The Brauer–Grothendieck Group

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