process

414 T. Geisserfrom higher Chow groups with Q l -coefficients to l-adic cohomology, for all X and i,CH 0 .X; i/ Ql ˚ CH 0 .X; i C 1/ Ql ! H i .X et ; yQ l /:Conjecture P.0/ can also be recovered from, and implies a structure theorem for higherChow groups of smooth affine schemes: For all smooth and affine schemes U ofdimension d over F q , the groups CH 0 .U; i/ are torsion for i 6D d, and the canonicalmap CH 0 .U; d/ Ql ! H d . xU et ; yQ l / Gal.Fq/ is an isomorphism.Finally, we reproduce an argument of Levine showing that if F is the absoluteFrobenius, the push-forward F acts like q n on CH n .X; i/, and the pull-back F actson motivic cohomology HM i .X; Z.n// like qn for all n. As a corollary, ConjectureP.0/ follows from finite dimensionality of smooth and projective schemes over finitefields in the sense of Kimura [13].Acknowledgements. This paper was inspired by the work of, and discussions with,U. Jannsen and S. Saito. We are indebted to the referee, whose careful reading helpedto improve the exposition.2 Parshin’s conjectureWe fix a perfect field k of characteristic p, and consider the category of separatedschemes of finite type over k. We recall some facts on Bloch’s higher Chow groups[1], see [3] for a survey. Let z n .X; i/ be the free abelian group generated by cycles ofdimension n C i on X k i which meet all faces properly, and let z n .X; / be thecomplex of abelian groups obtained by taking the alternating sum of intersection withface maps as differential. We define CH n .X; i/ as the ith homology of this complexand motivic Borel–Moore homology to beHi c .X; Z.n// D CH n.X; i 2n/:For a proper map f W X ! Y we have a push-forward z n .X; / ! z n .Y; /, for a flat,quasi-finite map f W X ! Y we have a pull-back z n .X; / ! z n .Y; /, and a closedembedding i W Z ! X with open complement j W U ! X induces a localizationsequence!Hi c .Z; Z.n// i ! Hi c .X; Z.n// j ! Hi c .U; Z.n// ! :If X is smooth of pure dimension d, then Hi c.X;Z.n// Š H 2d i .X; Z.d n//, wherethe right-hand side is Voevodsky’s motivic cohomology [15]. For a finitely generatedfield F over k, we define Hi c.F;Z.n// D colim U Hi c .U; Z.n//, where the colimitruns through U of finite type over k with field of functions F . For the reader who ismore familiar with motivic cohomology, we mention that Voevodsky’s theorem impliesthat for a field F of transcendence degree d over k, wehaveH ci .F; Z.n// Š H 2d i .F; Z.d n// Š(0; i < d C n;K M .F /;d ni D d C n: