The Stacks project

Lemma 37.13.5. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent

  1. $f$ is formally smooth,

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

Proof. This follows from Algebra, Proposition 10.138.8 and Lemma 37.11.10. $\square$

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