Homological Algebra (HomAlg)

Course content

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.


MSc Programme in Mathematics

MSc Programme in Mathematics with a minor subject

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 (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.

5 hours of lectures and 4 hours of exercises per week for 9 weeks.

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

Knowledge of advanced Linear Algebra corresponding to UCPH course AdVec, and knowledge of elementary group and ring theory corresponding to UCPH courses Alg1 and Alg2 is assumed. Basic point-set and algebraic topology (e.g., UCPH course Top and AlgTop) is helpful, but not assumed.

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
The grade will be determined by 3 "closed book" in-class exams: Two 90 minutes multiple choice exams during the first 8 weeks, and one 120 minutes written exam in week 9. The two multiple choice each count 25%, and the final 2-hour exam counts 50%.
Only certain aids allowed

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

Marking scale
7-point grading scale
Censorship form
External censorship

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 assessment

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

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 45
  • Preparation
  • 120
  • Theory exercises
  • 36
  • Exam
  • 5
  • English
  • 206


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 2
No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
Study Board of Mathematics and Computer Science
Contracting department
  • Department of Mathematical Sciences
Contracting faculty
  • Faculty of Science
Course Coordinators
  • Jesper Grodal   (2-6d6a437064776b316e7831676e)
  • Damian Longin Osajda   (2-6772437064776b316e7831676e)
Saved on the 14-02-2024

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