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

- Institution: IIT Madras
- Department: Mathematics
- Teacher: Dr. T.E. Venkata Balaji
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106097)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106097)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

## Lectures

1. [What is Algebraic Geometry?](https://invisiblenetwork.ai/learn/nptel/111106097/b950e1b51ca3) (Video)
2. [The Zariski Topology and Affine Space](https://invisiblenetwork.ai/learn/nptel/111106097/5cdf9b4828dc) (Video)
3. [Going back and forth between subsets and ideals](https://invisiblenetwork.ai/learn/nptel/111106097/e971599ea83f) (Video)
4. [Irreducibility in the Zariski Topology](https://invisiblenetwork.ai/learn/nptel/111106097/f81a1b4e7cec) (Video)
5. [Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime](https://invisiblenetwork.ai/learn/nptel/111106097/c4e81c5c6962) (Video)
6. [Understanding the Zariski Topology on the Affine Line; The Noetherian property in Topology and in Algebra](https://invisiblenetwork.ai/learn/nptel/111106097/64e906f92406) (Video)
7. [Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity](https://invisiblenetwork.ai/learn/nptel/111106097/43b61ba49a1e) (Video)
8. [Topological Dimension, Krull Dimension and Heights of Prime Ideals](https://invisiblenetwork.ai/learn/nptel/111106097/b3797d25e4af) (Video)
9. [The Ring of Polynomial Functions on an Affine Variety](https://invisiblenetwork.ai/learn/nptel/111106097/09db22c1bd61) (Video)
10. [Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces](https://invisiblenetwork.ai/learn/nptel/111106097/e02c197ec43a) (Video)
11. [Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?](https://invisiblenetwork.ai/learn/nptel/111106097/16c9d27d76d6) (Video)
12. [Capturing an Affine Variety Topologically From the Maximal Spectrum of its Ring of Functions](https://invisiblenetwork.ai/learn/nptel/111106097/221b54749609) (Video)
13. [Analyzing Open Sets and Basic Open Sets for the Zariski Topology](https://invisiblenetwork.ai/learn/nptel/111106097/ed860138420c) (Video)
14. [The Ring of Functions on a Basic Open Set in the Zariski Topology](https://invisiblenetwork.ai/learn/nptel/111106097/657c50f779a1) (Video)
15. [Quasi-Compactness in the Zariski Topology; Regularity of a Function at a point of an Affine Variety](https://invisiblenetwork.ai/learn/nptel/111106097/4fe8acd294d0) (Video)
16. [What is a Global Regular Function on a Quasi-Affine Variety?](https://invisiblenetwork.ai/learn/nptel/111106097/5f984517ee61) (Video)
17. [Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties](https://invisiblenetwork.ai/learn/nptel/111106097/93f437d95f60) (Video)
18. [Translating Morphisms into Affines as k-Algebra maps and the Grand Hilbert Nullstellensatz](https://invisiblenetwork.ai/learn/nptel/111106097/c1c1d4a03d5a) (Video)
19. [Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions](https://invisiblenetwork.ai/learn/nptel/111106097/a7e9b67272d5) (Video)
20. [The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to it](https://invisiblenetwork.ai/learn/nptel/111106097/bd7d0cef1d2c) (Video)
21. [Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture; The Punctured Plane is Not Affine](https://invisiblenetwork.ai/learn/nptel/111106097/e3b174666204) (Video)
22. [The Various Avatars of Projective n-space](https://invisiblenetwork.ai/learn/nptel/111106097/2800948f4335) (Video)
23. [Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology, Manifold Theory and Algebraic Geometry; The Key to the Definition of a Homogeneous Ideal](https://invisiblenetwork.ai/learn/nptel/111106097/d6bc5a38758c) (Video)
24. [Translating Projective Geometry into Graded Rings and Homogeneous Ideals](https://invisiblenetwork.ai/learn/nptel/111106097/b5b5676ee801) (Video)
25. [Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties](https://invisiblenetwork.ai/learn/nptel/111106097/eefc4d0f11ba) (Video)
26. [Translating Homogeneous Localisation into Geometry and Back](https://invisiblenetwork.ai/learn/nptel/111106097/a9443c255023) (Video)
27. [Adding a Variable is Undone by Homogenous Localization - What is the Geometric Significance of this Algebraic Fact ?](https://invisiblenetwork.ai/learn/nptel/111106097/fef07e5f5aa5) (Video)
28. [Doing Calculus Without Limits in Geometry ?](https://invisiblenetwork.ai/learn/nptel/111106097/42bdee647eb7) (Video)
29. [The Birth of Local Rings in Geometry and in Algebra](https://invisiblenetwork.ai/learn/nptel/111106097/6d4dce88945c) (Video)
30. [The Formula for the Local Ring at a Point of a Projective Variety Or Playing with Localisations, Quotients, Homogenisation and Dehomogenisation !](https://invisiblenetwork.ai/learn/nptel/111106097/03b804f24048) (Video)
31. [Global Regular Functions on Projective Varieties are Simply the Constants](https://invisiblenetwork.ai/learn/nptel/111106097/0714fe8386f1) (Video)
32. [The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety](https://invisiblenetwork.ai/learn/nptel/111106097/85e80b3d2e78) (Video)
33. [The Importance of Local Rings - A Morphism is an Isomorphism if it is a Homeomorphism and Induces Isomorphisms at the Level of Local Rings](https://invisiblenetwork.ai/learn/nptel/111106097/11243696ecb9) (Video)
34. [The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular](https://invisiblenetwork.ai/learn/nptel/111106097/e2d191ae407f) (Video)
35. [Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global](https://invisiblenetwork.ai/learn/nptel/111106097/26cf96f12418) (Video)
36. [Local Ring Isomorphism,Equals Function Field Isomorphism, Equals Birationality](https://invisiblenetwork.ai/learn/nptel/111106097/1e8b654daae0) (Video)
37. [Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!](https://invisiblenetwork.ai/learn/nptel/111106097/1dc974cc7271) (Video)
38. [How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry](https://invisiblenetwork.ai/learn/nptel/111106097/0d8fac96f7df) (Video)
39. [Any Variety is a Smooth Manifold with or without Non-Smooth Boundary](https://invisiblenetwork.ai/learn/nptel/111106097/784e6ea899b0) (Video)
40. [Any Variety is a Smooth Hypersurface On an Open Dense Subset](https://invisiblenetwork.ai/learn/nptel/111106097/503819bd51ec) (Video)