Lemma 37.13.7. 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 the definition of a smooth ring homomorphism (Algebra, Definition 10.137.1), Lemma 37.13.2, and the definition of a smooth morphism of schemes (Morphisms, Definition 29.34.1). We also use that finite locally free is the same as finite projective for modules over rings (Algebra, Lemma 10.78.2). $\square$

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