The Stacks project

Theorem 33.4.1. Let $k$ be a field. The category of varieties and dominant rational maps is equivalent to the category of finitely generated field extensions $K/k$.

Proof. Let $X$ and $Y$ be varieties with generic points $x \in X$ and $y \in Y$. Recall that dominant rational maps from $X$ to $Y$ are exactly those rational maps which map $x$ to $y$ (Morphisms, Definition 29.49.10 and discussion following). Thus given a dominant rational map $X \supset U \to Y$ we obtain a map of function fields

\[ k(Y) = \kappa (y) = \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x} = \kappa (x) = k(X) \]

Conversely, such a $k$-algebra map (which is automatically local as the source and target are fields) determines (uniquely) a dominant rational map by Morphisms, Lemma 29.49.2. In this way we obtain a fully faithful functor. To finish the proof it suffices to show that every finitely generated field extension $K/k$ is in the essential image. Since $K/k$ is finitely generated, there exists a finite type $k$-algebra $A \subset K$ such that $K$ is the fraction field of $A$. Then $X = \mathop{\mathrm{Spec}}(A)$ is a variety whose function field is $K$. $\square$

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