Galois cohomology:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Berlin [u.a.]
Springer
2002
|
| Edition: | Corr. 2. printing of the 1. Engl. ed. of 1997 |
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 210 S |
| ISBN: | 9783540421924 3540421920 |
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| 100 | 1 | |a Serre, Jean-Pierre |d 1926- |e Verfasser |0 (DE-588)142283126 |4 aut | |
| 240 | 1 | 0 | |a Cohomologie galoisienne |
| 245 | 1 | 0 | |a Galois cohomology |c Jean-Pierre Serre |
| 250 | |a Corr. 2. printing of the 1. Engl. ed. of 1997 | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2002 | |
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| 650 | 4 | |a Galois theory | |
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Record in the Search Index
| _version_ | 1804128752823697408 |
|---|---|
| adam_text | Table
of
Contents
Foreword
........................................................
V
Chapter I. Cohomology of
profinite
groups
§1.
Profinite
groups
............................................. 3
1.1
Definition
................................................ 3
1.2
Subgroups
................................................ 4
1.3
Indices
................................................... 5
1.4
Pro-p-groups and Sylow p-subgroups
......................... 6
1.5
Pro-p-groups
.............................................. 7
§2.
Cohomology
................................................. 10
2.1
Discrete G-modules
........................................ 10
2.2
Cochains, cocycles, cohomology
............................. 10
2.3
Low dimensions
........................................... 11
2.4
Punctoriality
.............................................. 12
2.5
Induced modules
.......................................... 13
2.6
Complements
............................................. 14
§3.
Cohomological dimension
.................................... 17
3.1
p-cohomological dimension
.................................. 17
3.2
Strict cohomological dimension
.............................. 18
3.3
Cohomological dimension of subgroups and extensions
......... 19
3.4
Characterization of the
profilate
groups
G
such that
cdp(G)
< 1 . 21
3.5
Dualizing modules
......................................... 24
§4.
Cohomology of pro-p-groups
................................. 27
4.1
Simple modules
........................................... 27
4.2
Interpretation of H1: generators
............................. 29
4.3
Interpretation of H2: relations
.............................. 33
4.4
A theorem of Shafarevich
................................... 34
4.5
Pomcaié
groups
........................................... 38
VIII Table of Contente
§5.
Nonabelian cohomology
..................................... 45
5.1 Definition
of
Я0
and of
Я1
................................. 45
5.2
Principal
homogeneous spaces over A
-
a new definition of
ЯЧС
A)
................................................. 46
5.3
Twisting
................................................. 47
5.4
The cohomology exact sequence associated to a subgroup
....... 50
5.5
Cohomology exact sequence associated to a normal subgroup
... 51
5.6
The case of an abelian normal subgroup
...................... 53
5.7
The case of a central subgroup
.............................. 54
5.8
Complements
............................................. 56
5.9
A property of groups with cohomological dimension
< 1........ 57
Bibliographic remarks for Chapter I
............................. 60
Appendix
1.
J.
Täte -
Some duality theorems
.................... 61
Appendix
2.
The Golod-Shafarevich inequality
................... 66
1.
The statement
............................................. 66
2.
Proof
..................................................... 67
Chapter II. Galois cohomology, the commutative case
§1.
Generalities
................................................. 71
1.1
Galois cohomology
......................................... 71
1.2
First examples
............................................ 72
§2.
Criteria for cohomological dimension
........................ 74
2.1
An auxiliary result
........................................ 74
2.2
Case when
ρ
is equal to the characteristic
..................... 75
2.3
Case when
ρ
differs from the characteristic
.................... 76
§3.
Fields of dimension <1
...................................... 78
3.1
Definition
................................................ 78
3.2
Relation with the property
(Ci)
............................. 79
3.3
Examples of fields of dimension
< 1.......................... 80
§4.
Transition theorems
......................................... 83
4.1
Algebraic extensions
....................................... 83
4.2
Transcendental extensions
.................................. 83
4.3
Local fields
............................................... 85
4.4
Cohomological dimension of the Galois group of an algebraic
number field
.............................................. 87
4.5
Property (Cr)
............................................. 87
Table
of Contents IX
§5.
p-adic fields
.................................................. 90
5.1
Summary of known results
.................................. 90
5.2
Cohomology of finite Gfc-modules
............................ 90
5.3
First applications
.......................................... 93
5.4
The
Euler-Poincaré
characteristic (elementary case)
............ 93
5.5
Unramified cohomology
.................................... 94
5.6
The Galois group of the maximal p-extension of A;
............. 95
5.7
Euler-Poincaré
characteristics
............................... 99
5.8
Groups of multiplicative type
............................... 102
§6.
Algebraic number fields
......................................105
6.1
Finite modules
-
definition of the groups
/*(*;,
А)
.............105
6.2
The finiteness theorem
.....................................106
6.3
Statements of the theorems of
Poitou
and
Tate
................107
Bibliographic remarks for Chapter II
............................109
Appendix. Galois cohomology of purely transcendental extensions
110
1.
An exact sequence
..........................................110
2.
The local case
.............................................
Ill
3.
Algebraic curves and function fields in one variable
.............112
4.
The case
К
=
k{T)
.........................................113
5.
Notation
..................................................114
6.
Killing by base change
......................................115
7.
Manin
conditions, weak approximation
and Schinzel s hypothesis
....................................116
8.
Sieve bounds
..............................................117
Chapter III. Nonabelian Galois cohomology
§1.
Forms
.......................................................121
1.1
Tensors
..................................................121
1.2
Examples
.................................................123
1.3
Varieties, algebraic groups, etc
...............................123
1.4
Example: the ifc-forms of the group SLn
......................125
§2.
Fields of dimension
< 1......................................128
2.1
Linear groups: summary of known results
.....................128
2.2
Vanishing of H1 for connected linear groups
..................130
2.3
Steinberg s theorem
........................................132
2.4
Rational points on homogeneous spaces
......................134
§3.
Fields of dimension
< 2......................................139
3.1
Conjecture II
.............................................139
3.2
Examples
.................................................140
X Table of Contents
§4.
Finiteness theorems
.........................................142
4.1
Condition (F)
.............................................142
4.2
Fields of type (F)
.........................................143
4.3
Finiteness of the cohomology of linear groups
.................144
4.4
Finiteness of orbits
........................................146
4.5
The case Jfc
=
R
...........................................147
4.6
Algebraic number fields
(Boreľs
theorem)
....................149
4.7
A counter-example to the
Hasse
principle
...................149
Bibliographic remarks for Chapter III
...........................154
Appendix
1.
Regular elements of
semisimple
groups (by R. Steinberg)
155
1.
Introduction and statement of results
.........................155
2.
Some recollections
..........................................158
3.
Some characterizations of regular elements
.....................160
4.
The existence of regular unipotent elements
....................163
5.
Irregular elements
..........................................166
6.
Class functions and the variety of regular classes
...............168
7.
Structure of
N.............................................172
8.
Proof of
1.4
and
1.5 ........................................176
9.
Rationality of
N...........................................178
10.
Some cohomological applications
.............................184
11.
Added in proof
.............................................185
Appendix
2.
Complements on Galois cohomology
...............187
1.
Notation
..................................................187
2.
The orthogonal case
........................................188
3.
Applications and examples
..................................189
4.
Injectivity problems
........................................192
5.
The trace form
.............................................193
6.
Bayer-Lenstra theory: self-dual normal bases
...................194
7.
Negligible cohomology classes
................................196
Bibliography
.....................................................199
Index
............................................................209
|
| any_adam_object | 1 |
| author | Serre, Jean-Pierre 1926- |
| author_GND | (DE-588)142283126 |
| author_facet | Serre, Jean-Pierre 1926- |
| author_role | aut |
| author_sort | Serre, Jean-Pierre 1926- |
| author_variant | j p s jps |
| building | Verbundindex |
| bvnumber | BV013910019 |
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| classification_tum | MAT 126f |
| ctrlnum | (OCoLC)48003273 (DE-599)BVBBV013910019 |
| dewey-full | 512/.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.74 |
| dewey-search | 512/.74 |
| dewey-sort | 3512 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Corr. 2. printing of the 1. Engl. ed. of 1997 |
| format | Book |
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| id | DE-604.BV013910019 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:54:15Z |
| institution | BVB |
| isbn | 9783540421924 3540421920 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009517570 |
| oclc_num | 48003273 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-19 DE-BY-UBM DE-11 DE-91G DE-BY-TUM DE-188 DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-19 DE-BY-UBM DE-11 DE-91G DE-BY-TUM DE-188 DE-83 |
| physical | X, 210 S |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Serre, Jean-Pierre 1926- Verfasser (DE-588)142283126 aut Cohomologie galoisienne Galois cohomology Jean-Pierre Serre Corr. 2. printing of the 1. Engl. ed. of 1997 Berlin [u.a.] Springer 2002 X, 210 S txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Algèbre homologique Galois, Théorie de Nombres algébriques, Théorie des Algebra, Homological Algebraic number theory Galois theory Galois-Gruppe (DE-588)4155897-2 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Galois-Kohomologie (DE-588)4019172-2 gnd rswk-swf Kohomologie (DE-588)4031700-6 gnd rswk-swf Galois-Theorie (DE-588)4155901-0 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Galois-Kohomologie (DE-588)4019172-2 s DE-604 Kohomologie (DE-588)4031700-6 s Galois-Gruppe (DE-588)4155897-2 s 1\p DE-604 Endliche Gruppe (DE-588)4014651-0 s 2\p DE-604 Zahlentheorie (DE-588)4067277-3 s 3\p DE-604 Algebraische Zahlentheorie (DE-588)4001170-7 s 4\p DE-604 Galois-Theorie (DE-588)4155901-0 s 5\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009517570&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Serre, Jean-Pierre 1926- Galois cohomology Algèbre homologique Galois, Théorie de Nombres algébriques, Théorie des Algebra, Homological Algebraic number theory Galois theory Galois-Gruppe (DE-588)4155897-2 gnd Endliche Gruppe (DE-588)4014651-0 gnd Galois-Kohomologie (DE-588)4019172-2 gnd Kohomologie (DE-588)4031700-6 gnd Galois-Theorie (DE-588)4155901-0 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4155897-2 (DE-588)4014651-0 (DE-588)4019172-2 (DE-588)4031700-6 (DE-588)4155901-0 (DE-588)4001170-7 (DE-588)4067277-3 |
| title | Galois cohomology |
| title_alt | Cohomologie galoisienne |
| title_auth | Galois cohomology |
| title_exact_search | Galois cohomology |
| title_full | Galois cohomology Jean-Pierre Serre |
| title_fullStr | Galois cohomology Jean-Pierre Serre |
| title_full_unstemmed | Galois cohomology Jean-Pierre Serre |
| title_short | Galois cohomology |
| title_sort | galois cohomology |
| topic | Algèbre homologique Galois, Théorie de Nombres algébriques, Théorie des Algebra, Homological Algebraic number theory Galois theory Galois-Gruppe (DE-588)4155897-2 gnd Endliche Gruppe (DE-588)4014651-0 gnd Galois-Kohomologie (DE-588)4019172-2 gnd Kohomologie (DE-588)4031700-6 gnd Galois-Theorie (DE-588)4155901-0 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Algèbre homologique Galois, Théorie de Nombres algébriques, Théorie des Algebra, Homological Algebraic number theory Galois theory Galois-Gruppe Endliche Gruppe Galois-Kohomologie Kohomologie Galois-Theorie Algebraische Zahlentheorie Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009517570&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT serrejeanpierre cohomologiegaloisienne AT serrejeanpierre galoiscohomology |