The Stacks project

Definition 44.4.1. Let $\mathit{Sch}_{fppf}$ be a big site as in Topologies, Definition 34.7.8. Let $f : X \to S$ be a morphism of this site. The Picard functor $\mathrm{Pic}_{X/S}$ is the fppf sheafification of the functor

\[ (\mathit{Sch}/S)_{fppf} \longrightarrow \textit{Sets},\quad T \longmapsto \mathop{\mathrm{Pic}}\nolimits (X_ T) \]

If this functor is representable, then we denote $\underline{\mathrm{Pic}}_{X/S}$ a scheme representing it.

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