2 7 Fe b 20 09 BIRATIONAL MOTIVES , I : PURE BIRATIONAL MOTIVES

In the preprint [19], we toyed with birational ideas in three areas of algebraic geometry: plain varieties, pure motives in the sense of Grothendieck, and triangulated motives in the sense of Voevodsky. These three themes are finally treated separately in revised versions. The first one was the object of [21]; the second one is the object of the present paper; we hope to complete the third one soon. We work over a field F . Recall that we introduced in [21] two “birational” categories. The first, place(F ), has for objects the function fields over F and for morphisms the F -places. The second one is the Gabriel-Zisman localisation of the category Sm(F ) of smooth F varieties obtained by inverting birational morphisms: we denoted this category by S b Sm(F ). We may also invert stable birational morphisms: those which are dominant and induce a purely transcendental extension of function