Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity


Lecture 1 - What is Algebraic Geometry?


Lecture 2 - The Zariski Topology and Affine Space


Lecture 3 - Going back and forth between subsets and ideals


Lecture 4 - Irreducibility in the Zariski Topology


Lecture 5 - Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime


Lecture 6 - Understanding the Zariski Topology on the Affine Line; The Noetherian property in Topology and in Algebra


Lecture 7 - Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity


Lecture 8 - Topological Dimension, Krull Dimension and Heights of Prime Ideals


Lecture 9 - The Ring of Polynomial Functions on an Affine Variety


Lecture 10 - Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces


Lecture 11 - Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?


Lecture 12 - Capturing an Affine Variety Topologically From the Maximal Spectrum of its Ring of Functions


Lecture 13 - Analyzing Open Sets and Basic Open Sets for the Zariski Topology


Lecture 14 - The Ring of Functions on a Basic Open Set in the Zariski Topology


Lecture 15 - Quasi-Compactness in the Zariski Topology; Regularity of a Function at a point of an Affine Variety


Lecture 16 - What is a Global Regular Function on a Quasi-Affine Variety?


Lecture 17 - Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties


Lecture 18 - Translating Morphisms into Affines as k-Algebra maps and the Grand Hilbert Nullstellensatz


Lecture 19 - Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions


Lecture 20 - The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to it


Lecture 21 - Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture; The Punctured Plane is Not Affine


Lecture 22 - The Various Avatars of Projective n-space


Lecture 23 - Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology, Manifold Theory and Algebraic Geometry; The Key to the Definition of a Homogeneous Ideal


Lecture 24 - Translating Projective Geometry into Graded Rings and Homogeneous Ideals


Lecture 25 - Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties


Lecture 26 - Translating Homogeneous Localisation into Geometry and Back


Lecture 27 - Adding a Variable is Undone by Homogenous Localization - What is the Geometric Significance of this Algebraic Fact ?


Lecture 28 - Doing Calculus Without Limits in Geometry ?


Lecture 29 - The Birth of Local Rings in Geometry and in Algebra


Lecture 30 - The Formula for the Local Ring at a Point of a Projective Variety Or Playing with Localisations, Quotients, Homogenisation and Dehomogenisation !


Lecture 31 - The Field of Rational Functions or Function Field of a Variety - The Local Ring at the Generic Point


Lecture 32 - Fields of Rational Functions or Function Fields of Affine and Projective Varieties and their Relationships with Dimensions


Lecture 33 - Global Regular Functions on Projective Varieties are Simply the Constants


Lecture 34 - The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety


Lecture 35 - The Importance of Local Rings - A Morphism is an Isomorphism if it is a Homeomorphism and Induces Isomorphisms at the Level of Local Rings


Lecture 36 - The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular


Lecture 37 - Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global


Lecture 38 - Local Ring Isomorphism,Equals Function Field Isomorphism, Equals Birationality


Lecture 39 - Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!


Lecture 40 - How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry


Lecture 41 - Any Variety is a Smooth Manifold with or without Non-Smooth Boundary


Lecture 42 - Any Variety is a Smooth Hypersurface On an Open Dense Subset