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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 the basic theory of rings, modules
INDUSTRIES SUPPORT: None
Computational Commutative Algebra - Intro video.
mod01lec01 - Definitions.
mod01lec02 - Homomorphisms.
mod01lec03 - Quotient rings.
mod01lec04 - Noetherian rings.
mod01lec05 - Monomials.
mod02lec06 - Initial ideals.
mod02lec07 - Division algorithm.
mod02lec08 - Grobner basis.
mod02lec09 - Solving Polynomial Equations.
mod02lec10 - Nullstellensatz - Part 1.
mod03lec11 - Nullstellensatz - Part 2.
mod03lec12 - Buchberger criterion.
mod03lec13 - Monomial basis.
mod03lec14 - Elimination.
mod03lec15 - Modules - Part 1.
mod04lec16 - Modules - Part 2.
mod04lec17 - Localisation.
mod04lec18 - Nakayama Lemma.
mod04lec19 - Spectrum - Part 1.
mod04lec20 - Spectrum - Part 2.
mod05lec21 - Associated primes.
mod05lec22 - Primary Decomposition.
mod05lec23 - Support of a module.
mod05lec24 - Associated primes.
mod05lec25 - Prime avoidance.
mod06lec26 - Saturation - Part 1.
mod06lec27 - saturation - Part 2.
mod06lec28 - saturation - Part 3.
mod06lec29 - Morphisms - Part 1.
mod06lec30 - Morphisms - Part 2.
mod07lec31 - Integral extensions.
mod07lec32 - Noether normalisation lemma.
mod07lec33 - Noether normalisation lemma.
mod07lec34 - Polynomial rings.
mod07lec35 - Going up theorem.
mod08lec36 - Artinian rings.
mod08lec37 - Graded modules.
mod08lec38 - Hilbert polynomial.
mod08lec39 - Hilbert-Samuel polynomial.
mod08lec40 - Artin Rees Lemma.
mod09lec41 - Degree of Hilbert-Samuel polynomial.
mod09lec42 - Dimension of noetherian local rings - Part 1.
mod09lec43 - Dimension of noetherian local rings - Part 2.
mod09lec44 - Dimension of polynomial rings.
mod09lec45 - Algebras over a field.
mod10lec46 - Graded rings - Part 1.
mod10lec47 - Graded rings - Part 2.
mod10lec48 - Polynomial rings over fields.
mod10lec49 - Hilbert series - Part 1.
mod10lec50 - Hilbert Series - Part 2.
mod11lec51 - Proj of a graded ring.
mod11lec52 - Homogenization - Part 1.
mod11lec53 - Homogenization - Part 2.
mod11lec54 - More on graded rings.
mod11lec55 - Free resolutions.
mod12lec56 - Computing syzygies.
mod12lec57 - Koszul complex.
mod12lec58 - More on Koszul complexes.
mod12lec59 - Castelnuovo Mumford regularity - Part 1.
mod12lec60 - Castelnuovo Mumford regularity - Part 2.
NPTEL-NOC IITM
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