Lemma 66.22.4. 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.