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 semisimple or Noetherian $R$algebra?
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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). 

