process

Parshin’s conjecture revisited 423Proposition 4.5. Conjecture P.0/ for all smooth and projective X implies the followingstatements:a) (Affine Gersten) For every smooth affine U of dimension d, the following sequenceis exact:CH 0 .U; d/ Q ,! MH d .k.x/; Q.d//! MH d 1 .k.x/; Q.d 1//! :x2U .0/ x2U .1/b) Let X D X d X d 1 X 1 X 0 be a filtration such that U i D X i X i 1is smooth and affine of dimension i. Then CH 0 .X; i/ Q is isomorphic to the ithhomology of the complex0 ! CH 0 .U d ;d/ Q ! CH 0 .U d 1 ;d 1/ Q !!CH 0 .U 0 ;0/ Q ! 0:The maps CH 0 .U i ;i/ Q ! CH 0 .X i 1 ;i 1/ Q ! CH 0 .U i 1 ;i 1/ Q arisefrom the localization sequence.Proof. a) follows because the spectral sequence (1) collapses, and b) by a diagramchase.Remark. If we fix a smooth scheme X of dimension d, and use cohomological notation,then by Proposition 2.1 and the Gersten resolution, we get that the rational motiviccomplex Q.d/ is conjecturally concentrated in degree d, say C d D H d .Q.d// DCH 0 . ;d/ Q D H W d. ; Q/. Then Conjecture L.0/ says that H i .U; C d / D 0 forU X affine and i>0. This is analog to the mod p situation, where the motiviccomplex agrees with the logarithmic de Rham–Witt sheaf Z=p.n/ Š d Œ d, andH i .U et ; d / D 0 for U X affine and i>0. The latter can be proved by writing d as the kernel of a map of coherent sheaves and using the vanishing of cohomologyof coherent sheaves on affine schemes. This suggest that one might try to do the samefor C d .4.1 Frobenius action. Let F W X ! X be the Frobenius morphism induced by theqth power map on the structure sheaf.Theorem 4.6. The push-forward F acts like q n on CH n .X; i/, and the pull-back F acts on H i .X; Z.n// as q n for all n.The theorem is well known, but we could not find a proof in the literature. Theproof of Soulé [14, Proposition 2] for Chow groups does not carry over to higherChow groups, because the Frobenius does not act on the simplices n , hence a cycleZ n X is not send to a multiple of itself by the Frobenius. We give an argumentdue to M. Levine.Proof. Let DM be Voevodsky’s derived category of bounded above complexes ofNisnevich sheaves with transfers with homotopy invariant cohomology sheaves. Then