  We are showing that the Deligne--Beilinson cohomology sheaves ${\cal
H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures
hold true for function fields. This result is `effective' for $q=2$; in this
case, by dealing with `arithmetic properties' of the presheaves of mixed Hodge
structures defined by singular cohomology, we are able to give a cohomological
characterization of the Albanese kernel for surfaces with $p_g=0$.
