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Unramified Brauer Group and Its Applications [Kietas viršelis]

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  • Formatas: Hardback, 200 pages, aukštis x plotis: 254x178 mm, weight: 535 g
  • Serija: Translations of Mathematical Monographs
  • Išleidimo metai: 01-Jun-2018
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470440725
  • ISBN-13: 9781470440725
Kitos knygos pagal šią temą:
Unramified Brauer Group and Its ApplicationsDidesnis paveikslėlis
  • Formatas: Hardback, 200 pages, aukštis x plotis: 254x178 mm, weight: 535 g
  • Serija: Translations of Mathematical Monographs
  • Išleidimo metai: 01-Jun-2018
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470440725
  • ISBN-13: 9781470440725
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781470440725.html
Raktažodžiai:
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and etale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.
Preface xi
Notation xv
Part I Preliminaries on Galois cohomology
1(44)
Chapter 1 Group Cohomology
3(16)
1.1 Definition and basic properties
3(8)
1.2 Behavior under change of group
11(5)
1.3 Cohomology of finite groups
16(1)
1.4 Permutation and stably permutation modules
17(2)
Chapter 2 Galois Cohomology
19(26)
2.1 Descent for fibered categories
19(7)
2.2 Forms and first Galois cohomology
26(5)
2.3 Cohomology of profinite groups
31(5)
2.4 Cohomology of the absolute Galois group
36(2)
2.5 Picard group as a stably permutation module
38(2)
2.6 Torsors
40(1)
2.7 Cohomology of the inverse limit
41(2)
2.8 Further reading
43(2)
Part II Brauer group
45(40)
Chapter 3 Brauer Group of a Field
47(18)
3.1 Definition and basic properties
47(9)
3.2 Brauer group and arithmetic properties of fields
56(2)
3.3 Brauer group and Severi-Brauer varieties
58(5)
3.4 Further reading
63(2)
Chapter 4 Residue Map on a Brauer Group
65(20)
4.1 Complete discrete valuation fields
65(3)
4.2 Brauer group of a complete discrete valuation field
68(5)
4.3 Unramified Brauer group of a function field
73(2)
4.4 Brauer group of a variety
75(3)
4.5 Geometric meaning of the residue map
78(5)
4.6 Further reading
83(2)
Part III Applications to rationality problems
85(52)
Chapter 5 Example of a Unirational Non-rational Variety
87(6)
5.1 Geometric data
87(1)
5.2 Construction of a group
88(3)
5.3 Further reading
91(2)
Chapter 6 Arithmetic of Two-dimensional Quadrics
93(8)
6.1 Invariants of quadrics
93(3)
6.2 Geometric meaning of invariants of quadrics
96(2)
6.3 Degenerations of quadrics
98(1)
6.4 Further reading
99(2)
Chapter 7 Non-rational Double Covers of P3
101(10)
7.1 More on the unramified Brauer group
101(1)
7.2 Families of two-dimensional quadrics
102(1)
7.3 Construction of a geometric example
103(2)
7.4 Some unirationality constructions
105(4)
7.5 Further reading
109(2)
Chapter 8 Weil Restriction and Algebraic Tori
111(16)
8.1 Weil restriction
111(4)
8.2 Algebraic tori
115(2)
8.3 Algebraic tori and Galois modules
117(2)
8.4 Universal torsor
119(1)
8.5 Chatelet surfaces and stably permutation modules
120(5)
8.6 Further reading
125(2)
Chapter 9 Example of a Non-rational Stably Rational Variety
127(10)
9.1 Plan of the construction
127(1)
9.2 The fields K, k', and K'
128(1)
9.3 Non-rational conic bundle
129(1)
9.4 Rational intersection of two quadrics
130(4)
9.5 Stable birational equivalence between X and V
134(2)
9.6 One more construction of stable rationality
136(1)
9.7 Further reading
136(1)
Part IV The Hasse principle and its failure
137(22)
Chapter 10 Minkowski-Hasse Theorem
139(8)
10.1 Preliminaries
139(1)
10.2 Quadrics over local fields
140(2)
10.3 Reduction to the case dim(Q) = 1
142(1)
10.4 The case dim(Q) ≤ 1
143(2)
10.5 Other examples of the Hasse principle
145(1)
10.6 Further reading
146(1)
Chapter 11 Brauer-Manin Obstruction
147(12)
11.1 Definition of the Brauer-Manin obstruction
147(2)
11.2 Computation of the Brauer Manin obstruction
149(5)
11.3 Brauer-Manin obstruction for a genus-one curve
154(3)
11.4 Further reading
157(2)
Appendix A Etale Cohomology 159(8)
A.1 Etale coverings
159(1)
A.2 Sheaves in the etale topology
159(1)
A.3 Cohomology of etale sheaves of abelian groups
160(1)
A.4 First etale cohomology with non-abelian coefficients
161(1)
A.5 Kummer sequence
162(2)
A.6 Brauer group
164(1)
A.7 The case of a complex algebraic variety
164(3)
Bibliography 167(10)
Index 177
Sergey Gorchinskiy, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.

Constantin Shramov, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.