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