A going down theorem for Grothendieck Chow motives
Résumé
Let X be a geometrically split, geometrically irreducible variety over a field F satisfying the nilpotence principle. Consider a motivic direct summand N of X and M a twisted motivic direct summand of another F-variety Y. We show that under some assumptions on a field extension E/F, if an indecomposable motive is both an outer direct summand of N_E and a direct summand of M_E, then an outer direct summand of N is also a direct summand of M.
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