Cohomology of number fields:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2008
|
| Edition: | 2. ed. |
| Series: | Grundlehren der mathematischen Wissenschaften
323 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 825 S. |
| ISBN: | 354037888X 9783540378884 |
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MARC
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| 100 | 1 | |a Neukirch, Jürgen |d 1937-1997 |e Verfasser |0 (DE-588)117716839 |4 aut | |
| 245 | 1 | 0 | |a Cohomology of number fields |c Jürgen Neukirch ; Alexander Schmidt ; Kay Wingberg |
| 250 | |a 2. ed. | ||
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Record in the Search Index
| _version_ | 1804136287352913920 |
|---|---|
| adam_text | Contents
Algebraic Theory
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
.............
Ill
§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
sed G
= 2 .................. 202
§7.
Poincaré
Groups
......................... 210
§8.
Filtrations
............................. 220
§9.
Generators and Relations
..................... 224
xiv Contents
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
Chapter VI: Galois Cohomology
....................337
§ 1.
Cohomology of the Additive Group
...............337
§2.
Hubert s
Satz 90.........................343
§3.
The
Brauer
Group
........................349
§4.
The Milnor if-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
Idèle
Class Group
..............425
§2.
The Connected Component of Ck
.................442
§3.
Restricted Ramification
......................451
§4.
The Global Duality Theorem
...................465
§5.
Local Cohomology of Global Galois Modules
..........471
§6.
Poitou-Tate Duality
........................479
§7.
The Global
Euler-Poincaré
Characteristic
............502
§8.
Duality for Unramified and Tamely Ramified Extensions
.....512
Contents
XV
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.
Šafarevič 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 oo
..................656
§7.
Dimension of ff^Gj, 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 fcoo
......722
§2.
Iwasawa Theory for p-adic Local Fields
.............731
§3.
The Maximal Abelianp-Extension of fcocUnramified Outside
5 . . 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
......................799
Literature
................................805
Index
..................................820
|
| adam_txt |
Contents
Algebraic Theory
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
.
Ill
§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
sed G
= 2 . 202
§7.
Poincaré
Groups
. 210
§8.
Filtrations
. 220
§9.
Generators and Relations
. 224
xiv Contents
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
Chapter VI: Galois Cohomology
.337
§ 1.
Cohomology of the Additive Group
.337
§2.
Hubert's
Satz 90.343
§3.
The
Brauer
Group
.349
§4.
The Milnor if-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
Idèle
Class Group
.425
§2.
The Connected Component of Ck
.442
§3.
Restricted Ramification
.451
§4.
The Global Duality Theorem
.465
§5.
Local Cohomology of Global Galois Modules
.471
§6.
Poitou-Tate Duality
.479
§7.
The Global
Euler-Poincaré
Characteristic
.502
§8.
Duality for Unramified and Tamely Ramified Extensions
.512
Contents
XV
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.
Šafarevič'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 oo
.656
§7.
Dimension of ff^Gj, 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 fcoo
.722
§2.
Iwasawa Theory for p-adic Local Fields
.731
§3.
The Maximal Abelianp-Extension of fcocUnramified Outside
5 . . 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
.799
Literature
.805
Index
.820 |
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| author | Neukirch, Jürgen 1937-1997 Schmidt, Alexander 1965- Wingberg, Kay 1949- |
| author_GND | (DE-588)117716839 (DE-588)132996952 (DE-588)142700975 |
| author_facet | Neukirch, Jürgen 1937-1997 Schmidt, Alexander 1965- Wingberg, Kay 1949- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512/.3 512/.74 |
| dewey-sort | 3512 13 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV022279483 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:48:54Z |
| indexdate | 2024-07-09T20:54:00Z |
| institution | BVB |
| isbn | 354037888X 9783540378884 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015489818 |
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| physical | XV, 825 S. |
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| publisher | Springer |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Neukirch, Jürgen 1937-1997 Verfasser (DE-588)117716839 aut Cohomology of number fields Jürgen Neukirch ; Alexander Schmidt ; Kay Wingberg 2. ed. Berlin [u.a.] Springer 2008 XV, 825 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 323 Algebraic fields Galois theory Homology theory Kohomologie (DE-588)4031700-6 gnd rswk-swf Zahlkörper (DE-588)4067273-6 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Kohomologie (DE-588)4031700-6 s Zahlkörper (DE-588)4067273-6 s DE-604 Algebraische Zahlentheorie (DE-588)4001170-7 s Schmidt, Alexander 1965- Verfasser (DE-588)132996952 aut Wingberg, Kay 1949- Verfasser (DE-588)142700975 aut Grundlehren der mathematischen Wissenschaften 323 (DE-604)BV000000395 323 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015489818&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Neukirch, Jürgen 1937-1997 Schmidt, Alexander 1965- Wingberg, Kay 1949- Cohomology of number fields Grundlehren der mathematischen Wissenschaften Algebraic fields Galois theory Homology theory Kohomologie (DE-588)4031700-6 gnd Zahlkörper (DE-588)4067273-6 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| subject_GND | (DE-588)4031700-6 (DE-588)4067273-6 (DE-588)4001170-7 |
| title | Cohomology of number fields |
| title_auth | Cohomology of number fields |
| title_exact_search | Cohomology of number fields |
| title_exact_search_txtP | Cohomology of number fields |
| title_full | Cohomology of number fields Jürgen Neukirch ; Alexander Schmidt ; Kay Wingberg |
| title_fullStr | Cohomology of number fields Jürgen Neukirch ; Alexander Schmidt ; Kay Wingberg |
| title_full_unstemmed | Cohomology of number fields Jürgen Neukirch ; Alexander Schmidt ; Kay Wingberg |
| title_short | Cohomology of number fields |
| title_sort | cohomology of number fields |
| topic | Algebraic fields Galois theory Homology theory Kohomologie (DE-588)4031700-6 gnd Zahlkörper (DE-588)4067273-6 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| topic_facet | Algebraic fields Galois theory Homology theory Kohomologie Zahlkörper Algebraische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015489818&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT neukirchjurgen cohomologyofnumberfields AT schmidtalexander cohomologyofnumberfields AT wingbergkay cohomologyofnumberfields |