The Stacks project

Lemma 29.47.9. Let $X$ be a scheme. The following are equivalent

  1. $X$ is seminormal,

  2. $X$ is equal to its own seminormalization, i.e., the morphism $X^{sn} \to X$ is an isomorphism,

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

Proof. The equivalence of (1) and (2) is clear from Lemma 29.47.7. If (3) holds, then $X^{sn} \to X$ is an isomorphism and we see that (2) holds.

Assume (2) holds and let $\pi : Y \to X$ be a universal homeomorphism inducing isomorphisms on residue fields with $Y$ reduced. Then there exists a factorization $X \to Y \to X$ of $\text{id}_ X$ by Lemma 29.47.7. Then $X \to Y$ is a closed immersion (by Schemes, Lemma 26.21.11 and the fact that $\pi $ is separated for example by Lemma 29.10.3). Since $X \to Y$ is also a bijection on points, the reducedness of $Y$ shows that it has to be an isomorphism. This finishes the proof. $\square$

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