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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 10.137.16. $\square$

Comments (2)

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.

Comment #10423 by on

The text above the lemma indicates this and the converse direction is the (trivial) combination of 3 lemmas following shortly after this one. Going to leave as is.

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  • 2 comment(s) on Section 29.34: Smooth morphisms

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