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

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

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Mod-15 Lec 41 Any Variety is a Smooth Manifold with or without Non-Smooth Boundary

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41 of 42

Mod-15 Lec 41 Any Variety is a Smooth Manifold with or without Non-Smooth Boundary

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Classroom Contents

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

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  1. 1 Mod-01 Lec-01 What is Algebraic Geometry?
  2. 2 Mod-01 Lec-02 The Zariski Topology and Affine Space
  3. 3 Mod-01 Lec-03 Going back and forth between subsets and ideals
  4. 4 Mod-02 Lec-04 Irreducibility in the Zariski Topology
  5. 5 Mod-02 Lec-05 Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime
  6. 6 Mod-03 Lec-06 Understanding the Zariski Topology on the Affine Line
  7. 7 Mod-03 Lec-07 The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties
  8. 8 Mod-04 Lec-08 Topological Dimension, Krull Dimension and Heights of Prime Ideals
  9. 9 Mod-04 Lec-09 The Ring of Polynomial Functions on an Affine Variety
  10. 10 Mod-04 Lec-10 Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces
  11. 11 Mod-05 Lec-11 Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?
  12. 12 Mod-05 Lec-12 Capturing an Affine Variety Topologically
  13. 13 Mod-06 Lec-13 Analyzing Open Sets and Basic Open Sets for the Zariski Topology
  14. 14 Mod-06 Lec-14 The Ring of Functions on a Basic Open Set in the Zariski Topology
  15. 15 Mod-07 Lec-15 Quasi-Compactness in the Zariski Topology
  16. 16 Mod-07 Lec-16 What is a Global Regular Function on a Quasi-Affine Variety?
  17. 17 Mod-08 Lec-17 Characterizing Affine Varieties
  18. 18 Mod-08 Lec-18 Translating Morphisms into Affines as k-Algebra maps
  19. 19 Mod-08 Lec-19 Morphisms into an Affine Correspond to k-Algebra Homomorphisms
  20. 20 Mod-08 Lec-20 The Coordinate Ring of an Affine Variety
  21. 21 Mod-08 Lec-21 Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture
  22. 22 Mod-09 Lec-22 The Various Avatars of Projective n-space
  23. 23 Mod-09 Lec-23 Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology
  24. 24 Mod-10 Lec-24 Translating Projective Geometry into Graded Rings and Homogeneous Ideals
  25. 25 Mod-10 Lec-25 Expanding the Category of Varieties
  26. 26 Mod-10 Lec-26 Translating Homogeneous Localisation into Geometry and Back
  27. 27 Mod-10 Lec-27 Adding a Variable is Undone by Homogenous Localization
  28. 28 Mod-11 Lec-28 Doing Calculus Without Limits in Geometry
  29. 29 Mod-11 Lec-29 The Birth of Local Rings in Geometry and in Algebra
  30. 30 Mod-11 Lec-30 The Formula for the Local Ring at a Point of a Projective Variety
  31. 31 Mod-12 Lec 31 The Field of Rational Functions or Function Field of a Variety
  32. 32 Mod-12 Lec 32 Fields of Rational Functions or Function Fields of Affine and Projective Varieties
  33. 33 Mod-13 Lec 33 Global Regular Functions on Projective Varieties are Simply the Constants
  34. 34 Mod-13 Lec 34 The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring
  35. 35 Mod-14 Lec 35 The Importance of Local Rings - A Morphism is an Isomorphism
  36. 36 Mod-14 Lec 36 The Importance of Local Rings
  37. 37 Mod-14 Lec 37 Geometric Meaning of Isomorphism of Local Rings
  38. 38 Mod-14 Lec 38 Local Ring Isomorphism, Equals Function Field Isomorphism, Equals Birationality
  39. 39 Mod-15 Lec 39 Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!
  40. 40 Mod-15 Lec 40 How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry
  41. 41 Mod-15 Lec 41 Any Variety is a Smooth Manifold with or without Non-Smooth Boundary
  42. 42 Mod-15 Lec 42 Any Variety is a Smooth Hypersurface On an Open Dense Subset

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