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NPTEL

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

NPTEL and Indian Institute of Technology Madras via YouTube

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Overview

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.

Syllabus

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

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