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Let $K$ be a field, $G$ a $K$-group scheme of finite type, and $X$ a $G$-torsor. Is the Picard functor $\mathrm{Pic}_{X/K}$ representable? I recall that $\mathrm{Pic}_{X/K}$ is the fppf sheafification of $S \mapsto \mathrm{Pic}(X\times_K S)$ (here $S$ is a variable $K$-scheme) and that the answer is 'yes' if $G$ is an abelian variety.

If it simplifies the question, feel free to assume that $K$ is perfect, $X = G$, and $G$ is connected and smooth.

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    $\begingroup$ This lacks motivation (beyond the proper case). Consider $P={\rm{Pic}}_{Y/K}$ defined by etale sheafification for smooth $Y$ over $K$. This satisfies the functorial criterion to be locally of finite presentation and $P(R)={\rm{Pic}}(Y_R)$ for strictly henselian local $R$ over $K$, so the tangent space at 1 is $\ker(P(K[\varepsilon])\rightarrow P(K))={\rm{H}}^1(Y,O_Y)$. For affine $Y$ this is 0, so if representable for such $Y$ it is the constant $K$-group associated to Pic($Y$), and hence ${\rm{Pic}}(Y_R)={\rm{Pic}}(Y)$ for strictly henselian local $R$ over $K$. False for $Y$ an affine line. $\endgroup$
    – user74230
    Commented Dec 6, 2014 at 11:07

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