### C2.2 Homological Algebra (2022-23)

General Prerequisites:
A3 Rings and Modules is essential: good understanding of modules over fields (aka vector spaces), polynomial rings, the ring of integers, and the ring of integers modulo n; good familiarity with module homomorphisms, submodules, and quotient modules.
Course Term: Michaelmas
Course Lecture Information: 16 lectures.
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:
Homological algebra is one of the most important tools in mathematics with application ranging from number theory and geometry to quantum physics. This course will introduce the basic concepts and tools of homological algebra with examples in module theory and group theory.
Learning Outcomes:
Students will learn about abelian categories and derived functors and will be able to apply these notions in different contexts. They will learn to compute Tor, Ext, and group cohomology and homology.
Course Synopsis:
Overview of category theory: adjoint functors, limits and colimits, Abelian categories (2 hours);

Chain complexes: complexes of R-modules and in an abelian category, operations on chain complexes, long exact sequences, chain homotopies, mapping cones and cylinders (3 hours);

Derived functors: delta functors, projective and injective resolutions, left and right derived functors, adjoint functors and exactness, balancing Tor and Ext (5 hours);

Tor and Ext: Tor and flatness, Ext and extensions, universal coefficients theorems, Kunneth formula, Koszul resolutions (3 hours);

Group homology and cohomology: definition, basic properties, cyclic groups, interpretation of $$H^1$$ and $$H^2$$, the Bar resolution (3 hours).