Lemma 68.22.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $f$ is birational and $V \to Y$ is an étale morphism with $V$ affine, then $X \times _ Y V$ is decent with finitely many irreducible components and $X \times _ Y V \to V$ is birational.

**Proof.**
The algebraic space $U = X \times _ Y V$ is decent (Lemma 68.6.6). The generic points of $V$ and $U$ are the elements of $|V|$ and $|U|$ which lie over generic points of $|Y|$ and $|X|$ (Lemma 68.20.1). Since $Y$ is decent we conclude there are finitely many generic points on $V$. Let $\xi \in |X|$ be a generic point of an irreducible component. By the discussion following Definition 68.22.1 we have a cartesian square

whose horizontal morphisms are monomorphisms identifying local rings and where the left vertical arrow is an isomorphism. It follows that in the diagram

the vertical arrow on the left is an isomorphism. The horizontal arrows have image contained in the schematic locus of $U$ and $V$ and identify local rings (some details omitted). Since the image of the horizontal arrows are the points of $|U|$, resp. $|V|$ lying over $\xi $, resp. $f(\xi )$ we conclude. $\square$

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