# Homological algebra seminar

#### Organizers: Bruce Bartlett and Gareth Boxall

*“…if I could only understand the beautiful consequence following from the concise proposition d*^{2}=0″. Henri Cartan
Homological algebra is the art of extracting invariants of diverse mathematical structures (eg. topological spaces, modules, rings, Lie algebras, sheaves) from the homology and cohomology of their associated chain complexes. As such it is a general set of tools with a great variety of applications in mathematics. This seminar will have two parts. The first part will cover the basics of homological algebra, along the lines of Moerdijk’s notes. The second part will be about the language of derived categories, a more modern approach to homological algebra, summarized in the maxim “Chain complexes good, homology bad.” [Lecture notes of Thomas]. The organizers will present the first few talks, and after that we will cycle amongst the participants. Everyone is encouraged to present the material in such a way as to make contact with his or her work.

**Texts**

- I. Moerdijk, Notes on homological algebra, lecture notes (2008).
- Gelfand and Manin, Methods of homological algebra, Springer lecture notes, 2nd edition (2010).
- C. Weibel, An introduction to homological algebra, Cambridge University Press (1994).
- R. Thomas, Derived categories for the working mathematician, lecture notes (2000).
- S. Mac Lane, Homology, Springer (1994).
- H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press (1956).

**Class notes**

1. 25 July 2013 (Bruce). General overview of homological algebra. Then covered 1.1 (Modules) and 1.2 (The Hom functor) of Moerdijk. Lecture notes (A, B, C).

2. 1 August (Gareth). 1.3 Tensor products and 1.4. Tensor-Hom adjunction. Class notes.

3. 8 August (Bruce). 1.5 Change of ring. Example of the augmentation map from the group ring to the integers. Example of map of groups leading to induced, coinduced and pull-back representations. Frobenius reciprocity. 1.6 Exact sequences. Left-exactness of Hom. Class notes.

4. 15 August (Gareth).1.7 Projective modules. Every projective module is a retract of a free module. Every module can be covered by a projective one. 1.8 Injective modules. Every R-module can be embedded into an injective one.

5. 22 August (Bruce). Recap; Baer’s criterion for injective modules. 1.9 Complexes. Examples: singular homology and cohomology with coefficients in an abelian group. De Rham cohomology. Vector fields in the plane. Short exact sequences of chain complexes gives a long exact sequence of homology groups. Class notes.