The Stacks project

Lemma 37.61.14. Let $f : X \to Y$ be a morphism of schemes. If $f$ is weakly ├ętale and a universal homeomorphism, it is an isomorphism.

Proof. Since $f$ is a universal homeomorphism, the diagonal $\Delta : X \to X \times _ Y X$ is a surjective closed immersion by Morphisms, Lemmas 29.45.4 and 29.10.2. Since $\Delta $ is also flat, we see that $\Delta $ must be an isomorphism by Morphisms, Lemma 29.26.1. In other words, $f$ is a monomorphism (Schemes, Lemma 26.23.2). Since $f$ is a universal homeomorphism it is certainly quasi-compact. Hence by Descent, Lemma 35.24.1 we find that $f$ is an isomorphism. $\square$

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