Subsections

MTH604: Homological and commutative algebra

[Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

Homological Algebra

• Categories and functors, Derived functors, Hom and functors, Flat, projective and injective modules, Resolutions of modules, Ext and Tor functors.
• Cohomology of groups.

Commutative algebra

• Recollection of rings, ideals, Spec and MaxSpec of rings, Zariski topology.
• Modules, Finitely generated modules, Nakayama lemma, Localisation of rings and modules.
• Chain conditions, Noetherian rings, Hilbert basis theorem, Artinian rings, Noetherian and Artinian modules.
• Associated primes and Primary decomposition, Integral extensions, Going up and going down theorems, Noether normalisation theorem, Hilbert Nullstellensatz and their geometric interpretations.
• Valuation rings and Dedekind domains, Ideal class group.
• Direct and inverse limits, Completions, Graded rings and modules, Artin-Rees lemma.
• Dimension theory, Hilbert and Samuel functions, Dimension theorem, Krull's principal ideal theorem.
• Regular sequences, Depth, Cohen-Macaulay rings, Gorenstein rings, Regular rings.