《數域的上同調（第2版）》是2017年世界圖書出版公司出版的著作，作者是[德] J.諾伊基希，A.施密特，K.溫伯格。
基本介紹
- 書名：《數域的上同調（第2版）
- 作者：[德] J.諾伊基希，A.施密特，K.溫伯格
- 出版社：世界圖書出版公司
- 出版時間：2017-05-01
內容簡介,目錄,
內容簡介
本書是一部教科書，適用於數論專業的學生和數學工作者。書中第1部分提供了代數的基礎理論，包括射有限群的上同調，對偶群，自由積，以及模的同調理論。第2部分詳述了局部域和全局域的伽羅瓦群，包括Tate二重性，局部域絕對伽羅瓦群的結構，限制分歧，Poitou-Tate二重性，Hasse原理，Grunwald-Wang定理，Leopoldt猜想，黎曼存在性定理，等等。本書是2008年版本的修訂版。
目錄
Algebraic Theory
Chapter Ⅰ：Cohomology of Profinite Groups
1.Profinite Spaces and Profinite Groups
2.Defirution of the Cohomology Groups
3.The Exact Cohomology Sequence
4.The Cup—Product
5.Change of the Group G
6.Basic Properties
7.Cohomology of Cyclic Groups
8.Cohomological Triviality
9.Tate Cohomology of Profinite Groups
Chapter Ⅱ：Some Homological Algebra
1.Spectral Sequences
2.Filtered Cochain Complexes
3.Degeneration of Spectral Sequences
4.The Hochschild—Serre Spectral Sequence
5.The Tate Spectral Sequence
6.Derived Functors
7.Continuous Cochain Cohomology
Chapter Ⅲ：Duality Properties of Profinite Groups
1.Duality for Class Formations
2.An Alternative Description of the Reciprocity Homomorphism
3.Cohomological Dimension
4.Dualizing Modules
5.Ptojective pro—c—groups
6.Profinite Groups of scd G=2
7.Poincare Groups
8.Filtrations
9.Generators and Relations
Chapter Ⅳ：Free Products of Profinite Groups
1.Free Products
2.Subgroups of Free Products
3.Generalized Free Products
Chapter Ⅴ：Iwasawa Modules
1.Modules up to Pseudo—Isomorphism
2.Complete Group Rings
3.Iwasawa Modules
4.Homotopy of Modules
5.Homotopy Invariants of Iwasawa Modules
6.Differential Modules and Presentations Arithmetic Theory
Chapter Ⅵ：Galois Cohomology
1.Cohomology of the Additive Group
2.Hilbert's Satz 90
3.The Brauer Group
4.The Milnor K—Groups
5.Dimension of Fields
Chapter Ⅶ：Cohomology of Local Fields
1.Cohomology of the Multiplicative Group
2.The Local Duality Theorem
3.The Local Euler—Poincare Characteristic
4.Galois Module Structure of the Multiplicative Group
5.Explicit Determination of Local Galois Groups
Chapter Ⅷ：Cohomology of Global Fields
1.Cohomology of the Idele Class Group
2.The Connected Component of Ck
3.Restricted Ramification
4.The Global Duality Theorem
5.Local Cohomology of Global Galois Modules
6.Poitou—Tate Duality
7.The Global Euler—Poincare Characteristic
8.Duality for Unramified and Tamely Ramified Extensions
Chapter Ⅸ：The Absolute Galois Group of a Global Field
1.The Hasse Principle
2.The Theorem of Grunwald—Wang
3.Construction of Cohomology Classes
4.Local Galois Groups in a Global Group
5.Solvable Groups as Galois Groups
6.Safarevic's Theorem
Chapter Ⅹ：Restricted Ranufication
1.The Function Field Case
2.First Observations on the Number Field Case
3.Leopoldt's Conjecture
4.Cohomology of Large Number Fields
5.Riemann's Existence Theorem
6.The Relation between 2 and ∞
7.Dimension of Hi（GTS，Z/pZ）
8.The Theorem of Kuz'min
9.Free Product Decomposition of Gs（P）
10.Class Field Towers
11.The Profinite Group Gs
Chapter Ⅺ：Iwasawa Theory of Number Fields
1.The Maximal Abelian Unramified p—Extension of k∞
2.Iwasawa Theory for p—adic Local Fields
3.The Maximal Abelianp—Extension of k∞ Unramified Outside S
4.Iwasawa Theory for Totally Real Fields and CM—Fields
5.Positively Ramified Extensions
6.The Main Conjecture
Chapter Ⅻ：Anabelian Geometry
1.Subgroups of Gk
2.The Neukirch—Uchida Theorem
3.Anabelian Conjectures
Literature
Index
Chapter Ⅰ：Cohomology of Profinite Groups
1.Profinite Spaces and Profinite Groups
2.Defirution of the Cohomology Groups
3.The Exact Cohomology Sequence
4.The Cup—Product
5.Change of the Group G
6.Basic Properties
7.Cohomology of Cyclic Groups
8.Cohomological Triviality
9.Tate Cohomology of Profinite Groups
Chapter Ⅱ：Some Homological Algebra
1.Spectral Sequences
2.Filtered Cochain Complexes
3.Degeneration of Spectral Sequences
4.The Hochschild—Serre Spectral Sequence
5.The Tate Spectral Sequence
6.Derived Functors
7.Continuous Cochain Cohomology
Chapter Ⅲ：Duality Properties of Profinite Groups
1.Duality for Class Formations
2.An Alternative Description of the Reciprocity Homomorphism
3.Cohomological Dimension
4.Dualizing Modules
5.Ptojective pro—c—groups
6.Profinite Groups of scd G=2
7.Poincare Groups
8.Filtrations
9.Generators and Relations
Chapter Ⅳ：Free Products of Profinite Groups
1.Free Products
2.Subgroups of Free Products
3.Generalized Free Products
Chapter Ⅴ：Iwasawa Modules
1.Modules up to Pseudo—Isomorphism
2.Complete Group Rings
3.Iwasawa Modules
4.Homotopy of Modules
5.Homotopy Invariants of Iwasawa Modules
6.Differential Modules and Presentations Arithmetic Theory
Chapter Ⅵ：Galois Cohomology
1.Cohomology of the Additive Group
2.Hilbert's Satz 90
3.The Brauer Group
4.The Milnor K—Groups
5.Dimension of Fields
Chapter Ⅶ：Cohomology of Local Fields
1.Cohomology of the Multiplicative Group
2.The Local Duality Theorem
3.The Local Euler—Poincare Characteristic
4.Galois Module Structure of the Multiplicative Group
5.Explicit Determination of Local Galois Groups
Chapter Ⅷ：Cohomology of Global Fields
1.Cohomology of the Idele Class Group
2.The Connected Component of Ck
3.Restricted Ramification
4.The Global Duality Theorem
5.Local Cohomology of Global Galois Modules
6.Poitou—Tate Duality
7.The Global Euler—Poincare Characteristic
8.Duality for Unramified and Tamely Ramified Extensions
Chapter Ⅸ：The Absolute Galois Group of a Global Field
1.The Hasse Principle
2.The Theorem of Grunwald—Wang
3.Construction of Cohomology Classes
4.Local Galois Groups in a Global Group
5.Solvable Groups as Galois Groups
6.Safarevic's Theorem
Chapter Ⅹ：Restricted Ranufication
1.The Function Field Case
2.First Observations on the Number Field Case
3.Leopoldt's Conjecture
4.Cohomology of Large Number Fields
5.Riemann's Existence Theorem
6.The Relation between 2 and ∞
7.Dimension of Hi（GTS，Z/pZ）
8.The Theorem of Kuz'min
9.Free Product Decomposition of Gs（P）
10.Class Field Towers
11.The Profinite Group Gs
Chapter Ⅺ：Iwasawa Theory of Number Fields
1.The Maximal Abelian Unramified p—Extension of k∞
2.Iwasawa Theory for p—adic Local Fields
3.The Maximal Abelianp—Extension of k∞ Unramified Outside S
4.Iwasawa Theory for Totally Real Fields and CM—Fields
5.Positively Ramified Extensions
6.The Main Conjecture
Chapter Ⅻ：Anabelian Geometry
1.Subgroups of Gk
2.The Neukirch—Uchida Theorem
3.Anabelian Conjectures
Literature
Index
