Comptes Rendus
no. 17-18
Homological Algebra/Lie Algebras
Cartan homotopy formulae and the bivariant Hochschild complex
[Formule homotopique de Cartan et le complexe de Hochschild bivariant]

Présenté par : Alain Connes

Abhishek Banerjee1
1 Department of Mathematics, Johns Hopkins University, 3400 N Charles St., 404 Krieger Hall, Baltimore, MD 21218, USA
Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 997-1000.

La formule LD=[eD+ED,b¯+B¯] (voir Goodwillie [Cyclic homology, derivations, and the free loopspace, Topology 24 (2) (1985) 187–215]) sur le complexe de Hochschild normalisé joue le rôle, en géométrie non commutative, de la formule homotopique de Cartan en homologie de Rham. Notre but est détendre cette formule au complexe de Hochschild bivariant normalisée.

The formula LD=[eD+ED,b¯+B¯] (see Goodwillie [Cyclic homology, derivations, and the free loopspace, Topology 24 (2) (1985) 187–215]) on the normalized Hochschild complex is the standard replacement in noncommutative geometry for the classical Cartan homotopy formula. Our purpose is to extend this formula to the normalized bivariant Hochschild complex.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.07.016
Abhishek Banerjee 1

1 Department of Mathematics, Johns Hopkins University, 3400 N Charles St., 404 Krieger Hall, Baltimore, MD 21218, USA 
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Abhishek Banerjee. Cartan homotopy formulae and the bivariant Hochschild complex. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 997-1000. doi : 10.1016/j.crma.2009.07.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.07.016/

[1] A. Connes Cohomologie cyclique et foncteurs Extn, C. R. Acad. Sci. Paris, Sér. I, Volume 296 (1983) no. 23, pp. 953-958

[2] A. Connes Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math., Volume 62 (1985), pp. 257-360

[3] T.G. Goodwillie Cyclic homology, derivations, and the free loopspace, Topology, Volume 24 (1985) no. 2, pp. 187-215

[4] J.D.S. Jones; C. Kassel Bivariant cyclic theory, K-Theory, Volume 3 (1989) no. 4, pp. 339-365

[5] J.-L. Loday Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, vol. 301, Springer-Verlag, Berlin, 1998

[6] G.S. Rinehart Differential forms on general commutative algebras, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 195-222

[7] B.L. Tsygan Homology of matrix Lie algebras over rings and the Hochschild homology, Russian Math. Surveys, Volume 38 (1983) no. 2, pp. 198-199

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