Parshin’s conjecture revisited

Arithmetic homology and an integral version of Kato's conjecture

“…) by ( 8). As a formal consequence of localization (10), we obtain Corollary 3.3 For n ≤ 0, there are spectral sequences…”

“…The first statement is equivalent to H i (k, Q(d)) = 0 for all fields k of transcendence degree d over F q and all i = d, or to CH 0 (X, i) Q ∼ = Hi (X, Q(0)) for all X and all i. The second statement is equivalent to the vanishing of Hd (X, Q(0)) = Hd−1 (X, Q(0)) = 0 for d = dim X > 1 [10].…”

“…Let Hi (X, Q(0)) = E 2 i,0 (X) be the homology of this complex. Then in [10], we used a theorem of Jannsen [14] to show that Conjecture P 0 (X) has two independent faces: Proposition 4.4 Assume resolution of singularities. Then conjecture P 0 (X) for all smooth and proper X is equivalent to the following statements for all smooth and proper X:…”

“…is the weight complex W (X) of Gillet-Soulé [13], see the discussion in [10]. By definition for X smooth and proper, H W 0 (X, Z) = Z, and H W i (X, Z) = 0 for i > 0.…”

“…The vanishing of the Chow groups is a special case of Parshin's conjecture K i (X) Q = 0 for i > 0, and in particular it follows from Tate's conjecture together with Beilinson's conjecture that rational and homological equivalence agree up to torsion [5] over finite fields, or from finite dimensionality of smooth and projective schemes over F q in the sense of Kimura-O'Sullivan [10].…”