TORSION BIRATIONAL MOTIVES OF SURFACES AND UNRAMIFIED COHOMOLOGY

Abstract Let S and T be smooth projective varieties over an algebraically closed field k . 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 $T \to S$ 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 $\mathbb {C}$ for which $S \times S$ violates the integral Hodge conjecture.