The Stacks project

Proof. Choose a surjective smooth morphism $U \to \mathcal{X}$ for some scheme $U$. Then $U \to S$ is locally of finite presentation, see Morphisms of Stacks, Section 101.27. We can write $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma 94.16.2. Since $U$ is locally of finite presentation over $S$ it follows that the algebraic space $R$ is locally of finite presentation over $S$. Recall that $[U/R]$ is the stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ obtained by stackyfying the category fibred in groupoids whose fibre category over $T$ is the groupoid $(U(T), R(T), s, t, c)$. Since $U$ and $R$ are limit preserving as functors (Limits of Spaces, Proposition 70.3.10) this category fibred in groupoids is limit preserving. Thus it suffices to show that fppf stackyfication preserves the property of being limit preserving. This is true (hint: use Topologies, Lemma 34.13.2). However, we give a direct proof below using that in this case we know what the stackyfication amounts to.

Let $T = \mathop{\mathrm{lim}}\nolimits T_\lambda $ be a directed limit of affine schemes over $S$. We have to show that the functor

is an equivalence of categories. Let us show this functor is essentially surjective. Let $x \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_ T)$. In Groupoids in Spaces, Lemma 78.24.1 the reader finds a description of the category $[U/R]_ T$. In particular $x$ corresponds to an fppf covering $\{ T_ i \to T\} _{i \in I}$ and a $[U/R]$-descent datum $(u_ i, r_{ij})$ relative to this covering. After refining this covering we may assume it is a standard fppf covering of the affine scheme $T$. By Topologies, Lemma 34.13.2 we may choose a $\lambda $ and a standard fppf covering $\{ T_{\lambda , i} \to T_\lambda \} _{i \in I}$ whose base change to $T$ is equal to $\{ T_ i \to T\} _{i \in I}$. For each $i$, after increasing $\lambda $, we can find a $u_{\lambda , i} : T_{\lambda , i} \to U$ whose composition with $T_ i \to T_{\lambda , i}$ is the given morphism $u_ i$ (this is where we use that $U$ is limit preserving). Similarly, for each $i, j$, after increasing $\lambda $, we can find a $r_{\lambda , ij} : T_{\lambda , i} \times _{T_\lambda } T_{\lambda , j} \to R$ whose composition with $T_{ij} \to T_{\lambda , ij}$ is the given morphism $r_{ij}$ (this is where we use that $R$ is limit preserving). After increasing $\lambda $ we can further assume that

and

In other words, we may assume that $(u_{\lambda , i}, r_{\lambda , ij})$ is a $[U/R]$-descent datum relative to the covering $\{ T_{\lambda , i} \to T_\lambda \} _{i \in I}$. Then we obtain a corresponding object of $[U/R]$ over $T_\lambda $ whose pullback to $T$ is isomorphic to $x$ as desired. The proof of fully faithfulness works in exactly the same way using the description of morphisms in the fibre categories of $[U/T]$ given in Groupoids in Spaces, Lemma 78.24.1. $\square$