Lemma 67.22.6. 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 67.6.6). The generic points of $V$ and $U$ are the elements of $|V|$ and $|U|$ which lie over generic points of $|Y|$ and $|X|$ (Lemma 67.20.1). 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 67.22.1 we have a cartesian square

$\xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi }) \ar[d] \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, f(\xi )}) \ar[r] & Y }$

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

$\xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi }) \times _ X U \ar[d] \ar[r] & U \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, f(\xi )}) \times _ Y V \ar[r] & V }$

the vertical arrow on the left is an isomorphism. The horizonal 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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