# Picard groups of reductive group schemes

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?

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Cristian, have you looked in Raynaud's thesis? One natural idea would be to see to what extent Pic(G) is entirely accounted for by central extensions by $\mathbf{G}_m$, which is what underlies the direct proof over a field (using derived group to split the extension in the simply connected case). I assume when you say $G$ is semisimple over a Dedekind base, you mean it is smooth and affine with all fibers connected semisimple (not just generic fiber). –  BCnrd Apr 26 '10 at 20:45
Hi, Cristian! Welcome to Math Overflow! –  Pete L. Clark Apr 26 '10 at 21:12

$\ \ \ \oplus_x \mathbb{Z}_x \to Pic \ G \to Pic \ G_K \to 0$
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.