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Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here. I am looking for a reference or explanation of the fact that is used in Mumford'...

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 ...