Lemma 10.166.3. Let $k$ be a field. Let $A \to B$ be a faithfully flat $k$-algebra map. If $B$ is geometrically regular over $k$, so is $A$.

** Geometric regularity descends through faithfully flat maps of algebras **

**Proof.**
Assume $B$ is geometrically regular over $k$. Let $k \subset k'$ be a finite, purely inseparable extension. Then $A \otimes _ k k' \to B \otimes _ k k'$ is faithfully flat as a base change of $A \to B$ (by Lemmas 10.30.3 and 10.39.7) and $B \otimes _ k k'$ is regular by our assumption on $B$ over $k$. Then $A \otimes _ k k'$ is regular by Lemma 10.164.4.
$\square$

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## Comments (1)

Comment #2110 by Matthew Emerton on