Birational motives, I, by Bruno Kahn and R. Sujatha

In this paper we define various "birational" categories over a fixed base field F (of characteristic 0 for simplicity): two nonadditive categories called the coarse and the fine birational categories, a tensor pseudo-abelian category of "birational Chow motives" and a tensor triangulated category of "birational geometric motives". Each of these categories maps to the next, in that order. The main result of the paper is that the last functor is fully faithful. In fact, for two smooth projective F-varieties X,Y, with birational triangulated motives \bar M(X) and \bar M(Y), and for an integer i, we get the formula

Hom(\bar M(X),\bar M(Y)[i]) =   CH_0(Y_{F(X)})  for i=0
                                0               for i \ne 0.

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