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Basic Algebraic Geometry

Basic Algebraic Geometry. Instructor: Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. This course is an introduction to Algebraic Geometry, whose aim is to study the geometry underlying the set of common zeros of a collection of polynomial equations. It sets up the language of varieties and of morphisms between them, and studies their topological and manifold-theoretic properties. Commutative Algebra is the "calculus" that Algebraic Geometry uses. Therefore a prerequisite for this course would be a course in Algebra covering basic aspects of commutative rings and some field theory, as also a course on elementary Topology. However, the necessary results from Commutative Algebra and Field Theory would be recalled as and when required during the course for the benefit of the students. (from nptel.ac.in)

Lecture 42 - Any Variety is a Smooth Hypersurface on an Open Dense Subset


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Unit 1: The Zariski Topology
Lecture 01 - What is Algebraic Geometry?
Lecture 02 - The Zariski Topology and Affine Space
Lecture 03 - Going Back and Forth between Subsets and Ideals
Unit 2: Irreducibility in the Zariski Topology
Lecture 04 - Irreducibility in the Zariski Topology
Lecture 05 - Irreducible Closed Subsets Correspond to Ideals whose Radicals are Prime
Unit 3: Noetherianness in the Zariski Topology
Lecture 06 - Understanding the Zariski Topology on the Affine Line; The Noetherian Property in Topology and in Algebra
Lecture 07 - Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
Unit 4: Dimension and Rings of Polynomial Functions
Lecture 08 - Topological Dimension, Krull Dimension and Heights of Prime Ideals
Lecture 09 - The Ring of Polynomial Functions on an Affine Variety
Lecture 10 - Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces
Unit 5: The Affine Coordinate Ring of an Affine Variety
Lecture 11 - Why should We Study Affine Coordinate Rings of Functions on Affine Varieties?
Lecture 12 - 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
Lecture 13 - Analyzing Open Sets and Basic Open Sets for the Zariski Topology
Lecture 14 - The Ring of Functions on a Basic Open Set in the Zariski Topology
Unit 7: Regular Functions in Affine Geometry
Lecture 15 - Quasi-Compactness in the Zariski Topology; Regularity of a Function at a Point of an Affine Variety
Lecture 16 - What is a Global Regular Function on a Quasi-Affine Variety?
Unit 8: Morphisms in Affine Geometry
Lecture 17 - Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties
Lecture 18 - Translating Morphisms into Affines as k-Algebra Maps and the Grand Hilbert Nullstellensatz
Lecture 19 - Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions
Lecture 20 - The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to It
Lecture 21 - 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
Lecture 22 - The Various Avatars of Projective n-Space
Lecture 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
Unit 10: Graded Rings, Homogeneous Ideals and Homogeneous Localization
Lecture 24 - Translating Projective Geometry into Graded Rings and Homogeneous Ideals
Lecture 25 - Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties
Lecture 26 - Translating Homogeneous Localization into Geometry and Back
Lecture 27 - Adding a Variable is Undone by Homogeneous Localization - What is the Geometric Significance of this Algebraic Fact?
Unit 11: The Local Ring of Germs of Functions at a Point
Lecture 28 - Doing Calculus without Limits in Geometry
Lecture 29 - The Birth of Local Rings in Geometry and in Algebra
Lecture 30 - The Formula for the Local Ring at a Point of a Projective Variety or Playing with Localizations, Quotients, Homogenization and Dehomogenization
Unit 12: The Function Field of Functions on Large Open Sets
Lecture 31 - The Field of Rational Functions or Function Field of a Variety - The Local Ring at the Generic Point
Lecture 32 - Fields of Rational Functions or Function Field of Affine and Projective Varieties and Their Relationships with Dimensions
Unit 13: Two Facts about Rings of Functions on Projective Varieties
Lecture 33 - Global Regular Functions on Projective Varieties are Simply the Constants
Unit 14: The Importance of Local Rings and Function Fields
Lecture 34 - The d-Uple Embedding and the Non-intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety
Lecture 35 - The Importance of Local Rings - A Morphism is an Isomorphism if it is a Homeomorphism and Induces Isomorphisms at the Level of Local Rings
Lecture 36 - The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular
Lecture 37 - Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global
Unit 15: Regular or Smooth Points and Manifold Varieties or Smooth Varieties
Lecture 38 - Local Ring Isomorphism - Equals Function Field Isomorphism - Equals Birationality
Lecture 39 - Why Local Rings Provide Calculus without Limits for Algebraic Geometry Pun Intended
Lecture 40 - How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry
Lecture 41 - Any Variety is a Smooth Manifold with or without Nonsmooth Boundary
Lecture 42 - Any Variety is a Smooth Hypersurface on an Open Dense Subset