NMAA05100U Homological Algebra (HomAlg)
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.
- 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.
In previous years Rotman: "An introduction to homological algebra" has been used.
Academic qualifications equivalent to a BSc degree is recommended.
- Theory exercises
- 7,5 ECTS
- Type of assessment
- Continuous assessmentThe grade will be determined by 3 "closed book" in-class exams: Two 90 minutes multiple choice exams in weeks 3 and 6, 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 and no electronic aids are allowed for the exams. Personally created
handwritten notes on paper are allowed
- 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 assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.