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Introduction to Algebraic Geometry and Commutative Algebra. Instructor: Prof. Dilip P. Patil, Department of Mathematics, IISc Bangalore.
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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)
Course Currilcum

 Lecture 01 – Motivation for Kalgebraic Sets Unlimited
 Lecture 02 – Definitions and Examples of Affine Algebraic Set Unlimited
 Lecture 03 – Rings and Ideals Unlimited
 Lecture 04 – Operation on Ideals Unlimited
 Lecture 05 – Prime Ideals and Maximal Ideals Unlimited

 Lecture 06 – Krull’s Theorem and Consequences Unlimited
 Lecture 07 – Module, Submodules and Quotient Modules Unlimited
 Lecture 08 – Algebras and Polynomial Algebras Unlimited
 Lecture 09 – Universal Property of Polynomial Algebra and Examples Unlimited
 Lecture 10 – Finite and Finite Type Algebras Unlimited

 Lecture 11 – Kspectrum (Krational Points) Unlimited
 Lecture 12 – Identity Theorem for Polynomial Functions Unlimited
 Lecture 13 – Basic Properties of Kalgebraic Sets Unlimited
 Lecture 14 – Examples of Kalgebraic Sets Unlimited
 Lecture 15 – KZariski Topology Unlimited

 Lecture 16 – The Map VL Unlimited
 Lecture 17 – Noetherian and Artinian Ordered Sets Unlimited
 Lecture 18 – Noetherian Induction and Transfinite Induction Unlimited
 Lecture 19 – Modules and Chain Conditions Unlimited
 Lecture 20 – Properties of Noetherian and Artinian Modules Unlimited

 Lecture 21 – Examples of Artinian and Noetherian Modules Unlimited
 Lecture 22 – Finite Modules over Noetherian Rings Unlimited
 Lecture 23 – Hilbert’s Basis Theorem (HBT) Unlimited
 Lecture 24 – Consequences of HBT Unlimited
 Lecture 25 – Free Modules and Rank Unlimited

 Lecture 26 – More on Noetherian and Artinian Modules Unlimited
 Lecture 27 – Ring of Fractions (Localization) Unlimited
 Lecture 28 – Nil Radical, Contraction of Ideals Unlimited
 Lecture 29 – Universal Property of S1A Unlimited
 Lecture 30 – Ideal Structure in S1A Unlimited

 Lecture 31 – Consequences of the Correspondence of Ideals Unlimited
 Lecture 32 – Consequences of the Correspondence of Ideals (cont.) Unlimited
 Lecture 33 – Modules of Fraction and Universal Properties Unlimited
 Lecture 34 – Exactness of the Functor S1 Unlimited
 Lecture 35 – Universal Property of Modules of Fractions Unlimited

 Lecture 36 – Further Properties of Modules and Module of Fractions Unlimited
 Lecture 37 – LocalGlobal Principle Unlimited
 Lecture 38 – Consequences of LocalGlobal Principle Unlimited
 Lecture 39 – Properties of Artinian Rings Unlimited
 Lecture 40 – KrullNakayama Lemma Unlimited

 Lecture 41 – Properties of IK and VL Maps Unlimited
 Lecture 42 – Hilbert’s Nullstellensatz Unlimited
 Lecture 43 – Hilbert’s Nullstellensatz (cont.) Unlimited
 Lecture 44 – Proof of Zariski’s Lemma (HNS 3) Unlimited
 Lecture 45 – Consequences of HNS Unlimited

 Lecture 46 – Consequences of HNS (cont.) Unlimited
 Lecture 47 – Jacobson Ring and Examples Unlimited
 Lecture 48 – Irreducible Subsets of Zariski Topology (Finite Type Kalgebra) Unlimited
 Lecture 49 – Spec Functor on Finite Type Kalgebras Unlimited
 Lecture 50 – Properties of Irreducible Topological Spaces Unlimited

 Lecture 51 – Zariski Topology on Arbitrary Commutative Rings Unlimited
 Lecture 52 – Spec Functor on Arbitrary Commutative Rings Unlimited
 Lecture 53 – Topological Properties of Spec A Unlimited
 Lecture 54 – Example to Support the Term Spectrum Unlimited
 Lecture 55 – Integral Extensions Unlimited

 Lecture 56 – Elementwise Characterization of Integral Extensions Unlimited
 Lecture 57 – Properties and Examples of Integral Extensions Unlimited
 Lecture 58 – Prime and Maximal Ideals in Integral Extensions Unlimited
 Lecture 59 – Lying over Theorem Unlimited
 Lecture 60 – CohenSiedelberg Theorem Unlimited