The Stacks project

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. $\square$

Comments (1)

Comment #4923 by Robot0079 on

Here is a conceptual proof.

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