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

Mod-01 Lec-01 What is Algebraic Geometry?.

Mod-01 Lec-02 The Zariski Topology and Affine Space.
Mod-01 Lec-03 Going back and forth between subsets and ideals.
Mod-02 Lec-04 Irreducibility in the Zariski Topology.
Mod-02 Lec-05 Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime.
Mod-03 Lec-06 Understanding the Zariski Topology on the Affine Line.
Mod-03 Lec-07 The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties.
Mod-04 Lec-08 Topological Dimension, Krull Dimension and Heights of Prime Ideals.
Mod-04 Lec-09 The Ring of Polynomial Functions on an Affine Variety.
Mod-04 Lec-10 Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces.
Mod-05 Lec-11 Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?.
Mod-05 Lec-12 Capturing an Affine Variety Topologically.
Mod-06 Lec-13 Analyzing Open Sets and Basic Open Sets for the Zariski Topology.
Mod-06 Lec-14 The Ring of Functions on a Basic Open Set in the Zariski Topology.
Mod-07 Lec-15 Quasi-Compactness in the Zariski Topology.
Mod-07 Lec-16 What is a Global Regular Function on a Quasi-Affine Variety?.
Mod-08 Lec-17 Characterizing Affine Varieties.
Mod-08 Lec-18 Translating Morphisms into Affines as k-Algebra maps.
Mod-08 Lec-19 Morphisms into an Affine Correspond to k-Algebra Homomorphisms.
Mod-08 Lec-20 The Coordinate Ring of an Affine Variety.
Mod-08 Lec-21 Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture.
Mod-09 Lec-22 The Various Avatars of Projective n-space.
Mod-09 Lec-23 Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology.
Mod-10 Lec-24 Translating Projective Geometry into Graded Rings and Homogeneous Ideals.
Mod-10 Lec-25 Expanding the Category of Varieties.
Mod-10 Lec-26 Translating Homogeneous Localisation into Geometry and Back.
Mod-10 Lec-27 Adding a Variable is Undone by Homogenous Localization.
Mod-11 Lec-28 Doing Calculus Without Limits in Geometry.
Mod-11 Lec-29 The Birth of Local Rings in Geometry and in Algebra.
Mod-11 Lec-30 The Formula for the Local Ring at a Point of a Projective Variety.
Mod-12 Lec 31 The Field of Rational Functions or Function Field of a Variety.
Mod-12 Lec 32 Fields of Rational Functions or Function Fields of Affine and Projective Varieties.
Mod-13 Lec 33 Global Regular Functions on Projective Varieties are Simply the Constants.
Mod-13 Lec 34 The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring.
Mod-14 Lec 35 The Importance of Local Rings - A Morphism is an Isomorphism.
Mod-14 Lec 36 The Importance of Local Rings.
Mod-14 Lec 37 Geometric Meaning of Isomorphism of Local Rings.
Mod-14 Lec 38 Local Ring Isomorphism, Equals Function Field Isomorphism, Equals Birationality.
Mod-15 Lec 39 Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!.
Mod-15 Lec 40 How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry.
Mod-15 Lec 41 Any Variety is a Smooth Manifold with or without Non-Smooth Boundary.Mod-15 Lec 42 Any Variety is a Smooth Hypersurface On an Open Dense Subset.