Generalized class formations and higher class field theory

Abstract

0 ) be an n-dimensional local field (whose last residue field is finite of characteristic p ).The following theorem can be viewed as a generalization to higher dimensional local fields of the fact Br(F ) → Q/Z for classical local fields F with finite residue field (see section 5). Theorem (Kato). There is a canonical isomorphismKato established higher local reciprocity map (see section 5 and [K1, Th. 2 of §6] (two-dimensional case), [K2, Th. II], [K3, §4]) using in particular this theorem.In this section we deduce the reciprocity map for higher local fields from this theorem and Bloch-Kato's theorem of section 4. Our approach which uses generalized class formations simplifies Kato's original argument.We use the notations of section 5. For a complex X • the shifted-by- Classical class formationsWe begin with recalling briefly the classical theory of class formations.A pair (G, C) consisting of a profinite group G and a discrete G-module C is called a class formation if (C1) H 1 (H, C) = 0 for every open subgroup H of G. (C2) There exists an isomorphism inv H : H 2 (H, C) → Q/Z for every open subgroup H of G.

An introduction to higher dimensional local fields and adeles

“…Y. Koya [42,43] proved higher dimensional local class field theory using the language of class formations and complexes of Galois modules. His methods were further developed by M. Speiss [69,70], who gave the first proof that if L/F is a finite Galois extension (not necessarily abelian) of n-dimensional local fields, then…”