Homologies d'algèbres Artin–Schelter régulières cubiques

Nicolas Marconnet

doi:10.1016/j.crma.2003.10.009

Algèbre

Homologies d'algèbres Artin–Schelter régulières cubiques
[Homologies of cubic Artin–Schelter regular algebras]

Presented by: Alain Connes

Nicolas Marconnet ¹

¹ LARAL, faculté des sciences et techniques, 23, rue P. Michelon, 42023 Saint-Etienne cedex, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 117-120.

Abstract
Résumé

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: 2003-06-10
Accepted: 2003-10-07
Published online: 2003-12-09

DOI: 10.1016/j.crma.2003.10.009

Author's affiliations:

Nicolas Marconnet ¹

¹ LARAL, faculté des sciences et techniques, 23, rue P. Michelon, 42023 Saint-Etienne cedex, France

@article{CRMATH_2004__338_2_117_0,
     author = {Nicolas Marconnet},
     title = {Homologies d'alg\`ebres {Artin{\textendash}Schelter} r\'eguli\`eres cubiques},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {117--120},
     publisher = {Elsevier},
     volume = {338},
     number = {2},
     year = {2004},
     doi = {10.1016/j.crma.2003.10.009},
     language = {fr},
}

TY  - JOUR
AU  - Nicolas Marconnet
TI  - Homologies d'algèbres Artin–Schelter régulières cubiques
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 117
EP  - 120
VL  - 338
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2003.10.009
LA  - fr
ID  - CRMATH_2004__338_2_117_0
ER  -

%0 Journal Article
%A Nicolas Marconnet
%T Homologies d'algèbres Artin–Schelter régulières cubiques
%J Comptes Rendus. Mathématique
%D 2004
%P 117-120
%V 338
%N 2
%I Elsevier
%R 10.1016/j.crma.2003.10.009
%G fr
%F CRMATH_2004__338_2_117_0

Nicolas Marconnet. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.009/

References
Cited by

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

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

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

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

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

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

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

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

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

Cited by Sources:

Comments - Policy