T. E. Venkata Balaji: Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity (IIT Madras)

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source: nptelhrd    2016年7月11日
Mathematics - Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity by Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://nptel.ac.in

01 What is Algebraic Geometry? 48:39
02 The Zariski Topology and Affine Space 51:42
03 Going back and forth between subsets and ideals 49:17
04 Irreducibility in the Zariski Topology 53:24
05 Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime 54:29
06 Understanding the Zariski Topology on the Affine Line 57:36
07 The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties 1:04:51
08 Topological Dimension, Krull Dimension and Heights of Prime Ideals 55:52
09 The Ring of Polynomial Functions on an Affine Variety 43:44
10 Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces 53:29
11 Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties? 49:07
12 Capturing an Affine Variety Topologically 49:11
13 Analyzing Open Sets and Basic Open Sets for the Zariski Topology 45:20
14 The Ring of Functions on a Basic Open Set in the Zariski Topology 47:17
15 Quasi-Compactness in the Zariski Topology 55:32
16 What is a Global Regular Function on a Quasi-Affine Variety? 55:15
17 Characterizing Affine Varieties 53:29
18 Translating Morphisms into Affines as k-Algebra maps 50:56
19 Morphisms into an Affine Correspond to k-Algebra Homomorphisms 55:01
20 The Coordinate Ring of an Affine Variety 53:24
21 Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture 1:07:11
22 The Various Avatars of Projective n-space 35:49
23 Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology 59:51
24 Translating Projective Geometry into Graded Rings and Homogeneous Ideals 58:28
25 Expanding the Category of Varieties 51:49
26 Translating Homogeneous Localisation into Geometry and Back 51:26
27 Adding a Variable is Undone by Homogenous Localization 44:26
28 Doing Calculus Without Limits in Geometry 52:12
29 The Birth of Local Rings in Geometry and in Algebra 51:56
30 The Formula for the Local Ring at a Point of a Projective Variety 48:28
31 The Field of Rational Functions or Function Field of a Variety 53:17
32 Fields of Rational Functions or Function Fields of Affine and Projective Varieties 55:34
33 Global Regular Functions on Projective Varieties are Simply the Constants 52:16
34 The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring 58:27
35 The Importance of Local Rings - A Morphism is an Isomorphism 47:43
36 The Importance of Local Rings 47:28
37 Geometric Meaning of Isomorphism of Local Rings 54:01
38 Local Ring Isomorphism, Equals Function Field Isomorphism, Equals Birationality 1:01:27
39 Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended! 59:35
40 How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry 57:09
41 Any Variety is a Smooth Manifold with or without Non-Smooth Boundary 38:49
42 Any Variety is a Smooth Hypersurface On an Open Dense Subset 43:17