{"id":"60d551a0-2266-4787-8e39-5ad1e30ddab6","arxiv_id":"1909.02200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Koszul dual of an exact Hochschild extension of a dg algebra is isomorphic to the deformed Calabi-Yau completion of the Koszul dual.","lead":"This paper introduces a new operation, the Hochschild extension, for differential graded algebras, and shows it naturally produces an A-infinity algebra. The main result is that the Koszul dual of an exact Hochschild extension is isomorphic to the deformed Calabi-Yau completion of the Koszul dual.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7 depends on [15, Thm. 8] and on the asserted Λ-module quasi-isomorphism ρ in §4.2; a failure there would break the exact-to-almost-exact bridge and the deformed CY completion isomorphism. Conditional acceptance pending independent verification remains appropriate.","rationale":"Reading the paper fairly, the internal structure is coherent: Theorem 1 and Theorem 2 have detailed proofs, Theorem 6 is concretely constructed, and the proposed Koszul-duality relationship between exact Hochschild extensions and deformed Calabi-Yau completions is natural and clearly stated. The concern is not a detected contradiction but a verification gap at exactly the point where the exact/almost-exact correspondence meets previously unpublished material. The Koszul duality isomorphisms are taken from [15], and Proposition 2 depends on an asserted dg Λ-module quasi-isomorphism ρ whose proof is only sketched. Because the almost-exactness of α† is necessary for applying Keller's Theorem 5 in Theorem 7(3), the central claim is contingent on this step. The reader identified the same weakest assumption, and I agree with that diagnosis. I see no reason to move away from the CONDITIONAL verdict: the paper should be accepted, if at all, on the condition that the missing proof or independent verification of Proposition 2 / [15, Thm. 8] is provided. No ad hominem is intended; the issue is purely epistemic support for a load-bearing external result.","tokens_in":27800,"tokens_out":12922,"duration_ms":134356,"concrete_test":"Write out ρ = ω_{A,BA} ∘ (ψ ⊗ id) ∘ ρ̃ in the notation of [15] and verify explicitly that it is a dg Λ-module quasi-isomorphism: the square in §4.2 must commute with B∨, and H(ρ) must be an isomorphism in every degree. Then test Proposition 2 on a minimal example, e.g. A = k[x]/(x^2) with n chosen so that A^{2−n}=0, and compare dim HC_i(A)^∨ with dim HN_{−i}(A†), together with the maps I^∨_n and P_{−n}; if the diagram in Proposition 2 fails in this example, the bridge used by Theorem 7 collapses. Alternatively, an independent full proof of [15, Thm. 8], or a formalized check of the Λ-module quasi-isomorphism, would settle the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Proposition 2 / Proposition 3: the exactness of [α] is translated into almost-exactness of [α†] only through the claimed isomorphism HC_n(A)^∨ ≅ HN_{−n}(A†). That isomorphism is the one part of the argument neither proved here nor supported by a published reference. The proof of Proposition 2 in §4.2 rests on the assertion that ρ := ω_{A,BA} ∘ (ψ ⊗ id) ∘ ρ̃ is a quasi-isomorphism of dg Λ-modules. The displayed diagram establishes at most compatibility with the operator B∨; it does not show that ρ induces isomorphisms on homology, nor that the Λ-action is preserved on the nose after dualization. Passing from ρ to cyclic and negative-cyclic homology uses a short exact sequence of k[[u]]-modules, and because C•(A)^∨ is the graded dual of a non-finite complex, the homology/dual interchange must be checked degree-wise. Theorem 7(3) then feeds the resulting α† into Keller's deformed Calabi-Yau completion and into Theorem 5. Consequently, a sign error, a missing finite-dimensionality check, or failure of ρ to be a Λ-module quasi-isomorphism would invalidate Proposition 3 and with it parts (2) and (3) of Theorem 7. Since Theorem 3 is cited from the same authors' unpublished preprint [15], this external dependence is the least secure point in the proof chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Hochschild extensions of augmented dg algebras by a dg bimodule and a Hochschild 2-cocycle, showing that these are naturally A∞-algebras (Theorem 1). It then focuses on extensions of a finite-dimensional complete typical dg K-algebra A by the shifted dual bimodule A^∨[-n], defines exactness of the defining Hochschild cohomology class, and proves that every exact Hochschild extension is an n-symmetric A∞-algebra (Theorem 2), recovering a result of Ohnuki–Takeda–Yamagata in the classical case. The main results are a Koszul duality statement: the Koszul dual of the trivial extension is the Calabi-Yau completion of the Koszul dual (Theorem 6), and the Koszul dual of an exact Hochschild extension is the deformed Calabi-Yau completion of the Koszul dual (Theorem 7). The paper also gives an application recovering Guo's theorem on trivial extensions and higher preprojective algebras (Corollary 1). The proofs are detailed in structure, but several load-bearing steps are either cited from the authors' unpublished preprint [15] or asserted without full verification.","tokens_in":28092,"tokens_out":11126,"duration_ms":102260,"significance":"If the results are correct, the paper makes a substantive contribution to the noncommutative geometry and representation theory literature by establishing a precise bridge between exact Hochschild extensions and deformed Calabi-Yau completions under Koszul duality. The construction of Hochschild extensions as A∞-algebras and the cohomological criterion for symmetry are potentially useful tools. The paper also provides a dg lift of Guo's isomorphism between higher preprojective algebras and Koszul duals of trivial extensions. However, the central theorems depend in an essential way on [15, Theorem 8] and [15, Theorem 7], an unpublished preprint by the same authors, and on a quasi-isomorphism ρ in §4.2 whose proof is only asserted. Until these gaps are closed, the main claims cannot be considered fully verified.","major_comments":[{"comment":"The proof of Proposition 2 rests on the assertion that ρ := ω_{A,BA} ∘ (ψ ⊗ id) ∘ ρ̃ is a quasi-isomorphism of dg Λ-modules. However, the displayed commutative diagram only shows compatibility with the Connes operator B∨; the components ρ̃, ψ, and ω are not defined in the manuscript, and no argument is given that ρ induces isomorphisms on homology or that it preserves the full Λ-action including the Hochschild differential b. Since Proposition 2 is used directly to prove Proposition 3 and hence Theorem 7(2) and (3), this gap is load-bearing.","section":"§4.2 (Proposition 2)"},{"comment":"The isomorphisms HH_n(A)^∨ ≅ HH_{-n}(A†) and HC_n(A)^∨ ≅ HN_{-n}(A†) are taken from [15, Theorem 8], an unpublished preprint by the same authors. The present paper does not reprove these results, and without them the exact/almost-exact correspondence in Proposition 3, and therefore Theorem 7(2), is unsupported. The authors should either provide self-contained proofs of these isomorphisms or replace the unpublished citation with a published reference, or at minimum include the full statements and proofs in an appendix.","section":"§4.1–4.2 (Theorem 3 and Proposition 2)"},{"comment":"The isomorphisms Φ : T_n(A)† → Π_n(A†) and Φ : T_n(A,α)† → Π_n(A†,α†) are established by asserting that both sides decompose into 'corresponding' direct summands and that the resulting bijection is 'compatible with differentials'. For the deformed case, this compatibility is the central point: the differential on T_n(A,α)† is determined by the A∞-structure involving α, while the differential on Π_n(A†,α†) is d + d_α with d_α determined by α†. The proof does not show that these differentials agree under the bijection. A full verification of the isomorphism, including the term-by-term identification and the differential compatibility, is required for both theorems.","section":"§5.2–5.3 (Theorems 6(3) and 7(3))"},{"comment":"The paper applies the functor (-)^∨ = Hom_k(-,k) to the Connes exact sequence and asserts that the dual sequence is exact, and in §4.2 it applies C_•(A)^∨[[u]] ⊗_{k[[u]]} - to a short exact sequence of k[[u]]-modules to obtain exact rows. Hom_k(-,k) is not exact on arbitrary complexes, and C_•(A)^∨[[u]] may fail to be flat over k[[u]] when C_•(A) is infinite-dimensional in each degree. The paper does not justify the required finite dimensionality or use continuous duals. This affects the definition of exactness (Definition 2) and the derivation of Proposition 2; the hypotheses under which these dual sequences are exact should be stated and proved.","section":"§3.2 and §4.2"}],"minor_comments":[{"comment":"In the sentence 'for a Hochschild class [α] ∈ HH^{n−2}(A), he introduced its deformed Calabi-Yau completion', the superscript should be a subscript: the class lies in Hochschild homology HH_{n−2}(A), as stated in Theorem 5 and Definition 6.","section":"Introduction"},{"comment":"The map ρ is denoted by different symbols (ρ, ρ̃, ̺) in consecutive lines, and the components ω_{A,BA}, ψ, and ρ̃ are not defined in the manuscript. Please fix the notation and define all components.","section":"§4.2"},{"comment":"The definition of the syzygy degree appears to contain a typo: ω(⟨c1|...|cn⟩) := ∑_{i=1}^n (|c1|+1) should read (|c_i|+1), with the index i inside the sum.","section":"Corollary 1 proof"},{"comment":"The sentence 'Then-trivial extension Tn(A) is the dg K-algebra...' should read 'The n-trivial extension...'.","section":"§5.2"},{"comment":"The description of the effect of s^{-n}˜q^∨(˜η1) — 'sending A in T to s^{-n}A^∨ in s^{-n}T^∨, and sending s^{-n}A^∨ in T to zero' — is confusing because φ0,0 must be an isomorphism. The total map should be described more explicitly to make the construction of the isomorphism transparent.","section":"Theorem 2 proof"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends heavily on the authors' own unpublished preprint [15] for several foundational results, including the Koszul duality isomorphisms that underlie the main theorem. The editor may wish to ask the authors to make [15] publicly available or to include the necessary proofs in the present paper. The topic fits the journal's scope, and the ideas are attractive, but the current level of verification is not sufficient for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this paper introduces the Hochschild extension of a dg algebra as an A-infinity algebra and then proves two structural identifications: the Koszul dual of a trivial extension is the Calabi-Yau completion (Theorem 6), and the Koszul dual of an exact Hochschild extension is the deformed Calabi-Yau completion (Theorem 7). These are real theorems, not just new names for old objects, and they generalize work of Guo, Grant-Iyama, and Keller to the dg setting. The paper also gives a clean cohomological criterion for when a Hochschild extension is symmetric (Theorem 2), extending Ohnuki-Takeda-Yamagata.\n\nThe writing is clear and the internal logic is mostly coherent. The proof of Theorem 2 is elaborate but the structure is there. Theorem 6's isomorphism is argued by decomposing both sides into matching direct summands, which is a bit sketchy but straightforward once you accept the setup. The recovery of Guo's theorem as a corollary is a nice sanity check.\n\nThe soft spot is real and load-bearing. Proposition 3, which converts exactness of [alpha] into almost-exactness of its Koszul dual [alpha dagger], relies on the isomorphism HC_n(A)^vee = HN_{-n}(A dagger) taken from the same authors' unpublished preprint [15, Theorem 8]. In the proof of Proposition 2 in Section 4.2, the map rho := omega_{A,BA} o (psi otimes id) o rho-tilde is asserted to be a quasi-isomorphism of dg Lambda-modules, but the displayed diagram only shows compatibility with the operator B^vee; it does not establish the homology isomorphism, nor that the Lambda-action is preserved on the nose after dualization. Since C_bullet(A)^vee is the graded dual of a non-finite complex, the homology/dual interchange needs a degree-wise check. If rho fails, then parts (2) and (3) of Theorem 7 break. This is not a fatal flaw by itself, but it is an external dependence that the authors need to fix, either by proving the Koszul duality isomorphisms in this paper or by pointing to a published, verifiable version. The proof of Theorem 2 also has a sketched step about constructing the A-infinity symmetric structure, but that looks like it can be filled in.\n\nWho should read this: anyone working on A-infinity algebras, Koszul duality, Calabi-Yau completions, or Hochschild extensions. It is a specialized but genuinely useful paper. I would send it to a serious referee, not desk reject, with instructions to focus on Section 4.2 and the dependence on [15]. If the authors supply the missing proof or a published reference, the paper is very likely correct and worth citing.","headline":"Genuinely new structural results linking exact Hochschild extensions to deformed Calabi-Yau completions, but the proof chain leans on an unpublished same-author preprint and a sketched quasi-isomorphism that need to be pinned down.","tokens_in":28627,"tokens_out":2958,"would_cite":true,"duration_ms":28080,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E40","16E45","18E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Exact Hochschild extensions of a finite dimensional complete typical dg algebra are symmetric $A_\\infty$-algebras, and their Koszul duals are deformed Calabi-Yau completions.","keywords":["Hochschild cohomology","Hochschild homology","Hochschild extension","A-infinity algebra","Koszul duality","deformed Calabi-Yau completion","trivial extension","cyclic homology"],"falsifier":"Check whether the map $\\rho\\colon C_{-\\bullet}(A^\\dagger)\\to C_\\bullet(A)^\\vee$ constructed in Section 4.2 is a quasi-isomorphism of dg $\\Lambda$-modules for a concrete example such as $A=k[x]/(x^2)$ with $\\deg x=1$. If $\\rho$ is not a quasi-isomorphism, or if some class that is exact on one side fails to be almost exact on the other, the main theorem is false.","tokens_in":27588,"feed_emoji":"🔄","tokens_out":10424,"duration_ms":84321,"temperature":0.7,"pith_summary":"This paper introduces Hochschild extensions of dg algebras: an $A_\\infty$-algebra structure on $A\\oplus M$ built from a dg algebra $A$, a dg bimodule $M$, and a Hochschild 2-cocycle $\\alpha$. It shows that when the extension is exact, meaning the cocycle class lies in the image of the map from cyclic to Hochschild cohomology, the resulting $A_\\infty$-algebra is symmetric. The central result is a Koszul-duality statement: the Koszul dual of an exact Hochschild extension is isomorphic to the deformed Calabi-Yau completion of the Koszul dual, and the zero-cocycle case identifies the Koszul dual of the trivial extension with the ordinary Calabi-Yau completion. If correct, this unifies the study of symmetric extensions with the study of Calabi-Yau completions and lifts earlier results on higher preprojective algebras to the dg setting.","feed_headline":"Exact Hochschild extensions become deformed CY completions","feed_subtitle":"An exact cohomology class yields a symmetric A∞-algebra, whose Koszul dual is an almost exact CY completion.","key_machinery":"The load-bearing object is the Hochschild extension $T_n(A,\\alpha)$, an augmented $A_\\infty$-algebra on $A\\oplus A^\\vee[-n]$ whose higher multiplications are recorded by a Hochschild 2-cocycle $\\alpha$. Exactness of $[\\alpha]$ is detected through Connes' long exact sequence: $[\\alpha]$ is exact when it lies in the image of $I^\\vee\\colon HC_\\bullet(A)^\\vee\\to HH_\\bullet(A)^\\vee$. The bridge to the Calabi-Yau side is Koszul duality $A^\\dagger=\\Omega(A^\\vee)$, together with the isomorphisms $HH_n(A)^\\vee\\cong HH_{-n}(A^\\dagger)$ and $HC_n(A)^\\vee\\cong HN_{-n}(A^\\dagger)$, which convert exact cohomology classes into almost exact homology classes. On that side, the deformed Calabi-Yau completion $\\Pi_n(A^\\dagger,\\alpha^\\dagger)$ is the tensor dg algebra over $A^\\dagger$ on the bimodule $A^\\dagger\\otimes s^{n-1}A\\otimes A^\\dagger$ with differential twisted by $\\alpha^\\dagger$. The final isomorphism is exhibited by decomposing both $T_n(A,\\alpha)^\\dagger$ and $\\Pi_n(A^\\dagger,\\alpha^\\dagger)$ into matching direct summands.","core_discovery":"Let $A$ be a finite dimensional complete typical dg $K$-algebra, $K=k^t$, and let $T_n(A,\\alpha)=A\\oplus A^\\vee[-n]$ be the Hochschild extension determined by a Hochschild 2-cocycle $\\alpha$ with $A^{2-n}=0$. The paper proves that if $[\\alpha]\\in H^2(A,A^\\vee[-n])$ is exact, i.e., lies in the image of $I^\\vee\\colon HC_{2-n}(A)^\\vee\\to HH_{2-n}(A)^\\vee$, then $T_n(A,\\alpha)$ is an $n$-symmetric $A_\\infty$-algebra: it is isomorphic to its graded dual shifted by $-n$ as an $A_\\infty$-bimodule. Writing $A^\\dagger=\\Omega(A^\\vee)$ for the Koszul dual and $[\\alpha^\\dagger]$ for the image of $[\\alpha]$ under the Koszul-duality isomorphism $HH^\\bullet(A)^\\vee\\cong HH_{-\\bullet}(A^\\dagger)$, the deformed Calabi-Yau completion $\\Pi_n(A^\\dagger,\\alpha^\\dagger)$ is isomorphic to the Koszul dual $T_n(A,\\alpha)^\\dagger$, and both are almost exact $n$-Calabi-Yau dg algebras. The case $\\alpha=0$ gives an isomorphism between the Calabi-Yau completion $\\Pi_n(A^\\dagger)$ and the Koszul dual of the trivial extension $T_n(A)$.","pith_inferences":["If the Koszul-duality isomorphisms from the authors' preprint hold for a wider class of dg algebras than typical ones, the same construction would attach deformed Calabi-Yau completions to exact Hochschild extensions beyond the finite-dimensional complete typical setting.","The exactness criterion gives a cohomological test for symmetry: a Hochschild extension is symmetric exactly when its defining class lifts to cyclic cohomology; nonsymmetric examples should correspond precisely to classes that fail this lift.","One could iterate the correspondence: applying Koszul duality to exact extensions of Koszul duals would produce a chain of Calabi-Yau completions, each matched with a symmetric extension one step back.","In quiver settings where almost exact classes come from superpotentials, the correspondence suggests that exact Hochschild 2-cocycles on $A$ should translate into superpotentials on $A^\\dagger$, giving a homological route from cocycles to potentials."],"forward_implications":["Every exact Hochschild extension of a finite dimensional complete typical dg algebra is an $n$-symmetric $A_\\infty$-algebra, extending the classical criterion for symmetry of Hochschild extensions from ordinary algebras to dg algebras.","The Koszul dual of the trivial extension is the Calabi-Yau completion, so constructions of Calabi-Yau completions can be translated back into trivial extensions.","The Koszul dual of an exact Hochschild extension is the deformed Calabi-Yau completion, which explains at the dg level why higher preprojective algebras appear as Koszul duals of trivial extensions.","Because the deformed completion is almost exact Calabi-Yau, the Koszul dual of an exact extension carries a canonical almost exact Calabi-Yau structure.","Corollary 1 recovers the known statement for Koszul $n$-homogeneous bound quiver algebras that the higher preprojective algebra of $A^!$ is the Koszul dual of the twisted trivial extension, now as a consequence of the dg-level isomorphism."],"supporting_citations":[{"why":"Supplies the central Koszul-duality isomorphisms between Hochschild and (negative) cyclic homology of $A$ and $A^\\dagger$, on which the exact/almost-exact correspondence rests.","marker":"[15]"},{"why":"Introduces Calabi-Yau and deformed Calabi-Yau completions and proves the exact Calabi-Yau property used in Theorem 6.","marker":"[25]"},{"why":"Erratum to the deformed Calabi-Yau completion paper, fixing the exact Calabi-Yau statement used here.","marker":"[26]"},{"why":"Provides the theorem that a deformed Calabi-Yau completion is almost exact Calabi-Yau when the deforming class is almost exact, used in Theorem 7(3).","marker":"[46]"},{"why":"Gives the classical symmetric Hochschild extension criterion that Theorem 2 generalizes to dg algebras.","marker":"[35]"},{"why":"The result on trivial extensions and higher preprojective algebras recovered as Corollary 1; serves as the baseline the dg lift must reproduce.","marker":"[13]"}],"fun_headline_variants":["Exact Hochschild extensions become symmetric A∞","Exact extensions' Koszul duals are deformed CY","From exact cohomology to deformed CY completion","Exact Hochschild extensions: CY via Koszul dual"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper leans on an earlier, not-reproved result that Hochschild and cyclic homology of a dg algebra match those of its Koszul dual; if that matching fails, the correspondence between exact extensions and deformed Calabi-Yau completions collapses.","fun_headline_variants_meta":{"raw":{"variants":["Exact Hochschild extensions become symmetric A∞","Exact extensions' Koszul duals are deformed CY","From exact cohomology to deformed CY completion","Exact Hochschild extensions: CY via Koszul dual"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00107,"raw_usage":{"total_tokens":4511,"prompt_tokens":1005,"completion_tokens":3506,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":3441}},"tokens_in":621,"tokens_out":3506,"duration_ms":30232,"temperature":1.0,"reasoning_tokens":3441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:56:26.890009+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the map $\\rho\\colon C_{-\\bullet}(A^\\dagger)\\to C_\\bullet(A)^\\vee$ constructed in Section 4.2 is a quasi-isomorphism of dg $\\Lambda$-modules for a concrete example such as $A=k[x]/(x^2)$ with $\\deg x=1$. If $\\rho$ is not a quasi-isomorphism, or if some class that is exact on one side fails to be almost exact on the other, the main theorem is false.","supporting_citations":[{"cited_title":"Hochschild (co)homologies of dg $K$-rings and their Koszul duals","cited_arxiv_id":"1810.05969","evidence_quote":"Supplies the central Koszul-duality isomorphisms between Hochschild and (negative) cyclic homology of $A$ and $A^\\dagger$, on which the exact/almost-exact correspondence rests."},{"cited_title":"Keller, Deformed Calabi-Yau completions, With an ap pendix by M","cited_arxiv_id":null,"evidence_quote":"Introduces Calabi-Yau and deformed Calabi-Yau completions and proves the exact Calabi-Yau property used in Theorem 6."},{"cited_title":"Yeung, Relative Calabi-Yau completion, arXiv:1612 .06352 [math.RT]","cited_arxiv_id":null,"evidence_quote":"Provides the theorem that a deformed Calabi-Yau completion is almost exact Calabi-Yau when the deforming class is almost exact, used in Theorem 7(3)."},{"cited_title":"Ohnuki, K","cited_arxiv_id":null,"evidence_quote":"Gives the classical symmetric Hochschild extension criterion that Theorem 2 generalizes to dg algebras."},{"cited_title":"On Trivial Extensions and Higher Preprojective Algebras","cited_arxiv_id":"1902.04772","evidence_quote":"The result on trivial extensions and higher preprojective algebras recovered as Corollary 1; serves as the baseline the dg lift must reproduce."}],"review_version":1}