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 k-rational 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?



Picard groups of reductive group schemes
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.)


