Pith. sign in

REVIEW 2 major objections 5 minor 51 references

On the finite generation of the cohomology of bosonizations

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper proves that bosonizations of suitable braided Hopf algebras by suitable finite-dimensional Hopf algebras have finitely generated cohomology, using deformation sequences of algebras and Hopf algebras.

desk verdict Genuinely new propagation theorems for fgc of smash products and bosonizations, with real applications; the one worrisome transfer in Section 4 is covered by Cartier duality and should be stated explicitly, but the paper deserves a serious referee. read the letter →

arxiv 2506.05267 v1 pith:MHNR73NQ submitted 2025-06-05 math.QA math.KTmath.RA

Nicolás Andruskiewitsch , David Jaklitsch , Van C. Nguyen , Amrei Oswald , Julia Plavnik , Anne V. Shepler , Xingting Wang This is my paper
classification math.QAmath.KTmath.RA MSC 16T0516E4018M05
keywords bosonizationsmashproductfinitegenerationofcohomologydeformationsequenceNicholsalgebrascocommutativeHopftensorcategories
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

The paper attacks a standing conjecture on finite tensor categories by asking when a bosonization, the Hopf algebra obtained by combining a braided Hopf algebra $R$ with a finite-dimensional Hopf algebra $K$, inherits finite generation of cohomology from its parts. It shows that if $R$ sits in a deformation sequence $Z \hookrightarrow Q \twoheadrightarrow R$, with $Z$ a smooth central subalgebra over which $Q$ is finite and flat and $Q$ of finite global dimension, then the smash product $R\rtimes K$ has finitely generated cohomology in three settings: $K$ semisimple; $K$ cocommutative and the sequence $K$-equivariant; or $K$ itself admitting a deformation sequence of Hopf algebras and $R$ admitting a compatible equivariant sequence. On the bosonization side, these results give $R\#K$ with finitely generated cohomology for Nichols algebras of quantum lines, quantum linear spaces, Cartan-type diagonal data, the restricted Jordan plane, and direct sums of Jordan blocks with points labeled by $1$, recovering some known cases and adding new ones. A positive answer to the paper's Question 1.1 follows in these classes.

What carries the argument

The central mechanism is the deformation sequence, a triple $Z \hookrightarrow Q \twoheadrightarrow R$ in which $Z$ is a smooth finitely generated central subalgebra, $Q$ is finitely generated and flat over $Z$ with finite global dimension, and $R$ is the finite-dimensional quotient at the augmentation ideal of $Z$. The paper also uses formal, $K$-equivariant, and Hopf-algebra versions of this notion. The key fact, carried over from [34], is that any finite-dimensional augmented algebra admitting such a sequence has finitely generated cohomology: the formal completion converts the sequence into a setup where a symmetric algebra $A_Z$ controls all $\operatorname{Ext}$ modules. In the equivariant setting, the dg algebra $T = \operatorname{REnd}_{K\otimes_Z K^{\mathrm{op}}}(\mathcal K)$, with cohomology $A_Z$, acts as the finite-generation witness, and Proposition 5.8 shows that smashing a Hopf-algebra deformation sequence with an equivariant one again yields a deformation sequence.

What would settle it

Verify Lemma 4.8 for a finite-dimensional cocommutative Hopf algebra outside the finite-group-scheme class, such as the restricted enveloping algebra of a non-abelian restricted Lie algebra in positive characteristic; if the cohomology of $T$ is not affine or the module $\operatorname{RHom}_{\mathcal K_Q}(M,N)$ is not Noetherian over $H(T)$, then the transfer step in Theorem 4.11 fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that finite generation of cohomology for smash products can be proved by finding a deformation sequence for the algebra $R$ rather than by computing its cohomology ring. Theorem 3.8 shows that if $K$ is semisimple and $R$ is a finite-dimensional augmented $K$-module algebra admitting a deformation sequence, then $R\rtimes K$ has finitely generated cohomology. Theorem 4.11 relaxes the hypothesis on $K$ to cocommutativity at the cost of requiring the deformation sequence to be $K$-equivariant, adapting the equivariant dg-algebra construction of [33]. Theorem 5.9 treats $K$ admitting a deformation sequence of Hopf algebras $C$ together with a $C$-equivariant deformation sequence for $R$; Proposition 5.8 assembles these into a single deformation sequence $Z\otimes W \hookrightarrow Q\rtimes H \twoheadrightarrow R\rtimes K$, so that the deformation-sequence criterion applies directly. Because bosonizations are smash products, the same statements prove finite generation for $R\#K$, and the Nichols-algebra applications follow by exhibiting such deformation sequences through pre-Nichols algebras.

Load-bearing premise

The load-bearing premise is that the equivariant dg-algebra machinery proven for finite group schemes transfers word-for-word to arbitrary finite-dimensional cocommutative Hopf algebras; Lemmas 4.6 and 4.8 assert this transfer rather than prove it in detail.

Editorial extensions

If this is right

  • When $K$ is semisimple and $R$ admits a deformation sequence, the smash product $R\rtimes K$ has finitely generated cohomology, and the same holds for the bosonization $R\#K$ (Theorem 3.8 and Corollary 3.9).
  • When $K$ is cocommutative and $R$ admits a $K$-equivariant deformation sequence, $R\rtimes K$ and $R\#K$ have finitely generated cohomology (Theorem 4.11 and Corollary 4.12).
  • When $K$ admits a deformation sequence of Hopf algebras $C$ and $R$ admits a $C$-equivariant deformation sequence, $R\rtimes K$ and $R\#K$ have finitely generated cohomology (Theorem 5.9 and Corollary 5.10).
  • Question 1.1 now has a positive answer in these three classes, giving new evidence for the finite-generation conjecture for finite tensor categories.
  • The applications recover known finite-generation results for bosonizations of quantum lines, quantum linear spaces, and the restricted Jordan plane, and add new cases for Cartan-type diagonal Nichols algebras and direct sums of Jordan blocks with points labeled by $1$.

Reading between the lines

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

  • Editorial inference: the paper's strategy suggests a general recipe for Question 1.1, namely prove finite generation for $R$ by finding any deformation sequence and then look for a compatible action of $K$; cohomology computations are replaced by deformation-theoretic data.
  • Editorial inference: the paper's discussion of diagonal type indicates that extending the method to super or modular types would require different deformation sequences, because a nilpotent degree-one generator forces infinite global dimension in the distinguished pre-Nichols algebra.
  • Editorial inference: Theorem 4.11 should plausibly hold for any power reductive finite-dimensional Hopf algebra, not only cocommutative ones, since power reductivity is what Proposition 4.9 actually uses; re-checking Lemmas 4.6 and 4.8 under that hypothesis is a testable extension.
  • Editorial inference: the category of deformation sequences of Hopf algebras, defined only in passing in the paper, may itself carry homological information; understanding which finite-dimensional Hopf algebras admit such sequences would sharpen the boundary of Question 1.1.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops a deformation-sequence framework, extending work of Negron and Pevtsova, to prove finite generation of cohomology (fgc) for smash products R⋊K under three sets of hypotheses: K semisimple and R admits a deformation sequence (Theorem 3.8); K cocommutative and R admits a K-equivariant deformation sequence (Theorem 4.11); and K admits a deformation sequence of Hopf algebras while R admits a compatible equivariant deformation sequence (Theorem 5.9). In each case the corresponding bosonizations R#K for R a Hopf algebra in the Yetter–Drinfeld category are shown to have fgc, giving new positive instances of Question 1.1. Applications include quantum lines, quantum linear spaces, Nichols algebras of Cartan type, the restricted Jordan plane, and a class of Nichols algebras in positive characteristic.

Significance. If the results are correct, they provide a substantial extension of the known cases of the Etingof–Ostrik conjecture and of Question 1.1, unifying several earlier results through a single deformation-sequence technique. The paper contains detailed proofs of the main framework (Propositions 4.9 and 5.8 are new and reasonably self-contained) and a rich set of applications that recover old theorems and produce genuinely new examples. The main theorems are clearly stated and the paper is generally well organized. The most valuable contributions are the equivariant deformation-sequence machinery (Theorem 4.11) and the deformation-sequence-of-Hopf-algebras technique (Theorem 5.9), both of which are likely to stimulate further work.

major comments (2)
  1. The proof of Theorem 3.8 applies Lemma 2.13 to the graded algebra A = H(R,k), but no justification is given that H(R,k) is a K-module algebra for an arbitrary finite-dimensional augmented K-module algebra R. For a general Hopf algebra K, and even a semisimple one, the cup product on Hochschild cohomology is not automatically K-linear; this property is usually established when R is a bialgebra in the Yetter–Drinfeld category (as in Lemma 2.14) or when K is cocommutative. Without this structure the smash product H(R,k)⋊K is not defined and Lemma 2.13 cannot be invoked. The main bosonization applications are safe because Corollary 3.9 assumes R is a Hopf algebra in K KYD, so Lemma 2.14 applies, but the statement of Theorem 3.8 as a general smash-product result is not supported by the proof as written. Please either prove that H(R,k) is a K-module algebra under the stated hypotheses, add the necessary hypothesis, or restrict the theorem to the setting in which the module-algebra structure is available.
  2. The transfer of [33, Lemma 2.4] and [33, Theorem 5.4] from finite group schemes to arbitrary finite-dimensional cocommutative Hopf algebras is asserted without proof. Since Lemma 4.8 is load-bearing for Theorem 4.11 and for the applications in Sections 6.5 and 6.6, the paper should include a short reduction: over an algebraically closed field, a finite-dimensional cocommutative Hopf algebra K is isomorphic to the group algebra kG of a finite group scheme G (take G = Spec(K^*)), so Rep(K) is equivalent to Rep(G) and Negron's results apply verbatim. Without this remark, the proof of Theorem 4.11 relies on an unverified extension of cited results.
minor comments (5)
  1. The dgK-module algebra in Lemma 4.7 is denoted by the same symbol K as the Hopf algebra K, which is confusing in Section 4; please use a different notation (for example a script K or the symbol Κ).
  2. In the proof of (iv)⇒(v), the claim that H(S) is Noetherian relies on Lemma 2.9(i); this use should be stated explicitly for readability.
  3. In verifying condition (e) of Definition 3.1, the application of Corollary A.2 is terse; it would be helpful to explicitly note that conditions (k) and (l) of Definition 5.7 guarantee the required restrictions of the twisting map to bijections.
  4. There is a typo in the statement: “admiting” should be “admitting”.
  5. Equation (6.1) states “q is a root of 1, and q ≠ 1 when chark = 0”, but the subsequent definition of n also covers q = 1 in positive characteristic; the phrasing is slightly ambiguous and could be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems propagate finite generation from external deformation-sequence machinery, with the equivariant transfer resting on Negron's independent work.

full rationale

After walking the derivation chain, I find no circular step. The central mechanism is external: Theorem 3.7 (fgc for augmented algebras admitting a deformation sequence) is quoted from Negron-Pevtsova [34] with a proof sketch, and Theorem 3.8 reduces smash-product fgc to that theorem together with invariant-ring arguments (Lemmas 2.12-2.13) that are proven in the text. The equivariant results in Section 4 rely on Negron's [33] for the dg K-module algebra constructions (Lemmas 4.6-4.8); [33] is by Cris Negron, not by the present authors, and the adaptation from finite group schemes to cocommutative Hopf algebras is justified by Cartier duality, so this is a legitimate transfer rather than an assumption of the conclusion. Lemma 4.10 is attributed to [36], which overlaps with two of the present authors, but its proof is sketched in the text via the Lyndon-Hochschild-Serre spectral sequence, so the content is not merely assumed. Section 5 composes a deformation sequence of Hopf algebras with a C-equivariant deformation sequence (Proposition 5.8) and then applies Theorem 3.7; the proof of Proposition 5.8 checks the conditions of Definition 3.1 directly. Section 6 supplies explicit deformation sequences (e.g., M_q: k[X^n] -> k[X] -> k[X]/(X^n)) and verifies equivariance via YD-pairs; these are concrete constructions, not fitted parameters. No prediction is statistically forced, no target is defined in terms of itself, and no load-bearing claim rests solely on a self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its logical inputs are the deformation-sequence criterion from [34], the equivariant dg-algebra results from [33], and known structural facts about pre-Nichols algebras. The main hypotheses, such as semisimplicity or cocommutativity of K and the existence of deformation sequences, are domain assumptions taken from prior literature.

assumptions (5)
  • domain assumption A finite-dimensional augmented algebra admitting a deformation sequence has fgc [34, Theorem 3.7].
    Used as the engine in Theorems 3.8, 4.11, and 5.9; cited from Negron and Pevtsova rather than proved in this paper.
  • domain assumption The equivariant dg algebra construction and its finite-generation consequences hold for finite-dimensional cocommutative Hopf algebras as in [33, Lemmas 3.1, 5.2, Theorem 5.4].
    Lemmas 4.7 and 4.8 transfer these results from finite group schemes; the proof of the transfer is not reproduced.
  • domain assumption Finite-dimensional cocommutative Hopf algebras are power reductive and have fgc [23, 47].
    These facts anchor Theorem 4.11 and Proposition 4.9; they are cited from the literature.
  • domain assumption For Cartan-type q satisfying (6.11), the distinguished pre-Nichols algebra eB(V_q) has finite global dimension and (6.10) is a deformation sequence [6, Subsection 4.5].
    Used in Proposition 6.10; verification is referenced rather than carried out in the paper.
  • domain assumption The Jordan plane J and the pre-Nichols algebra for sums of blocks admit deformation sequences into their Nichols quotients [4, 19].
    Used in Sections 6.5 and 6.6; finite global dimension and the deformation sequence are asserted rather than fully proved in the text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the finite generation of the cohomology of bosonizations." pith.science (2026). https://pith.science/paper/MHNR73NQ

@misc{pith2026250605267,
  author       = {Pith},
  title        = {Pith review of: On the finite generation of the cohomology of bosonizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHNR73NQ}},
  note         = {Machine review of arXiv:2506.05267}
}
read the original abstract

We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated cohomology. In fact, our results are shown in more generality for smash products. As applications, we prove the bosonizations of some Nichols algebras (such as Nichols algebras of diagonal type, the restricted Jordan plane, Nichols algebras of direct sums of Jordan blocks plus points labeled with 1), by some suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as well as providing new examples.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

51 extracted references · 49 canonical work pages

  1. [3]

    Nicolás Andruskiewitsch, Iván Angiono, and István Heckenberger,Examples of finite-dimensional pointed Hopf algebras in positive characteristic, Representation theory, mathematical physics, and integrable systems, (2021), pp. 1–38

    work page 2021

  2. [1]

    Nicolás Andruskiewitsch,An introduction to Nichols algebras, Quantization, geometry and noncom- mutative structures in mathematics and physics, 2017, pp. 135–195

    work page 2017

  3. [2]

    Nicolás Andruskiewitsch and Iván Angiono,On finite dimensional Nichols algebras of diagonal type, Bull. Math. Sci.7(2017), no. 3, 353–573

    work page 2017

  4. [4]

    Nicolás Andruskiewitsch, Iván Angiono, and István Heckenberger,On finite GK-dimensional Nichols algebras over abelian groups, Mem. Amer. Math. Soc.271(2021), no. 1329, ix+125. MR4298502

    work page 2021

  5. [5]

    Nicolás Andruskiewitsch, Iván Angiono, Julia Pevtsova, and Sarah Witherspoon,Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups, Res. Math. Sci.9(2022), no. 1, Paper No. 12, 132

    work page 2022

  6. [6]

    Math.428(2023), Paper No

    Nicolás Andruskiewitsch, Iván Angiono, and Milen Yakimov,Poisson orders on large quantum groups, Adv. Math.428(2023), Paper No. 109134, 66

    work page 2023

  7. [7]

    1, 22–61

    Nicolás Andruskiewitsch and Jorge Devoto,Extensions of Hopf algebras, Algebra i Analiz7(1995), no. 1, 22–61

    work page 1995

  8. [8]

    Pure Appl

    Nicolás Andruskiewitsch and Sonia Natale,Braided Hopf algebras arising from matched pairs of groups, J. Pure Appl. Algebra182(2003), no. 2-3, 119–149. MR1903050

    work page 2003

Show all 51 references
  1. [9]

    Nicolás Andruskiewitsch and Sonia Natale,On the finite generation of the cohomology of abelian extensions of Hopf algebras, Doc. Math. (2025). To appear

    2025

  2. [10]

    Nicolás Andruskiewitsch and Guillermo Sanmarco,Finite GK-dimensional pre-Nichols algebras of quantum linear spaces and of Cartan type, Trans. Amer. Math. Soc. Ser. B8(2021), 296–329. 32

    2021

  3. [11]

    Algebra209(1998), no

    Nicolás Andruskiewitsch and Hans-Jürgen Schneider,Lifting of quantum linear spaces and pointed Hopf algebras of orderp3, J. Algebra209(1998), no. 2, 658–691

    1998

  4. [12]

    Nicolás Andruskiewitsch and Hans-Jürgen Schneider,Pointed Hopf algebras, New directions in Hopf algebras, 2002, pp. 1–68

    2002

  5. [13]

    Reine Angew

    Iván Angiono,On Nichols algebras of diagonal type, J. Reine Angew. Math.683(2013), 189–251

    2013

  6. [14]

    Groups21(2016), no

    Iván Angiono,Distinguished pre-Nichols algebras, Transform. Groups21(2016), no. 1, 1–33

    2016

  7. [15]

    Noncommut

    Iván Angiono, Emiliano Campagnolo, and Guillermo Sanmarco,Finite GK-dimensional pre-Nichols algebras of (super)modular and unidentified type, J. Noncommut. Geom.17(2023), no. 2, 499–525

    2023

  8. [16]

    Atiyah and Ian G

    Michael F. Atiyah and Ian G. Macdonald,Introduction to commutative algebra, Student economy edition, Boulder: Westview Press, 2016

    2016

  9. [17]

    Roman Bezrukavnikov and Victor Ginzburg,On deformations of associative algebras, Ann. of Math. (2)166(2007), no. 2, 533–548

    2007

  10. [18]

    Andreas Cap, Hermann Schichl, and Jiří Vanžura,On twisted tensor products of algebras, Comm. Alg. 23(1995), no. 12, 4701–4735

    1995

  11. [19]

    Claude Cibils, Aaron Lauve, and Sarah Witherspoon,Hopf quivers and Nichols algebras in positive characteristic., Proc. Am. Math. Soc.137(2009), no. 12, 4029–4041

    2009

  12. [20]

    Categ.41(2024), Paper No

    Javier Cóppola and Andrea Solotar,Graded braided commutativity in Hochschild cohomology, Theory Appl. Categ.41(2024), Paper No. 46, 1596–1643

    2024

  13. [21]

    Pavel Etingof and Viktor Ostrik,Finite tensor categories, Mosc. Math. J.4(2004), no. 3, 627–654, 782–783

    2004

  14. [22]

    Leonard Evens,The cohomology ring of a finite group, Trans. Am. Math. Soc.101(1961), 224–239

    1961

  15. [23]

    Friedlander and Andrei Suslin,Cohomology of finite group schemes over a field, Invent

    Eric M. Friedlander and Andrei Suslin,Cohomology of finite group schemes over a field, Invent. Math. 127(1997), no. 2, 209–270

    1997

  16. [24]

    Math.220(2009), no

    István Heckenberger,Classification of arithmetic root systems, Adv. Math.220(2009), no. 1, 59–124

    2009

  17. [25]

    arXiv:2503.12268

    Huan Jia and Yinhuo Zhang,Affineness on Noetherian graded rings, algebras and Hopf algebras(2025). arXiv:2503.12268

    2025 arXiv

  18. [26]

    Kirkman, James J

    Ellen E. Kirkman, James J. Kuzmanovich, and James J. Zhang,Gorenstein subrings of invariants under Hopf algebra actions, J. Algebra322(2009), no. 10, 3640–3669

    2009

  19. [27]

    Krause and Thomas H

    Günter R. Krause and Thomas H. Lenagan,Growth of algebras and Gelfand-Kirillov dimension., Revised ed., Grad. Stud. Math., vol. 22, Providence, RI: American Mathematical Society, 2000

    2000

  20. [28]

    Lorenz and Martin Lorenz,On crossed products of Hopf algebras, Proc

    Maria E. Lorenz and Martin Lorenz,On crossed products of Hopf algebras, Proc. of AMS 123123 (1995), 33–38

    1995

  21. [29]

    Mitja Mastnak, Julia Pevtsova, Peter Schauenburg, and Sarah Witherspoon,Cohomology of finite- dimensional pointed Hopf algebras., Proc. Lond. Math. Soc. (3)100(2010), no. 2, 377–404

    2010

  22. [30]

    McConnell and J

    John C. McConnell and J. Chris Robson,Noncommutative Noetherian rings. With the cooperation of L. W. Small., Reprinted with corrections from the 1987 original, Grad. Stud. Math., vol. 30, Providence, RI: American Mathematical Society (AMS), 2001

    1987

  23. [31]

    Milne,Algebraic groups

    James S. Milne,Algebraic groups. The theory of group schemes of finite type over a field, Camb. Stud. Adv. Math., vol. 170, Cambridge: Cambridge University Press, 2017

    2017

  24. [32]

    82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993

    Susan Montgomery,Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993

    1993

  25. [33]

    Cris Negron,Finite generation of cohomology for drinfeld doubles of finite group schemes, Selecta Mathematica27(2021), 1–20

    2021

  26. [34]

    3, 1882–1958

    Cris Negron and Julia Pevtsova,Support for integrable Hopf algebras via noncommutative hypersur- faces, International Mathematics Research Notices2023(2023), no. 3, 1882–1958

    2023

  27. [35]

    Nguyen, Xingting Wang, and Sarah Witherspoon,Finite generation of some cohomology rings via twisted tensor product and Anick resolutions, J

    Van C. Nguyen, Xingting Wang, and Sarah Witherspoon,Finite generation of some cohomology rings via twisted tensor product and Anick resolutions, J. Pure Appl. Algebra223(2019), no. 1, 316–339

    2019

  28. [36]

    Nguyen, Xingting Wang, and Sarah Witherspoon,New approaches to finite generation of cohomology rings, J

    Van C. Nguyen, Xingting Wang, and Sarah Witherspoon,New approaches to finite generation of cohomology rings, J. Algebra587(2021), 390–428

    2021

  29. [37]

    Math.291(2016), 584–620

    Dragoş Ştefan and Cristian Vay,The cohomology ring of the 12-dimensional Fomin-Kirillov algebra, Adv. Math.291(2016), 584–620. 33

    2016

  30. [38]

    Radford,Hopf algebras, Series on Knots and Everything, vol

    David E. Radford,Hopf algebras, Series on Knots and Everything, vol. 49, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012

    2012

  31. [39]

    Algebra 152(1992), no

    Hans-Jürgen Schneider,Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152(1992), no. 2, 289–312

    1992

  32. [40]

    Algebra21 (1993), no

    Hans-Jürgen Schneider,Some remarks on exact sequences of quantum groups, Comm. Algebra21 (1993), no. 9, 3337–3357

    1993

  33. [41]

    Shepler and Sarah Witherspoon,Twisted tensor products: Alexander-Whitney and Eilenberg- Zilber maps(2024).arXiv:2411.04102

    Anne V. Shepler and Sarah Witherspoon,Twisted tensor products: Alexander-Whitney and Eilenberg- Zilber maps(2024).arXiv:2411.04102

    2024 arXiv

  34. [42]

    Math.183(2004), no

    Serge Skryabin,Invariants of finite Hopf algebras, Adv. Math.183(2004), no. 2, 209–239

    2004

  35. [43]

    Algebra306(2006), no

    Serge Skryabin,New results on the bijectivity of antipode of a Hopf algebra, J. Algebra306(2006), no. 2, 622–633

    2006

  36. [44]

    Serge Skryabin,Flatness over PI coideal subalgebras, Isr. J. Math.245(2021), no. 2, 735–772

    2021

  37. [45]

    Mariano Suarez-Alvarez,The Hilton-Heckmann argument for the anti-commutativity of cup products, Proc. Am. Math. Soc.132(2004), no. 8, 2241–2246

    2004

  38. [46]

    Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, La Falda, Sierras de Córdoba, Argentina, August 9–13, 1999, 2000, pp

    Mitsuhiro Takeuchi,Survey of braided Hopf algebras, New trends in Hopf Algebra Theory. Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, La Falda, Sierras de Córdoba, Argentina, August 9–13, 1999, 2000, pp. 301–323

    1999

  39. [47]

    Math., New Ser.32(2021), no

    Wilberd van der Kallen,Reductivity properties over an affine base, Indag. Math., New Ser.32(2021), no. 5, 961–967

    2021

  40. [48]

    Charles Weibel,An introduction to homological algebra, Camb. Stud. Adv. Math., vol. 38, Cambridge: Cambridge University Press, 1994

    1994

  41. [49]

    Wu and James J

    Quanshui S. Wu and James J. Zhang,Some homological invariants of local PI algebras, J. Algebra 225(2000), no. 2, 904–935

    2000

  42. [50]

    Amnon Yekutieli,Derived categories, Camb. Stud. Adv. Math., vol. 183, Cambridge: Cambridge Uni- versity Press, 2020

    2020

  43. [51]

    Algebra180(1996), no

    Shenglin Zhu,Integrality of module algebras over its invariants, J. Algebra180(1996), no. 1, 187–205. F aMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina Email address:nicolas.andruskiewitsch@unc.edu.ar Depa...

    1996

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.