Lemma 39.9.3. Let $k$ be a field. Let $A$ be an abelian variety over $k$. For any field extension $K/k$ the base change $A_ K$ is an abelian variety over $K$.

**Proof.**
Omitted. Note that this is why we insisted on $A$ being geometrically integral; without that condition this lemma (and many others below) would be wrong.
$\square$

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