Lemma 38.15.8. In Situation 38.15.1 the following are equivalent

1. 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 $s$,

2. 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

3. 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 $t$.

Proof. As an étale morphism is locally quasi-finite it is clear that (2) implies (3). We have seen that (3) implies (2) in Lemma 38.15.5. We have seen that (1) implies (2) in Lemma 38.15.7. Finally, if $(T \to S, t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}$ above $s$ such that $(T, t) \to (S, s)$ is an elementary étale neighbourhood, then we can choose a factorization $S^ h \to T \to S$ of the structure morphism $S^ h \to S$. Choose any point $s' \in S^ h$ mapping to $t'$ and choose any $\xi ' \in X_{s'}$ mapping to $\xi \in X_{t'}$. Then $(S^ h \to S, s' \leadsto s, \xi ')$ is an impurity of $\mathcal{F}$ above $s$. We omit the details. $\square$

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