NMAA05100U  Homological Algebra (HomAlg)

Volume 2017/2018

MSc Programme in Mathematics


Basic notions in module theory, tensor products of modules, exact sequences. Categories, functors, natural transformations, adjoint functors. Chain complexes and homology, resolutions, exactness of functors and derived functors.

Learning Outcome
  • Knowledge: To display knowledge of the course topics and content.
  • Skills: To be able to use the acquired knowledge to perform computations.
  • Competences: At the end of the course the student should
    • Be well versed in the basic theory of modules over a ring (direct sums and products, tensor products, exact sequences, free, projective, injective and flat modules.)
    • Understand the basic methods of category theory and be able to apply these in module categories (isomorphisms of functors, exactness properties of functors, adjoint functors, pushouts and pullbacks).
    • Have a thorough understanding of constructions within the category of chain complexes (homology, homotopy, connecting homomorphism, tensor products, Hom-complexes, mapping cones).
    • Have ability to perform calculations of derived functors by constructing resolutions (Ext and Tor).
    • Be able to interpret properties of rings and modules in terms of derived functors (homological dimensions, regularity).
    • Have ability to solve problems in other areas of mathematics, such as commutative algebra, group theory or topology, using methods from homological algebra.
Algebra 2 (Alg2), Topologi (Top)
5 hours of lectures and 4 hours of exercises per week for 9 weeks.
7,5 ECTS
Type of assessment
Continuous assessment
Submission of 3 exercise sets; each set counts for 1/3 of the final grade.
All aids allowed
Marking scale
7-point grading scale
Censorship form
External censorship

30 minutes oral examination with 30 min. preparation time. All aids allowed during the preparation time, but no aids allowed during the examination.

Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.

  • Category
  • Hours
  • Lectures
  • 45
  • Theory exercises
  • 36
  • Preparation
  • 125
  • Total
  • 206