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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

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