Lemma 37.13.8. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent

1. $f$ is étale, and

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

Proof. This follows from the definition of an étale ring homomorphism (Algebra, Definition 10.143.1), Lemma 37.13.2, and the definition of an étale morphism of schemes (Morphisms, Definition 29.36.1). $\square$

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