Tame class field theory for arithmetic schemes

Abstract

Takagi's class field theory gave a decription of the abelian extensions of a number field in terms of ideal groups in . In the 1980s, and [``Global class field theory of arithmetic schemes". Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 255--331 (1986; Zbl 0614.14001)] were able to generalize class field theory to higher dimensional fields, and to describe their abelian extensions using a generalized idèle class group whose definition is quite involved. In the case of unramified extensions, however, the class fields can be described geometrically using Chow groups. For fields of positive characteristic, a similarly geometric description for tamely ramified extensions was obtained by the author and [J. Reine Angew. Math. 527, 13--36 (2000; Zbl 0961.14013)]. In this article, an analogous result is proved for the case of mixed characteristic.