20 credits / 30 lectures
Homological algebra was originally seen as a collection of algebraic tools which was used in homology theory of topological spaces. Today, it has already more than 50 year history of existence as an independent part of abstract algebra; it is a kind of a specific direction in generalized group/ring/module theory. In fact it is about exact sequences, complexes, homology, and derived functors, which are or can be used in the study of all kinds of algebraic, geometric, and other mathematical structures.
- Modules over rings, homomorphisms, kernels and cokernels, direct sums and products of modules, Hom modules
- Exact sequences of modules, classical homological lemmas
- Free, projective, and injective modules
- Categories, functors, and natural transformations. Hom functors. Products and coproducts in categories
- Additive and abelian categories
- Chain complexes, homology, homotopy of chain transformations, long exact sequence of homology modules
- Theory of derived functors and satellites
- Some applications of the above-listed constructions in algebraic topology and classical algebra, and their categorical generalizations.
Although the course is essentially self-contained, familiarity with groups and rings, and with their basic properties, as presented in undergraduate courses in algebra, is required.