Introduction to Algebraic Geometry

Preface
Introduction
Contents
1 Beginning Concepts
1 The Definition of Algebraic Varieties
1.1 Affine Algebraic Sets
1.1.1 Complex Numbers
1.1.2 Nullstellensatz
1.2 Zariski Topology
1.2.1 Topology
1.2.2 Zariski and Analytic Topology
1.3 Affine and Projective Varieties
1.3.1 Affine and Quasi-Affine Varieties
1.3.2 Projective and Quasi-Projective Varieties
1.4 Regular Functions on Different Types of Varieties
1.4.1 Regular Functions on a Quasiaffine and Quasiprojective Variety
1.4.2 Regular Functions on AnC
1.4.3 Regular Functions on an Affine Variety
1.4.4 Regular Functions on the Complement of a Set of Zeros Z(f)
1.4.5 Regular Functions on a Quasiaffine Variety
1.4.6 Regular Functions on a Projective Variety
1.5 Sheaves
2 Categories, and the Category of Algebraic Varieties
2.1 Categories, Functors and Algebraic Structures
2.1.1 The Definition of a Category, and an Example: The Category of Sets
2.1.2 Categories of Algebraic Structures
2.1.3 Isomorphisms
2.1.4 Functors and Natural Transformations
2.2 Categories: Topological Spaces and Algebraic Varieties
2.2.1 The Category of Topological Spaces
2.2.2 The Category of Algebraic Varieties
2.3 The Morphisms into an Affine Variety
2.3.1 Quasiaffine Varieties which are not Isomorphic to Affine Varieties
2.3.2 Quasiaffine Varieties which are Isomorphic to Affine Varieties
3 Rational Maps, Smooth Maps and Dimension
3.1 Definition of a Rational Map
3.2 The Field of Rational Functions
3.2.1 The Category of Varieties and Dominant Rational Maps
3.3 Standard Smooth Homomorphisms of Commutative Rings
3.4 Smooth Morphisms of Varieties
3.4.1 Smooth Varieties
3.5 Regular Rings and Dimension
3.5.1 Smooth Varieties and Regular Rings
4 Computing with Polynomials
4.1 Divisibility of Polynomials
4.2 Gröbner Basis
4.3 Nullstellensatz
5 Introduction to Commutative Algebra
5.1 Primary Decomposition
5.2 Artinian Rings
5.3 Dimension
5.4 Regular Local Rings
5.5 Dimension of Affine Varieties
6 Exercises
2 Schemes
1 Sheaves and Schemes
1.1 Sheaves Revisited
1.2 Ringed Spaces and Locally Ringed Spaces
1.3 Schemes
1.4 Category Theory Revisited: Adjoints, Limits and Colimits, Universality
1.5 Abelian Categories: Abelian Sheaves
2 Beginning Properties and Examples of Schemes
2.1 Connected, Irreducible, Reduced and Integral Schemes
2.2 Properties of Affine Schemes
2.3 Open and Closed Subschemes
2.4 Limits of Schemes
2.5 Gluing of Schemes: Colimits
2.6 A Diagram of Schemes Which Does Not Have a Limit
2.7 Proj Schemes
2.8 The Affine and Projective Space Over a Scheme: Projective Schemes
3 Finiteness Properties of Schemes and Morphisms of Schemes
3.1 Quasicompactness
3.2 Noetherian Schemes
3.3 Morphisms Locally of Finite Type and Morphisms of Finite Type
3.4 Finite Morphisms
4 Exercises
3 Properties of Schemes
1 Separated Schemes and Morphisms
1.1 Hausdorff and T1 Topological Spaces
1.2 The Product of Topological Spaces: Reformulating the Hausdorff Property
1.3 Separated Schemes and Morphisms
2 Universally Closed Schemes and Morphisms
2.1 More Facts About Compact and Hausdorff Spaces
2.2 Universally Closed Schemes and Morphisms
2.3 Specialization
2.4 Valuation Rings
2.5 Valuation Criteria for Universally Closed and Separated Morphisms
2.6 More Observations on Universally Closed and Separated Morphisms
3 Regular Schemes and Smooth Morphisms
3.1 Regular Schemes
3.2 Smooth Morphisms
3.3 Smooth Morphisms Over Regular Schemes
3.4 Étale Schemes Over a Field
4 Abstract Varieties
4.1 The Definition of an Algebraic Variety
4.2 A Classification of Smooth Curves
4.3 The Role of Closed Points
5 The Galois Group and the Fundamental Group
5.1 Varieties Over Perfect Fields and G-Sets
5.2 Some Details on G-Sets
5.3 Closed Points as Galois Group Orbits
5.4 The Absolute Galois Group: Profinite Groups
5.5 The Fundamental Group of a Topological Space
5.6 From Coverings to the Fundamental Group: The Étale Fundamental Group
5.6.1 Regular Coverings
5.6.2 The Étale Case
5.7 Finite Fields and the Weil Conjectures
6 Exercises
4 Sheaves of Modules
1 Sheaves of Modules
1.1 Presheaves and Sheaves Valued in a Category
1.1.2 An alternate characterization of sheafification
1.2 The Effect of Continuous Maps on Sheaves
1.3 Sheaves of Modules
1.4 The Effect of Continuous Maps on Sheaves of Modules
1.5 Biproduct, Tensor Product and Hom of Sheaves of Modules
2 Quasicoherent and Coherent Sheaves
2.1 Invertible Sheaves, Picard Group, Locally Free Sheaves, Algebraic Vector Bundles, Algebraic K-Theory
2.2 Quasicoherent and Coherent Sheaves of Modules
2.3 Sheaves of Ideals
2.4 Quasicoherent Sheaves on Proj Schemes
3 Divisors
3.1 First Cohomology
3.2 Cartier Divisors
3.3 Weil Divisors
3.3.9 Example: Divisors on Dedekind Domains
3.4 Examples and Calculations
3.5 Very Ample Line Bundles
3.6 Blow-ups
4 Exercises
5 Introduction to Cohomology
1 De Rham Cohomology in Analysis
1.1 Smooth and Complex Manifolds
1.2 Differential Forms
1.3 De Rham Cohomology
1.3.3 The Difference Between Vector Fields and 1-Forms
1.4 The de Rham Complex of a Complex Manifold
1.4.1 The Holomorphic de Rham Complex
2 Derived Categories and Sheaf Cohomology
2.1 Derived Categories
2.2 Properties of Abelian Categories
2.3 The Derived Category of an Abelian Category
2.3.1 Projective and Injective Objects
2.3.2 The Homotopy Category of Chain Complexes
2.3.5 Introduction to homological algebra
2.3.9 Cell Chain Complexes
2.3.11 Proof of (1) and (2) of Theorem 2.3.4
2.3.12 Proof of (3) and (4) of Theorem 2.3.4
2.3.15 Derived Functors
2.3.18 Derived Functors in an Abelian Category
2.4 Examples of Abelian Categories
2.4.1 Abelian Groups and Modules
2.4.3 Definition of TorR(M,N)
2.5 Sheaf Cohomology
2.5.2 Functoriality of Sheaf Cohomology
2.5.3 Flasque and Soft Sheaves
2.5.7 Proof of Proposition 2.5.4
2.6 A Cohomological Criterion for Regular Local Rings
2.6.7 Proof of Theorem 2.6.1
3 Singular Homology and Cohomology
3.1 The Singular Chain and Cochain Complex
3.2 Eilenberg-Steenrod Axioms
3.3 Proof of the Homotopy and Excision Axioms
3.4 The Homology of Spheres
3.5 Universal Coefficients and Künneth Theorem
3.6 CW-Homology
3.6.4 Example 3.6.1 Continued
3.7 Spectral Sequences
3.7.1 The Spectral Sequence of a Filtered Chain Complex
3.7.2 Proof of Theorem 3.6.3
3.8 Singular Cohomology vs. Sheaf Cohomology
3.8.2 Sheaves Do Not Have Enough Projectives
4 Exercises
6 Cohomology in Algebraic Geometry
1 Hodge Theory
1.1 Dolbeault Cohomology
1.2 Riemann and Hermitian Metrics
1.3 Hodge Theorem
1.4 Kähler Manifolds
1.5 The Lefschetz Decomposition
1.6 Examples
1.6.1 The Fubini-Study Metric
1.6.3 The Cohomology of the Complex Projective Space
1.6.4 Elliptic Curves over C
2 Algebraic de Rham Cohomology
2.1 The Algebraic de Rham Complex
2.1.2 Kähler Differentials
2.1.4 Algebraic de Rham Cohomology
2.1.5 The Non-smooth Case
2.2 Quasicoherent Cohomology
2.2.4 Quasicoherent Čech Cohomology
2.3 Algebraic de Rham Cohomology of Projective Varieties—GAGA
2.3.3 Serre's GAGA Theorem
2.4 Algebraic and Analytic de Rham Cohomology of Smooth Varieties over C
2.5 Examples
2.5.1 The Affine and Projective Lines
2.5.2 Algebraic Dolbeault and de Rham Cohomology of the Projective Space
2.5.3 The Algebraic de Rham Cohomology of an Elliptic Curve
2.5.4 Non-canonicity over R
3 Crystalline Cohomology
3.1 Witt Vectors
3.1.5 Truncated Witt Vectors
3.1.6 The Witt Vectors of Fp
3.2 The de Rham-Witt Complex and Crystalline Cohomology
3.2.1 Verschiebung and Frobenius
3.3 Crystalline Cohomology of Pn
4 Étale Cohomology
4.1 Grothendieck Topology
4.2 Geometric Points
4.2.3 Stalks of the Structure Sheaf
4.2.4 The Kummer Exact Sequence
4.2.5 The Étale Topology on Spec(k)
4.3 Étale Cohomology
4.3.1 Étale Cohomology of Spec(k) for a Field k
4.4 Étale Cohomology of Curves
4.4.2 Cohomology of Function Fields of Transcendence Degree 1
5 Motivic Cohomology
5.1 A1-Homotopy Invariance
5.1.3 Proof of Proposition 5.1.1
5.2 Finite Correspondences and the Definition of Motivic Cohomology
5.3 Some Computations of Motivic Cohomology
5.3.2 Milnor K-Theory
5.4 Relation with étale Cohomology—Voevodsky's Theorem
6 Exercises
Bibliography
Index