Lemma 38.15.7. In Situation 38.15.1. If there exists an impurity $(S^ h \to S, s' \leadsto s, \xi )$ of $\mathcal{F}$ above $s$ then there exists an impurity $(T \to S, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $s$ where $(T, t) \to (S, s)$ is an elementary étale neighbourhood.
Proof. We may replace $S$ by an affine neighbourhood of $s$. Say $S = \mathop{\mathrm{Spec}}(A)$ and $s$ corresponds to the prime $\mathfrak p \subset A$. Then $\mathcal{O}_{S, s}^ h = \mathop{\mathrm{colim}}\nolimits _{(T, t)} \Gamma (T, \mathcal{O}_ T)$ where the limit is over the opposite of the cofiltered category of affine elementary étale neighbourhoods $(T, t)$ of $(S, s)$, see More on Morphisms, Lemma 37.35.5 and its proof. Hence $S^ h = \mathop{\mathrm{lim}}\nolimits _ i T_ i$ and we win by Lemma 38.15.4. $\square$
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