Lemma 35.32.9. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. Given a commutative diagram of schemes
\[ \vcenter { \xymatrix{ X' \ar[d]_{g'} \ar[r]_{f'} & Y' \ar[d]^ g \\ X \ar[r]^ f & Y } } \quad \text{with points}\quad \vcenter { \xymatrix{ x' \ar[d] \ar[r] & y' \ar[d] \\ x \ar[r] & y } } \]
such that $g'$ is étale at $x'$ and $g$ is étale at $y'$, then $x \in W(f) \Leftrightarrow x' \in W(f')$ where $W(-)$ is as in Lemma 35.26.3.
Proof.
Lemma 35.26.3 applies since $\mathcal{P}$ is étale local on the source by Lemma 35.32.4.
Assume $x \in W(f)$. Let $U' \subset X'$ and $V' \subset Y'$ be open neighbourhoods of $x'$ and $y'$ such that $f'(U') \subset V'$, $g'(U') \subset W(f)$ and $g'|_{U'}$ and $g|_{V'}$ are étale. Then $f \circ g'|_{U'} = g \circ f'|_{U'}$ has $\mathcal{P}$ by property (1) of Definition 35.32.3. Then $f'|_{U'} : U' \to V'$ has property $\mathcal{P}$ by (4) of Lemma 35.32.4. Then by (3) of Lemma 35.32.4 we conclude that $f'_{U'} : U' \to Y'$ has $\mathcal{P}$. Hence $U' \subset W(f')$ by definition. Hence $x' \in W(f')$.
Assume $x' \in W(f')$. Let $U' \subset X'$ and $V' \subset Y'$ be open neighbourhoods of $x'$ and $y'$ such that $f'(U') \subset V'$, $U' \subset W(f')$ and $g'|_{U'}$ and $g|_{V'}$ are étale. Then $U' \to Y'$ has $\mathcal{P}$ by definition of $W(f')$. Then $U' \to V'$ has $\mathcal{P}$ by (4) of Lemma 35.32.4. Then $U' \to Y$ has $\mathcal{P}$ by (3) of Lemma 35.32.4. Let $U \subset X$ be the image of the étale (hence open) morphism $g'|_ U' : U' \to X$. Then $\{ U' \to U\} $ is an étale covering and we conclude that $U \to Y$ has $\mathcal{P}$ by (1) of Lemma 35.32.4. Thus $U \subset W(f)$ by definition. Hence $x \in W(f)$.
$\square$
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