The Stacks project

Lemma 97.11.3. Let $S$ be a scheme. Let $X' \to X \to Z \to B$ be morphisms of algebraic spaces over $S$. Assume

  1. $X' \to X$ is étale, and

  2. $Z \to B$ is finite locally free.

Then $\text{Res}_{Z/B}(X') \to \text{Res}_{Z/B}(X)$ is representable by algebraic spaces and étale. If $X' \to X$ is also surjective, then $\text{Res}_{Z/B}(X') \to \text{Res}_{Z/B}(X)$ is surjective.

Proof. Let $U$ be a scheme and let $\xi = (a, b)$ be an element of $\text{Res}_{Z/B}(X)(U)$. We have to prove that the functor

\[ h_ U \times _{\xi , \text{Res}_{Z/B}(X)} \text{Res}_{Z/B}(X') \]

is representable by an algebraic space étale over $U$. Set $Z_ U = U \times _{a, B} Z$ and $W = Z_ U \times _{b, X} X'$. Then $W \to Z_ U \to U$ is as in Lemma 97.9.2 and the sheaf $F$ defined there is identified with the fibre product displayed above. Hence the first assertion of the lemma. The second assertion follows from this and Lemma 97.9.1 which guarantees that $F \to U$ is surjective in the situation above. $\square$

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