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Mackey Functors and Abelian Class Field Theories

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Mackey Functors and Abelian Class Field Theories Mackey functors and abelian class field theories Ulrich Thiel¤ November 2, 2018 Abstract 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. arXiv:1111.0942v1 [math.NT] 3 Nov 2011 ¤University of Kaiserslautern. Email: thiel at mathematik.uni-kl.de 2 Contents 1 Introduction 3 2 RIC-functors 6 2.1 Definition and basic properties of RIC-functors . .7 2.2 G-subgroup systems . 11 2.3 Discrete G-modules . 17 2.4 Abelianizations . 23 3 Abelian class field theories 30 3.1 Two-dimensional G-subgroup systems . 32 3.2 Tautological class field theories . 34 3.3 Induction representations . 36 3.4 Definition and basic properties of abelian class field theories . 37 3.5 The case of induction representations . 45 3.6 Class field theories for profinite groups . 48 4 Fesenko–Neukirch class field theories 50 4.1 Abstract ramification theory for compact groups . 52 4.2 Abstract valuation theory for compact groups . 55 4.3 Class field theories for unramified extensions . 56 4.4 Class field theories for all coabelian extensions . 59 5 Discrete valuation fields of higher rank 76 5.1 Basic facts about valuations on fields . 76 5.2 Ramification theory . 78 5.3 Discrete valuation fields of higher rank . 80 5.4 Milnor K-theory . 87 5.5 Sequential topologies . 89 6 Fesenko–Neukirch class field theory for certain types of discrete valuation fields 93 6.1 Local fields . 93 6.2 Local fields of higher rank and of positive characteristic . 95 A Topological groups 98 A.1 Filters . 98 A.2 Topological groups . 100 A.3 Quasi-compact and compact groups . 102 A.4 Transversals . 103 A.5 Topological abelian groups . 104 A.6 Abelianization . 105 B Projective limits of topological groups 107 B.1 Basic facts . 107 B.2 Approximations . 107 B.3 Complete proto-C groups for classes of compact groups . 111 B.4 Profinite groups . 116 B.5 Procyclic groups . 118 C Miscellaneous 122 C.1 Category theory . 122 C.2 Double cosets . 122 C.3 G-modules . 123 C.4 Normal cores . 125 C.5 The compact-open topology . 125 References 128 3 1. Introduction The author’s first contact with class field theories was Jürgen Neukirch’s presentation of this concept in his famous book [Neu99, chapter IV] on algebraic number theory. Unfortunately, and this may be explained by the author’s lack of knowledge and intuition in algebraic number theory, the author did not understand what the goal of all the investigations was until the very end of the presentation where the main results were summarized in a theorem. Surprised by the fact that this theorem was not presented at the beginning as a motivation for all further considerations, the author started to think about the abstract core of this theorem and tried to define what exactly a class field theory is. Such a single definition did not exist in the literature and it seemed that this was an obvious concept for people working in algebraic number the- ory. However, the author’s abstract considerations were further motivated by the exercises in this chapter where Neukirch proposes a generalization of his approach, mentioning that this generalization has been applied by Ivan Fesenko in [Fes92]. This thesis is the result of trying to understand what a class field theory is and trying to give a single-line definition (see 3.4.4) which captures certain aspects of several explicit class field theories. Although no new theorems are proven and the reader familiar with class field theories will not find anything surprising in this thesis, the author still hopes to communicate at least a certain point of view which is not presented in this form in the existing literature. To give an overview of the contents, we first have to present one result of the above considerations, namely what the idea of a class field theory is (we will give a more detailed motivation at the beginning of chapter 3). The essence of an abelian class field theory (ACFT for short) is to provide for a field k a description of the finite abelian extensions of finite separable extensions K k. There may exist of course several formaliza- j tions of this fuzzy concept and in this thesis we will develop one particular formalization. We understand an ACFT as consisting of three parts: a group-theoretical part, a functorial or compatibility part and an arithmetic part. In the first place, an ACFT should model for each finite separable extension K k the finite Galois extensions j of K as certain subgroups of some abelian group C(K) in such a way that this model is faithful on the lattice of abelian extensions of K and such that abelianized Galois groups can be calculated in this model. More explicitly, there should exist a map ©(K, ): E f(K) E a(C(K)) from the set E f(K) of all finite Galois ¡ ! extensions of K to the set E a(C(K)) of all subgroups of C(K) such that the restriction of ©(K, ) to the ¡ lattice L (K) E f(K) of finite abelian extensions is an injective morphism of lattices and there should exist ⊆ an isomorphism, called reciprocity morphism, ab ½L K : Gal(L K) C(K)/©(K,L) j j ! for each L E f(K). We can depict the passage from the field internal theory to its model as the following 2 scheme K C(K) L E f(K) ©(K,L) C(K) 2 · Gal(L K) C(K)/©(K,L). j These data are what we will refer to as the group-theoretical part of an ACFT. The functorial part of an ACFT is the requirement that this passage should satisfy several compatibility relations. It is an essential part of this thesis to precisely define these compatibility relations and to introduce objects that capture these relations. These objects will be the RIC-functors introduced in chapter2. The abbreviation RIC stands for Restriction-Induction-Conjugation, three natural operations which occur in several places of mathematics. In particular, such a triplet of operations exists on the abelianizations above and therefore the abelian groups C(K) should also be connected with such operations, that is, they should form a RIC- functor C and the reciprocity morphisms should be compatible with these operations, that is, the reciprocity morphism should be a morphism of RIC-functors. The arithmetic part of an ACFT will remain to be a fuzzy concept and can be described as the condition that an ACFT should be tied to the “arithmetic” of k. The meaning of this is two-fold. On the one hand, it means that the groups C(K) should be obtained directly from k itself. By what we have said about ACFTs, this would imply that all the abelian extensions and their Galois groups can be computed by data directly connected to k. Claude Chevalley explains this philosophy in [Che40] as follows: L’objet de la théorie du corps de classes est de montrer comment les extensions abeliennes d’un corps de nombres algébriques k peuvent êtres déterminées par des éléments tirès de la connais- sance de k lui-même ; ou, si l’on veut présenter les choses en termes dialectiques, comment un corps possêde en soi les éléments de son propre dépassement (et ce, sans aucune contradiction 4 interne !).1 To get an idea of this, we note that for local (respectively global) fields there exists an ACFT in the sense discussed so far, called local class field theory (respectively global class field theory). For a local field k the RIC-functor C of the local class field theory is given by the multiplicative groups C(K) GL1(K) with Æ the obvious conjugation, restriction and induction morphisms. For a global field k the RIC-functor C of the global class field theory is given by the idèle class groups C(K) GL1(AK )/GL1(K) with the canonical Æ conjugation, restriction and induction morphisms, where AK is the adèle ring of K. In both local and global class field theory it is evident that C is more or less directly obtained from k.
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