The Stacks project

Lemma 35.14.5. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

  1. $f$ is surjective, flat, and locally of finite presentation,

  2. $p$ is smooth (resp. étale).

Then $q$ is smooth (resp. étale).

Proof. Assume (1) and that $p$ is smooth. By Lemma 35.14.3 we see that $q$ is locally of finite presentation. By Morphisms, Lemma 29.25.13 we see that $q$ is flat. Hence now it suffices to show that the fibres of $q$ are smooth, see Morphisms, Lemma 29.34.3. Apply Varieties, Lemma 33.25.9 to the flat surjective morphisms $X_ s \to Y_ s$ for $s \in S$ to conclude. We omit the proof of the étale case. $\square$

Comments (2)

Comment #774 by Kestutis Cesnavicius on

I think it would be helpful to have this stated also for algebraic spaces (e.g., it is useful for showing that a quotient of a smooth scheme by a free action of an fppf group scheme is a smooth alg. space). The proof immediately reduces to the scheme case by replacing , then , and finally by etale covers by schemes.

Comment #796 by on

OK, yes, I added this to the chapter on Descent on Spaces. It does not have a tag yet, but it will later today. See this commit.

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