250030 VO Homological Algebra (2024S)

Homological Algebra has its cources in works of Poincare and Hilbert. Poincare in his investigations of topological
spaces introduced homology groups which he obtained from simplicial complexes attached to the topological space.
Hilbert following his solution of the basis problem (Hilbert basis Theorem) and with the aim to make his solution
explicit introduced resolutions of ideals in polynomial rings which again are certain complexes.

The idea of investigating objects by looking at resolutions turned out to be a general abstract principle with wide applications in mathematics.
Homological Algebra is the study of this principle in a general and abstract setting. In fact Homological algebra can be
formulated and applied in arbitrary abelian categories and it may be seen as the study of abelian categories and their functors.

In the course we want to explain the abstract principle and touch on some of the techniques for computing (Co)Homolgy of
objects.

Prerequisites are very basic knowledge of modules (definition, morphisms, Sub- and Quotient modules,...) and of categories
(essentially the definition of category and functor will suffice; if we need more we will review this material).