Lemma 15.47.5. Let $R \to S$ be a ring map. Assume that

1. $R$ is a Noetherian domain and J-0,

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

3. $S$ is a domain, and

4. the induced extension of fraction fields is separable.

Then $S$ is J-0.

Proof. We may replace $R$ by a principal localization and assume $R$ is a regular ring. By Algebra, Lemma 10.140.9 the ring map $R \to S$ is smooth at $(0)$. Hence after replacing $S$ by a principal localization we may assume that $S$ is smooth over $R$. Then $S$ is regular too, see Algebra, Lemma 10.163.10. $\square$

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