Lemma 39.8.4. Let $k$ be a perfect field of characteristic $p > 0$ (see Lemma 39.8.2 for the characteristic zero case). Let $G$ be a locally algebraic group scheme over $k$. If $G$ is reduced then the structure morphism $G \to \mathop{\mathrm{Spec}}(k)$ is smooth, i.e., $G$ is a smooth group scheme.

**Proof.**
By Lemma 39.6.3 the sheaf $\Omega _{G/k}$ is free. Hence the lemma follows from Varieties, Lemma 33.25.2.
$\square$

## 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Your email address will not be published.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.

## Comments (0)