44.5 A representability criterion

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

1. $G$ satisfies the sheaf property for the Zariski topology,

2. there exists a subfunctor $F \subset G$ such that

1. $F$ is representable,

2. $F \subset G$ is representable by open immersion,

3. 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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