Lemma 10.163.6 (0C21)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.163: Ascending properties
Lemma 10.163.6 (cite)

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Lemma 10.163.6. 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 reduced,
$R$ is reduced.

Then $S$ is reduced.

Proof. For Noetherian rings reduced is the same as having properties $(S_1)$ and $(R_0)$ , see Lemma 10.157.3. 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 $(S_1)$ and $(R_0)$ , in other words reduced by Lemma 10.157.3 again. $\square$

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