### 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.

A much more detailed summary is given in the introduction.

Bruno Kahn <kahn@math.jussieu.fr>

R. Sujatha <sujatha@math.tifr.res.in>