Lemma 37.8.5. Let f : X \to S be a morphism of schemes. Let U \subset X and V \subset S be open subschemes such that f(U) \subset V. If f is formally étale, so is f|_ U : U \to V.
Proof. Consider a solid diagram
\xymatrix{ U \ar[d]_{f|_ U} & T \ar[d]^ i \ar[l]^ a \\ V & T' \ar[l] \ar@{-->}[lu] }
as in Definition 37.8.1. If f is formally ramified, then there exists exactly one S-morphism a' : T' \to X such that a'|_ T = a. Since |T'| = |T| we conclude that a'(T') \subset U which gives our unique morphism from T' into U. \square
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