Lemma 76.21.6 (0D10)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.21: The naive cotangent complex
Lemma 76.21.6 (cite)

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Lemma 76.21.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . The following are equivalent

$f$ is formally smooth,
$H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = 0$ and $H^0(\mathop{N\! L}\nolimits _{X/Y}) = \Omega _{X/Y}$ is locally projective.

Proof. This follows from Lemma 76.19.10, Lemma 76.21.2, Lemma 76.21.3 and the case of schemes which is More on Morphisms, Lemma 37.13.5. $\square$

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