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# Homologies d'algèbres Artin-Schelter régulières cubiques

Abstract : The Hochschild homology of cubic Artin-Schelter regular algebras of type A with generic coefficients is computed. Let $A$ be such an algebra. We follow the method used by Van den Bergh (K-Theory 8 (1994) 213-230) in the quadratic case and consider this algebra as a deformation of a polynomial algebra with remarkable Poisson bracket. We compute the Poisson homology and show that the Brylinski spectral sequence degenerates. We use the fact that this algebra is generalized Koszul in the sense of R. Berger (J. Algebra 239 (2001) 705-734) and we give a new quasi-isomorphism between the Koszul resolution of $A$ by bimodules and the bar-resolution of $A$. The de Rham cohomology, cyclic and periodic cyclic homologies are deduced from the Hochschild homology using standard results. The Koszul property allows us to give an explicit quasi-isomorphism between the Hochschild cochain complex and the Hochschild chain complex. We get a Poincaré duality. Then we can deduce the Hochschild cohomology from the Hochschild homology of $A$. As a result, we get the center of $A$, which was not known. We give several complements. In particular we give an explicit injection from the Koszul resolution of $A$ by bimodules to the bar-resolution of $A$, available for every generalized Koszul algebra $A$.
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https://tel.archives-ouvertes.fr/tel-00007763
Contributor : Nicolas Marconnet <>
Submitted on : Wednesday, December 15, 2004 - 2:54:06 PM
Last modification on : Wednesday, November 20, 2019 - 2:38:18 AM
Long-term archiving on: : Friday, April 2, 2010 - 9:07:44 PM

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• HAL Id : tel-00007763, version 1

### Citation

Nicolas Marconnet. Homologies d'algèbres Artin-Schelter régulières cubiques. Mathématiques [math]. Université Jean Monnet - Saint-Etienne, 2004. Français. ⟨tel-00007763⟩

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