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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)

 Introduction

 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

 References 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.