of the non-liftability example appears. This is a load-bearing advertised contribution. Either provide the example and its proof or remove/qualify the claim.
[§7.2, null-homotopy of the map (7.13)] The proof that the Massey bimodule cohomology vanishes contains a sign inconsistency. The text displays: { {m} }·x = [{ {m} },{ {δ/d} }·x] = [{ {m} },{ {δ/d} }]·x − δ/d·[{ {m} },x], and then rewrites this as { {m} }·x = [{ {m} },{ {δ/d} }·x] + δ/d·[{ {m} },x]. The first equality of the displayed chain is false; the Gerstenhaber relation gives [{ {m} },{ {δ/d} }·x] = { {m} }·x − δ/d·[{ {m} },x], so the final plus-sign equation is the correct null-homotopy condition, but the intervening displayed identity is not a valid derivation. Since the vanishing of EM is needed to apply Theorem 6.16, this step needs to be corrected and re-verified.

minor comments (3)

[§4.4, Definition-Proposition 4.65] In formula (4.66), the inputs of the cochain are written inconsistently: the left-hand side has x_1,...,x_q in the last block, while the right-hand side and the surrounding text use x_1,...,x_p. Please correct the typo.
[§6.10 and §7.2] The notation { {m^{A⋉M}_{d+2}} } is introduced for the bimodule UMP, but in Definition 6.10 the reference to 'Definition 6.10' for the cocycle statement is confusing; the relevant definition of the Massey bimodule complex is Definition 6.12. Please align the cross-references.
[Throughout] There are a number of typographical slips ('condtions', 'biomdule', 'Hochchild', 'vanihsing', and similar). They do not affect the mathematics but should be cleaned up in the final version.

Circularity Check

0 steps flagged

No significant circularity: the CY variant is proved from the previous correspondence plus a disclosed companion theorem; the derivation is not forced by its own definitions.

full rationale

The paper's central Theorem D is a restriction of the bijection in [JKM22, Theorem A] to Calabi-Yau objects. The new content is the verification that the bimodule right Calabi-Yau property transfers across that known correspondence. This verification uses Theorem E, whose key input is Theorem 6.16 from the companion article [JM25]. That is a self-citation, but it is not circular: Theorem 6.16 is a Kadeishvili-type uniqueness statement about minimal A-infinity-bimodules with independent hypotheses (vanishing Massey bimodule cohomology and equality of bimodule UMP), and nothing in the present paper's target result is assumed there. The conditions (2c) in Theorem D are not definitions of (1c): the implication (2c) to (1c) runs through Theorem E and requires a substantive quasi-isomorphism lift A[n] ≃ DA. The injectivity direction is inherited from the earlier bijection, not manufactured. There is an apparent sign inconsistency in Section 7.2 in the null-homotopy for the Massey bimodule differential, where the displayed identity has a minus sign and the rewritten 'null-homotopy equation' has a plus sign; this threatens the proof of the required vanishing EM = 0, but it is a correctness risk, not a circularity. No fitted parameters are renamed as predictions, and no known result is repackaged under new coordinates. Accordingly the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numerical fitting or invented physical entities; the paper introduces new cohomology theories as mathematical tools but they are not 'entities' in the schema sense. The central claim rests on the listed prior theorems, especially [JKM22, Theorem A] and [JM25, Theorem 6.2.7].

axioms (5)

domain assumption Ground field k is perfect and Hom-finite, so that [JKM22, Theorem A] applies
Main theorems require perfect field (stated before Theorem C and D); needed for the bijective correspondence in [JKM22, Theorem A].
domain assumption [JKM22, Theorem A] bijective correspondence between dg algebras with dZ-cluster tilting object and twisted (d+2)-periodic Frobenius algebras
Used as the backbone of Theorem D; without it, Theorem D reduces to an empty statement. Stated in the proof of Theorem D.
domain assumption [JM25, Theorem 6.2.7] (Theorem 6.16) on uniqueness of minimal A∞-bimodule structures given vanishing Massey bimodule cohomology
Key technical input for Theorem E and Theorem D; quoted from companion paper, not proved here.
standard math Standard existence and uniqueness of minimal models for dg algebras and dg bimodules via homotopy transfer (Kadeishvili, Markl, etc.)
Used in Section 5.6 to construct minimal models and in Definition 6.24-6.25 for universal Massey products.
standard math The cohomological identification H•(DA) ≅ DH•(A) for degree-wise finite-dimensional dg bimodules
Used in Remark 2.39 and Proposition 6.27 to pass from bimodule quasi-isomorphism to graded isomorphism.

pith-pipeline@v1.3.0-alltime-deepseek · 78121 in / 10811 out tokens · 124732 ms · 2026-08-04T14:48:03.481533+00:00 · methodology

0 comments

read the original abstract

Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras (dg) with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\ZZ$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. In this article, we prove a variant of our general correspondence for bimodule right Calabi--Yau dg algebras. A novel ingredient is a new cohomology theory which contains obstructions to the existence and uniqueness of minimal $A_\infty$-bimodule structures on a graded bimodule. As an application of our results, we obtain, to our knowledge, the first example of an algebraic triangulated category with a triangulated Calabi--Yau structure that cannot be lifted to a bimodule right Calabi--Yau structure on any of its dg enhancements.

discussion (0)

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arXiv: 2012.11927. [Che+21] X. Chen, Y. Chen, F. Eshmatov, and S. Yang. “Poisson cohomol- ogy, Koszul duality, and Batalin-Vilkovisky algebras”.J. Noncommut. Geom.15.3 (2021), pp. 889–918. [Cho08] C.-H. Cho. “Strong homotopy inner product of anA ∞-algebra”.Int. Math. Res. Not. IMRN13 (2008), Art. ID rnn041,

Pith/arXiv arXiv 2012
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Dualit´ e de Van den Bergh et structure de Batalin-Vilkoviski ˘i sur les alg` ebres de Calabi-Yau

arXiv:2111.10771. [Lam10] T. Lambre. “Dualit´ e de Van den Bergh et structure de Batalin-Vilkoviski ˘i sur les alg` ebres de Calabi-Yau”.J. Noncommut. Geom.4.3 (2010), pp. 441–457. [LH03] K. Lef` evre-Hasegawa. “Sur lesA-infini cat´ egories” (2003). arXiv:math/ 0310337. [LM08] V. Lyubashenko and O. Manzyuk. “A ∞-bimodules and SerreA ∞- functors”.Geometry ...

Pith/arXiv arXiv 2010
[2022]

The Donovan–Wemyss Conjecture via the Derived Auslander–Iyama Correspondence

arXiv:2208.14413. [JKM24] G. Jasso, B. Keller, and F. Muro. “The Donovan–Wemyss Conjecture via the Derived Auslander–Iyama Correspondence”.Triangulated Cat- egories in Representation Theory and Beyond. Vol

Pith/arXiv arXiv
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Trivial extension DG-algebras, unitally positiveA ∞-algebras, and applications

arXiv:2311.06538. [KL W24] J. Karmazyn, E. Lepri, and M. Wemyss. “Trivial extension DG-algebras, unitally positiveA ∞-algebras, and applications” (Nov. 2024). arXiv: 2411.17359 [math.AG]. REFERENCES 89 [Kon93] M. Kontsevich. “Formal (non)commutative symplectic geometry”.The Gel′fand Mathematical Seminars, 1990–1992. Birkh¨ auser Boston, Boston, MA, 1993, ...

Pith/arXiv arXiv 2024
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The algebraic structure in the homology of an A(∞)-algebra

arXiv:2507.17568. [Kad82] T. V. Kadeishvili. “The algebraic structure in the homology of an A(∞)-algebra”.Soobshch. Akad. Nauk Gruzin. SSR108.2 (1982), 249– 252 (1983). [Kad88] T. V. Kadeishvili. “The structure of theA(∞)-algebra, and the Hochschild and Harrison cohomologies”.Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR91 (1988), pp. 19–27. [...

Pith/arXiv arXiv 1982
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Homological mirror symmetry for the quartic surface

[Sei08] P. Seidel.Fukaya categories and Picard-Lefschetz theory. Zurich Lec- tures in Advanced Mathematics. European Mathematical Society (EMS), Z¨ urich, 2008, pp. viii+326. [Sei15] P. Seidel. “Homological mirror symmetry for the quartic surface”.Mem. Amer. Math. Soc.236.1116 (2015), pp. vi+129. [SY11] A. Skowro´ nski and K. Yamagata.Frobenius algebras. ...

2008
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EnhancedA ∞-obstruction theory

[Mur20] F. Muro. “EnhancedA ∞-obstruction theory”.J. Homotopy Relat. Struct. 15.1 (2020), pp. 61–112. [Mur22] F. Muro. “Enhanced Finite Triangulated Categories”.J. Inst. Math. Jussieu21.3 (2022), pp. 741–783. [OT12] S. Oppermann and H. Thomas. “Higher-dimensional cluster combina- torics and representation theory”.J. Eur. Math. Soc. (JEMS)14.6 (2012), pp. ...

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Lecture Notes in Phys. Springer, Berlin, 2006, pp. 153–219. [KS08] M. Kontsevich and Y. Soibelman. “Stability structures, motivic Donaldson- Thomas invariants and cluster transformations” (Nov. 2008). arXiv: 0811.2435 [math.AG]. [KW21] B. Keller and Y. Wang.An introduction to relative Calabi-Yau struc- tures

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[BKS04] D. Benson, H. Krause, and S. Schwede. “Realizability of modules over Tate cohomology”.Trans. Amer. Math. Soc.356.9 (2004), pp. 3621–

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[HIO14] M. Herschend, O. Iyama, and S. Oppermann. “n-representation infinite algebras”.Adv. Math.252 (2014), pp. 292–342. [HK24] Z. Hua and B. Keller. “Cluster categories and rational curves”.Geom. Topol.28.6 (2024), pp. 2569–2634. [IJ17] O. Iyama and G. Jasso. “Higher Auslander correspondence for dualiz- ingR-varieties”.Algebr. Represent. Theory20.2 (201...

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Poisson cohomol- ogy, Koszul duality, and Batalin-Vilkovisky algebras

arXiv: 2012.11927. [Che+21] X. Chen, Y. Chen, F. Eshmatov, and S. Yang. “Poisson cohomol- ogy, Koszul duality, and Batalin-Vilkovisky algebras”.J. Noncommut. Geom.15.3 (2021), pp. 889–918. [Cho08] C.-H. Cho. “Strong homotopy inner product of anA ∞-algebra”.Int. Math. Res. Not. IMRN13 (2008), Art. ID rnn041,

Pith/arXiv arXiv 2012

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Dualit´ e de Van den Bergh et structure de Batalin-Vilkoviski ˘i sur les alg` ebres de Calabi-Yau

arXiv:2111.10771. [Lam10] T. Lambre. “Dualit´ e de Van den Bergh et structure de Batalin-Vilkoviski ˘i sur les alg` ebres de Calabi-Yau”.J. Noncommut. Geom.4.3 (2010), pp. 441–457. [LH03] K. Lef` evre-Hasegawa. “Sur lesA-infini cat´ egories” (2003). arXiv:math/ 0310337. [LM08] V. Lyubashenko and O. Manzyuk. “A ∞-bimodules and SerreA ∞- functors”.Geometry ...

Pith/arXiv arXiv 2010

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The Donovan–Wemyss Conjecture via the Derived Auslander–Iyama Correspondence

arXiv:2208.14413. [JKM24] G. Jasso, B. Keller, and F. Muro. “The Donovan–Wemyss Conjecture via the Derived Auslander–Iyama Correspondence”.Triangulated Cat- egories in Representation Theory and Beyond. Vol

Pith/arXiv arXiv

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Trivial extension DG-algebras, unitally positiveA ∞-algebras, and applications

arXiv:2311.06538. [KL W24] J. Karmazyn, E. Lepri, and M. Wemyss. “Trivial extension DG-algebras, unitally positiveA ∞-algebras, and applications” (Nov. 2024). arXiv: 2411.17359 [math.AG]. REFERENCES 89 [Kon93] M. Kontsevich. “Formal (non)commutative symplectic geometry”.The Gel′fand Mathematical Seminars, 1990–1992. Birkh¨ auser Boston, Boston, MA, 1993, ...

Pith/arXiv arXiv 2024

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The algebraic structure in the homology of an A(∞)-algebra

arXiv:2507.17568. [Kad82] T. V. Kadeishvili. “The algebraic structure in the homology of an A(∞)-algebra”.Soobshch. Akad. Nauk Gruzin. SSR108.2 (1982), 249– 252 (1983). [Kad88] T. V. Kadeishvili. “The structure of theA(∞)-algebra, and the Hochschild and Harrison cohomologies”.Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR91 (1988), pp. 19–27. [...

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Homological mirror symmetry for the quartic surface

[Sei08] P. Seidel.Fukaya categories and Picard-Lefschetz theory. Zurich Lec- tures in Advanced Mathematics. European Mathematical Society (EMS), Z¨ urich, 2008, pp. viii+326. [Sei15] P. Seidel. “Homological mirror symmetry for the quartic surface”.Mem. Amer. Math. Soc.236.1116 (2015), pp. vi+129. [SY11] A. Skowro´ nski and K. Yamagata.Frobenius algebras. ...

2008

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EnhancedA ∞-obstruction theory

[Mur20] F. Muro. “EnhancedA ∞-obstruction theory”.J. Homotopy Relat. Struct. 15.1 (2020), pp. 61–112. [Mur22] F. Muro. “Enhanced Finite Triangulated Categories”.J. Inst. Math. Jussieu21.3 (2022), pp. 741–783. [OT12] S. Oppermann and H. Thomas. “Higher-dimensional cluster combina- torics and representation theory”.J. Eur. Math. Soc. (JEMS)14.6 (2012), pp. ...

2020

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On the derived category of an algebra over an operad

[BM09] C. Berger and I. Moerdijk. “On the derived category of an algebra over an operad”.Georgian Math. J.16.1 (2009), pp. 13–28. [Boc08] R. Bocklandt. “Graded Calabi Yau algebras of dimension 3”.J. Pure Appl. Algebra212.1 (2008), pp. 14–32. [Bol84] M. L. Bolla. “Isomorphisms between endomorphism rings of progener- ators”.J. Algebra87.1 (1984), pp. 261–28...

2009