Algebraic Varieties and MorphismsIntroduces projective, quasi-projective, and algebraic varieties, emphasizing the importance of regular functions in defining morphisms.
Morphisms and Local RingsExplores morphisms, regularity, varieties, local rings, and evaluation morphism in algebraic geometry.
Field of Rational FunctionsExplores the construction of the field of rational functions on a variety and its connection with local rings and dimensions.
Dimension of Algebraic VarietyExplores the dimension of algebraic varieties, including the (Krull)-dimension of rings and computing dimensions using tools from commutative algebra.
Affine Plane CurvesCovers the study of affine plane curves and their properties, including the concept of multiplicity and its applications.
Local Rings and ResiduesCovers the proof of theorem 4.2 on multiplicities and the special structure of local rings at a simple point of a plane.
Discrete Valuation RingsExplores discrete valuation rings, their properties, uniqueness of representation, and relationship with principal ideal domains.
