The Stacks project

Lemma 35.19.1. If $f : X \to Y$ is a flat and surjective morphism of schemes and $X$ is reduced, then $Y$ is reduced.

Proof. The result follows by looking at local rings (Schemes, Definition 26.12.1) and Algebra, Lemma 10.164.2. $\square$

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View Lemma 35.19.1 as pdf