Description of the étale schemes over fields and fibres of étale morphisms.

Lemma 29.36.7. Fibres of étale morphisms.

1. Let $X$ be a scheme over a field $k$. The structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is étale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$.

2. If $f : X \to S$ is an étale morphism, then for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$.

Proof. You can deduce this from Lemma 29.35.11 via Lemma 29.36.5 above. Here is a direct proof.

We will use Algebra, Lemma 10.143.4. Hence it is clear that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Conversely, suppose that $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Then for any affine open $U \subset X$ we see that $U$ is a finite disjoint union of spectra of finite separable field extensions of $k$. Hence all points of $X$ are closed points (see Lemma 29.20.2 for example). Thus $X$ is a discrete space and we win. $\square$

Comment #1039 by Jakob Scholbach on

Suggested slogan: Description of the étale site of a field.

Comment #7507 by Peng Du on

Don't we need the disjoint union to be finite (in both statement and proof)?

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