A few localisation theorems
  Given a functor $T:C \to D$ carrying a class of morphisms $S\subset C$ into a class $S'\subset D$, we give sufficient conditions in order that $T$ induces an equivalence on the localised categories. These conditions are in the spirit of Quillen's theorem A. We give some applications in algebaic and birational geometry. 
Bruno Kahn (IMJ), R. Sujatha (TIFR)