NMAK25001U Homological Algebra (HomAlg)

Volume 2025/2026
Education

MSc Programme in Mathematics

MSc Programme in Mathematics with a minor subject

Content

Categories, functors, natural transformations, adjoint functors. Chain complexes and homology, resolutions, exactness of functors and derived functors. Applications of the above to group theory, module theory, and commutative algebra.

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 theory of modules over a ring (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 (e.g. 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.
Literature

In previous years Rotman: "An introduction to homological algebra" has been used.

Knowledge of basic module theory and ring theory (including e.g., tensor products) e.g., obtained through the UCPH MSc course "Commutative Algebra" is assumed.

Knowledge of advanced Linear Algebra e.g., obtained through the UCPH MSc course "Advanced Vector Spaces" is assumed.

Knowledge of Algebraic topology (e.g., obtained through MSc course AlgTop) is helpful, but not assumed.

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures and 4 hours of exercises per week for 9 weeks.
  • Category
  • Hours
  • Lectures
  • 45
  • Preparation
  • 122
  • Theory exercises
  • 36
  • Exam
  • 3
  • Total
  • 206
Written
Oral
Individual
Collective
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Weekly homework counting 50 % towards the grade and a 3 hours 'closed-book' final in-class problem set counting 50 % of the grade.
Aid
Only certain aids allowed (see description below)

No books, notes, or electronic aids are allowed for the in-class exam, except for one personally created one-sided A4 page of handwritten notes.

Marking scale
7-point grading scale
Censorship form
No external censorship
Re-exam

30 minutes oral examination with no aids or preparation time. The oral examination will cover the entire material of the course, including the exercise sets.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.