process

420 T. Geisser1 0, henceConjecture P.n/ follows. Conversely, Conjecture P.n/ implies Proposition 2.1 c),then 3.3 c), and finally 3.4 c) by using 2.1 a) and 3.3 a).Remark. Propositions 2.1, 3.4, 3.3 as well as Theorem 3.5 remain true if we restrictourselves to schemes of dimension at most N , and to fields of transcendence degree atmost N , for a fixed integer N .Remark. Gillet announced that one can obtain a rational version of weight complexesby using de Jong’s theorem on alterations instead of resolution of singularities. Thesame argument should then give the generalization [11, Theorem 5.13]. In this case,all arguments of this section hold true rationally, except the proof of c) ) b) in Propositions3.3 and 3.4, and the proof of P.n/ ) A.n/; B.n/; C.n/ in Theorem 3.5, whichrequire that every finitely generated field over F q has a smooth and projective model.4 The case n D 0Since H ci .X; Q.0// D CH 0.X; i/ Q , we use higher Chow groups in this section.Proposition 4.1. We have CH 0 .X; i/ Q Š zHi c .X; Q.0// for i 2, and the mapCH 0 .X; 3/ Q ! zH3 c .X; Q.0// is surjective for all X. In particular, A.0/ holds indimensions at most 2.Proof. Since H i .k.x/; Q.1// D 0 for i 6D 1 and H i .k.x/; Q.0// D 0 for i 6D 0, thisfollows from an inspection of the niveau spectral sequence.Jannsen [11] defines a variant of weight homology with coefficients A, HiW .X; A/,as the homology of the complex Hom.CH 0 .W.X//; A/. Note that HiW .X; Q/ DHi W .X; Q.0// because Hom.CH 0 .X/; Q/ Š CH 0 .X/ Q for smooth and projective X,in a functorial way. Indeed, a map of connected, smooth and projective varieties inducesthe identity pull-back on CH 0 and the identity push-forward on CH 0 .