# A sheaf-theoretic reformulation of the Tate conjecture

Research paper by **Bruno Kahn**

Indexed on: **06 Jan '98**Published on: **06 Jan '98**Published in: **Mathematics - Algebraic Geometry**

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#### Abstract

Let p be a prime number. We give a conjecture of a sheaf-theoretic nature
which is equivalent to the strong form of the Tate conjecture for smooth,
projective varieties X over F_p: for all n>0, the order of pole of the
Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic
cycles of codimension n modulo numerical equivalence. Our main result is that
this conjecture implies other well-known conjectures in characteristic p, among
which:
- The (weak) Tate conjecture for smooth, projective varieties X over any
finitely generated field of characteristic p: given a prime l different from p,
the geometric cycle map from algebraic cycles over X to the Galois invariants
of the l-adic cohomology of the geometric fibre of X, tensored by Q_l, is
surjective.
- For X as above, the algebraicity of the Kunneth components of the diagonal
and the hard Lefschetz theorem for cycles modulo numerical equivalence.
- For X as above, the existence of a filtration conjectured by Beilinson on
the Chow groups of X.
- The rational Bass conjecture: for any smooth variety X over F_p, the
algebraic K-groups of X have finite rank.
- The Bass-Tate conjecture: for F a field of characteristic p, of absolute
transcendence degree d, the i-th Milnor K-group of F is torsion for i>d.
- Soule's conjecture: given a quasi-projective variety over F_p, the order of
the zero of its Hasse-Weil zeta function at an integer n is given by the
alternating sum of the ranks of the weight n part of its algebraic K'-groups.