Lemma 79.12.8 (04QE)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 79: More on Groupoids in Spaces
Section 79.12: The finite part of a morphism
Lemma 79.12.8 (cite)

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Lemma 79.12.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Suppose that $U$ is a scheme, $U \to Y$ is an étale morphism and $Z \subset U \times _ Y X$ is an open subspace finite over $U$ . Then the induced morphism $U \to (X/Y)_{fin}$ is étale.

Proof. This is formal from the description of the diagonal in Lemma 79.12.7 but we write it out since it is an important step in the development of the theory. We have to check that for any scheme $T$ over $S$ and a morphism $T \to (X/Y)_{fin}$ the projection map

$T \times _{(X/Y)_{fin}} U \longrightarrow T$

is étale. Note that

$T \times _{(X/Y)_{fin}} U = (X/Y)_{fin} \times _{((X/Y)_{fin} \times _ Y (X/Y)_{fin})} (T \times _ Y U)$

Applying the result of Lemma 79.12.7 we see that $T \times _{(X/Y)_{fin}} U$ is represented by an open subscheme of $T \times _ Y U$ . As the projection $T \times _ Y U \to T$ is étale by Morphisms of Spaces, Lemma 67.39.4 we conclude. $\square$

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