Introduction to homological algebra

The contents of the course will depend on the audience' background and interests. I plan to start with exact sequences of abelian groups, complexes and their cohomology, the snake lemma, 5-lemma, and chain homotopy. Then we will discuss nonexactness of Hom in the category of modules over a (noncommutative) ring, projective and injective modules, resolutions, and the Ext functor. This should be enough for the first lecture, and then we will see. Other topics to be covered include additive and abelian categories, additive functors and their derived functors, the homotopy and derived categories of abelian categories, triangulated categories and the Verdier localization, and semiorthogonal decompositions arising in connection with the injective and projective resolutions. Spectral sequences may or may not be covered.
Bibliography: before starting on homological algebra, it may be instructive to learn a bit of basic algebraic topology. So the audience is encouraged to look into the first chapters of the book by Fuchs and Fomenko, where they discuss the basic properties of the homology and homotopy groups of topological spaces. Fuchs-Fomenko also have an excellent discussion of spectral sequences. Concerning homological algebra proper, there are introductory textbooks by Rotman and Weibel, and a much more advanced book by Gelfand-Manin. One can also read about triangulated categories in Verdier's article in SGA 4 1/2.