Adelic constructions of direct images for differentials and symbols

Let f be a projective morphism from a smooth algebraic surface X to a smooth algebraic curve S over a perfect field k . Using the adelic language we give some relative constructions of residues and symbols and apply them to the Gysin morphism for differentials and algebraic cycles. The first section of this article is devoted to various constructions of relative residue maps from differentials of 2-dimensional local fields to differentials of 1-dimensional local fields. (See definitions 4, 5.) For these maps we prove some reciprocity laws. (See propositions 6 and 7.) Then using the adelic resolutions of sheafs ΩX and Ω 1 S we apply these residue maps to the construction of Gysin maps from H(X,ΩX) to H (S,ΩS) (n = 1, 2 ). (See propositions 9, 10.) In the second section of this article we assume chark = 0 and construct relative maps f∗( , )x,C from K2 -groups of 2-dimensional local fields Kx,C on the surface X , associated with pairs: an irredicuble curve C ⊂ X and a point x ∈ C , to multiplicative groups of complete local fields Ks of points s on the curve S . (See theorems 1 and 2.) These relative symbol maps are directly connected to the other local maps on the surface X and the curve S : such as the 2-dimensional tame symbol, 2-dimensional residue map and so forth. Also, we prove some relative reciprocity laws. (See corollaries from theorem 1.) Let us remark, if C is not a fibre of morphism f , then the required symbol map is the usual tame symbol. If C is in the fibre of morphism f , then the required symbol was originally introduced by K. Kato in [3]. We give another proofs of all theorems, we need on this symbol. Moreover, in theorem 2 we give an explicit formula for this symbol when chark = 0 . The third section of this article is similar to the end of the first section. In this part we apply the constructed symbol maps for a construction of direct image maps from H(X,K2(X)) to H (S,K1(S)) (n = 1, 2 ). (See proposition 19.) (Here K2(X) (correspondingly K1(S) ) is the sheaf on the surface X (corr. on the curve S ) associated to the presheaf {U 7→ K2(U)} (corr. {U 7→ K1(U)} ).) If n = 2 , then this map is the Gysin map from CH(X) to CH(S) . (See proposition 20.) For this goal we construct a K2 -adelic resolution of the sheaf K2(X) . (See theorem 3.) Note also that all constructions in this paper are presented by means of explicit expressions and for almost all statements we give variants of their proofs which don’t use