## 15.42 Ascending properties along regular ring maps

This section is the analogue of Algebra, Section 10.163 but where the ring map $R \to S$ is regular.

Lemma 15.42.1. Let $\varphi : R \to S$ be a ring map. Assume

1. $\varphi$ is regular,

2. $S$ is Noetherian, and

3. $R$ is Noetherian and reduced.

Then $S$ is reduced.

Proof. For Noetherian rings being reduced is the same as having properties $(S_1)$ and $(R_0)$, see Algebra, Lemma 10.157.3. Hence we may apply Algebra, Lemmas 10.163.4 and 10.163.5. $\square$

Lemma 15.42.2. Let $\varphi : R \to S$ be a ring map. Assume

1. $\varphi$ is regular,

2. $S$ is Noetherian, and

3. $R$ is Noetherian and normal.

Then $S$ is normal.

Proof. For Noetherian rings being normal is the same as having properties $(S_2)$ and $(R_1)$, see Algebra, Lemma 10.157.4. Hence we may apply Algebra, Lemmas 10.163.4 and 10.163.5. $\square$

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