Lemma 39.8.2. Let $k$ be a field of characteristic $0$. Let $G$ be a locally algebraic group scheme over $k$. 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 module of differentials of $G$ over $k$ is free. Hence smoothness follows from Varieties, Lemma 33.25.1.
$\square$

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