The Brauer–Grothendieck Group

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Preface Contents Notation Chapter 1 Galois cohomology 1.1 Quaternion algebras and conics 1.1.1 Quaternions 1.1.2 Conics 1.2 The language of central simple algebras 1.2.1 Central simple algebras 1.2.2 Cyclic algebras 1.2.3 C1-fields 1.3 The language of Galois cohomology 1.3.1 Group cohomology and Galois cohomology 1.3.2 Galois descent 1.3.3 Cohomological description of the Brauer group 1.3.4 Cyclic algebras, cup-products and the Kummer sequence 1.4 Galois cohomology of discretely valued fields 1.4.1 Serre residue 1.4.2 Extensions of rings 1.4.3 Witt residue 1.4.4 Compatibility of residues 1.5 The Faddeev exact sequences Chapter 2 Étale cohomology 2.1 Topologies, sites, sheaves 2.1.1 Grothendieck topologies 2.1.2 Presheaves and sheaves 2.1.3 Direct and inverse images 2.1.4 Sheaves on the small étale site 2.2 Cohomology 2.2.1 Definition and basic properties 2.2.2 Passing to the limit 2.2.3 Étale and Galois cohomology 2.2.4 Standard spectral sequences 2.3 Cohomological purity 2.3.1 Absolute purity with torsion coefficients 2.3.2 The Gysin exact sequence 2.3.3 Cohomology of henselian discrete valuation rings 2.3.4 Gysin residue and functoriality 2.4 H1 with coefficients Z and Gm 2.5 The Picard group and the Picard scheme 2.6 Excellent rings Chapter 3 Brauer groups of schemes 3.1 The Brauer–Azumaya group 3.2 The Brauer–Grothendieck group 3.2.1 The Kummer exact sequence 3.2.2 The Mayer–Vietoris exact sequence 3.2.3 Passing to the reduced subscheme 3.3 Comparing the two Brauer groups, I 3.4 Localising elements of the Brauer group 3.5 Going over to the generic point 3.6 Schemes of dimension 1 3.6.1 Regular schemes of dimension 1 3.6.2 Singular schemes of dimension 1 3.7 Purity for the Brauer group 3.8 The Brauer group and finite morphisms Chapter 4 Comparing the two Brauer groups, II 4.1 The language of stacks 4.1.1 Fibred categories 4.1.2 Stacks 4.1.3 Algebraic spaces and algebraic stacks 4.1.4 Gerbes 4.1.5 Twisted sheaves 4.2 de Jong's proof of Gabber's theorem Chapter 5 Varieties over a field 5.1 The Picard group of a variety 5.1.1 Picard variety 5.1.2 Albanese variety and Albanese torsor 5.2 The geometric Brauer group 5.3 The Tate module of the Brauer group as a Galois representation 5.4 Algebraic and transcendental Brauer groups 5.4.1 The Picard group and the algebraic Brauer group 5.4.2 Geometric interpretation of differentials 5.4.3 Galois invariants of the geometric Brauer group 5.5 Projective varieties with Hi(X,OX)=0 5.6 The Picard and Brauer groups of curves 5.7 The Picard and Brauer groups of a product 5.7.1 The Picard group of a product 5.7.2 Topological Künneth formula in degrees 1 and 2 5.7.3 Künneth formula for étale cohomology in degrees 1 and 2 Chapter 6 Birational invariance 6.1 Affine and projective spaces 6.2 The unramified Brauer group 6.3 Examples of unramified classes 6.4 Zero-cycles and the Brauer group Chapter 7 Severi–Brauer varieties and hypersurfaces 7.1 Severi–Brauer varieties 7.1.1 Two applications of Severi–Brauer varieties 7.1.2 Torsors for tori as birational models of Severi–Brauer varieties 7.1.3 Morphisms to Severi–Brauer varieties 7.2 Projective quadrics 7.3 Some affine hypersurfaces Chapter 8 Singular schemes and varieties 8.1 The Brauer–Grothendieck group is not always a torsion group 8.2 Isolated singularities 8.3 Intersections of hypersurfaces 8.4 Projective cones 8.5 Singular curves and their desingularisation 8.6 Some examples Chapter 9 Varieties with a group action 9.1 Tori 9.2 Simply connected semisimple groups 9.3 Theorems of Bogomolov and Saltman 9.4 Homogeneous spaces over an arbitrary field Chapter 10 Schemes over local rings and fields 10.1 Split varieties and split fibres 10.1.1 Split varieties 10.1.2 Split fibres 10.2 Quadrics over a discrete valuation ring 10.2.1 Conics 10.2.2 Quadric surfaces 10.3 Schemes of dimension 2 10.4 Smooth proper schemes over a henselian discrete valuation ring 10.5 Varieties over a local field 10.5.1 Evaluation at rational and closed points 10.5.2 The index of a variety over a p-adic field 10.5.3 Finiteness results for the Brauer group 10.5.4 Unramified Brauer classes and evaluation at points Chapter 11 The Brauer group and families of varieties 11.1 The vertical Brauer group 11.2 Families of split varieties 11.3 Conic bundles 11.3.1 Conic bundles over a curve 11.3.2 Conic bundles over a complex surface 11.3.3 Variations on the Artin–Mumford example 11.4 Double covers 11.5 The universal family of cyclic twists Chapter 12 Rationality in a family 12.1 The specialisation method 12.1.1 Main theorem 12.1.2 Irrational conic bundles with smooth ramification 12.2 Quadric bundles over the complex plane 12.2.1 A special quadric bundle 12.2.2 Rationality is not deformation invariant Chapter 13 The Brauer–Manin set and the formal lemma 13.1 Number fields 13.1.1 Primes and approximation 13.1.2 Class field theory and the Brauer group Adèles and adelic points 13.2 The Hasse principle and approximation 13.3 The Brauer–Manin obstruction 13.3.1 The Brauer–Manin set 13.3.2 The structure of the Brauer–Manin set 13.3.3 Examples of Brauer–Manin obstruction 13.3.4 The Brauer–Manin set of a product 13.4 Harari's formal lemma Chapter 14 Rational points in the Brauer–Manin set? 14.1 Rationally connected varieties: a conjecture 14.2 Schinzel's hypothesis and additive combinatorics 14.2.1 Applications of Schinzel's hypothesis 14.2.2 Additive combinatorics enters 14.2.3 Hypothesis of Harpaz and Wittenberg 14.2.4 Main steps of the proof of Theorem 14.2.14 14.2.5 Fibrations with two non-split fibres and ramified descent 14.3 Beyond the Brauer–Manin obstruction 14.3.1 Insufficiency of the Brauer–Manin obstruction 14.3.2 Quadric bundles over a curve, I 14.3.3 Distinguished subsets of the adelic space 14.3.4 Quadric bundles over a curve, II 14.3.5 Curves, K3 surfaces, Enriques surfaces Chapter 15 The Brauer–Manin obstruction for zero-cycles 15.1 Local-to-global principles for zero-cycles 15.2 From rational points to zero-cycles 15.3 Salberger's method 15.4 A fibration theorem for zero-cycles Chapter 16 The Tate conjecture, abelian varieties and K3 surfaces 16.1 Tate conjecture for divisors 16.2 Abelian varieties 16.3 Varieties dominated by products 16.4 K3 surfaces 16.5 Kuga–Satake variety 16.6 Moduli spaces of K3 surfaces and Shimura varieties 16.7 Tate conjecture and Brauer group of K3 surfaces 16.8 Diagonal surfaces References Index List of symbols