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

$f$ is an open immersion,

$f$ is universally injective and étale, and

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

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

$f$ is an open immersion,

$f$ is universally injective and étale, and

$f$ 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 35.22.1 shows that $f$ is an isomorphism and we win. $\square$

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