MAGIC103: Cohomology of groups

An introduction to cohomology via derived functors and the theory of connected δ-functors leading to a theory that applies to abelian categories (for example of modules over a ring or sheaves over a space or scheme) with either enough projectives or enough injectives. The statement and outline of the proof of the Universal Property of such theories and its application to concrete calculations. The application of the ideas specifically to the \ext and \tor groups for modules over a ring leading to long exact sequences in both variables and two dimension shifting strategies. The specific application to cohomology groups. The connection between first cohomology and conjugacy classes of complements in split extensions. The connection between second cohomology and group extensions. Applications to abstract group theory.

A basic course in rings and modules. Some homological algebra would also be helpful: for example a first course in homology of simplicial complexes.

Short and long exact sequences of modules over a ring. Diagram chasing arguments. The Five Lemma, The Snake Lemma, The 4 Sequence Lemma, The Horseshoe Lemma
The notions projective and injective modules and the fact that there are enough of them in the category of modules over an associative ring. Remarks on other abelian categories.
Fundamental adjunctions. The induction and coinduction functors for modules over group rings. Comparison with the induction functor in the representation theory of algebraic groups and group schemes.
The axioms for a cohomology theory. The long exact sequence axiom. The coeffaceability axiom. The construction of theories that satisfy these axioms.
Statement and proof of the fundamental universal property of cohomology theories that satisfy both axioms.
Applications to the theory of extensions and conjugacy classes in the theory of groups. Derivations and factor sets.
Application to the structure of crystallographic groups.
Cohomological dimension and finiteness conditions related to it. Cohomological dimension of abelian groups. Statement (but not the proof) of the Stallings-Swan theorem.
Mayer-Vietoris sequences for groups actions on trees.
(Time permitting) Remarks on non-abelian cohomology, Galois cohomology, the Brauer group, the classification of real Lie algebras, the classification of quadratic forms.

MAGIC103: Cohomology of groups

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Professor Peter Kropholler

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