# NMAA05100U  Homological algebra (HomAlg)

Volume 2014/2015
Education
MSc programme in Mathematics
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.

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 (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.
Algebra 2 (Alg2), Topologi (Top)
5 hours of lectures and 4 hours of exercises per week for 9 weeks.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Submission of 3 exercise sets.
Marking scale
Censorship form
External censorship
Re-exam
30 minutes oral examination with time for preparation. All aids allowed during the preparation time.
##### Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.

• Category
• Hours
• Lectures
• 45
• Theory exercises
• 36
• Preparation
• 125
• Total
• 206