Lemma 94.9.1. Let $Z \to U$ be a finite morphism of schemes. Let $W$ be an algebraic space and let $W \to Z$ be a surjective étale morphism. Then there exists a surjective étale morphism $U' \to U$ and a section

$\sigma : Z_{U'} \to W_{U'}$

of the morphism $W_{U'} \to Z_{U'}$.

Proof. We may choose a separated scheme $W'$ and a surjective étale morphism $W' \to W$. Hence after replacing $W$ by $W'$ we may assume that $W$ is a separated scheme. Write $f : W \to Z$ and $\pi : Z \to U$. Note that $f \circ \pi : W \to U$ is separated as $W$ is separated (see Schemes, Lemma 26.21.13). Let $u \in U$ be a point. Clearly it suffices to find an étale neighbourhood $(U', u')$ of $(U, u)$ such that a section $\sigma$ exists over $U'$. Let $z_1, \ldots , z_ r$ be the points of $Z$ lying above $u$. For each $i$ choose a point $w_ i \in W$ which maps to $z_ i$. We may pick an étale neighbourhood $(U', u') \to (U, u)$ such that the conclusions of More on Morphisms, Lemma 37.36.5 hold for both $Z \to U$ and the points $z_1, \ldots , z_ r$ and $W \to U$ and the points $w_1, \ldots , w_ r$. Hence, after replacing $(U, u)$ by $(U', u')$ and relabeling, we may assume that all the field extensions $\kappa (u) \subset \kappa (z_ i)$ and $\kappa (u) \subset \kappa (w_ i)$ are purely inseparable, and moreover that there exist disjoint union decompositions

$Z = V_1 \amalg \ldots \amalg V_ r \amalg A, \quad W = W_1 \amalg \ldots \amalg W_ r \amalg B$

by open and closed subschemes with $z_ i \in V_ i$, $w_ i \in W_ i$ and $V_ i \to U$, $W_ i \to U$ finite. After replacing $U$ by $U \setminus \pi (A)$ we may assume that $A = \emptyset$, i.e., $Z = V_1 \amalg \ldots \amalg V_ r$. After replacing $W_ i$ by $W_ i \cap f^{-1}(V_ i)$ and $B$ by $B \cup \bigcup W_ i \cap f^{-1}(Z \setminus V_ i)$ we may assume that $f$ maps $W_ i$ into $V_ i$. Then $f_ i = f|_{W_ i} : W_ i \to V_ i$ is a morphism of schemes finite over $U$, hence finite (see Morphisms, Lemma 29.42.14). It is also étale (by assumption), $f_ i^{-1}(\{ z_ i\} ) = w_ i$, and induces an isomorphism of residue fields $\kappa (z_ i) = \kappa (w_ i)$ (because both are purely inseparable extensions of $\kappa (u)$ and $\kappa (z_ i) \subset \kappa (w_ i)$ is separable as $f$ is étale). Hence by Étale Morphisms, Lemma 41.14.2 we see that $f_ i$ is an isomorphism in a neighbourhood $V_ i'$ of $z_ i$. Since $\pi : Z \to U$ is closed, after shrinking $U$, we may assume that $W_ i \to V_ i$ is an isomorphism. This proves the lemma. $\square$

Comment #4923 by Robot0079 on

Here is a conceptual proof.

We call an etale sheaf $F$ over scheme $S$ surjective, if the structure map $F \to S$ is surjective. Here we identify etale sheaves with etale algebraic spaces over $S$. Note that this is equivalent to require stalks of $F$ is nonempty at every (geometric) point. Another equivalent condition is $F/S$ has sections locally.

Then our lemma says that direct image of finite morphism preserve surjectivity.

Now that we have formula for stalk of finite direct image functor, this is obvious.

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