Lemma 38.15.5. In Situation 38.15.1. If there exists an impurity (g : T \to S, t' \leadsto t, \xi ) of \mathcal{F} above s with g quasi-finite at t, then there exists an impurity (g : T \to S, t' \leadsto t, \xi ) such that (T, t) \to (S, s) is an elementary étale neighbourhood.
Proof. Let (g : T \to S, t' \leadsto t, \xi ) be an impurity of \mathcal{F} above s such that g is quasi-finite at t. After shrinking T we may assume that g is locally of finite type. Apply More on Morphisms, Lemma 37.41.1 to T \to S and t \mapsto s. This gives us a diagram
where (U, u) \to (S, s) is an elementary étale neighbourhood and V \subset T \times _ S U is an open neighbourhood of v = (t, u) such that V \to U is finite and such that v is the unique point of V lying over u. Since the morphism V \to T is étale hence flat we see that there exists a specialization v' \leadsto v such that v' \mapsto t'. Note that \kappa (t') \subset \kappa (v') is finite separable. Pick any point \zeta \in X_{v'} mapping to \xi \in X_{t'}. By Divisors, Lemma 31.7.3 we see that \zeta \in \text{Ass}_{X_ V/V}(\mathcal{F}_ V). Moreover, the closure \overline{\{ \zeta \} } does not meet the fibre X_ v as by assumption the closure \overline{\{ \xi \} } does not meet X_ t. In other words (V \to S, v' \leadsto v, \zeta ) is an impurity of \mathcal{F} above S.
Next, let u' \in U' be the image of v' and let \theta \in X_ U be the image of \zeta . Then \theta \mapsto u' and u' \leadsto u. By Divisors, Lemma 31.7.3 we see that \theta \in \text{Ass}_{X_ U/U}(\mathcal{F}). Moreover, as \pi : X_ V \to X_ U is finite we see that \pi \big (\overline{\{ \zeta \} }\big ) = \overline{\{ \pi (\zeta )\} }. Since v is the unique point of V lying over u we see that X_ u \cap \overline{\{ \pi (\zeta )\} } = \emptyset because X_ v \cap \overline{\{ \zeta \} } = \emptyset . In this way we conclude that (U \to S, u' \leadsto u, \theta ) is an impurity of \mathcal{F} above s and we win. \square
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