The Stacks project

Lemma 99.14.6. The functor ( defines a $1$-morphism

\[ \mathcal{P}\! \mathit{olarized}\to \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \]

of stacks in groupoids over $\mathit{Sch}_{fppf}$ which is algebraic in the sense of Criteria for Representability, Definition 97.8.1.

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

\[ \mathcal{X} = (\mathit{Sch}/S)_{fppf} \times _{\xi , \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}} \mathcal{P}\! \mathit{olarized} \]

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

\[ \mathcal{X} \subset \mathcal{P}\! \mathit{ic}_{X/S} \]

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. $\square$

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