In [Kim], Kimura introduced the notion of a “finite dimensional” motive (which we will refer to as “Kimura-finite” motive) and he conjectured that all Q-linear motives modulo rational equivalence are Kimura-finite. The same notion was introduced independently in a different context by O’Sullivan. Kimura-finiteness has been the subject of several articles recently ([GP02], [GP], [AK02]). In [GP]...