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.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05B5. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 35.14.5 as pdf