Lemma 39.9.4. Let $k$ be a field. Let $A$ be an abelian variety over $k$. Then $A$ is smooth over $k$.

Proof. If $k$ is perfect then this follows from Lemma 39.8.2 (characteristic zero) and Lemma 39.8.4 (positive characteristic). We can reduce the general case to this case by descent for smoothness (Descent, Lemma 35.23.27) and going to the perfect closure using Lemma 39.9.3. $\square$

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