The Stacks project

Lemma 35.33.5. Consider a commutative diagram of morphisms of schemes

\[ \xymatrix{ U' \ar[r] \ar[d] & V' \ar[d] \\ U \ar[r] & V } \]

with étale vertical arrows and a point $v' \in V'$ mapping to $v \in V$. Then the morphism of fibres $U'_{v'} \to U_ v$ is étale.

Proof. Note that $U'_ v \to U_ v$ is étale as a base change of the étale morphism $U' \to U$. The scheme $U'_ v$ is a scheme over $V'_ v$. By Morphisms, Lemma 29.36.7 the scheme $V'_ v$ is a disjoint union of spectra of finite separable field extensions of $\kappa (v)$. One of these is $v' = \mathop{\mathrm{Spec}}(\kappa (v'))$. Hence $U'_{v'}$ is an open and closed subscheme of $U'_ v$ and it follows that $U'_{v'} \to U'_ v \to U_ v$ is étale (as a composition of an open immersion and an étale morphism, see Morphisms, Section 29.36). $\square$

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