Lemma 98.3.2 (07WK)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 98: Artin's Axioms
Section 98.3: Predeformation categories
Lemma 98.3.2 (cite)

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Lemma 98.3.2. 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

$F$ is formally smooth on objects (Criteria for Representability, Section 97.6),
$F$ is representable by algebraic spaces and formally smooth, or
$F$ 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 (98.3.1.1) is smooth in the sense of Formal Deformation Theory, Definition 90.8.1.

Proof. Case (1) is a matter of unwinding the definitions. Assumption (2) implies (1) by Criteria for Representability, Lemma 97.6.3. Assumption (3) implies (2) by More on Morphisms of Spaces, Lemma 76.19.6 and the principle of Algebraic Stacks, Lemma 94.10.9. $\square$

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