The Stacks project

Lemma 29.45.5. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent:

  1. $f$ is a universal homeomorphism, and

  2. $f$ is integral, universally injective and surjective.

Proof. Assume $f$ is a universal homeomorphism. By Lemma 29.45.4 we see that $f$ is affine. Since $f$ is clearly universally closed we see that $f$ is integral by Lemma 29.44.7. It is also clear that $f$ is universally injective and surjective.

Assume $f$ is integral, universally injective and surjective. By Lemma 29.44.7 $f$ is universally closed. Since it is also universally bijective (see Lemma 29.9.4) we see that it is a universal homeomorphism. $\square$

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