Galois Cohomology and Class Field Theory

Preface
Notation and Conventions
Contents
Part I Group Cohomology and Galois Cohomology: Generalities
1 Cohomology of Finite Groups: Basic Properties
	1.1 The Notion of G-Module
	1.2 The Category of G-Modules
	1.3 The Cohomology Groups HGA
	1.4 Computation of Cohomology Using the Cochains
	1.5 Change of Group: Restriction, Corestriction, the Hochschild–Serre …
	1.6 Corestriction; Applications
	1.7 Exercises
2 Groups Modified à la Tate, Cohomology of Cyclic Groups
	2.1 Tate Modified Cohomology Groups.
	2.2 Change of Group. Transfer
	2.3 Cohomology of a Cyclic Group
	2.4 Herbrand Quotient
	2.5 Cup-Products
	2.6 Cup-Products for the Modified Cohomology
	2.7 Exercises
3 p-Groups, the Tate–Nakayama Theorem
	3.1 Cohomologically Trivial Modules
	3.2 The Tate–Nakayama Theorem
	3.3 Exercises
4 Cohomology of Profinite Groups
	4.1 Basic Facts About Profinite Groups
	4.2 G-Modules
	4.3 Cohomology of a Discrete G-Module
	4.4 Exercises
5 Cohomological Dimension
	5.1 Definitions, First Examples
	5.2 Properties of the Cohomological Dimension
	5.3 Exercises
6 First Notions of Galois Cohomology
	6.1 Generalities
	6.2 Hilbert\'s Theorem 90 and Applications
	6.3 Brauer Group of a Field
	6.4 Cohomological Dimension of a Field
	6.5 C1
	6.6 Exercises
Part II Local Fields
7 Basic Facts About Local Fields
	7.1 Discrete Valuation Rings
	7.2 Complete Field for a Discrete Valuation
	7.3 Extensions of Complete Fields
	7.4 Galois Theory of a Complete Field for a Discrete Valuation
	7.5 Structure Theorem; Filtration of the Group of Units
	7.6 Exercises
8 Brauer Group of a Local Field
	8.1 Local Class Field Axiom
	8.2 Computation of the Brauer Group
	8.3 Cohomological Dimension; Finiteness Theorem
	8.4 Exercises
9 Local Class Field Theory: The Reciprocity Map
	9.1 Definition and Main Properties
	9.2 The Existence Theorem: Preliminary Lemmas and the Case of a p-adic Field
	9.3 Exercises
10 The Tate Local Duality Theorem
	10.1 The Dualising Module
	10.2 The Local Duality Theorem
	10.3 The Euler–Poincaré Characteristic
	10.4 Unramified Cohomology
	10.5 From the Duality Theorem to the Existence Theorem
	10.6 Exercises
11 Local Class Field Theory: Lubin–Tate Theory
	11.1 Formal Groups
	11.2 Change of the Uniformiser
	11.3 Fields Associated to Torsion Points
	11.4 Computation of the Reciprocity Map
	11.5 The Existence Theorem (the General Case)
	11.6 Exercises
Part III Global Fields
12 Basic Facts About Global Fields
	12.1 Definitions, First Properties
	12.2 Galois Extensions of a Global Field
	12.3 Idèles, Strong Approximation Theorem
	12.4 Some Complements in the Case of a Function Field
	12.5 Exercises
13 Cohomology of the Idèles: The Class Field Axiom
	13.1 Cohomology of the Idèle Group
	13.2 The Second Inequality
	13.3 Kummer Extensions
	13.4 First Inequality and the Axiom of Class Field Theory
	13.5 Proof of the Class Field Axiom for a Function Field
	13.6 Exercises
14 Reciprocity Law and the Brauer–Hasse–Noether Theorem
	14.1 Existence of a Neutralising Cyclic Extension
	14.2 The Global Invariant and the Norm Residue Symbol
	14.3 Exercises
15 The Abelianised Absolute Galois Group of a Global Field
	15.1 Reciprocity Map and the Idèle Class Group
	15.2 The Global Existence Theorem
	15.3 The Case of a Function Field
	15.4 Ray Class Fields; Hilbert Class Field
	15.5 Galois Groups with Restricted Ramification
	15.6 Exercises
Part IV Duality Theorems
16 Class Formations
	16.1 Notion of Class Formation
	16.2 The Spectral Sequence of the Ext
	16.3 The Duality Theorem for a Class Formation
	16.4 P-Class Formation
	16.5 From the Existence Theorem to the Duality Theorem for a p-adic Field
	16.6 Complements
	16.7 Exercises
17 Poitou–Tate Duality
	17.1 The P-Class Formation Associated to a Galois Group with Restricted Ramification
	17.2 The Groups PSM
	17.3 Statement of the Poitou–Tate Theorems
	17.4 Proof of Poitou–Tate Theorems (I): Computation  of the Ext Groups
	17.5 Proof of the Poitou–Tate Theorems (II): Computation of the Ext with Values in IS and End of the Proof
	17.6 Exercises
18 Some Applications
	18.1 Triviality of Some of the Sha
	18.2 The Strict Cohomological Dimension  of a Number Field
	18.3 Exercises
Appendix A Some Results from Homological Algebra
A.1 Generalities on Categories
A.2 Functors
A.3 Abelian Categories
A.4 Categories of Modules
A.5 Derived Functors
A.6 Ext and Tor
A.7 Spectral Sequences
Appendix B A Survey of Analytic Methods
B.1 Dirichlet Series
B.2 Dedekind ζ Function; Dirichlet L-Functions
B.3 Complements on the Dirichlet Density
B.4 The First Inequality
B.5 Class Field Theory in Terms of Ideals
B.6 Proof of the Čebotarev Theorem
Index