Exact Hochschild extensions and deformed Calabi-Yau completions

Kai Wang; Xin Liu; Yang Han

REVIEW 4 major objections 5 minor 46 references

Exact Hochschild extensions and deformed Calabi-Yau completions

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Pith's one-line read 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.

desk verdict 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results linking exact Hochschild extensions to deformed Calabi-Yau completions, but the proof chain leans on an unpublished same-author preprint and a sketched quasi-isomorphism that need to be pinned down. read the letter →

arxiv 1909.02200 v1 pith:CMEOR7MC submitted 2019-09-05 math.RA math.KTmath.RT

Yang Han , Xin Liu , Kai Wang This is my paper · ORCID

classification math.RAmath.KTmath.RT MSC 16E4016E4518E30

keywords HochschildcohomologyhomologyextensionA-infinityalgebraKoszuldualitydeformedCalabi-Yaucompletiontrivialcyclic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $T_n(A)$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

You 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.

The 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.

The 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.

Who 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[§4.2 (Proposition 2)] 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.
[§4.1–4.2 (Theorem 3 and Proposition 2)] 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.
[§5.2–5.3 (Theorems 6(3) and 7(3))] 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.
[§3.2 and §4.2] 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.

minor comments (5)

[Introduction] 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.
[§4.2] 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.
[Corollary 1 proof] 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.
[§5.2] The sentence 'Then-trivial extension Tn(A) is the dg K-algebra...' should read 'The n-trivial extension...'.
[Theorem 2 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is a direct structural comparison; the main external dependence on the authors' preprint [15] is load-bearing but independent support under the stated rules, and the asserted quasi-isomorphism rho is a correctness risk, not a circular step.

full rationale

The paper contains no fitted parameters, no data-dependent predictions, and no definitional equivalence between its inputs and its conclusions. The construction of T_n(A,alpha) as an A_infinity-algebra (Theorem 1) and the proof that exactness makes it n-symmetric (Theorem 2) proceed by explicit cocycle-lifting and dual-basis arguments; exactness is used through the definitional existence of a lift of [alpha] to HC, and the symmetric bimodule map is then built, so no conclusion is assumed as input. The exact/almost-exact correspondence (Proposition 3) is a diagram chase from Proposition 2, whose proof in Section 4.2 rests on the assertion that rho := omega_{A,BA} composed with (psi tensor id) composed with rho-tilde is a quasi-isomorphism of dg Lambda-modules, taken from the proof of [15, Theorem 8]; this is an unproved step and a real correctness risk, but it is a dependence on a prior stated theorem, not a reduction of the present result to its own assumption. The isomorphisms HH_n(A)^vee is isomorphic to HH_{-n}(A-dagger) and HC_n(A)^vee is isomorphic to HN_{-n}(A-dagger) are cited from [15], a same-author preprint; under the review rules these are parameter-free theorems with stated assumptions that do not include the target isomorphism, so they count as independent support and do not raise the circularity score. The isomorphisms in Theorems 6 and 7 are obtained by decomposing both sides as tensor algebras on s^{-1}A^vee plus s^{n-1}A and checking differential compatibility; this is a structural identification close to a construction, but the definitions of Keller's (deformed) Calabi-Yau completion and of the Koszul dual are independent, and the Calabi-Yau conclusions use external Theorems 4 and 5. Overall no step reduces by construction to its own input, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The paper introduces new definitions (Hochschild extension of a dg algebra as an A-infinity algebra) but these are mathematical constructions, not empirical postulated entities requiring independent experimental evidence. The main external assumptions are the domain conditions (typicality, degree vanishing) and the cited Koszul duality isomorphisms from the same-authors' preprint [15].

assumptions (4)

domain assumption A is a finite dimensional complete typical dg K-algebra
This assumption is used in Theorems 2, 6 and 7. It guarantees that the bar construction is locally finite and that (BA)^∨ ≅ Ω(A^∨), which is needed for Koszul duality.
domain assumption The dg bimodule M satisfies M_{-2}=0
Theorem 1 requires M_{-2}=0 to avoid a curved A-infinity algebra. Remark 2 explicitly notes this condition is necessary.
domain assumption A^{2-n}=0 for the integer n
Theorems 2 and 7 assume A^{2-n}=0, which is a cohomological degree condition needed for the symmetric structure and for the deformed completion construction to behave well.
domain assumption The Koszul duality isomorphisms from [15, Theorem 8] hold for complete typical dg algebras
The paper relies on HH_n(A)^∨ ≅ HH_{−n}(A†) and the compatibility with cyclic homology from [15, Theorem 8]. This is an external result by the same authors, cited rather than reproved, and it underpins Proposition 3 and hence part (2) of Theorem 7.

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Pith. "Pith review of Exact Hochschild extensions and deformed Calabi-Yau completions." pith.science (2026). https://pith.science/paper/CMEOR7MC

@misc{pith2026190902200,
  author       = {Pith},
  title        = {Pith review of: Exact Hochschild extensions and deformed Calabi-Yau completions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMEOR7MC}},
  note         = {Machine review of arXiv:1909.02200}
}

abstract

We introduce the Hochschild extensions of dg algebras, which are $A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric $A_\infty$-algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact Hochschild extension of a finite dimensional complete typical dg algebra is the deformed Calabi-Yau completion of its Koszul dual.

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