Schur finiteness and nilpotency

Alessio Del Padrone; Carlo Mazza

doi:10.1016/j.crma.2005.07.010

Algebra/Algebraic Geometry

Schur finiteness and nilpotency

Presented by: Pierre Deligne

Alessio Del Padrone ¹; Carlo Mazza 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

Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 283-286.

Abstract
Résumé

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 ${CH}^{n i} {(X^{i} \times X^{i})}_{num}$ for all i (where $n = \dim X$ ) are all equivalent.

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 ${CH}^{n i} {(X^{i} \times X^{i})}_{num}$ pour tous i, sont équivalentes.

Received: 2005-04-08
Accepted: 2005-07-18
Published online: 2005-08-25

DOI: 10.1016/j.crma.2005.07.010

Author's affiliations:

Alessio Del Padrone ¹; Carlo Mazza 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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     title = {Schur finiteness and nilpotency},
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     pages = {283--286},
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     doi = {10.1016/j.crma.2005.07.010},
     language = {en},
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Alessio Del Padrone; Carlo Mazza. Schur finiteness and nilpotency. Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 283-286. doi : 10.1016/j.crma.2005.07.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.010/

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