To prove the Picard functor is representable we will use the following criterion.
Lemma 44.5.1. Let k be a field. Let G : (\mathit{Sch}/k)^{opp} \to \textit{Groups} be a functor. With terminology as in Schemes, Definition 26.15.3, assume that
G satisfies the sheaf property for the Zariski topology,
there exists a subfunctor F \subset G such that
F is representable,
F \subset G is representable by open immersion,
for every field extension K of k and g \in G(K) there exists a g' \in G(k) such that g'g \in F(K).
Then G is representable by a group scheme over k.
Proof. This follows from Schemes, Lemma 26.15.4. Namely, take I = G(k) and for i = g' \in I take F_ i \subset G the subfunctor which associates to T over k the set of elements g \in G(T) with g'g \in F(T). Then F_ i \cong F by multiplication by g'. The map F_ i \to G is isomorphic to the map F \to G by multiplication by g', hence is representable by open immersions. Finally, the collection (F_ i)_{i \in I} covers G by assumption (2)(c). Thus the lemma mentioned above applies and the proof is complete. \square
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