There exists information on the Picard (and Brauer) group of a reductive algebraic group over a number field k. For example, Sansuc shows (in his big Crelle paper of 1980) that if G is connected and semisimple over a number field k, then Pic G is the group of krational points of the character group of the fundamental group of G. In particular, if G is semisimple and simply connected, then Pic G=0. My question is: do there exist results of this type over more general bases? For example, a natural generalization of the equality "Pic G=0 if G semisimple and simply connected over a number field" would be "Pic G=Pic U if G is a semisimple and simply connected group scheme over a Dedekind scheme U". Is the latter true?
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Assuming you know that Pic of the generic fibre is trivial, this seems to follow immediately from the localisation sequence: since G is a group there is a section, so the map Pic U to Pic G is an injection. On the other hand (since G is smooth so Pic = Cl) there is an exact sequence: $\ \ \ \oplus_x \mathbb{Z}_x \to Pic \ G \to Pic \ G_K \to 0 $ where X runs over the closed points of U and $G_K$ denotes the generic fibre. Since $Pic\ G_K$ is trivial it follows from this that the map Pic U to Pic G is surjective. (Note that the only property of the closed fibres that is used is that they are reduced and irreducible.) 

