2
$\begingroup$

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 $\dim X-\dim Y$.

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ 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)]. $\endgroup$
    – Francesco Polizzi
    Commented Apr 27, 2021 at 18:44
  • 2
    $\begingroup$ Another reference: Matsumura, Commutative Ring Theory, Theorem 23.1. $\endgroup$
    – Laurent Moret-Bailly
    Commented Apr 27, 2021 at 18:51
  • 1
    $\begingroup$ Thank you, @FrancescoPolizzi! I did know the book of Matsumura, but not the text of Ravi Vakil.... $\endgroup$
    – Mikhail Borovoi
    Commented Apr 27, 2021 at 19:42

1 Answer 1

Reset to default
1
$\begingroup$

Let me expand my comment into an answer, so that the question will not appear as unanswered anymore.

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)].

References.

[Vakil] R. Vakil, The Rising Sea: Foundations of Algebraic Geometry, November 18, 2017 version.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .