Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity EF0X0EV4YJ
| URI | m2://res/EF0X0EV4YJ |
|---|---|
| Kind | course |
| Creators | T.E. Venkata Balaji (instructor) |
| Date | 2016-07-11 |
| MSC | 14-XX |
| Scope | in |
| Ids | course 03v0r5n49:111106097 |
Expressions
e1: edition
| Language | en |
|---|---|
| Date | 2016-07-11 |
- m1: page, IIT Madras (record). nptel 111106097
- m2: page, 2016-07-11 (record). nptel 111106097
Contents
- part 1: Unit 1: The Zariski Topology
- session 1: What is Algebraic Geometry?
- session 2: The Zariski Topology and Affine Space
- session 3: Going back and forth between subsets and ideals
- part 2: Unit 2: Irreducibility in the Zariski Topology
- session 4: Irreducibility in the Zariski Topology
- session 5: Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime
- part 3: Unit 3: Noetherianness in the Zariski Topology
- session 6: Understanding the Zariski Topology on the Affine Line; The Noetherian property in Topology and in Algebra
- session 7: Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
- part 4: Unit 4: Dimension and Rings of Polynomial Functions
- session 8: Topological Dimension, Krull Dimension and Heights of Prime Ideals
- session 9: The Ring of Polynomial Functions on an Affine Variety
- session 10: Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces
- part 5: Unit 5: The Affine Coordinate Ring of an Affine Variety
- session 11: Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?
- session 12: Capturing an Affine Variety Topologically From the Maximal Spectrum of its Ring of Functions
- part 6: Unit 6: Open sets in the Zariski Topology and Functions on such sets
- session 13: Analyzing Open Sets and Basic Open Sets for the Zariski Topology
- session 14: The Ring of Functions on a Basic Open Set in the Zariski Topology
- part 7: Unit 7: Regular Functions in Affine Geometry
- session 15: Quasi-Compactness in the Zariski Topology; Regularity of a Function at a point of an Affine Variety
- session 16: What is a Global Regular Function on a Quasi-Affine Variety?
- part 8: Unit 8: Morphisms in Affine Geometry
- session 17: Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties
- session 18: Translating Morphisms into Affines as k-Algebra maps and the Grand Hilbert Nullstellensatz
- session 19: Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions
- session 20: The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to it
- session 21: Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture; The Punctured Plane is Not Affine
- part 9: Unit 9: The Zariski Topology on Projective Space and Projective Varieties
- session 22: The Various Avatars of Projective n-space
- session 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
- part 10: Unit 10: Graded Rings, Homogeneous Ideals and Homogeneous Localisation
- session 24: Translating Projective Geometry into Graded Rings and Homogeneous Ideals
- session 25: Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties
- session 26: Translating Homogeneous Localisation into Geometry and Back
- session 27: Adding a Variable is Undone by Homogenous Localization - What is the Geometric Significance of this Algebraic Fact ?
- part 11: Unit 11: The Local Ring of Germs of Functions at a Point
- session 28: Doing Calculus Without Limits in Geometry ?
- session 29: The Birth of Local Rings in Geometry and in Algebra
- session 30: The Formula for the Local Ring at a Point of a Projective Variety Or Playing with Localisations, Quotients, Homogenisation and Dehomogenisation !
- part 12: Unit 13: Two Facts about Rings of Functions on Projective Varieties
- session 31: Global Regular Functions on Projective Varieties are Simply the Constants
- session 32: The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety
- part 13: Unit 14: The Importance of Local Rings and Function Fields
- session 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
- session 34: The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular
- session 35: Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global
- session 36: Local Ring Isomorphism,Equals Function Field Isomorphism, Equals Birationality
- part 14: Unit 15: Regular or Smooth Points and Manifold Varieties or Smooth Varieties
- session 37: Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!
- session 38: How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry
- session 39: Any Variety is a Smooth Manifold with or without Non-Smooth Boundary
- session 40: Any Variety is a Smooth Hypersurface On an Open Dense Subset
Catalogue records
- nptel: 111106097 (manifestation)
- nptel_video: 111106097 (manifestation)