Lemma 15.42.1. Let $\varphi : R \to S$ be a ring map. Assume
$\varphi $ is regular,
$S$ is Noetherian, and
$R$ is Noetherian and reduced.
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.
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
$\varphi $ is regular,
$S$ is Noetherian, and
$R$ 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. $\square$
Lemma 15.42.3. Let $\varphi : R \to S$ be a ring map. Assume
$\varphi $ is regular,
$S$ is Noetherian, and
$R$ is Noetherian and regular.
Then $S$ is regular.
Proof. For Noetherian rings being regular is the same as having properties $(R_ k)$ for all $k$. Hence we may apply Algebra, Lemma 10.163.5. $\square$
Lemma 15.42.4. Let $\varphi : R \to S$ be a ring map. Assume
$\varphi $ is regular,
$S$ is Noetherian, and
$R$ is Noetherian and Cohen-Macaulay.
Then $S$ is Cohen-Macaulay.
Proof. For Noetherian rings being Cohen-Macaulay is the same as having properties $(S_ k)$ for all $k$. Hence we may apply Algebra, Lemma 10.163.4. $\square$
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