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

Lectures

Unit 1: The Zariski Topology

1. What is Algebraic Geometry? Video
2. The Zariski Topology and Affine Space Video
3. Going back and forth between subsets and ideals Video

Unit 2: Irreducibility in the Zariski Topology

4. Irreducibility in the Zariski Topology Video
5. Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime Video

Unit 3: Noetherianness in the Zariski Topology

6. Understanding the Zariski Topology on the Affine Line; The Noetherian property in Topology and in Algebra Video
7. Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity Video

Unit 4: Dimension and Rings of Polynomial Functions

8. Topological Dimension, Krull Dimension and Heights of Prime Ideals Video
9. The Ring of Polynomial Functions on an Affine Variety Video
10. Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces Video

Unit 5: The Affine Coordinate Ring of an Affine Variety

11. Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ? Video
12. Capturing an Affine Variety Topologically From the Maximal Spectrum of its Ring of Functions Video

Unit 6: Open sets in the Zariski Topology and Functions on such sets

13. Analyzing Open Sets and Basic Open Sets for the Zariski Topology Video
14. The Ring of Functions on a Basic Open Set in the Zariski Topology Video

Unit 7: Regular Functions in Affine Geometry

15. Quasi-Compactness in the Zariski Topology; Regularity of a Function at a point of an Affine Variety Video
16. What is a Global Regular Function on a Quasi-Affine Variety? Video

Unit 8: Morphisms in Affine Geometry

17. Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties Video
18. Translating Morphisms into Affines as k-Algebra maps and the Grand Hilbert Nullstellensatz Video
19. Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions Video
20. The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to it Video
21. Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture; The Punctured Plane is Not Affine Video

Unit 9: The Zariski Topology on Projective Space and Projective Varieties

22. The Various Avatars of Projective n-space Video
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 Video

Unit 10: Graded Rings, Homogeneous Ideals and Homogeneous Localisation

24. Translating Projective Geometry into Graded Rings and Homogeneous Ideals Video
25. Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties Video
26. Translating Homogeneous Localisation into Geometry and Back Video
27. Adding a Variable is Undone by Homogenous Localization - What is the Geometric Significance of this Algebraic Fact ? Video

Unit 11: The Local Ring of Germs of Functions at a Point

28. Doing Calculus Without Limits in Geometry ? Video
29. The Birth of Local Rings in Geometry and in Algebra Video
30. The Formula for the Local Ring at a Point of a Projective Variety Or Playing with Localisations, Quotients, Homogenisation and Dehomogenisation ! Video

Unit 13: Two Facts about Rings of Functions on Projective Varieties

31. Global Regular Functions on Projective Varieties are Simply the Constants Video
32. The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety Video

Unit 14: The Importance of Local Rings and Function Fields

33. The Importance of Local Rings - A Morphism is an Isomorphism if it is a Homeomorphism and Induces Isomorphisms at the Level of Local Rings Video
34. The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular Video
35. Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global Video
36. Local Ring Isomorphism,Equals Function Field Isomorphism, Equals Birationality Video

Unit 15: Regular or Smooth Points and Manifold Varieties or Smooth Varieties

37. Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended! Video
38. How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry Video
39. Any Variety is a Smooth Manifold with or without Non-Smooth Boundary Video
40. Any Variety is a Smooth Hypersurface On an Open Dense Subset Video