Lemma 10.164.2. Let $R \to S$ be a ring map. Assume that

1. $R \to S$ is faithfully flat, and

2. $S$ is reduced.

Then $R$ is reduced.

Proof. This is clear as $R \to S$ is injective. $\square$

Comment #873 by Dylan on

Suggested slogan: If you're covered (faithfully and flatly) by a reduced guy, then you're a reduced guy.

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