The Stacks project

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

  1. $f$ is locally of finite type and $Y$ reduced (i.e., integral),

  2. $f$ 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 66.20.4 we may assume that $X$ and $Y$ are schemes. In this case the result is Morphisms, Lemma 29.49.5. $\square$


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