Lemma 32.8.9. Notation and assumptions as in Situation 32.8.1. If

$f$ is smooth,

$f_0$ is locally of finite presentation,

then $f_ i$ is smooth for some $i \geq 0$.

Lemma 32.8.9. Notation and assumptions as in Situation 32.8.1. If

$f$ is smooth,

$f_0$ is locally of finite presentation,

then $f_ i$ is smooth for some $i \geq 0$.

**Proof.**
Being smooth is local on the source and the target (Morphisms, Lemma 29.34.2) hence we may assume $S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is Algebra, Lemma 10.168.8.
$\square$

Your email address will not be published.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.

## Comments (0)

There are also: