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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.
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

Citations:

Zbl 0495.14011
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