process

Parshin’s conjecture revisited 415The latter isomorphism is due to Nesterenko–Suslin and Totaro. It follows formallyfrom localization that there are spectral sequencesEs;t 1 .X/ D MHsCt c .k.x/; Z.n// ) H sCt c .X; Z.n//: (1)x2X .s/Here X .s/ denotes points of X of dimension s. Since Hi c .F; Z.n// D 0 for i 0. If Tate’s conjecture holds and rational equivalenceand homological equivalence agree up to torsion for all X, then Parshin’s conjectureholds by [2]. Since K i .X/ Q D L n CH n.X; i/ Q , it follows that Parshin’s conjectureis equivalent to the following conjecture for all n.Conjecture P.n/. For all smooth and projective schemes X over the finite field F q ,the groups Hi c .X; Q.n// vanish for i 6D 2n.We will refer to the following equivalent statements as Conjecture A.n/:Proposition 2.1. For a fixed integer n, the following statements are equivalent:a) For all schemes X=F q and all i, ˛ induces an isomorphism Hi c .X; Q.n// ŠzHi c .X; Q.n//.b) For all finitely generated fields k=F q with d WD trdeg k=F q , and all i 6D d C n,we have Hi c .k; Q.n// D 0.c) For all smooth and projective X over F q and all i > dim X C n, we haveHi c .X; Q.n// D 0.d) For all smooth and affine schemes U over F q and all i>dim U C n, we have.U; Q.n// D 0.H ciProof. a) ) c), d): The complex (2) is concentrated in degrees Œ2n; d C n.c) ) b): This is proved by induction on the transcendence degree. By de Jong’stheorem, we find a smooth and proper model X of a finite extension of k. Lookingat the niveau spectral sequence (1), we see that the induction hypothesis impliesH ci.X; Q.n// D 0 for i>dC n (see [2] for details).