NOC:Commutative Algebra

Lecture 1 - Zariski Topology and K-Spectrum

Lecture 2 - Algebraic Varieties and Classical Nullstelensatz

Lecture 3 - Motivation for Krulls Dimension

Lecture 4 - Chevalleys dimension

Lecture 5 - Associated Prime Ideals of a Module

Lecture 6 - Support of a Module

Lecture 7 - Primary Decomposition

Lecture 8 - Primary Decomposition (Continued...)

Lecture 9 - Uniqueness of Primary Decomposition

Lecture 10 - Modules of Finite Length

Lecture 11 - Modules of Finite Length (Continued...)

Lecture 12 - Introduction to Krull’s Dimension

Lecture 13 - Noether Normalization Lemma (Classical Version)

Lecture 14 - Consequences of Noether Normalization Lemma

Lecture 15 - Nil Radical and Jacobson Radical of Finite type Algebras over a Field and digression of Integral Extension

Lecture 16 - Nagata’s version of NNL

Lecture 17 - Dimensions of Polynomial ring over Noetherian rings

Lecture 18 - Dimension of Polynomial Algebra over arbitrary Rings

Lecture 19 - Dimension Inequalities

Lecture 20 - Hilbert’s Nullstelensatz

Lecture 21 - Computational rules for Poincaré Series

Lecture 22 - Graded Rings, Modules and Poincaré Series

Lecture 23 - Hilbert-Samuel Polynomials

Lecture 24 - Hilbert-Samuel Polynomials (Continued...)

Lecture 25 - Numerical Function of polynomial type

Lecture 26 - Hilbert-Samuel Polynomial of a Local ring

Lecture 27 - Filtration on a Module

Lecture 28 - Artin-Rees Lemma

Lecture 29 - Dimension Theorem

Lecture 30 - Dimension Theorem (Continued...)

Lecture 31 - Consequences of Dimension Theorem

Lecture 32 - Generalized Krull’s Principal Ideal Theorem

Lecture 33 - Second proof of Krull’s Principal Ideal Theorem

Lecture 34 - The Spec Functor

Lecture 35 - Prime ideals in Polynomial rings

Lecture 36 - Characterization of Equidimensional Affine Algebra

Lecture 37 - Connection between Regular local rings and associated graded rings

Lecture 38 - Statement of the Jacobian Criterion for Regularity

Lecture 39 - Hilbert function for Affine Algebra

Lecture 40 - Hilbert Serre Theorem

Lecture 41 - Jacobian Matrix and its Rank

Lecture 42 - Jacobian Matrix and its Rank (Continued...)

Lecture 43 - Proof of Jacobian Critrerion

Lecture 44 - Proof of Jacobian Critrerion (Continued...)

Lecture 45 - Preparation for Homological Dimension

Lecture 46 - Complexes of Modules and Homology

Lecture 47 - Projective Modules

Lecture 48 - Homological Dimension and Projective module

Lecture 49 - Global Dimension

Lecture 50 - Homological characterization of Regular Local Rings (RLR)

Lecture 51 - Homological characterization of Regular Local Rings (Continued...)

Lecture 52 - Homological Characterization of Regular Local Rings (Continued...)

Lecture 53 - Regular Local Rings are UFD

Lecture 54 - RLR-Prime ideals of height 1

Lecture 55 - Discrete Valuation Ring

Lecture 56 - Discrete Valuation Ring (Continued...)

Lecture 57 - Dedekind Domains

Lecture 58 - Fractionary Ideals and Dedekind Domains

Lecture 59 - Characterization of Dedekind Domain

Lecture 60 - Dedekind Domains and prime factorization of ideals