The Stacks project

Lemma 65.37.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. There is a maximal open subspace $U \subset X$ such that $f|_ U : U \to Y$ is smooth. Moreover, formation of this open commutes with base change by

  1. morphisms which are flat and locally of finite presentation,

  2. flat morphisms provided $f$ is locally of finite presentation.

Proof. The existence of $U$ follows from the fact that the property of being smooth is Zariski (and even ├ętale) local on the source, see Lemma 65.37.4. Moreover, this lemma allows us to translate properties (1) and (2) into the case of morphisms of schemes. The case of schemes is Morphisms, Lemma 29.34.15. Some details omitted. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 65.37: Smooth morphisms

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Your email address will not be published.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 0DZI. Beware of the difference between the letter 'O' and the digit '0'.