The Stacks project

Proof. The key ingredient of the proof is More on Morphisms, Lemma 37.12.1 which (almost) says that a morphism of schemes of finite type over $S$ satisfying (98.13.2.1) is a smooth morphism. The other arguments of the proof are essentially bookkeeping.

Let $V$ be a scheme over $S$ and let $y$ be an object of $\mathcal{Y}$ over $V$. Let $Z$ be an algebraic space representing the $2$-fibre product $\mathcal{Z} = \mathcal{X} \times _{f, \mathcal{X}, y} (\mathit{Sch}/V)_{fppf}$. We have to show that the projection morphism $Z \to V$ is smooth, see Algebraic Stacks, Definition 94.10.1. In fact, it suffices to do this when $V$ is an affine scheme locally of finite presentation over $S$, see Criteria for Representability, Lemma 97.5.6. Then $(\mathit{Sch}/V)_{fppf}$ is limit preserving by Lemma 98.11.3. Hence $Z \to S$ is locally of finite presentation by Lemmas 98.11.2 and 98.11.3. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then $W$ is locally of finite presentation over $S$.

Since $f$ satisfies (98.13.2.1) we see that so does $\mathcal{Z} \to (\mathit{Sch}/V)_{fppf}$, see Lemma 98.13.5. Next, we see that $(\mathit{Sch}/W)_{fppf} \to \mathcal{Z}$ satisfies (98.13.2.1) by Lemma 98.13.6. Thus the composition

satisfies (98.13.2.1) by Lemma 98.13.4. More on Morphisms, Lemma 37.12.1 shows that the composition $W \to Z \to V$ is smooth at every finite type point $w_0$ of $W$. Since the smooth locus is open we conclude that $W \to V$ is a smooth morphism of schemes by Morphisms, Lemma 29.16.7. Thus we conclude that $Z \to V$ is a smooth morphism of algebraic spaces by definition. $\square$