Masters seminar on Triangulated Categories

The aim of this seminar is to expand and deepen our understanding of triangulated categories, and in particular derived categories, beyond what is achieved in the course MA3204 Homological Algebra.

In particular having taken that course, or having a similar background in homological algebra, is a necessary prerequisite for participating in this seminar.

The precise content of the seminar is not fixed, and will depend on the interest of the participants. However I expect that repeating the definition of derived categories and looking a bit more carefully at the construction of localizations involved in that definition may be a natural first subject.

Participants will take turns presenting the courses content to each other. Here it is not vital that the presenter has a complete overview of the matter at hand, but rather we can discuss in the group if some aspects remain unclear.

I tried to write down approximately what we planned today. Please let me know if I misrepresented something (or misremembered who volunteered for what). Also please let me know if you have further ideas, want to be assigned to a specific topic, or similar.

1. Jacob: Definitions, basic properties
2. Håvard: Derived categories of hereditary algebras
3. Tor: Verdier localization
4. Johannes, Ole: Brown representability for compactly generated triangulated categories
5. Sebastian: Injective and projective resolutions of infinite complexes
6. Didrik: Derived Functors between derived categories
7. Endre, Odin, Christina: Algebraic triangulated categories – DG categories
8. Erlend Bergtun, Peter, Sigurd, Torgeir: Topological triangulated categories – Model categories
9. Erlend Børve, Kristoffer: Derivators

The following are some useful sources to get us started. We will expand the list in the course of the semester depending on which direction we are going, and what other sources we find and deem useful.

• Henning Krause, Derived categories, resolutions, and Brown representability
• Bernhard Keller, On the construction of triangle equivalences