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. $\square$

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