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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