The Stacks project

Proof. We prove $\mathcal{P}\! \mathit{olarized}$ is a stack in groupoids over $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ by checking conditions (1), (2), and (3) of Stacks, Definition 8.5.1. We have already seen (1) in Lemma 99.14.2.

A covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ comes about in the following manner: Let $X \to S$ be an object of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$. Suppose that $\{ S_ i \to S\} _{i \in I}$ is a covering of $\mathit{Sch}_{fppf}$. Set $X_ i = S_ i \times _ S X$. Then $\{ (X_ i \to S_ i) \to (X \to S)\} _{i \in I}$ is a covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ and every covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ is isomorphic to one of these. Set $S_{ij} = S_ i \times _ S S_ j$ and $X_{ij} = S_{ij} \times _ S X$ so that $(X_{ij} \to S_{ij}) = (X_ i \to S_ i) \times _{(X \to S)} (X_ j \to S_ j)$. Next, suppose that $\mathcal{L}, \mathcal{N}$ are ample invertible sheaves on $X/S$ so that $(X \to S, \mathcal{L})$ and $(X \to S, \mathcal{N})$ are two objects of $\mathcal{P}\! \mathit{olarized}$ over the object $(X \to S)$. To check descent for morphisms, we assume we have morphisms $(\text{id}, \text{id}, \varphi _ i)$ from $(X_ i \to S_ i, \mathcal{L}|_{X_ i})$ to $(X_ i \to S_ i, \mathcal{N}|_{X_ i})$ whose base changes to morphisms from $(X_{ij} \to S_{ij}, \mathcal{L}|_{X_{ij}})$ to $(X_{ij} \to S_{ij}, \mathcal{N}|_{X_{ij}})$ agree. Then $\varphi _ i : \mathcal{L}|_{X_ i} \to \mathcal{N}|_{X_ i}$ are isomorphisms of invertible modules over $X_ i$ such that $\varphi _ i$ and $\varphi _ j$ restrict to the same isomorphisms over $X_{ij}$. By descent for quasi-coherent sheaves (Descent on Spaces, Proposition 74.4.1) we obtain a unique isomorphism $\varphi : \mathcal{L} \to \mathcal{N}$ whose restriction to $X_ i$ recovers $\varphi _ i$.

Decent for objects is proved in exactly the same manner. Namely, suppose that $\{ (X_ i \to S_ i) \to (X \to S)\} _{i \in I}$ is a covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ as above. Suppose we have objects $(X_ i \to S_ i, \mathcal{L}_ i)$ of $\mathcal{P}\! \mathit{olarized}$ lying over $(X_ i \to S_ i)$ and a descent datum

satisfying the obvious cocycle condition over $(X_{ijk} \to S_{ijk})$ for every triple of indices. Then by descent for quasi-coherent sheaves (Descent on Spaces, Proposition 74.4.1) we obtain a unique invertible $\mathcal{O}_ X$-module $\mathcal{L}$ and isomorphisms $\mathcal{L}|_{X_ i} \to \mathcal{L}_ i$ recovering the descent datum $\varphi _{ij}$. To show that $(X \to S, \mathcal{L})$ is an object of $\mathcal{P}\! \mathit{olarized}$ we have to prove that $\mathcal{L}$ is ample. This follows from Descent on Spaces, Lemma 74.13.1.

Since we already have seen that $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$ (Lemma 99.13.3) it now follows formally that $\mathcal{P}\! \mathit{olarized}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$. See Stacks, Lemma 8.10.6. $\square$