Let S and T be smooth projective varieties over an algebraically closed
field. Suppose that S is a surface admitting a decomposition of the diagonal.
We show that, away from the characteristic of k, if an algebraic
correspondence TβS acts trivially on the unramified cohomology, then it
acts trivially on any normalized, birational, and motivic functor. This
generalizes Kahn's result on the torsion order of S. We also exhibit an
example of S over C for which SΓS violates the integral
Hodge conjecture.Comment: 25 page