Lemma 10.163.8 (0C22)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.163: Ascending properties
Lemma 10.163.8 (cite)

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Lemma 10.163.8. Let $\varphi : R \to S$ be a ring map. Assume

$R$ is Noetherian,
$S$ is Noetherian,
$\varphi$ is flat,
the fibre rings $S \otimes _ R \kappa (\mathfrak p)$ are normal, and
$R$ is normal.

Then $S$ is normal.

Proof. For a Noetherian ring being normal is the same as having properties $(S_2)$ and $(R_1)$ , see Lemma 10.157.4. Thus we know $R$ and the fibre rings have these properties. Hence we may apply Lemmas 10.163.4 and 10.163.5 and we see that $S$ is $(S_2)$ and $(R_1)$ , in other words normal by Lemma 10.157.4 again. $\square$

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