Pith. sign in

REVIEW 3 major objections 4 minor 21 references

Homological smoothness of Hopf-Galois extensions

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Homological smoothness is inherited by any faithfully flat Hopf-Galois extension whose Hopf algebra and coinvariant subalgebra are homologically smooth.

desk verdict A clean, likely correct generalization of smoothness ascent to faithfully flat Hopf-Galois extensions, with the main risk being quoted results from Stefan's 1995 paper that a referee should verify. read the letter →

arxiv 2412.04365 v1 pith:J4PABN7J submitted 2024-12-05 math.KT

Julian Le Clainche This is my paper · ORCID
classification math.KT MSC 16E4016T0518G40
keywords homologicallysmoothalgebraHopf-GaloisextensionHochschildhomologytypeFP∞StefanspectralsequencecohomologicaldimensionquantumenvelopingBieri-Eckmanncriterion
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

Hopf-Galois extensions are the noncommutative analogue of principal bundles: a Hopf algebra $H$ coacts on an algebra $A$, and the extension of its coinvariant subalgebra $B \subset A$ is controlled by a bijectivity condition on a canonical map $\beta : A \otimes_B A \to A \otimes H$. This paper proves that homological smoothness—the noncommutative analogue of regularity, meaning that the algebra has a finite resolution by finitely generated projective bimodules—ascends in such extensions. Concretely, if $H$ has bijective antipode, $B \subset A$ is $H$-Galois, $A$ is faithfully flat as a left and right $B$-module, and both $H$ and $B$ are homologically smooth, then $A$ is homologically smooth. The proof also gives the quantitative bound $\mathrm{cd}(A) \leq \mathrm{cd}(B) + \mathrm{cd}(H)$ on cohomological dimension, and the final section applies the theorem to a family of quantum algebras $U_q^{B,b}$ built from a smooth commutative base, yielding smoothness with $\mathrm{cd} \leq \mathrm{cd}(B) + 3$.

What carries the argument

The load-bearing object is Stefan's spectral sequence for Hopf-Galois extensions, a Leray-Serre-type spectral sequence for Hochschild homology: for an $A$-bimodule $M$ it reads $E^2_{p,q}=\mathrm{Tor}_p^H(k_\varepsilon, \mathrm{HH}_q(B,M)) \Rightarrow \mathrm{HH}_{p+q}(A,M)$ when $A$ is projective as a $B$-module, a hypothesis obtained here from faithful flatness. The collapse is engineered by pairing this spectral sequence with the Bieri-Eckmann Tor criterion (Proposition 2.4), which says a module is of type $FP_\infty$ exactly when Tor against arbitrary direct products of copies of the algebra vanishes in positive degrees and commutes with products in degree zero. Smoothness of $B$ and $H$ supplies exactly the vanishing needed to kill all $E^2$ terms except $(0,0)$ for the test module $M=\prod A^e$.

What would settle it

Compute $\mathrm{HH}_1(A,\prod A^e)$ for any faithfully flat Hopf-Galois extension with $B$ and $H$ homologically smooth (for instance a strong group grading or a smash product): if it is ever nonzero, the Bieri-Eckmann criterion forces $A$ not to be $FP_\infty$ and Theorem 1.1 is false. A more targeted check is to find a projective $A$-bimodule $P$ for which $\mathrm{HH}_0(B,P)$ is not projective as an $H$-module, which would directly refute the quoted lemma [16, Proposition 4.4] used to kill the $p>0$ rows.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: homological smoothness is inherited by the total algebra of a Hopf-Galois extension from the Hopf algebra and the coinvariant subalgebra, provided the extension is faithfully flat. The proof runs by testing the Bieri-Eckmann characterization of type $FP_\infty$ on the $A$-bimodule $M=\prod A^e$. Because $B$ is smooth, its Hochschild homology with coefficients in $M$ vanishes in positive degrees and commutes with the direct product; because $H$ is smooth, the $H$-Tor groups against the coinvariant space $\mathrm{HH}_0(B,A^e)$ vanish in positive degrees. Stefan's homology spectral sequence therefore collapses at $E^2$ with only the $(0,0)$-term possibly nonzero, giving $\mathrm{HH}_n(A,M)=0$ for $n>0$ and the required product isomorphism in degree zero. With finite cohomological dimension supplied by Proposition 4.1, the $FP_\infty$ conclusion upgrades to homological smoothness.

Load-bearing premise

The proof leans on three quoted facts from Stefan's paper—flatness of $A^e$ over $B^e$, projectivity of $\mathrm{HH}_0(B,P)$ as an $H$-module for projective $A$-bimodules $P$, and the identification of $\mathrm{HH}_0(A,M)$ with $H$-coinvariants of $\mathrm{HH}_0(B,M)$—and on the collapse of the spectral sequence they produce; if any of these was misstated or does not apply, the $FP_\infty$ conclusion no longer follows.

Editorial extensions

If this is right

  • Every faithfully flat Hopf-Galois extension of a homologically smooth algebra by a homologically smooth Hopf algebra with bijective antipode is itself homologically smooth.
  • The cohomological dimension of the total algebra is at most the sum of the dimensions of the base and the Hopf algebra: $\mathrm{cd}(A) \leq \mathrm{cd}(B) + \mathrm{cd}(H)$.
  • The theorem applies in particular to smash products and exact sequences of Hopf algebras, where the extension is free and hence faithfully flat, and to strong group gradings when faithful flatness holds.
  • For the family $U_q^{B,b}$ over a smooth commutative base $B$, the result gives homological smoothness with $\mathrm{cd}(U_q^{B,b}) \leq \mathrm{cd}(B) + 3$, so in particular the quantum enveloping algebra $U_q(\mathfrak{sl}_2)$ is smooth.
  • In the Galois-object case $B=k$ the theorem reduces to the statement that any Hopf-Galois object over a homologically smooth Hopf algebra is homologically smooth, recovering the earlier Calabi-Yau result as a special case.

Reading between the lines

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

  • Editorial inference: the collapse argument is degree-by-degree, so the same proof should yield a finite-presentability version—if $B$ and $H$ are $FP_\infty$ and the relevant Tor vanish, then $A$ is $FP_\infty$ as an $A$-bimodule—separating finiteness from finite dimension.
  • Editorial inference: the author's suspicion that $\mathrm{cd}(A)=\mathrm{cd}(B)+\mathrm{cd}(H)$ could be tested by computing the top Hochschild cohomology $\mathrm{HH}^{\mathrm{cd}(B)+\mathrm{cd}(H)}(A,A^e)$ in the quantum examples; a nonzero answer would make the dimension bound a principal-bundle-style dimension formula.
  • Editorial inference: because the only input from the example is that the extension is a free Hopf-Galois extension, the theorem should extend to multiparameter deformations and other cleft extensions over smooth bases, giving a broad source of new homologically smooth algebras.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proves an ascent theorem for homological smoothness along faithfully flat Hopf-Galois extensions. Specifically, Theorem 1.1 states that if H is a Hopf algebra with bijective antipode, B ⊂ A is an H-Galois extension with A faithfully flat as a left and right B-module, and both H and B are homologically smooth, then A is homologically smooth. The proof uses Stefan's spectral sequences (Theorem 3.7) and a Bieri-Eckmann criterion for FP∞ modules (Proposition 2.4). Proposition 4.1 adds a quantitative bound cd(A) ≤ cd(B)+cd(H). The final Section 5 gives an example involving quantum enveloping algebras U^{B,b}_q.

Significance. If correct, the main theorem is a useful and natural generalisation of known smoothness results for smash products and Galois objects, providing a uniform framework for producing new homologically smooth algebras from principal-bundle-like extensions. The proof is concise and elegantly combines a spectral-sequence collapse with the Bieri-Eckmann product criterion. The paper also points to the quantitative bound on cohomological dimension as a separate contribution. The example in Section 5 shows the theorem applies to familiar quantum algebras. However, the proof's reliance on several quoted results from [16], especially Proposition 4.4, means that the paper is not fully self-contained and the central argument is only as solid as those external statements.

major comments (3)
  1. [Section 4, proof of Theorem 1.1, after the H-linearity check] The collapse of the spectral sequence for M = ∏Ae depends on the assertion that Tor^H_p(kε, HH_0(B, Ae)) = 0 for p ≥ 1, which is quoted as [16, Proposition 4.4] with the only justification being that Ae is projective as an A-bimodule. The proposition is not stated in the paper, and the paper does not verify that its hypotheses cover the non-finitely generated projective bimodule Ae. If [16, Proposition 4.4] carries any finite-generation or other finiteness hypothesis, the vanishing of E^2_{p,0} for p ≥ 1 would not follow from the product commutation, and the argument yielding that A is FP∞ would break exactly at this step. This is load-bearing for Theorem 1.1. Please state the quoted proposition explicitly, including all hypotheses, and check that Ae satisfies them, or provide a direct proof.
  2. [Section 4, proof of Proposition 4.1] The proof asserts that for p > cd(H), the term E^{pq}_2 = Ext^p_{H^op}(kε, HH^q(B, M)) vanishes. This implicitly requires that the projective dimension of kε as a right H-module is at most cd(H). The paper does not prove or cite this fact; Theorem 2.7 only states an equivalence between homological smoothness of H and type-FPness of kε, not an equality (or inequality) of dimensions. Since Proposition 4.1 is used to conclude that cd(A) is finite, this is a load-bearing step. Please either prove that pdim_{H^op}(kε) ≤ cd(H), or provide a precise reference, or reformulate the proof using the projective dimension of kε as the relevant quantity.
  3. [Section 4, final paragraph of the proof of Theorem 1.1] The proof concludes that the composite isomorphism HH_0(A, ∏Ae) ≅ ∏ HH_0(A, Ae) is the natural map appearing in Proposition 2.4(iii), but only states 'It is not difficult to check' without providing the check. This identification is necessary to apply the Bieri-Eckmann criterion, so it is part of the load-bearing reasoning. A short commutative diagram or an explicit verification of the naturality of the composite would make the proof complete.
minor comments (4)
  1. [Section 4, proof of Theorem 1.1] The proof relies on several results from [16]—Lemma 2.1, Proposition 4.2, and Proposition 4.4—without stating them. Including their precise statements would make the paper substantially more readable and easier to verify.
  2. [Section 5, Proposition 5.2] The sentence 'If B is an homologically smooth commutative algebra' contains a grammatical error; it should be 'a homologically smooth'. Also, the reference [6, Proposition 3.2.1] is invoked without explaining exactly what it provides; a one-sentence clarification would help.
  3. [Remark 4.2] The forward reference to a forthcoming paper for the equality cd(A) = cd(B) + cd(H) is acceptable, but the sentence could be phrased more cautiously, e.g., 'we conjecture that equality holds in general'.
  4. [Throughout] There are minor typographical issues, such as the inconsistent use of 'FP' versus 'type FP' and the abbreviation 'F P∞' appearing with a space in one place. A careful proofreading pass would improve the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is an external-spectral-sequence argument; the only self-reference is a non-load-bearing forward pointer in Remark 4.2.

full rationale

The derivation chain is not circular. Theorem 1.1 is proven by applying Stefan's spectral sequence [16, Theorem 3.7] to the test bimodule M = ∏ Ae; the vanishing of E^2_{p,q} off (0,0) relies on [16, Lemma 2.1] (flatness of Ae over Be), [16, Proposition 4.4] (projectivity of HH_0(B, P) as an H-module for projective P), and [16, Proposition 4.2] (identification of the HH_0 terms). These are quoted prior results of other authors, not consequences of the theorem being proved, so any question about their validity is a correctness risk, not circularity. The paper does verify in the text the one isomorphism whose H-linearity is needed for the product commutation, namely HH_0(B, ∏ Ae) ≅ ∏ HH_0(B, Ae). Proposition 2.4 (Bieri–Eckmann) is an external criterion and is not equivalent to the desired smoothness conclusion. There is no fitting of parameters, no renaming of a known empirical pattern, and no uniqueness theorem imported from the author's own prior work. The only self-reference is a forward pointer in Remark 4.2 to a forthcoming paper on equality cd(A) = cd(B) + cd(H) in the exact-sequence case; that equality is not needed for Theorem 1.1, and the remark itself states the general equality remains open. Thus no load-bearing step reduces to its own input.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

Pure mathematics paper: no free parameters (no data fitting) and no invented entities. The ledger lists the unproved background results the central claim rests on, all quoted from published literature (Stefan 1995, Schauenburg-Schneider 2005, Wang-Yu-Zhang 2017, Bieri-Eckmann 1974). The paper contributes the synthesis: applying the Bieri-Eckmann product criterion to Stefan's spectral sequences. The only self-reference is a forward pointer to a forthcoming paper in Remark 4.2, which does not load the proof.

assumptions (9)
  • domain assumption Stefan's spectral sequences (Theorem 3.7): for an H-Galois extension B ⊂ A, if A is flat over B there is a spectral sequence Ext^p_{H^op}(kε, HH^q(B, M)) ⇒ HH^{p+q}(A, M), and if A is projective over B there is Tor^H_p(kε, HH_q(B, M)) ⇒ HH_{p+q}(A, M).
    Invoked in Proposition 4.1 (cohomology version) and in the proof of Theorem 1.1 (homology version); quoted from [16] without proof.
  • domain assumption Schauenburg-Schneider projectivity theorem (Remark 3.8, from [14, Theorem 4.10]): if H has bijective antipode and B ⊂ A is an H-Galois extension with A faithfully flat as left and right B-modules, then A is projective as left and right B-modules.
    Converts the faithful flatness hypothesis of Theorem 1.1 into the projectivity required to invoke the homology spectral sequence.
  • domain assumption [16, Lemma 2.1]: the enveloping algebra Ae = A ⊗ A^op is flat as a left and right Be-module.
    Used to conclude HH_q(B, Ae) = Tor^{Be}_q(Ae, B) = 0 for q ≥ 1, a key vanishing in the collapse of the spectral sequence.
  • domain assumption [16, Proposition 4.4]: for a projective A-bimodule P, the H-module HH_0(B, P) is projective, so Tor^H_p(kε, HH_0(B, P)) = 0 for p ≥ 1.
    The critical vanishing forcing E^2_{p,0} = 0 for p ≥ 1; quoted, not proved.
  • domain assumption [16, Proposition 4.2]: natural identification of HH_0(A, M) with Tor^H_0(kε, HH_0(B, M)) for A-bimodules M.
    Used in the final step of the proof of Theorem 1.1 to identify the (0,0) term and to compare with ∏HH_0(A, Ae).
  • domain assumption Theorem 2.7 (from [19, Proposition A.2]): a Hopf algebra H is homologically smooth if and only if the right H-module kε and the left H-module εk are of type FP.
    Provides the type FP∞ of kε, which is what makes Tor^H_p(kε, −) commute with direct products in the proof.
  • standard math Bieri-Eckmann criterion (Proposition 2.4 in the paper): a module is of type FP∞ if and only if Tor with it commutes with arbitrary direct products (equivalently, the product conditions in item (iii) hold).
    The operative criterion used to prove A is of type FP∞ as an Ae-module; stated in the paper with references [2] and [4].
  • standard math Characterization of type FP (Proposition 2.3): a module is of type FP if and only if it is of type FP∞ and has finite projective dimension.
    Reduces homological smoothness of A to finite cohomological dimension plus FP∞; cited from [5].
  • standard math Ground conventions: all algebras are unital algebras over a field k; standard homological algebra (Tor, Ext, spectral sequence convergence, faithful flatness) is assumed.
    Baseline assumptions stated in the Notations paragraph and Section 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Homological smoothness of Hopf-Galois extensions." pith.science (2026). https://pith.science/paper/J4PABN7J

@misc{pith2026241204365,
  author       = {Pith},
  title        = {Pith review of: Homological smoothness of Hopf-Galois extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4PABN7J}},
  note         = {Machine review of arXiv:2412.04365}
}
abstract

We show that if $H$ is a Hopf algebra with bijective antipode and $B \subset A$ is a faithfully flat $H$-Galois extension, then $A$ is homologically smooth if $H$ and $B$ are.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages

  1. [16]

    Hochschild cohomology on Hopf Galois extensions

    Stef an, D. Hochschild cohomology on Hopf Galois extensions. J. Pure Appl. Algebra 103 , 2 (1995), 221–233

    work page 1995

  2. [1]

    Extensions of Hopf algebras

    Andruskiewitsch, N., and Devoto, J. Extensions of Hopf algebras. St. Petersburg Math. J. 7 (1996), 17–52

    work page 1996

  3. [2]

    Homological dimension of discrete groups

    Bieri, R. Homological dimension of discrete groups . Queen Mary College, 1983

    work page 1983

  4. [3]

    Groups with homological duality

    Bieri, R., and Eckmann, B. Groups with homological duality. Invent. Math. 20 , 2 (1973), 103–124

    work page 1973

  5. [4]

    Finiteness properties of duality groups

    Bieri, R., and Eckmann, B. Finiteness properties of duality groups. Comment. Math. Helv. 49 (1974), 74–83

    work page 1974

  6. [5]

    Brown, K. S. Cohomology of groups , vol. 87 of Graduate Texts in Mathematics . Springer- Verlag, 1982

    work page 1982

  7. [6]

    Rigid dualizing complex for quantum enveloping algebras an d algebras of general- ized differential operators

    Chemla, S. Rigid dualizing complex for quantum enveloping algebras an d algebras of general- ized differential operators. J. Algebra 276 , 1 (2004), 80–102

    work page 2004

  8. [7]

    On the computation of Hopf 2-cocycles with an example of diagonal type

    García Iglesias, A., and Sánchez, J. On the computation of Hopf 2-cocycles with an example of diagonal type. Glasg. Math. J. 1965 , 1 (2023), 141–169

    work page 2023

Show all 21 references
  1. [8]

    Calabi-Yau algebras

    Ginzburg, V. Calabi-Yau algebras. arXiv:math/0612139

    arXiv

  2. [9]

    Crossed products for pointed hopf algebras

    Günther, R. Crossed products for pointed hopf algebras. Comm. Algebra 27 , 9 (1999), 4389– 4410

    1999

  3. [10]

    F., and Takeuchi, M

    Kreimer, H. F., and Takeuchi, M. Hopf algebras and Galois extensions of an algebra. Indiana Univ. Math. J. 30 , 5 (1981), 675–692

    1981

  4. [11]

    Smash products of Calabi–Yau algebras by Hopf algebras

    Le Meur, P. Smash products of Calabi–Yau algebras by Hopf algebras. J. Noncommut. Geom. 13, 3 (2019), 887–961

    2019

  5. [12]

    Hopf algebras and their actions on rings , vol

    Montgomery, S. Hopf algebras and their actions on rings , vol. 82 of CBMS Regional Confer- ence Series in Mathematics . American Mathematical Society, Providence, RI, 1993. 8

    1993

  6. [13]

    Hopf-Galois and bi-Galois extensions

    Schauenburg, P. Hopf-Galois and bi-Galois extensions. In Galois theory, Hopf algebras, and semiabelian categories, vol. 43 of Fields Inst. Commun. Amer. Math. Soc., Providence, RI, 2004, pp. 469–515

    2004

  7. [14]

    On generalized Hopf galois extensions

    Schauenburg, P., and Schneider, H. On generalized Hopf galois extensions. J. Pure Appl. Algebra 202, 1-3 (2005), 168–194

    2005

  8. [15]

    Representation theory of Hopf galois extensions

    Schneider, H. Representation theory of Hopf galois extensions. Israel J. Math. 72 , 1-2 (1990), 196–231

    1990

  9. [17]

    A relation between Hochschild homology and cohomology for g orenstein rings

    V an Den Bergh, M. A relation between Hochschild homology and cohomology for g orenstein rings. Proc. Amer. Math. Soc. 126 , 5 (1998), 1345–1348

    1998

  10. [18]

    A relation between Hochschild homology and coh omology for gorenstein rings

    V an Den Bergh, M. Erratum to : "A relation between Hochschild homology and coh omology for gorenstein rings". Proc. Amer. Math. Soc 130 , no 8 (2002), 2809–2810

    2002

  11. [19]

    Calabi-Yau property under monoidal Morita-Takeuchi equivalence

    W ang, X., Yu, X., and Zhang, Y. Calabi-Yau property under monoidal Morita-Takeuchi equivalence. Pac. J. Math. 290 , 2 (2017), 481–510

    2017

  12. [20]

    Introduction to homological algebra

    Weibel, C. Introduction to homological algebra . Cambridge University Press, 2008

    2008

  13. [21]

    Hopf-Galois objects of Calabi-Yau Hopf algebras

    Yu, X. Hopf-Galois objects of Calabi-Yau Hopf algebras. J. Algebra Appl. 15 , 10 (2016), 1650194, 19. Université Clermont Auvergne, CNRS, LMBP, F-63000 CLERMON T-FERRAND, FRANCE 9

    2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.