Definition 66.22.1. Let $S$ be a scheme. Let $X$ and $Y$ algebraic spaces over $S$. Assume $X$ and $Y$ are decent and that $|X|$ and $|Y|$ have finitely many irreducible components. We say a morphism $f : X \to Y$ is *birational* if

$|f|$ induces a bijection between the set of generic points of irreducible components of $|X|$ and the set of generic points of the irreducible components of $|Y|$, and

for every generic point $x \in |X|$ of an irreducible component the local ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an isomorphism (see clarification below).

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