Cohomology of Number Fields

by Neukirch, Jurgen; Schmidt, Alexander; Wingberg, Kay

• ISBN13: 9783540378884
• ISBN10: 354037888X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-04-18
• Publisher: Springer Nature

Summary

The present second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. The first part provides algebraic background: cohomology of profinite groups, duality groups, free products, and homotopy theory of modules, with new sections on spectral sequences and on Tate cohomology of profinite groups. The second part deals with Galois groups of local and global fields: Tate duality, structure of absolute Galois groups of local fields, extensions with restricted ramification, Poitou-Tate duality, Hasse principles, theorem of Grunwald-Wang, Leopoldt's conjecture, Riemann's existence theorem, the theorems of Iwasawa and of Å afarevic on solvable groups as Galois groups, Iwasawa theory, and anabelian principles. New material is introduced here on duality theorems for unramified and tamely ramified extensions, a careful analysis of 2-extensions of real number fields and a complete proof of Neukirch's theorem on solvable Galois groups with given local conditions.

Table of Contents

Algebraic Theory
Cohomology of Profinite Groups	p. 3
Profinite Spaces and Profinite Groups	p. 3
Definition of the Cohomology Groups	p. 12
The Exact Cohomology Sequence	p. 25
The Cup-Product	p. 36
Change of the Group G	p. 45
Basic Properties	p. 60
Cohomology of Cyclic Groups	p. 74
Cohomological Triviality	p. 80
Tate Cohomology of Profinite Groups	p. 83
Some Homological Algebra	p. 97
Spectral Sequences	p. 97
Filtered Cochain Complexes	p. 101
Degeneration of Spectral Sequences	p. 107
The Hochschild-Serre Spectral Sequence	p. 111
The Tate Spectral Sequence	p. 120
Derived Functors	p. 127
Continuous Cochain Cohomology	p. 136
Duality Properties of Profinite Groups	p. 147
Duality for Class Formations	p. 147
An Alternative Description of the Reciprocity Homomorphism	p. 164
Cohomological Dimension	p. 171
Dualizing Modules	p. 181
Projective pro-c-groups	p. 189
Profinite Groups of scdG = 2	p. 202
Poincaré Groups	p. 210
Filtrations	p. 220
Generators and Relations	p. 224 xivChapt
Free Products	p. 245
Subgroups of Free Products	p. 252
Generalized Free Products	p. 256
Iwasawa Modules	p. 267
Modules up to Pseudo-Isomorphism	p. 268
Complete Group Rings	p. 273
Iwasawa Modules	p. 289
Homotopy of Modules	p. 301
Homotopy Invariants of Iwasawa Modules	p. 312
Differential Modules and Presentations	p. 321
Arithmetic Theory
Galois Cohomology	p. 337
Cohomology of the Additive Group	p. 337
Hilbert's Satz 90	p. 343
The Brauer Group	p. 349
The Milnor K-Groups	p. 356
Dimension of Fields	p. 360
Cohomology of Local Fields	p. 371
Cohomology of the Multiplicative Group	p. 371
The Local Duality Theorem	p. 378
The Local Euler-Poincare Characteristic	p. 391
Galois Module Structure of the Multiplicative Group	p. 401
Explicit Determination of Local Galois Groups	p. 409
Cohomology of Global Fields	p. 425
Cohomology of the Idele Class Group	p. 425
The Connected Component of Ck	p. 442
Restricted Ramification	p. 451
The Global Duality Theorem	p. 465
Local Cohomology of Global Galois Modules	p. 471
Poitou-Tate Duality	p. 479
The Global Euler-Poincare Characteristic	p. 502
Duality for Unramified and Tamely Ramified Extensions	p. 512
The Absolute Galois Group of a Global Field	p. 521
The Hasse Principle	p. 522
The Theorem of Grunwald-Wang	p. 536
Construction of Cohomology Classes	p. 543
Local Galois Groups in a Global Group	p. 553
Solvable Groups as Galois Groups	p. 557
èafarevic's Theorem	p. 574
Restricted Ramification	p. 599
The Function Field Case	p. 602
First Observations on the Number Field Case	p. 618
Leopoldt's Conjecture	p. 624
Cohomology of Large Number Fields	p. 642
Riemann's Existence Theorem	p. 647
The Relation between 2 and 8	p. 656
Dimension of Hi ($$$$)	p. 666
The Theorem of Kuz'min	p. 678
Free Product Decomposition of GS(p)	p. 686
Class Field Towers	p. 697
The Profinite Group GS	p. 706
Iwasawa Theory of Number Fields	p. 721
The Maximal Abelian Unramified p-Extension of k&infty;	p. 722
Iwasawa Theory for p-adic Local Fields	p. 731
The Maximal Abelian p-Extension of k&infty; UnramifiedOutsideS	p. 735
Iwasawa Theory for Totally Real Fields and CM-Fields	p. 751
Positively Ramified Extensions	p. 763
The Main Conjecture	p. 771
Anabelian Geometry	p. 785
Subgroups of Gk	p. 785
The Neukirch-Uchida Theorem	p. 791
Anabelian Conjectures	p. 799
Literature	p. 805
Index	p. 820
Table of Contents provided by Publisher. All Rights Reserved.