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

  • Bruno Kahn

  • R. Sujatha


We explore connections between places of function fields over a base field FF and birational morphisms between smooth FF-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 Sb1Sm(F)S_b^{-1}\bold{Sm}(F), where Sm(F)\bold{Sm}(F) denotes the usual category of smooth varieties over FF and SbS_b is the set of birational morphisms, the set of morphisms between two objects XX and YY, with YY proper, is the set of RR-equivalence classes Y(F(X))/RY(F(X))/R.