Lemma 29.34.11. Fibres of unramified morphisms.

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

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

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