The Stacks project

Lemma 76.21.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. 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 Lemma 76.19.10, Lemma 76.21.2, Lemma 76.21.3 and the case of schemes which is More on Morphisms, Lemma 37.13.5. $\square$

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