Lemma 68.22.4. Let $S$ be a scheme. Let $f : X \to Y$ be a birational morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. Assume one of the following conditions is satisfied
$f$ is locally of finite type and $Y$ reduced (i.e., integral),
$f$ is locally of finite presentation.
Then there exist dense opens $U \subset X$ and $V \subset Y$ such that $f(U) \subset V$ and $f|_ U : U \to V$ is an isomorphism.
Proof. By Lemma 68.20.4 we may assume that $X$ and $Y$ are schemes. In this case the result is Morphisms, Lemma 29.50.5. $\square$
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