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Course11110813662 lectures

Introduction to Algebraic Geometry and Commutative Algebra

Prof. Dilip P. Patil · IISc Bangalore

Start with lecture 1 At NPTEL

Lectures

Week 1 Algebraic Preliminaries 1- Rings and Ideals

  1. Motivation for K-algebraic sets Video
  2. Definitions and examples of Affine Algebraic Set Video
  3. Rings and Ideals Video
  4. Operation on Ideals Video
  5. Prime Ideals and Maximal Ideals Video

Week 2 Algebraic Preliminaries 2-Modules and Algebras

  1. Krull's Theorem and consequences Video
  2. Module, submodules and quotient modules Video
  3. Algebras and polynomial algebras Video
  4. Universal property of polynomial algebra and examples Video
  5. Finite and Finite type algebras Video

Week 3 The K-Spectrum of a K-algebra and Affine algebraic sets

  1. K-Spectrum (K-rational points) Video
  2. Identity theorem for Polynomial functions Video
  3. Basic properties of K-algebraic sets Video
  4. Examples of K-algebraic sets Video
  5. K-Zariski Topology Video

Week 4 Noetherian and Artinian Modules

  1. The map V L Video
  2. Noetherian and Artinian Ordered sets Video
  3. Noetherian induction and Transfinite induction Video
  4. Modules with Chain Conditions Video
  5. Properties of Noetherian and Artinian Modules Video

Week 5 Hilbert’s Basis Theorem and consequences

  1. Examples of Artinian and Noetherian Modules Video
  2. Finite modules over Noetherian Rings Video
  3. Hilbert's Basis Theorem(HBT) Video
  4. Consequences of HBT Video
  5. Free Modules and rank Video

Week 6 Rings of Fractions

  1. More on Noetherian and Artinian modules Video
  2. Ring of Fractions(Localization) Video
  3. Nil radical, contraction of ideals Video
  4. Universal property of S -1 A Video
  5. Ideal structure in S -1 A Video

Week 7 Modules of fractions

  1. Consequences of the Correspondence of Ideals Video
  2. Consequences of the Correspondence of Ideals(Contd.) Video
  3. Modules of Fraction and universal properties Video
  4. Exactness of the functor S -1 Video
  5. Universal property of Modules of Fractions Video

Week 8 Local Global Principle and Consequences

  1. Further properties of Modules and Module of Fractions Video
  2. Local-Global Principle Video
  3. Consequences of Local-Global Principle Video
  4. Properties of Artinian Rings Video
  5. Krull-Nakayama Lemma Video

Week 9 Hilbert's Nullstellensatz and its equivalent formulations

  1. Properties of I K and V L maps Video
  2. Hilbert's Nullstelensatz Video
  3. Hilbert's Nullstelensatz(Contd.) Video
  4. Proof of Zariski's Lemma(HNS 3) Video
  5. Consequences of HNS Video

Week 10 Consequences of HNS

  1. Consequences of HNS(Contd.) Video
  2. Jacobson Ring and examples Video
  3. Irreducible subsets of Zariski Topology(Finite type K-algebra) Video
  4. Spec functor on Finite type K-algebras Video
  5. Properties of Irreducible topological spaces Video

Week 11 Zariski Topology

  1. Zariski Topology on arbitrary commutative rings Video
  2. Spec functor on arbitrary commutative rings Video
  3. Topological properties of Spec A Video
  4. Example to support the term Spectrum- Video
  5. Integral Extensions Video

Week 12 Integral Extensions

  1. Elementwise characterization of Integral extensions Video
  2. Properties and examples of Integral extensions Video
  3. Prime and Maximal ideals in integral extensions Video
  4. Lying over Theorem Video
  5. Cohen-Siedelberg Theorem Video

Live Session

  1. Live Session 17-03-2022 Video
  2. Live Session 03-03-2025 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mathematics
Teacher
Prof. Dilip P. Patil
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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