Algebraic Geometry

  1. Affine Algebraic Geometry
    1. Affine Space
      1. Definition of Affine n-Space
        1. Coordinate Functions
          1. Points over Different Fields
            1. Morphisms between Affine Spaces
            2. Algebraic Sets in Affine Space
              1. Definition of Algebraic Sets
                1. Zero Sets of Polynomials
                  1. Solution Sets of Polynomial Systems
                    1. Examples and Basic Properties
                    2. The Ideal of an Algebraic Set
                      1. Vanishing Ideals
                        1. Generators of Vanishing Ideals
                          1. Properties of Vanishing Ideals
                          2. Radical Ideals
                            1. Definition and Properties
                              1. Radical of an Ideal
                                1. Characterizations
                                2. Hilbert's Nullstellensatz
                                  1. Weak Nullstellensatz
                                    1. Statement
                                      1. Proof Outline
                                        1. Consequences
                                        2. Strong Nullstellensatz
                                          1. Statement
                                            1. Proof Strategy
                                            2. Correspondence between Algebraic Sets and Radical Ideals
                                              1. Bijective Correspondence
                                                1. Geometric and Algebraic Perspectives
                                              2. The Zariski Topology
                                                1. Definition on Affine Space
                                                  1. Closed Sets and Basic Properties
                                                    1. Open Sets and Basis
                                                      1. Topological Properties
                                                        1. Non-Hausdorff Nature
                                                          1. Compactness Properties
                                                          2. Irreducibility
                                                            1. Characterizations
                                                          3. Irreducible Algebraic Sets
                                                            1. Definition and Examples
                                                              1. Decomposition into Irreducible Components
                                                                1. Uniqueness of Decomposition
                                                                  1. Connection to Prime Ideals
                                                                  2. Affine Varieties
                                                                    1. Examples of Affine Varieties
                                                                      1. Irreducible Components
                                                                      2. The Coordinate Ring
                                                                        1. Definition and Construction
                                                                          1. Polynomial Functions on Varieties
                                                                            1. Properties as k-algebra
                                                                              1. Finitely Generated k-algebras
                                                                                1. Correspondence with Affine Varieties
                                                                                2. Morphisms of Affine Varieties
                                                                                  1. Regular Functions
                                                                                    1. Ring of Regular Functions
                                                                                    2. Polynomial Maps
                                                                                      1. Definition and Examples
                                                                                        1. Induced Maps on Coordinate Rings
                                                                                        2. Morphisms between Varieties
                                                                                          1. Functoriality
                                                                                            1. Composition of Morphisms
                                                                                            2. Isomorphisms
                                                                                              1. Definition and Criteria
                                                                                            3. Dimension Theory for Affine Varieties
                                                                                              1. Krull Dimension
                                                                                                1. Chains of Prime Ideals
                                                                                                2. Transcendence Degree
                                                                                                  1. Field Extensions
                                                                                                    1. Connection to Dimension
                                                                                                    2. Dimension of Intersections
                                                                                                      1. Dimension Formula
                                                                                                        1. Applications and Examples

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                                                                                                    3. Gröbner Bases

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                                                                                                    5. Projective Algebraic Geometry