The Stacks project

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].

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. $\square$

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

  1. $\pi $ is finite,

  2. $\pi $ is étale,

  3. $\pi ^{-1}(\{ s\} ) = \{ x\} $, and

  4. $\kappa (s) \subset \kappa (x)$ 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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