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Course Co-ordinated by :
IIT Madras
Course Available from :
29-August-2016
NPTEL
Mathematics
Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity (Video)
What is Algebraic Geometry?
Modules / Lectures
Unit 1: The Zariski Topology
What is Algebraic Geometry?
The Zariski Topology and Affine Space
Going back and forth between subsets and ideals
Unit 2: Irreducibility in the Zariski Topology
Irreducibility in the Zariski Topology
Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime
Unit 3: Noetherianness in the Zariski Topology
Understanding the Zariski Topology on the Affine Line; The Noetherian property in Topology and in Algebra
Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
Unit 4: Dimension and Rings of Polynomial Functions
Topological Dimension, Krull Dimension and Heights of Prime Ideals
The Ring of Polynomial Functions on an Affine Variety
Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces
Unit 5: The Affine Coordinate Ring of an Affine Variety
Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?
Capturing an Affine Variety Topologically From the Maximal Spectrum of its Ring of Functions
Unit 6: Open sets in the Zariski Topology and Functions on such sets
Analyzing Open Sets and Basic Open Sets for the Zariski Topology
The Ring of Functions on a Basic Open Set in the Zariski Topology
Unit 7: Regular Functions in Affine Geometry
Quasi-Compactness in the Zariski Topology; Regularity of a Function at a point of an Affine Variety
What is a Global Regular Function on a Quasi-Affine Variety?
Unit 8: Morphisms in Affine Geometry
Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties
Translating Morphisms into Affines as k-Algebra maps and the Grand Hilbert Nullstellensatz
Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions
The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to it
Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture; The Punctured Plane is Not Affine
Unit 9: The Zariski Topology on Projective Space and Projective Varieties
The Various Avatars of Projective n-space
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
Unit 10: Graded Rings, Homogeneous Ideals and Homogeneous Localisation
Translating Projective Geometry into Graded Rings and Homogeneous Ideals
Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties
Translating Homogeneous Localisation into Geometry and Back
Adding a Variable is Undone by Homogenous Localization - What is the Geometric Significance of this Algebraic Fact ?
Unit 11: The Local Ring of Germs of Functions at a Point
Doing Calculus Without Limits in Geometry ?
The Birth of Local Rings in Geometry and in Algebra
The Formula for the Local Ring at a Point of a Projective Variety Or Playing with Localisations, Quotients, Homogenisation and Dehomogenisation !
Unit 12: The Function Field of Functions on Large Open Sets
The Field of Rational Functions or Function Field of a Variety - The Local Ring at the Generic Point
Fields of Rational Functions or Function Fields of Affine and Projective Varieties and their Relationships with Dimensions
Unit 13: Two Facts about Rings of Functions on Projective Varieties
Global Regular Functions on Projective Varieties are Simply the Constants
Unit 14: The Importance of Local Rings and Function Fields
The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety
The Importance of Local Rings - A Morphism is an Isomorphism if it is a Homeomorphism and Induces Isomorphisms at the Level of Local Rings
The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular
Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global
Unit 15: Regular or Smooth Points and Manifold Varieties or Smooth Varieties
Local Ring Isomorphism,Equals Function Field Isomorphism, Equals Birationality
Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!
How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry
Any Variety is a Smooth Manifold with or without Non-Smooth Boundary
Any Variety is a Smooth Hypersurface On an Open Dense Subset
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