Learn · Introduction to Algebraic Geometry and Commutative Algebra

Lecture 15 of 62Week 3 The K-Spectrum of a K-algebra and Affine algebraic sets

K-Zariski Topology

Prof. Dilip P. Patil · IISc Bangalore

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From
Introduction to Algebraic Geometry and Commutative Algebra, IISc Bangalore (at NPTEL)
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CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
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All the lectures

  1. Motivation for K-algebraic sets Video
  2. Definitions and examples of Affine Algebraic Set Video
  3. Rings and Ideals Video
  4. Operation on Ideals Video
  5. Prime Ideals and Maximal Ideals Video
  6. Krull's Theorem and consequences Video
  7. Module, submodules and quotient modules Video
  8. Algebras and polynomial algebras Video
  9. Universal property of polynomial algebra and examples Video
  10. Finite and Finite type algebras Video
  11. K-Spectrum (K-rational points) Video
  12. Identity theorem for Polynomial functions Video
  13. Basic properties of K-algebraic sets Video
  14. Examples of K-algebraic sets Video
  15. K-Zariski Topology Video
  16. The map V L Video
  17. Noetherian and Artinian Ordered sets Video
  18. Noetherian induction and Transfinite induction Video
  19. Modules with Chain Conditions Video
  20. Properties of Noetherian and Artinian Modules Video
  21. Examples of Artinian and Noetherian Modules Video
  22. Finite modules over Noetherian Rings Video
  23. Hilbert's Basis Theorem(HBT) Video
  24. Consequences of HBT Video
  25. Free Modules and rank Video
  26. More on Noetherian and Artinian modules Video
  27. Ring of Fractions(Localization) Video
  28. Nil radical, contraction of ideals Video
  29. Universal property of S -1 A Video
  30. Ideal structure in S -1 A Video
  31. Consequences of the Correspondence of Ideals Video
  32. Consequences of the Correspondence of Ideals(Contd.) Video
  33. Modules of Fraction and universal properties Video
  34. Exactness of the functor S -1 Video
  35. Universal property of Modules of Fractions Video
  36. Further properties of Modules and Module of Fractions Video
  37. Local-Global Principle Video
  38. Consequences of Local-Global Principle Video
  39. Properties of Artinian Rings Video
  40. Krull-Nakayama Lemma Video
  41. Properties of I K and V L maps Video
  42. Hilbert's Nullstelensatz Video
  43. Hilbert's Nullstelensatz(Contd.) Video
  44. Proof of Zariski's Lemma(HNS 3) Video
  45. Consequences of HNS Video
  46. Consequences of HNS(Contd.) Video
  47. Jacobson Ring and examples Video
  48. Irreducible subsets of Zariski Topology(Finite type K-algebra) Video
  49. Spec functor on Finite type K-algebras Video
  50. Properties of Irreducible topological spaces Video
  51. Zariski Topology on arbitrary commutative rings Video
  52. Spec functor on arbitrary commutative rings Video
  53. Topological properties of Spec A Video
  54. Example to support the term Spectrum- Video
  55. Integral Extensions Video
  56. Elementwise characterization of Integral extensions Video
  57. Properties and examples of Integral extensions Video
  58. Prime and Maximal ideals in integral extensions Video
  59. Lying over Theorem Video
  60. Cohen-Siedelberg Theorem Video
  61. Live Session 17-03-2022 Video
  62. Live Session 03-03-2025 Video