Algebraic Geometry 1: Schemes. With Examples and Exercises
By Ulrich Görtz, Torsten Wedhorn
1 Prevarieties
Affine algebraic sets
Affine algebraic sets as spaces with functions
Prevarieties
Projective varieties
Exercises
2 Spectrum of a Ring
Spectrum of a ring as a topological space
Excursion: Sheaves
Spectrum of a ring as a locally ringed space
Exercises
3 Schemes
Schemes
Examples of schemes
Basic properties of schemes and morphisms of schemes
Prevarieties as Schemes
Subschemes and Immersions
Exercises
4 Fiber products
Schemes as functors
Fiber products of schemes
Base change, Fibers of a morphism
Exercises
5 Schemes over fields
Schemes over a field which is not algebraically closed
Dimension of schemes over a field
Schemes over fields and extensions of the base field
Intersections of plane curves
Exercises
6 Local Properties of Schemes
The tangent space
Smooth morphisms
Regular schemes
Normal schemes
Exercises
7 Quasi-coherent modules
Excursion: OX-modules
Quasi-coherent modules on a scheme
Properties of quasi-coherent modules
Exercises
8 Representable Functors
Representable Functors
The example of the Grassmannian
Brauer-Severi schemes
Exercises
9 Separated morphisms
Diagonal of scheme morphisms and separated morphisms
Rational maps and function fields
Exercises
10 Finiteness Conditions
Finiteness conditions (noetherian case)
Finiteness conditions in the non-noetherian case
Schemes over inductive limits of rings
Constructible properties
Exercises
11 Vector bundles
Vector bundles and locally free modules
Flattening stratification for modules
Divisors
Vector bundles on P1
Exercises
12 Affine and proper morphisms
Affine morphisms
Finite and quasi-finite morphisms
Serre’s and Chevalley’s criteria to be affine
Normalization
Proper morphisms
Zariski’s main theorem
Exercises
13 Projective morphisms
Projective spectrum of a graded algebra
Embeddings into projective space
Blowing-up
Exercises
14 Flat morphisms and dimension
Flat morphisms
Properties of flat morphisms
Faithfully flat descent
Dimension and fibers of morphisms
Dimension and regularity conditions
Hilbert schemes
Exercises
15 One-dimensional schemes
Morphisms into and from one-dimensional schemes
Valuative criteria
Curves over fields
Divisors on curves
Exercises
16 Examples
Determinantal varieties
Cubic surfaces and a Hilbert modular surface
Cyclic quotient singularities
Abelian varieties
Exercises
A The language of categories
B Commutative Algebra
C Permanence for properties of morphisms of schemes
D Relations between properties of morphisms of schemes
E Constructible and open properties
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By Ulrich Görtz, Torsten Wedhorn
- Publisher: Vieweg+Teubner Verlag
- Number Of Pages: 615
- Publication Date: 2010-06
- ISBN-10 / ASIN: 3834806765
- ISBN-13 / EAN: 9783834806765
1 Prevarieties
Affine algebraic sets
Affine algebraic sets as spaces with functions
Prevarieties
Projective varieties
Exercises
2 Spectrum of a Ring
Spectrum of a ring as a topological space
Excursion: Sheaves
Spectrum of a ring as a locally ringed space
Exercises
3 Schemes
Schemes
Examples of schemes
Basic properties of schemes and morphisms of schemes
Prevarieties as Schemes
Subschemes and Immersions
Exercises
4 Fiber products
Schemes as functors
Fiber products of schemes
Base change, Fibers of a morphism
Exercises
5 Schemes over fields
Schemes over a field which is not algebraically closed
Dimension of schemes over a field
Schemes over fields and extensions of the base field
Intersections of plane curves
Exercises
6 Local Properties of Schemes
The tangent space
Smooth morphisms
Regular schemes
Normal schemes
Exercises
7 Quasi-coherent modules
Excursion: OX-modules
Quasi-coherent modules on a scheme
Properties of quasi-coherent modules
Exercises
8 Representable Functors
Representable Functors
The example of the Grassmannian
Brauer-Severi schemes
Exercises
9 Separated morphisms
Diagonal of scheme morphisms and separated morphisms
Rational maps and function fields
Exercises
10 Finiteness Conditions
Finiteness conditions (noetherian case)
Finiteness conditions in the non-noetherian case
Schemes over inductive limits of rings
Constructible properties
Exercises
11 Vector bundles
Vector bundles and locally free modules
Flattening stratification for modules
Divisors
Vector bundles on P1
Exercises
12 Affine and proper morphisms
Affine morphisms
Finite and quasi-finite morphisms
Serre’s and Chevalley’s criteria to be affine
Normalization
Proper morphisms
Zariski’s main theorem
Exercises
13 Projective morphisms
Projective spectrum of a graded algebra
Embeddings into projective space
Blowing-up
Exercises
14 Flat morphisms and dimension
Flat morphisms
Properties of flat morphisms
Faithfully flat descent
Dimension and fibers of morphisms
Dimension and regularity conditions
Hilbert schemes
Exercises
15 One-dimensional schemes
Morphisms into and from one-dimensional schemes
Valuative criteria
Curves over fields
Divisors on curves
Exercises
16 Examples
Determinantal varieties
Cubic surfaces and a Hilbert modular surface
Cyclic quotient singularities
Abelian varieties
Exercises
A The language of categories
B Commutative Algebra
C Permanence for properties of morphisms of schemes
D Relations between properties of morphisms of schemes
E Constructible and open properties
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