Mathematics - Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity (English)

   Saturday, May 17, 2025   

Starting a new Lecture Notes Series on Mathematics - Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity

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Mathematics - Basic Algebraic Geometry : Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity By Lecture Notes together!

Lecture 1: Mod-01 Lec-01 What is Algebraic Geometry?

Lecture 2: Mod-01 Lec-02 The Zariski Topology and Affine Space

Lecture 3: Mod-01 Lec-03 Going back and forth between subsets and ideals

Lecture 4: Mod-02 Lec-04 Irreducibility in the Zariski Topology

Lecture 5: Mod-02 Lec-05 Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime

Lecture 6: Mod-03 Lec-06 Understanding the Zariski Topology on the Affine Line

Lecture 7: Mod-03 Lec-07 The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties

Lecture 8: Mod-04 Lec-08 Topological Dimension, Krull Dimension and Heights of Prime Ideals

Lecture 9: Mod-04 Lec-09 The Ring of Polynomial Functions on an Affine Variety

Lecture 10: Mod-04 Lec-10 Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces

Lecture 11: Mod-05 Lec-11 Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?

Lecture 12: Mod-05 Lec-12 Capturing an Affine Variety Topologically

Lecture 13: Mod-06 Lec-13 Analyzing Open Sets and Basic Open Sets for the Zariski Topology

Lecture 14: Mod-06 Lec-14 The Ring of Functions on a Basic Open Set in the Zariski Topology

Lecture 15: Mod-07 Lec-15 Quasi-Compactness in the Zariski Topology

Lecture 16: Mod-07 Lec-16 What is a Global Regular Function on a Quasi-Affine Variety?

Lecture 17: Mod-08 Lec-17 Characterizing Affine Varieties

Lecture 18: Mod-08 Lec-18 Translating Morphisms into Affines as k-Algebra maps

Lecture 19: Mod-08 Lec-19 Morphisms into an Affine Correspond to k-Algebra Homomorphisms

Lecture 20: Mod-08 Lec-20 The Coordinate Ring of an Affine Variety

Lecture 21: Mod-08 Lec-21 Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture

Lecture 22: Mod-09 Lec-22 The Various Avatars of Projective n-space

Lecture 23: Mod-09 Lec-23 Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology

Lecture 24: Mod-10 Lec-24 Translating Projective Geometry into Graded Rings and Homogeneous Ideals

Lecture 25: Mod-10 Lec-25 Expanding the Category of Varieties

Lecture 26: Mod-10 Lec-26 Translating Homogeneous Localisation into Geometry and Back

Lecture 27: Mod-10 Lec-27 Adding a Variable is Undone by Homogenous Localization

Lecture 28: Mod-11 Lec-28 Doing Calculus Without Limits in Geometry

Lecture 29: Mod-11 Lec-29 The Birth of Local Rings in Geometry and in Algebra

Lecture 30: Mod-11 Lec-30 The Formula for the Local Ring at a Point of a Projective Variety

Lecture 31: Mod-12 Lec 31 The Field of Rational Functions or Function Field of a Variety

Lecture 32: Mod-12 Lec 32 Fields of Rational Functions or Function Fields of Affine and Projective Varieties

Lecture 33: Mod-13 Lec 33 Global Regular Functions on Projective Varieties are Simply the Constants

Lecture 34: Mod-13 Lec 34 The d-Uple Embedding and the Non-Intrinsic Nature of the Homogeneous Coordinate Ring

Lecture 35: Mod-14 Lec 35 The Importance of Local Rings - A Morphism is an Isomorphism

Lecture 36: Mod-14 Lec 36 The Importance of Local Rings

Lecture 37: Mod-14 Lec 37 Geometric Meaning of Isomorphism of Local Rings

Lecture 38: Mod-14 Lec 38 Local Ring Isomorphism, Equals Function Field Isomorphism, Equals Birationality

Lecture 39: Mod-15 Lec 39 Why Local Rings Provide Calculus Without Limits for Algebraic Geometry Pun Intended!

Lecture 40: Mod-15 Lec 40 How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry

Lecture 41: Mod-15 Lec 41 Any Variety is a Smooth Manifold with or without Non-Smooth Boundary

Lecture 42: Mod-15 Lec 42 Any Variety is a Smooth Hypersurface On an Open Dense Subset