Comptes Rendus
no. 3
Algebraic geometry/Homological algebra
A case of the deformational Hodge conjecture via a pro Hochschild–Kostant–Rosenberg theorem
[Un cas de conjecture de Hodge infinitésimale via un théorème de Hochschild–Kostant–Rosenberg pro]

Présenté par : Claire Voisin

Matthew Morrow1
1 Hausdorff Center for Mathematics, Endenicher Allee 60, 53115 Bonn, Germany
Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 173-177.

En suivant des idées de Bloch, Esnault et Kerz, nous établissons la partie formelle de la conjecture de Hodge variationnelle pour les schémas propres et lisses sur K〚t〛, où K est une extension algébrique de Q. L'outil principal est un théorème de Hochschild–Kostant–Rosenberg pro pour l'homologie de Hochschild.

Following ideas of Bloch, Esnault, and Kerz, we establish the deformational part of Grothendieck's variational Hodge conjecture for proper, smooth schemes over K〚t〛, where K is an algebraic extension of Q. The main tool is a pro Hochschild–Kostant–Rosenberg theorem for Hochschild homology.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.01.008
Matthew Morrow 1

1 Hausdorff Center for Mathematics, Endenicher Allee 60, 53115 Bonn, Germany 
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Matthew Morrow. A case of the deformational Hodge conjecture via a pro Hochschild–Kostant–Rosenberg theorem. Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 173-177. doi : 10.1016/j.crma.2014.01.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.01.008/

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[4] S. Bloch; H. Esnault; M. Kerz Deformation of algebraic cycle classes in characteristic zero, 2013 | arXiv

[5] G. Cortiñas; C. Haesemeyer; C.A. Weibel Infinitesimal cohomology and the Chern character to negative cyclic homology, Math. Ann., Volume 344 (2009) no. 4, pp. 891-922

[6] T.G. Goodwillie Relative algebraic K-theory and cyclic homology, Ann. of Math. (2), Volume 124 (1986) no. 2, pp. 347-402

[7] A. Grothendieck Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math., Volume 11 (1961), p. 167

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[11] C. Weibel Le caractère de Chern en homologie cyclique périodique, C. R. Acad. Sci. Paris, Ser. I, Volume 317 (1993) no. 9, pp. 867-871

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