The Stacks project

Smooth morphisms are locally standard smooth.

Lemma 29.33.11. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent

  1. The morphism $f$ is smooth at $x$.

  2. There exist an affine open $U \subset X$, with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_ U : U \to V$ is standard smooth.

Proof. Follows from the definitions and Algebra, Lemmas 10.136.7 and 10.136.10. $\square$

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