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I am interested in certain properties Chow motives that seem to be (more or less) "classical" (so, I mostly need references; the base field may be assumed to be algebraically closed).

So, consider the full subcategories $C_{\le 1}$ and $C_{\ge 2}$ of effective Chow motives with rational coefficients consisting of objects whose ($\mathbb{Q}_l$-adic for l being a fixed prime distinct from the characteristic of the base field) etale cohomology is concentrated in degrees at most 1 and at least 2, respectively. Then the well-known results on the existence of the so-called Chow-Kunneth summands of motives of projective varieties corresponding to cohomology degrees 0 and 1 seem to yield that there are only zero Chow-morphisms from $C_{\ge 2}$ into $C_{\le 1}$, and (as a consequence) any effective Chow motive can be presented as the sum of an object of $C_{\le 1}$ and an object of $C_{\ge 2}$.

Is this true? Where may facts of this sort be found? I was not able to find a reference in "pure motivic" texts, whereas the treatise "On the derived category of 1-motives" (by Barbieri-Viale and Kahn) does not seem to include the formulations of "well-known" facts necessary for my purposes.

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