The Stacks project

Smooth morphisms are locally standard smooth.

Lemma 29.34.11. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent

  1. The morphism $f$ is smooth at $x$.

  2. There exists an affine open $U \subset X$, with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_ U : U \to V$ is standard smooth.

Proof. Follows from the definitions and Algebra, Lemmas 10.137.7 and 10.137.10. $\square$


Comments (1)

There are also:

  • 2 comment(s) on Section 29.34: 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Your email address will not be published.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 01V7. Beware of the difference between the letter 'O' and the digit '0'.