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 $k'/k$ 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #2110 by Matthew Emerton on