I was wondering when the Kunneth formula holds for motivic cohomology:
$$
 H^p(X,A(\alpha)) = \bigoplus\_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma)) 
$$
where $H^p(X,A(\alpha))$ is defined as you wish: by higher Chow groups, Hom groups in $DM(X)$, etc... The case I'm most interested in is $A= \mathbb{Q}$ and $M(X) \in DM(\mathbb{Q})\_{\mathbb{Q}}$ in the thick sub-triangagulated category generated by the $\mathbb{Q}(n)$, $n\in \mathbb{Z}$. 



Kunneth formula for motivic cohomology
I now remember a nice argument, why there's no Kunneth formula for Chow groups of $X \times X$ unless $X$ has a Tate motive. Let $X$ be smooth projective of dimension $d$. 
We start with a decomposition of a diagonal:
$$
[\Delta] = \sum\_{i,j} \alpha^i\_j \beta^{d-i}\_j \in \oplus\_i CH^i(X) \otimes CH^{d-i}(X)
$$
We can assume $\alpha^i\_j$ are linearly independent.
In this case we can show that $\alpha^i\_j$ form a basis of Chow groups and 
$\beta^{d-i}\_j$ is the dual basis.


Indeed, as a correspondence $[\Delta]$ acts as identity on Chow groups,
so for any class c, $$c = [\Delta]c = \sum\_{i,j} \alpha^i\_j deg(\beta^{d-i}\_j \cup c),$$
and the claim follows if we substitute $c = \alpha^i\_j$.


Now $CH\_i(X) = Hom(\mathbb Z(i)[2i], M(X))$ and we can consider the set of $\alpha^i\_j$ as a morphism of motives $$\oplus\_{i,j}\mathbb Z(i)[2i] \to M(X).$$ 
A simple computation shows that it is an isomorphism with the inverse given by $\beta^i\_j$.


And of course, on the other hand, if $X$ has a Tate motive, then Kunneth formula for Chow groups follows (it doesn't answer the question, since I only consider smooth projective varieties).


