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I Basic Definitions
I.1 Affine Schemes
I.1.1 Schemes as Sets
I.1.2 Schemes as Topological Spaces
I.1.3 An Interlude on Sheaf Theory References for the Theory of Sheaves
I.1.4 Schemes as Schemes (Structure Sheaves)
I.2 Schemes in General
I.2.1 Subschemes
I.2.2 The Local Ring at a Point
I.2.3 Morphisms
I.2.4 The Gluing Construction Projective Space
I.3 Relative Schemes
I.3.1 Fibered Products
I.3.2 The Category of S-Schemes
I.3.3 Global Spec
I.4 The Functor of Points
II Examples
II.1 Reduced Schemes over Algebraically Closed Fields
II. 1.1 Affine Spaces
II.1.2 Local Schemes
II.2 Reduced Schemes over Non-Algebraically Closed Fields
II.3 Nonreduced Schemes
II.3.1 Double Points
II.3.2 Multiple Points Degree and Multiplicity
II.3.3 Embedded Points Primary Decomposition
II.3.4 Flat Families of Schemes
Limits
Examples
Flatness
II.3.5 Multiple Lines
II.4 Arithmetic Schemes
II.4.1 Spec Z
II.4.2 Spec of the Ring of Integers in a Number Field
II.4.3 Affine Spaces over Spec Z
II.4.4 A Conic over Spec Z
II.4.5 Double Points in Al
III Projective Schemes
III.1 Attributes of Morphisms
III.1.1 Finiteness Conditions
III.1.2 Properness and Separation
III.2 Proj of a Graded Ring
III.2.1 The Construction of Proj S
III.2.2 Closed Subschemes of Proj R
III.2.3 Global Proj
Proj of a Sheaf of Graded 0x-Algebras
The Projectivization P(¦Å) of a Coherent Sheaf ¦Å
III.2.4 Tangent Spaces and Tangent Cones
Affine and Projective Tangent Spaces
Tangent Cones
III.2.5 Morphisms to Projective Space
III.2.6 Graded Modules and Sheaves
III.2.7 Grassmannians
III.2.8 Universal Hypersurfaces
III.3 Invariants of Projective Schemes
III.3.1 Hilbert Functions and Hilbert Polynomials
1II.3.2 Flatness Il: Families of Projective Schemes
III.3.3 Free Resolutions
III.3.4 Examples
Points in the Plane
Examples: Double Lines in General and in p3
III.3.5 BEzout's Theorem
Multiplicity of Intersections
III.3.6 Hilbert Series
IV Classical Constructions
V Local Constructions
VI Schemes and Functors
References
Index