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**Unramified class field theory of arithmetical surfaces.**
*(English)*
Zbl 0562.14011

For X a scheme of finite type over some base ring R, the group \(\pi_ 1^{ab}(X)\) classifies the abelian unramified coverings of X. As in the class field theory of number fields, the primary goal for schemes is to obtain a description of \(\pi_ 1^{ab}(X)\) solely in terms of X itself, via some kind of ”reciprocity” map. When R is a finite field, Lang in 1956, defined a reciprocity map from \(Z_ 0(X)\), the free abelian group on the closed points, and showed that for normal X the map had a dense image. Subsequent work led to the study of \(CH_ 0(X)\), the 0- dimensional Chow group of X, which for smooth projective varieties over a field is \(Z_ 0(X)\) modulo rational equivalence, and which in general can be defined using algebraic K-theory. It turns out that for X proper over \({\mathbb{Z}}\), there is a reciprocity map from \(CH_ 0(X)\) to the quotient \({\tilde \pi}{}_ 1^{ab}(X)\) of \(\pi_ 1^{ab}(X)\) which classifies the unramified abelian covers that split completely over any real valued point of X. (For schemes over finite fields, this quotient is just \(\pi_ 1^{ab}(X).)\)

The main part of the current paper studies arithmetical surfaces, that is, proper smooth surfaces over finite fields, or connected regular surfaces which are proper and flat over \({\mathbb{Z}}\). For surfaces over finite fields, the degree map exhibits \(CH_ 0(X)\) as an extension of \({\mathbb{Z}}\) by \(CH_ 0(X)^ 0\), the subgroup of degree 0 cycle classes. Correspondingly, there is a map from \(\pi_ 1^{ab}(X)\) onto \({\hat {\mathbb{Z}}}\), the Galois group of the algebraic closure of the ground field. Call \(\pi_ 1^{ab}(X)^ 0\) the kernel of this map. In 1981 N. Katz and S. Lang [Enseign. Math., II. Sér. 27, 285-319 (1981; Zbl 0495.14011)] showed that \(\pi_ 1^{ab}(X)^ 0\) is finite, while work of Bloch and Milne established the finiteness of \(CH_ 0(X)^ 0\). Milne further established the p-primary injectivity of the reciprocity map \(CH_ 0(X)\to \pi_ 1^{ab}(X),\) assuming a certain condition on X. In this paper the authors show that for any smooth projective geometrically irreducible scheme over a finite field, the reciprocity map is always injective, and induces an isomorphism of the finite groups \(CH_ 0(X)^ 0\) and \(\pi_ 1^{ab}(X)^ 0\) so that, roughly speaking, \(\pi_ 1^{ab}(X)\) is obtained from \(CH_ 0(X)\) by replacing \({\mathbb{Z}}\) by \({\hat {\mathbb{Z}}}\). According to the authors, they had originally obtained these results for surfaces, and an elegant induction argument of Colliot-Thélène established the general case.

For X a regular connected surface proper and flat over \({\mathbb{Z}}\), the main result established in this paper is that the reciprocity map \(CH_ 0(X)\to {\tilde \pi}_ 1^{ab}(X)\) is an isomorphism of finite abelian groups. Earlier, Bloch had proved this result in the case X is smooth over the ring of integers in a number field. To establish the unramified class field theory of arithmetical surfaces X in general, the authors concentrate on a fibration \(X\to C\), where C is a one dimensional scheme, and where the fibration may have singular fibres. In essence, the authors are able to improve on the earlier results of Bloch and Milne because they have a technique for treating the class field theory of a singular curve over a local field.

The main part of the current paper studies arithmetical surfaces, that is, proper smooth surfaces over finite fields, or connected regular surfaces which are proper and flat over \({\mathbb{Z}}\). For surfaces over finite fields, the degree map exhibits \(CH_ 0(X)\) as an extension of \({\mathbb{Z}}\) by \(CH_ 0(X)^ 0\), the subgroup of degree 0 cycle classes. Correspondingly, there is a map from \(\pi_ 1^{ab}(X)\) onto \({\hat {\mathbb{Z}}}\), the Galois group of the algebraic closure of the ground field. Call \(\pi_ 1^{ab}(X)^ 0\) the kernel of this map. In 1981 N. Katz and S. Lang [Enseign. Math., II. Sér. 27, 285-319 (1981; Zbl 0495.14011)] showed that \(\pi_ 1^{ab}(X)^ 0\) is finite, while work of Bloch and Milne established the finiteness of \(CH_ 0(X)^ 0\). Milne further established the p-primary injectivity of the reciprocity map \(CH_ 0(X)\to \pi_ 1^{ab}(X),\) assuming a certain condition on X. In this paper the authors show that for any smooth projective geometrically irreducible scheme over a finite field, the reciprocity map is always injective, and induces an isomorphism of the finite groups \(CH_ 0(X)^ 0\) and \(\pi_ 1^{ab}(X)^ 0\) so that, roughly speaking, \(\pi_ 1^{ab}(X)\) is obtained from \(CH_ 0(X)\) by replacing \({\mathbb{Z}}\) by \({\hat {\mathbb{Z}}}\). According to the authors, they had originally obtained these results for surfaces, and an elegant induction argument of Colliot-Thélène established the general case.

For X a regular connected surface proper and flat over \({\mathbb{Z}}\), the main result established in this paper is that the reciprocity map \(CH_ 0(X)\to {\tilde \pi}_ 1^{ab}(X)\) is an isomorphism of finite abelian groups. Earlier, Bloch had proved this result in the case X is smooth over the ring of integers in a number field. To establish the unramified class field theory of arithmetical surfaces X in general, the authors concentrate on a fibration \(X\to C\), where C is a one dimensional scheme, and where the fibration may have singular fibres. In essence, the authors are able to improve on the earlier results of Bloch and Milne because they have a technique for treating the class field theory of a singular curve over a local field.

Reviewer: Lawrence G.Roberts

### MSC:

14J20 | Arithmetic ground fields for surfaces or higher-dimensional varieties |

14E20 | Coverings in algebraic geometry |

14C05 | Parametrization (Chow and Hilbert schemes) |

14H30 | Coverings of curves, fundamental group |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |