This course note introduces
the reader to the language of categories and to present the basic notions of
homological algebra, first from an elementary point of view, with the notion of
derived functors, next with a more sophisticated approach, with the introduction
of triangulated and derived categories.
Covered topics are: General
manipulations of complexes, More on Koszul complexes, General manipulations
applied to projective resolutions, Tor, Regular rings, review of Krull
dimension, Regular sequences and Tor, Cohen–Macaulay rings and modules,
Injective and divisible modules, Injective resolutions, A definition of Ext
using injective resolutions, Duality and injective hulls, Gorenstein rings, Bass
This note covers the following topics: Reflections , Adjoint functors ,
Chain complexes, Homology , Homological algebra, First approximation to derived
functors, Bar resolutions and the classical theory of derived functors, Double
complexes, Long exact sequences, Diagrammatics , The third fundamental lemma,
Generators and cogenerators.