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Adele constructions of direct images of differentials and symbols. (English. Russian original) Zbl 0909.14011
Sb. Math. 188, No. 5, 697-723 (1997); translation from Mat. Sb. 188, No. 5, 59-84 (1997).
For a projective morphism of an algebraic surface $$X$$ onto an algebraic curve $$S$$, both given over a field $$k$$, we construct the Gysin map in two cases: from $$H^2 (X,\Omega^2_X)$$ to $$H^1 (S,\Omega^1_S)$$ and from $$\text{CH}^2(X)$$ to $$\text{CH}^1(S)$$ when $$\text{char} k=0$$ or when $$k$$ is a finite field.
To do this in the first case we use the adele representation of $$H^2(X, \Omega^2_X)$$ and $$H^1(S, \Omega^1_S)$$ and reduce the Gysin map to direct image maps of differentials of two-dimensional local fields to differentials of one-dimensional local fields.
In the second case we prove a theorem on representation of $$\text{CH}^2(X)$$ by means of $$K_2$$-adeles and thus reduce the Gysin map to the construction of symbols from the two-dimensional local fields associated with the pairs $$x\in C \subset X$$ to the one-dimensional local fields associated with discrete valuations on the curve $$S$$. We prove reciprocity laws for these maps. It should be mentioned that the symbol constructed turns out to be the well-known tame symbol in the case of the two-dimensional local field associated with a point and a curve not coinciding with a fibre of the map from $$X$$ to $$S$$. However, the situation is more complicated in the case of the two-dimensional local field associated with a point and a curve that is a fibre of the map from $$X$$ to S.-K. Kato proved the existence of the desired symbol, though without giving an explicit formula for it [K. Kato in Galois groups and their representations, Proc. Symp., Nagoya 1981, Adv. Stud. Pure Math. 2, 153-172 (1983; Zbl 0586.12011)]. Here we produce a concrete formula for this symbol in the case when $$\text{char} k =0$$.
All the constructions and proofs in this paper are carried out with the help of explicit formulae and do not use, in particular, the higher Quillen $$K$$-theory.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 11S70 $$K$$-theory of local fields 11R56 Adèle rings and groups 14B15 Local cohomology and algebraic geometry 19C20 Symbols, presentations and stability of $$K_2$$ 55N99 Homology and cohomology theories in algebraic topology 12J10 Valued fields
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