{"id":"47f20571-ef68-42d2-9359-1eb546ca0c9b","arxiv_id":"2412.04365","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A faithfully flat Hopf-Galois extension of a homologically smooth algebra by a homologically smooth Hopf algebra is homologically smooth.","lead":"This mathematics paper proves that homological smoothness, a finiteness property of algebras at the heart of noncommutative geometry, is inherited by Hopf-Galois extensions: if the base algebra and the symmetry Hopf algebra are smooth, so is the total algebra. The result unifies two known special cases and yields the dimension bound cd(A) ≤ cd(B) + cd(H), producing new smooth algebras such as quantum algebras over smooth commutative rings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The collapse step for M=∏Ae depends on unverified [16, Prop. 4.4]; if that quoted projectivity statement fails for the non-finitely-generated bimodule Ae, Theorem 1.1 is unsupported.","rationale":"The paper is coherent and the main theorem is plausible; my stress test found no internal contradiction. The proof is a clean reduction to three quoted statements from Stefan's paper, and the risk is that one of them, especially [16, Prop. 4.4] with P = Ae, may not apply as stated. This is exactly the kind of missing verification that makes a paper correct-but-provisional: the theorem should be accepted once the quoted results are confirmed, and once the final naturality check is written down. Because the reader already identified the external quotes as the weakest assumption and assigned CONDITIONAL, my read does not change the verdict.","tokens_in":8275,"tokens_out":16982,"duration_ms":178532,"concrete_test":"Verify Stefan [16, Prop. 4.4] in the original and re-derive it for the non-finitely-generated projective bimodule P = Ae, using the explicit H-action of §3.2 in the model case A = B#H (e.g. H = kG with G finite, B smooth commutative). If the proposition requires finite generation, or if the computation yields a non-projective H-module for some index set I, then the collapse in §4 needs a new argument and the proof of Theorem 1.1 does not go through as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive point in §4 is the collapse of Stefan's spectral sequence for M = ∏_{i∈I} Ae. The paper verifies that HH_0(B,∏Ae) ≅ ∏ HH_0(B,Ae) is H-linear, but the two vanishing facts needed for E^2_{p,q} = 0 off (0,0) are imported from [16]: for q ≥ 1, Ae is flat over Be ([16, Lem. 2.1]); for p ≥ 1, q = 0, HH_0(B,Ae) is projective as an H-module ([16, Prop. 4.4]). The latter is the most load-bearing: P = Ae is a projective A-bimodule, but not a finitely generated one, and the paper neither states nor proves the proposition. If [16, Prop. 4.4] carries an unstated finite-generation or other finiteness hypothesis, then the product commutation does not force E^2_{p,0} to vanish, and the collapse fails exactly at the step that yields FP∞ of A. The final identification of the HH_0 isomorphism with the Bieri–Eckmann natural map is also left as 'It is not difficult to check', but that is secondary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8395,"tokens_out":16139,"duration_ms":141634,"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":[{"comment":"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.","section":"Section 4, proof of Theorem 1.1, after the H-linearity check"},{"comment":"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.","section":"Section 4, proof of Proposition 4.1"},{"comment":"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.","section":"Section 4, final paragraph of the proof of Theorem 1.1"}],"minor_comments":[{"comment":"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.","section":"Section 4, proof of Theorem 1.1"},{"comment":"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.","section":"Section 5, Proposition 5.2"},{"comment":"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'.","section":"Remark 4.2"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and useful question, and the overall strategy is sound if the quoted results from [16] are correct in the required generality. The main concern is that the proof rests on several statements from a 1995 paper that are not reproduced, and at least one of them (Proposition 4.4) is used in a potentially delicate situation (non-finitely generated projective bimodule). The author should be encouraged to make the paper self-contained regarding those quoted results. This is not a rejection-level issue, but it does require a substantive revision before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper proves a clean, useful theorem—homological smoothness ascends along faithfully flat Hopf-Galois extensions, with the additive cohomological-dimension bound. It genuinely covers the earlier special cases (Yu's Galois objects, Le Meur's smash products) as instances of one argument, and the proof strategy is a nice combination of Stefan's spectral sequences with the Bieri–Eckmann Tor-criterion. I followed the proof as written and the internal steps are coherent; the H-linearity check for the product isomorphism is explicit, and the paper is honest that equality cd(A)=cd(B)+cd(H) is not proved in general (Remark 4.2).\n\nThe soft spot is exactly what the stress-test note flags. The collapse of the spectral sequence for M=∏Ae depends on two quoted results from Stefan's 1995 paper: [16, Lemma 2.1] (flatness of Ae over Be) and, more importantly, [16, Proposition 4.4] (projectivity of HH_0(B,P) as an H-module for a projective A-bimodule P). Neither is stated or proved in the text. If Proposition 4.4 carries an unstated finiteness hypothesis that the non-finitely-generated bimodule Ae fails, then the vanishing of E^2_{p,0} for p≥1 is not justified and the FP∞ argument collapses. I cannot check the original paper from here, so the conditional verdict is appropriate. This is standard citation practice, not a red flag, but it is load-bearing enough that a referee should check it explicitly. The 'It is not difficult to check' final naturality is a minor point; a referee could ask for a sentence.\n\nThe example in Section 5 (quantized enveloping algebra families) is a genuine illustration and correctly uses the theorem. No self-citation circularity; the forward pointer in Remark 4.2 is not used.\n\nBottom line: the result is likely correct and worth having in the literature. A serious referee should verify that [16, Proposition 4.4] applies to arbitrary projective bimodules, and ask the author to include the precise statement. If that check passes, accept. This is a good paper for people in Hopf-Galois theory, Calabi-Yau algebras, and homological finiteness.","headline":"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.","tokens_in":9118,"tokens_out":3524,"would_cite":true,"duration_ms":34600,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E40","16T05","18G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Homological smoothness is inherited by any faithfully flat Hopf-Galois extension whose Hopf algebra and coinvariant subalgebra are homologically smooth.","keywords":["homologically smooth algebra","Hopf-Galois extension","Hochschild homology","type FP∞","Stefan spectral sequence","cohomological dimension","quantum enveloping algebra","Bieri-Eckmann criterion"],"falsifier":"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.","tokens_in":7893,"feed_emoji":"","tokens_out":10316,"duration_ms":96859,"temperature":0.7,"pith_summary":"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 $\\mathrm{cd} \\leq \\mathrm{cd}(B) + 3$.","feed_headline":"Smoothness ascends Hopf-Galois extensions from base and Hopf algebra","feed_subtitle":"A spectral-sequence collapse lifts smoothness to A and bounds its cohomological dimension by cd(B)+cd(H).","key_machinery":"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 $M=\\prod A^e$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Stefan spectral sequences and the three cited facts (flatness of $A^e$ over $B^e$, projectivity of $\\mathrm{HH}_0(B,P)$ for projective $P$, and identification of $\\mathrm{HH}_0(A,M)$) that collapse the sequence.","marker":"[16]"},{"why":"Corollary 1.6 gives the Tor-against-products criterion for type $FP_\\infty$ that is used to test the bimodule $A$.","marker":"[2]"},{"why":"Proves the implication that vanishing of Tor against arbitrary products forces $FP_\\infty$, the key direction of Proposition 2.4.","marker":"[4]"},{"why":"Shows a faithfully flat Hopf-Galois extension is projective as a left and right $B$-module, allowing the homology spectral sequence Theorem 3.7(ii) to be used.","marker":"[14]"},{"why":"Provides Theorem 2.7, characterizing homological smoothness of a Hopf algebra by the type $FP$ of the trivial module $k_\\varepsilon$, which is how smoothness of $H$ enters.","marker":"[19]"},{"why":"Proves that $B \\subset U_q^{B,b}$ is a cleft, in particular free, $U_q(\\mathfrak{sl}_2)$-Galois extension, making the example an instance of Theorem 1.1.","marker":"[9]"},{"why":"Gives the rigid dualizing complex computation for $U_q(\\mathfrak{sl}_2)$ used to bound the cohomological dimension in the example.","marker":"[6]"}],"fun_headline_variants":["Smoothness climbs Hopf-Galois extensions","Hopf-Galois: smooth H and B force smooth A","When H and B are smooth, so is the Hopf-Galois total algebra","Faithful Hopf-Galois extensions inherit homological smoothness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Smoothness climbs Hopf-Galois extensions","Hopf-Galois: smooth H and B force smooth A","When H and B are smooth, so is the Hopf-Galois total algebra","Faithful Hopf-Galois extensions inherit homological smoothness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000861,"raw_usage":{"total_tokens":3665,"prompt_tokens":802,"completion_tokens":2863,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":418,"completion_tokens_details":{"reasoning_tokens":2787}},"tokens_in":418,"tokens_out":2863,"duration_ms":20383,"temperature":1.0,"reasoning_tokens":2787,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:34:55.674907+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hochschild cohomology on Hopf Galois extensions","cited_arxiv_id":null,"evidence_quote":"Supplies the Stefan spectral sequences and the three cited facts (flatness of $A^e$ over $B^e$, projectivity of $\\mathrm{HH}_0(B,P)$ for projective $P$, and identification of $\\mathrm{HH}_0(A,M)$) that collapse the sequence."},{"cited_title":"Homological dimension of discrete groups","cited_arxiv_id":null,"evidence_quote":"Corollary 1.6 gives the Tor-against-products criterion for type $FP_\\infty$ that is used to test the bimodule $A$."},{"cited_title":"Finiteness properties of duality groups","cited_arxiv_id":null,"evidence_quote":"Proves the implication that vanishing of Tor against arbitrary products forces $FP_\\infty$, the key direction of Proposition 2.4."},{"cited_title":"On generalized Hopf galois extensions","cited_arxiv_id":null,"evidence_quote":"Shows a faithfully flat Hopf-Galois extension is projective as a left and right $B$-module, allowing the homology spectral sequence Theorem 3.7(ii) to be used."},{"cited_title":"Calabi-Yau property under monoidal Morita-Takeuchi equivalence","cited_arxiv_id":null,"evidence_quote":"Provides Theorem 2.7, characterizing homological smoothness of a Hopf algebra by the type $FP$ of the trivial module $k_\\varepsilon$, which is how smoothness of $H$ enters."},{"cited_title":"Crossed products for pointed hopf algebras","cited_arxiv_id":null,"evidence_quote":"Proves that $B \\subset U_q^{B,b}$ is a cleft, in particular free, $U_q(\\mathfrak{sl}_2)$-Galois extension, making the example an instance of Theorem 1.1."},{"cited_title":"Rigid dualizing complex for quantum enveloping algebras an d algebras of general- ized differential operators","cited_arxiv_id":null,"evidence_quote":"Gives the rigid dualizing complex computation for $U_q(\\mathfrak{sl}_2)$ used to bound the cohomological dimension in the example."}],"review_version":1}