Is there some kind of universal coefficient theorem for motivic cohomology?
In particular, suppose we have a ring morphism $R\to S$, then I would like to know when 
$$ H^{\star\star}(-,S)\simeq H^{\star\star}(-,R)\otimes\_{R}S\; ?$$
Does this for example hold when $R$ is a field? In particular, does it hold for $R=\mathbb{Q}$?
Or do we need additional assumption on $S$ as well? E.g. that $S$ is a semi-simple or Noetherian $R$-algebra?



Is there a universal coefficient theorem for motivic cohomology?
Yes, there is a universal coefficient theorem: the corresponding object of the derived category (of $S$-modules) could be obtained by tensoring by $S$. This is easy, since motivic cohomology is defined as the cohomology of a complex of free modules (over $R$ and $S$, respectively).


