Course Homological Algebra

Homology groups and cohomology groups have been proven to be suitable answering questions in, for example, differential and algebraic topology. It is very striking to show that two topological spaces are non-homeomorphic in computing singular homology groups. In differential topology computing certain cohomology groups (de Rham, compact de Rham or compact vertical de Rham) is used to study manifolds and vector bundles. Those theories (homology and cohomology) have a lot in common in their formalism. In this course we are going to study homology and cohomology theories for abelian categories and tools for computations like spectral sequences. The syllabus can be found here.

Topics:

Chain and cochain complexes, derived functors, Leray-Serre spectral sequences, dimension theory, derived categories.

Literature:

1. Textbook: Charles Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994, ISBN 0-521-43500-5
2. Joseph J. Rotman, An introduction to homological algebra, Sringer Science+Business Media, LLC 2009, second edition, ISBN 978-0-387-24527-0
3. Sergei I. Gelfand and Yuri I. Manin, Methods of Homological Algebra, Springer-Verlag Berlin Heidelberg, 2003, second edition, ISBN 978-3-642-07813-2

Lecture time: Mondays (odd) and every Friday: 15:00-16:40 IMS 507

Lecture notes: Week1 Week2 Week3 Week4 Week5 Week6 Week7

Problem sheets: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5  Sheet 6