Lift étale neighbourhood of point on fibre to total space.

Lemma 37.35.6. Let $X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $(V, v) \to (X_ s, x)$ be an étale neighbourhood. Then there exists an étale neighbourhood $(U, u) \to (X, x)$ such that there exists a morphism $(U_ s, u) \to (V, v)$ of étale neighbourhoods of $(X_ s, x)$ which is an open immersion.

Proof. We may assume $X$, $V$, and $S$ affine. Say the morphism $X \to S$ is given by $A \to B$ the point $x$ by a prime $\mathfrak q \subset B$, the point $s$ by $\mathfrak p = A \cap \mathfrak q$, and the morphism $V \to X_ s$ by $B \otimes _ A \kappa (\mathfrak p) \to C$. Since $\kappa (\mathfrak p)$ is a localization of $A/\mathfrak p$ there exists an $f \in A$, $f \not\in \mathfrak p$ and an étale ring map $B \otimes _ A (A/\mathfrak p)_ f \to D$ such that

$C = (B \otimes _ A \kappa (\mathfrak p)) \otimes _{B \otimes _ A (A/\mathfrak p)_ f} D$

See Algebra, Lemma 10.143.3 part (9). After replacing $A$ by $A_ f$ and $B$ by $B_ f$ we may assume $D$ is étale over $B \otimes _ A A/\mathfrak p = B/\mathfrak p B$. Then we can apply Algebra, Lemma 10.143.10. This proves the lemma. $\square$

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