Lemma 59.80.6. Let X be an affine integral normal scheme with separably closed function field. Let Z \subset X be a closed subscheme. Let V \to Z be an étale morphism with V affine. Then V is a finite disjoint union of open subschemes of Z. If V \to Z is surjective and finite étale, then V \to Z has a section.
Proof. By Algebra, Lemma 10.143.10 we can lift V to an affine scheme U étale over X. Apply Lemma 59.80.4 to U \to X to get the first statement.
The final statement is a consequence of the first. Let V = \coprod _{i = 1, \ldots , n} V_ i be a finite decomposition into open and closed subschemes with V_ i \to Z an open immersion. As V \to Z is finite we see that V_ i \to Z is also closed. Let U_ i \subset Z be the image. Then we have a decomposition into open and closed subschemes
Z = \coprod \nolimits _{(A, B)} \bigcap \nolimits _{i \in A} U_ i \cap \bigcap \nolimits _{i \in B} U_ i^ c
where the disjoint union is over \{ 1, \ldots , n\} = A \amalg B where A has at least one element. Each of the strata is contained in a single U_ i and we find our section. \square
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