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Course11110111660 lectures

Commutative Algebra

Prof. Dilip Patil · IIT Bombay

Start with lecture 1 At NPTEL

Lectures

Week 1

  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

Week 2

  1. Lecture 6 : Support of a Module Video
  2. Lecture 7 : Primary Decomposition Video
  3. Lecture 8 : Primary Decomposition(Contd) Video
  4. Lecture 9 : Uniqueness of Primary Decomposition Video
  5. Lecture 10 : Modules of Finite Length Video

Week 3

  1. Lecture 11 : Modules of Finite Length(Contd) Video
  2. Lecture 12 : Introduction to Krull's Dimension Video
  3. Lecture 13 : Noether Normalization Lemma(Classical Version) Video
  4. Lecture 14 : Consequences of Noether Normalization Lemma Video
  5. Lecture 15 : Nil Radical and Jacobson Radical of Finite type Algebras over a Field and digression of Integral Extension Video

Week 4

  1. Lecture 16 : Nagata's version of NNL Video
  2. Lecture 17 : Dimensions of Polynomial ring over Noetherian rings Video
  3. Lecture 18 : Dimension of Polynomial Algebra over arbitrary Rings Video
  4. Lecture 19 : Dimension Inequalities Video
  5. Lecture 20 : Hilbert's Nullstelensatz Video

Week 5

  1. Lecture 21 : Computational rules for Poincaré Series Video
  2. Lecture 22 : Graded Rings, Modules and Poincaré Series Video
  3. Lecture 23 : Hilbert-Samuel Polynomials Video
  4. Lecture 24 : Hilbert-Samuel Polynomials(Contd) Video
  5. Lecture 25 : Numerical Function of polynomial type Video

Week 6

  1. Lecture 26 : Hilbert-Samuel Polynomial of a Local ring Video
  2. Lecture 27 : Filtration on a Module Video
  3. Lecture 28 : Artin-Rees Lemma Video
  4. Lecture 29 : Dimension Theorem Video
  5. Lecture 30 : Dimension Theorem(Continued) Video

Week 7

  1. Lecture 31 : Consequences of Dimension Theorem Video
  2. Lecture 32 : Generalized Krull's Principal Ideal Theorem Video
  3. Lecture 33 : Second proof of Krull's Principal Ideal Theorem Video
  4. Lecture 34 : The Spec Functor Video
  5. Lecture 35 : Prime ideals in Polynomial rings Video

Week 8

  1. Lecture 36 : Characterization of Equidimensional Affine Algebra Video
  2. Lecture 37 : Connection between Regular local rings and associated graded rings Video
  3. Lecture 38 : Statement of the Jacobian Criterion for Regularity Video
  4. Lecture 39 : Hilbert function for Affine Algebra Video
  5. Lecture 40 : Hilbert Serre Theorem Video

Week 9

  1. Lecture 41 : Jacobian Matrix and its Rank Video
  2. Lecture 42 : Jacobian Matrix and its Rank(Contd) Video
  3. Lecture 43 : Proof of Jacobian Critrerion Video
  4. Lecture 44 : Proof of Jacobian Critrerion(Contd) Video
  5. Lecture 45 : Preparation for Homological Dimension Video

Week 10

  1. Lecture 46 : Complexes of Modules and Homology Video
  2. Lecture 47 : Projective Modules Video
  3. Lecture 48 : Homological Dimension and Projective module Video
  4. Lecture 49 : Global Dimension Video
  5. Lecture 50 : Homological characterization of Regular Local Ring(RLR) Video

Week 11

  1. Lecture 51 : Homological characterization of Regular Local Ring(Contd) Video
  2. Lecture 52 : Homological Characterization of Regular Local Rings(Contd) Video
  3. Lecture 53 : Regular Local Rings are UFD Video
  4. Lecture 54 : RLR-Prime ideals of height 1 Video
  5. Lecture 55 : Discrete Valuation Ring Video

Week 12

  1. Lecture 56 : Discrete Valuation Ring(Contd) Video
  2. Lecture 57 : Dedekind Domains Video
  3. Lecture 58 : Fractionary Ideals and Dedekind Domains Video
  4. Lecture 59 : Characterization of Dedekind Domain Video
  5. Lecture 60 : Dedekind Domains and prime factorization of ideals Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Bombay
Department
Mathematics
Teacher
Prof. Dilip Patil
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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