The Geisser-Levine method revisited and algebraic cycles over a finite field, by Bruno Kahn

This is a thoroughly revised and expanded version of preprint #438. We axiomatize the method of Geisser and Levine further; in the present version, a new notion of a "malleable" complex of Zariski sheaves plays a central role. We conclude with a proof that three conjectures are equivalent for vareties over Fp:

(1) For any smooth projective variety X and any integer n greater than or equal to 0,

  • (Tate) the order of the pole of zeta(X,s) at s=n equals the rank of the group of cycles of codimension n modulo numerical equivalence, and
  • (Beilinson) rational equivalence equals numerical equivalence.
  • (2) A certain modified cycle class map from a version of motivic cohomology to continuous étale l-adic cohomology is a quasi-isomorphism.

    (3) Continuous étale cohomology with coefficients Ql(n) is malleable for n>0.

    Here, l is a prime number different from p.

    The equivalence between (1) and (2) had been proven earlier in preprint 247, under resolution of singularities.

    Bruno Kahn <>