Article

# A going down theorem for Grothendieck Chow motives

01/2010;
Source: arXiv

ABSTRACT 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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### Keywords

base field

field extension E/F

field F satisfying Rost nilpotence principle

finite field K

geometrically irreducible variety

inner type E_6

motivic tool

Petrov

so-called outer motives

Springer