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