The Stacks project

Proof. Let $T$ be a scheme over $B$ and let $s, t \in \mathrm{Pic}_{X/B}(T)$. We want to show that there exists a locally closed subscheme $Z \subset T$ such that $s|_ Z = t|_ Z$ and such that a morphism $T' \to T$ factors through $Z$ if and only if $s|_{T'} = t|_{T'}$.

We first reduce the general problem to the case where $s$ and $t$ come from invertible modules on $X_ T$. We suggest the reader skip this step. Choose an fppf covering $\{ T_ i \to T\} _{i \in I}$ such that $s|_{T_ i}$ and $t|_{T_ i}$ come from $\mathop{\mathrm{Pic}}\nolimits (X_{T_ i})$ for all $i$. Suppose that we can show the result for all the pairs $s|_{T_ i}, t|_{T_ i}$. Then we obtain locally closed subschemes $Z_ i \subset T_ i$ with the desired universal property. It follows that $Z_ i$ and $Z_ j$ have the same scheme theoretic inverse image in $T_ i \times _ T T_ j$. This determines a descend datum on $Z_ i/T_ i$. Since $Z_ i \to T_ i$ is locally quasi-finite, it follows from More on Morphisms, Lemma 37.57.1 that we obtain a locally quasi-finite morphism $Z \to T$ recovering $Z_ i \to T_ i$ by base change. Then $Z \to T$ is an immersion by Descent, Lemma 35.24.1. Finally, because $\mathrm{Pic}_{X/B}$ is an fppf sheaf, we conclude that $s|_ Z = t|_ Z$ and that $Z$ satisfies the universal property mentioned above.

Assume $s$ and $t$ come from invertible modules $\mathcal{V}$, $\mathcal{W}$ on $X_ T$. Set $\mathcal{L} = \mathcal{V} \otimes \mathcal{W}^{\otimes -1}$ We are looking for a locally closed subscheme $Z$ of $T$ such that $T' \to T$ factors through $Z$ if and only if $\mathcal{L}_{X_{T'}}$ is the pullback of an invertible sheaf on $T'$, see Lemma 99.11.3. Hence the existence of $Z$ follows from More on Morphisms of Spaces, Lemma 76.53.1. $\square$