Two dimensional class field theory. (English) Zbl 0544.12011

Kato, Kazuya; Saito, Shuji

Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103-152 (1983).

[For the entire collection see Zbl 0516.00012.]
Let K be a two-dimensional arithmetic field. The authors show how to describe \(Gal(K^{ab}/K)\) in terms of algebraic K-theory \((K{}_ 2\) in fact). If X is a scheme (regular and connected), with function field K, which is proper over \({\mathbb{Z}}\), they construct ”idèle class groups” \(\bar C_ m(X)\) and \(C_ m(X)\) associated with a modulus, m, on X. These groups are defined in terms of algebraic K-theory. Their class field theorem asserts, for example, \[ (*)\quad Gal(K^{ab}/K)\quad \cong \quad \lim_{\overset \leftarrow m}\bar C_ m(X) \] as m ranges over admissible moduli on X.
To prove (*), the authors develop their two dimensional local and global class field theories. In addition, they describe how their results are related to the class field theory of S. Lang [cf., e.g., N. M. Katz and S. Lang, Enseign. Math., II. Sér. 27, 285-314; 315-319 (1981; Zbl 0495.14011)].

Reviewer: V.P.Snaith

MSC:

11S31	Class field theory; \(p\)-adic formal groups
11R37	Class field theory
14K15	Arithmetic ground fields for abelian varieties
11R70	\(K\)-theory of global fields
11S70	\(K\)-theory of local fields
12F20	Transcendental field extensions
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
14E20	Coverings in algebraic geometry

Keywords:

proper scheme over \({\mathbb{Z}}\); two dimensional class field theories; two- dimensional arithmetic field; \(K_ 2\); idèle class groups; algebraic K-theory

Citations:

Zbl 0516.00012; Zbl 0495.14011