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The Stacks project

Lemma 29.34.3. Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and all fibres $X_ s$ are smooth, then $f$ is smooth.

Proof. Follows from Algebra, Lemma 10.137.17. $\square$

Comments (1)

Comment #9881 by on

This implication is in fact an equivalence, and it may be worth stating explicitly, especially since these conditions are often given as the definition of smoothness.

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