Galois cohomology /

Galois cohomology /

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Bibliographic Details
Author / Creator:Serre, Jean Pierre. 1926-
Uniform title:Cohomologie galoisienne. English
Imprint:Berlin ; New York : Springer, c1997.
Description:x, 210 p. ; 25 cm.
Language:English
Subject:
Algebraic number theory.
Galois theory.
Algebra, Homological.
Algebra, Homological.
Algebraic number theory.
Galois theory.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/2759103
Hidden Bibliographic Details
ISBN:3540619909 (hardcover : alk. paper)
Notes:Includes bibliographical references (p. [199]-207) and index.
Table of Contents:
  • Foreword
  • Chapter I. Cohomology of profinite groups
  •  1. Profinite groups
  • 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. Functoriality
  • 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 cd p (G) ≤ 1
  • 3.5. Dualizing modules
  •  4. Cohomology of pro-p-groups
  • 4.1. Simple modules
  • 4.2. Interpretation of H 1 : generators
  • 4.3. Interpretation of H 2 : relations
  • 4.4. A theorem of Shafarevich
  • 4.5. Poincaré groups
  •  5. Nonabelian cohomology
  • 5.1. Definition of H 0 and of H 1
  • 5.2. Principal homogeneous spaces over A - a new definition of H 1 (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 &leq 1
  • Bibliographic remarks for Chapter I
  • Appendix 1. J. Tate - Some duality theorems
  • Appendix 2. The Golod-Shafarevich inequality
  • 1. The statement
  • 2. Proof
  • Chapter 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 (C 1 )
  • 3.3. Examples of fields of dimension &leq 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 (C r )
  •  5. p-adic fields
  • 5.1. Summary of known results
  • 5.2. Cohomology of finite G k -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 P i (k, A)
  • 6.2. The finiteness theorem
  • 6.3. Statements of the theorems of Poitou and Tate
  • Bibliographic remarks for Chapter II
  • Appendix. Galois cohomology of purely transcendental extensions
  • 1. An exact sequence
  • 2. The local case
  • 3. Algebraic curves and function fields in one variable
  • 4. The case K= k(T)
  • 5. Notation
  • 6. Killing by base change
  • 7. Manin conditions, weak approximation and Schinzel's hypothesis
  • 8. Sieve bounds
  • Chapter 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 SL n
  •  2. Fields of dimension ≤ 1
  • 2.1. Linear groups: summary of known results
  • 2.2. Vanishing of H 1 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"
  • Bibliographic remarks for Chapter III
  • Appendix 1. Regular elements of semisimple groups
  • 1. Introduction and statement of results
  • 2. Some recollections
  • 3. Some characterizations of regular elements
  • 4. The existence of regular unipotent elements
  • 5. Irregular elements
  • 6. Class functions and the variety of regular classes
  • 7. Structure of N
  • 8. Proof of 1.4 and 1.5
  • 9. Rationality of N
  • 10. Some cohomological applications184
  • 11. Added in proof
  • Appendix 2. Complements on Galois cohomology
  • 1. Notation
  • 2. The orthogonal case
  • 3. Applications and examples
  • 4. Injectivity problems
  • 5. The trace form
  • 6. Bayer-Lenstra theory: self-dual normal bases
  • 7. Negligible cohomology classes
  • Bibliography
  • Index

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