Lemma 76.21.7 (0D11)—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 76.21.7 (cite)

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Lemma 76.21.7. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent

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

Proof. Assume (1). A formally étale morphism is a formally smooth morphism. Thus $H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = 0$ by Lemma 76.21.6. On the other hand, a formally étale morphism if formally unramified hence we have $\Omega _{X/Y} = 0$ by Lemma 76.14.6. Conversely, if (2) holds, then $f$ is formally smooth by Lemma 76.21.6 and formally unramified by Lemma 76.14.6 and hence formally étale by Lemmas 76.19.4. $\square$

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