process

418 T. GeisserProof. The statement is clear for X smooth and projective. We proceed by induction onthe dimension of X. Using the localization sequence for both theories, we can assumethat X is proper. Let f W X 0 ! X be a resolution of singularities of X, Z be the closedsubscheme (of lower dimension) where f is not an isomorphism, and Z 0 D Z X X 0 .Then we conclude by comparing localization sequencesH2n c .Z0 ; Z.n// H2n c .Z; Z.n// ˚ H 2n c .X 0 ; Z.n// H2n c .X; Z.n// 0H2n W .Z0 ; Z.n// HW2n.Z; Z.n// ˚ H2n W .X 0 ; Z.n// HW2n.X; Z.n// 0.The map for fields induces a map ˇ W zHi c.X;Q.n// ! zHiW .X; Q.n//, which fitsinto the (non-commutative) diagramH czH cii .X; Z.n// ˛ˇ.X; Z.n//HiW.X; Z.n// zHiW .X; Z.n//.We now return to the situation k D F q , and compare weight homology to higherChow groups using their niveau spectral sequences. We saw that the niveau spectralsequence for higher Chow groups is concentrated above the line t D n, and that theniveau spectral sequence for weight homology is concentrated below the line t D n.Our aim is to show that Parshin’s conjecture is equivalent to both being rationallyconcentrated on this line, and that the resulting complexes are isomorphic.The following statements will be referred to as Conjecture B.n/:Proposition 3.3. For a fixed integer n, the following statements are equivalent:a) The map ˇ induces an isomorphism zH i .X; Q.n// Š zHiW .X; Q.n// for allschemes X and all i.b) The map induces an isomorphism H c dCn .k; Q.n// Š H W .k; Q.n// for alldCnfinitely generated fields k=F q , where d D trdeg k=F q .c) For every smooth and projective X over F q the following holds: if d D dim X>n C 1, then zH c dCn .X; Q.n// D zH c .X; Q.n// D 0, and if dim X D n C 1,dCn 1then zH2nC1 c .X; Q.n// D 0.Note that assuming the conjecture A.n/, c) is equivalent to H c .X; Q.n// DdCnH c dCn 1 .X; Q.n// D 0 and H 2nC1 c .X; Q.n// D 0 for all smooth and projective X ofdimension d>nC 1 and d D n C 1, respectively, hence are part of Conjecture P.n/.