The Stacks project

Proof. There exists a surjective, flat, finitely presented morphism $B' \to B$ of algebraic spaces such that the base change $X' = X \times _ B B'$ over $B'$ has a section: namely, we can take $B' = X$. Observe that $\mathrm{Pic}_{X'/B'} = B' \times _ B \mathrm{Pic}_{X/B}$. Hence $\mathrm{Pic}_{X'/B'} \to \mathrm{Pic}_{X/B}$ is representable by algebraic spaces, surjective, flat, and finitely presented. Hence, if we can show that $\mathrm{Pic}_{X'/B'}$ is an algebraic space, then it follows that $\mathrm{Pic}_{X/B}$ is an algebraic space by Bootstrap, Theorem 80.10.1. In this way we reduce to the case described in the next paragraph.

In addition to the assumptions of the proposition, assume that we have a section $\sigma : B \to X$. By Proposition 99.10.2 we see that $\mathcal{P}\! \mathit{ic}_{X/B}$ is an algebraic stack. By Lemma 99.11.6 and Algebraic Stacks, Lemma 94.15.4 we see that $\mathcal{P}\! \mathit{ic}_{X/B, \sigma }$ is an algebraic stack. By Lemma 99.11.7 and Algebraic Stacks, Lemma 94.8.2 we see that $T \mapsto \mathop{\mathrm{Ker}}(\sigma _ T^* : \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathop{\mathrm{Pic}}\nolimits (T))$ is an algebraic space. By Lemma 99.11.4 this functor is the same as $\mathrm{Pic}_{X/B}$. $\square$