The Stacks project

Lemma 59.80.4. Let $X$ be an integral normal scheme with separably closed function field.

  1. A separated étale morphism $U \to X$ is a disjoint union of open immersions.

  2. All local rings of $X$ are strictly henselian.

Proof. Let $R$ be a normal domain whose fraction field is separably algebraically closed. Let $R \to A$ be an étale ring map. Then $A \otimes _ R K$ is as a $K$-algebra a finite product $\prod _{i = 1, \ldots , n} K$ of copies of $K$. Let $e_ i$, $i = 1, \ldots , n$ be the corresponding idempotents of $A \otimes _ R K$. Since $A$ is normal (Algebra, Lemma 10.163.9) the idempotents $e_ i$ are in $A$ (Algebra, Lemma 10.37.12). Hence $A = \prod Ae_ i$ and we may assume $A \otimes _ R K = K$. Since $A \subset A \otimes _ R K = K$ (by flatness of $R \to A$ and since $R \subset K$) we conclude that $A$ is a domain. By the same argument we conclude that $A \otimes _ R A \subset (A \otimes _ R A) \otimes _ R K = K$. It follows that the map $A \otimes _ R A \to A$ is injective as well as surjective. Thus $R \to A$ defines an open immersion by Morphisms, Lemma 29.10.2 and Étale Morphisms, Theorem 41.14.1.

Let $f : U \to X$ be a separated étale morphism. Let $\eta \in X$ be the generic point and let $f^{-1}(\{ \eta \} ) = \{ \xi _ i\} _{i \in I}$. The result of the previous paragraph shows the following: For any affine open $U' \subset U$ whose image in $X$ is contained in an affine we have $U' = \coprod _{i \in I} U'_ i$ where $U'_ i$ is the set of point of $U'$ which are specializations of $\xi _ i$. Moreover, the morphism $U'_ i \to X$ is an open immersion. It follows that $U_ i = \overline{\{ \xi _ i\} }$ is an open and closed subscheme of $U$ and that $U_ i \to X$ is locally on the source an isomorphism. By Morphisms, Lemma 29.49.7 the fact that $U_ i \to X$ is separated, implies that $U_ i \to X$ is injective and we conclude that $U_ i \to X$ is an open immersion, i.e., (1) holds.

Part (2) follows from part (1) and the description of the strict henselization of $\mathcal{O}_{X, x}$ as the local ring at $\overline{x}$ on the étale site of $X$ (Lemma 59.33.1). It can also be proved directly, see Fundamental Groups, Lemma 58.12.2. $\square$

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