Lemma 76.21.8 (0D12)—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 76.21.8 (cite)

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Lemma 76.21.8. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent

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

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 37.13.7. $\square$

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