Algèbre
Homologies d'algèbres Artin–Schelter régulières cubiques
[Homologies of cubic Artin–Schelter regular algebras]
Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 117-120.

Hochschild homology of cubic Artin–Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by Van den Bergh (K-Theory 8 (1994) 213–230) in the quadratic case, by considering these algebras as deformations of a polynomial algebra, with remarkable Poisson brackets. A new quasi-isomorphism is introduced. De Rham cohomology, cyclic and periodic cyclic homologies, and Hochschild cohomology are also computed.

Nous calculons l'homologie de Hochschild des algèbres Artin–Schelter régulières cubiques de type A à coefficients génériques. Nous suivons la méthode employée par Van den Bergh (K-Theory 8 (1994) 213–230) dans le cas quadratique, en considérant ces algèbres comme déformations d'une algèbre de polynômes, avec crochets de Poisson remarquables. Un nouveau quasi-isomorphisme est introduit. Nous calculons aussi la cohomologie de de Rham, l'homologie cyclique, l'homologie cyclique périodique et la cohomologie de Hochschild.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.009
Marconnet, Nicolas 1

1 LARAL,  faculté des sciences et techniques, 23, rue P. Michelon, 42023 Saint-Etienne cedex, France
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Marconnet, Nicolas. Homologies d'algèbres Artin–Schelter régulières cubiques. Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 117-120. doi : 10.1016/j.crma.2003.10.009. http://www.numdam.org/articles/10.1016/j.crma.2003.10.009/

[1] Artin, M.; Schelter, W. Graded algebras of global dimension 3, Adv. in Math., Volume 66 (1987), pp. 171-216

[2] Artin, M.; Tate, J.; Van den Bergh, M. Some algebras associated to automorphisms of elliptic curves (Cartier, P. et al., eds.), The Grothendieck Festschrift, vol. 1, Birkhäuser, 1990, pp. 33-85

[3] Berger, R. Koszulity for nonquadratic algebras, J. Algebra, Volume 239 (2001), pp. 705-734

[4] Berger, R. Koszulity for nonquadratic algebras II | arXiv

[5] Berger, R.; Dubois-Violette, M.; Wambst, M. Homogeneous algebras, J. Algebra, Volume 261 (2003), pp. 172-185

[6] Brylinski, J.-L. A differential complex for Poisson manifolds, J. Differential Geom., Volume 28 (1988), pp. 93-114

[7] Connes, A.; Dubois-Violette, M. Yang–Mills algebra, Lett. Math. Phys., Volume 61 (2002), pp. 149-158

[8] Van den Bergh, M. Non-commutative homology of some three-dimensional quantum spaces, K-Theory, Volume 8 (1994), pp. 213-230

[9] Van den Bergh, M. A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc., Volume 126 (1998) no. 5, pp. 1345-1348

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