  Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to \Gamma
H^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored
with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r))
is singular cohomology. We study the image of cl_{r,m} in terms of kernels of
Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the
cokernel of cl_{r,m} at the generic point of U is the same for integral or
rational coefficients.
