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Comptes Rendus. Mathématique
Algebra
On the monoidal invariance of the cohomological dimension of Hopf algebras
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 561-582.

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber–Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

Nous étudions la question de l’invariance monoïdale de la dimension globale des algèbres de Hopf : si deux algèbres de Hopf ont des catégories de comodules monoïdalement équivalentes, ont-elles même dimension globale ? Nous apportons plusieurs nouvelles réponses positives dans les cas d’algèbres de Hopf lisses, cosemisimples ou de dimension finie. Nous comparons également la dimension globale et la dimension cohomologique de Gerstenhaber–Schack dans le cas cosemisimple. Un outil important pour obtenir ces divers résultats est la nouvelle notion de foncteur séparable twisté.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.329
Classification: 16T05,  16E40,  16E10
Julien Bichon 1

1 Université Clermont Auvergne, CNRS, LMBP, 63000 Clermont-Ferrand, France
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Julien Bichon. On the monoidal invariance of the cohomological dimension of Hopf algebras. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 561-582. doi : 10.5802/crmath.329. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.329/

[1] Eli Aljadeff; Pavel Etingof; Shlomo Gelaki; Dmitri Nikshych On twisting of finite-dimensional Hopf algebras, J. Algebra, Volume 256 (2002) no. 2, pp. 484-501 | Article | MR: 1939116 | Zbl: 1053.16026

[2] Teodor Banica; Roland Vergnioux Fusion rules for quantum reflection groups, J. Noncommut. Geom., Volume 3 (2009) no. 3, pp. 327-359 | Article | MR: 2511633 | Zbl: 1203.46048

[3] Julien Bichon Free wreath product by the quantum permutation group, Algebr. Represent. Theory, Volume 7 (2004) no. 4, pp. 343-362 | Article | MR: 2096666 | Zbl: 1112.46313

[4] Julien Bichon Co-representation theory of universal co-sovereign Hopf algebras, J. Lond. Math. Soc., Volume 75 (2007) no. 1, pp. 83-98 | Article | MR: 2302731 | Zbl: 1138.16018

[5] Julien Bichon Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit, Compos. Math., Volume 149 (2013) no. 4, pp. 658-678 | Article | MR: 3049699 | Zbl: 1365.16016

[6] Julien Bichon Hopf-Galois objects and cogroupoids, Rev. Unión Mat. Argent., Volume 55 (2014) no. 2, pp. 11-69 | MR: 3285340 | Zbl: 1322.16021

[7] Julien Bichon Gerstenhaber-Schack and Hochschild cohomologies of Hopf algebras, Doc. Math., Volume 21 (2016), pp. 955-986 | MR: 3548138 | Zbl: 1385.16029

[8] Julien Bichon Cohomological dimensions of universal cosovereign Hopf algebras, Publ. Mat., Barc., Volume 62 (2018) no. 2, pp. 301-330 | Article | MR: 3815282 | Zbl: 1427.16023

[9] Julien Bichon; Uwe Franz; Malte Gerhold Homological properties of quantum permutation algebras, New York J. Math., Volume 23 (2017), pp. 1671-1695 | MR: 3741856 | Zbl: 1386.16015

[10] Kenneth S. Brown Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1994, x+306 pages (Corrected reprint of the 1982 original) | MR: 1324339

[11] Stefaan Caenepeel; Gigel Militaru; Shenglin Zhu Crossed modules and Doi–Hopf modules, Isr. J. Math., Volume 100 (1997), pp. 221-247 | Article | MR: 1469112 | Zbl: 0888.16018

[12] Stefaan Caenepeel; Gigel Militaru; Shenglin Zhu Frobenius and separable functors for generalized module categories and nonlinear equations, Lecture Notes in Mathematics, 1787, Springer, 2002, xiv+354 pages | Article | MR: 1926102

[13] Alexandru Chirvasitu Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras, Algebra Number Theory, Volume 8 (2014) no. 5, pp. 1179-1199 | Article | MR: 3263140 | Zbl: 1346.16026

[14] Alexandru Chirvasitu Relative Fourier transforms and expectations on coideal subalgebras, J. Algebra, Volume 516 (2018), pp. 271-297 | Article | MR: 3863479 | Zbl: 1433.16039

[15] Alexandru Chirvasitu; Chelsea Walton; Xingting Wang Gelfand-Kirillov dimension of cosemisimple Hopf algebras, Proc. Am. Math. Soc., Volume 147 (2019) no. 11, pp. 4665-4672 | Article | MR: 4011503 | Zbl: 1481.16024

[16] Warren Dicks; Martin J. Dunwoody Groups acting on graphs, Cambridge Studies in Advanced Mathematics, 17, Cambridge University Press, 1989, xvi+283 pages | MR: 1001965

[17] Yukio Doi Braided bialgebras and quadratic bialgebras, Commun. Algebra, Volume 21 (1993) no. 5, pp. 1731-1749 | MR: 1213985 | Zbl: 0779.16015

[18] Pavel Etingof; Shlomo Gelaki On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, Int. Math. Res. Not. (1998) no. 16, pp. 851-864 | Article | MR: 1643702 | Zbl: 0918.16027

[19] Pavel Etingof; Shlomo Gelaki; Dmitri Nikshych; Victor Ostrik Tensor categories, Mathematical Surveys and Monographs, 205, American Mathematical Society, 2015, xvi+343 pages | Article | MR: 3242743

[20] Pierre Fima; Lorenzo Pittau The free wreath product of a compact quantum group by a quantum automorphism group, J. Funct. Anal., Volume 271 (2016) no. 7, pp. 1996-2043 | Article | MR: 3535324 | Zbl: 1355.46062

[21] Murray Gerstenhaber; Samuel D. Schack Bialgebra cohomology, deformations, and quantum groups, Proc. Natl. Acad. Sci. USA, Volume 87 (1990) no. 1, pp. 478-481 | Article | MR: 1031952 | Zbl: 0695.16005

[22] Victor Ginzburg; Shrawan Kumar Cohomology of quantum groups at roots of unity, Duke Math. J., Volume 69 (1993) no. 1, pp. 179-198 | MR: 1201697 | Zbl: 0774.17013

[23] André Joyal; Ross Street An introduction to Tannaka duality and quantum groups, Category theory (Como, 1990) (Lecture Notes in Mathematics), Volume 1488, Springer, 1991, pp. 413-492 | Article | MR: 1173027 | Zbl: 0745.57001

[24] Anatoli Klimyk; Konrad Schmüdgen Quantum groups and their representations, Texts and Monographs in Physics, Springer, 1997, xx+552 pages | Article | MR: 1492989

[25] Richard G. Larson Characters of Hopf algebras, J. Algebra, Volume 17 (1971), pp. 352-368 | Article | MR: 283054 | Zbl: 0217.33801

[26] Richard G. Larson; David E. Radford Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra, Volume 117 (1988) no. 2, pp. 267-289 | Article | MR: 957441 | Zbl: 0649.16005

[27] François Lemeux; Pierre Tarrago Free wreath product quantum groups: the monoidal category, approximation properties and free probability, J. Funct. Anal., Volume 270 (2016) no. 10, pp. 3828-3883 | Article | MR: 3478874 | Zbl: 1356.46057

[28] Maria E. Lorenz; Martin Lorenz On crossed products of Hopf algebras, Proc. Am. Math. Soc., Volume 123 (1995) no. 1, pp. 33-38 | Article | MR: 1227522 | Zbl: 0826.16037

[29] Akira Masuoka Cleft extensions for a Hopf algebra generated by a nearly primitive element, Commun. Algebra, Volume 22 (1994) no. 11, pp. 4537-4559 | Article | MR: 1284344 | Zbl: 0809.16046

[30] Akira Masuoka; David Wigner Faithful flatness of Hopf algebras, J. Algebra, Volume 170 (1994) no. 1, pp. 156-164 | Article | MR: 1302835 | Zbl: 0820.16034

[31] Susan Montgomery Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, American Mathematical Society, 1993, xiv+238 pages | Article | MR: 1243637

[32] Colin Mrozinski Quantum groups of GL (2) representation type, J. Noncommut. Geom., Volume 8 (2014) no. 1, pp. 107-140 | Article | MR: 3275027 | Zbl: 1292.16027

[33] Colin Mrozinski Quantum automorphism groups and SO(3)-deformations, J. Pure Appl. Algebra, Volume 219 (2015) no. 1, pp. 1-32 | Article | MR: 3240820 | Zbl: 1342.17013

[34] Constantin Năstăsescu; Michel Van den Bergh; Freddy Van Oystaeyen Separable functors applied to graded rings, J. Algebra, Volume 123 (1989) no. 2, pp. 397-413 | Article | MR: 1000494 | Zbl: 0673.16026

[35] Piotr Podleś; E. Müller Introduction to quantum groups, Rev. Math. Phys., Volume 10 (1998) no. 4, pp. 511-551 | Article | MR: 1629723 | Zbl: 0918.17005

[36] David E. Radford Minimal quasitriangular Hopf algebras, J. Algebra, Volume 157 (1993) no. 2, pp. 285-315 | Article | MR: 1220770 | Zbl: 0787.16028

[37] Theo Raedschelders; Michel Van den Bergh The Manin Hopf algebra of a Koszul Artin-Schelter regular algebra is quasi-hereditary, Adv. Math., Volume 305 (2017), pp. 601-660 | Article | MR: 3570144 | Zbl: 1405.16044

[38] M. D. Rafael Separable functors revisited, Commun. Algebra, Volume 18 (1990) no. 5, pp. 1445-1459 | Article | MR: 1059740 | Zbl: 0713.18002

[39] Peter Schauenburg Hopf modules and Yetter-Drinfelʼd modules, J. Algebra, Volume 169 (1994) no. 3, pp. 874-890 | Article | MR: 1302122 | Zbl: 0810.16037

[40] Peter Schauenburg Hopf bi-Galois extensions, Commun. Algebra, Volume 24 (1996) no. 12, pp. 3797-3825 | Article | MR: 1408508

[41] Peter Schauenburg Bi-Galois objects over the Taft algebras, Isr. J. Math., Volume 115 (2000), pp. 101-123 | Article | MR: 1749674 | Zbl: 0949.16040

[42] Peter Schauenburg Hopf-Galois and bi-Galois extensions, Galois theory, Hopf algebras, and semiabelian categories (Fields Institute Communications), Volume 43, American Mathematical Society, 2004, pp. 469-515 | MR: 2075600 | Zbl: 1091.16023

[43] Steven Shnider; Shlomo Sternberg Quantum groups from coalgebras to Drinfeld algebras. A guided tour, International Press, 1993, xxii+496 pages | MR: 1287162 | Zbl: 0845.17015

[44] Dragoş Ştefan Hochschild cohomology on Hopf Galois extensions, J. Pure Appl. Algebra, Volume 103 (1995) no. 2, pp. 221-233 | Article | MR: 1358765 | Zbl: 0838.16008

[45] Rachel Taillefer Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product, J. Algebra, Volume 276 (2004) no. 1, pp. 259-279 | Article | MR: 2054397 | Zbl: 1063.16048

[46] Shuzhou Wang Quantum symmetry groups of finite spaces, Commun. Math. Phys., Volume 195 (1998) no. 1, pp. 195-211 | Article | MR: 1637425 | Zbl: 1013.17008

[47] Xingting Wang; Xiaolan Yu; Yinhuo Zhang Calabi-Yau property under monoidal Morita-Takeuchi equivalence, Pac. J. Math., Volume 290 (2017) no. 2, pp. 481-510 | Article | MR: 3681116 | Zbl: 1405.16015

[48] Charles A. Weibel An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1994, xiv+450 pages | Article | MR: 1269324

[49] Stanisław L. Woronowicz Compact matrix pseudogroups, Commun. Math. Phys., Volume 111 (1987) no. 4, pp. 613-665 | Article | MR: 901157 | Zbl: 0627.58034

[50] Xiaolan Yu Hopf-Galois objects of Calabi-Yau Hopf algebras, J. Algebra Appl., Volume 15 (2016) no. 10, 1650194, 19 pages | MR: 3575984 | Zbl: 1373.16060

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