process

Parshin’s conjecture revisited 421Theorem 4.2 (Jannsen). Under resolution of singularities, HaW .k; A/ D 0 for a 6Dtrdeg k=F q , hence HiW .X; A/ is the homology of the complexM0 H0 W .k.x/; A/ MHs W .k.x/; A/ (5)x2X .0/ x2X .s/for all schemes X. In particular, Conjecture C.0/ holds.This is proved in [11, Proposition 5.4, Theorem 5.10]. The proof only works forn D 0, because it uses the bijectivity of CH 0 .Y / ! CH 0 .X/ for a map of connectedsmooth and projective schemes X ! Y . The second statement follows using the niveauspectral sequence, which exists because HiW .X; A/ satisfies the localization propertyby property (4) of weight complexes.Let Z c .0/ be the complex of etale sheaves z 0 . ; /. For any prime l, considerl-adic cohomologyH i .X et ; yQ l / WD Q ˝Z lim H i .X et ; Z c =l r .0//:In [4], we showed that for every positive integer m, and every scheme f W X ! k overa perfect field, there is a quasi-isomorphism Z c =m.0/ Š Rf Š Z=m. In particular, theabove definition agrees with the usual definition of l-adic homology if l 6D p D char F q .If xX D X FqxF q and yG D Gal.xF q =F q /, then there is a short exact sequence0 ! H iC1 . xX et ; yQ l / O G ! H i.X et ; yQ l / ! H i . xX et ; yQ l / OG ! 0and for U affine and smooth, H i .U et ; yQ l / vanishes for i 6D d;d 1 and l 6D p bythe affine Lefschetz theorem and a weight argument [12, Theorem 3a)]. The map fromZariski-hypercohomology of Z c =m.0/ to etale-hypercohomology of Z c =m.0/ inducesa functorial maphence in the limit a mapCH 0 .X; i/=m ! CH 0 .X; i; Z=m/ ! H i .X et ; Z c =m.0//;! W CH 0 .X; i/ Ql ! H i .X et ; yQ l /:Similarly, the map x! W CH 0 . xX;i/ Ql! H i . xX et ; yQ l / induces a map W CH 0 .X; i C 1/ Ql.CH0 . xX;i C 1/ Ql / O G ! H iC1. xX et ; yQ l / O G ! H i.X et ; yQ l /(the left map is an isomorphism by a trace argument). The sum' i X W CH 0.X; i/ Ql ˚ CH 0 .X; i C 1/ Ql ! H i .X et ; yQ l / (6)is compatible with localization sequences, because all maps involved in the definitionare. The following proposition shows that Parshin’s conjecture can be recovered fromand implies a structure theorem for higher Chow groups of smooth affine schemes;compare to Jannsen [11, Conjecture 12.4b)].