Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity

Course Description

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.

Algebraic Geometry in its generality is connected to various areas of Mathematics such as Complex Analysis, PDE, Complex Manifolds, Homological Algebra, Field and Galois Theory, Sheaf Theory and Cohomology, Algebraic Topology, Number Theory, QuadraticForms, Representation Theory, Combinatorics, Commutative Ring Theory etc and also to areas of Physics like String Theory and Cosmology. Many of the Fields Medals awarded till date are for research in areas connected in a non-trivial way to Algebraic Geometry directly or indirectly. The Taylor-Wiles proof of Fermat's Last Theorem used the full machinery and power of the language of Schemes, the most sophisticated language of Algebraic Geometry developed over a couple of decades from the 1960s by Alexander Grothendieck in his voluminous expositions running to several thousand pages. The foundations laid in this course will help in a further study of the language of schemes.

Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity
Commutative Algebra is the "calculus" that Algebraic Geometry uses. Here is the universal mapping property (UMP) of the residue class ring. (Image by MIT OpenCourseWare.)
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Video Lectures & Study Materials

Visit the official course website for more study materials: http://nptel.ac.in/syllabus/111106097/

# Lecture Play Lecture
I. The Zariski Topology
1 What is Algebraic Geometry? Play Video
2 The Zariski Topology and Affine Space Play Video
3 Going back and forth between subsets and ideals Play Video
II. Irreducibility in the Zariski Topology
4 Irreducibility in the Zariski Topology Play Video
5 Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime Play Video
III. Noetherianness in the Zariski Topology
6 Understanding the Zariski Topology on the Affine Line Play Video
7 The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties Play Video
IV. Dimension and Rings of Polynomial Functions
8 Topological Dimension, Krull Dimension and Heights of Prime Ideals Play Video
9 The Ring of Polynomial Functions on an Affine Variety Play Video
10 Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces Play Video
V. The Affine Coordinate Ring of an Affine Variety
11 Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ? Play Video
12 Capturing an Affine Variety Topologically Play Video
VI. Open sets in the Zariski Topology and Functions on such sets
13 Analyzing Open Sets and Basic Open Sets for the Zariski Topology Play Video
14 The Ring of Functions on a Basic Open Set in the Zariski Topology Play Video
VII. Regular Functions in Affine Geometry
15 Quasi-Compactness in the Zariski Topology Play Video
16 What is a Global Regular Function on a Quasi-Affine Variety? Play Video
VIII. Morphisms in Affine Geometry
17 Characterizing Affine Varieties Play Video
18 Translating Morphisms into Affines as k-Algebra maps Play Video
19 Morphisms into an Affine Correspond to k-Algebra Homomorphisms Play Video
20 The Coordinate Ring of an Affine Variety Play Video
21 Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture Play Video
IX. The Zariski Topology on Projective Space and Projective Varieties
22 The Various Avatars of Projective n-space Play Video
23 Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology Play Video
X. Graded Rings, Homogeneous Ideals and Homogeneous Localisation
24 Translating Projective Geometry into Graded Rings and Homogeneous Ideals Play Video
25 Expanding the Category of Varieties Play Video
26 Translating Homogeneous Localisation into Geometry and Back Play Video
27 Adding a Variable is Undone by Homogenous Localization Play Video
XI. The Local Ring of Germs of Functions at a Point
28 Doing Calculus Without Limits in Geometry Play Video
29 The Birth of Local Rings in Geometry and in Algebra Play Video
30 The Formula for the Local Ring at a Point of a Projective Variety Play Video

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