The Stacks project

Lemma 29.47.10. Let $X$ be a scheme. The following are equivalent

  1. $X$ is absolutely weakly normal,

  2. $X$ is equal to its own absolute weak normalization, i.e., the morphism $X^{awn} \to X$ is an isomorphism,

  3. if $\pi : Y \to X$ is a universal homeomorphism with $Y$ reduced, then $\pi $ is an isomorphism.

Proof. This is proved in exactly the same manner as Lemma 29.47.9. $\square$

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