The Stacks project

Lemma 10.163.6. Let $\varphi : R \to S$ be a ring map. Assume

  1. $R$ is Noetherian,

  2. $S$ is Noetherian

  3. $\varphi $ is flat,

  4. the fibre rings $S \otimes _ R \kappa (\mathfrak p)$ are reduced,

  5. $R$ 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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