Adelic constructions of direct images for differentials and symbols

Abstract

For a projective morphism of an smooth algebraic surface $X$ onto a smooth algebraic curve $S$, both given over a perfect field $k$, we construct the direct image morphism in two cases: from $H^i(X,\Omega^2_X)$ to $H^{i-1}(S,\Omega^1_S)$ and when $char k =0$ from $H^i(X,K_2(X))$ to $H^{i-1}(S,K_1(S))$. (If i=2, then the last map is the Gysin map from $CH^2(X)$ to $CH^1(S)$.) To do this in the first case we use the known adelic resolution for sheafs $\Omega^2_X$ and $\Omega^1_S$. In the second case we construct a $K_2$-adelic resolution of a sheaf $K_2(X)$. And thus we reduce the direct image morphism to the construction of some residues and symbols from differentials and symbols of 2-dimensional local fields associated with pairs $x \in C$ ($x$ is a closed point on an irredicuble curve $C \in X$) to 1-dimensional local fields associated with closed points on the curve $S$. We prove reciprocity laws for these maps.