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Galois cohomology:

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Bibliographic Details
Main Author:	Serre, Jean-Pierre 1926- (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:	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
Links:	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
Physical Description:	X, 210 S
ISBN:	9783540421924 3540421920

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Record in the Search Index

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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
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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

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Call Number:	0040 MAT 126f 2011 A 8129 Floor plan
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