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Computational Commutative Algebra

Prof. Manoj Kummini · Chennai Mathematical Institute

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About

This is an introductory course in computational commutative algebra. Topics in a typical first course in commutative algebra are developed along with computations in Macaulay2. The emphasis will be on concrete computations, more than on giving complete proofs of theorems.

INTENDED AUDIENCE : Advanced undergraduate / post-graduate students

PREREQUISITES : Introduction to basic theory of rings, modules

Lectures

Week 1

  1. Definitions YouTube
  2. Homomorphisms YouTube
  3. Quotient rings YouTube
  4. Noetherian rings YouTube
  5. Monomials YouTube

Week 2

  1. Initial ideals YouTube
  2. Division algorithm YouTube
  3. Grobner basis YouTube
  4. Solving Polynomial Equations YouTube
  5. Nullstellensatz - Part 1 YouTube

Week 3

  1. Nullstellensatz - Part 2 YouTube
  2. Buchberger criterion YouTube
  3. Monomial basis YouTube
  4. Elimination YouTube
  5. Modules - Part 1 YouTube

Week 4

  1. Modules - Part 2 YouTube
  2. Localisation YouTube
  3. Nakayama Lemma YouTube
  4. Spectrum - Part 1 YouTube
  5. Spectrum - Part 2 YouTube

Week 5

  1. Associated primes YouTube
  2. Primary Decomposition YouTube
  3. Support of a module YouTube
  4. Associated primes YouTube
  5. Prime avoidance YouTube

Week 6

  1. Saturation - Part 1 YouTube
  2. saturation - Part 2 YouTube
  3. saturation - Part 3 YouTube
  4. Morphisms - Part 1 YouTube
  5. Morphisms - Part 2 YouTube

Week 7

  1. Integral extensions YouTube
  2. Noether normalisation lemma YouTube
  3. Noether normalisation lemma YouTube
  4. Polynomial rings YouTube
  5. Going up theorem YouTube

Week 8

  1. Artinian rings YouTube
  2. Graded modules YouTube
  3. Hilbert polynomial YouTube
  4. Hilbert-Samuel polynomial YouTube
  5. Artin Rees Lemma YouTube

Week 9

  1. Degree of Hilbert-Samuel polynomial YouTube
  2. Dimension of noetherian local rings - Part 1 YouTube
  3. Dimension of noetherian local rings - Part 2 YouTube
  4. Dimension of polynomial rings YouTube
  5. Algebras over a field YouTube

Week 10

  1. Graded rings - Part 1 YouTube
  2. Graded rings - Part 2 YouTube
  3. Polynomial rings over fields YouTube
  4. Hilbert series - Part 1 YouTube
  5. Hilbert series - Part 2 YouTube

Week 11

  1. Proj of a graded ring YouTube
  2. Homogenization - Part 1 YouTube
  3. Homogenization - Part 2 YouTube
  4. More on graded rings YouTube
  5. Free resolutions YouTube

Week 12

  1. Computing syzygies YouTube
  2. Koszul complex YouTube
  3. More on Koszul complexes YouTube
  4. Castelnuovo Mumford regularity YouTube
  5. Castelnuovo Mumford regularity YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
Chennai Mathematical Institute
Department
Mathematics
Teacher
Prof. Manoj Kummini
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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