The Stacks project

Lemma 97.13.7. Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume

  1. $f$ is representable by algebraic spaces,

  2. $f$ satisfies (97.13.2.1),

  3. $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects, and

  4. $\mathcal{Y}$ is limit preserving.

Then $f$ is smooth.

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 (97.13.2.1) 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 93.10.1. In fact, it suffices to do this when $V$ is an affine scheme locally of finite presentation over $S$, see Criteria for Representability, Lemma 96.5.6. Then $(\mathit{Sch}/V)_{fppf}$ is limit preserving by Lemma 97.11.3. Hence $Z \to S$ is locally of finite presentation by Lemmas 97.11.2 and 97.11.3. Choose a scheme $W$ and a surjective ├ętale morphism $W \to Z$. Then $W$ is locally of finite presentation over $S$.

Since $f$ satisfies (97.13.2.1) we see that so does $\mathcal{Z} \to (\mathit{Sch}/V)_{fppf}$, see Lemma 97.13.5. Next, we see that $(\mathit{Sch}/W)_{fppf} \to \mathcal{Z}$ satisfies (97.13.2.1) by Lemma 97.13.6. Thus the composition

\[ (\mathit{Sch}/W)_{fppf} \to \mathcal{Z} \to (\mathit{Sch}/V)_{fppf} \]

satisfies (97.13.2.1) by Lemma 97.13.4. 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. $\square$


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