In the first part of the course classical homological algebra is developed within the framework of abelian categories. This involves the following topics: chain complexes, homotopy and homology, injective and projective resolutions, derived functors, Tor and Ext (especially in the context of abelian groups and modules).
The second part of the course consists of the in-depth study of homological methods in some specific topological, algebraic or geometric contexts. Possible topics include:
- homological dimensions in algebra
- group (co)homology, Lie algebra (co)homology
- Hochschild (co)homology of algebras
- simplicial methods and the link with algebraic topology
- sheaf cohomology in de algebraic geometry
- derived and triangualted categories