The Stacks project

Lemma 29.35.12. Let $f : X \to S$ be a morphism of schemes.

  1. If $f$ is unramified then for any $x \in X$ the field extension $\kappa (f(x)) \subset \kappa (x)$ is finite separable.

  2. If $f$ is locally of finite type, and for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$ then $f$ is unramified.

  3. If $f$ is locally of finite presentation, and for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$ then $f$ is G-unramified.


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