Seminar: Homologische Algebra

Dr. Elena Martinengo (email)



The aim of the seminar is to define the cohomology of groups and study its properties.
At first we will need some tools from basic (co)-homology theory and classical functor theory, that we will carry out through elementary and concrete examples.We will in particular focus on injective/projective resolutions of modules and on the Tor and Ext functors.
We define the (co)-homology of a group G as the (co)-homology of the standard resolution of Z over Z[G], while the (co)-homology of a group G with coefficients in a Z[G]-module M will be defined using injective/projective resolutions and calculating the Tor and Ext groups. The functoriality properties of these derived functors will assure they are well defined, the existence of a long exact sequence of (co)-homology groups and some other functoriality properties.
Since we aim to remain concrete as much as possible, we will calculate a lot of examples: 0-groups and 1-groups (for trivial modules) are easy to be calculated and there are explict results of finite cyclic groups too.
Then the main three aims of the seminar are to explain the meaning of the cohomology of a group in terms of derivations, semidirect products and extensions with abelian kernel. We will develop this part presenting examples too.
Depending on the audience we can decide the last topics.

(Draft of the) List of the talks

Depending on the intersts of the audience: