MAGIC103: Cohomology of groups

Course details

A specialist MAGIC course


Spring 2023
Monday, January 23rd to Friday, March 31st


Live lecture hours
Recorded lecture hours
Total advised study hours


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.


  1. 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
  2. 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.
  3. 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.
  4. The axioms for a cohomology theory. The long exact sequence axiom. The coeffaceability axiom. The construction of theories that satisfy these axioms.
  5. Statement and proof of the fundamental universal property of cohomology theories that satisfy both axioms.
  6. Applications to the theory of extensions and conjugacy classes in the theory of groups. Derivations and factor sets.
  7. Application to the structure of crystallographic groups.
  8. Cohomological dimension and finiteness conditions related to it. Cohomological dimension of abelian groups. Statement (but not the proof) of the Stallings-Swan theorem.
  9. Mayer-Vietoris sequences for groups actions on trees.
  10. (Time permitting) Remarks on non-abelian cohomology, Galois cohomology, the Brauer group, the classification of real Lie algebras, the classification of quadratic forms.


  • PK

    Professor Peter Kropholler

    University of Southampton


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 1st May 2023 and is due in by Friday 12th May 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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