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.

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 |