Motivic splitting lemma

Abstract

Let $M$ be a Chow motive over a field $F$. Let $X$ be a smooth projective variety over $F$ and $N$ be a direct summand of the motive of $X$. Assume that over the generic point of $X$ the motives $M$ and $N$ become isomorphic to a direct sum of twisted Tate motives. The main result of the paper says that if a morphism $f:M\to N$ splits over the generic point of $X$ then it splits over $F$, i.e., $N$ is a direct summand of $M$. We apply this result to various examples of motives of projective homogeneous varieties.