Useful Links
Mathematics
Algebraic Geometry
1. Foundations of Algebraic Geometry
2. Polynomial Rings and Commutative Algebra
3. Gröbner Bases
4. Affine Algebraic Geometry
5. Projective Algebraic Geometry
6. Rational Maps and Birational Geometry
7. Local Properties and Singularities
8. Introduction to Schemes
9. Coherent Sheaves and Cohomology
10. Divisors and Line Bundles
11. Algebraic Curves
12. Algebraic Surfaces
13. Advanced Topics
Introduction to Schemes
Motivation for Schemes
Limitations of Classical Algebraic Geometry
Arithmetic Applications
Functorial Viewpoint
Sheaves and Presheaves
Presheaves on Topological Spaces
Definition and Examples
Morphisms of Presheaves
Sheaves
Definition and Sheaf Axioms
Examples of Sheaves
Sheafification Process
Stalks of Sheaves
Definition and Properties
Local Properties
Ringed Spaces
Definition of Ringed Space
Morphisms of Ringed Spaces
Locally Ringed Spaces
Definition and Examples
Local Rings in Stalks
Affine Schemes
The Spectrum of a Ring
Points as Prime Ideals
Motivation and Examples
Zariski Topology on Spec(A)
Closed Sets and Basic Opens
Properties of the Topology
Structure Sheaf on Spec(A)
Definition via Localization
Sections over Open Sets
Affine Schemes as Locally Ringed Spaces
General Schemes
Definition by Gluing Affine Schemes
Examples of Schemes
Affine Space over Integers
Projective Space as Scheme
Spec of Fields and Other Rings
Morphisms of Schemes
Affine Morphisms
Properties of Schemes
Reduced and Integral Schemes
Noetherian Schemes
Separated Schemes
Proper Schemes
Dimension of Schemes
The Functor of Points
Yoneda Lemma Perspective
Schemes as Functors
Representable Functors
Previous
7. Local Properties and Singularities
Go to top
Next
9. Coherent Sheaves and Cohomology