ℓ-Adic class field theory for regular local rings

Abstract

In this paper, we prove the -adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is an -adic variant of a result of Matsumi (Class field theory for F q [[X 1 , . . . , X n ]], preprint, 2002).

Ideles in Higher Dimension

“…I would like to thank Uwe Jannsen, Shuji Saito and Alexander Schmidt for numerous discussions on higher class field theory. Reading Kanetomo Sato's article [16] was very enlightening for the preparation of the present note. Alexander Schmidt made helpful suggestions for improving the text.…”

“…x P (X P , K M d X (O X P , I P )). This local Kato-Saito class group was also studied by Sato in [16]. Note that it gives a useful definition for higher class field theory only if P is a Parshin chain, i.e.…”