Learn · Commutative Algebra

Lecture 18 of 60Week 4

Lecture 18 : Dimension of Polynomial Algebra over arbitrary Rings

Prof. Dilip Patil · IIT Bombay

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Commutative Algebra, IIT Bombay (at NPTEL)
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All the lectures

  1. Lecture 1 : Zariski Topology and K-Spectrum Video
  2. Lecture 2 : Algebraic Varieties and Classical Nullstelensatz Video
  3. Lecture 3 : Motivation for Krulls Dimension Video
  4. Lecture 4 : Chevalleys dimension Video
  5. Lecture 5 : Associated Prime Ideals of a Module Video
  6. Lecture 6 : Support of a Module Video
  7. Lecture 7 : Primary Decomposition Video
  8. Lecture 8 : Primary Decomposition(Contd) Video
  9. Lecture 9 : Uniqueness of Primary Decomposition Video
  10. Lecture 10 : Modules of Finite Length Video
  11. Lecture 11 : Modules of Finite Length(Contd) Video
  12. Lecture 12 : Introduction to Krull's Dimension Video
  13. Lecture 13 : Noether Normalization Lemma(Classical Version) Video
  14. Lecture 14 : Consequences of Noether Normalization Lemma Video
  15. Lecture 15 : Nil Radical and Jacobson Radical of Finite type Algebras over a Field and digression of Integral Extension Video
  16. Lecture 16 : Nagata's version of NNL Video
  17. Lecture 17 : Dimensions of Polynomial ring over Noetherian rings Video
  18. Lecture 18 : Dimension of Polynomial Algebra over arbitrary Rings Video
  19. Lecture 19 : Dimension Inequalities Video
  20. Lecture 20 : Hilbert's Nullstelensatz Video
  21. Lecture 21 : Computational rules for Poincaré Series Video
  22. Lecture 22 : Graded Rings, Modules and Poincaré Series Video
  23. Lecture 23 : Hilbert-Samuel Polynomials Video
  24. Lecture 24 : Hilbert-Samuel Polynomials(Contd) Video
  25. Lecture 25 : Numerical Function of polynomial type Video
  26. Lecture 26 : Hilbert-Samuel Polynomial of a Local ring Video
  27. Lecture 27 : Filtration on a Module Video
  28. Lecture 28 : Artin-Rees Lemma Video
  29. Lecture 29 : Dimension Theorem Video
  30. Lecture 30 : Dimension Theorem(Continued) Video
  31. Lecture 31 : Consequences of Dimension Theorem Video
  32. Lecture 32 : Generalized Krull's Principal Ideal Theorem Video
  33. Lecture 33 : Second proof of Krull's Principal Ideal Theorem Video
  34. Lecture 34 : The Spec Functor Video
  35. Lecture 35 : Prime ideals in Polynomial rings Video
  36. Lecture 36 : Characterization of Equidimensional Affine Algebra Video
  37. Lecture 37 : Connection between Regular local rings and associated graded rings Video
  38. Lecture 38 : Statement of the Jacobian Criterion for Regularity Video
  39. Lecture 39 : Hilbert function for Affine Algebra Video
  40. Lecture 40 : Hilbert Serre Theorem Video
  41. Lecture 41 : Jacobian Matrix and its Rank Video
  42. Lecture 42 : Jacobian Matrix and its Rank(Contd) Video
  43. Lecture 43 : Proof of Jacobian Critrerion Video
  44. Lecture 44 : Proof of Jacobian Critrerion(Contd) Video
  45. Lecture 45 : Preparation for Homological Dimension Video
  46. Lecture 46 : Complexes of Modules and Homology Video
  47. Lecture 47 : Projective Modules Video
  48. Lecture 48 : Homological Dimension and Projective module Video
  49. Lecture 49 : Global Dimension Video
  50. Lecture 50 : Homological characterization of Regular Local Ring(RLR) Video
  51. Lecture 51 : Homological characterization of Regular Local Ring(Contd) Video
  52. Lecture 52 : Homological Characterization of Regular Local Rings(Contd) Video
  53. Lecture 53 : Regular Local Rings are UFD Video
  54. Lecture 54 : RLR-Prime ideals of height 1 Video
  55. Lecture 55 : Discrete Valuation Ring Video
  56. Lecture 56 : Discrete Valuation Ring(Contd) Video
  57. Lecture 57 : Dedekind Domains Video
  58. Lecture 58 : Fractionary Ideals and Dedekind Domains Video
  59. Lecture 59 : Characterization of Dedekind Domain Video
  60. Lecture 60 : Dedekind Domains and prime factorization of ideals Video