Lemma 97.9.1. 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
of the morphism W_{U'} \to Z_{U'}.
Lemma 97.9.1. 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
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 37.41.5 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 (z_ i)/\kappa (u) and \kappa (w_ i)/\kappa (u) are purely inseparable, and moreover that there exist disjoint union decompositions
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 29.44.14). 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 (w_ i)/\kappa (z_ 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
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