Birational geometry and localisation of categories (with appendices by Jean-Louis Colliot-Thélène and Ofer Gabber)

Abstract

We explore connections between places of function fields over a base field $F$ and birational morphisms between smooth $F$-varieties. This is done by considering various categories of fractions involving function fields or varieties as objects, and constructing functors between these categories. The main result is that in the localised category $S_b^{-1}\mathbf{Sm}(F)$, where $\mathbf{Sm}(F)$ denotes the usual category of smooth varieties over $F$ and $S_b$ is the set of birational morphisms, the set of morphisms between two objects $X$ and $Y$, with $Y$ proper, is the set of $R$-equivalence classes $Y(F(X))/R$.