Arithmetic duality theorems / J.S. Milne.

Format:

Book

Author/Creator:

Milne, J. S., 1942-

Series:

Perspectives in mathematics ; vol. 1.

Perspectives in mathematics ; vol. 1

Language:

English

Subjects (All):

Algebraic fields.

Homology theory.

Duality theory (Mathematics).

Physical Description:

x, 419 pages ; 24 cm.

Place of Publication:

Boston : Academic Press, [1986]

Summary:

This volume presents for the first time complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. Chapter 1 is devoted to an exposition of these theorems in the Galois cohomology of number fields announced by Tate in 1962 and describes later work in the same area. The discussion assumes only a knowledge of basic Galois cohomology and class field theory.

Chapter 2 focuses on the work of Artin and Verdier who re-interpreted and developed Tate's ideas in the framework of etale cohomology; some of the more recent developments in this area are also covered.

Finally, in Chapter 3, which contains a number of new results, it is shown how flat cohomology is needed in order to prove and to apply duality theorems in the case of groups which have torsion of order divisible by one of the residue characteristics.

Contents:

Chapter I Galois Cohomology 1

1 Duality relative to a class formation 19

2 Local fields 31

3 Abelian varieties over local fields 49

4 Global fields 60

5 Global Euler-Poincare characteristics 81

6 Abelian varieties over global fields 89

7 An application to the conjecture of Birch and Swinnerton-Dyer 114

8 Abelian class field theory, in the sense of Langlands 123

9 Other applications 143

Appendix A Class field theory for function fields 154

Chapter II Etale Cohomology 171

1 Local results 181

2 Global results: preliminary calculations 200

3 Global results: the main theorem 216

4 Global results: complements 230

5 Global results: abelian schemes 242

6 Global results: singular schemes 251

7 Global results: higher dimensions 254

Chapter III Flat Cohomology 267

1 Local results: mixed characteristic, finite group schemes 285

2 Local results: mixed characteristic, abelian varieties 303

3 Global results: number field case 311

4 Local results: mixed characteristic, perfect residue field 318

5 Two exact sequences 329

6 Local fields of characteristic p 337

7 Local results: equicharacteristic 348

8 Global results: curves over finite fields, finite sheaves 360

9 Global results: curves over finite fields, Neron models 367

10 Local results: equicharacteristic, perfect residue field 374

11 Global results: curves over perfect fields 379

Appendix A Embedding finite group schemes 383

Appendix B Extending finite group schemes 390

Appendix C Biextensions and Neron models 395.

Notes:

Includes index.

Bibliography: pages [410]-419.

ISBN:

0124980406

OCLC:

14356210