Lemma 38.2.2. Let

be a commutative diagram of schemes with $X' \to X$ and $S' \to S$ étale. Let $s' \in S'$ be a point. Then

is étale.

Lemma 38.2.2. Let

\[ \xymatrix{ X \ar[d] & X' \ar[l] \ar[d] \\ S & S' \ar[l] } \]

be a commutative diagram of schemes with $X' \to X$ and $S' \to S$ étale. Let $s' \in S'$ be a point. Then

\[ X' \times _{S'} \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \longrightarrow X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

is étale.

**Proof.**
This is true because $X' \to X_{S'}$ is étale as a morphism of schemes étale over $X$, see Morphisms, Lemma 29.36.18 and the base change of an étale morphism is étale, see Morphisms, Lemma 29.36.4.
$\square$

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