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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