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Viser: Galois Cohomology
Galois Cohomology
Jean-Pierre Serre, N. N. Dutta, Friederike Hammar, D. Haralampidis, N. Ganesh Karanth og P. Ion
(2001)
Sprog: Engelsk
Detaljer om varen
- 1. Udgave
- Hardback
- Udgiver: Springer Berlin / Heidelberg (Oktober 2001)
- Forfattere: Jean-Pierre Serre, N. N. Dutta, Friederike Hammar, D. Haralampidis, N. Ganesh Karanth og P. Ion
- ISBN: 9783540421924
1.1 Definition.-
1.2 Subgroups.-
1.3 Indices.-
1.4 Pro-p-groups and Sylow p-subgroups.-
1.5 Pro-p-groups.- §2. Cohomology.-
2.1 Discrete G-modules.-
2.2 Cochains, cocycles, cohomology.-
2.3 Low dimensions.-
2.4 Fimctoriality.-
2.5 Induced modules.-
2.6 Complements.- §3. Cohomological dimension.-
3.1 p-cohomological dimension.-
3.2 Strict cohomological dimension.-
3.3 Cohomological dimension of subgroups and extensions.-
3.4 Characterization of the profinite groups G such that cdp(G) ?
1.-
3.5 Dualizing modules.- §4. Cohomology of pro-p-groups.-
4.1 Simple modules.-
4.2 Interpretation of H1: generators.-
4.3 Interpretation of H2: relations.-
4.4 A theorem of Shafarevich.-
4.5 Poincaré groups.- §5. Nonabelian cohomology.-
5.1 Definition of H0 and of H1.-
5.2 Principal homogeneous spaces over A -- a new definition of H1(G,A).-
5.3 Twisting.-
5.4 The cohomology exact sequence associated to a subgroup.-
5.5 Cohomology exact sequence associated to a normal subgroup.-
5.6 The case of an abelian normal subgroup.-
5.7 The case of a central subgroup.-
5.8 Complements.-
5.9 A property of groups with cohomological dimension ?
1.- II. Galois cohomology, the commutative case.- §1. Generalities.-
1.1 Galois cohomology.-
1.2 First examples.- §2. Criteria for cohomological dimension.-
2.1 An auxiliary result.-
2.2 Case when p is equal to the characteristic.-
2.3 Case when p differs from the characteristic.- §3. Fields of dimension ?1.-
3.1 Definition.-
3.2 Relation with the property (C1).-
3.3 Examples of fields of dimension ?
1.- §4. Transition theorems.-
4.1 Algebraic extensions.-
4.2 Transcendental extensions.-
4.3 Local fields.-
4.4 Cohomological dimension of the Galois group of an algebraic number field.-
4.5 Property (Cr).- §5. p-adic fields.-
5.1 Summary of known results.-
5.2 Cohomology of finite Gk-modules.-
5.3 First applications.-
5.4 The Euler-Poincaré characteristic (elementary case).-
5.5 Unramified cohomology.-
5.6 The Galois group of the maximal p-extension of k.-
5.7 Euler-Poincaré characteristics.-
5.8 Groups of multiplicative type.- §6. Algebraic number fields.-
6.1 Finite modules -- definition of the groups Pi(k, A).-
6.2 The finiteness theorem.-
6.3 Statements of the theorems of Poitou and Tate.- III. Nonabelian Galois cohomology.- §1. Forms.-
1.1 Tensors.-
1.2 Examples.-
1.3 Varieties, algebraic groups, etc.-
1.4 Example: the k-forms of the group SLn.- §2. Fields of dimension ?
1.-
2.1 Linear groups: summary of known results.-
2.2 Vanishing of H1 for connected linear groups.-
2.3 Steinberg's theorem.-
2.4 Rational points on homogeneous spaces.- §3. Fields of dimension ?
2.-
3.1 Conjecture II.-
3.2 Examples.- §4. Finiteness theorems.-
4.1 Condition (F).-
4.2 Fields of type (F).-
4.3 Finiteness of the cohomology of linear groups.-
4.4 Finiteness of orbits.-
4.5 The case k = R.-
4.6 Algebraic number fields (Borel's theorem).-
4.7 A counter-example to the "Hasse principle".