Adelic constructions for direct images of differentials and symbols

Let X be a smooth algebraic surface over a perfect field k. Consider pairs x ∈ C , x is a closed point of X , C is either an irreducible curve on X which is smooth at x, or an irreducible analytic branch near x of an irreducible curve on X . As in the previous section 1 for every such pair x ∈ C we get a two-dimensional local field Kx,C . If X is a projective surface, then from the adelic description of Serre duality on X there is a local decomposition for the trace map H(X,ΩX ) → k by using a two-dimensional residue map resKx,C/k(x): ΩKx,C/k(x) → k(x) (see [P1]). From the adelic interpretation of the divisors intersection index on X there is a similar local decomposition for the global degree map from the group CH2(X) of algebraic cycles of codimension 2 on X modulo the rational equivalence to Z by means of explicit maps from K2(Kx,C ) to Z (see [P3]). Now we pass to the relative situation. Further assume that X is any smooth surface, but there are a smooth curve S over k and a smooth projective morphism f :X → S with connected fibres. Using two-dimensional local fields and explicit maps we describe in this section a local decomposition for the maps