Geodesic
Gerstenhaber-schack and Hochschild cohomologies of Hopf algebras
Documenta mathematica, Tome 21 (2016), pp. 955-986
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We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum permutation group are 3.
DOI : 10.4171/dm/550
Classification : 16E10, 16E40, 16T05
Mots-clés : Hopf algebra, cohomological dimension, cohomology, Yetter-Drinfeld module 
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     author = {Julien Bichon},
     title = {Gerstenhaber-schack and {Hochschild} cohomologies of {Hopf} algebras},
     journal = {Documenta mathematica},
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     year = {2016},
     volume = {21},
     doi = {10.4171/dm/550},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/550/}
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Julien Bichon. Gerstenhaber-schack and Hochschild cohomologies of Hopf algebras. Documenta mathematica, Tome 21 (2016), pp. 955-986. doi: 10.4171/dm/550

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