目錄
Preface
1 Schemes and sheaves: definitions
1.1 Spec(R)
1.2 M
1.3 Schemes
1.4 Products
1.5 Quasi-coherent sheaves
1.6 The functor of points
1.7 Relativization
1.8 Defining schemes as functors
Appendix: Theory of sheaves
Exercises
2 Exploring the world of schemes
2.1 Classical varieties as schemes
2.2 The properties: reduced, irreducible and finite type
2.3 Closed subschemes and primary decompositions
2.4 Separated schemes
2.5 Proj R
2.6 Proper morphisms
Exercises
3 Elementary global study of Proj R
3.1 Invertible sheaves and twists
3.2 The functor of Proj R
3.3 Blowups
3.4 Quasi-coherent sheaves on Proj R
3.5 Ample invertible sheaves
3.6 Invertible sheaves via cocycles, divisors, line bundles
Exercises
4 Ground fields and base rings
4.1 Kronecker's big picture
4.2 Galois theory and schemes
4.3 The Frobenius morphism
4.4 Flatness and specialization
4.5 Dimension of fibres of a morphism
4.6 Hensel's lemma
Exercises
5 Singular vs. non-singular
5.1 Regularity
5.2 Kahler differential
5.3 Smooth morphisms
5.4 Criteria for smoothness
5.5 Normality
5.6 Zariski's Main Theorem
5.7 Multiplicities following Well
Exercises
6 Group schemes and applications
6.1 Group schemes
6.2 Lang's theorems over finite fields
Exercises
7 The cohomology of coherent sheaves
7.1 Basic Cech cohomology
7.2 The case of schemes: Serre's theorem
7.3 Higher direct images and Leray's spectral sequence
7.4 Computing cohomology (1): Push f" into a huge acyclic sheaf
7.5 Computing cohomology (2): Directly via the Cech complex .
7.6 Computing cohomology (3): Generate Jr by "known" sheaves
7.7 Computing cohomology (4): Push 5r into a coherent acyclic sheaf
7.8 Serre's criterion for ampleness
7.9 Functorial properties of ampleness
7.10 The Euler characteristic
7.11 Intersection numbers
7.12 The criterion of Nakai-Moishezon
7.13 Seshadri constants
Exercises
8 Applications of cohomology
8.1 The Riemann-Roch theorem
Appendix: Residues of differentials on curves
8.2 Comparison of algebraic with analytic cohomology
8.3 de Rham cohomology
8.4 Characteristic p phenomena
8.5 Deformation theory
Exercises
9 Two deeper results
9.1 Mori's existence theorem of rational curves .
9.2 Belyi's three-point theorem
References
Index
[