2011/11/03 by Ulrich Thiel, Thiel, Ulrich
Mathematics · #11S31 #11S70 #19F05 #20E18 #22C05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1111.0942
openalex publication_date 2011/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the work of Jürgen Neukirch and Ivan Fesenko we propose a general definition of an abelian class field theory from a purely group-theoretical and functorial point of view. This definition allows a modeling of abelian extensions of a field inside more general objects than the invariants of a discrete module over the absolute Galois group of the field. The main objects serving as such models are cohomological Mackey functors as they have enough structure to make several reduction theorems of classical approaches work in this generalized setting and, as observed by Fesenko, they even have enough structure to make Neukirch's approach to class field theories via Frobenius lifts work. This approach is discussed in full detail and in its most general setting, including the pro-P setting proposed by Neukirch. As an application and justification of this generalization we describe Fesenko's approach to class field theory of higher local fields of positive characteristic, where the modeling of abelian extensions takes place inside the cohomological Mackey functor formed by the Milnor-Paršin K-groups. The motivation for this work (which is the author's Diplom thesis) was the attempt to understand what a class field theory is and to give a single-line definition which captures certain common aspects of several instances of class field theories. We do not claim to prove any new theorem here, but we think that our general and uniform approach offers a point of view not discussed in this form in the existing literature.