Lemma 33.25.1. Let $k$ be a field. Let $X$ be a scheme over $k$. Assume

1. $X$ is locally of finite type over $k$,

2. $\Omega _{X/k}$ is locally free, and

3. $k$ has characteristic zero.

Then the structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is smooth.

Proof. This follows from Algebra, Lemma 10.140.7. $\square$

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