750 - Homological Algebra

Mid-20-th century it was observed that many constructions specific to algebraic topology (homology, cohomology, etc.) can be done in a more general context, using instead of topological spaces modules over rings. This led to the development of homological algebra, which turned out to be one of the most powerful tools developed in recent history.

The course will be an introduction to these topics, and should prove useful to students of algebra, algebraic geometry, topology, among others. We'll begin with some categorical concepts, and follow up with a study of complexes, homology, Tor and Ext, derived functors, progressing to the definitions of derived category and, if time allows, some more advanced topics like dg- and A_infinity-categories and functors.

The course should be accessible to anyone with basic knowledge of module theory, but some familiarity with algebraic topology will be useful (though not required) to motivate the constructions.