process

Parshin’s conjecture revisited 419Proof. b) ) a) is trivial, and a) ) b) follows by a colimit argument because(colim zH i c .X; Q.n// Š H c .k.X/; Q.n//;dCn i D d C nIU X0; otherwise.a) ) c) follows because for X smooth and projective of dimension d, the cohomologyof the complex (2) tensored with Q equals zHi c.X;Q.n// D zHiW .X; Q.n// fori D d C n and i D d C n 1 (or i D 2n C 1 in case d D n C 1). An inspection ofthe niveau spectral sequence (3) shows that this is a subgroup of Hi W .X; Q.n// D 0.c) ) b): For n > d, both sides vanish, whereas for d D n, both sides arecanonically isomorphic to Q. For nnC 1. The upper sequence is exact by hypothesis, and an inspection of (3) showsthat the lower sequence is exact because HiW .X; Q.n// D 0 for i>2n.We refer to the following statements as Conjecture C.n/:Proposition 3.4. For a fixed integer n, the following statements are equivalent:a) For all schemes X over F q and for all i, the map induces an isomorphismzHiW .X; Q.n/ Š HiW .X; Q.n//.b) For all finitely generated fields k=F q and for all i 6D trdeg k=F q C n, we haveHi W .k; Q.n// D 0.c) For all smooth and projective X, the map induces an isomorphismzH iW .X; Q.n// D(0; i > 0ICH n .X/ Q ; i D 2n:Proof. b) ) a) is trivial and a) ) b) follows by a colimit argument.a) ) c) is trivial for i>2n, and Lemma 3.2 for i D 2n.c) ) b): This is proved like Proposition 2.1, by induction on the transcendencedegree of k. Let X be a smooth and projective model of k. The induction hypothesisimplies that the niveau spectral sequence (3) collapses to the horizontal line t D nand the vertical line s D d. Since it converges to HiW .X; Q.n//, which is zero fori>2n, we obtain isomorphisms H W dCniC1 .k.X/; Q.n// d i! zH W .X; Q.n// fordCn i