First-year knowledge of groups and modules.
This will be an introduction to the subject of Homological Algebra, which is a tool used in many branches of mathematics, especially in Algebra, Topology and Algebraic Geometry.
The first part of the course will cover Chain Complexes, Projective and Injective Modules, Derived Functors, Ext and Tor, Universal Coefficients. In addition, some basic notions of Category Theory will be presented: adjoint functors, abelian categories, natural transformations, limits and colimits.
The second part of the course will cover topics determined by the interests of the students in the class. Possible topics are: Spectral Sequences, Homology of Groups and Lie Algebras, Derived Categories, Model Categories, Sheaf cohomology.