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# Introduction to Algebraic Geometry and Commutative Algebra

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### Overview

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 Poincar/’e 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 rigour. 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 arewidely used in algebraic geometry and are the basic technical tools for the definition of scheme theory — a generalization of algebraic geometry introduced by Grothendieck.INTENDED AUDIENCE : BS / ME / MSc / PhDPREREQUISITES : Linear Algebra ;Algebra – First CourseINDUSTRY SUPPORT : R & D Departments of IBM / Microsoft Research Labs SAP /TCS / Wipro / Infosys

### Syllabus

Week 01 :Algebraic Preliminaries 1 --- Rings and Ideals
Week 02 :Algebraic Preliminaries 2 --- Modules and Algebras Week 03 :The K -Spectrum of a K -algebra and Affine algebraic sets Week 04 : Noetherian and Artinian Modules Week 05 :Hilbert's Basis Theorem and Consequences Week 06 :Rings of Fractions Week 07 :Modules of Fractions Week 08 : Local Global Principle and Consequences Week 09 :Hilbert’s Nullstellensatz and its equivalent formulations Week 10 : Consequences of HNS Week 11 :Zariski Topology Week 12 :Integral Extensions

### Taught by

Prof. Dilip P. Patil

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