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.
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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