Algebraic Geometry

  1. Polynomial Rings and Commutative Algebra
    1. Polynomial Rings
      1. Definition and Basic Properties
        1. Universal Property of Polynomial Rings
          1. Properties of k[x₁, ..., xₙ]
            1. Structure as a Commutative Ring
              1. Graded Structure
                1. Degree Functions
                2. Monomials and Term Orderings
                  1. Monomial Orderings
                    1. Lexicographic Order
                      1. Graded Lexicographic Order
                        1. Degree Reverse Lexicographic Order
                        2. Division Algorithm for Polynomials
                          1. Multivariate Division
                            1. Remainder and Quotient
                          2. Ideals in Polynomial Rings
                            1. Definition and Examples
                              1. Operations on Ideals
                                1. Sum and Product
                                  1. Intersection
                                    1. Quotient of Ideals
                                    2. Generators and Bases
                                      1. Principal Ideals
                                        1. Quotient Rings
                                          1. Construction
                                            1. Universal Property
                                          2. Prime and Maximal Ideals
                                            1. Definitions and Characterizations
                                              1. Examples in Polynomial Rings
                                                1. Existence Theorems
                                                  1. Zorn's Lemma Applications
                                                  2. Properties of Prime Ideals
                                                    1. Correspondence with Points
                                                      1. Localization at Prime Ideals
                                                      2. Noetherian Rings
                                                        1. Ascending Chain Condition
                                                          1. Equivalent Characterizations
                                                            1. Finiteness Properties
                                                              1. Examples and Non-examples
                                                              2. Hilbert's Basis Theorem
                                                                1. Statement and Proof
                                                                  1. Consequences for Polynomial Rings
                                                                    1. Applications to Algebraic Geometry
                                                                    2. Modules over Rings
                                                                      1. Definition and Basic Properties
                                                                        1. Examples of Modules
                                                                          1. Submodules and Quotient Modules
                                                                            1. Homomorphisms of Modules
                                                                              1. Finitely Generated Modules
                                                                                1. Free Modules
                                                                                  1. Exact Sequences
                                                                                  2. Localization
                                                                                    1. Multiplicative Sets
                                                                                      1. Construction of Localizations
                                                                                        1. Universal Property
                                                                                          1. Local Properties
                                                                                            1. Local Rings
                                                                                              1. Definition and Examples
                                                                                                1. Maximal Ideal
                                                                                                  1. Local Homomorphisms
                                                                                                2. Tensor Products
                                                                                                  1. Definition and Universal Property
                                                                                                    1. Construction and Examples
                                                                                                      1. Properties of Tensor Products
                                                                                                        1. Flatness
                                                                                                          1. Characterizations

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                                                                                                      1. Foundations of Algebraic Geometry

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                                                                                                      3. Gröbner Bases