The Stacks project

Lemma 68.22.5. 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

  1. either $f$ is quasi-compact or $f$ is separated, and

  2. either $f$ is locally of finite type and $Y$ is reduced or $f$ is locally of finite presentation.

Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is an isomorphism.

Proof. By Lemma 68.20.4 we may assume $Y$ is a scheme. By Lemma 68.21.4 we may assume that $f$ is finite. Then $X$ is a scheme too and the result follows from Morphisms, Lemma 29.51.6. $\square$

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