Lemma 10.166.4. Let $k$ be a field. Let $A \to B$ be a smooth ring map of $k$-algebras. If $A$ is geometrically regular over $k$, then $B$ is geometrically regular over $k$.

**Proof.**
Let $k'/k$ be a finitely generated field extension. Then $A \otimes _ k k' \to B \otimes _ k k'$ is a smooth ring map (Lemma 10.137.4) and $A \otimes _ k k'$ is regular. Hence $B \otimes _ k k'$ is regular by Lemma 10.163.10.
$\square$

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