Lemma 38.15.8. In Situation 38.15.1 the following are equivalent

there exists an impurity $(S^ h \to S, s' \leadsto s, \xi )$ of $\mathcal{F}$ above $s$ where $S^ h$ is the henselization of $S$ at $s$,

there exists an impurity $(T \to S, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $s$ such that $(T, t) \to (S, s)$ is an elementary étale neighbourhood, and

there exists an impurity $(T \to S, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $s$ such that $T \to S$ is quasi-finite at $t$.

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