Lemma 10.164.4. Let $R \to S$ be a ring map. Assume that

$R \to S$ is faithfully flat, and

$S$ is a regular ring.

Then $R$ is a regular ring.

Lemma 10.164.4. Let $R \to S$ be a ring map. Assume that

$R \to S$ is faithfully flat, and

$S$ is a regular ring.

Then $R$ is a regular ring.

**Proof.**
We see that $R$ is Noetherian by Lemma 10.164.1. Let $\mathfrak p \subset R$ be a prime. Choose a prime $\mathfrak q \subset S$ lying over $\mathfrak p$. Then Lemma 10.110.9 applies to $R_\mathfrak p \to S_\mathfrak q$ and we conclude that $R_\mathfrak p$ is regular. Since $\mathfrak p$ was arbitrary we see $R$ is regular.
$\square$

Your email address will not be published.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 (0)