A going-down theorem for Chow-Grothendieck motives
Résumé
Let $(M(X),p)$ be a direct summand of the motive associated with a geometrically split, geometrically irreducible variety over a field $F$ satisfying the nilpotence principle. We show that under some conditions, if $(M(X_E),p_E)$ is a direct summand of another motive $M_E$ over a field extension $E$, then $(M(X),p)$ is a direct summand of $M$ over $F$.
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