Lemma 15.47.4. Let $R \to S$ be a ring map. Assume that

$R$ is a Noetherian domain,

$R \to S$ is injective and of finite type, and

$S$ is a domain and J-0.

Then $R$ is J-0.

Lemma 15.47.4. Let $R \to S$ be a ring map. Assume that

$R$ is a Noetherian domain,

$R \to S$ is injective and of finite type, and

$S$ is a domain and J-0.

Then $R$ is J-0.

**Proof.**
After replacing $S$ by $S_ g$ for some nonzero $g \in S$ we may assume that $S$ is a regular ring. By generic flatness we may assume that also $R \to S$ is faithfully flat, see Algebra, Lemma 10.118.1. Then $R$ is regular by Algebra, Lemma 10.164.4.
$\square$

Your email address will not be published.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)

There are also: