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Galois Cohomology [Hardback]

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  • Formāts: Hardback, 228 pages, height x width x depth: 234x156x14 mm, weight: 499 g, bibliography
  • Izdošanas datums: 01-Aug-1997
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540619909
  • ISBN-13: 9783540619901
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Galois CohomologyPalielināt
  • Formāts: Hardback, 228 pages, height x width x depth: 234x156x14 mm, weight: 499 g, bibliography
  • Izdošanas datums: 01-Aug-1997
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540619909
  • ISBN-13: 9783540619901
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783540619901.html
This is an updated English translation of "Cohomologie Galoisien-ne" published more than 30 years ago, as one of the very first "Lecture Notes in Mathematics". It includes a reproduction of an influential paper of R. Steinberg, together with some new material and an expanded bibliography.
Foreword V
Chapter I. Cohomology of profinite groups
3(68)
1. Profinite groups
3(7)
1.1 Definition
3(1)
1.2 Subgroups
4(1)
1.3 Indices
5(1)
1.4 Pro-p-groups and Sylow p-subgroups
6(1)
1.5 Pro-p-groups
7(3)
2. Cohomology
10(7)
2.1 Discrete G-modules
10(1)
2.2 Cochains, cocycles, cohomology
10(1)
2.3 Low dimensions
11(1)
2.4 Functoriality
12(1)
2.5 Induced modules
13(1)
2.6 Complements
14(3)
3. Cohomological dimension
17(10)
3.1 p-cohomological dimension
17(1)
3.2 Strict cohomological dimension
18(1)
3.3 Cohomological dimension of subgroups and extensions
19(2)
3.4 Characterization of the profinite groups G such that cd(p)(G) XXX 1
21(3)
3.5 Dualizing modules
24(3)
4. Cohomology of pro-p-groups
27(18)
4.1 Simple modules
27(2)
4.2 Interpretation of H(1): generators
29(4)
4.3 Interpretation of H(2): relations
33(1)
4.4 A theorem of Shafarevich
34(4)
4.5 Poincare groups
38(7)
5. Nonabelian cohomology
45(15)
5.1 Definition of H(0) and of H(1)
45(1)
5.2 Principal homogeneous spaces over A - a new definition of H(1)(G, A)
46(1)
5.3 Twisting
47(3)
5.4 The cohomology exact sequence associated to a subgroup
50(1)
5.5 Cohomology exact sequence associated to a normal subgroup
51(2)
5.6 The case of an abelian normal subgroup
53(1)
5.7 The case of a central subgroup
54(2)
5.8 Complements
56(1)
5.9 A property of groups with cohomological dimension XXX 1
57(3)
Bibliographic remarks for
Chapter I
60(1)
Appendix
1. J. Tate - Some duality theorems
61(5)
Appendix
2. The Golod-Shafarevich inequality
66(5)
1. The statement
66(1)
2. Proof
67(4)
Chapter II. Galois cohomology, the commutative case
71(50)
1. Generalities
71(3)
1.1 Galois cohomology
71(1)
1.2 First examples
72(2)
2. Criteria for cohomological dimension
74(4)
2.1 An auxiliary result
74(1)
2.2 Case when p is equal to the characteristic
75(1)
2.3 Case when p differs from the characteristic
76(2)
3. Fields of dimension XXX 1
78(5)
3.1 Definition
78(1)
3.2 Relation with the property (C(1))
79(1)
3.3 Examples of fields of dimension XXX 1
80(1)
4. Transition theorems
83(7)
4.1 Algebraic extensions
83(1)
4.2 Transcendental extensions
83(2)
4.3 Local fields
85(2)
4.4 Cohomological dimension of the Galois group of an algebraic number field
87(1)
4.5 Property (C(r))
87(3)
5. p-adic fields
90(15)
5.1 Summary of known results
90(1)
5.2 Cohomology of finite G(k)-modules
90(3)
5.3 First applications
93(1)
5.4 The Euler-Poincare characteristic (elementary case)
93(1)
5.5 Unramified cohomology
94(1)
5.6 The Galois group of the maximal p-extension of k
95(1)
5.7 Euler-Poincare characteristics
99(4)
5.8 Groups of multiplicative type
102(3)
6. Algebraic number fields
105(4)
6.1 Finite modules - definition of the groups P(i)(k, A)
105(1)
6.2 The finiteness theorem
106(1)
6.3 Statements of the theorems of Poitou and Tate
107(2)
Bibliographic remarks for
Chapter II
109(1)
Appendix. Galois cohomology of purely transcendental extensions
110(11)
1. An exact sequence
110(1)
2. The local case
111(1)
3. Algebraic curves and function fields in one variable
112(1)
4. The case K = k(T)
113(1)
5. Notation
114(1)
6. Killing by base change
115(1)
7. Manin conditions, weak approximation and Schinzel's hypothesis
116(1)
8. Sieve bounds
117(4)
Chapter III. Nonabelian Galois cohomology
121(78)
1. Forms
121(7)
1.1 Tensors
121(2)
1.2 Examples
123(1)
1.3 Varieties, algebraic groups, etc
123(2)
1.4 Example: the k-forms of the group SL(n)
125(3)
2. Fields of dimension XXX 1
128(11)
2.1 Linear groups: summary of known results
128(2)
2.2 Vanishing of H(1) for connected linear groups
130(2)
2.3 Steinberg's theorem
132(2)
2.4 Rational points on homogeneous spaces
134(5)
3. Fields of dimension XXX 2
139(3)
3.1 Conjecture II
139(1)
3.2 Examples
140(2)
4. Finiteness theorems
142(12)
4.1 Condition (F)
142(1)
4.2 Fields of type (F)
143(1)
4.3 Finiteness of the cohomology of linear groups
144(1)
4.4 Finiteness of orbits
146(2)
4.5 The case k = R
147(2)
4.6 Algebraic number fields (Borel's theorem)
149(1)
4.7 A counter-example to the "Hasse principle"
149(5)
Bibliographic remarks for
Chapter III
154(1)
Appendix
1. Regular elements of semisimple groups (by R. Steinberg)
155(32)
1. Introduction and statement of results
155(3)
2. Some recollections
158(2)
3. Some characterizations of regular elements
160(3)
4. The existence of regular unipotent elements
163(3)
5. Irregular elements
166(2)
6. Class functions and the variety of regular classes
168(4)
7. Structure of N
172(4)
8. Proof of 1.4 and 1.5
176(2)
9. Rationality of N
178(6)
10. Some cohomological applications
184(1)
11. Added in proof
185(2)
Appendix
2. Complements on Galois cohomology
187(12)
1. Notation
187(1)
2. The orthogonal case
188(1)
3. Applications and examples
189(3)
4. Injectivity problems
192(1)
5. The trace form
193(1)
6. Bayer-Lenstra theory: self-dual normal bases
194(2)
7. Negligible cohomology classes
196(3)
Bibliography 199(10)
Index 209