Definition 29.49.1. Let $X$, $Y$ be schemes.

Let $f : U \to Y$, $g : V \to Y$ be morphisms of schemes defined on dense open subsets $U$, $V$ of $X$. We say that $f$ is

*equivalent*to $g$ if $f|_ W = g|_ W$ for some $W \subset U \cap V$ dense open in $X$.A

*rational map from $X$ to $Y$*is an equivalence class for the equivalence relation defined in (1).If $X$, $Y$ are schemes over a base scheme $S$ we say that a rational map from $X$ to $Y$ is an

*$S$-rational map from $X$ to $Y$*if there exists a representative $f : U \to Y$ of the equivalence class which is an $S$-morphism.

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