# C2.2 Homological Algebra - Material for the year 2020-2021

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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.

16 lectures.

### Assessment type:

- Written Examination

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.

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.

Chain complexes: complexes of R-modules, operations on chain complexes, long exact sequences, chain homotopies, mapping cones and cylinders (4 hours)

Derived functors: delta functors, projective and injective resolutions, left and right derived functors (5 hours)

Tor and Ext: Tor and flatness, Ext and extensions, universal coefficients theorems, Koszul resolutions (4 hours)

Group homology and cohomology: definition, interpretation of $H^1$ and $H^2$, universal central extensions, the Bar resolution (3 hours).

- Weibel, Charles
*An introduction to Homological algebra*(see Google Books)