The Stacks project

Theorem 41.14.1. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent:

  1. $f$ is an open immersion,

  2. $f$ is universally injective and étale, and

  3. $f$ is a flat monomorphism, locally of finite presentation.

Proof. An open immersion is universally injective since any base change of an open immersion is an open immersion. Moreover, it is étale by Morphisms, Lemma 29.36.9. Hence (1) implies (2).

Assume $f$ is universally injective and étale. Since $f$ is étale it is flat and locally of finite presentation, see Morphisms, Lemmas 29.36.12 and 29.36.11. By Lemma 41.7.1 we see that $f$ is a monomorphism. Hence (2) implies (3).

Assume $f$ is flat, locally of finite presentation, and a monomorphism. Then $f$ is open, see Morphisms, Lemma 29.25.10. Thus we may replace $Y$ by $f(X)$ and we may assume $f$ is surjective. Then $f$ is open and bijective hence a homeomorphism. Hence $f$ is quasi-compact. Hence Descent, Lemma 35.25.1 shows that $f$ is an isomorphism and we win. $\square$

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