The Stacks project

Lemma 96.3.2. 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 either

  1. $F$ is formally smooth on objects (Criteria for Representability, Section 95.6),

  2. $F$ is representable by algebraic spaces and formally smooth, or

  3. $F$ is representable by algebraic spaces and smooth.

Then for every finite type field $k$ over $S$ and object $x_0$ of $\mathcal{X}$ over $k$ the functor (96.3.1.1) is smooth in the sense of Formal Deformation Theory, Definition 88.8.1.

Proof. Case (1) is a matter of unwinding the definitions. Assumption (2) implies (1) by Criteria for Representability, Lemma 95.6.3. Assumption (3) implies (2) by More on Morphisms of Spaces, Lemma 74.19.6 and the principle of Algebraic Stacks, Lemma 92.10.9. $\square$


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