Course11110616875 lectures
Introduction To Algebraic Geometry
Prof. Arijit Dey · IIT Madras
About
Commutative algebra and Algebraic geometry are twin topics. They are very much interrelated. Giving a course in algebraic geometry requires good grasp of commutative algebra.. This course will motivate students to learn more advanced topics in commutative algebra and algebraic geometry. I am hoping person who are interested to learn commutative algebra and algebraic geometry will see a beautiful mixture of these two important topics in this course.
Prerequisites: Knowledge in basic algebra (groups/rings/fields/modules). Litttle knowledge in commutative algebra for e.g. prime and maximal ideals of a ring, tensor product of modules, localisation. A little knowledge in general topology is also required.
Lectures
Week 1
- Topics in Commutative Algebra - Part 1
- Topics in Commutative Algebra - Part 2
- Topics in Commutative Algebra - Part 3
- Topics in Commutative Algebra - Part 4
- Topics in Commutative Algebra - Part 5
- Tutorial 1: Cayley-Hamilton Theorem, Nakayama's Lemma
Week 2
- Topics in Commutative Algebra: Part 6
- Topics in Commutative Algebra: Part 7
- Topics in Commutative Algebra: Part 8
- Affine Algebraic Sets - Part 1
- Affine Algebraic Sets - Part 2
- Tutorial 2: Noether Normalization Lemma, Some Important Results in Dimension Theory
Week 3
- Regular Morphisms
- Abstract Algebraic Sets
- Zariski Topology on Affine Space
- Irreducible Affine Algebraic Sets
- Ring of Regular Functions
- Projective Space
- Tutorial 3: Some Applications of Dimension Theory
Week 4
- Zariski Topology on Projective Space
- Affine Open Cover of Projective Space
- Projective & Quasi-Projective Varieties
- Regular Functions on Quasi-Projective Varieties
- Presheaves & Sheaves
- Morphism of Presheaves/Sheaves
- Tutorial 4: More Applications of Dimension Theory
Week 5
- A Brief Overview of Sheaf Theory - Part 1
- A Brief Overview of Sheaf Theory - Part 2
- A Brief Overview of Sheaf Theory - Part 3
- Prevarieties
- Sheaf of Regular Functions
- Ring of Germs of Regular Functions at a point, Field of Rational Functions
- Tutorial 5: Sheafification
Week 6
- Ring of Regular Functions, Local Ring at a Point, and Field of Rational Functions of an Affine Variety
- Equivalence of Categories of the Category of Affine Varieties over a Field k and the Category of Finitely Generated Integral Domains over k
- Equivalence of Categories of the Category of Affine Varieties over a Field k and the Category of Finitely Generated Integral Domains over k (Contd.)
- Some Examples, Open Immersions & Closed Immersions
- Product of Quasi-affine Varieties
- Diagonal Morphisms, Abstract Varieties
- Tutorial 6: Normal Varieties & Normalization of a Variety
Week 7
- Projective Varieties Revisited: Part 1
- Projective Varieties Revisited: Part 2
- Global Regular Functions on Projective Varieties are Constants : Part 1
- Global Regular Functions on Projective Varieties are Constants : Part 2
- Product of Prevarieties : Part 1
- Product of Prevarieties : Part 2
- Tutorial 7 : A Result on Tensor Products of k-algebras
Week 8
- Morphisms of Prevarieties : Part 1
- Morphisms of Prevarieties : Part 2
- Finite Morphisms : Part 1
- Finite Morphisms : Part 2
- Fiber Products
- Tutorial 8 : Finite Morphisms
Week 9
- Immersions
- Fiber Products, Separatedness
- Criterion of Separatedness
- Proper Morphisms & Complete Varieties
- Tutorial 9 : Closed Immersions & Graph of a Morphism
Week 10
- Projective Varieties are Complete
- Zariski Tangent Space, Singular & Nonsingular Points
- Smooth Points Form a Non-empty Open Subset
- Blow-Ups, Rational Maps & Birational Maps
- Tutorial 10 : Zariski Tangent Space at a Point of an Affine Variety
Week 11
- Blow-Ups ( Contd.)
- Smooth Morphisms
- Bertini's Theorem
- Sard's Theorem
- Tutorial 11 : Dimension of fiber of a morphism
Week 12
- Introduction to Affine Schemes : Spectrum of a Ring
- Introduction to Affine Schemes : Topology on Spec A
- Introduction to Affine Schemes : Topology on Spec A ( Contd. )
- Introduction to Affine Schemes : Sheaf Structure on Spec A
- Abstract Non-singular Curves : Part 1
- Abstract Non-singular Curves : Part 2
- Tutorial 12 : Extension of Regular Functions
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Madras
- Department
- Mathematics
- Teacher
- Prof. Arijit Dey
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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