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Pré-Publication, Document De Travail Année : 2009

A going down theorem for Grothendieck Chow motives

Charles de Clercq
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Résumé

Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of X_E with coefficients in K to be lifted to the base field. This going down result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and Springer.
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Dates et versions

hal-00443691 , version 1 (03-01-2010)
hal-00443691 , version 2 (05-04-2011)
hal-00443691 , version 3 (06-01-2012)
hal-00443691 , version 4 (13-09-2012)

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Charles de Clercq. A going down theorem for Grothendieck Chow motives. 2009. ⟨hal-00443691v3⟩
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