# Smoothness of a morphism of smooth varieties with smooth fibres

I am asking for a reference for the following lemma (for which I know a proof).

Lemma. Let $$f\colon X\to Y$$ be a surjective morphism of irreducible smooth complex algebraic varieties (separated, reduced, irreducible schemes of finite type over $$\Bbb C$$) with smooth fibres over closed points of $$Y$$. The $$f$$ is smooth if and only if all these fibres have the same dimension $$\dim X-\dim Y$$.

• If $f$ is smooth, then $f$ is flat and the fibres are automatically smooth and equidimensional, see [Vakil, thm. 25.2.2]. This provides one implication of your desired statement. The converse implication is [Vakil, Exercise 25.2.F (a)]. Apr 27 at 18:44
• Another reference: Matsumura, Commutative Ring Theory, Theorem 23.1. Apr 27 at 18:51
• Thank you, @FrancescoPolizzi! I did know the book of Matsumura, but not the text of Ravi Vakil.... Apr 27 at 19:42

If $$f$$ is smooth, then $$f$$ is flat and its fibres are automatically smooth and equidimensional, see [Vakil, Theorem. 25.2.2]. This provides one implication of your statement. The converse implication is [Vakil, Exercise 25.2.F (a)].