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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