InfoCoBuild

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)

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


Go to the Course Home or watch other lectures:

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