The Stacks project

Lemma 99.14.4. Let $T$ be an algebraic space over $\mathbf{Z}$. Let $\mathcal{S}_ T$ denote the corresponding algebraic stack (Algebraic Stacks, Sections 94.7, 94.8, and 94.13). We have an equivalence of categories

\[ \left\{ \begin{matrix} (X \to T, \mathcal{L})\text{ where }X \to T\text{ is a morphism} \\ \text{of algebraic spaces, is proper, flat, and of} \\ \text{finite presentation and }\mathcal{L}\text{ ample on }X/T \end{matrix} \right\} \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathit{Sch}_{fppf}}(\mathcal{S}_ T, \mathcal{P}\! \mathit{olarized}) \]

Proof. Omitted. Hints: Argue exactly as in the proof of Lemma 99.13.4 and use Descent on Spaces, Proposition 74.4.1 to descent the invertible sheaf in the construction of the quasi-inverse functor. The relative ampleness property descends by Descent on Spaces, Lemma 74.13.1. $\square$

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