Unramified Brauer Group and Its Applications

Cover......Page 1
Title page......Page 4
Contents......Page 8
Preface......Page 12
Notation......Page 16
Part I . Preliminaries on Galois cohomology......Page 20
1.1. Definition and basic properties......Page 22
1.2. Behavior under change of group......Page 30
1.3. Cohomology of finite groups......Page 35
1.4. Permutation and stably permutation modules......Page 36
2.1. Descent for fibered categories......Page 38
2.2. Forms and first Galois cohomology......Page 45
2.3. Cohomology of profinite groups......Page 50
2.4. Cohomology of the absolute Galois group......Page 55
2.5. Picard group as a stably permutation module......Page 57
2.6. Torsors......Page 59
2.7. Cohomology of the inverse limit......Page 60
2.8. Further reading......Page 62
Part II . Brauer group......Page 64
3.1. Definition and basic properties......Page 66
3.2. Brauer group and arithmetic properties of fields......Page 75
3.3. Brauer group and Severi–Brauer varieties......Page 77
3.4. Further reading......Page 82
4.1. Complete discrete valuation fields......Page 84
4.2. Brauer group of a complete discrete valuation field......Page 87
4.3. Unramified Brauer group of a function field......Page 92
4.4. Brauer group of a variety......Page 94
4.5. Geometric meaning of the residue map......Page 97
4.6. Further reading......Page 102
Part III . Applications to rationality problems......Page 104
5.1. Geometric data......Page 106
5.2. Construction of a group......Page 107
5.3. Further reading......Page 110
6.1. Invariants of quadrics......Page 112
6.2. Geometric meaning of invariants of quadrics......Page 115
6.3. Degenerations of quadrics......Page 117
6.4. Further reading......Page 118
7.1. More on the unramified Brauer group......Page 120
7.2. Families of two-dimensional quadrics......Page 121
7.3. Construction of a geometric example......Page 122
7.4. Some unirationality constructions......Page 124
7.5. Further reading......Page 128
8.1. Weil restriction......Page 130
8.2. Algebraic tori......Page 134
8.3. Algebraic tori and Galois modules......Page 136
8.4. Universal torsor......Page 138
8.5. Châtelet surfaces and stably permutation modules......Page 139
8.6. Further reading......Page 144
9.1. Plan of the construction......Page 146
9.2. The fields ��, ��’, and ��’......Page 147
9.3. Non-rational conic bundle......Page 148
9.4. Rational intersection of two quadrics......Page 149
9.5. Stable birational equivalence between �� and ��......Page 153
9.7. Further reading......Page 155
Part IV . The Hasse principle and its failure......Page 156
10.1. Preliminaries......Page 158
10.2. Quadrics over local fields......Page 159
10.3. Reduction to the case dim(��)=1......Page 161
10.4. The case dim(��)⩽1......Page 162
10.5. Other examples of the Hasse principle......Page 164
10.6. Further reading......Page 165
11.1. Definition of the Brauer–Manin obstruction......Page 166
11.2. Computation of the Brauer–Manin obstruction......Page 168
11.3. Brauer–Manin obstruction for a genus-one curve......Page 173
11.4. Further reading......Page 176
A.2. Sheaves in the étale topology......Page 178
A.3. Cohomology of étale sheaves of abelian groups......Page 179
A.4. First étale cohomology with non-abelian coefficients......Page 180
A.5. Kummer sequence......Page 181
A.7. The case of a complex algebraic variety......Page 183
Bibliography......Page 186
Index......Page 196
Back Cover......Page 201