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Algebraic Theory 1

Chapter Ⅰ：Cohomology of Profinite Groups 3

1．Profinite Spaces and Profinite Groups 3

2．Definition of the Cohomology Groups 12

3．The Exact Cohomology Sequence 25

4．The Cup-Product 36

5．Change of the Group G 45

6．Basic Properties 60

7．Cohomology of Cyclic Groups 74

8．Cohomological Triviality 80

9．Tate Cohomology of Profinite Groups 83

Chapter Ⅱ：Some Homological Algebra 97

1．Spectral Sequences 97

2．Filtered Cochain Complexes 101

3．Degeneration of Spectral Sequences 107

4．The Hochschild-Serre Spectral Sequence 111

5．The Tate Spectral Sequence 120

6．Derived Functors 127

7．Continuous Cochain Cohomology 136

Chapter Ⅲ：Duality Properties of Profinite Groups 147

1．Duality for Class Formations 147

2．An Alternative Description of the Reciprocity Homomorphism 164

3．Cohomological Dimension 171

4．Dualizing Modules 181

5．Projective pro-c-groups 189

6．Profinite Groups of scd G=2 202

7．Poincaré Groups 210

8．Filtrations 220

9．Generators and Relations 224

Chapter Ⅳ：Free Products of Profinite Groups 245

1．Free Products 245

2．Subgroups of Free Products 252

3．Generalized Free Products 256

Chapter Ⅴ：Iwasawa Modules 267

1．Modules up to Pseudo-Isomorphism 268

2．Complete Group Rings 273

3．Iwasawa Modules 289

4．Homotopy of Modules 301

5．Homotopy Invariants of Iwasawa Modules 312

6．Differential Modules and Presentations 321

Arithmetic Theory 335

Chapter Ⅵ：Galois Cohomology 337

1．Cohomology of the Additive Group 337

2．Hilbert's Satz 90 343

3．The Brauer Group 349

4．The Milnor K-Groups 356

5．Dimension of Fields 360

Chapter Ⅶ：Cohomology of Local Fields 371

1．Cohomology of the Multiplicative Group 371

2．The Local Duality Theorem 378

3．The Local Euler-Poincaré Characteristic 391

4．Galois Module Structure of the Multiplicative Group 401

5．Explicit Determination of Local Galois Groups 409

Chapter Ⅷ：Cohomology of Global Fields 425

1．Cohomology of the Idè1e Class Group 425

2．The Connected Component of Ck 443

3．Restricted Ramification 452

4．The Global Duality Theorem 466

5．Local Cohomology of Global Galois Modules 472

6．Poitou-Tate Duality 480

7．The Global Euler-PoincaréCharacteristic 503

8．Duality for Unramified and Tamely Ramified Extensions 513

Chapter Ⅸ：The Absolute Galois Group of a Global Field 521

1．The Hasse Principle 522

2．The Theorem of Grunwald-Wang 536

3．Construction of Cohomology Classes 543

4．Local Galois Groups in a Global Group 553

5．Solvable Groups as Galois Groups 557

6．？afarevi？'s Theorem 574

Chapter Ⅹ：Restricted Ramification 599

1．The Function Field Case 602

2．First Observations on the Number Field Case 618

3．Leopoldt's Conjecture 624

4．Cohomology of Large Number Fields 642

5．Riemann's Existence Theorem 647

6．The Relation between 2 and ∞ 656

7．Dimension of Hi(G？,Z/pZ) 666

8．The Theorem of Kuz'min 678

9．Free Product Decomposition of Gs(p) 686

10．Class Field Towers 697

11．The Profinite Group Gs 706

Chapter Ⅺ：Iwasawa Theory of Number Fields 721

1．The Maximal Abelian Unramified p-Extension of k∞ 722

2．Iwasawa Theory for p-adic Local Fields 731

3．The Maximal Abelian p-Extension of k∞ Unramified Outside S 735

4．Iwasawa Theory for Totally Real Fields and CM-Fields 751

5．Positively Ramified Extensions 763

6．The Main Conjecture 771

Chapter Ⅻ：Anabelian Geometry 785

1．Subgroups of Gk 785

2．The Neukirch-Uchida Theorem 791

3．Anabelian Conjectures 798

Literature 805

Index 821