Homological Algebra (HomAlg)
Course 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.
MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject
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.
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 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.
- ECTS
- 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 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
- 122
- Theory exercises
- 36
- Exam
- 3
- English
- 206
Kursusinformation
- Language
- English
- Course number
- NMAK25001U
- ECTS
- 7,5 ECTS
- Programme level
- Full Degree Master
- Duration
-
1 block
- Placement
- Block 3
- Schedulegroup
-
C
- Capacity
- No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
- Studyboard
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Contracting faculty
- Faculty of Science
Course Coordinator
- Jesper Grodal (2-6e6b447165786c326f7932686f)
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