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Last updated:

September 23, 2023

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Unlimited Duration

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Description

Basic Algebraic Geometry. Instructor: Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. This course is an introduction to Algebraic Geometry, whose aim is to study the geometry underlying the set of common zeros of a collection of polynomial equations.

It sets up the language of varieties and of morphisms between them, and studies their topological and manifold-theoretic properties. Commutative Algebra is the "calculus" that Algebraic Geometry uses. Therefore a prerequisite for this course would be a course in Algebra covering basic aspects of commutative rings and some field theory, as also a course on elementary Topology. However, the necessary results from Commutative Algebra and Field Theory would be recalled as and when required during the course for the benefit of the students. (from nptel.ac.in)

Course Curriculum

    • Lecture 01 – What is Algebraic Geometry? Unlimited
    • Lecture 02 – The Zariski Topology and Affine Space Unlimited
    • Lecture 03 – Going Back and Forth between Subsets and Ideals Unlimited
    • Lecture 04 – Irreducibility in the Zariski Topology Unlimited
    • Lecture 05 – Irreducible Closed Subsets Correspond to Ideals whose Radicals are Prime Unlimited
    • Lecture 06 – Understanding the Zariski Topology on the Affine Line Unlimited
    • Lecture 07 – Basic Algebraic Geometry: Varieties, Morphisms, Local Rings … Unlimited
    • Lecture 08 – Topological Dimension, Krull Dimension and Heights of Prime Ideals Unlimited
    • Lecture 09 – The Ring of Polynomial Functions on an Affine Variety Unlimited
    • Lecture 10 – Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces Unlimited
    • Lecture 11 – Why should We Study Affine Coordinate Rings of Functions on Affine Varieties? Unlimited
    • Lecture 12 – Capturing an Affine Variety Topologically from the Maximal Spectrum of … Unlimited
    • Lecture 13 – Analyzing Open Sets and Basic Open Sets for the Zariski Topology Unlimited
    • Lecture 14 – The Ring of Functions on a Basic Open Set in the Zariski Topology Unlimited
    • Lecture 15 – Quasi-Compactness in the Zariski Topology Unlimited
    • Lecture 16 – What is a Global Regular Function on a Quasi-Affine Variety? Unlimited
    • Lecture 17 – Characterizing Affine Varieties Unlimited
    • Lecture 18 – Translating Morphisms into Affines as k-Algebra Maps and … Unlimited
    • Lecture 19 – Morphisms into an Affine Correspond to k-Algebra Homomorphisms from … Unlimited
    • Lecture 20 – The Coordinate Ring of an Affine Variety Determines the Affine Variety and … Unlimited
    • Lecture 21 – Automorphisms of Affine Spaces and of Polynomial Rings Unlimited
    • Lecture 22 – The Various Avatars of Projective n-Space Unlimited
    • Lecture 24 – Translating Projective Geometry into Graded Rings and Homogeneous Ideals Unlimited
    • Lecture 25 – Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties Unlimited
    • Lecture 26 – Translating Homogeneous Localization into Geometry and Back Unlimited
    • Lecture 27 – Adding a Variable is Undone by Homogeneous Localization Unlimited
    • Lecture 28 – Doing Calculus without Limits in Geometry Unlimited
    • Lecture 29 – The Birth of Local Rings in Geometry and in Algebra Unlimited
    • Lecture 30 – The Formula for the Local Ring at a Point of a Projective Variety or … Unlimited
    • Lecture 31 – The Field of Rational Functions or Function Field of a Variety Unlimited
    • Lecture 32 – Fields of Rational Functions or Function Field of Affine and Projective Varieties Unlimited
    • Lecture 33 – Global Regular Functions on Projective Varieties are Simply the Constants Unlimited
    • Lecture 34 – The d-Uple Embedding and the Non-intrinsic Nature of the Homogeneous … Unlimited
    • Lecture 35 – The Importance of Local Rings – A Morphism is an Isomorphism … Unlimited
    • Lecture 36 – The Importance of Local Rings – A Rational Function in Every Local Ring is … Unlimited
    • Lecture 37 – Geometric Meaning of Isomorphism of Local Rings – Local Rings are Almost Global Unlimited
    • Lecture 38 – Local Ring Isomorphism – Equals Function Field Isomorphism – Equals Birationality Unlimited
    • Lecture 39 – Why Local Rings Provide Calculus without Limits for Algebraic Geometry Pun Intended Unlimited
    • Lecture 40 – How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry Unlimited
    • Lecture 41 – Any Variety is a Smooth Manifold with or without Nonsmooth Boundary Unlimited
    • Lecture 42 – Any Variety is a Smooth Hypersurface on an Open Dense Subset Unlimited

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