# 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.

Education

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 basic group and ring theory (e.g, UCPH courses Alg1 and Alg2) and a solid knowledge of Linear Algebra (e.g., UCPH course LinAlg, and ideally also AdVec) 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.

Written
Oral
Individual
Collective
Continuous feedback during the course of the semester
ECTS
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%.
Aid
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
Censorship form
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
• 120
• Theory exercises
• 36
• Exam
• 5
• English
• 206

### Kursusinformation

Language
English
Course number
NMAA05100U
ECTS
7,5 ECTS
Programme level
Full Degree Master
Duration

1 block

Placement
Block 2
Schedulegroup
C
Capacity
No limit
The number of seats may be reduced in the late registration period
Studyboard
Study Board of Mathematics and Computer Science
##### Contracting department
• Department of Mathematical Sciences
##### Contracting faculty
• Faculty of Science
##### Course Coordinators
• Jesper Grodal   (2-79764f7c7083773d7a843d737a)
• Damian Longin Osajda   (2-737e4f7c7083773d7a843d737a)
Saved on the 28-02-2023

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