Lemma 37.13.7. 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.

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