Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme.


Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially on $S$.
I was wondering what is $Pic(BG)$. 


Is it true that $Pic(BG)= H^1(k,G)$ when $S=Speck$ the spectrum of a field?


What can we say in general?



Picard group of classifying stack
$\mathrm{Pic}([S/G])$ is the group of line bundles on $S$ together with a $G$-linearization. When $S=\mathrm{Spec}(k)$, any line bundle on $S$ is trivial, and a $G$-linearization is given by a character $G\rightarrow \mathbb{G}\_m$, so $\mathrm{Pic}[S/G]=\mathrm{Hom}(G,\mathbb{G}\_m)$. No relation with $H^1(k,G)$, which is not a group when $G$ is not abelian.


