The Stacks project

Definition 67.47.1. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.

  1. Let $f : U \to Y$, $g : V \to Y$ be morphisms of algebraic spaces over $S$ defined on dense open subspaces $U$, $V$ of $X$. We say that $f$ is equivalent to $g$ if $f|_ W = g|_ W$ for some dense open subspace $W \subset U \cap V$.

  2. A rational map from $X$ to $Y$ is an equivalence class for the equivalence relation defined in (1).

  3. Given morphisms $X \to B$ and $Y \to B$ of algebraic spaces over $S$ we say that a rational map from $X$ to $Y$ is a $B$-rational map from $X$ to $Y$ if there exists a representative $f : U \to Y$ of the equivalence class which is a morphism over $B$.

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