Homological Algebra (MAST90068)

Homological algebra is a set of tools designed to linearise problems from geometry and topology, algebra, mathematical physics, and other areas of mathematics. This subject provides an introduction to this fascinating field as well as to basic notions of category theory. 

The subject covers categories, functors, limits and colimits, adjoint pairs, and abelian categories; chain complexes and homology, chain maps and chain homotopies, resolutions, and derived functors; the functors Hom and tensor product and their derived functors.

Intended learning outcomes

• illustrate core definitions in category theory and homological algebra, including categories and functors, chain complexes and their maps, and derived functors, by providing and analysing examples;
• construct and compute resolutions and derived functors;
• explain the proofs of important results in homological algebra and identify their core ideas and techniques;
• independently find and write proofs of simple results in category theory and homological algebra.

Generic skills

• problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
• analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
• collaborative skills: the ability to work in a team;
• time-management skills: the ability to meet regular deadlines while balancing competing commitments.