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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.