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.
41.14 Topological properties of étale morphisms
We present a few of the topological properties of étale and unramified morphisms. First, we give what Grothendieck calls the fundamental property of étale morphisms, see [Exposé I.5, SGA1].
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. $\square$
Here is another result of a similar flavor.
Lemma 41.14.2. Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that
$\pi $ is finite,
$\pi $ is étale,
$\pi ^{-1}(\{ s\} ) = \{ x\} $, and
$\kappa (s) \subset \kappa (x)$ is purely inseparable1.
Then there exists an open neighbourhood $U$ of $s$ such that $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is an isomorphism.
Proof. By Lemma 41.7.3 there exists an open neighbourhood $U$ of $s$ such that $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is a closed immersion. But a morphism which is étale and a closed immersion is an open immersion (for example by Theorem 41.14.1). Hence after shrinking $U$ we obtain an isomorphism. $\square$
Lemma 41.14.3. Let $U \to X$ be an étale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \to U \times _{X, F_ X} X$ is an isomorphism.
Proof. The morphism $F_{U/X}$ is a universal homeomorphism by Varieties, Lemma 33.36.6. The morphism $F_{U/X}$ is étale as a morphism between schemes étale over $X$ (Morphisms, Lemma 29.36.18). Hence $F_{U/X}$ is an isomorphism by Theorem 41.14.1. $\square$
[1] In view of condition (2) this is equivalent to $\kappa (s) = \kappa (x)$.
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