no. 5
Algebra/Algebraic Geometry
Schur finiteness and nilpotency
[Finitude de Schur et nilpotence]

Présenté par : Pierre Deligne

Del Padrone, Alessio1 ; Mazza, Carlo2
1 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
2 Institute for Advanced Study, 1, Einstein Drive, 08854 Princeton, NJ, USA
Comptes Rendus. Mathématique, Tome 341 (2005) no. 5, pp. 283-286.

Soit A une catégorie tensorielle rigide pseudo-abélienne Q-lineaire. Une notion de finitude de Kimura et (indépendamment) O'Sullivan garantit que l'idéal des endomorphismes numériquement triviaux d'un objet est nilpotent. Nous généralisons ce résultat à certains objets Schur-finis. En particulier, dans la catégorie des motifs de Chow, si X est une variété projective lisse purement de dimension n qui satisfait la conjecture homologique de signe, alors la finitude de Kimura, l'annulation du motif de X par un certain foncteur de Schur, et la nilpotence de CHni(Xi×Xi)num pour tous i, sont équivalentes.

Let A be a Q-linear pseudo-Abelian rigid tensor category. A notion of finiteness due to Kimura and (independently) O'Sullivan guarantees that the ideal of numerically trivial endomorphism of an object is nilpotent. We generalize this result to special Schur-finite objects. In particular, in the category of Chow motives, if X is a smooth projective variety which satisfies the homological sign conjecture, then Kimura-finiteness, a special Schur-finiteness, and the nilpotency of CHni(Xi×Xi)num for all i (where n=dimX) are all equivalent.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.07.010
Del Padrone, Alessio 1 ; Mazza, Carlo 2

1 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
2 Institute for Advanced Study, 1, Einstein Drive, 08854 Princeton, NJ, USA 
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Del Padrone, Alessio; Mazza, Carlo. Schur finiteness and nilpotency. Comptes Rendus. Mathématique, Tome 341 (2005) no. 5, pp. 283-286. doi : 10.1016/j.crma.2005.07.010. http://archive.numdam.org/articles/10.1016/j.crma.2005.07.010/

[1] André, Y. Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, Panoramas and Syntheses, vol. 17, Société Mathématique de France, Paris, 2004

[2] André, Y.; Kahn, B. Nilpotence, radicaux et structures monoïdales, Rend. Sem. Mat. Univ. Padova, Volume 108 (2002), pp. 107-291 (with an appendix by P. O'Sullivan)

[3] Deligne, P. Catégories tensorielles, Moscow. Math. J., Volume 2 (2002) no. 2, pp. 227-248 (Dedicated to Yuri I. Manin on the occasion of his 65th birthday)

[4] Fulton, W.; Harris, J. Representation Theory, Graduate Texts in Math., vol. 129, Springer-Verlag, New York, 1991 (A first course, Readings in Mathematics)

[5] V. Guletskiĭ, A remark on nilpotent correspondences, Preprint, January 27, 2004, K-theory Preprint Archives, http://www.math.uiuc.edu/K-theory/0651/

[6] Guletskiĭ, V.; Pedrini, C. Finite-dimensional motives and the conjectures of Beilinson and Murre, K-Theory, Volume 30 (2003) no. 3, pp. 243-263 (Special issue in honor of Hyman Bass on his seventieth birthday. Part III)

[7] Kimura, S.-I. Chow groups are finite dimensional, in some sense, Math. Ann., Volume 331 (2005) no. 1, pp. 173-201

[8] Mazza, C. Schur functors and motives, K-Theory, Volume 33 (2004) no. 2, pp. 89-106

[9] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4, 2004, http://www.gap-system.org

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