The Stacks project

Proof. By Lemmas 99.13.3 and 99.14.3 the statement makes sense. To prove it, we choose a scheme $S$ and an object $\xi = (X \to S)$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ over $S$. We have to show that

is an algebraic stack over $S$. Observe that an object of $\mathcal{X}$ is given by a pair $(T/S, \mathcal{L})$ where $T$ is a scheme over $S$ and $\mathcal{L}$ is an invertible $\mathcal{O}_{X_ T}$-module which is ample on $X_ T/T$. Morphisms are defined in the obvious manner. In particular, we see immediately that we have an inclusion

of categories over $(\mathit{Sch}/S)_{fppf}$, inducing equality on morphism sets. Since $\mathcal{P}\! \mathit{ic}_{X/S}$ is an algebraic stack by Proposition 99.10.2 it suffices to show that the inclusion above is representable by open immersions. This is exactly the content of Descent on Spaces, Lemma 74.13.2. $\square$