The Stacks project

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

  1. $f$ is smooth at $u_0$,

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

Proof. This is a restatement of More on Morphisms, Lemma 37.12.1. $\square$

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