Commutative Algebra (Prof. Dilip P. Patil, IIT Bombay) | Mathematics

Commutative Algebra

Commutative Algebra. Instructor: Prof. Dilip P. Patil, Department of Mathematics, IIT Bombay. The main purpose of this course is to provide important workhorses of commutative algebra assuming only basic course on commutative algebra. Special efforts are made to present the concepts at the center of the field in a coherent, tightly knit way, streamlined proofs and a focus on the coreresults. Virtually all concepts and results of commutative algebra have natural interpretations. It is the geometric view point that brings out the true meaning of the theory. (from nptel.ac.in)

Introduction

Lecture 01 - Zariski Topology and K-Spectrum

Lecture 02 - Algebraic Varieties and Classical Nullstelensatz

Lecture 03 - Motivation for Krull's Dimension

Lecture 04 - Chevalley's Dimension

Lecture 05 - Associated Prime Ideals of a Module

Lecture 06 - Support of a Module

Lecture 07 - Primary Decomposition

Lecture 08 - Primary Decomposition (cont.)

Lecture 09 - Uniqueness of Primary Decomposition

Lecture 10 - Modules of Finite Length

Lecture 11 - Modules of Finite Length (cont.)

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

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 Poincare Series

Lecture 22 - Graded Rings, Modules and Poincare Series

Lecture 23 - Hilbert-Samuel Polynomials

Lecture 24 - Hilbert-Samuel Polynomials (cont.)

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 (cont.)

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 (cont.)

Lecture 43 - Proof of Jacobian Criterion

Lecture 44 - Proof of Jacobian Criterion (cont.)

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 Ring (RLR)

Lecture 51 - Homological Characterization of Regular Local Ring (cont.)

Lecture 52 - Homological Characterization of Regular Local Ring (cont.)

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 (cont.)

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

References

Commutative Algebra
Instructor: Prof. Dilip P. Patil, Department of Mathematics, IIT Bombay. The main purpose of this course is to provide important workhorses of commutative algebra assuming only basic course on commutative algebra.