{"id":"da14b6b6-2fbe-4610-8c4e-cdc873f54d86","arxiv_id":"2509.22625","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Auslander-Iyama correspondence is extended to bimodule right Calabi-Yau dg algebras via a new Massey bimodule cohomology, with an obstruction (BV operator on universal Massey product) and a first non-liftable example.","lead":"This paper characterizes the bimodule Calabi-Yau property for differential graded algebras that have a cluster tilting object, reducing it to data of a Frobenius algebra and an obstruction class in a new cohomology theory. It is the sequel to the authors' earlier correspondence and provides a counterexample showing that not every triangulated Calabi-Yau category lifts to a dg enhancement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem D hinges on external Theorem 6.16 (gauge A∞-isomorphism for minimal A∞-bimodules) from [JM25]; unless that theorem and its hypotheses in the d-sparse diagonal/char-2 setting are verified, the cohomological-to-quasi-isomorphism lift in Theorem E—and hence Theorem D—does not follow.","rationale":"The reader’s weakest assumption—that Theorem D depends on the unproved Theorem 6.16 from [JM25]—is exactly the load-bearing concern I find. The paper’s proof of Theorem D does supply a significant part of the needed hypotheses: the vanishing of Massey bimodule cohomology for the diagonal Λ-bimodule is proved in Section 7.2, and the equality of bimodule UMPs is reduced to the BV-operator vanishing via Proposition 6.20. But the final inference from those hypotheses to the required gauge A∞-isomorphism is outsourced to the companion paper. If Theorem 6.16 is false, or if its hypotheses are not met in the d-sparse diagonal case (e.g., because of characteristic-2 subtleties in the Gerstenhaber square or because the Massey bimodule differential used here differs from the one in [JM25] at bidegree (d+1,-d)), then Theorem E fails, and Theorem D collapses. I also flag an internal sign inconsistency in the null-homotopy proof: the two displayed equations in Section 7.2 are not algebraic rewrites of one another, and only the plus version is consistent with the paper’s own Gerstenhaber relation. This is likely a typo, but it should be corrected because the vanishing of EM depends on it. I did not find an independent internal contradiction in the statement of Theorem D beyond this dependency, and the extensive structural work in Sections 2–6 is coherent as far as I can tell. The abstract’s advertised first non-liftable example is not constructed anywhere in the body, which is a separate gap in presentation, but it is not the load-bearing step for Theorem D itself. Therefore the reader’s CONDITIONAL verdict stands; my concern reinforces it rather than moving it.","tokens_in":78427,"tokens_out":11528,"duration_ms":85950,"concrete_test":"Recompute the null-homotopy in §7.2 using the stated Gerstenhaber relation (Definition 3.10) with m = {{mΛ_{d+2}}}, δ/d, and x arbitrary: the relation gives [{m}, δ/d·x] = [{m},δ/d]·x − δ/d·[{m},x] because |sm|=1 and |δ/d|=1. Since [{δ/d},{m}] = −{m}, we get [{m},δ/d] = {m}, hence [{m}, δ/d·x] = {m}·x − δ/d·[{m},x], so the null-homotopy is {m}·x = [{m}, δ/d·x] + δ/d·[{m},x], i.e. f = D h + h D. Check which of the two displayed equations in §7.2 is used; if the minus version is used, Theorem D’s proof of EM^{p+1,-p}=0 is invalid. Also read [JM25, Theorem 6.2.7] to confirm it covers perfect fields of characteristic 2 and applies to the diagonal bimodule of a d-sparse minimal A∞-algebra with the differential of Definition 6.12.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem D’s surjectivity runs through Theorem E, whose key step is Theorem 6.21 (the A∞-version of the Calabi–Yau lift). Theorem 6.21 is proved by invoking Theorem 6.16, stated verbatim from [JM25, Theorem 6.2.7]: given two minimal A∞-bimodule structures with the same underlying graded bimodule and equal bimodule UMP, vanishing of Massey bimodule cohomology EM^{p+1,-p} = 0 for p > d forces them to be gauge A∞-isomorphic. This is the black box that converts the cohomological isomorphism φ: H•(A)(n) ≅ DH•(A) plus the BV-operator vanishing into an actual quasi-isomorphism A[n] ≃ DA. The present paper neither proves Theorem 6.16 nor checks that its hypotheses are met in the relevant diagonal d-sparse setting, especially in characteristic 2 at the delicate bidegree (d+1,-d), where the algebra and bimodule Hochschild–Massey differentials differ (see Remark 6.22). Additionally, the proof of the required vanishing EM = 0 in §7.2 contains an internal sign inconsistency: the intermediate identity displayed before the null-homotopy has a minus sign where the following 'rewrite' has a plus sign; only the plus version satisfies f = D h + h D for the stated Gerstenhaber bracket. If the sign error is taken at face value, the null-homotopy claim fails, so the vanishing of EM is not established. Together these make Theorem D conditional on the companion paper and on a sign correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":78894,"tokens_out":7504,"duration_ms":62775,"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":[{"comment":"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.","section":"§6.4, Theorem 6.16; §6.7, proof of Theorem E"},{"comment":"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.","section":"Abstract; §1.3–§1.4"},{"comment":"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.","section":"§7.2, null-homotopy of the map (7.13)"}],"minor_comments":[{"comment":"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.","section":"§4.4, Definition-Proposition 4.65"},{"comment":"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.","section":"§6.10 and §7.2"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The dependence on [JM25] is disclosed and appears to be genuine prior work rather than circularity. For a journal submission, I would want the editor to confirm that the companion paper is available or that its key theorem is reproduced. The advertised 'first example' should either be supplied or removed; as written, the abstract overstates what the body establishes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":79304,"tokens_out":5739,"would_cite":true,"duration_ms":41615,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G80","18N40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single cohomological vanishing condition decides when a cluster-tilting dg algebra is bimodule Calabi–Yau.","keywords":["triangulated categories","differential graded algebras","Hochschild cohomology","A∞-algebras","A∞-bimodules","Massey products","Calabi–Yau algebras","cluster tilting objects"],"falsifier":"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.","tokens_in":78366,"feed_emoji":"🔗","tokens_out":5298,"duration_ms":39042,"temperature":0.7,"pith_summary":"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.","feed_headline":"Vanishing BV obstruction characterizes Calabi–Yau dg algebras","feed_subtitle":"New bijection with twisted periodic Frobenius algebras, plus the first CY category with no dg lift.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["First CY category with no dg lift from bimodule CY algebras","Bijection links bimodule CY dg algebras to twisted Frobenius","Obstruction theory yields new CY category with no dg lift","BV obstruction decides bimodule CY structures"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["First CY category with no dg lift from bimodule CY algebras","Bijection links bimodule CY dg algebras to twisted Frobenius","Obstruction theory yields new CY category with no dg lift","BV obstruction decides bimodule CY structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000645,"raw_usage":{"total_tokens":2822,"prompt_tokens":786,"completion_tokens":2036,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":1972}},"tokens_in":530,"tokens_out":2036,"duration_ms":11636,"temperature":1.0,"reasoning_tokens":1972,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T14:48:03.481533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}