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Introduction to Algebraic Geometry and Commutative Algebra

Introduction to Algebraic Geometry and Commutative Algebra. Instructor: Prof. Dilip P. Patil, Department of Mathematics, IISc Bangalore. Algebraic geometry played a central role in 19th century math. The deepest results of Abel, Riemann, Weierstrass, and the most important works of Klein and Poincare were part of this subject. In the middle of the 20th century algebraic geometry had been through a large reconstruction. The domain of application of its ideas had grown tremendously, in the direction of algebraic varieties over arbitrary fields and more general complex manifolds. Many of the best achievements of algebraic geometry could be cleared of the accusation of incomprehensibility or lack of rigor. The foundation for this reconstruction was (commutative) algebra. In the 1950s and 60s have brought substantial simplifications to the foundation of algebraic geometry, which significantly came closer to the ideal combination of logical transparency and geometric intuition. Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers in number fields constitute an important class of commutative rings - the Dedekind domains. This has led to the notions of integral extensions and integrally closed domains. The notion of localization of a ring (in particular the localization with respect to a prime ideal leads to an important class of commutative rings - the local rings. The set of the prime ideals of a commutative ring is naturally equipped with a topology - the Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory - a generalization of algebraic geometry introduced by Grothendiek. (from nptel.ac.in)

Lecture 38 - Consequences of Local-Global Principle


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Algebraic Preliminaries: Rings and Ideals
Lecture 01 - Motivation for K-algebraic Sets
Lecture 02 - Definitions and Examples of Affine Algebraic Set
Lecture 03 - Rings and Ideals
Lecture 04 - Operation on Ideals
Lecture 05 - Prime Ideals and Maximal Ideals
Algebraic Preliminaries: Modules and Algebras
Lecture 06 - Krull's Theorem and Consequences
Lecture 07 - Module, Submodules and Quotient Modules
Lecture 08 - Algebras and Polynomial Algebras
Lecture 09 - Universal Property of Polynomial Algebra and Examples
Lecture 10 - Finite and Finite Type Algebras
The K-spectrum of a K-algebra and Affine Algebraic Sets
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
Noetherian and Artinian Modules
Lecture 16 - The Map VL
Lecture 17 - Noetherian and Artinian Ordered Sets
Lecture 18 - Noetherian Induction and Transfinite Induction
Lecture 19 - Modules and Chain Conditions
Lecture 20 - Properties of Noetherian and Artinian Modules
Hilbert's Basis Theorem and Consequences
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
Rings of Fractions
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-1A
Lecture 30 - Ideal Structure in S-1A
Modules of Fractions
Lecture 31 - Consequences of the Correspondence of Ideals
Lecture 32 - Consequences of the Correspondence of Ideals (cont.)
Lecture 33 - Modules of Fraction and Universal Properties
Lecture 34 - Exactness of the Functor S-1
Lecture 35 - Universal Property of Modules of Fractions
Local Global Principle and Consequences
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
Hilbert's Nullstellensatz and its Equivalent Formulations
Lecture 41 - Properties of IK and VL Maps
Lecture 42 - Hilbert's Nullstellensatz
Lecture 43 - Hilbert's Nullstellensatz (cont.)
Lecture 44 - Proof of Zariski's Lemma (HNS 3)
Lecture 45 - Consequences of HNS
Consequences of HNS
Lecture 46 - Consequences of HNS (cont.)
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
Zariski Topology
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
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