Abstract

We propose and study a generalised Kawada–Satake method for Mackey functors in the class field theory of positive characteristic. The root of this method is in the use of explicit pairings, such as the Artin–Schreier–Witt pairing, for groups describing abelian extensions. We separate and simplify the algebraic component of the method and discuss a relation between the existence theorem in class field theory and topological reflexivity with respect to the explicit pairing. We apply this method to derive higher local class field theory of positive characteristic, using advanced properties of topological Milnor K-groups of such fields.

Highlights

  • We propose and study a generalised Kawada–Satake method for Mackey functors in the class field theory of positive characteristic

  • From the axioms one derives the existence of the reciprocity homomorphism ΦH : AH −→ Hab for every open subgroup H of G

  • On AH one can introduce a translation invariant topology in which open neighbourhoods of zero are NH/KAK where K runs through all open subgroups K of H, call it the normic topology

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Summary

CLASSICAL WITT THEORY

For the convenience of the reader we include a brief summary of classical Witt theory. It is the inverse limit of the rings of truncated Witt vectors Wm(k) = {(w0, . There is a bijective correspondence between subgroups V in Wm(k) that contain ℘(Wm(k)) and abelian extensions l/k of exponent pi, i m: V ↔ l = k ℘−1(V ) , where k(℘−1(V )) is the compositum of the fields k(w0, . Inaba theory seems to be virtually unknown to mathematicians working with non-abelian extensions of fields of positive characteristic. It is isomorphic to the group of characters of the direct limit of groups Wm(k)/℘(Wm(k)) with respect to appropriate morphisms, which we describe. The group W (k) is the direct limit of Wm(k) with respect to the canonical injective morphisms im : Wm(k) → Wm+1(k). In the category of topological groups, Witt theory implies isomorphisms of discrete and compact groups vk : W (k) −→ Gab,p(k)◦, wk : Gab,p(k) −→ W (k)◦. For the existence theorem in this class field theory see [27, sect.1]

GENERALISED KAWADA–SATAKE CLASS FIELD THEORY MECHANISM FOR MACKEY FUNCTORS
CLASS FIELD THEORY OF HIGHER LOCAL FIELDS OF POSITIVE CHARACTERISTIC
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