Lemma 97.12.4. Let $S$ be a locally Noetherian scheme. Let $f : U \to V$ be a morphism of schemes locally of finite type over $S$. Let $u_0 \in U$ be a finite type point. The following are equivalent

$f$ is smooth at $u_0$,

$f$ viewed as an object of $(\mathit{Sch}/V)_{fppf}$ over $U$ is versal at $u_0$.

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