NOC:Introduction to Algebraic Geometry and Commutative Algebra

Lecture 1 - Motivation for K-algebraic sets

Lecture 2 - Definitions and examples of Affine Algebraic Set

Lecture 3 - Rings and Ideals

Lecture 4 - Operation on Ideals

Lecture 5 - Prime Ideals and Maximal Ideals

Lecture 6 - Krull's Theorem and consequences

Lecture 7 - Module, submodules and quotient modules

Lecture 8 - Algebras and polynomial algebras

Lecture 9 - Universal property of polynomial algebra and examples

Lecture 10 - Finite and Finite type algebras

Lecture 11 - K-Spectrum (K-rational points)

Lecture 12 - Identity theorem for Polynomial functions

Lecture 13 - Basic properties of K-algebraic sets

Lecture 14 - Examples of K-algebraic sets

Lecture 15 - K-Zariski Topology

Lecture 16 - The map V L

Lecture 17 - Noetherian and Artinian Ordered sets

Lecture 18 - Noetherian induction and Transfinite induction

Lecture 19 - Modules with Chain Conditions

Lecture 20 - Properties of Noetherian and Artinian Modules

Lecture 21 - Examples of Artinian and Noetherian Modules

Lecture 22 - Finite modules over Noetherian Rings

Lecture 23 - Hilbert’s Basis Theorem (HBT)

Lecture 24 - Consequences of HBT

Lecture 25 - Free Modules and rank

Lecture 26 - More on Noetherian and Artinian modules

Lecture 27 - Ring of Fractions (Localization)

Lecture 28 - Nil radical, contraction of ideals

Lecture 29 - Universal property of S -1 A

Lecture 30 - Ideal structure in S -1 A

Lecture 31 - Consequences of the Correspondence of Ideals

Lecture 32 - Consequences of the Correspondence of Ideals (Continued...)

Lecture 33 - Modules of Fraction and universal properties

Lecture 34 - Exactness of the functor S -1

Lecture 35 - Universal property of Modules of Fractions

Lecture 36 - Further properties of Modules and Module of Fractions

Lecture 37 - Local-Global Principle

Lecture 38 - Consequences of Local-Global Principle

Lecture 39 - Properties of Artinian Rings

Lecture 40 - Krull-Nakayama Lemma

Lecture 41 - Properties of I K and V L maps

Lecture 42 - Hilbert’s Nullstelensatz

Lecture 43 - Hilbert’s Nullstelensatz (Continued...)

Lecture 44 - Proof of Zariski’s Lemma (HNS 3)

Lecture 45 - Consequences of HNS

Lecture 46 - Consequences of HNS (Continued...)

Lecture 47 - Jacobson Ring and examples

Lecture 48 - Irreducible subsets of Zariski Topology (Finite type K-algebra)

Lecture 49 - Spec functor on Finite type K-algebras

Lecture 50 - Properties of Irreducible topological spaces

Lecture 51 - Zariski Topology on arbitrary commutative rings

Lecture 52 - Spec functor on arbitrary commutative rings

Lecture 53 - Topological properties of Spec A

Lecture 54 - Example to support the term Spectrum

Lecture 55 - Integral Extensions

Lecture 56 - Elementwise characterization of Integral extensions

Lecture 57 - Properties and examples of Integral extensions

Lecture 58 - Prime and Maximal ideals in integral extensions

Lecture 59 - Lying over Theorem

Lecture 60 - Cohen-Siedelberg Theorem