The Stacks project

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

  1. $f$ is smooth, and

  2. $f$ 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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